helm/www/lambdadelta/xslt/basic_2_sum.xsl
helm/www/lambdadelta/xslt/static_2_src.xsl
helm/www/lambdadelta/xslt/static_2_sum.xsl
+helm/www/lambdadelta/xslt/basic_2A_src.xsl
+helm/www/lambdadelta/xslt/basic_2A_sum.xsl
helm/www/lambdadelta/xslt/ground_2_src.xsl
helm/www/lambdadelta/xslt/ground_2_sum.xsl
helm/www/lambdadelta/xslt/basic_1_blk.xsl
SHELL := $(shell which bash)
H=@
-TAGS = up-html up-html-ix up-home html html-ix home ldw ldw-ix
-
TAGS = www up-html up-css up-images up-download \
lint-xml index lddl install-xml \
test-html html install-html \
install-jed install-bib install-2 install-1 install-coq \
install-automath install-v install-matita \
+TAGS = up-html up-html-ix up-home html html-ix home ldw ldw-ix install-2
+
LDWEBURL = http://helm.cs.unibo.it/lambdadelta/
SITEDIR = html
ldw-ix: $(LDWIXSTAMP)
+# INSTALL λδ-2 PACKAGES ######################################################
+
+install-2: PKGS = $(shell find -L $(CONTRIBDIR) -name lambdadelta_*.tar.bz2)
+
+install-2: $(PKGS)
+ @echo " INSTALL $(PKGS:$(CONTRIBDIR)/lambdadelta_%.tar.bz2=%)"
+ $(H)scp $(PKGS) $(DOWNDIR)
+ $(H)$(RM) $(PKGS)
+
##############################################################################
lint-xml: $(XMLS:%=$(XMLDIR)/%)
$(H)scp $< $(DOWNDIR)
$(H)scp $< $(DOWNDIR)/$(BIB:%.bib=%.txt)
-install-2: $(CONTRIB:%=$(CONTRIBDIR)/%)
- @echo " INSTALL $(CONTRIB)"
- $(H)scp $< $(DOWNDIR)
-
install-automath: $(HELENADIR)/$(AUTOMATH)
@echo " INSTALL $(notdir $<)"
$(H)scp $< $(DOWNDIR)
is available in the following formats.
</body>
- <body>
- Informational pages on the parts of the specification:
- <rlink to="html/ground_2.html">Background</rlink>,
- <rlink to="html/static_2.html">Syntax</rlink>,
- <rlink to="html/basic_2.html">Core</rlink>,
- <rlink to="html/apps_2.html">Applications</rlink>.
- </body>
-
- <body>
- <notice class="alpha" text="Notice:"/>
- The scripts are grouped in directories, first by part, then by component.
- </body>
-
<body>
<notice class="alpha" text="Notice:"/>
the scripts are checked by the latest version of Matita from
<topitem name="source2B">
<body>
<rlink to="download/lambdadelta_2B.tar.bz2">lambdadelta_2B for Matita 0.99.4</rlink>
- (revised <notice class="gamma" text="2019-11"/>).
- Source scripts [Git revision: 2019-11-19 20:45:15].
+ (revised <notice class="gamma" text="2020-02"/>).
+ Source scripts.
<rlink to="html/documentation.html#ldJ2a">Documentation (J2a)</rlink>.
+ <list><item>
+ <notice class="gamma" text="2020-02-27."/>
+ repackaged without λδ-ground-2.
+ </item><item>
+ <notice class="gamma" text="2019-11-19."/>
+ released [Git revision: 2019-11-19 20:45:15].
+ </item></list>
</body>
+ <body>
+ Informational pages on the parts of the specification:
+ <rlink to="html/ground_2.html">Background</rlink>,
+ <rlink to="html/static_2.html">Syntax</rlink>,
+ <rlink to="html/basic_2.html">Core</rlink>,
+ <rlink to="html/apps_2.html">Applications</rlink>.
+ </body>
</topitem>
<topitem name="source2A">
<body>
<rlink to="download/lambdadelta_2A.tar.bz2">lambdadelta_2A for Matita 0.99.2</rlink>
- (revised <notice class="delta" text="2019-11"/>).
- Source scripts [Git revision: 2014-10-28 17:46:26].
+ (revised <notice class="delta" text="2020-02"/>).
+ Source scripts.
<rlink to="html/documentation.html#ldR2c">Documentation (R2c)</rlink>.
<list><item>
- <notice class="delta" text="2019 November 20."/>
- repackaging (was lambdadelta_2A1).
+ <notice class="delta" text="2020-02-27."/>
+ repackaged on the basis of λδ-ground-2.
+ </item><item>
+ <notice class="delta" text="2019-11-20."/>
+ repackaged (was lambdadelta_2A1).
+ </item><item>
+ <notice class="gamma" text="2014-10-28."/>
+ released [Git revision: 2014-10-28 17:46:26].
+ </item></list>
+ </body>
+ <body>
+ Informational pages on the parts of the specification:
+ <rlink to="html/basic_2A.html">Core</rlink>.
+ </body>
+ </topitem>
+
+ <topitem name="source2g">
+ <body>
+ <rlink to="download/lambdadelta_ground.tar.bz2">lambdadelta_ground for Matita 0.99.4</rlink>
+ (revised <notice class="gamma" text="2020-02"/>).
+ Source scripts.
+ <list><item>
+ <notice class="gamma" text="2020-02-27."/>
+ released [Git revision: 2014-10-28 17:46:26].
</item></list>
</body>
+ <body>
+ Informational pages on the parts of the specification:
+ <rlink to="html/ground_2.html">Core</rlink>.
+ </body>
</topitem>
+
<!-- VERSION 1 =========================================================== -->
<subsection name="v1"><img logo="ld1"/>λδ version 1 (superseded)</subsection>
is available in the following formats.
</body>
- <body>
- Informational pages on the parts of the specification:
- <rlink to="html/ground_1.html">Background</rlink>,
- <rlink to="html/basic_1.html">Core</rlink>.
- </body>
-
- <body>
- <notice class="alpha" text="Notice:"/>
- The scripts are grouped in directories, one for each part.
- </body>
-
<topitem name="source1A">
<body>
<rlink to="download/lambdadelta_1A.tar.bz2">lambdadelta_1A for Coq 7.3.1</rlink>
Source scripts.
<rlink to="html/documentation.html#ldJ1a">Documentation (J1a)</rlink>.
<list><item>
- <notice class="delta" text="2019 November 20."/>
+ <notice class="delta" text="2019-11-20."/>
repackaging (was lambdadelta_1).
</item><item>
- <notice class="delta" text="2015 January 15."/>
+ <notice class="delta" text="2015-01-15."/>
17 new lemmas and former lemma "eq_nat_dec" renamed as "nat_dec_neg".
</item></list>
</body>
+ <body>
+ Informational pages on the parts of the specification:
+ <rlink to="html/ground_1.html">Background</rlink>,
+ <rlink to="html/basic_1.html">Core</rlink>.
+ </body>
</topitem>
<topitem name="static1A">
ORIG := . ./orig.sh
ORIGS := basic_2/basic_1.orig
-CONTRIB := lambdadelta_2B
-
WWW := ../../../../helm/www/lambdadelta
-TAGS := all names xoa orig elim deps top leaf stats tbls odeps trim contrib clean \
- home up-home
+TAGS := all names xoa orig elim deps top leaf stats tbls odeps trim clean \
+ pack-ground pack-2a pack-2b \
+ home up-home \
-PACKAGES := ground_2 static_2 basic_2 apps_2 alpha_1
-XPACKAGES := ground_2 static_2 basic_2
+PACKAGES := ground_2 basic_2A static_2 basic_2 apps_2 alpha_1
LDWS := $(shell find -name "*.ldw.xml")
TBLS := $(shell find -name "*.tbl")
$(foreach PKG, $(PACKAGES), $(eval $(call MAS_TEMPLATE,$(PKG))))
-# XMAS #######################################################################
-
-define XMAS_TEMPLATE
- XMAS += $$(MAS_$(1))
-endef
-
-$(foreach PKG, $(XPACKAGES), $(eval $(call XMAS_TEMPLATE,$(PKG))))
-
# names ######################################################################
NAMES = basic_1A basic_2A static_2 basic_2
trim: $(TRIMS:%=%.trimmed)
-# contrib ####################################################################
+# package ground #############################################################
+
+pack-ground: PKG = lambdadelta_ground
+
+pack-ground: DIRS = ground_2
+
+pack-ground: PMAS = $(shell find $(DIRS) -name *.ma)
+
+pack-ground:
+ @echo " TAR -cjf $(PKG).tar.bz2 root $(DIRS)"
+ $(H)tar -cjf $(PKG).tar.bz2 ../lambdadelta/root $(PMAS:%=../lambdadelta/%)
+
+# package 2A ###################################################################
+
+pack-2a: PKG = lambdadelta_2A
+
+pack-2a: DIRS = basic_2A
+
+pack-2a: PMAS = $(shell find $(DIRS) -name *.ma)
+
+pack-2a:
+ @echo " TAR -cjf $(PKG).tar.bz2 $(DIRS)"
+ $(H)tar -cjf $(PKG).tar.bz2 $(PMAS:%=../lambdadelta/%)
+
+# package 2B ###################################################################
+
+pack-2b: PKG = lambdadelta_2B
+
+pack-2b: DIRS = static_2 basic_2
+
+pack-2b: PMAS = $(shell find $(DIRS) -name *.ma)
-contrib:
- @echo " TAR -cjf $(CONTRIB).tar.bz2 root $(XPACKAGES)"
- $(H)tar -cjf $(CONTRIB).tar.bz2 ../lambdadelta/root $(XMAS:%=../lambdadelta/%)
+pack-2b:
+ @echo " TAR -cjf $(PKG).tar.bz2 $(DIRS)"
+ $(H)tar -cjf $(PKG).tar.bz2 $(PMAS:%=../lambdadelta/%)
# clean ######################################################################
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/crr_append.ma".
-include "basic_2/reduction/cir.ma".
-
-(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************)
-
-(* Advanved properties ******************************************************)
-
-lemma cir_labst_last: ∀G,L,T,W. ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, ⋆.ⓛW @@ L⦄ ⊢ ➡ 𝐈⦃T⦄.
-/3 width=2 by crr_inv_labst_last/ qed.
-
-lemma cir_tif: ∀G,T,W. ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐈⦃T⦄.
-/3 width=2 by crr_inv_trr/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cir_inv_append_sn: ∀G,L,K,T. ⦃G, K @@ L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
-/3 width=1/ qed-.
-
-lemma cir_inv_tir: ∀G,T,W. ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃T⦄.
-/3 width=1/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/crx_append.ma".
-include "basic_2/reduction/cix.ma".
-
-(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cix_inv_append_sn: ∀h,g,G,L,K,T. ⦃G, K @@ L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
-/3 width=1 by crx_append_sn/ qed-.
-
-lemma cix_inv_tix: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
-/3 width=1 by trx_crx/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_append.ma".
-
-(* SHIFT OF A CLOSURE *******************************************************)
-
-let rec shift L T on L ≝ match L with
-[ LAtom ⇒ T
-| LPair L I V ⇒ shift L (-ⓑ{I} V. T)
-].
-
-interpretation "shift (closure)" 'Append L T = (shift L T).
-
-(* Basic properties *********************************************************)
-
-lemma shift_append_assoc: ∀L,K. ∀T:term. (L @@ K) @@ T = L @@ K @@ T.
-#L #K elim K -K // normalize //
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma shift_inj: ∀L1,L2. ∀T1,T2:term. L1 @@ T1 = L2 @@ T2 → |L1| = |L2| →
- L1 = L2 ∧ T1 = T2.
-#L1 elim L1 -L1
-[ * normalize /2 width=1/
- #L2 #I2 #V2 #T1 #T2 #_ <plus_n_Sm #H destruct
-| #L1 #H1 #V1 #IH * normalize
- [ #T1 #T2 #_ <plus_n_Sm #H destruct
- | #L2 #I2 #V2 #T1 #T2 #H1 #H2
- elim (IH … H1) -IH -H1 /2 width=1/ -H2 #H1 #H2 destruct /2 width=1/
- ]
-]
-qed-.
+++ /dev/null
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
-
-lemma cpr_append: ∀G. l_appendable_sn … (cpr G).
-#G #K #T1 #T2 #H elim H -G -K -T1 -T2
-/2 width=3 by cpr_bind, cpr_flat, cpr_zeta, cpr_tau, cpr_beta, cpr_theta/
-#G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpr_delta … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
-qed.
-
-lemma cpr_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpr_inv_bind1 … H) -H *
- [ #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
-
+++ /dev/null
-lemma cpx_append: ∀h,g,G. l_appendable_sn … (cpx h g G).
-#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-/2 width=3 by cpx_sort, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_ti, cpx_beta, cpx_theta/
-#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpx_delta … I … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
-qed.
-
-lemma cpx_fwd_shift1: ∀h,g,G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#h #g #G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpx_inv_bind1 … H) -H *
- [ #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
+++ /dev/null
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
-
-lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
-#G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
-/2 width=1 by cpy_atom, cpy_bind, cpy_flat/
-#I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
-qed-.
-
-lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #d #e #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X #d #e
- >shift_append_assoc normalize #H
- elim (cpy_inv_bind1 … H) -H
- #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
- /2 width=3 by trans_eq/
-]
-qed-.
+++ /dev/null
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
-
-lemma cpys_append: ∀G. l_appendable_sn … (cpys G).
-#G #K #T1 #T2 #H elim H -G -K -T1 -T2
-/2 width=3 by cpys_bind, cpys_flat/
-#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpys_delta … I … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
-qed.
-
-lemma cpys_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X >shift_append_assoc normalize
- #H elim (cpys_inv_bind1 … H) -H
- #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
-]
-qed-.
+++ /dev/null
-lemma cpys_append: ∀G,d,e. l_appendable_sn … (cpys d e G).
-#G #d #e #K #T1 #T2 #H @(cpys_ind … H) -T2
-/3 width=3 by cpys_strap1, cpy_append/
-qed-.
-
-lemma cpys_fwd_shift1: ∀G,L,L1,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L #L1 #T1 #T #d #e #H @(cpys_ind … H) -T
-[ /2 width=4 by ex2_2_intro/
-| #T #X #_ #HX * #L0 #T0 #HL10 #H destruct
- elim (cpy_fwd_shift1 … HX) -HX #L2 #T2 #HL02 #H destruct
- /2 width=4 by ex2_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_append.ma".
-include "basic_2/reduction/crr.ma".
-
-(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
-
-(* Advanved properties ******************************************************)
-
-lemma crr_append_sn: ∀G,L,K,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, K @@ L⦄ ⊢ ➡ 𝐑⦃T⦄.
-#G #L #K0 #T #H elim H -L -T /2 width=1/
-#L #K #V #i #HLK
-lapply (ldrop_fwd_length_lt2 … HLK) #Hi
-lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=2/ -Hi /2 width=3/
-qed.
-
-lemma trr_crr: ∀G,L,T. ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄.
-#G #L #T #H lapply (crr_append_sn … H) //
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact crr_inv_labst_last_aux: ∀G,L1,T,W. ⦃G, L1⦄ ⊢ ➡ 𝐑⦃T⦄ →
- ∀L2. L1 = ⋆.ⓛW @@ L2 → ⦃G, L2⦄ ⊢ ➡ 𝐑⦃T⦄.
-#G #L1 #T #W #H elim H -L1 -T /2 width=1/ /3 width=1/
-[ #L1 #K1 #V1 #i #HLK1 #L2 #H destruct
- lapply (ldrop_fwd_length_lt2 … HLK1)
- >append_length >commutative_plus normalize in ⊢ (??% → ?); #H
- elim (le_to_or_lt_eq i (|L2|)) /2 width=1/ -H #Hi destruct
- [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2
- lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi
- normalize #H destruct /2 width=3/
- | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // <minus_n_n #H
- lapply (ldrop_inv_O2 … H) -H #H destruct
- ]
-| #a #I #L1 #V #T #HI #_ #IHT #L2 #H destruct /3 width=1/
-]
-qed.
-
-lemma crr_inv_labst_last: ∀G,L,T,W. ⦃G, ⋆.ⓛW @@ L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄.
-/2 width=4/ qed-.
-
-lemma crr_inv_trr: ∀G,T,W. ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃T⦄.
-/2 width=4/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_append.ma".
-include "basic_2/reduction/crx.ma".
-
-(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************)
-
-(* Advanved properties ******************************************************)
-
-lemma crx_append_sn: ∀h,g,G,L,K,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⦃G, K @@ L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
-#h #g #G #L #K0 #T #H elim H -L -T
-/2 width=2 by crx_sort, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/
-#I #L #K #V #i #HLK
-lapply (ldrop_fwd_length_lt2 … HLK) #Hi
-lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=4 by crx_delta, lt_to_le/
-qed.
-
-lemma trx_crx: ∀h,g,G,L,T. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
-#h #g #G #L #T #H lapply (crx_append_sn … H) //
-qed.
+++ /dev/null
-lemma lpr_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ➡ K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L1 @@ K1⦄ ⊢ ➡ L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpr_append/ qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpr_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡ L →
- ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡ K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpr_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ➡ K2 @@ L2 →
- ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡ K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
+++ /dev/null
-lemma lpx_append: ∀h,g,G,K1,K2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ⦃G, L1 @@ K1⦄ ⊢ ➡[h, g] L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpx_append/ qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpx_fwd_append1: ∀h,g,G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡[h, g] L →
- ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpx_fwd_append2: ∀h,g,G,L,K2,L2. ⦃G, L⦄ ⊢ ➡[h, g] K2 @@ L2 →
- ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
+++ /dev/null
-lemma lpx_sn_append: ∀R. l_appendable_sn R →
- ∀K1,K2. lpx_sn R K1 K2 → ∀L1,L2. lpx_sn R L1 L2 →
- lpx_sn R (L1 @@ K1) (L2 @@ K2).
-#R #HR #K1 #K2 #H elim H -K1 -K2 /3 width=1 by lpx_sn_pair/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpx_sn_fwd_append1: ∀R,L1,K1,L. lpx_sn R (K1 @@ L1) L →
- ∃∃K2,L2. lpx_sn R K1 K2 & L = K2 @@ L2.
-#R #L1 elim L1 -L1
-[ #K1 #K2 #HK12
- @(ex2_2_intro … K2 (⋆)) // (* explicit constructor, /2 width=4/ does not work *)
-| #L1 #I #V1 #IH #K1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
- elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #H destruct
- @(ex2_2_intro … (L2.ⓑ{I}V2) HK12) // (* explicit constructor, /2 width=4/ does not work *)
-]
-qed-.
-
-lemma lpx_sn_fwd_append2: ∀R,L2,K2,L. lpx_sn R L (K2 @@ L2) →
- ∃∃K1,L1. lpx_sn R K1 K2 & L = K1 @@ L1.
-#R #L2 elim L2 -L2
-[ #K2 #K1 #HK12
- @(ex2_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=4/ does not work *)
-| #L2 #I #V2 #IH #K2 #X #H
- elim (lpx_sn_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
- elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #H destruct
- @(ex2_2_intro … (L1.ⓑ{I}V1) HK12) // (* explicit constructor, /2 width=4/ does not work *)
-]
-qed-.
+++ /dev/null
-lemma lpys_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ▶*× K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 →
- ⦃G, L1 @@ K1⦄ ⊢ ▶*× L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpys_append/ qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpys_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ▶*× L →
- ∃∃K2,L2. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpys_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ▶*× K2 @@ L2 →
- ∃∃K1,L1. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpys_fwd_shift1_ext: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
- ∃∃L2,T2. ⦃G, L @@ L1⦄ ⊢ ▶*× L @@ L2 & ⦃G, L @@ L1⦄ ⊢ T1 ▶*× T2 &
- T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #HT1 @ex3_2_intro
- [3: // |4,5: // |1,2: skip ] (**) (* auto does not work *)
-| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
- elim (cpys_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
- lapply (lpys_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1 by lpys_append, lpys_pair/ #HL12
- @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3 by trans_eq/ | skip ] <append_assoc // (**) (* explicit constructor *)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubstnormal_5.ma".
-include "basic_2/relocation/cpy.ma".
-
-(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION *****************)
-
-definition cny: ∀d,e. relation3 genv lenv term ≝
- λd,e,G,L. NF … (cpy d e G L) (eq …).
-
-interpretation
- "normality for context-sensitive extended substitution (term)"
- 'PSubstNormal G L T d e = (cny d e G L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cny_inv_lref: ∀G,L,d,e,i. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄ →
- ∨∨ yinj i < d | d + e ≤ yinj i | |L| ≤ i.
-#G #L #d #e #i #H elim (ylt_split i d) /2 width=1 by or3_intro0/
-#Hdi elim (ylt_split i (d+e)) /2 width=1 by or3_intro1/
-#Hide elim (lt_or_ge i (|L|)) /2 width=1 by or3_intro2/
-#Hi elim (ldrop_O1_lt L i) //
-#I #K #V #HLK elim (lift_total V 0 (i+1))
-#W #HVW lapply (H W ?) -H /2 width=5 by cpy_subst/ -HLK
-#H destruct elim (lift_inv_lref2_be … HVW) -L -d -e //
-qed-.
-
-lemma cny_inv_bind: ∀a,I,G,L,V,T,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓑ{a,I}V.T⦄ →
- ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ▶[⫯d, e] 𝐍⦃T⦄.
-#a #I #G #L #V1 #T1 #d #e #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (ⓑ{a,I}V2.T1) ?) -HVT1
-| #T2 #HT2 lapply (HVT1 (ⓑ{a,I}V1.T2) ?) -HVT1
-]
-/2 width=1 by cpy_bind/ #H destruct //
-qed-.
-
-lemma cny_inv_flat: ∀I,G,L,V,T,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓕ{I}V.T⦄ →
- ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ ∧ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T⦄.
-#I #G #L #V1 #T1 #d #e #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (ⓕ{I}V2.T1) ?) -HVT1
-| #T2 #HT2 lapply (HVT1 (ⓕ{I}V1.T2) ?) -HVT1
-]
-/2 width=1 by cpy_flat/ #H destruct //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cny_conf: ∀G,d,e.
- ∀L1,T. ⦃G, L1⦄ ⊢ ▶[d, e] 𝐍⦃T⦄ →
- ∀L2. L1 ⊑×[d, e] L2 → ⦃G, L2⦄ ⊢ ▶[d, e] 𝐍⦃T⦄.
-#G #d #e #L1 #T1 #HT1 #L2 #HL12 #T2 #HT12
-@HT1 /3 width=3 by lsuby_cpy_trans/
-qed-.
-
-lemma cny_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃⋆k⦄.
-#G #L #d #e #k #X #H elim (cpy_inv_sort1 … H) -H //
-qed.
-
-lemma cny_lref_free: ∀G,L,d,e,i. |L| ≤ i → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
-#G #L #d #e #i #Hi #X #H elim (cpy_inv_lref1 … H) -H // *
-#I #K #V #_ #_ #HLK #_ lapply (ldrop_fwd_length_lt2 … HLK) -HLK
-#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
-qed.
-
-lemma cny_lref_atom: ∀G,L,d,e,i. ⇩[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
-#G #L #d #e #i #HL @cny_lref_free >(ldrop_fwd_length … HL) -HL //
-qed.
-
-lemma cny_lref_top: ∀G,L,d,e,i. d+e ≤ yinj i → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
-#G #L #d #e #i #Hdei #X #H elim (cpy_inv_lref1 … H) -H // *
-#I #K #V #_ #H elim (ylt_yle_false … H) //
-qed.
-
-lemma cny_lref_skip: ∀G,L,d,e,i. yinj i < d → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
-#G #L #d #e #i #Hid #X #H elim (cpy_inv_lref1 … H) -H // *
-#I #K #V #H elim (ylt_yle_false … H) //
-qed.
-
-lemma cny_gref: ∀G,L,d,e,p. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃§p⦄.
-#G #L #d #e #p #X #H elim (cpy_inv_gref1 … H) -H //
-qed.
-
-lemma cny_bind: ∀G,L,V,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ →
- ∀I,T. ⦃G, L.ⓑ{I}V⦄ ⊢ ▶[⫯d, e] 𝐍⦃T⦄ →
- ∀a. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓑ{a,I}V.T⦄.
-#G #L #V1 #d #e #HV1 #I #T1 #HT1 #a #X #H
-elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
->(HV1 … HV12) -V2 >(HT1 … HT12) -T2 //
-qed.
-
-lemma cny_flat: ∀G,L,V,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ →
- ∀T. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T⦄ →
- ∀I. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓕ{I}V.T⦄.
-#G #L #V1 #d #e #HV1 #T1 #HT1 #I #X #H
-elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
->(HV1 … HV12) -V2 >(HT1 … HT12) -T2 //
-qed.
-
-lemma cny_narrow: ∀G,L,T,d1,e1. ⦃G, L⦄ ⊢ ▶[d1, e1] 𝐍⦃T⦄ →
- ∀d2,e2. d1 ≤ d2 → d2 + e2 ≤ d1 + e1 → ⦃G, L⦄ ⊢ ▶[d2, e2] 𝐍⦃T⦄.
-#G #L #T1 #d1 #e1 #HT1 #d2 #e2 #Hd12 #Hde21 #T2 #HT12
-@HT1 /2 width=5 by cpy_weak/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cpy_lift.ma".
-include "basic_2/relocation/cny.ma".
-
-(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION *****************)
-
-(* Properties on relocation *************************************************)
-
-lemma cny_lift_le: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → dt + et ≤ d → ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdetd #U2 #HU12
-elim (cpy_inv_lift1_le … HU12 … HLK … HTU1) // -L -Hdetd #T2 #HT12
->(HT1 … HT12) -K /2 width=5 by lift_mono/
-qed-.
-
-lemma cny_lift_be: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → dt ≤ d → yinj d ≤ dt + et → ⦃G, L⦄ ⊢ ▶[dt, et+e] 𝐍⦃U⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdtd #Hddet #U2 #HU12
-elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hdtd -Hddet #T2
->yplus_minus_inj #HT12 >(HT1 … HT12) -K /2 width=5 by lift_mono/
-qed-.
-
-lemma cny_lift_ge: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → d ≤ dt → ⦃G, L⦄ ⊢ ▶[dt+e, et] 𝐍⦃U⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #U2 #HU12
-elim (cpy_inv_lift1_ge … HU12 … HLK … HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hddt #T2
->yplus_minus_inj #HT12 >(HT1 … HT12) -K /2 width=5 by lift_mono/
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma cny_inv_lift_le: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → dt + et ≤ d → ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdetd #T2 #HT12
-elim (lift_total T2 d e) #U2 #HTU2
-lapply (cpy_lift_le … HT12 … HLK … HTU1 … HTU2 ?) // -K -Hdetd #HU12
-lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
-qed-.
-
-lemma cny_inv_lift_be: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → dt ≤ d → yinj d + e ≤ dt + et → ⦃G, K⦄ ⊢ ▶[dt, et-e] 𝐍⦃T⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdtd #Hdedet #T2 #HT12
-lapply (yle_fwd_plus_ge_inj … Hdedet) // #Heet
-elim (yle_inv_plus_inj2 … Hdedet) -Hdedet #Hddete #Hedet
-elim (lift_total T2 d e) #U2 #HTU2
-lapply (cpy_lift_be … HT12 … HLK … HTU1 … HTU2 ? ?) // [ >yplus_minus_assoc_inj // ] -K -Hdtd -Hddete
->ymax_pre_sn // -Heet #HU12
-lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
-qed-.
-
-lemma cny_inv_lift_ge: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → yinj d + e ≤ dt → ⦃G, K⦄ ⊢ ▶[dt-e, et] 𝐍⦃T⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdedt #T2 #HT12
-elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #Hddte #Hedt
-elim (lift_total T2 d e) #U2 #HTU2
-lapply (cpy_lift_ge … HT12 … HLK … HTU1 … HTU2 ?) // -K -Hddte
->ymax_pre_sn // -Hedt #HU12
-lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma cny_inv_lift_ge_up: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
- ⦃G, K⦄ ⊢ ▶[d, dt + et - (yinj d + e)] 𝐍⦃T⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hddt #Hdtde #Hdedet
-lapply (cny_narrow … HU1 (d+e) (dt+et-(d+e)) ? ?) -HU1 [ >ymax_pre_sn_comm ] // #HU1
-lapply (cny_inv_lift_ge … HU1 … HLK … HTU1 ?) // -L -U1
->yplus_minus_inj //
-qed-.
-
-lemma cny_inv_lift_subst: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i+1] L ≡ K → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
- ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
-#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW
-lapply (cny_inv_lift_ge_up … HW … HLK … HVW ? ? ?) //
->yplus_O1 <yplus_inj >yplus_SO2
-[ /2 width=1 by ylt_fwd_le_succ1/
-| /2 width=3 by yle_trans/
-| >yminus_succ2 //
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Note: this should be applicable in a forward manner *)
-lemma cny_lift_ge_up: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[yinj d, dt + et - (yinj d + yinj e)] 𝐍⦃T⦄ →
- ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U →
- yinj d ≤ dt → dt ≤ yinj d + yinj e → yinj d + yinj e ≤ dt + et →
- ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄.
-#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #Hdtde #Hdedet
-lapply (cny_lift_be … HT1 … HLK … HTU1 ? ?) // -K -T1
-#HU1 @(cny_narrow … HU1) -HU1 // (**) (* auto fails *)
-qed-.
-
-lemma cny_lift_subst: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i+1] L ≡ K → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
- ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
-#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW
-@(cny_lift_ge_up … HLK … HVW) // >yplus_O1 <yplus_inj >yplus_SO2
-[ >yminus_succ2 //
-| /2 width=3 by yle_trans/
-| /2 width=1 by ylt_fwd_le_succ1/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/delta_equivalence/cpzs.ma".
-include "basic_2/reduction/cpx.ma".
-
-fact destruct_tsort_tsort: ∀k1,k2. ⋆k1 = ⋆k2 → k1 = k2.
-#k1 #k2 #H destruct //
-qed-.
-
-axiom cpzs_inv_subst: ∀I,G,L,K,V1,V2,W2,i.
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ W2 →
- ⦃G, L⦄ ⊢ #i ◆*[O, ∞] W2 → ⦃G, K⦄⊢ V1 ◆*[O, ∞] V2.
-
-axiom cpzs_subst: ∀I,G,L,K,V1,V2,W2,i.
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ W2 →
- ⦃G, K⦄⊢ V1 ◆*[O, ∞] V2 → ⦃G, L⦄ ⊢ #i ◆*[O, ∞] W2.
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-(* Forward lemmas on delta-equivalence for terms ****************************)
-
-lemma cpx_fwd_cpys_cpzs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
- ∀d,e. ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2 ↔ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2
-[ /2 width=1 by conj/
-| #G #L #k #l #_ #d #e @conj #H lapply (next_lt h k)
- [ <(cpzs_inv_sort … H)
- | lapply (cpys_inv_sort1 … H) -H #H >(destruct_tsort_tsort … H)
- ] -H #H elim (lt_refl_false … H)
-| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #d #e @conj #H
-(*
- [ @(cpys_subst … HLK … HVW2) // >yminus_Y_inj /3 width=7 by cpzs_inv_subst/
- | elim (cpys_inv_lref1_ldrop … H … HLK … HVW2) -H /3 width=7 by cpzs_subst/
- ]
-*)
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e
- elim (IHV12 d e) -IHV12 elim (IHT12 (⫯d) e) -IHT12
- #IHTdx #IHTsn #IHVdx #IHVsn @conj #H
- [ elim (cpzs_inv_bind … H) -H /3 width=1 by cpys_bind/
- | elim (cpys_inv_bind1 … H) -H #X1 #X2 #H1 #H2 #H destruct /3 width=1 by cpzs_bind/
- ]
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e
- elim (IHV12 d e) -IHV12 elim (IHT12 (d) e) -IHT12
- #IHTdx #IHTsn #IHVdx #IHVsn @conj #H
- [ elim (cpzs_inv_flat … H) -H /3 width=1 by cpys_flat/
- | elim (cpys_inv_flat1 … H) -H #X1 #X2 #H1 #H2 #H destruct /3 width=1 by cpzs_flat/
- ]
-| #G #L #V #U1 #U2 #T2 #_ #HTU2 #_ #d #e @conj #H
-| #G #L #V1 #T1 #T2 #_ #_ #d #e @conj #H
-| #G #L #V1 #V2 #T1 #HV12 #_ #d #e @conj #H
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #d #e @conj #H
- [ elim (cpzs_inv_flat_bind … H)
- | elim (cpys_inv_flat1 … H) -H #X1 #X2 #H1 #H2 #H destruct
- ]
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #d #e @conj #H
- [ elim (cpzs_inv_flat_bind … H)
- | elim (cpys_inv_flat1 … H) -H #X1 #X2 #H1 #H2 #H destruct
- ]
-]
-
-
\ No newline at end of file
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubsteval_6.ma".
-include "basic_2/relocation/cny.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
-
-definition cpye: ynat → ynat → relation4 genv lenv term term ≝
- λd,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 ∧ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T2⦄.
-
-interpretation "evaluation for context-sensitive extended substitution (term)"
- 'PSubstEval G L T1 T2 d e = (cpye d e G L T1 T2).
-
-(* Basic_properties *********************************************************)
-
-(* Note: this should go in subconversion *)
-lemma leqy_cpye_trans: ∀G,L2,T1,T2,d,e. ⦃G, L2⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ →
- ∀L1. L1 ⊑×[d, e] L2 → L2 ⊑×[d, e] L1 → ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
-#G #L2 #T1 #T2 #d #e *
-/4 width=8 by lsuby_cpys_trans, lsuby_cny_conf, conj/
-qed-.
-
-lemma cpye_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃⋆k⦄.
-/3 width=5 by cny_sort, conj/ qed.
-
-lemma cpye_free: ∀G,L,d,e,i. |L| ≤ i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
-/3 width=6 by cny_lref_free, conj/ qed.
-
-lemma cpye_top: ∀G,L,d,e,i. d + e ≤ yinj i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
-/3 width=6 by cny_lref_top, conj/ qed.
-
-lemma cpye_skip: ∀G,L,d,e,i. yinj i < d → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
-/3 width=6 by cny_lref_skip, conj/ qed.
-
-lemma cpye_gref: ∀G,L,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃§p⦄.
-/3 width=5 by cny_gref, conj/ qed.
-
-lemma cpye_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃ⓑ{a,I}V2.T2⦄.
-#G #L #V1 #V2 #d #e * #HV12 #HV2 #I #T1 #T2 *
-/5 width=8 by cpys_bind, cny_bind, lsuby_cny_conf, lsuby_succ, conj/
-qed.
-
-lemma cpye_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃ⓕ{I}V2.T2⦄.
-#G #L #V1 #V2 #d #e * #HV12 #HV2 #T1 #T2 *
-/3 width=7 by cpys_flat, cny_flat, conj/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpye_inv_sort1: ∀G,L,X,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃X⦄ → X = ⋆k.
-#G #L #X #d #e #k * /2 width=5 by cpys_inv_sort1/
-qed-.
-
-lemma cpye_inv_gref1: ∀G,L,X,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃X⦄ → X = §p.
-#G #L #X #d #e #p * /2 width=5 by cpys_inv_gref1/
-qed-.
-
-lemma cpye_inv_bind1: ∀a,I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ &
- X = ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_bind1 … H1) -H1
-#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_bind … H2) -H2
-/5 width=8 by lsuby_cny_conf, lsuby_succ, ex3_2_intro, conj/
-qed-.
-
-lemma cpye_inv_flat1: ∀I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ &
- X = ⓕ{I}V2.T2.
-#I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_flat1 … H1) -H1
-#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_flat … H2) -H2
-/3 width=5 by ex3_2_intro, conj/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubstevalalt_6.ma".
-include "basic_2/substitution/cpye_lift.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
-
-(* Note: alternative definition of cpye *)
-inductive cpyea: ynat → ynat → relation4 genv lenv term term ≝
-| cpyea_sort : ∀G,L,d,e,k. cpyea d e G L (⋆k) (⋆k)
-| cpyea_free : ∀G,L,d,e,i. |L| ≤ i → cpyea d e G L (#i) (#i)
-| cpyea_top : ∀G,L,d,e,i. d + e ≤ yinj i → cpyea d e G L (#i) (#i)
-| cpyea_skip : ∀G,L,d,e,i. yinj i < d → cpyea d e G L (#i) (#i)
-| cpyea_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d+e →
- ⇩[i] L ≡ K.ⓑ{I}V1 → cpyea (yinj 0) (⫰(d+e-yinj i)) G K V1 V2 →
- ⇧[0, i+1] V2 ≡ W2 → cpyea d e G L (#i) W2
-| cpyea_gref : ∀G,L,d,e,p. cpyea d e G L (§p) (§p)
-| cpyea_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
- cpyea d e G L V1 V2 → cpyea (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
- cpyea d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpyea_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
- cpyea d e G L V1 V2 → cpyea d e G L T1 T2 →
- cpyea d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation
- "evaluation for context-sensitive extended substitution (term) alternative"
- 'PSubstEvalAlt G L T1 T2 d e = (cpyea d e G L T1 T2).
-
-(* Main properties **********************************************************)
-
-theorem cpye_cpyea: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄.
-#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
-#Z #Y #X #IH #G #L * *
-[ #k #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_sort1 … H) -H //
-| #i #HG #HL #HT #T2 #d #e #H destruct
- elim (cpye_inv_lref1 … H) -H *
- /4 width=7 by cpyea_subst, cpyea_free, cpyea_top, cpyea_skip, fqup_lref/
-| #p #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_gref1 … H) -H //
-| #a #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
- elim (cpye_inv_bind1 … H) -H /3 width=1 by cpyea_bind/
-| #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
- elim (cpye_inv_flat1 … H) -H /3 width=1 by cpyea_flat/
-]
-qed.
-
-(* Main inversion properties ************************************************)
-
-theorem cpyea_inv_cpye: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-/2 width=7 by cpye_subst, cpye_flat, cpye_bind, cpye_skip, cpye_top, cpye_free/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma cpye_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
- (∀G,L,d,e,k. R d e G L (⋆k) (⋆k)) →
- (∀G,L,d,e,i. |L| ≤ i → R d e G L (#i) (#i)) →
- (∀G,L,d,e,i. d + e ≤ yinj i → R d e G L (#i) (#i)) →
- (∀G,L,d,e,i. yinj i < d → R d e G L (#i) (#i)) →
- (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[yinj O, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ →
- ⇧[O, i+1] V2 ≡ W2 → R (yinj O) (⫰(d+e-yinj i)) G K V1 V2 → R d e G L (#i) W2
- ) →
- (∀G,L,d,e,p. R d e G L (§p) (§p)) →
- (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
- R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
- R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L T1 T2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #d #e #G #L #T1 #T2 #H elim (cpye_cpyea … H) -G -L -T1 -T2 -d -e
-/3 width=8 by cpyea_inv_cpye/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_cny.ma".
-include "basic_2/substitution/cpys_cpys.ma".
-include "basic_2/substitution/cpye.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
-
-(* Advanced properties ******************************************************)
-
-lemma cpye_cpys_conf: ∀G,L,T,T2,d,e. ⦃G, L⦄ ⊢ T ▶*[d, e] 𝐍⦃T2⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ▶*[d, e] T1 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-#G #L #T #T2 #d #e * #H2 #HT2 #T1 #H1 elim (cpys_conf_eq … H1 … H2) -T
-#T0 #HT10 #HT20 >(cpys_inv_cny1 … HT2 … HT20) -T2 //
-qed-.
-
\ No newline at end of file
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cny_lift.ma".
-include "basic_2/substitution/fqup.ma".
-include "basic_2/substitution/cpys_lift.ma".
-include "basic_2/substitution/cpye.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
-
-(* Advanced properties ******************************************************)
-
-lemma cpye_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ →
- ⇧[O, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃W2⦄.
-#I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK *
-/4 width=13 by cpys_subst, cny_lift_subst, ldrop_fwd_drop2, conj/
-qed.
-
-lemma cpye_total: ∀G,L,T1,d,e. ∃T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
-#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
-#Z #Y #X #IH #G #L * *
-[ #k #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
-| #i #HG #HL #HT #d #e destruct
- elim (ylt_split i d) /3 width=2 by cpye_skip, ex_intro/
- elim (ylt_split i (d+e)) /3 width=2 by cpye_top, ex_intro/
- elim (lt_or_ge i (|L|)) /3 width=2 by cpye_free, ex_intro/
- #Hi #Hide #Hdi elim (ldrop_O1_lt L i) // -Hi
- #I #K #V1 #HLK elim (IH G K V1 … 0 (⫰(d+e-i))) -IH /2 width=2 by fqup_lref/
- #V2 elim (lift_total V2 0 (i+1)) /3 width=8 by ex_intro, cpye_subst/
-| #p #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
-| #a #I #V1 #T1 #HG #HL #HT #d #e destruct
- elim (IH G L V1 … d e) // elim (IH G (L.ⓑ{I}V1) T1 … (⫯d) e) //
- /3 width=2 by cpye_bind, ex_intro/
-| #I #V1 #T1 #HG #HL #HT #d #e destruct
- elim (IH G L V1 … d e) // elim (IH G L T1 … d e) //
- /3 width=2 by cpye_flat, ex_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpye_inv_lref1: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
- ∨∨ |L| ≤ i ∧ T2 = #i
- | d + e ≤ yinj i ∧ T2 = #i
- | yinj i < d ∧ T2 = #i
- | ∃∃I,K,V1,V2. d ≤ yinj i & yinj i < d + e &
- ⇩[i] L ≡ K.ⓑ{I}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
- ⇧[O, i+1] V2 ≡ T2.
-#G #L #T2 #i #d #e * #H1 #H2 elim (cpys_inv_lref1 … H1) -H1
-[ #H destruct elim (cny_inv_lref … H2) -H2
- /3 width=1 by or4_intro0, or4_intro1, or4_intro2, conj/
-| * #I #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
- @or4_intro3 @(ex5_4_intro … HLK … HVT2) (**) (* explicit constructor *)
- /4 width=13 by cny_inv_lift_subst, ldrop_fwd_drop2, conj/
-]
-qed-.
-
-lemma cpye_inv_lref1_free: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
- (∨∨ |L| ≤ i | d + e ≤ yinj i | yinj i < d) → T2 = #i.
-#G #L #T2 #d #e #i #H * elim (cpye_inv_lref1 … H) -H * //
-#I #K #V1 #V2 #Hdi #Hide #HLK #_ #_ #H
-[ elim (lt_refl_false i) -d
- @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/ (**) (* full auto slow: 19s *)
-]
-elim (ylt_yle_false … H) //
-qed-.
-
-lemma cpye_inv_lref1_lget: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
- ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 →
- ∨∨ d + e ≤ yinj i ∧ T2 = #i
- | yinj i < d ∧ T2 = #i
- | ∃∃V2. d ≤ yinj i & yinj i < d + e &
- ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
- ⇧[O, i+1] V2 ≡ T2.
-#G #L #T2 #d #e #i #H #I #K #V1 #HLK elim (cpye_inv_lref1 … H) -H *
-[ #H elim (lt_refl_false i) -T2 -d
- @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/
-| /3 width=1 by or3_intro0, conj/
-| /3 width=1 by or3_intro1, conj/
-| #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
- lapply (ldrop_mono … HLY … HLK) -HLY -HLK #H destruct
- /3 width=3 by or3_intro2, ex4_intro/
-]
-qed-.
-
-lemma cpye_inv_lref1_subst_ex: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
- ∀I,K,V1. d ≤ yinj i → yinj i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V1 →
- ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
- ⇧[O, i+1] V2 ≡ T2.
-#G #L #T2 #d #e #i #H #I #K #V1 #Hdi #Hide #HLK
-elim (cpye_inv_lref1_lget … H … HLK) -H * /2 width=3 by ex2_intro/
-#H elim (ylt_yle_false … H) //
-qed-.
-
-lemma cpye_inv_lref1_subst: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
- ∀I,K,V1,V2. d ≤ yinj i → yinj i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ T2 →
- ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄.
-#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2
-elim (cpye_inv_lref1_subst_ex … H … HLK) -H -HLK //
-#X2 #H0 #HXT2 lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma cpye_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdetd
-elim (cpys_inv_lift1_le … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
-lapply (cny_inv_lift_le … HU2 … HLK … HTU2 ?) -L
-/3 width=3 by ex2_intro, conj/
-qed-.
-
-lemma cpye_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet
-elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
-lapply (cny_inv_lift_be … HU2 … HLK … HTU2 ? ?) -L
-/3 width=3 by ex2_intro, conj/
-qed-.
-
-lemma cpye_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- yinj d + e ≤ dt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdedt
-elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
-lapply (cny_inv_lift_ge … HU2 … HLK … HTU2 ?) -L
-/3 width=3 by ex2_intro, conj/
-qed-.
-
-lemma cpye_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] 𝐍⦃T2⦄ &
- ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
-lapply (cny_inv_lift_ge_up … HU2 … HLK … HTU2 ? ? ?) -L
-/3 width=3 by ex2_intro, conj/
-qed-.
-
-lemma cpye_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] 𝐍⦃W2⦄ →
- ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
- d ≤ yinj i → i < d + e →
- ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ & ⇧[O, i+1] V2 ≡ W2.
-#G #L #W1 #W2 #d #e * #HW12 #HW2 #K #V1 #i #HLK #HVW1 #Hdi #Hide
-elim (cpys_inv_lift1_subst … HW12 … HLK … HVW1) -W1 // #V2 #HV12 #HVW2
-lapply (cny_inv_lift_subst … HLK HW2 HVW2) -L
-/3 width=3 by ex2_intro, conj/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cny.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Inversion lemmas on normality for extended substitution ******************)
-
-lemma cpys_inv_cny1: ∀G,L,T1,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → T1 = T2.
-#G #L #T1 #d #e #HT1 #T2 #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct <(HT1 … HT2) -T //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/pdeltaconvstar_6.ma".
-include "basic_2/substitution/cpye_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED DELTA-EQUIVALENCE FOR TERMS *******************)
-
-definition cpzs: ynat → ynat → relation4 genv lenv term term ≝
- λd,e,G,L,T1,T2.
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T⦄ & ⦃G, L⦄ ⊢ T2 ▶*[d, e] 𝐍⦃T⦄.
-
-interpretation "context-sensitive extended delta-equivalence (term)"
- 'PDeltaConvStar G L T1 d e T2 = (cpzs d e G L T1 T2).
-
-(* Basic properties **********************************************************)
-
-lemma cpye_div: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T2 ▶*[d, e] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2.
-/2 width=3 by ex2_intro/ qed.
-
-lemma cpzs_refl: ∀G,L,d,e. reflexive … (cpzs d e G L).
-#G #L #d #e #T elim (cpye_total G L T d e) /2 width=3 by cpye_div/
-qed.
-
-lemma cpzs_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ◆*[⫯d, e] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ◆*[d, e] ⓑ{a,I}V2.T2.
-#G #L #V1 #V2 #d #e * #V #HV1 #HV2 #I #T1 #T2 *
-/5 width=10 by cpye_div, cpye_bind, leqy_cpye_trans, cny_bind, lsuby_succ/
-qed.
-
-lemma cpzs_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ◆*[d, e] ⓕ{I}V2.T2.
-#G #L #V1 #V2 #d #e * #V #HV1 #HV2 #T1 #T2 *
-/3 width=5 by cpye_div, cpye_flat, cny_flat/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpzs_inv_sort: ∀G,L,d,e,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ◆*[d, e] ⋆k2 → k1 = k2.
-#G #L #d #e #k1 #k2 * #X #H1 #H2
-lapply (cpye_inv_sort1 … H1) -H1 #H1
-lapply (cpye_inv_sort1 … H2) -H2 #H2
-destruct //
-qed-.
-
-lemma cpzs_inv_bind: ∀a1,a2,I1,I2,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ ⓑ{a1,I1}V1.T1 ◆*[d, e] ⓑ{a2,I2}V2.T2 →
- ∧∧ a1 = a2 & I1 = I2
- & ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 & ⦃G, L.ⓑ{I1}V1⦄ ⊢ T1 ◆*[⫯d, e] T2.
-#a1 #a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2
-elim (cpye_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1
-elim (cpye_inv_bind1 … H2) -H2 #W2 #U2 #HW12 #HU12 #H2
-destruct /5 width=8 by cpye_div, leqy_cpye_trans, lsuby_succ, and4_intro/
-qed-.
-
-lemma cpzs_inv_flat: ∀I1,I2,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ ⓕ{I1}V1.T1 ◆*[d, e] ⓕ{I2}V2.T2 →
- ∧∧ I1 = I2
- & ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 & ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2.
-#I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2
-elim (cpye_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1
-elim (cpye_inv_flat1 … H2) -H2 #W2 #U2 #HW12 #HU12 #H2
-destruct /3 width=3 by cpye_div, and3_intro/
-qed-.
-
-lemma cpzs_inv_flat_bind: ∀a2,I1,I2,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ ⓕ{I1}V1.T1 ◆*[d, e] ⓑ{a2,I2}V2.T2 → ⊥.
-#a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2
-elim (cpye_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1
-elim (cpye_inv_bind1 … H2) -H2 #W2 #U2 #HW12 #HU12 #H2
-destruct
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpye_lift.ma".
-include "basic_2/substitution/lleq_alt.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Forward lemmas on evaluation for extended substitution *******************)
-
-lemma lleq_fwd_cpye: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → ∀G,T1. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] 𝐍⦃T1⦄ →
- ∀T2. ⦃G, L2⦄ ⊢ T ▶*[d, ∞] 𝐍⦃T2⦄ → T1 = T2.
-#L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d
-[ #L1 #L2 #d #k #_ #G #T1 #H1 #T2 #H2
- >(cpye_inv_sort1 … H1) -H1 >(cpye_inv_sort1 … H2) -H2 //
-| #L1 #L2 #d #i #_ #Hid #G #T1 #H1 #T2 #H2
- >(cpye_inv_lref1_free … H1) -H1 [ >(cpye_inv_lref1_free … H2) -H2 ]
- /2 width=1 by or3_intro2/
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #_ #IHV #G #T1 #H1 #T2 #H2
- elim (cpye_inv_lref1_subst_ex … H1 … HLK1) -H1 -HLK1 //
- elim (cpye_inv_lref1_subst_ex … H2 … HLK2) -H2 -HLK2 //
- >yminus_Y_inj #V2 #HV2 #HVT2 #V1 #HV1 #HVT1
- lapply (IHV … HV1 … HV2) -IHV -HV1 -HV2 #H destruct /2 width=5 by lift_mono/
-| #L1 #L2 #d #i #_ #HL1 #HL2 #G #T1 #H1 #T2 #H2
- >(cpye_inv_lref1_free … H1) -H1 [ >(cpye_inv_lref1_free … H2) -H2 ]
- /2 width=1 by or3_intro0/
-| #L1 #L2 #d #p #_ #G #T1 #H1 #T2 #H2
- >(cpye_inv_gref1 … H1) -H1 >(cpye_inv_gref1 … H2) -H2 //
-| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2
- elim (cpye_inv_bind1 … H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct
- elim (cpye_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /3 width=3 by eq_f2/
-| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2
- elim (cpye_inv_flat1 … H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct
- elim (cpye_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /3 width=3 by eq_f2/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpye_cpye.ma".
-include "basic_2/reduction/lpx_cpys.ma".
-
-axiom cpx_cpys_conf_lpx: ∀h,g,G,d,e.
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h, g] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, g] L1 →
- ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*[d, e] T2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡[h, g] T.
-
-(* SN EXTENDED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *********************)
-
-(* Forward lemmas on evaluation for extended substitution *******************)
-
-lemma cpx_cpys_cpye_fwd_lpx: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] U1 →
- ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
- ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2.
-#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e #HTU1
-elim (cpx_cpys_conf_lpx … HT12 … HL12 … HTU1) -T1
-/3 width=9 by cpx_cpys_trans_lpx, cpye_cpys_conf/
-qed-.
-
-lemma cpx_cpye_fwd_lpx: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
- ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
- ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2.
-#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e *
-/2 width=9 by cpx_cpys_cpye_fwd_lpx/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpye_lift.ma".
-include "basic_2/reduction/lpx_cpye.ma".
-include "basic_2/computation/cpxs_cpxs.ma".
-include "basic_2/computation/lpxs.ma".
-
-(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
-
-(* Forward lemmas on evaluation for extended substitution *******************)
-
-lemma cpx_cpye_fwd_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
- ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
- ⦃G, L1⦄ ⊢ U1 ➡*[h, g] U2.
-#h #g #G #L1 #L2 #H @(lpxs_ind_dx … H) -L1
-[ /3 width=9 by cpx_cpxs, cpx_cpye_fwd_lpx/
-| #L1 #L #HL1 #_ #IHL2 #T1 #T2 #HT12 #U1 #d #e #HTU1 #U2 #HTU2
- elim (cpye_total G L T2 d e) #X2 #HTX2
- lapply (cpx_cpye_fwd_lpx … HT12 … HL1 … HTU1 … HTX2) -T1
- /4 width=9 by lpx_cpxs_trans, cpxs_strap2/ (**) (* full auto too long: 41s *)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ◆ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PDeltaConvStar $G $L $T1 $d $e $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PSubstEval $G $L $T1 $T2 $d $e }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PSubstEvalAlt $G $L $T1 $T2 $d $e }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ▶ break [ term 46 d , break term 46 e ] 𝐍 break ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'PSubstNormal $G $L $T $d $e }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/cofreestar_4.ma".
-include "basic_2/relocation/lift_neg.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-definition cofrees: relation4 ynat nat lenv term ≝
- λd,i,L,U1. ∀U2. ⦃⋆, L⦄ ⊢ U1 ▶*[d, ∞] U2 → ∃T2. ⇧[i, 1] T2 ≡ U2.
-
-interpretation
- "context-sensitive exclusion from free variables (term)"
- 'CoFreeStar L i d T = (cofrees d i L T).
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cofrees_fwd_lift: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
-/2 width=1 by/ qed-.
-
-lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
-#a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
-#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_bind_dx: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
-#a #I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓑ{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1
-#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_flat_sn: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
-#I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓕ{I}W2.U)) /2 width=1 by cpys_flat/ -W1
-#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_flat_dx: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-#I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓕ{I}W.U2)) /2 width=1 by cpys_flat/ -U1
-#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cofrees_inv_gen: ∀L,U,U0,d,i. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 → (∀T. ⇧[i, 1] T ≡ U0 → ⊥) →
- L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥.
-#L #U #U0 #d #i #HU0 #HnU0 #HU elim (HU … HU0) -L -U -d /2 width=2 by/
-qed-.
-
-lemma cofrees_inv_lref_eq: ∀L,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃#i⦄ → ⊥.
-#L #d #i #H elim (H (#i)) -H //
-#X #H elim (lift_inv_lref2_be … H) -H //
-qed-.
-
-lemma cofrees_inv_bind: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
-/3 width=8 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-.
-
-lemma cofrees_inv_flat: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-/3 width=7 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-.
-
-(* Basic Properties *********************************************************)
-
-lemma cofrees_lsuby_conf: ∀L1,U,d,i. L1 ⊢ i ~ϵ 𝐅*[d]⦃U⦄ →
- ∀L2. L1 ⊆[d, ∞] L2 → L2 ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-/3 width=3 by lsuby_cpys_trans/ qed-.
-
-lemma cofrees_sort: ∀L,d,i,k. L ⊢ i ~ϵ 𝐅*[d]⦃⋆k⦄.
-#L #d #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_gref: ∀L,d,i,p. L ⊢ i ~ϵ 𝐅*[d]⦃§p⦄.
-#L #d #i #p #X #H >(cpys_inv_gref1 … H) -X /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_bind: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d] ⦃V⦄ →
- ∀I,T. L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ →
- ∀a. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄.
-#L #W1 #d #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 … H) -H
-#W2 #U2 #HW12 #HU12 #H destruct
-elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/
-qed.
-
-lemma cofrees_flat: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ∀T. L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ →
- ∀I. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄.
-#L #W1 #d #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 … H) -H
-#W2 #U2 #HW12 #HU12 #H destruct
-elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/
-qed.
-
-lemma cofrees_cpy_trans: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
- ∀i. L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄.
-/3 width=3 by cpys_strap2/ qed-.
-
-axiom cofrees_dec: ∀L,T,d,i. Decidable (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄).
-
-(* Basic negated properties *************************************************)
-
-lemma frees_cpy_div: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
- ∀i. (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → ⊥).
-/3 width=7 by cofrees_cpy_trans/ qed-.
-
-(* Basic negated inversion lemmas *******************************************)
-
-lemma frees_inv_bind: ∀a,I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄ → ⊥) →
- (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ → ⊥).
-#a #I #L #W #U #d #i #H elim (cofrees_dec L W d i)
-/4 width=9 by cofrees_bind, or_intror, or_introl/
-qed-.
-
-lemma frees_inv_flat: ∀I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄ → ⊥) →
- (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥).
-#I #L #W #U #d #H elim (cofrees_dec L W d)
-/4 width=8 by cofrees_flat, or_intror, or_introl/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cpy_nlift.ma".
-include "basic_2/substitution/cofrees_lift.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-(* Alternative definition of frees_ge ***************************************)
-
-lemma nlift_frees: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
-#L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
-qed-.
-
-lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
- (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
- ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
- (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
-#L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
-#U1 #U2 #HU12 #HU2 *
-[ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
- * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
-| * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
- * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
- lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
- #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
- #HK12
- @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
- @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
- >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
-]
-qed-.
-
-lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
- (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
- (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
- ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
-#R #IH1 #IH2 #d #i #L #U
-generalize in match d; -d generalize in match i; -i
-@(f2_ind … rfw … L U) -L -U
-#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
--IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_lift.ma".
-include "basic_2/substitution/cofrees.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cofrees_inv_lref_be: ∀L,d,i,j. L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → d ≤ yinj j → j < i →
- ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #i #j #Hj #Hdj #Hji #I #K #W1 #HLK #W2 #HW12 elim (lift_total W2 0 (j+1))
-#X2 #HWX2 elim (Hj X2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
-#Z2 #HZX2 elim (lift_div_le … HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/
->minus_plus <plus_minus_m_m //
-qed-.
-
-lemma cofrees_inv_be: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∀j. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
- ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → d ≤ yinj j → j < i → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH #k #_ #d #i #_ #j #H elim (H (⋆k)) -H //
-| -IH #j #_ #d #i #Hi0 #j0 #H <(nlift_inv_lref_be_SO … H) -j0
- /2 width=9 by cofrees_inv_lref_be/
-| -IH #p #_ #d #i #_ #j #H elim (H (§p)) -H //
-| #a #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
- elim (cofrees_inv_bind … H1) -H1 #HW #HU
- elim (nlift_inv_bind … H2) -H2 [ -HU /3 width=9 by/ ]
- -HW #HnU lapply (IH … HU … HnU I K V ? ? ?)
- /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/ -a -I -J -L -W -U -d
- >minus_plus_plus_l //
-| #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
- elim (cofrees_inv_flat … H1) -H1 #HW #HU
- elim (nlift_inv_flat … H2) -H2 [ /3 width=9 by/ ]
- #HnU @(IH … HU … HnU … HLK) // (**) (* full auto fails *)
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cofrees_lref_skip: ∀L,d,i,j. j < i → yinj j < d → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I #K #W1 #W2 #Hdj elim (ylt_yle_false … Hdj) -i -I -L -K -W1 -W2 -X //
-]
-qed.
-
-lemma cofrees_lref_lt: ∀L,d,i,j. i < j → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
-| * #I #K #V1 #V2 #_ #_ #_ #H -I -L -K -V1 -d
- elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
-]
-qed.
-
-lemma cofrees_lref_gt: ∀I,L,K,W,d,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
- K ⊢ (i-j-1) ~ϵ 𝐅*[O]⦃W⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
- #H destruct elim (HW1 … HW12) -I -K -W1 -d
- #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
- >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
-]
-qed.
-
-lemma cofrees_lref_free: ∀L,d,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#L #d #i #j #Hj #Hji #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I #K #W1 #W2 #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
- #H elim (lt_refl_false j) -d -i -K -W1 -W2 -X /2 width=3 by lt_to_le_to_lt/
-]
-qed.
-
-(* Advanced negated inversion lemmas ****************************************)
-
-lemma frees_inv_lref_gt: ∀L,d,i,j. j < i → (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) →
- ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*[0]⦃W⦄ → ⊥) & d ≤ yinj j.
-#L #d #i #j #Hji #H elim (ylt_split j d) #Hjd
-[ elim H -H /2 width=6 by cofrees_lref_skip/
-| elim (lt_or_ge j (|L|)) #Hj
- [ elim (ldrop_O1_lt … Hj) -Hj /4 width=10 by cofrees_lref_gt, ex3_3_intro/
- | elim H -H /2 width=6 by cofrees_lref_free/
- ]
-]
-qed-.
-
-lemma frees_inv_lref_free: ∀L,d,i,j. (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) → |L| ≤ j → j = i.
-#L #d #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
-#Hij elim H -H /2 width=6 by cofrees_lref_lt, cofrees_lref_free/
-qed-.
-
-lemma frees_inv_gen: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
- ∃∃U0. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH #k #_ #d #i #H elim H -H //
-| #j #Hn #d #i #H elim (lt_or_eq_or_gt i j)
- [ -n #Hij elim H -H /2 width=5 by cofrees_lref_lt/
- | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
- | #Hji elim (frees_inv_lref_gt … H) // -H
- #I #K #W1 #HLK #H #Hdj elim (IH … H) /2 width=3 by ldrop_fwd_rfw/ -H -n
- #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
- #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
- #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
- >minus_plus <plus_minus_m_m //
- ]
-| -IH #p #_ #d #i #H elim H -H //
-| #a #I #W #U #Hn #d #i #H elim (frees_inv_bind … H) -H
- #H elim (IH … H) // -H -n
- /4 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn, ex2_intro/
-| #I #W #U #Hn #d #i #H elim (frees_inv_flat … H) -H
- #H elim (IH … H) // -H -n
- /4 width=9 by cpys_flat, nlift_flat_dx, nlift_flat_sn, ex2_intro/
-]
-qed-.
-
-lemma frees_ind: ∀L,d,i. ∀R:predicate term.
- (∀U1. (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) → R U1) →
- (∀U1,U2. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 → (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → R U2 → R U1) →
- ∀U. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U.
-#L #d #i #R #IH1 #IH2 #U1 #H elim (frees_inv_gen … H) -H
-#U2 #H #HnU2 @(cpys_ind_dx … H) -U1 /4 width=8 by cofrees_inv_gen/
-qed-.
-
-(* Advanced negated properties **********************************************)
-
-lemma frees_be: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
- ∀i. j < i → (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) →
- ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
- ∀d. d ≤ yinj j → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
-/4 width=11 by cofrees_inv_be/ qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma cofrees_lift_be: ∀d0,e0,i. d0 ≤ i → i ≤ d0 + e0 →
- ∀L,K,s. ⇩[s, d0, e0+1] L ≡ K → ∀T,U. ⇧[d0, e0+1] T ≡ U →
- ∀d. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-#d0 #e0 #i #Hd0i #Hide0 #L #K #s #HLK #T1 #U1 #HTU1 #d #U2 #HU12
-elim (yle_split d0 d) #H1
-[ elim (yle_split d (d0+e0+1)) #H2
- [ letin cpys_inv ≝ cpys_inv_lift1_ge_up
- | letin cpys_inv ≝ cpys_inv_lift1_ge
- ]
-| letin cpys_inv ≝ cpys_inv_lift1_be
-]
-elim (cpys_inv … HU12 … HLK … HTU1) // #T2 #_ #HTU2 -s -L -K -U1 -T1 -d
-elim (lift_split … HTU2 i e0) /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_lift_ge: ∀d0,e0,i. d0 + e0 ≤ i →
- ∀L,K,s. ⇩[s, d0, e0] L ≡ K → ∀T,U. ⇧[d0, e0] T ≡ U →
- ∀d. K ⊢ i-e0 ~ϵ 𝐅*[d-yinj e0]⦃T⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-#d0 #e0 #i #Hde0i #L #K #s #HLK #T1 #U1 #HTU1 #d #HT1 #U2 #HU12
-elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i
-elim (yle_split d0 d) #H1
-[ elim (yle_split d (d0+e0)) #H2
- [ elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) // >yplus_inj >yminus_Y_inj #T2 #HT12
- lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12
- | elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) // #T2
- ]
-| elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) // >yminus_Y_inj #T2 #HT12
- lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) // -HT12
-]
--s -L #HT12 #HTU2
-elim (HT1 … HT12) -T1 #V2 #HVT2
-elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_intro/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L ⊢ break term 46 i ~ ϵ 𝐅 * [ break term 46 d ] ⦃ break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'CoFreeStar $L $i $d $T }.
+++ /dev/null
-lemma cpcs_beta_dx: ∀a,L,V1,V2,W1,W2,T1,T2.
- L ⊢ V1 ⬌* V2 → L ⊢ W1 ⬌* W2 → L.ⓛW2 ⊢ T1 ⬌* T2 →
- L ⊢ ⓐV1.ⓛ{a}W1.T1 ⬌* ⓓ{a}ⓝW2.V2.T2.
-#a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12
-@(cpcs_cpr_strap1 … (ⓐV2.ⓛ{a}W2.T2)) /2 width=1/ /3 width=1/
-qed.
-
-lemma cpcs_beta_sn: ∀a,L,V1,V2,W1,W2,T1,T2.
- L ⊢ V1 ⬌* V2 → L ⊢ W1 ⬌* W2 → L.ⓛW1 ⊢ T1 ⬌* T2 →
- L ⊢ ⓐV1.ⓛ{a}W1.T1 ⬌* ⓓ{a}ⓝW2.V2.T2.
-#a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12
-lapply (lsubr_cpcs_trans … HT12 (L.ⓓⓝW1.V1) ?) /2 width=1/ #H2T12
-@(cpcs_cpr_strap2 … (ⓓ{a}ⓝW1.V1.T1)) /2 width=1/ -HT12 /3 width=1/
-qed.
+++ /dev/null
-(* Advanced properties ******************************************************)
-
-lemma cpds_cpes_dx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T2.
-#h #g #G #L #T1 #T2 #l * /3 width=6 by cpcs_cprs_dx, ex4_2_intro/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpes_inv_abst2: ∀h,g,a,G,L,W1,T1,T,l. ⦃G, L⦄ ⊢ T •*⬌*[h, g, l] ⓛ{a}W1.T1 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ T •*➡*[h, g, l] ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
-#h #g #a #G #L #W1 #T1 #T #l2 * #T0 #l1 #Hl21 #HT #HT0 #H
-elim (cpcs_inv_abst2 … H) -H /3 width=6 by ex4_2_intro, ex2_2_intro/
-qed-.
-
-(****************************************************************************)
-
-lemma sta_cpcs_cpes: ∀h,g,G,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h] T →
- ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, 1] T2.
-/3 width=6 by sta_lstas, ex4_2_intro/ qed.
-
-lemma lstas_cpes: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T2.
-/2 width=6 by ex4_2_intro/ qed.
-
-lemma cpes_strap1: ∀h,g,G,L,T1,T,T2,l.
- ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T2.
-#h #g #G #L #T1 #T #T2 #l * /3 width=8 by cpcs_strap1, ex4_2_intro/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_dx_ldrop.ma".
-include "basic_2/static/aaa_lift.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about dx parallel unfold **************************************)
-
-lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #H elim H -L1 -T1 -A
-[ #L1 #k #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H //
-| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lt_or_ge i d) #Hdi
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- | elim (lt_or_ge i (d + e)) #Hide
- [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- | -Hdi
- lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/
- ]
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
-| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-]
-qed.
-
-lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/3 width=7/ qed.
-
-lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
-/2 width=7/ qed.
-
-lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A.
-/2 width=7/ qed.
-
-lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A.
-/2 width=7/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/static/aaa_ltpss_dx.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma aaa_ltpss_sn_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
-#L1 #T #A #HT #L2 #d #e #HL12
-lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
-@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/
-qed.
-
-lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/3 width=5/ qed.
-
-lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/3 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cif.ma".
-include "basic_2/reducibility/cnf_lift.ma".
-
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
-
-(* Main properties **********************************************************)
-
-lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
- elim (cif_inv_delta … HLK ?) //
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
- elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
- lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
- elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
-]
-qed.
-
-lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 #HT1
-lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
-qed.
-
-lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#T1 #T2 #H elim H -T1 -T2
-[ //
-| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
- [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
- >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
- | elim (cif_inv_ri2 … H) /2 width=1/
- ]
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
- elim (cif_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
- [ lapply (tps_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
- elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
- lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
- lapply (IHT1 … HT1) -IHT1 #H destruct
- lapply (tps_cif_eq … HT2 ?) -HT2 //
- | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
- elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
- ]
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
- elim (cif_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #V1 #T1 #T #T2 #_ #_ #_ #L #H
- elim (cif_inv_ri2 … H) /2 width=1/
-| #V1 #T1 #T2 #_ #_ #L #H
- elim (cif_inv_ri2 … H) /2 width=1/
-]
-qed.
-
-lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
-#L #T1 #T2 * #T0 #HT10 #HT02 #HT1
-lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
-qed.
-
-theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
-/3 width=3/ qed.
-
-(* Note: this property is unusual *)
-lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
-#L #T #H elim H -L -T
-[ #L #K #V #i #HLK #H
- elim (cnf_inv_delta … HLK H)
-| #L #V #T #_ #IHV #H
- elim (cnf_inv_appl … H) -H /2 width=1/
-| #L #V #T #_ #IHT #H
- elim (cnf_inv_appl … H) -H /2 width=1/
-| #I #L #V #T * #H1 #H2 destruct
- [ elim (cnf_inv_zeta … H2)
- | elim (cnf_inv_tau … H2)
- ]
-|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
- [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
- |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
- ]
-| #a #L #V #W #T #H
- elim (cnf_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #a #L #V #W #T #H
- elim (cnf_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-]
-qed.
-
-theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
-/2 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cprs_ltpss_dx.ma".
-include "basic_2/equivalence/cpcs_cpcs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Properties concerning dx partial unfold on local environments ************)
-
-lemma cpcs_ltpss_dx_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
- ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
-elim (cprs_ltpss_dx_conf … HT1 … HL12) -HT1 #U1 #HTU1 #H1
-elim (cprs_ltpss_dx_conf … HT2 … HL12) -L1 #U2 #HTU2 #H2
-elim (tpss_conf_eq … H1 … H2) -T #U #HU1 #HU2
-lapply (cprs_tpss_trans … HTU1 … HU1) -U1
-lapply (cprs_tpss_trans … HTU2 … HU2) -U2 /2 width=3/
-qed-.
-
-lemma cpcs_ltpss_dx_tpss_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
- ∀T,T2. L1 ⊢ T ⬌* T2 →
- ∀T1. L2 ⊢ T ▶* [d, e] T1 →
- L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #d #e #HL12 #T #T2 #HT2 #T1 #HT1
-lapply (cpcs_ltpss_dx_conf … HL12 … HT2) -L1 #HT2
-lapply (cpcs_tpss_conf … HT1 … HT2) -T //
-qed-.
-
-lemma cpcs_ltpss_dx_tpss2_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
- ∀T1,T2. L1 ⊢ T1 ⬌* T2 →
- ∀T3. L2 ⊢ T1 ▶* [d, e] T3 →
- ∀T4. L2 ⊢ T2 ▶* [d, e] T4 →
- L2 ⊢ T3 ⬌* T4.
-#L1 #L2 #d #e #HL12 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24
-lapply (cpcs_ltpss_dx_tpss_conf … HL12 … HT12 … HT13) -L1 -T1 #HT32
-lapply (cpcs_tpss_strap1 … HT32 … HT24) -T2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/equivalence/cpcs_ltpss_dx.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Properties concerning sn partial unfold on local environments ************)
-
-lemma cpcs_ltpss_sn_conf: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL1 #T1 #U1 #HTU1
-lapply (IHL1 … HTU1) -IHL1 -HTU1 #HTU1
-lapply (cpcs_ltpss_dx_conf … HL2 … HTU1) -HTU1 -HL2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Basic_1: includes: pr2_delta1 *)
-definition cpr: lenv → relation term ≝
- λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2.
-
-interpretation
- "context-sensitive parallel reduction (term)"
- 'PRed L T1 T2 = (cpr L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed-.
-
-(* Basic_1: was by definition: pr2_free *)
-lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
-/2 width=3/ qed.
-
-lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed.
-
-lemma cpr_refl: ∀L,T. L ⊢ T ➡ T.
-/2 width=1/ qed.
-
-(* Note: new property *)
-(* Basic_1: was only: pr2_thin_dx *)
-lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
- L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-lemma cpr_cast: ∀L,V,T1,T2.
- L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2.
-#L #V #T1 #T2 * /3 width=3/
-qed.
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-(* Basic_1: was only: pr2_change *)
-lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
-lapply (tpss_lsubr_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: pr2_gen_csort *)
-lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2.
-#T1 #T2 * #T #HT normalize #HT2
-<(tpss_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-(* Basic_1: was: pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
-#L #T2 #k * #X #H
->(tpr_inv_atom1 … H) -H #H
->(tpss_inv_sort1 … H) -H //
-qed-.
-
-(* Basic_1: was: pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → (
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U2 = ⓝV2. T2
- ) ∨ L ⊢ T1 ➡ U2.
-#L #V1 #T1 #U2 * #X #H #HU2
-elim (tpr_inv_cast1 … H) -H /3 width=3/
-* #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
- ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
-#I #L #V1 #T1 #T * #X #H1 #H2 #b
-elim (tpr_fwd_bind1_minus … H1 b) -H1 #V0 #T0 #HT10 #H destruct
-elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
-qed-.
-
-lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L #L1 #T1 #T * #X #H1 #H2
-elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct
-elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/
-qed-.
-
-(* Basic_1: removed theorems 6:
- pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
- pr2_gen_ctail pr2_ctail
- Basic_1: removed local theorems 3:
- pr2_free_free pr2_free_delta pr2_delta_delta
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_aaa.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_cpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A.
-#L #T1 #A #HT1 #T2 * /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/tpr_tpr.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpr_bind_sn: ∀a,I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
- L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
-@ex2_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *)
-qed.
-
-(* Basic_1: was only: pr2_gen_cbind *)
-lemma cpr_bind_dx: ∀a,I,L,V1,V2,T1,T2. V1 ➡ V2 → L. ⓑ{I} V2 ⊢ T1 ➡ T2 →
- L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
-elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
-lapply (tpss_lsubr_trans … HT0 (⋆. ⓑ{I} V2) ?) -HT0 /2 width=1/ #HT0
-lapply (tpss_inv_SO2 … HT0) -HT0 #HT0
-@ex2_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
-qed.
-
-(* Basic_1: was only: pr2_head_1 *)
-lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
- L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
-* /2 width=1/ /3 width=1/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @@ T1 ➡ L @@ T2.
-#L elim L -L
-[ #T1 #T2 #HT12 @(cpr_inv_atom … HT12)
-| normalize /3 width=1/
-].
-qed-.
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: pr2_confluence *)
-theorem cpr_conf: ∀L,U0,T1,T2. L ⊢ U0 ➡ T1 → L ⊢ U0 ➡ T2 →
- ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
-#L #U0 #T1 #T2 * #U1 #HU01 #HUT1 * #U2 #HU02 #HUT2
-elim (tpr_conf … HU01 HU02) -U0 #U #HU1 #HU2
-elim (ltpr_tpr_tpss_conf ? L … HU1 … HUT1) -U1 // #U1 #HTU1 #HU1
-elim (ltpr_tpr_tpss_conf ? L … HU2 … HUT2) -U2 // #U2 #HTU2 #HU2
-elim (tpss_conf_eq … HU1 … HU2) -U /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/reducibility/tpr_lift.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
- ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
- ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
-#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
-lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
-@ex2_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
-qed.
-
-lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
- ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
-#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 #a #I
-lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
-lapply (tpss_lsubr_trans … HT02 (L.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
-@(ex2_intro … (ⓑ{a,I}V0.T0)) /2 width=1/ (* explicit constructors *)
-qed.
-
-lemma cpr_beta: ∀a,L,V1,V2,W,T1,T2.
- L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
-#a #L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
-lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
-lapply (tpss_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
-@(ex2_intro … (ⓓ{a}V.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
-qed.
-
-lemma cpr_beta_dx: ∀a,L,V1,V2,W,T1,T2.
- V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
-/3 width=1/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was: pr2_gen_lref *)
-lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
- T2 = #i ∨
- ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
- ⇧[0, i + 1] T1 ≡ T2 &
- i < |L|.
-#L #T2 #i * #X #H
->(tpr_inv_atom1 … H) -H #H
-elim (tpss_inv_lref1 … H) -H /2 width=1/
-* /3 width=6/
-qed-.
-
-(* Basic_1: was pr2_gen_abbr *)
-lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡ U2 →
- (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 &
- L. ⓓV ⊢ T1 ➡ T2 &
- U2 = ⓓ{a}V2. T2
- ) ∨
- ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2 & a = true.
-#a #L #V1 #T1 #Y * #X #H1 #H2
-elim (tpr_inv_abbr1 … H1) -H1 *
-[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct
- elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct
- lapply (tps_lsubr_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
- lapply (tps_weak_full … HT0) -HT0 #HT0
- lapply (tpss_lsubr_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02
- lapply (tpss_weak_full … HT02) -HT02 #HT02
- lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/
-| #T2 #HT12 #HXT2 #H destruct
- elim (lift_total Y 0 1) #Z #HYZ
- lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H
- lapply (cpr_intro … HT12 … H) -T2 /3 width=3/
-]
-qed-.
-
-(* Basic_1: was: pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡ U2 → ∀I,W.
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
-#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W
-elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
-lapply (tpss_lsubr_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
-qed-.
-
-(* Basic_1: was pr2_gen_appl *)
-lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2
- | ∃∃a,V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2.
-#L #V1 #U0 #Y * #X #H1 #H2
-elim (tpr_inv_appl1 … H1) -H1 *
-[ #V #U #HV1 #HU0 #H destruct
- elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
-| #a #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
- elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=9/
-| #a #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
- elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
- elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
- elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
- lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=13/
-]
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U = ⓐV2. T2.
-#L #V1 #T1 #U #H #HT1
-elim (cpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
- ∃∃V2,T2. L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
- U2 = ⓛ{a}V2.T2.
-#a #L #V1 #T1 #U2 * #U #H #HU2 #b #I #W
-elim (tpr_fwd_abst1 … H b I W) -H #V #T #HT1 #H destruct
-elim (tpss_inv_bind1 … HU2) -HU2 #V2 #T2 #_ #HT2
-lapply (tpss_lsubr_trans … HT2 (L.ⓑ{I}W) ?) -HT2 /2 width=1/ /4 width=5/
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr2_lift *)
-lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
- K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
-elim (lt_or_ge (|K|) d) #HKd
-[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
-| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
-]
-qed.
-
-(* Basic_1: was: pr2_gen_lift *)
-lemma cpr_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
-elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #T #HTU #T1
-elim (lt_or_ge (|L|) d) #HLd
-[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK /2 width=2/
- /3 width=7 by ex2_intro, cpr_intro/
-| elim (lt_or_ge (|L|) (d + e)) #HLde
- [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // /2 width=2/
- /3 width=7 by ex2_intro, cpr_intro/
- | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK //
- /3 width=7 by ex2_intro, cpr_intro/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_tpss.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning parallel unfold on terms ***************************)
-
-(* Note: we could invoke tpss_weak_full instead of ltpr_fwd_length *)
-(* Basic_1: was only: pr2_subst1 *)
-lemma cpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#L1 #L2 #HL12 #T1 #T2 * #T #HT1 #HT2 #d #e #U1 #HTU1
-elim (ltpr_tpr_tpss_conf … HL12 … HT1 … HTU1) -L1 -HT1 #U #HU1 #HTU
-elim (tpss_conf_eq … HT2 … HTU) -T /3 width=3/
-qed.
-
-lemma cpr_ltpr_conf_eq: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ➡ L2 →
- ∃∃T. L2 ⊢ T1 ➡ T & T2 ➡ T.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
->(ltpr_fwd_length … HL12) in HT2; #HT2
-elim (ltpr_tpr_tpss_conf … HL12 … HT2) -L1 /3 width=3/
-qed.
-
-lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1
-elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2
-elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2
-lapply (tpss_weak_full … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *)
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning dx partial unfold on local environments ************)
-
-lemma ltpss_dx_cpr_conf: ∀L1,T,U1. L1 ⊢ T ➡ U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. L2 ⊢ T ➡ U2 & L2 ⊢ U1 ▶* [d, e] U2.
-#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
-lapply (ltpss_dx_fwd_length … HL12) #H >H in HU1; -H #HU1
-elim (ltpss_dx_tpss_conf … HU1 … HL12) -L1 /3 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning sn partial unfold on local environments ************)
-
-lemma ltpss_sn_cpr_trans: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 *
-lapply (ltpss_sn_weak_full … HL12)
-<(ltpss_sn_fwd_length … HL12) -HL12 /3 width=5/
-qed.
-
-lemma ltpss_sn_cpr_conf: ∀L1,T,U1. L1 ⊢ T ➡ U1 →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∃∃U2. L2 ⊢ T ➡ U2 & L1 ⊢ U1 ▶* [d, e] U2.
-#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
-lapply (ltpss_sn_fwd_length … HL12) #H >H in HU1; -H #HU1
-elim (ltpss_sn_tpss_conf … HU1 … HL12) -HU1 -HL12 /3 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_tpss.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties on partial unfold for terms ***********************************)
-
-lemma cpr_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡ T →
- ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2
-lapply (tpss_weak_full … HT2) -HT2 #HT2
-lapply (tpss_trans_eq … HT0 HT2) -T /2 width=3/
-qed.
-
-lemma cpr_tps_trans: ∀L,T1,T. L ⊢ T1 ➡ T →
- ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed.
-
-lemma cpr_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡ T1 →
- ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 →
- ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡ T.
-#L #T0 #T1 * #U0 #HTU0 #HU0T1 #T2 #d #e #HT02
-elim (tpr_tpss_conf … HTU0 … HT02) -T0 #T0 #HT20 #HUT0
-elim (tpss_conf_eq … HU0T1 … HUT0) -U0 /3 width=5/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_ltpr.ma".
-include "basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties concerning parallel unfold on terms ***************************)
-
-(* Basic_1: was only: pr3_subst1 *)
-lemma cprs_tpss_ltpr: ∀L1,T1,U1,d,e. L1 ⊢ T1 ▶* [d, e] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
- ∃∃U2. L2 ⊢ U1 ➡* U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2
-[ #T2 #HT12
- elim (cpr_tpss_ltpr … HL12 … HT12 … HTU1) -L1 -T1 /3 width=3/
-| #T #T2 #_ #HT2 * #U #HU1 #HTU
- elim (cpr_tpss_ltpr … HT2 … HTU) -L1 -T // /3 width=3/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_ltpss_dx.ma".
-include "basic_2/computation/cprs_tpss.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties concerning dx partial unfold on local environments ************)
-
-lemma cprs_ltpss_dx_conf: ∀L1,T,U1. L1 ⊢ T ➡* U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. L2 ⊢ T ➡* U2 & L2 ⊢ U1 ▶* [d, e] U2.
-#L1 #T #U1 #H @(cprs_ind … H) -U1 /2 width=3/
-#T1 #U1 #_ #HTU1 #IHT1 #L2 #d #e #HL12
-elim (IHT1 … HL12) -IHT1 #U #HTU #HT1U
-elim (ltpss_dx_cpr_conf … HTU1 … HL12) -L1 #U0 #HT1U0 #HU10
-elim (cpr_tpss_conf … HT1U0 … HT1U) -T1 #U2 #HU02 #HU2
-lapply (tpss_trans_eq … HU10 HU02) -U0 /3 width=3/
-qed-.
-
-lemma cprs_ltpss_dx_tpss_conf: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. L2 ⊢ T2 ➡* U2 & L2 ⊢ U1 ▶* [d, e] U2.
-#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
-elim (cprs_ltpss_dx_conf … HTU1 … HL12) -L1 #U #HT1U #HU1
-elim (cprs_tpss_conf … HT1U … HT12) -T1 #T #HUT #HT2
-lapply (tpss_trans_eq … HU1 HUT) -U /2 width=3/
-qed-.
-
-lemma cprs_ltpss_dx_tpss2_conf: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∀U2. L2 ⊢ U1 ▶* [d, e] U2 →
- ∃∃U. L2 ⊢ T2 ➡* U & L2 ⊢ U2 ▶* [d, e] U.
-#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 #U2 #HU12
-elim (cprs_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -L1 -T1 #U #HT2U #HU1
-elim (tpss_conf_eq … HU12 … HU1) -U1 #U0 #HU20 #HU0
-lapply (cprs_tpss_trans … HT2U … HU0) -U /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/computation/cprs_ltpss_dx.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties concerning sn partial unfold on local environments ************)
-
-lemma cprs_ltpss_sn_conf: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T,U1. L1 ⊢ T ➡* U1 →
- ∃∃U2. L2 ⊢ T ➡* U2 & L1 ⊢ U1 ▶* [d, e] U2.
-#L1 #L2 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
-#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1
-lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
-lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
-elim (IHL1 … HTU1) -IHL1 -HTU1 #U #HTU #HU1
-elim (cprs_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #HTU2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_tpss.ma".
-include "basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties on partial unfold for terms ***********************************)
-
-lemma cprs_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T →
- ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡* T2.
-#L #T1 #T #H @(cprs_ind … H) -T /2 width=3/ /3 width=5/
-qed.
-
-lemma cprs_tps_trans: ∀L,T1,T. L ⊢ T1 ➡* T →
- ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡* T2.
-/3 width=5 by inj, cprs_tpss_trans/ qed. (**) (* auto too slow without trace *)
-
-lemma cprs_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 →
- ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 →
- ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡* T.
-#L #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3/
-#T #T1 #_ #HT1 #IHT0 #T2 #d #e #HT02
-elim (IHT0 … HT02) -T0 #T0 #HT0 #HT20
-elim (cpr_tpss_conf … HT1 … HT0) -T /3 width=5/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unwind/sstas_ltpss_dx.ma".
-include "basic_2/computation/cprs_ltpss_dx.ma".
-include "basic_2/computation/dxprs.ma".
-
-(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
-
-(* Properties about dx parallel unfold **************************************)
-
-lemma dxprs_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
-elim (sstas_ltpss_dx_conf … HTU … HL12) -HTU #U0 #HTU0 #HU0
-elim (cprs_ltpss_dx_conf … HU1 … HL12) -L1 #U2 #HU2 #HU12
-elim (cprs_tpss_conf … HU2 … HU0) -U #U #HU2 #HU0
-lapply (tpss_trans_eq … HU12 HU2) -U2 /3 width=3/
-qed-.
-
-lemma dxprs_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*➡*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*➡*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #d #e #HT12
-elim (sstas_tpss_conf … HTW1 … HT12) -T1 #W2 #HTW2 #HW12
-elim (cprs_tpss_conf … HWU1 … HW12) -W1 /3 width=3/
-qed-.
-
-lemma dxprs_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
-elim (dxprs_ltpss_dx_conf … HTU1 … HL12) -L1 #U #HT1U #HU1
-elim (dxprs_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/computation/dxprs_ltpss_dx.ma".
-
-(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma dxprs_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #L2 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 [ /2 width=3/ ]
-#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1
-lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
-lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
-elim (IHL1 … HTU1) -IHL1 -HTU1 #U #HTU #HU1
-elim (dxprs_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #HTU2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lsubr_lbotr.ma".
-include "basic_2/relocation/ldrop_ldrop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Inversion lemmas about local env. full refinement for substitution *******)
-
-(* Note: ldrop_ldrop not needed *)
-lemma lbotr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ⊒[d, e] L →
- d ≤ i → i < d + e → I = Abbr.
-#I #L elim L -L
-[ #K #V #i #H
- lapply (ldrop_inv_atom1 … H) -H #H destruct
-| #L #J #W #IHL #K #V #i #H
- elim (ldrop_inv_O1 … H) -H *
- [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
- lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
- lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
- elim (lsubr_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
- | #Hi #HLK #d @(nat_ind_plus … d) -d
- [ #e #H #_ #Hide
- elim (lbotr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
- @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/
- | #d #_ #e #H #Hdi #Hide
- lapply (lbotr_inv_skip … H ?) -H // #HL
- @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/
- ]
- ]
-]
-qed-.
-
-(* Properties about local env. full refinement for substitution *************)
-
-(* Note: ldrop_ldrop not needed *)
-lemma lbotr_ldrop: ∀L,d,e.
- (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
- ⊒[d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d @(nat_ind_plus … d) -d
-[ #e @(nat_ind_plus … e) -e //
- #e #_ #H0
- >(H0 I L V 0 ? ? ?) //
- /5 width=6 by lbotr_abbr, ldrop_ldrop, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
-| #d #_ #e #H0
- /5 width=6 by lbotr_skip, ldrop_ldrop, le_S_S, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
-]
-qed.
-
-lemma lbotr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
- dd + ee ≤ d → ⊒[dd, ee] L2.
-#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
-@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
-lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
-elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
-elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
-@(lbotr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
-qed.
-
-lemma lbotr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
- ∀dd,ee. ⊒[dd, ee] L1 →
- dd ≤ d + e → d + e ≤ dd + ee →
- ⊒[d, dd + ee - d - e] L2.
-#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
-@lbotr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
-lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
->commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
-lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi #HL1K2
-@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
-qed.
-
-lemma lbotr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
- d + e ≤ dd → ⊒[dd - e, ee] L2.
-#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
-@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
-elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
->plus_minus in Hiddee; // #Hiddee
-lapply (transitive_le … Hdde Hddi) -Hdde #Hid
-lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
-@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_px.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Properties on pointwise extension ****************************************)
-
-lemma lpx_deliftable_dropable: ∀R. t_deliftable_sn R → dropable_sn (lpx R).
-#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #X #H
- elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
- elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 /3 width=3/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (HR … HV12 … HWV1) -V1
- elim (IHLK1 … HL12) -L1 /3 width=5/
-]
-qed.
-
-lemma lpx_liftable_dedropable: ∀R. reflexive ? R →
- t_liftable R → dedropable_sn (lpx R).
-#R #H1R #H2R #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #X #H
- elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
- elim (IHLK1 … HK12) -K1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (H2R … HW12 … HWV1 … HWV2) -W1
- elim (IHLK1 … HK12) -K1 /3 width=5/
-]
-qed.
-
-fact lpx_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx R L1 L2 →
- d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx R K1 K2.
-#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #X #H >(lpx_inv_atom2 … H) -H /2 width=3/
-| #K2 #I #V2 #X #H
- elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
-| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
- elim (lpx_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
-| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
- >commutative_plus normalize #H destruct
-]
-qed-.
-
-lemma lpx_dropable: ∀R. dropable_dx (lpx R).
-/2 width=5 by lpx_dropable_aux/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_append.ma".
-
-(* POINTWISE EXTENSION OF A CONTEXT-FREE REALTION FOR TERMS *****************)
-
-inductive lpx (R:relation term): relation lenv ≝
-| lpx_stom: lpx R (⋆) (⋆)
-| lpx_pair: ∀I,K1,K2,V1,V2.
- lpx R K1 K2 → R V1 V2 → lpx R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lpx_inv_atom1_aux: ∀R,L1,L2. lpx R L1 L2 → L1 = ⋆ → L2 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_inv_atom1: ∀R,L2. lpx R (⋆) L2 → L2 = ⋆.
-/2 width=4 by lpx_inv_atom1_aux/ qed-.
-
-fact lpx_inv_pair1_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #L1 #L2 * -L1 -L2
-[ #J #K1 #V1 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
-]
-qed-.
-
-lemma lpx_inv_pair1: ∀R,I,K1,V1,L2. lpx R (K1. ⓑ{I} V1) L2 →
- ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_inv_pair1_aux/ qed-.
-
-fact lpx_inv_atom2_aux: ∀R,L1,L2. lpx R L1 L2 → L2 = ⋆ → L1 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_inv_atom2: ∀R,L1. lpx R L1 (⋆) → L1 = ⋆.
-/2 width=4 by lpx_inv_atom2_aux/ qed-.
-
-fact lpx_inv_pair2_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #L1 #L2 * -L1 -L2
-[ #J #K2 #V2 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
-]
-qed-.
-
-lemma lpx_inv_pair2: ∀R,I,L1,K2,V2. lpx R L1 (K2. ⓑ{I} V2) →
- ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_inv_pair2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpx_fwd_length: ∀R,L1,L2. lpx R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L1 -L2 normalize //
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lpx_inv_append1: ∀R,L1,K1,L. lpx R (K1 @@ L1) L →
- ∃∃K2,L2. lpx R K1 K2 & lpx R L1 L2 & L = K2 @@ L2.
-#R #L1 elim L1 -L1 normalize
-[ #K1 #K2 #HK12
- @(ex3_2_intro … K2 (⋆)) // (**) (* explicit constructor, /2 width=5/ does not work *)
-| #L1 #I #V1 #IH #K1 #X #H
- elim (lpx_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
- elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #HL12 #H destruct
- @(ex3_2_intro … HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *)
-]
-qed-.
-
-lemma lpx_inv_append2: ∀R,L2,K2,L. lpx R L (K2 @@ L2) →
- ∃∃K1,L1. lpx R K1 K2 & lpx R L1 L2 & L = K1 @@ L1.
-#R #L2 elim L2 -L2 normalize
-[ #K2 #K1 #HK12
- @(ex3_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=5/ does not work *)
-| #L2 #I #V2 #IH #K2 #X #H
- elim (lpx_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
- elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #HL12 #H destruct
- @(ex3_2_intro … HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *)
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R).
-#R #HR #L elim L -L // /2 width=1/
-qed.
-
-lemma lpx_sym: ∀R. symmetric ? R → symmetric … (lpx R).
-#R #HR #L1 #L2 #H elim H -H // /3 width=1/
-qed.
-
-lemma lpx_trans: ∀R. Transitive ? R → Transitive … (lpx R).
-#R #HR #L1 #L #H elim H -L //
-#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
-elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
-qed.
-
-lemma lpx_conf: ∀R. confluent ? R → confluent … (lpx R).
-#R #HR #L0 #L1 #H elim H -L1
-[ #X #H >(lpx_inv_atom1 … H) -X /2 width=3/
-| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H
- elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
- elim (IHK01 … HK02) -K0 #K #HK1 #HK2
- elim (HR … HV01 … HV02) -HR -V0 /3 width=5/
-]
-qed.
-
-lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2.
-#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
-qed.
-
-lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L →
- ∀L2. lpx R L L2 → lpx (TC … R) L1 L2.
-#R #L1 #L #H elim H -L /2 width=1/
-#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
-elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
-qed.
-
-lemma TC_lpx_pair_dx: ∀R. reflexive ? R →
- ∀I,K,V1,V2. TC … R V1 V2 →
- TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2).
-#R #HR #I #K #V1 #V2 #H elim H -V2
-/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
-qed.
-
-lemma TC_lpx_pair_sn: ∀R. reflexive ? R →
- ∀I,V,K1,K2. TC … (lpx R) K1 K2 →
- TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V).
-#R #HR #I #V #K1 #K2 #H elim H -K2
-/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
-qed.
-
-lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2.
-#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/
-qed.
-
-lemma lpx_inv_TC: ∀R. reflexive ? R →
- ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/
-qed.
-
-lemma lpx_append: ∀R,K1,K2. lpx R K1 K2 → ∀L1,L2. lpx R L1 L2 →
- lpx R (L1 @@ K1) (L2 @@ K2).
-#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 𝟙 break term 46 T2 )"
- non associative with precedence 45
- for @{ 'RTop $T1 $T2 }.
-
-include "basic_2/grammar/lenv_px.ma".
-
-(* POINTWISE EXTENSION OF TOP RELATION FOR TERMS ****************************)
-
-definition ttop: relation term ≝ λT1,T2. True.
-
-definition ltop: relation lenv ≝ lpx ttop.
-
-interpretation
- "top reduction (environment)"
- 'RTop L1 L2 = (ltop L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ltop_refl: reflexive … ltop.
-/2 width=1/ qed.
-
-lemma ltop_sym: symmetric … ltop.
-/2 width=1/ qed.
-
-lemma ltop_trans: transitive … ltop.
-/2 width=3/ qed.
-
-lemma ltop_append: ∀K1,K2. K1 𝟙 K2 → ∀L1,L2. L1 𝟙 L2 → L1 @@ K1 𝟙 L2 @@ K2.
-/2 width=1/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ltop_inv_atom1: ∀L2. ⋆ 𝟙 L2 → L2 = ⋆.
-/2 width=2 by lpx_inv_atom1/ qed-.
-
-lemma ltop_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 𝟙 L2 →
- ∃∃K2,V2. K1 𝟙 K2 & L2 = K2. ⓑ{I} V2.
-#K1 #I #V1 #L2 #H
-elim (lpx_inv_pair1 … H) -H /2 width=4/
-qed-.
-
-lemma ltop_inv_atom2: ∀L1. L1 𝟙 ⋆ → L1 = ⋆.
-/2 width=2 by lpx_inv_atom2/ qed-.
-
-lemma ltop_inv_pair2: ∀L1,K2,I,V2. L1 𝟙 K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 𝟙 K2 & L1 = K1. ⓑ{I} V1.
-#L1 #K2 #I #V2 #H
-elim (lpx_inv_pair2 … H) -H /2 width=4/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltop_fwd_length: ∀L1,L2. L1 𝟙 L2 → |L1| = |L2|.
-/2 width=2 by lpx_fwd_length/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1 ⦄ ➡ break ⦃ term 46 L2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRed $L1 $L2 }.
-
-include "basic_2/unfold/ltpss_sn.ma".
-include "basic_2/reducibility/ltpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-
-definition lfpr: relation lenv ≝
- λL1,L2. ∃∃L. L1 ➡ L & L ⊢ ▶* [0, |L|] L2
-.
-
-interpretation
- "focalized parallel reduction (environment)"
- 'FocalizedPRed L1 L2 = (lfpr L1 L2).
-
-(* Basic properties *********************************************************)
-
-(* Note: lemma 250 *)
-lemma lfpr_refl: ∀L. ⦃L⦄ ➡ ⦃L⦄.
-/2 width=3/ qed.
-
-lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄.
-/3 width=5/ qed.
-
-lemma ltpr_lfpr: ∀L1,L2. L1 ➡ L2 → ⦃L1⦄ ➡ ⦃L2⦄.
-/3 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆.
-#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_sn_inv_atom1 … HL2) -HL2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_aaa.ma".
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_lfpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → L2 ⊢ T ⁝ A.
-#L1 #T #A #HT #L2 * /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_ldrop.ma".
-include "basic_2/reducibility/cpr.ma".
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-
-(* Advanced properties ******************************************************)
-
-lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
- ∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄.
-#L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
-<(ltpss_sn_fwd_length … HL2) #V #HV1 #HV2 #I
-lapply (ltpss_sn_tpss_trans_eq … HV2 … HL2) -HV2 #V2
-@(ex2_intro … (L.ⓑ{I}V)) /2 width=1/ (**) (* explicit constructor *)
-qed.
-
-(* Properties on supclosure *************************************************)
-(*
-lamma fsub_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃L1⦄ ➡ ⦃L⦄ & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #HT12 #U2 * #T #H1 #H2
-elim (fsub_tpr_trans … HT12 … H1) -T2 #L #U #HL1 #HT1U #HUT
-elim (fsup_tpss_trans_full … HUT … H2) -T -HUT -H2 #L #U #HL1 #HT1U #HUT
-
-
-
-
-
-
- #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
-elim (lift_total T d e) #U #HTU
-elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
-lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-qed-.
-*)
\ No newline at end of file
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_ltpss_sn.ma".
-include "basic_2/reducibility/ltpr_ltpr.ma".
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Main properties **********************************************************)
-
-theorem lfpr_conf: ∀L0,L1,L2. ⦃L0⦄ ➡ ⦃L1⦄ → ⦃L0⦄ ➡ ⦃L2⦄ →
- ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄.
-#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2
-lapply (ltpr_fwd_length … HK01) #H
->(ltpr_fwd_length … HK02) in H; #H
-elim (ltpr_conf … HK01 … HK02) -K0 #K #HK1 #HK2
-lapply (ltpss_sn_fwd_length … HKL1) #H1
-lapply (ltpss_sn_fwd_length … HKL2) #H2
->H1 in HKL1 H; -H1 #HKL1
->H2 in HKL2; -H2 #HKL2 #H
-elim (ltpr_ltpss_sn_conf … HKL1 … HK1) -K1 #K1 #HK1 #HLK1
-elim (ltpr_ltpss_sn_conf … HKL2 … HK2) -K2 #K2 #HK2 #HLK2
-elim (ltpss_sn_conf … HK1 … HK2) -K #K #HK1 #HK2
-lapply (ltpr_fwd_length … HLK1) #H1
-lapply (ltpr_fwd_length … HLK2) #H2
-/3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/ltprs.ma".
-include "basic_2/computation/lfprs.ma".
-
-(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
-
-(* Properties on context-free parallel computation for local environments ***)
-
-lemma ltprs_lfprs: ∀L1,L2. L1 ➡* L2 → ⦃L1⦄ ➡* ⦃L2⦄.
-/3 width=3/ qed.
+++ /dev/null
-definition t_liftable: relation term → Prop ≝
- λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2.
-
-definition t_deliftable_sn: relation term → Prop ≝
- λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2.
-
-lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R).
-#R #HR #T1 #T2 #H elim H -T2
-[ /3 width=7/
-| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2
- elim (lift_total T d e) /3 width=9/
-]
-qed.
-
-lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R).
-#R #HR #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #T1 #d #e #HTU1
- elim (HR … HU12 … HTU1) -U1 /3 width=3/
-| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1
- elim (IHU1 … HTU1) -U1 #T #HTU #HT1
- elim (HR … HU2 … HTU) -U /3 width=5/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L1 break ⊑ [ term 46 d , break term 46 e ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'SubEq $L1 $d $e $L2 }.
-
-include "basic_2/grammar/lenv_length.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-inductive lsubr: nat → nat → relation lenv ≝
-| lsubr_sort: ∀d,e. lsubr d e (⋆) (⋆)
-| lsubr_OO: ∀L1,L2. lsubr 0 0 L1 L2
-| lsubr_abbr: ∀L1,L2,V,e. lsubr 0 e L1 L2 →
- lsubr 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
-| lsubr_abst: ∀L1,L2,I,V1,V2,e. lsubr 0 e L1 L2 →
- lsubr 0 (e + 1) (L1. ⓑ{I}V1) (L2. ⓛV2)
-| lsubr_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
- lsubr d e L1 L2 → lsubr (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
-.
-
-interpretation
- "local environment refinement (substitution)"
- 'SubEq L1 d e L2 = (lsubr d e L1 L2).
-
-definition lsubr_trans: ∀S. (lenv → relation S) → Prop ≝ λS,R.
- ∀L2,s1,s2. R L2 s1 s2 →
- ∀L1,d,e. L1 ⊑ [d, e] L2 → R L1 s1 s2.
-
-(* Basic properties *********************************************************)
-
-lemma lsubr_bind_eq: ∀L1,L2,e. L1 ⊑ [0, e] L2 → ∀I,V.
- L1. ⓑ{I} V ⊑ [0, e + 1] L2.ⓑ{I} V.
-#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
-qed.
-
-lemma lsubr_abbr_lt: ∀L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
- L1. ⓓV ⊑ [0, e] L2.ⓓV.
-#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma lsubr_abst_lt: ∀L1,L2,I,V1,V2,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
- L1. ⓑ{I}V1 ⊑ [0, e] L2. ⓛV2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma lsubr_skip_lt: ∀L1,L2,d,e. L1 ⊑ [d - 1, e] L2 → 0 < d →
- ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 ⊑ [d, e] L2. ⓑ{I2} V2.
-#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
-qed.
-
-lemma lsubr_bind_lt: ∀I,L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
- L1. ⓓV ⊑ [0, e] L2. ⓑ{I}V.
-* /2 width=1/ qed.
-
-lemma lsubr_refl: ∀d,e,L. L ⊑ [d, e] L.
-#d elim d -d
-[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
-| #d #IHd #e #L elim L -L // /2 width=1/
-]
-qed.
-
-lemma TC_lsubr_trans: ∀S,R. lsubr_trans S R → lsubr_trans S (λL. (TC … (R L))).
-#S #R #HR #L1 #s1 #s2 #H elim H -s2
-[ /3 width=5/
-| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
- lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubr_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L1 = ⋆ →
- L2 = ⋆ ∨ (d = 0 ∧ e = 0).
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ /2 width=1/
-| /3 width=1/
-| #L1 #L2 #W #e #_ #H destruct
-| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
-| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
-]
-qed.
-
-lemma lsubr_inv_atom1: ∀L2,d,e. ⋆ ⊑ [d, e] L2 →
- L2 = ⋆ ∨ (d = 0 ∧ e = 0).
-/2 width=3/ qed-.
-
-fact lsubr_inv_skip1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀I1,K1,V1. L1 = K1.ⓑ{I1}V1 → 0 < d →
- ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #I1 #K1 #V1 #H destruct
-| #L1 #L2 #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
-]
-qed.
-
-lemma lsubr_inv_skip1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑ [d, e] L2 → 0 < d →
- ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=5/ qed-.
-
-fact lsubr_inv_atom2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L2 = ⋆ →
- L1 = ⋆ ∨ (d = 0 ∧ e = 0).
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ /2 width=1/
-| /3 width=1/
-| #L1 #L2 #W #e #_ #H destruct
-| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
-| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
-]
-qed.
-
-lemma lsubr_inv_atom2: ∀L1,d,e. L1 ⊑ [d, e] ⋆ →
- L1 = ⋆ ∨ (d = 0 ∧ e = 0).
-/2 width=3/ qed-.
-
-fact lsubr_inv_abbr2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀K2,V. L2 = K2.ⓓV → d = 0 → 0 < e →
- ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #K1 #V #H destruct
-| #L1 #L2 #K1 #V #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #W #e #HL12 #K1 #V #H #_ #_ destruct /2 width=3/
-| #L1 #L2 #I #W1 #W2 #e #_ #K1 #V #H destruct
-| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #K1 #V #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubr_inv_abbr2: ∀L1,K2,V,e. L1 ⊑ [0, e] K2.ⓓV → 0 < e →
- ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
-/2 width=5/ qed-.
-
-fact lsubr_inv_skip2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀I2,K2,V2. L2 = K2.ⓑ{I2}V2 → 0 < d →
- ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #I1 #K1 #V1 #H destruct
-| #L1 #L2 #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
-]
-qed.
-
-lemma lsubr_inv_skip2: ∀I2,L1,K2,V2,d,e. L1 ⊑ [d, e] K2.ⓑ{I2}V2 → 0 < d →
- ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=5/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-fact lsubr_fwd_length_full1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- d = 0 → e = |L1| → |L1| ≤ |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubr_fwd_length_full1: ∀L1,L2. L1 ⊑ [0, |L1|] L2 → |L1| ≤ |L2|.
-/2 width=5/ qed-.
-
-fact lsubr_fwd_length_full2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- d = 0 → e = |L2| → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubr_fwd_length_full2: ∀L1,L2. L1 ⊑ [0, |L2|] L2 → |L2| ≤ |L1|.
-/2 width=5/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⊒ [ term 46 d , break term 46 e ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'SubEqBottom $d $e $L2 }.
-
-include "basic_2/relocation/lsubr.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-(* bottom element of the refinement *)
-definition lbotr: nat → nat → predicate lenv ≝
- λd,e. NF_sn … (lsubr d e) (lsubr d e …).
-
-interpretation
- "local environment full refinement (substitution)"
- 'SubEqBottom d e L = (lbotr d e L).
-
-(* Basic properties *********************************************************)
-
-lemma lbotr_atom: ∀d,e. ⊒[d, e] ⋆.
-#d #e #L #H
-elim (lsubr_inv_atom2 … H) -H
-[ #H destruct //
-| * #H1 #H2 destruct //
-]
-qed.
-
-lemma lbotr_OO: ∀L. ⊒[0, 0] L.
-// qed.
-
-lemma lbotr_abbr: ∀L,V,e. ⊒[0, e] L → ⊒[0, e + 1] L.ⓓV.
-#L #V #e #HL #K #H
-elim (lsubr_inv_abbr2 … H ?) -H // <minus_plus_m_m #X #HLX #H destruct
-lapply (HL … HLX) -HL -HLX /2 width=1/
-qed.
-
-lemma lbotr_abbr_O: ∀L,V. ⊒[0,1] L.ⓓV.
-#L #V
-@(lbotr_abbr … 0) //
-qed.
-
-lemma lbotr_skip: ∀I,L,V,d,e. ⊒[d, e] L → ⊒[d + 1, e] L.ⓑ{I}V.
-#I #L #V #d #e #HL #K #H
-elim (lsubr_inv_skip2 … H ?) -H // <minus_plus_m_m #J #X #W #HLX #H destruct
-lapply (HL … HLX) -HL -HLX /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lbotr_inv_bind: ∀I,L,V,e. ⊒[0, e] L.ⓑ{I}V → 0 < e →
- ⊒[0, e - 1] L ∧ I = Abbr.
-#I #L #V #e #HL #He
-lapply (HL (L.ⓓV) ?) /2 width=1/ #H
-elim (lsubr_inv_abbr2 … H ?) -H // #K #_ #H destruct
-@conj // #L #HKL
-lapply (HL (L.ⓓV) ?) -HL /2 width=1/ -HKL #H
-elim (lsubr_inv_abbr2 … H ?) -H // -He #X #HLX #H destruct //
-qed-.
-
-lemma lbotr_inv_skip: ∀I,L,V,d,e. ⊒[d, e] L.ⓑ{I}V → 0 < d → ⊒[d - 1, e] L.
-#I #L #V #d #e #HL #Hd #K #HLK
-lapply (HL (K.ⓑ{I}V) ?) -HL /2 width=1/ -HLK #H
-elim (lsubr_inv_skip2 … H ?) -H // -Hd #J #X #W #HKX #H destruct //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_px.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-definition ltpr: relation lenv ≝ lpx tpr.
-
-interpretation
- "context-free parallel reduction (environment)"
- 'PRed L1 L2 = (ltpr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ltpr_refl: reflexive … ltpr.
-/2 width=1/ qed.
-
-lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2.
-/2 width=1/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: wcpr0_gen_sort *)
-lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
-/2 width=2 by lpx_inv_atom1/ qed-.
-
-(* Basic_1: was: wcpr0_gen_head *)
-lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
- ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
-/2 width=1 by lpx_inv_pair1/ qed-.
-
-lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
-/2 width=2 by lpx_inv_atom2/ qed-.
-
-lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-/2 width=1 by lpx_inv_pair2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
-/2 width=2 by lpx_fwd_length/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L →
- ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2.
-/2 width=1 by lpx_inv_append1/ qed-.
-
-lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 →
- ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1.
-/2 width=1 by lpx_inv_append2/ qed-.
-
-(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa_ltpss_dx.ma".
-include "basic_2/static/lsuba_aaa.ma".
-include "basic_2/reducibility/ltpr_ldrop.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 →
- ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
-#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * *
-[1,2,3:
- #i #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
- >(tpr_inv_atom1 … H2) -Y
-|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
-]
-[ >(aaa_inv_sort … H1) -X //
-| elim (aaa_inv_lref … H1) #I #K1 #V1 #HLK1 #HA
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #Y #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (IH … HKV1 … HA … HK12 … HV12) -L1 -K1 -V1 /2 width=5/
-| elim (aaa_inv_gref … H1)
-| elim (aaa_inv_abbr … H1) -H1 #B #HB #HA
- elim (tpr_inv_abbr1 … H2) -H2 *
- [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
- lapply (tps_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2
- lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
- lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/
- | -B #T #HT1 #HXT #H destruct
- lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA
- @(aaa_inv_lift … HA … HXT) /2 width=1/
- ]
-| elim (aaa_inv_abst … H1) -H1 #B #A #HB #HA #H destruct
- elim (tpr_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
- lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/
-| elim (aaa_inv_appl … H1) -H1 #B #HB #HA
- elim (tpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
- lapply (IH … HA … HL12 … HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/
- | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
- elim (aaa_inv_abst … HA) -HA #B0 #A0 #HB0 #HA0 #H destruct
- lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
- lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
- lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 // /2 width=1/ -T0 -L1 -V1 /4 width=7/
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
- elim (aaa_inv_abbr … HA) -HA #B0 #HW0 #HT0
- lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
- lapply (IH … HB … HL12 … HV10) -HB -HV10 /width=5/ #HV0
- lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2
- @(aaa_abbr … HW2) -HW2
- @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
- ]
-| elim (aaa_inv_cast … H1) -H1 #HV1 #HT1
- elim (tpr_inv_cast1 … H2) -H2
- [ * #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
- lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
- | -HV1 #HT1X
- lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
- ]
-]
-qed.
-
-lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A.
-/2 width=5/ qed.
-
-lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A.
-/2 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_lpx.ma".
-include "basic_2/substitution/fsup.ma".
-include "basic_2/reducibility/tpr_lift.ma".
-include "basic_2/reducibility/ltpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properies on local environment slicing ***********************************)
-
-(* Basic_1: was: wcpr0_drop *)
-lemma ltpr_ldrop_conf: dropable_sn ltpr.
-/3 width=3 by lpx_deliftable_dropable, tpr_inv_lift1/ qed.
-
-(* Basic_1: was: wcpr0_drop_back *)
-lemma ldrop_ltpr_trans: dedropable_sn ltpr.
-/2 width=3/ qed.
-
-lemma ltpr_ldrop_trans_O1: dropable_dx ltpr.
-/2 width=3/ qed.
-
-(* Properties on supclosure *************************************************)
-
-lemma fsub_tpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. T2 ➡ U2 →
- ∃∃L,U1. L1 ➡ L & T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
-elim (lift_total T d e) #U #HTU
-elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
-lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/tpr_tpr.ma".
-include "basic_2/reducibility/ltpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Main properties **********************************************************)
-
-theorem ltpr_conf: ∀L0:lenv. ∀L1. L0 ➡ L1 → ∀L2. L0 ➡ L2 →
- ∃∃L. L1 ➡ L & L2 ➡ L.
-#L0 #L1 #H elim H -L0 -L1 /2 width=3/
-#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H
-elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
-elim (IHK01 … HK02) -K0 #K #HK1 #HK2
-elim (tpr_conf … HV01 HV02) -V0 /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties concerning dx parallel unfold on local environments ***********)
-
-lemma ltpr_ltpss_dx_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2.
-#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ /2 width=3/
-| #L1 #I #V1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
- elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
- elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /3 width=5/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/reducibility/ltpr_ltpss_dx.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties on sn parallel unfold on local environments *******************)
-
-(* Note: this can also be proved like ltpr_ltpss_dx_conf *)
-lemma ltpr_ltpss_sn_conf: ∀L1,K1,d,e. L1 ⊢ ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. L2 ⊢ ▶* [d, e] K2 & K1 ➡ K2.
-#L1 #K1 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H
-@(ltpssa_ind … H) -K1 /2 width=3/
-#K #K1 #_ #HK1 #IHK #L2 #HL12
-elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
-elim (ltpr_ltpss_dx_conf … HK1 … HK2) -K /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_ldrop.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties concerning parallel substitution on terms *********************)
-
-(* Basic_1: was: pr0_subst1_fwd *)
-lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 →
- ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ /2 width=3/
-| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
- elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
- elim (lift_total V2 0 (i+1)) #W2 #HVW2
- lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
-| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
- elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
- elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
-| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
- elim (IHV12 … HL12) -IHV12
- elim (IHT12 … HL12) -L1 /3 width=5/
-]
-qed-.
-
-(* Basic_1: was: pr0_subst1_back *)
-lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 →
- ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2.
-#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
-[ /2 width=3/
-| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
- elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
- elim (lift_total V1 0 (i+1)) #W1 #HVW1
- lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
-| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
- elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
-| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (IHV12 … HL12) -IHV12
- elim (IHT12 … HL12) -L2 /3 width=5/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
-include "basic_2/reducibility/ltpr_tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties on partial unfold for terms ***********************************)
-
-(* Basic_1: was: pr0_subst1 *)
-lemma ltpr_tpr_tps_conf: ∀T1,T2. T1 ➡ T2 →
- ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 →
- ∀L2. L1 ➡ L2 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#T1 #T2 #H elim H -T1 -T2
-[ #I #L1 #d #e #U1 #H #L2 #HL12
- elim (ltpr_tps_conf … H … HL12) -L1 /3 width=3/
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV12 … HVW1 … HL12) -V1
- elim (IHT12 … HTU1 … HL12) -T1 -HL12 /3 width=5/
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
- elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- lapply (tpss_lsubr_trans … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/
-| #a #I #V1 #V2 #T1 #T #T2 #HV12 #_ #HT2 #IHV12 #IHT1 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV12 … HVW1 … HL12) -V1 #W2 #HW12 #HVW2
- elim (IHT1 … HTU1 (L2. ⓑ{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
- elim (tpss_strip_neq … HTU … HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2
- lapply (tps_lsubr_trans … HU2 (L2. ⓑ{I} V2) ?) -HU2 /2 width=1/ #HU2
- elim (ltpss_dx_tps_conf … HU2 (L2. ⓑ{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23
- lapply (tps_lsubr_trans … HU3 (⋆. ⓑ{I} W2) ?) -HU3 /2 width=1/ #HU3
- lapply (tpss_lsubr_trans … HU23 (L2. ⓑ{I} W2) ?) -HU23 /2 width=1/ #HU23
- lapply (tpss_trans_eq … HTU2 … HU23) -U2 /3 width=5/
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
- elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
- elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- elim (lift_total VV2 0 1) #VV #H2VV
- lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV
- @ex2_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_bind1 … H) -H #W #U1 #_ #HTU1 #H destruct -V
- elim (IHT1 … HTU1 (L2.ⓓW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
- elim (tpss_inv_lift1_ge … HTU L2 … HT2 ?) -T <minus_plus_m_m /3 width=3/
-| #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
- elim (IHT12 … HTT1 … HL12) -T1 -HL12 /3 width=3/
-]
-qed-.
-
-lemma tpr_tps_conf_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
- ⋆. ⓑ{I} V1 ⊢ T1 ▶ [0, 1] U1 →
- ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 ▶ [0, 1] U2.
-#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
-elim (ltpr_tpr_tps_conf … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ -V1 #U2 #HU12 #HTU2
-lapply (tpss_inv_SO2 … HTU2) -HTU2 /2 width=3/
-qed-.
-
-lemma ltpr_tpr_tpss_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
-[ /2 width=3/
-| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
- elim (ltpr_tpr_tps_conf … HUT … HU1 … HL12) -U -HL12 #U2 #HU12 #HTU2
- lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
-]
-qed-.
-
-lemma tpr_tpss_conf: ∀T1,T2. T1 ➡ T2 →
- ∀L,U1,d,e. L ⊢ T1 ▶* [d, e] U1 →
- ∃∃U2. U1 ➡ U2 & L ⊢ T2 ▶* [d, e] U2.
-/2 width=5 by ltpr_tpr_tpss_conf/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr.ma".
-include "basic_2/computation/tprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
-
-definition ltprs: relation lenv ≝ TC … ltpr.
-
-interpretation
- "context-free parallel computation (environment)"
- 'PRedStar L1 L2 = (ltprs L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma ltprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 ➡* L → L ➡ L2 → R L → R L2) →
- ∀L2. L1 ➡* L2 → R L2.
-#L1 #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma ltprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. L1 ➡ L → L ➡* L2 → R L → R L1) →
- ∀L1. L1 ➡* L2 → R L1.
-#L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltprs_refl: reflexive … ltprs.
-/2 width=1/ qed.
-
-lemma ltpr_ltprs: ∀L1,L2. L1 ➡ L2 → L1 ➡* L2.
-/2 width=1/ qed.
-
-lemma ltprs_strap1: ∀L1,L,L2. L1 ➡* L → L ➡ L2 → L1 ➡* L2.
-/2 width=3/ qed.
-
-lemma ltprs_strap2: ∀L1,L,L2. L1 ➡ L → L ➡* L2 → L1 ➡* L2.
-/2 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ltprs_inv_atom1: ∀L2. ⋆ ➡* L2 → L2 = ⋆.
-#L2 #H @(ltprs_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL1 destruct
->(ltpr_inv_atom1 … HL2) -L2 //
-qed-.
-
-lemma ltprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ➡* L2 →
- ∃∃K2,V2. K1 ➡* K2 & V1 ➡* V2 & L2 = K2. ⓑ{I} V2.
-#I #K1 #L2 #V1 #H @(ltprs_ind … H) -L2 /2 width=5/
-#L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
-elim (ltpr_inv_pair1 … HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
-qed-.
-
-lemma ltprs_inv_atom2: ∀L1. L1 ➡* ⋆ → L1 = ⋆.
-#L1 #H @(ltprs_ind_dx … H) -L1 //
-#L1 #L #HL1 #_ #IHL2 destruct
->(ltpr_inv_atom2 … HL1) -L1 //
-qed-.
-
-lemma ltprs_inv_pair2: ∀I,L1,K2,V2. L1 ➡* K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ➡* K2 & V1 ➡* V2 & L1 = K1. ⓑ{I} V1.
-#I #L1 #K2 #V2 #H @(ltprs_ind_dx … H) -L1 /2 width=5/
-#L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
-elim (ltpr_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltprs_fwd_length: ∀L1,L2. L1 ➡* L2 → |L1| = |L2|.
-#L1 #L2 #H @(ltprs_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL1
->IHL1 -L1 >(ltpr_fwd_length … HL2) -HL2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 ➡ ➡ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRedStarAlt $T1 $T2 }.
-
-include "basic_2/computation/ltprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
-
-(* alternative definition of ltprs *)
-definition ltprsa: relation lenv ≝ lpx tprs.
-
-interpretation
- "context-free parallel computation (environment) alternative"
- 'PRedStarAlt L1 L2 = (ltprsa L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ltprs_ltprsa: ∀L1,L2. L1 ➡* L2 → L1 ➡➡* L2.
-/2 width=1/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ltprsa_ltprs: ∀L1,L2. L1 ➡➡* L2 → L1 ➡* L2.
-/2 width=1/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_ldrop.ma".
-include "basic_2/computation/ltprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
-
-lemma ltprs_ldrop_conf: dropable_sn ltprs.
-/2 width=3/ qed.
-
-lemma ldrop_ltprs_trans: dedropable_sn ltprs.
-/2 width=3/ qed.
-
-lemma ltprs_ldrop_trans_O1: dropable_dx ltprs.
-/2 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_ltpr.ma".
-include "basic_2/computation/ltprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
-
-(* Advanced properties ******************************************************)
-
-lemma ltprs_strip: ∀L1. ∀L:lenv. L ➡* L1 → ∀L2. L ➡ L2 →
- ∃∃L0. L1 ➡ L0 & L2 ➡* L0.
-/3 width=3/ qed.
-
-(* Main properties **********************************************************)
-
-theorem ltprs_conf: confluent … ltprs.
-/3 width=3/ qed.
-
-theorem ltprs_trans: Transitive … ltprs.
-/2 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Basic_1: includes: csubst1_bind *)
-inductive ltpss_dx: nat → nat → relation lenv ≝
-| ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆)
-| ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
-| ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e.
- ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
- ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-| ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e.
- ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
- ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-.
-
-interpretation "parallel unfold (local environment, dx variant)"
- 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
-[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
- >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
-]
-qed.
-
-lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
-/2 width=4/ qed-.
-
-fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2.
- L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
- ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K1 #V1 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
- ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3/ qed-.
-
-fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2.
- L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K2 #V2 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e.
- L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
- L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e.
- L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
- L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2 width=1/
-qed.
-
-lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2 width=1/
-qed.
-
-lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-(* Basic_1: was by definition: csubst1_refl *)
-lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L.
-#L elim L -L //
-#L #I #V #IHL * /2 width=1/ * /2 width=1/
-qed.
-
-lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
-#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
-[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
- lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
- >plus_plus_comm_23 in Hde12; #Hde12
- elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
- [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
- lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
- | -Hd21 normalize in Hde12;
- lapply (lt_to_le_to_lt 0 … Hde12) // #He2
- lapply (le_plus_to_minus_r … Hde12) -Hde12
- /3 width=5 by ltpss_dx_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
- ]
-]
-qed.
-
-lemma ltpss_dx_weak_full: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-// /3 width=2/ /3 width=3/
-qed.
-
-fact ltpss_dx_append_le_aux: ∀K1,K2,d,x. K1 ▶* [d, x] K2 → x = |K1| - d →
- ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ▶* [d, x + e] L2 @@ K2.
-#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
-[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_n_O_to_eq … H2) -H2 #H destruct //
-| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
- lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
- /4 width=5 by ltpss_dx_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_plus_to_le_r … H2) -H2 #Hd
- /4 width=5 by ltpss_dx_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-]
-qed-.
-
-lemma ltpss_dx_append_le: ∀K1,K2,d. K1 ▶* [d, |K1| - d] K2 →
- ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ▶* [d, |K1| - d + e] L2 @@ K2.
-/2 width=1 by ltpss_dx_append_le_aux/ qed.
-
-lemma ltpss_dx_append_zero: ∀K1,K2. K1 ▶* [0, |K1|] K2 →
- ∀L1,L2,e. L1 ▶* [0, e] L2 →
- L1 @@ K1 ▶* [0, |K1| + e] L2 @@ K2.
-/2 width=1/ qed.
-
-lemma ltpss_dx_append_ge: ∀K1,K2,d,e. K1 ▶* [d, e] K2 →
- ∀L1,L2. L1 ▶* [d - |K1|, e] L2 → |K1| ≤ d →
- L1 @@ K1 ▶* [d, e] L2 @@ K2.
-#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
-[ #d #e #L1 #L2 <minus_n_O //
-| #K #I #V #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
- normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
- lapply (le_plus_to_le_r … H) -H /3 width=1/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpss_dx_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-normalize //
-qed-.
-
-(* Basic_1: removed theorems 28:
- csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
- csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
- csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
- csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
- csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
- csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
- csubst1_head csubst1_flat csubst1_gen_head
- csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
- fsubst0_gen_base
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fsup.ma".
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/unfold/ltpss_dx.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma ltpss_dx_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_dx_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_dx_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ltpss_dx_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ldrop_ltpss_dx_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d1 ≤ d2 →
- ∃∃L2. L1 ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
- elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
- elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
- elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_ge … HW12 … HLK2 HWV1 … HWV2) -W1 // -Hd12
- <le_plus_minus_comm // /4 width=5/
-]
-qed-.
-
-lemma ldrop_ltpss_dx_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ▶* [d2, e2] K2 →
- d2 ≤ d1 → d1 ≤ d2 + e2 →
- ∃∃L2. L1 ▶* [d2, e1 + e2] L2 &
- ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
- >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
- lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
- elim (IHLK1 … HK12 H1 H2) -K1 -H2
- lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
- elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
- [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
- | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 // /2 width=1/
- >plus_minus // >commutative_plus /4 width=5/
- | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // ] /2 width=1/
- >commutative_plus /3 width=5/
- ]
-]
-qed-.
-
-lemma ldrop_ltpss_dx_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
- ∃∃L2. L1 ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
- elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
- elim (IHLK1 … HK12 H) -K1
- elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
- >(ltpss_dx_inv_refl_O2 … HL12) -L1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
- elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
- [ -IHLK1 -Hd21 <(ltpss_dx_inv_refl_O2 … H) -X /3 width=5/
- | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 /2 width=1/ /3 width=5/
- | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
- ]
-]
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fsup_tps_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶[0,|L2|] U2 →
- ∃∃L,U1. L1 ▶*[0,|L|] L & L ⊢ T1 ▶[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-elim (le_or_ge d (|K|)) #Hd
-[ elim (ldrop_ltpss_dx_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tps_lift_be … HT1 … HL2K … HTU1 HTU … Hd) // -HT1 -HTU1 #HU1
-| elim (ldrop_ltpss_dx_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tps_lift_le … HT1 … HL2K … HTU1 HTU Hd) -HT1 -HTU1 #HU1
-]
-lapply (ltpss_dx_weak_full … HL12) -HL12 #HL12
-lapply (tps_weak_full … HU1) -HU1 #HU1
-@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_dx_tpss.ma".
-
-(* DX PARTIAL UNFOLD ON LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_dx_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
- L1 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind … H) -U2 /2 width=3/
-#U #U2 #_ #HU2 * #X2 #HTX2 #HUX2
-elim (ltpss_dx_tps_conf … HU2 … HL01) -L0 #X1 #HUX1 #HU2X1
-elim (tpss_strip_eq … HUX2 … HUX1) -U #X #HX2 #HX1
-lapply (tpss_trans_eq … HU2X1 … HX1) -X1 /3 width=3/
-qed.
-
-lemma ltpss_dx_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
- L1 ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L0 ⊢ T ▶* [d1, e1] U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
-[ /2 width=3/
-| #U #U2 #_ #HU2 * #T #HT2 #HTU
- elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
- elim (ltpss_dx_tps_trans … HTU … HL10) -HTU -HL10 #X #HTX #HXU
- lapply (tpss_trans_eq … HXU HU2) -U /3 width=3/
-]
-qed.
-
-lemma ltpss_dx_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
- ∀L0. L0 ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
-#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
-[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
- elim (tpss_inv_atom1 … H) -H // *
- #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
- lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
- elim (ltpss_dx_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
- elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
- lapply (tpss_fwd_tw … HV01) #H2
- lapply (transitive_le (♯{K1} + ♯{V0}) … H1) -H1 /2 width=1/ -H2 #H
- lapply (tpss_trans_eq … HV01 HV12) -V1 #HV02
- lapply (IH … HV02 … HK01) -IH -HV02 -HK01
- [ normalize /2 width=1/ | /2 width=6/ ]
-| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
- [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
- lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
- lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
- lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
- | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (IH … HVW2 … HL0) -HVW2 //
- lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ▶* [d, e] L1 →
- L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
-/3 width=3/ qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpss_dx_conf: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ▶* [d1, e1] L.
-#L0 @(f_ind … lw … L0) -L0 #n #IH *
-[ #_ #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 -n
- >(ltpss_dx_inv_atom1 … H1) -L1
- >(ltpss_dx_inv_atom1 … H2) -L2 /2 width=3/
-| #K0 #I0 #V0 #Hn #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 destruct
- elim (eq_or_gt d1) #Hd1 [ elim (eq_or_gt e1) #He1 ] destruct
- [ lapply (ltpss_dx_inv_refl_O2 … H1) -H1 #H1
- | elim (ltpss_dx_inv_tpss21 … H1 … He1) -H1 #K1 #V1 #HK01 #HV01 #H1
- | elim (ltpss_dx_inv_tpss11 … H1 … Hd1) -H1 #K1 #V1 #HK01 #HV01 #H1
- ] destruct
- elim (eq_or_gt d2) #Hd2 [1,3,5: elim (eq_or_gt e2) #He2 ] destruct
- [1,3,5: lapply (ltpss_dx_inv_refl_O2 … H2) -H2 #H2
- |2,4,6: elim (ltpss_dx_inv_tpss21 … H2 … He2) -H2 #K2 #V2 #HK02 #HV02 #H2
- |7,8,9: elim (ltpss_dx_inv_tpss11 … H2 … Hd2) -H2 #K2 #V2 #HK02 #HV02 #H2
- ] destruct
- [1: -IH /2 width=3/
- |2,3,4,7: -IH /3 width=5/
- |5,6,8,9:
- elim (IH … HK01 … HK02) // -K0 #K #HK1 #HK2
- elim (ltpss_dx_tpss_conf … HV01 … HK1) -HV01 #W1 #HW1 #HVW1
- elim (ltpss_dx_tpss_conf … HV02 … HK2) -HV02 #W2 #HW2 #HVW2
- elim (tpss_conf_eq … HW1 … HW2) -V0 #V #HW1 #HW2
- lapply (tpss_trans_eq … HVW1 HW1) -W1
- lapply (tpss_trans_eq … HVW2 HW2) -W2 /3 width=5/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_dx_ldrop.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial substitution on terms **********************)
-
-lemma ltpss_dx_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
-
-lemma ltpss_dx_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_dx_tps.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial unfold on terms ****************************)
-
-lemma ltpss_dx_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_dx_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-(* Basic_1: was: subst1_subst1_back *)
-lemma ltpss_dx_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L1 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_dx_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_dx_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2
- elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
-]
-qed.
-
-lemma ltpss_dx_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_dx_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-(* Basic_1: was: subst1_subst1 *)
-lemma ltpss_dx_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ▶* [d1, e1] L0 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L0 ⊢ T ▶* [d1, e1] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_dx_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_dx_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2
- elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 break ⊢ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarSn $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-inductive ltpss_sn: nat → nat → relation lenv ≝
-| ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆)
-| ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
-| ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e.
- ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 →
- ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-| ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e.
- ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 →
- ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-.
-
-interpretation "parallel unfold (local environment, sn variant)"
- 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
-[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
- >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
-]
-qed.
-
-lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2.
-/2 width=4/ qed-.
-
-fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2.
- L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e →
- ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K1 #V1 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d →
- ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3/ qed-.
-
-fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2.
- L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K2 #V2 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e.
- L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 →
- L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e.
- L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 →
- L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2 width=1/
-qed.
-
-lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2 width=1/
-qed.
-
-lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L.
-#L elim L -L //
-#L #I #V #IHL * /2 width=1/ * /2 width=1/
-qed.
-
-lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2.
-#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
-[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
- lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
- >plus_plus_comm_23 in Hde12; #Hde12
- elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
- [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
- lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
- | -Hd21 normalize in Hde12;
- lapply (lt_to_le_to_lt 0 … Hde12) // #He2
- lapply (le_plus_to_minus_r … Hde12) -Hde12
- /3 width=5 by ltpss_sn_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
- ]
-]
-qed.
-
-lemma ltpss_sn_weak_full: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [0, |L1|] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-// /3 width=2/ /3 width=3/
-qed.
-
-fact ltpss_sn_append_le_aux: ∀K1,K2,d,x. K1 ⊢ ▶* [d, x] K2 → x = |K1| - d →
- ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ⊢ ▶* [d, x + e] L2 @@ K2.
-#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
-[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_n_O_to_eq … H2) -H2 #H destruct //
-| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
- lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
- /4 width=5 by ltpss_sn_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_plus_to_le_r … H2) -H2 #Hd
- /4 width=5 by ltpss_sn_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-]
-qed-.
-
-lemma ltpss_sn_append_le: ∀K1,K2,d. K1 ⊢ ▶* [d, |K1| - d] K2 →
- ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ⊢ ▶* [d, |K1| - d + e] L2 @@ K2.
-/2 width=1 by ltpss_sn_append_le_aux/ qed.
-
-lemma ltpss_sn_append_ge: ∀K1,K2,d,e. K1 ⊢ ▶* [d, e] K2 →
- ∀L1,L2. L1 ⊢ ▶* [d - |K1|, e] L2 → |K1| ≤ d →
- L1 @@ K1 ⊢ ▶* [d, e] L2 @@ K2.
-#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
-[ #d #e #L1 #L2 <minus_n_O //
-| #K #I #V #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
- normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
- lapply (le_plus_to_le_r … H) -H /3 width=1/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpss_sn_fwd_length: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-normalize //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }.
-
-include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
-include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* alternative definition of ltpss_sn *)
-definition ltpssa: nat → nat → relation lenv ≝
- λd,e. TC … (ltpss_dx d e).
-
-interpretation "parallel unfold (local environment, sn variant) alternative"
- 'PSubstStarSnAlt L1 d e L2 = (ltpssa d e L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma ltpssa_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 ⊢ ▶▶* [d, e] L → L ▶* [d, e] L2 → R L → R L2) →
- ∀L2. L1 ⊢ ▶▶* [d, e] L2 → R L2.
-#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma ltpssa_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. L1 ▶* [d, e] L → L ⊢ ▶▶* [d, e] L2 → R L → R L1) →
- ∀L1. L1 ⊢ ▶▶* [d, e] L2 → R L1.
-#d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpssa_refl: ∀L,d,e. L ⊢ ▶▶* [d, e] L.
-/2 width=1/ qed.
-
-lemma ltpssa_tpss2: ∀I,L1,V1,V2,e. L1 ⊢ V1 ▶*[0, e] V2 →
- ∀L2. L1 ⊢ ▶▶* [0, e] L2 →
- L1.ⓑ{I}V1 ⊢ ▶▶* [O, e + 1] L2.ⓑ{I}V2.
-#I #L1 #V1 #V2 #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
-[ /3 width=1/ | /3 width=5/ ]
-qed.
-
-lemma ltpssa_tpss1: ∀I,L1,V1,V2,d,e. L1 ⊢ V1 ▶*[d, e] V2 →
- ∀L2. L1 ⊢ ▶▶* [d, e] L2 →
- L1.ⓑ{I}V1 ⊢ ▶▶* [d + 1, e] L2.ⓑ{I}V2.
-#I #L1 #V1 #V2 #d #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
-[ /3 width=1/ | /3 width=5/ ]
-qed.
-
-lemma ltpss_sn_ltpssa: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶▶* [d, e] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // /2 width=1/
-qed.
-
-lemma ltpss_sn_dx_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
- ∀L2. L ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L #d #e #H elim H -L1 -L -d -e
-[ #d #e #X #H
- lapply (ltpss_dx_inv_atom1 … H) -H #H destruct //
-| #L #I #V #X #H
- lapply (ltpss_dx_inv_refl_O2 … H) -H #H destruct //
-| #L1 #L #I #V1 #V #e #_ #HV1 #IHL1 #X #H
- elim (ltpss_dx_inv_tpss21 … H ?) -H // <minus_plus_m_m
- #L2 #V2 #HL2 #HV2 #H destruct
- lapply (IHL1 … HL2) -L #HL12
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
- lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
-| #L1 #L #I #V1 #V #d #e #_ #HV1 #IHL1 #X #H
- elim (ltpss_dx_inv_tpss11 … H ?) -H // <minus_plus_m_m
- #L2 #V2 #HL2 #HV2 #H destruct
- lapply (IHL1 … HL2) -L #HL12
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
- lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_sn_trans_eq: ∀L1,L,d,e. L1 ▶* [d, e] L →
- ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-/3 width=3/ qed.
-
-lemma ltpssa_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
-/3 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ltpssa_ltpss_sn: ∀L1,L2,d,e. L1 ⊢ ▶▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L2 #d #e #H @(ltpssa_ind … H) -L2 // /2 width=3/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_sn_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
-#L0 #L1 #d1 #e1 #H #L2 #d2 #e2 #HL02
-lapply (ltpss_sn_ltpssa … H) -H #HL01
-elim (ltpssa_strip … HL01 … HL02) -L0
-/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
-qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-lemma ltpss_sn_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
- L0 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #HTU2 #L1 #d1 #e1 #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L1 /2 width=3/ -HTU2
-#L #L1 #H #HL1 * #T #HT2 #HU2T
-lapply (ltpssa_ltpss_sn … H) -H #HL0
-lapply (ltpss_sn_dx_trans_eq … HL0 … HL1) -HL0 #HL01
-elim (ltpss_dx_tpss_conf … HT2 … HL1) -HT2 -HL1 #T0 #HT20 #HT0
-lapply (ltpss_sn_tpss_trans_eq … HT0 … HL01) -HT0 -HL01 #HT0
-lapply (tpss_trans_eq … HU2T HT0) -T /2 width=3/
-qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-lemma ltpss_sn_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
- L1 ⊢ ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L1 ⊢ T ▶* [d1, e1] U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #H #HTU2
-lapply (ltpss_sn_ltpssa … H) -H #HL10
-@(ltpssa_ind_dx … HL10) -L1 /2 width=3/ -HTU2
-#L1 #L #HL1 #_ * #T #HT2 #HTU2
-elim (ltpss_dx_tpss_trans_down … HL1 HT2) -HT2 // #T0 #HT20 #HT0 -Hde2d1
-lapply (tpss_trans_eq … HT0 HTU2) -T #HT0U2
-lapply (ltpss_dx_tpss_trans_eq … HT0U2 … HL1) -HT0U2 -HL1 /2 width=3/
-qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpssa_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ⊢ ▶▶* [d2, e2] L2 →
- ∃∃L. L1 ⊢ ▶▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
-/3 width=3/ qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-theorem ltpss_sn_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ⊢ ▶* [d2, e2] L2 →
- ∃∃L. L1 ⊢ ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
-#L0 #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2
-lapply (ltpss_sn_ltpssa … H1) -H1 #HL01
-lapply (ltpss_sn_ltpssa … H2) -H2 #HL02
-elim (ltpssa_conf … HL01 … HL02) -L0
-/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fsup.ma".
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/unfold/ltpss_sn.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma ltpss_sn_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-]
-qed.
-
-lemma ltpss_sn_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
-]
-qed.
-
-lemma ltpss_sn_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L2 ⊢ ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_sn_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L ⊢ ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_sn_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 ⊢ ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ltpss_sn_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L ⊢ ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ldrop_ltpss_sn_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d1 ≤ d2 →
- ∃∃L2. L1 ⊢ ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
- elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
- elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
- elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_ge … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // -Hd12
- <le_plus_minus_comm // /4 width=5/
-]
-qed-.
-
-lemma ldrop_ltpss_sn_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 →
- d2 ≤ d1 → d1 ≤ d2 + e2 →
- ∃∃L2. L1 ⊢ ▶* [d2, e1 + e2] L2 &
- ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
- >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
- lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
- elim (IHLK1 … HK12 H1 H2) -K1 -H2
- lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
- elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
- [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
- | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // /2 width=1/
- >plus_minus // >commutative_plus /4 width=5/
- | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // ] /2 width=1/
- >commutative_plus /3 width=5/
- ]
-]
-qed-.
-
-lemma ldrop_ltpss_sn_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
- ∃∃L2. L1 ⊢ ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
- elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
- elim (IHLK1 … HK12 H) -K1
- elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
- >(ltpss_sn_inv_refl_O2 … HL12) -L1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
- elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
- [ -IHLK1 -Hd21 <(ltpss_sn_inv_refl_O2 … H) -X /3 width=5/
- | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 /2 width=1/ /3 width=5/
- | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
- ]
-]
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fsup_tpss_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶*[0,|L2|] U2 →
- ∃∃L,U1. L1 ⊢ ▶*[0,|L1|] L & L ⊢ T1 ▶*[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-lapply (ltpss_sn_fwd_length … HK1) #H >H in HK1; -H #HK1
-elim (le_or_ge d (|K|)) #Hd
-[ elim (ldrop_ltpss_sn_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tpss_lift_be … HT1 … Hd HL2K HTU1 … HTU) // -HT1 -HTU1 #HU1
-| elim (ldrop_ltpss_sn_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tpss_lift_le … HT1 … Hd HL2K HTU1 … HTU) -HT1 -HTU1 #HU1
-]
-lapply (ltpss_sn_weak_full … HL12) -HL12 #HL12
-lapply (tpss_weak_full … HU1) -HU1 #HU1
-@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_sn_tpss.ma".
-
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_sn_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
- ∀L0. L0 ⊢ ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
-#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
-[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
- elim (tpss_inv_atom1 … H) -H // *
- #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
- lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
- elim (ltpss_sn_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
- elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
- lapply (IH … HV12 … HK01) -IH -HV12 -HK01 [ normalize /2 width=1/ ] #HV12
- lapply (tpss_trans_eq … HV01 HV12) -V1 /2 width=6/
-| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
- [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
- lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
- lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
- lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
- | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (IH … HVW2 … HL0) -HVW2 //
- lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
-]
-qed.
-
-lemma ltpss_sn_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ⊢ ▶* [d, e] L1 →
- L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
-/3 width=3/ qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpss_sn_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
- ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L #d #e #H elim H -L1 -L -d -e //
-[ #L1 #L #I #V1 #V #e #HL1 #HV1 #IHL1 #X #H
- elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
- lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
-| #L1 #L #I #V1 #V #d #e #HL1 #HV1 #IHL1 #X #H
- elim (ltpss_sn_inv_tpss11 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
- lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
-]
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶ [d, e] T →
- ∃∃L2,T2. L @@ L1 ⊢ ▶* [d + |L1|, e] L @@ L2 &
- L @@ L2 ⊢ T1 ▶ [d + |L1|, e] T2 &
- T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #d #e #HT1
- @ex3_2_intro [3: // |5: // |4: normalize /2 width=1/ |1,2: skip ] (**) (* /2 width=4/ does not work *)
-| #I #L1 #V1 #IH #L #T1 #T #d #e >shift_append_assoc #H
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
- <append_assoc >append_length <associative_plus
- lapply (ltpss_sn_trans_eq (L.ⓑ{I}V1@@L1) … HL12) -HL12 /3 width=1/ #HL12
- @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_ldrop.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial substitution on terms **********************)
-
-lemma ltpss_sn_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
-
-lemma ltpss_sn_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_tps.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial unfold on terms ****************************)
-
-lemma ltpss_sn_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_sn_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-lemma ltpss_sn_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L0 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_sn_ldrop_conf_le … HL01 … HLK0 ?) /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // >minus_plus <plus_minus_m_m // /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_sn_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 // /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H
- lapply (tpss_lsubr_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2
- elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
-]
-qed.
-
-lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L1 ⊢ T ▶* [d1, e1] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_sn_ldrop_trans_be … HL10 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H
- lapply (tpss_lsubr_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2
- elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/dxprs_ltpss_dx.ma".
-include "basic_2/equivalence/cpcs_ltpss_dx.ma".
-include "basic_2/dynamic/snv_lift.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties about dx parallel unfold **************************************)
-
-lemma snv_ltpss_dx_tpss_conf: ∀h,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
-#h #g #L1 #T1 #H elim H -L1 -T1
-[ #L1 #k #L2 #d #e #_ #X #H
- >(tpss_inv_sort1 … H) -X //
-| #I #L1 #K1 #V1 #i #HLK1 #HV1 #IHV1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ | elim (lt_or_ge i (d + e)) #Hide ]
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 …) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H …) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- lapply (IHV1 … HK12 … HV12) -IHV1 -HK12 -HV12 /2 width=5/
- | elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 …) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H …) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- lapply (IHV1 … HK12 … HV12) -IHV1 -HK12 -HV12 /2 width=5/
- | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=5/
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 …) -L1 // /2 width=2/ #X #H1 #H2
- elim (ltpss_dx_inv_tpss21 … H1 …) -H1 /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … H2 … HLK2) -H2 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
- lapply (IHV1 … HK12 … HV1W2) -IHV1 -HK12 -HV1W2 /2 width=7/
- ]
-| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IHV1 … HL12 … HV12) -IHV1 #HV2
- lapply (IHT1 … HT12) -IHT1 -HT12 /2 width=1/
-| #a #L1 #V1 #W1 #W0 #T1 #U1 #l #_ #_ #HVW1 #HW10 #HTU1 #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IHV1 … HL12 … HV12) -IHV1 #HV2
- lapply (IHT1 … HT12) -IHT1 /2 width=1/ #HT2
- elim (ssta_ltpss_dx_tpss_conf … HVW1 … HL12 … HV12) -V1 #W2 #HVW2 #HW12
- elim (dxprs_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -T1 #X #HTU2 #H
- elim (tpss_inv_bind1 … H) -H #W #U2 #HW0 #_ #H -U1 destruct
- elim (cprs_ltpss_dx_tpss2_conf … HW10 … HL12 … HW12 … HW0) -L1 -W1 -W0 #W0 #HW20 #HW0
- lapply (dxprs_strap1 … (ⓛ{a}W0.U2) HTU2 ?) -HTU2 /3 width=3/ -HW0 /2 width=8/
-| #L1 #W1 #T1 #U1 #l #_ #_ #HTU1 #HUW1 #IHW1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
- lapply (IHW1 … HL12 … HW12) -IHW1 #HW2
- lapply (IHT1 … HL12 … HT12) -IHT1 #HT2
- elim (ssta_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -T1 #U2 #HTU2 #HU12
- lapply (cpcs_ltpss_dx_tpss2_conf … HL12 … HUW1 … HU12 … HW12) -L1 -W1 -U1 /2 width=4/
-]
-qed-.
-
-lemma snv_ltpss_dx_conf: ∀h,g,L1,T. ⦃h, L1⦄ ⊢ T ¡[g] →
- ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T ¡[g].
-#h #g #L1 #T #HT #L2 #d #e #HL12
-@(snv_ltpss_dx_tpss_conf … HT … HL12) //
-qed-.
-
-lemma snv_tpss_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊢ T1 ¡[g] →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 ¡[g].
-#h #g #L #T1 #HT1 #T2 #d #e #HT12
-@(snv_ltpss_dx_tpss_conf … HT1 … HT12) //
-qed-.
-
-lemma snv_tps_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊢ T1 ¡[g] →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 ¡[g].
-#h #g #L #T1 #HT1 #T2 #d #e #HT12
-@(snv_tpss_conf … HT1 T2) /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snv_ltpss_dx.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma snv_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T. ⦃h, L1⦄ ⊢ T ¡[g] → ⦃h, L2⦄ ⊢ T ¡[g].
-#h #g #L1 #L2 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL1 #T1 #HT1
-lapply (IHL1 … HT1) -IHL1 -HT1 #HT1
-lapply (snv_ltpss_dx_conf … HT1 … HL2) -HT1 -HL2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_dx_tpss.ma".
-include "basic_2/static/ssta_lift.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about dx parallel unfold **************************************)
-
-(* Note: apparently this was missing in basic_1 *)
-lemma ssta_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
-[ #L1 #k1 #l1 #Hkl1 #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H /3 width=3/
-| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
- [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
- elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (ssta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
- lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HWV1 ]
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
- [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
- ]
-| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
-| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
-| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
-]
-qed.
-
-lemma ssta_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=5/ qed.
-
-lemma ssta_ltpss_dx_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma ssta_tpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma ssta_tps_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/static/ssta_ltpss_dx.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma ssta_ltpss_sn_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T #U1 #l #HTU1 #L2 #d #e #HL12
-lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
-@(ltpssa_ind … HL12) -L2 [ /2 width=3/ ] -HTU1
-#L #L2 #HL1 #HL2 * #U #HTU #HU1
-lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
-elim (ssta_ltpss_dx_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
-lapply (ltpss_dx_tpss_trans_eq … HU2 … HL2) -HU2 -HL2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL1) -HU2 -HL1 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma ssta_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #l #HTU1 #L2 #d #e #HL12 #T2 #HT12
-elim (ssta_ltpss_sn_conf … HTU1 … HL12) -HTU1 #U #HT1U #HU1
-elim (ssta_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma ssta_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L1 ⊢ U1 ▶* [d, e] U2.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_ltpss.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************)
-
-(* Properties about parallel unfold *****************************************)
-
-lemma sstas_ltpss_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #H @(sstas_ind_alt … H) -T1
-[ #T1 #HUT1 #L2 #d #e #HL12 #U2 #HU12
- elim (ssta_ltpss_tpss_conf … HUT1 … HL12 … HU12) -HUT1 -HL12 /3 width=3/
-| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
- elim (ssta_ltpss_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
- elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
-]
-qed.
-
-lemma sstas_ltpss_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=5/ qed.
-
-lemma sstas_ltpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_ltpss_dx.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Properties about dx parallel unfold **************************************)
-
-lemma sstas_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
-#T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
-elim (ssta_ltpss_dx_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
-elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
-qed.
-
-lemma sstas_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=7 by step, sstas_ltpss_dx_tpss_conf/ qed. (**) (* auto fails without trace *)
-
-lemma sstas_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/unwind/sstas_ltpss_dx.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ************************)
-
-(* Properties about sn parallel unfold *****************************************)
-
-lemma sstas_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∃∃U2. L1 ⊢ U1 ▶* [d, e] U2 & ⦃h, L2⦄ ⊢ T •*[g] U2.
-#h #g #L1 #L2 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
-#L #L2 #HL1 #HL2 #IHL1 #T #U1 #H1
-lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
-lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
-elim (IHL1 … H1) -IHL1 -H1 #U #HU1 #HTU
-elim (sstas_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #H2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma sstas_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
-elim (sstas_ltpss_sn_conf … HL12 … HTU1) -HTU1 #U #HU1 #HT1U
-elim (sstas_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma sstas_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L1 ⊢ U1 ▶* [d, e] U2.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_tpss.ma".
-include "basic_2/static/sta_lift.ma".
-
-(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
-
-(* Properties about parallel unfold *****************************************)
-
-lemma sta_ltpss_tpss_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 • U2 & L2 ⊢ U1 ▶* [d, e] U2.
-#h #L1 #T1 #U #H elim H -L1 -T1 -U
-[ #L1 #k1 #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H /2 width=3/
-| #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
- elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (sta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
- lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HWV1 ]
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
- ]
-| #I #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
-| #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
-| #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
-]
-qed.
-
-lemma sta_ltpss_tps_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 • U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=5/ qed.
-
-lemma sta_ltpss_conf: ∀h,L1,T,U1. ⦃h, L1⦄ ⊢ T • U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T • U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sta_tpss_conf: ∀h,L,T1,U1. ⦃h, L⦄ ⊢ T1 • U1 →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 • U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sta_tps_conf: ∀h,L,T1,U1. ⦃h, L⦄ ⊢ T1 • U1 →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 • U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 ➡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRed $T1 $T2 }.
-
-include "basic_2/substitution/tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Basic_1: includes: pr0_delta1 *)
-inductive tpr: relation term ≝
-| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I})
-| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
- tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| tpr_beta : ∀a,V1,V2,W,T1,T2.
- tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
-| tpr_delta: ∀a,I,V1,V2,T1,T,T2.
- tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 →
- tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2.
- tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
- tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2)
-| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2
-| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2
-.
-
-interpretation
- "context-free parallel reduction (term)"
- 'PRed T1 T2 = (tpr T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-/2 width=3/ qed.
-
-(* Basic_1: was by definition: pr0_refl *)
-lemma tpr_refl: reflexive … tpr.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}.
-#U1 #U2 * -U1 -U2
-[ //
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
-| #V #T1 #T #T2 #_ #_ #k #H destruct
-| #V #T1 #T2 #_ #k #H destruct
-]
-qed.
-
-(* Basic_1: was: pr0_gen_sort pr0_gen_lref *)
-lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}.
-/2 width=3/ qed-.
-
-fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-#U1 #U2 * -U1 -U2
-[ #J #a #I #V #T #H destruct
-| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct
-| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct
-| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/
-| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct
-| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/
-| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr.
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_abbr *)
-lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓓ{a}V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
-#a #V1 #T1 #U2 #H
-elim (tpr_inv_bind1 … H) -H * /3 width=7/
-qed-.
-
-fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-#U1 #U2 * -U1 -U2
-[ #I #J #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/
-| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/
-| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct
-| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
-]
-qed.
-
-lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_appl *)
-lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2.
-#V1 #U0 #U2 #H
-elim (tpr_inv_flat1 … H) -H *
-/3 width=5/ /3 width=9/ /3 width=13/
-#_ #H destruct
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
- U = ⓐV2. T2.
-#V1 #T1 #U #H #HT1
-elim (tpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Basic_1: was: pr0_gen_cast *)
-lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 →
- (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2)
- ∨ T1 ➡ U2.
-#V1 #T1 #U2 #H
-elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2
-[ #_ #_ #_ #_ #H destruct
-| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i →
- ∨∨ T1 = #i
- | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
-#T1 #T2 * -T1 -T2
-[ #I #i #H destruct /2 width=1/
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
-| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/
-| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/
-]
-qed.
-
-lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
- ∨∨ T1 = #i
- | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
-/2 width=3/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tpr_fwd_bind1_minus: ∀I,V1,T1,T. -ⓑ{I}V1.T1 ➡ T → ∀b.
- ∃∃V2,T2. ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
-#I #V1 #T1 #T #H #b elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct /3 width=4/
-| #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #T1 #X
- >shift_append_assoc normalize #H
- elim (tpr_inv_bind1 … H) -H *
- [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct
- elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct
- elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct
- >append_length >HL1 >HL2 -L1 -L
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
-
-(* Basic_1: removed theorems 3:
- pr0_subst0_back pr0_subst0_fwd pr0_subst0
-*)
-(* Basic_1: removed local theorems: 1: pr0_delta_tau *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/tps_lift.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr0_lift *)
-lemma tpr_lift: t_liftable tpr.
-#T1 #T2 #H elim H -T1 -T2
-[ * #i #U1 #d #e #HU1 #U2 #HU2
- lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
- [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
- | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
- | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
- ]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
- elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
- elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
- elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
- elim (lift_total T (d + 1) e) #U #HTU
- @tpr_delta
- [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *)
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
- elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
- elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
- elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2
- elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V
- elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/
-| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/
-]
-qed.
-
-(* Basic_1: was: pr0_gen_lift *)
-lemma tpr_inv_lift1: t_deliftable_sn tpr.
-#T1 #T2 #H elim H -T1 -T2
-[ * #i #X #d #e #HX
- [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/
- | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/
- ]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
- elim (IHV12 … HV01) -V1
- elim (IHT12 … HT01) -T1 /3 width=5/
-| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HV01) -V1
- elim (IHT12 … HT01) -T1 /3 width=5/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX
- elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct
- elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
- elim (IHT1 … HUT1) -T1 #U #HUT #HU1
- elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2
- @ex2_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *)
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
- elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
- elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
- elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
- @ex2_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX
- elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct
- elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31
- elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/
-| #V #T1 #T2 #_ #IHT12 #X #d #e #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
- elim (IHT12 … HT01) -T1 /3 width=3/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was pr0_gen_abst *)
-lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
-#a #V1 #T1 #U2 #H elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T #T2 #HV12 #HT1 #HT2
- lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /2 width=5/
-| #T2 #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma tpr_fwd_abst1: ∀a,V1,T1,U2. ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
- ∃∃V2,T2. ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
- U2 = ⓛ{a}V2.T2.
-#a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T #T2 #HV12 #HT1 #HT2
- lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /3 width=4/
-| #T2 #_ #_ #_ #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Confluence lemmas ********************************************************)
-
-fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
-/2 width=3/ qed.
-
-fact tpr_conf_flat_flat:
- ∀I,V0,V1,T0,T1,V2,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
- ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0.
-#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/
-qed.
-
-fact tpr_conf_flat_beta:
- ∀a,V0,V1,T1,V2,W0,U0,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 →
- U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 →
- ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X.
-#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
-elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/
-qed.
-
-(* Basic-1: was:
- pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
- pr0_cong_upsilon_cong pr0_cong_upsilon_delta
-*)
-fact tpr_conf_flat_theta:
- ∀a,V0,V1,T1,V2,V,W0,W2,U0,U2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V →
- W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 →
- ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X.
-#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2
-elim (lift_total VV 0 1) #VVV #HVV
-lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
-elim (tpr_inv_abbr1 … H) -H *
-(* case 1: delta *)
-[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct
- elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
- elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
- elim (tpr_tps_conf_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1
- @ex2_intro
- [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
- |1:skip
- |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
- ] (**) (* /5 width=14/ is too slow *)
-(* case 3: zeta *)
-| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1
- elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2
- elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct
- @(ex2_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *)
-]
-qed.
-
-fact tpr_conf_flat_cast:
- ∀X2,V0,V1,T0,T1. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 →
- ∃∃X. ⓝV1. T1 ➡ X & X2 ➡ X.
-#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/
-qed.
-
-fact tpr_conf_beta_beta:
- ∀a. ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
- ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X.
-#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/
-qed.
-
-(* Basic_1: was: pr0_cong_delta pr0_delta_delta *)
-fact tpr_conf_delta_delta:
- ∀a,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
- ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 →
- ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 →
- ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X.
-#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
-elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2
-elim (tpr_tps_conf_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1
-elim (tpr_tps_conf_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
-elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2
-@ex2_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
-qed.
-
-fact tpr_conf_delta_zeta:
- ∀X2,V0,V1,T0,T1,TT1,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 →
- T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 →
- ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X.
-#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2
-elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2
-elim (tpr_tps_conf_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT
-elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2
-lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /3 width=3/
-qed.
-
-(* Basic_1: was: pr0_upsilon_upsilon *)
-fact tpr_conf_theta_theta:
- ∀a,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
- ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 →
- ⇧[O, 1] V1 ≡ VV1 → ⇧[O, 1] V2 ≡ VV2 →
- ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X.
-#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
-elim (lift_total V 0 1) #VV #HVV
-lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
-lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2
-@ex2_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
-qed.
-
-fact tpr_conf_zeta_zeta:
- ∀V0:term. ∀X2,TT0,T0,T1,TT2. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{TT0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 →
- TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 →
- ∃∃X. T1 ➡ X & X2 ➡ X.
-#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2
-elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2
-elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10
-elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2
-lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/
-qed.
-
-fact tpr_conf_tau_tau:
- ∀V0,T0:term. ∀X2,T1. (
- ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
- ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
- ∃∃X. X1 ➡ X & X2 ➡ X
- ) →
- T0 ➡ T1 → T0 ➡ X2 →
- ∃∃X. T1 ➡ X & X2 ➡ X.
-#V0 #T0 #X2 #T1 #IH #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/
-qed.
-
-(* Confluence ***************************************************************)
-
-(* Basic_1: was: pr0_confluence *)
-theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 →
- ∃∃T. T1 ➡ T & T2 ➡ T.
-#T0 @(f_ind … tw … T0) -T0 #n #IH *
-[ #I #_ #X1 #X2 #H1 #H2 -n
- >(tpr_inv_atom1 … H1) -X1
- >(tpr_inv_atom1 … H2) -X2
-(* case 1: atom, atom *)
- //
-| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2
- [ elim (tpr_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (tpr_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #U2 #HV02 #HT02 #HTU2 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
-(* case 2: delta, delta *)
- [ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
-(* case 3: zeta, delta (repeated) *)
- | @ex2_commute /3 width=10 by tpr_conf_delta_zeta/
-(* case 4: delta, zeta *)
- | /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
-(* case 5: zeta, zeta *)
- | /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
- ]
- | elim (tpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #b1 #V1 #W1 #U1 #T1 #HV01 #HUT1 #H11 #H12 #H13
- | #b1 #V1 #Y1 #W1 #Z1 #U1 #T1 #HV01 #HWZ1 #HUT1 #HVY1 #H11 #H12 #H13
- | #HX1 #H1
- ]
- elim (tpr_inv_flat1 … H2) -H2 *
- [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
- |2,6,10,14: #b2 #V2 #W2 #U2 #T2 #HV02 #HUT2 #H21 #H22 #H23
- |3,7,11,15: #b2 #V2 #Y2 #W2 #Z2 #U2 #T2 #HV02 #HWZ2 #HUT2 #HVY2 #H21 #H22 #H23
- |4,8,12,16: #HX2 #H2
- ] destruct
-(* case 6: flat, flat *)
- [ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
-(* case 7: beta, flat (repeated) *)
- | @ex2_commute /3 width=8 by tpr_conf_flat_beta/
-(* case 8: theta, flat (repeated) *)
- | @ex2_commute /3 width=11 by tpr_conf_flat_theta/
-(* case 9: tau, flat (repeated) *)
- | @ex2_commute /3 width=6 by tpr_conf_flat_cast/
-(* case 10: flat, beta *)
- | /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
-(* case 11: beta, beta *)
- | /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
-(* case 12: flat, theta *)
- | /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
-(* case 13: theta, theta *)
- | /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
-(* case 14: flat, tau *)
- | /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
-(* case 15: tau, tau *)
- | /3 width=5 by tpr_conf_tau_tau/
- ]
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 ➡ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRedStar $T1 $T2 }.
-
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
-
-(* Basic_1: includes: pr1_pr0 *)
-definition tprs: relation term ≝ TC … tpr.
-
-interpretation "context-free parallel computation (term)"
- 'PRedStar T1 T2 = (tprs T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma tprs_ind: ∀T1. ∀R:predicate term. R T1 →
- (∀T,T2. T1 ➡* T → T ➡ T2 → R T → R T2) →
- ∀T2. T1 ➡* T2 → R T2.
-#T1 #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-lemma tprs_ind_dx: ∀T2. ∀R:predicate term. R T2 →
- (∀T1,T. T1 ➡ T → T ➡* T2 → R T → R T1) →
- ∀T1. T1 ➡* T2 → R T1.
-#T2 #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tprs_refl: reflexive … tprs.
-/2 width=1/ qed.
-
-lemma tpr_tprs: ∀T1,T2. T1 ➡ T2 → T2 ➡* T2.
-/2 width=1/ qed.
-
-lemma tprs_strap1: ∀T1,T,T2. T1 ➡* T → T ➡ T2 → T1 ➡* T2.
-/2 width=3/ qed.
-
-lemma tprs_strap2: ∀T1,T,T2. T1 ➡ T → T ➡* T2 → T1 ➡* T2.
-/2 width=3/ qed.
-
-(* Basic_1: was only: pr1_head_1 *)
-lemma tprs_pair_sn: ∀I,T1,T2. T1 ➡ T2 → ∀V1,V2. V1 ➡* V2 →
- ②{I} V1. T1 ➡* ②{I} V2. T2.
-* [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind … H) -V2
-[1,3: /3 width=1/
-|2,4: #V #V2 #_ #HV2 #IHV1
- @(tprs_strap1 … IHV1) -IHV1 /2 width=1/
-]
-qed.
-
-(* Basic_1: was only: pr1_head_2 *)
-lemma tprs_pair_dx: ∀I,V1,V2. V1 ➡ V2 → ∀T1,T2. T1 ➡* T2 →
- ②{I} V1. T1 ➡* ②{I} V2. T2.
-* [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind … H) -T2
-[1,3: /3 width=1/
-|2,4: #T #T2 #_ #HT2 #IHT1
- @(tprs_strap1 … IHT1) -IHT1 /2 width=1/
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tprs_inv_atom1: ∀U2,k. ⋆k ➡* U2 → U2 = ⋆k.
-#U2 #k #H @(tprs_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU1 destruct
->(tpr_inv_atom1 … HU2) -HU2 //
-qed-.
-
-lemma tprs_inv_cast1: ∀W1,T1,U2. ⓝW1.T1 ➡* U2 → T1 ➡* U2 ∨
- ∃∃W2,T2. W1 ➡* W2 & T1 ➡* T2 & U2 = ⓝW2.T2.
-#W1 #T1 #U2 #H @(tprs_ind … H) -U2 /3 width=5/
-#U #U2 #_ #HU2 * /3 width=3/ *
-#W #T #HW1 #HT1 #H destruct
-elim (tpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
-#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/tpr_lift.ma".
-include "basic_2/computation/tprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tprs_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡* U2 →
- ∃∃V2,T2. V1 ➡* V2 & T1 ➡* T2 & U2 = ⓛ{a}V2. T2.
-#a #V1 #T1 #U2 #H @(tprs_ind … H) -U2 /2 width=5/
-#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
-elim (tpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
-qed-.
-
-lemma tprs_inv_abst: ∀a,V1,V2,T1,T2. ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 →
- V1 ➡* V2 ∧ T1 ➡* T2.
-#a #V1 #V2 #T1 #T2 #H
-elim (tprs_inv_abst1 … H) -H #V #T #HV1 #HT1 #H destruct /2 width=1/
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Note: this was missing in basic_1 *)
-lemma tprs_lift: t_liftable tprs.
-/3 width=7/ qed.
-
-(* Note: this was missing in basic_1 *)
-lemma tprs_inv_lift1: t_deliftable_sn tprs.
-/3 width=3 by tpr_inv_lift1, t_deliftable_sn_TC/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/tpr_tpr.ma".
-include "basic_2/computation/tprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was: pr1_strip *)
-lemma tprs_strip: ∀T1,T. T ➡* T1 → ∀T2. T ➡ T2 →
- ∃∃T0. T1 ➡ T0 & T2 ➡* T0.
-/3 width=3 by TC_strip1, tpr_conf/ qed.
-
-(* Main propertis ***********************************************************)
-
-(* Basic_1: was: pr1_confluence *)
-theorem tprs_conf: confluent … tprs.
-/3 width=3/ qed.
-
-(* Basic_1: was: pr1_t *)
-theorem tprs_trans: Transitive … tprs.
-/2 width=3/ qed.
-
-(* Basic_1: was: pr1_comp *)
-lemma tprs_pair: ∀I,V1,V2. V1 ➡* V2 → ∀T1,T2. T1 ➡* T2 →
- ②{I} V1. T1 ➡* ②{I} V2. T2.
-#I #V1 #V2 #H @(tprs_ind … H) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12
-@(tprs_trans … (②{I}V.T2)) /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubst $L $T1 $d $e $T2 }.
-
-include "basic_2/substitution/ldrop_append.ma".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-inductive tps: nat → nat → lenv → relation term ≝
-| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
-| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV → ⇧[0, i + 1] V ≡ W → tps d e L (#i) W
-| tps_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
- tps d e L V1 V2 → tps (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- tps d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
- tps d e L V1 V2 → tps d e L T1 T2 →
- tps d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "parallel substritution (term)"
- 'PSubst L T1 d e T2 = (tps d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tps_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶ [d, e] T2.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ //
-| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tps_refl: ∀T,L,d,e. L ⊢ T ▶ [d, e] T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic_1: was: subst1_ex *)
-lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
- ∃∃T2,T. L ⊢ T1 ▶ [d, 1] T2 & ⇧[d, 1] T ≡ T2.
-#K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK /2 width=4/
- elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i ? ? ?) // /3 width=4/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
- ]
-]
-qed.
-
-lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶ [d1, e1] T2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- L ⊢ T1 ▶ [d2, e2] T2.
-#L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1
-[ //
-| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
- lapply (transitive_le … Hd12 … Hid1) -Hd12 -Hid1 #Hid2
- lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2 width=4/
-| /4 width=3/
-| /4 width=1/
-]
-qed.
-
-lemma tps_weak_top: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [d, |L| - d] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
- lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
- lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
-| normalize /2 width=1/
-| /2 width=1/
-]
-qed.
-
-lemma tps_weak_full: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [0, |L|] T2.
-#L #T1 #T2 #d #e #HT12
-lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
-lapply (tps_weak_top … HT12) //
-qed.
-
-lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 ▶ [d, i - d] T & L ⊢ T ▶ [i, d + e - i] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ /2 width=3/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j)
- [ -Hide -Hjde
- >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/
- | -Hdi -Hdj #Hij
- lapply (plus_minus_m_m … Hjde) -Hjde /3 width=8/
- ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi -Hide /3 width=5/
-]
-qed.
-
-lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 ▶ [i, d + e - i] T &
- L ⊢ T ▶ [d, i - d] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ /2 width=3/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j)
- [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=8/
- | -Hdi -Hdj
- >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
- ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi -Hide /3 width=5/
-]
-qed.
-
-lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 →
- ∀L. L @@ K ⊢ T1 ▶ [d, e] T2.
-#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
-#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
-lapply (ldrop_fwd_ldrop2_length … HK0) #H
-@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
-[ #L #I #d #e #J #H destruct /2 width=1/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-]
-qed.
-
-lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶ [d, e] T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-/2 width=3/ qed-.
-
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶ [d, e] T2 → T2 = ⋆k.
-#L #T2 #k #d #e #H
-elim (tps_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: subst1_gen_lref *)
-lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶ [d, e] T2 →
- T2 = #i ∨
- ∃∃K,V. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2.
-#L #T2 #i #d #e #H
-elim (tps_inv_atom1 … H) -H /2 width=1/
-* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
-qed-.
-
-lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶ [d, e] T2 → T2 = §p.
-#L #T2 #p #d #e #H
-elim (tps_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
- L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #a #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
-| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
-]
-qed.
-
-lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶ [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
- L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-/2 width=3/ qed-.
-
-fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-]
-qed.
-
-lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶ [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-/2 width=3/ qed-.
-
-fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → e = 0 → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
- lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
- elim (lt_refl_false … Hdd)
-| /3 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 0] T2 → T1 = T2.
-/2 width=6/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
-/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
-qed-.
-
-lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #d #e #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X #d #e
- >shift_append_assoc normalize #H
- elim (tps_inv_bind1 … H) -H
- #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
-]
-qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_ldrop.ma".
-include "basic_2/substitution/tps.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
-#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
-[ //
-| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
- elim (lt_or_ge i (d+1)) #HiSd
- [ -Hide1 -HV0
- lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
- lapply (ldrop_mono … HLK0 … HLK) #H destruct
- | -V -Hdi /2 width=4/
- ]
-| /4 width=3/
-| /3 width=3/
-]
-qed.
-
-lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
-/2 width=3/ qed-.
-
-lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #HT12 #K #V #HLK
-lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
-lapply (tps_inv_refl_O2 … HT12) -HT12 //
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
- ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- dt + et ≤ d →
- L ⊢ U1 ▶ [dt, et] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
- lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
- elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-]
-qed.
-
-lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
- ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- dt ≤ d → d ≤ dt + et →
- L ⊢ U1 ▶ [dt, et + e] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
- elim (lift_inv_lref1 … H) -H * #Hid #H destruct
- [ -Hdtd
- lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
- elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
- | -Hdti
- lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
- ]
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
- ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-]
-qed.
-
-(* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
- ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- d ≤ dt →
- L ⊢ U1 ▶ [dt + e, et] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
- lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
- lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
- lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
- /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ?) -V1 //
- elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
- lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ -Hdtd -Hidet
- lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
- | -Hdti -Hdedet
- lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
- elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
- @ex2_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
- ]
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
- /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ? ?) -V1 //
- elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d + e ≤ dt →
- ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
- lapply (transitive_le … Hdedt … Hdti) #Hdei
- elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
- elim (le_inv_plus_l … Hdei) #Hdie #Hei
- lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
- #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
- @ex2_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (le_inv_plus_l … Hdetd) #_ #Hedt
- elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
- <plus_minus // /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ?) -V1 //
- elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
- L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
- elim (lift_inv_lref2 … H) -H * #H
- [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
- elim (lt_refl_false … H)
- | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
- elim (lt_refl_false … H)
- ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >IHV12 // >IHT12 //
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >IHV12 // >IHT12 //
-]
-qed.
-(*
- Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (le (plus d h) i) ->
- (EX u1 | (subst0 (minus i h) v u1 u2) &
- t1 = (lift h d u1)
- ).
-
-
- Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (le d i) -> (lt i (plus d h)) ->
- (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
-*)
-lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
-qed.
-
-lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
-elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
-qed.
-
-lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
-elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
-elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
-lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
-lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/tps_lift.ma".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶ [d1, e1] T1 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶ [d1, e1] T.
-#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
-[ /2 width=3/
-| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
- elim (tps_inv_lref1 … H) -H
- [ #HX destruct /3 width=6/
- | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
- ]
-| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
- elim (IHV01 … HV02) -V0 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
-| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -V0
- elim (IHT01 … HT02) -T0 /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_confluence_neq *)
-theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T.
-#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
-[ /2 width=3/
-| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
- elim (tps_inv_lref1 … H1) -H1
- [ #H destruct /3 width=6/
- | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
- [ -Hd1 -Hde2
- lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
- lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
- elim (lt_refl_false … H)
- | -Hd2 -Hde1
- lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
- lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
- elim (lt_refl_false … H)
- ]
- ]
-| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
- elim (IHT01 … HT02 ?) -T0
- [ -H #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubr_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
- | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H
- [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *)
- ]
-| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0
- elim (IHT01 … HT02 H) -T0 -H /3 width=5/
-]
-qed.
-
-(* Note: the constant 1 comes from tps_subst *)
-(* Basic_1: was: subst1_trans *)
-theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 →
- ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e →
- L ⊢ T1 ▶ [d, e] T2.
-#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e
-[ #L #I #d #e #T2 #H #He
- elim (tps_inv_atom1 … H) -H
- [ #H destruct //
- | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
- ]
-| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
- <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
-| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
- lapply (IHT10 … HT02 He) -T0 #HT12
- lapply (tps_lsubr_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
-| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
-]
-qed.
-
-theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
-#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
-[ /2 width=3/
-| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
- lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
- <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=8/
-| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
- elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
- elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
-| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV10 … HV02 ?) -V0 //
- elim (IHT10 … HT02 ?) -T0 // /3 width=6/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $L $T1 $d $e $T2 }.
-
-include "basic_2/substitution/tps.ma".
-
-(* PARTIAL UNFOLD ON TERMS **************************************************)
-
-definition tpss: nat → nat → lenv → relation term ≝
- λd,e,L. TC … (tps d e L).
-
-interpretation "partial unfold (term)"
- 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
- ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
-#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
- ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
-#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-/2 width=1/ qed.
-
-lemma tpss_strap1: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-/2 width=3/ qed.
-
-lemma tpss_strap2: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-/2 width=3/ qed.
-
-lemma tpss_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
-/3 width=3/ qed.
-
-lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
-/2 width=1/ qed.
-
-lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
- L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
-#L #V1 #V2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
- [ /3 width=5/
- | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
- ]
-| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
- lapply (tpss_lsubr_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
- lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
-]
-qed.
-
-lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
- L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 →
- L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2.
-#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #HT12 elim HT12 -T2
- [ /3 width=1/
- | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
- ]
-| #V #V2 #_ #HV12 #IHV #HT12
- lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
-]
-qed.
-
-lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- L ⊢ T1 ▶* [d2, e2] T2.
-#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT12 #IHT
- lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
-]
-qed.
-
-lemma tpss_weak_top: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT12 #IHT
- lapply (tps_weak_top … HT12) -HT12 /2 width=3/
-]
-qed.
-
-lemma tpss_weak_full: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2.
-#L #T1 #T2 #d #e #HT12
-lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
-lapply (tpss_weak_top … HT12) //
-qed.
-
-lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
- ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
-#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k.
-#L #T2 #k #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT destruct
- >(tps_inv_sort1 … HT2) -HT2 //
-]
-qed-.
-
-(* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
-#L #T2 #p #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT destruct
- >(tps_inv_gref1 … HT2) -HT2 //
-]
-qed-.
-
-lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 &
- L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
-[ /2 width=5/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
-]
-qed-.
-
-lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
-[ /2 width=5/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/
-]
-qed-.
-
-lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2.
-#L #T1 #T2 #d #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1
-lapply (tps_fwd_tw … HT2) -HT2 #HT2
-@(transitive_le … IHT1) //
-qed-.
-
-lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
-[ /2 width=4/
-| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
- elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarAlt $L $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss_lift.ma".
-
-(* PARALLEL UNFOLD ON TERMS *************************************************)
-
-(* alternative definition of tpss *)
-inductive tpssa: nat → nat → lenv → relation term ≝
-| tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I})
-| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 →
- ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2
-| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
- tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e.
- tpssa d e L V1 V2 → tpssa d e L T1 T2 →
- tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "parallel unfold (term) alternative"
- 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tpssa_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ //
-| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T →
- ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
-#L #T1 #T #d #e #H elim H -L -T1 -T -d -e
-[ #L #I #d #e #X #H
- elim (tps_inv_atom1 … H) -H // * /2 width=6/
-| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
- lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
- lapply (tps_weak … H 0 (d+e) ? ?) -H // #H
- elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/
-| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tps_lsubr_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
- lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
- lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
- lapply (tpssa_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
-| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/
-]
-qed.
-
-lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/
-qed-.
-
-lemma tpss_ind_alt: ∀R:nat→nat→lenv→relation term.
- (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) →
- (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 →
- ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
- ) →
- (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 →
- R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 →
- R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2.
-#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e
-// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/tps_lift.ma".
-include "basic_2/unfold/tpss.ma".
-
-(* PARTIAL UNFOLD ON TERMS **************************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma tpss_subst: ∀L,K,V,U1,i,d,e.
- d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 →
- ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2.
-#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1
-[ /3 width=4/
-| #U #U1 #_ #HU1 #IHU #U2 #HU12
- elim (lift_total U 0 (i+1)) #U0 #HU0
- lapply (IHU … HU0) -IHU #H
- lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
- lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
-]
-qed.
-
-(* Advanced inverion lemmas *************************************************)
-
-lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V1,V2,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV1 &
- K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-#L #T2 #I #d #e #H @(tpss_ind … H) -T2
-[ /2 width=1/
-| #T #T2 #_ #HT2 *
- [ #H destruct
- elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
- | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
- lapply (ldrop_fwd_ldrop2 … HLK) #H
- elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
- @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
- ]
-]
-qed-.
-
-lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶* [d, e] T2 →
- T2 = #i ∨
- ∃∃K,V1,V2. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV1 &
- K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #d #e #H
-elim (tpss_inv_atom1 … H) -H /2 width=1/
-* #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
-qed-.
-
-lemma tpss_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e + 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶* [d + 1, e] T2.
-#L #T1 #T2 #d #e #H #K #V #HLK @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT
-lapply (tps_inv_S2 … HT2 … HLK) -HT2 -HLK /2 width=3/
-qed-.
-
-lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
-qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
- ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ U1 ▶* [dt, et] U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
- ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
- ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
- ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
- ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ U1 ▶* [dt + e, et] U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d + e ≤ dt →
- ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e.
- L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU destruct
-<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
-qed.
-
-lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
- ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/tps_tps.ma".
-include "basic_2/unfold/tpss_lift.ma".
-
-(* PARTIAL UNFOLD ON TERMS **************************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tpss_inv_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2.
-#L #T1 #T2 #d #H @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1
-lapply (tps_trans_ge … IHT1 … HT2 ?) //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
-/3 width=3/ qed.
-
-lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
-/3 width=3/ qed.
-
-lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2.
-#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2
-[ /2 width=3/
-| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
- elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
- elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ]
- /3 width=7 by ex2_intro, step/ (**) (* just /3 width=7/ is too slow *)
-]
-qed.
-
-lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
- ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
-qed.
-
-(* Main properties **********************************************************)
-
-theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
- ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 →
- ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 ▶* [d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 ▶* [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶* [d, e] T2 →
- L ⊢ T1 ▶* [d, e] T2.
-/2 width=3/ qed.
-
-theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubststar_4.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/lsuby.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* avtivate genv *)
-inductive cpys: relation4 genv lenv term term ≝
-| cpys_atom : ∀I,G,L. cpys G L (⓪{I}) (⓪{I})
-| cpys_delta: ∀I,G,L,K,V,V2,W2,i.
- ⇩[i] L ≡ K.ⓑ{I}V → cpys G K V V2 →
- ⇧[0, i + 1] V2 ≡ W2 → cpys G L (#i) W2
-| cpys_bind : ∀a,I,G,L,V1,V2,T1,T2.
- cpys G L V1 V2 → cpys G (L.ⓑ{I}V1) T1 T2 →
- cpys G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpys_flat : ∀I,G,L,V1,V2,T1,T2.
- cpys G L V1 V2 → cpys G L T1 T2 →
- cpys G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation
- "context-sensitive extended multiple substitution (term)"
- 'PSubstStar G L T1 T2 = (cpys G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpys_trans: ∀G. lsub_trans … (cpys G) lsuby.
-#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
-[ //
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsuby_ldrop_trans … HL12 … HLK1) -HL12 -HLK1 *
- /3 width=7 by cpys_delta/
-| /4 width=1 by lsuby_pair, cpys_bind/
-| /3 width=1 by cpys_flat/
-]
-qed-.
-
-(* Note: this is "∀L. reflexive … (cpys L)" *)
-lemma cpys_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ▶* T.
-#G #T elim T -T // * /2 width=1 by cpys_bind, cpys_flat/
-qed.
-
-lemma cpys_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶* V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ▶* ②{I}V2.T.
-* /2 width=1 by cpys_bind, cpys_flat/
-qed.
-
-lemma cpys_bind_ext: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶* V2 →
- ∀J,T1,T2. ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 →
- ∀a,I. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* ⓑ{a,I}V2.T2.
-/4 width=4 by lsuby_cpys_trans, cpys_bind, lsuby_pair/ qed.
-
-lemma cpys_delift: ∀I,G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓑ{I}V) →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
-#I #G #K #V #T1 elim T1 -T1
-[ * /2 width=4 by cpys_atom, lift_sort, lift_gref, ex2_2_intro/
- #i #L #d elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i) /3 width=7 by cpys_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by cpys_bind, ldrop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpys_flat, lift_flat, ex2_2_intro/
- ]
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpys_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶* T2 → ∀J. T1 = ⓪{J} →
- T2 = ⓪{J} ∨
- ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
-#G #L #T1 #T2 * -L -T1 -T2
-[ #I #G #L #J #H destruct /2 width=1 by or_introl/
-| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by ex4_5_intro, or_intror/
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-]
-qed-.
-
-lemma cpys_inv_atom1: ∀J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ▶* T2 →
- T2 = ⓪{J} ∨
- ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
-/2 width=3 by cpys_inv_atom1_aux/ qed-.
-
-lemma cpys_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ▶* T2 → T2 = ⋆k.
-#G #L #T2 #k #H elim (cpys_inv_atom1 … H) -H // *
-#I #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-lemma cpys_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶* T2 →
- T2 = #i ∨
- ∃∃I,K,V,V2. ⇩[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#G #L #T2 #i #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-#I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
-qed-.
-
-lemma cpys_inv_lref1_ge: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶* T2 → |L| ≤ i → T2 = #i.
-#G #L #T2 #i #H elim (cpys_inv_lref1 … H) -H // *
-#I #K #V1 #V2 #HLK #_ #_ #HL -V2 lapply (ldrop_fwd_length_lt2 … HLK) -K -I -V1
-#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
-qed-.
-
-lemma cpys_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ▶* T2 → T2 = §p.
-#G #L #T2 #p #H elim (cpys_inv_atom1 … H) -H // *
-#I #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-fact cpys_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶* U2 →
- ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 &
- U2 = ⓑ{a,J}V2.T2.
-#G #L #U1 #U2 * -L -U1 -U2
-[ #I #G #L #b #J #W #U1 #H destruct
-| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
-]
-qed-.
-
-lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶* T2 &
- U2 = ⓑ{a,I}V2.T2.
-/2 width=3 by cpys_inv_bind1_aux/ qed-.
-
-lemma cpys_inv_bind1_ext: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* U2 → ∀J.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 &
- U2 = ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #T1 #U2 #H #J elim (cpys_inv_bind1 … H) -H
-#V2 #T2 #HV12 #HT12 #H destruct
-/4 width=5 by lsuby_cpys_trans, lsuby_pair, ex3_2_intro/
-qed-.
-
-fact cpys_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ▶* U2 →
- ∀J,V1,U1. U = ⓕ{J}V1.U1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L⦄ ⊢ U1 ▶* T2 &
- U2 = ⓕ{J}V2.T2.
-#G #L #U #U2 * -L -U -U2
-[ #I #G #L #J #W #U1 #H destruct
-| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Note: lemma 1250 *)
-lemma cpys_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ▶* U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L⦄ ⊢ U1 ▶* T2 &
- U2 = ⓕ{I}V2.T2.
-/2 width=3 by cpys_inv_flat1_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpys_fwd_bind1: ∀a,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* T → ∀b,J.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,J}V1.T1 ▶* ⓑ{b,J}V2.T2 &
- T = ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #T1 #T #H #b #J elim (cpys_inv_bind1_ext … H J) -H
-#V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpys_bind, ex2_2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fqup.ma".
-include "basic_2/substitution/lpys_ldrop.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Main properties **********************************************************)
-
-theorem cpys_antisym: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶* T2 → ⦃G, L⦄ ⊢ T2 ▶* T1 → T1 = T2.
-#G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
-[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #_ #HW2 lapply (ldrop_fwd_drop2 … HLK) -I -V1
- #HLK elim (cpys_inv_lift1 … HW2 … HLK … HVW2) -L -HVW2
- #X #H #_ elim (lift_inv_lref2_be … H) -G -K -V2 -W2 -X //
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_bind1 … H) -H
- #V #T #HV2 #HT2 #H destruct
- lapply (IHV12 HV2) #H destruct -IHV12 -HV2 /3 width=1 by eq_f2/
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_flat1 … H) -H
- #V #T #HV2 #HT2 #H destruct /3 width=1 by eq_f2/
-]
-qed-.
-
-theorem cpys_trans_lpys: ∀G. lpx_sn_transitive (cpys G) (cpys G).
-#G0 #L0 #T0 @(fqup_wf_ind_eq … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #G1 #L1 * [|*]
-[ #I #HG #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
- elim (cpys_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpys_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #I2 #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
- elim (lpys_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpys_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
- ]
- | * #I1 #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- elim (lpys_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpys_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
- elim (cpys_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
- ]
-| #a #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpys_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpys_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /4 width=5 by cpys_bind, lpys_pair/
-| #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpys_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpys_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /3 width=5 by cpys_flat/
-]
-qed-.
-
-theorem cpys_trans: ∀G,L. Transitive … (cpys G L).
-/2 width=5 by cpys_trans_lpys/ qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lpys_cpys_trans: ∀G. lsub_trans … (cpys G) (lpys G).
-/2 width=5 by cpys_trans_lpys/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/fqus_alt.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Relocation properties ****************************************************)
-
-lemma cpys_lift: ∀G. l_liftable (cpys G).
-#G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #I #G #K #L #s #d #e #_ #U1 #H1 #U2 #H2
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
- elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #HKV #HVY #H destruct /3 width=10 by cpys_delta/
- | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpys_delta, ldrop_inv_gen/
- ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpys_bind, ldrop_skip/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpys_flat/
-]
-qed.
-
-lemma cpys_inv_lift1: ∀G. l_deliftable_sn (cpys G).
-#G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #G #L #i #K #s #d #e #_ #T1 #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpys_atom, lift_sort, ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpys_atom, lift_gref, ex2_intro/
- ]
-| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
- elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
- elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpys_delta, ex2_intro/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpys_delta, ex2_intro/
- ]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
- elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpys_bind, ldrop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -V1
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpys_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fqu_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpys_pair_sn, cpys_bind, cpys_flat, ex2_intro/
-[ #I #G #L #V2 #U2 #HVU2
- elim (lift_total U2 0 1)
- /4 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_drop, ex2_intro/
-| #G #L #K #T1 #U1 #e #HLK1 #HTU1 #T2 #HTU2
- elim (lift_total T2 0 (e+1))
- /3 width=11 by cpys_lift, fqu_drop, ex2_intro/
-]
-qed-.
-
-lemma fquq_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpys_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
-| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma fqup_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpys_trans … H12 … HTU2) -T2
- /3 width=3 by fqu_fqup, ex2_intro/
-| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
- elim (fqu_cpys_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
- elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
-]
-qed-.
-
-lemma fqus_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
-[ #HT12 elim (fqup_cpys_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
-| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma fqu_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
- #U2 #HVU2 @(ex3_intro … U2)
- [1,3: /3 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_drop/
- | #H destruct /2 width=7 by lift_inv_lref2_be/
- ]
-| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
- [1,3: /2 width=4 by fqu_pair_sn, cpys_pair_sn/
- | #H0 destruct /2 width=1 by/
- ]
-| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
- [1,3: /2 width=4 by fqu_bind_dx, cpys_bind/
- | #H0 destruct /2 width=1 by/
- ]
-| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
- [1,3: /2 width=4 by fqu_flat_dx, cpys_flat/
- | #H0 destruct /2 width=1 by/
- ]
-| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
- #U2 #HTU2 @(ex3_intro … U2)
- [1,3: /2 width=10 by cpys_lift, fqu_drop/
- | #H0 destruct /3 width=5 by lift_inj/
-]
-qed-.
-
-lemma fquq_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
-[ #H12 elim (fqu_cpys_trans_neq … H12 … HTU2 H) -T2
- /3 width=4 by fqu_fquq, ex3_intro/
-| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
-]
-qed-.
-
-lemma fqup_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
-[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpys_trans_neq … H12 … HTU2 H) -T2
- /3 width=4 by fqu_fqup, ex3_intro/
-| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
- #U1 #HTU1 #H #H12 elim (fqu_cpys_trans_neq … H1 … HTU1 H) -T1
- /3 width=8 by fqup_strap2, ex3_intro/
-]
-qed-.
-
-lemma fqus_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
-[ #H12 elim (fqup_cpys_trans_neq … H12 … HTU2 H) -T2
- /3 width=4 by fqup_fqus, ex3_intro/
-| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubstsnstar_3.ma".
-include "basic_2/relocation/lpx_sn.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
-
-definition lpys: relation3 genv lenv lenv ≝ λG. lpx_sn (cpys G).
-
-interpretation
- "extended multiple substitution (local environment, sn variant)"
- 'PSubstSnStar G L1 L2 = (lpys G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpys_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ▶* L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-lemma lpys_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpys_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ▶* ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpys_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ▶* K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-lemma lpys_inv_pair: ∀I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ▶* L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ▶* L2 & ⦃G, L1⦄ ⊢ V1 ▶* V2 & I1 = I2.
-/2 width=1 by lpx_sn_inv_pair/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpys_refl: ∀G,L. ⦃G, L⦄ ⊢ ▶* L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpys_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ▶* K2 → ⦃G, K1⦄ ⊢ V1 ▶* V2 →
- ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* K2.ⓑ{I}V2.
-/2 width=1 by lpx_sn_pair/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpys_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ▶* L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lpx_sn_ldrop.ma".
-include "basic_2/substitution/cpys_lift.ma".
-include "basic_2/substitution/lpys.ma".
-
-(* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma lpys_ldrop_conf: ∀G. dropable_sn (lpys G).
-/3 width=6 by lpx_sn_deliftable_dropable, cpys_inv_lift1/ qed-.
-
-lemma ldrop_lpys_trans: ∀G. dedropable_sn (lpys G).
-/3 width=10 by lpx_sn_liftable_dedropable, cpys_lift/ qed-.
-
-lemma lpys_ldrop_trans_O1: ∀G. dropable_dx (lpys G).
-/2 width=3 by lpx_sn_dropable/ qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fqu_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpys_pair, ex3_2_intro/
-[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpys_inv_pair1 … H) -H
- #K2 #W2 #HLK2 #HVW2 #H destruct
- /3 width=5 by fqu_fquq, cpys_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
- elim (ldrop_lpys_trans … HLK1 … HK12) -HK12
- /3 width=7 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma fquq_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_lpys_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpys_fqu_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpys_pair, ex3_2_intro/
-[ #I #G1 #L1 #V1 #X #H elim (lpys_inv_pair2 … H) -H
- #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
- /4 width=7 by cpys_delta, fqu_drop, ldrop_drop, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HL01
- elim (lpys_ldrop_trans_O1 … HL01 … HLK1) -L1
- /3 width=5 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma lpys_fquq_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (lpys_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_cpys.ma".
-
-(* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
-
-(* Main properties **********************************************************)
-
-theorem lpys_trans: ∀G. Transitive … (lpys G).
-/3 width=5 by lpx_sn_trans, cpys_trans_lpys/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 break ⊆ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEq $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lrsubeq_2.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-inductive lsuby: relation lenv ≝
-| lsuby_atom: ∀L. lsuby L (⋆)
-| lsuby_pair: ∀I1,I2,L1,L2,V. lsuby L1 L2 → lsuby (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
-.
-
-interpretation
- "local environment refinement (extended substitution)"
- 'LRSubEq L1 L2 = (lsuby L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_refl: ∀L. L ⊆ L.
-#L elim L -L /2 width=1 by lsuby_pair/
-qed.
-
-lemma lsuby_sym: ∀L1,L2. L1 ⊆ L2 → |L1| = |L2| → L2 ⊆ L1.
-#L1 #L2 #H elim H -L1 -L2
-[ #L1 #H >(length_inv_zero_dx … H) -L1 //
-| #I1 #I2 #L1 #L2 #V #_ #IHL12 #H lapply (injective_plus_l … H) -H
- /3 width=1 by lsuby_pair/
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊆ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2 //
-#I1 #I2 #L1 #L2 #V #_ #H destruct
-qed-.
-
-lemma lsuby_inv_atom1: ∀L2. ⋆ ⊆ L2 → L2 = ⋆.
-/2 width=3 by lsuby_inv_atom1_aux/ qed-.
-
-fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊆ L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
- L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
-#L1 #L2 * -L1 -L2
-[ #L #J1 #K1 #W #H destruct /2 width=1 by or_introl/
-| #I1 #I2 #L1 #L2 #V #HL12 #J1 #K1 #W #H destruct /3 width=4 by ex2_2_intro, or_intror/
-]
-qed-.
-
-lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊆ L2 →
- L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
-/2 width=4 by lsuby_inv_pair1_aux/ qed-.
-
-fact lsuby_inv_pair2_aux: ∀L1,L2. L1 ⊆ L2 → ∀J2,K2,W. L2 = K2.ⓑ{J2}W →
- ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
-#L1 #L2 * -L1 -L2
-[ #L #J2 #K2 #W #H destruct
-| #I1 #I2 #L1 #L2 #V #HL12 #J2 #K2 #W #H destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lsuby_inv_pair2: ∀I2,L1,K2,W. L1 ⊆ K2.ⓑ{I2}W →
- ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
-/2 width=4 by lsuby_inv_pair2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsuby_fwd_length: ∀L1,L2. L1 ⊆ L2 → |L2| ≤ |L1|.
-#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
-qed-.
-
-lemma lsuby_ldrop_trans: ∀L1,L2. L1 ⊆ L2 →
- ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
- ∃∃I1,K1. K1 ⊆ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
-#L1 #L2 #H elim H -L1 -L2
-[ #L #J2 #K2 #W #s #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #s #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=4 by ldrop_pair, ex2_2_intro/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_drop_lt, ex2_2_intro/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lsuby.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-(* Main properties **********************************************************)
-
-theorem lsuby_trans: Transitive … lsuby.
-#L1 #L #H elim H -L1 -L
-[ #L1 #X #H lapply (lsuby_inv_atom1 … H) -H //
-| #I1 #I #L1 #L #V #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H // *
- #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ▶ * break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PSubstSnStar $G $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_alt.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-(* Properties on local environment refinement for extended substitution *****)
-
-lemma lsuby_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) (lsuby 0 (∞)).
-#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
-[ //
-| /2 width=2 by cpx_sort/
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsuby_ldrop_trans_be … HL12 … HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
-|4,9: /4 width=1 by cpx_bind, cpx_beta, lsuby_pair_O_Y/
-|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
-|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsuby_pair_O_Y/
-]
-qed-.
-
-(* Properties on context-sensitive extended multiple substitution for terms *)
-
-lemma cpys_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L -T1 -T2 -d -e
-/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
-qed.
-
-lemma cpy_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-/3 width=3 by cpy_cpys, cpys_cpx/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_alt.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-(* Properties on context-sensitive extended multiple substitution for terms *)
-
-lemma cpys_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L -T1 -T2 -d -e
-/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
-qed.
-
-lemma cpy_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-/3 width=3 by cpy_cpys, cpys_cpx/ qed.
-
-lemma cpx_cpy_trans: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
- ∀T2,d,e. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T #H elim H -G -L -T1 -T
-[ #I #G #L #X #d #e #H elim (cpy_inv_atom1 … H) //
- * /2 width=3 by cpy_cpx/
-| #G #L #k #l #Hkl #X #d #e #H >(cpy_inv_sort1 … H) -X /2 width=2 by cpx_sort/
-| #I #G #L #K #V1 #V #W #i #HLK #_ #HVW #IHV1 #X #d #e #H
- lapply (ldrop_fwd_drop2 … HLK) #H0
- lapply (cpy_weak … H 0 (∞) ? ?) -H // #H
- elim (cpy_inv_lift1_be … H … H0 … HVW) -H -H0 -HVW
- /3 width=7 by cpx_delta/
-| #a #I #G #L #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #X #d #e #H elim (cpy_inv_bind1 … H) -H
- #V2 #T2 #HV2 #HT2 #H destruct
- /5 width=7 by cpx_bind, lsuby_cpy_trans, lsuby_succ/
-| #I #G #L #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #X #d #e #H elim (cpy_inv_flat1 … H) -H
- #V2 #T2 #HV2 #HT2 #H destruct /3 width=3 by cpx_flat/
-| #G #L #V1 #U1 #U #T #_ #HTU #IHU1 #T2 #d #e #HT2
- lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
- elim (lift_total T2 0 1) #U2 #HTU2
- lapply (cpy_lift_be … HT2 (L.ⓓV1) … (Ⓕ) … HTU … HTU2 ? ?) -T
- /3 width=3 by cpx_zeta, ldrop_drop/
-| /3 width=3 by cpx_tau/
-| /3 width=3 by cpx_ti/
-| #a #G #L #V1 #V #W1 #W #T1 #T #_ #_ #_ #IHV1 #IHW1 #IHT1 #X #d #e #HX
- elim (cpy_inv_bind1 … HX) -HX #Y #T2 #HY #HT2 #H destruct
- elim (cpy_inv_flat1 … HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
- /5 width=7 by cpx_beta, lsuby_cpy_trans, lsuby_succ/
-| #a #G #L #V1 #V #U #W1 #W #T1 #T #_ #HVU #_ #_ #IHV1 #IHW1 #IHT1 #X #d #e #HX
- elim (cpy_inv_bind1 … HX) -HX #W2 #Y #HW2 #HY #H destruct
- elim (cpy_inv_flat1 … HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
- lapply (cpy_weak … HU2 0 (∞) ? ?) -HU2 // #HU2
- elim (cpy_inv_lift1_be … HU2 L … HVU) -U
- /5 width=7 by cpx_theta, lsuby_cpy_trans, lsuby_succ, ldrop_drop/
-]
-qed-.
-
-lemma cpx_cpys_trans: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
- ∀T2,d,e. ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T #HT1 #T2 #d #e #H @(cpys_ind … H) -T2
-/2 width=5 by cpx_cpy_trans/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cpx_cpys.ma".
-include "basic_2/computation/cpxs.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-
-(* Properties on local environment refinement for extended substitution *****)
-
-lemma lsuby_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) (lsuby 0 (∞)).
-/3 width=5 by lsuby_cpx_trans, LTC_lsub_trans/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cpy_lift.ma".
-include "basic_2/substitution/cpys.ma".
-include "basic_2/reduction/lpx_ldrop.ma".
-
-(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
-
-(* Properties on context-sensitive extended substitution for terms **********)
-
-lemma cpx_cpy_trans_lpx: ∀h,g,G,L1,T1,T. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ∀T2,d,e. ⦃G, L2⦄ ⊢ T ▶[d, e] T2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L1 #T1 #T #H elim H -G -L1 -T1 -T
-[ #J #G #L1 #L2 #HL12 #T2 #d #e #H elim (cpy_inv_atom1 … H) -H //
- * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct
- elim (lpx_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpx_inv_pair2 … H) -H #K1 #V1 #_ #HV12 #H destruct
- /2 width=7 by cpx_delta/
-| #G #L1 #k #l #Hkl #L2 #_ #X #d #e #H >(cpy_inv_sort1 … H) -X /2 width=2 by cpx_sort/
-| #I #G #L1 #K1 #V1 #V #T #i #HLK1 #_ #HVT #IHV1 #L2 #HL12 #T2 #d #e #HT2
- elim (lpx_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
- lapply (ldrop_fwd_drop2 … HLK2) -V0 #HLK2
- lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
- elim (cpy_inv_lift1_be … HT2 … HLK2 … HVT) -HT2 -HLK2 -HVT
- /3 width=7 by cpx_delta/
-| #a #I #G #L1 #V1 #V #T1 #T #HV1 #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_bind1 … H) -H
- #V2 #T2 #HV2 #HT2 #H destruct /4 width=5 by lpx_pair, cpx_bind/
-| #I #G #L1 #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_flat1 … H) -H
- #V2 #T2 #HV2 #HT2 #H destruct /3 width=5 by cpx_flat/
-| #G #L1 #V1 #U1 #U #T #_ #HTU #IHU1 #L2 #HL12 #T2 #d #e #HT2
- lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
- elim (lift_total T2 0 1) #U2 #HTU2
- lapply (cpy_lift_be … HT2 (L2.ⓓV1) … (Ⓕ) … HTU … HTU2 ? ?) -T
- /4 width=5 by cpx_zeta, lpx_pair, ldrop_drop/
-| /3 width=5 by cpx_tau/
-| /3 width=5 by cpx_ti/
-| #a #G #L1 #V1 #V #W1 #W #T1 #T #HV1 #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
- elim (cpy_inv_bind1 … HX) -HX #Y #T2 #HY #HT2 #H destruct
- elim (cpy_inv_flat1 … HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
- /5 width=11 by lpx_pair, cpx_beta, lsuby_cpy_trans, lsuby_succ/
-| #a #G #L1 #V1 #V #U #W1 #W #T1 #T #_ #HVU #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
- elim (cpy_inv_bind1 … HX) -HX #W2 #Y #HW2 #HY #H destruct
- elim (cpy_inv_flat1 … HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
- lapply (cpy_weak … HU2 0 (∞) ? ?) -HU2 // #HU2
- elim (cpy_inv_lift1_be … HU2 L2 … HVU) -U
- /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/
-]
-qed-.
-
-lemma cpx_cpys_trans_lpx: ∀h,g,G,L1,T1,T. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ∀T2,d,e. ⦃G, L2⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind … H) -T2
-/2 width=7 by cpx_cpy_trans_lpx/
-qed-.
+++ /dev/null
-(*
-lemma lsuby_weak: ∀L1,L2,d1,e1. L1 ⊑×[d1, e1] L2 →
- ∀d2,e2. d1 ≤ d2 → e2 ≤ e1 → L1 ⊑×[d2, e2] L2.
-#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #_ #d2 #e2 #_ #He21
- >(yle_inv_O2 … He21) -He21
- /4 width=3 by lsuby_fwd_length, lsuby_O1, monotonic_le_plus_l/
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #d2 #e2 #_ #He21
- elim (ynat_cases e2) /4 width=3 by lsuby_fwd_length, lsuby_O1, monotonic_le_plus_l/
- * #e0 #H destruct lapply (yle_inv_succ … He21) -He21 #He21
- elim (ynat_cases d2) /3 width=1 by lsuby_pair/
- * #d0 #H destruct @lsuby_succ @IHL12 //
- [ destruct
-
-*)
+++ /dev/null
-(* Basic forward lemmas *****************************************************)
-
-lemma csup_fwd_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ →
- ∃i. ⇩[0, i] L1 ≡ L2 ∨ ⇩[0, i] L2 ≡ L1.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=2/ /4 width=2/
-#I #L1 #K1 #V1 #i #HLK1
-lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /3 width=2/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lift_csup_trans_eq: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀L,U2. ⦃L, U1⦄ > ⦃L, U2⦄ →
- ∃T2. ⇧[d, e] T2 ≡ U2.
-#T1 #U1 #d #e * -T1 -U1 -d -e
-[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #_ #L #X #H
- elim (csup_inv_bind1 … H) -H *
- [ #_ #H destruct /2 width=2/
- | #H elim (discr_lpair_x_xy … H)
- ]
-|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H
- elim (csup_inv_flat1 … H) -H #_ * #H destruct /2 width=2/
-]
-#i #d #e [2,3: #_ ] #L #X #H
-elim (csup_inv_atom1 … H) -H #I #j #HL #H destruct
-lapply (ldrop_pair2_fwd_cw … HL X) -HL #H
-elim (lt_refl_false … H)
-qed-.
-(*
-lemma lift_csup_trans_gt: ∀L1,L2,U1,U2. ⦃L1, U1⦄ > ⦃L2, U2⦄ →
- ⇩[0, 1] L2 ≡ L1 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∃T2. ⇧[d + 1, e] T2 ≡ U2.
-#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2
-[ #I #L1 #K1 #V #i #HLK1 #HKL1
- lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
- lapply (ldrop_fwd_lw … HKL1) -HKL1 #HKL1
- lapply (transitive_le … HLK1 HKL1) -L1 normalize #H
-
-
-| #a
-| #a
-]
-#I #L1 #W1 #U1 #HL1
-
-
-
- #X #d #e #H
- lapply (ldrop_inv_refl … HL1) -HL1
-| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H
- lapply (ldrop_inv_ldrop1 … HL1)
-
- elim (lift_inv_bind2 … H) -H #W2 #U2 #HW21 #HU21 #H destruct
-
-
- /3 width=2/ /4 width=2/
-
-*)
-
-
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma csup_inv_lref2_be: ∀L,U,i. ⦃L, U⦄ > ⦃L, #i⦄ →
- ∀T,d,e. ⇧[d, e] T ≡ U → d ≤ i → i < d + e → ⊥.
-#L #U #i #H #T #d #e #HTU #Hdi #Hide
-elim (lift_csup_trans_eq … HTU … H) -H -T #T #H
-elim (lift_inv_lref2_be … H ? ?) //
-qed-.
-
-
-fact csup_inv_all4_refl_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → L1 = L2 →
- ∨∨ ∃∃a,I,U. T1 = ⓑ{a,I}T2.U
- | ∃∃I,W. T1 = ⓕ{I}W.T2
- | ∃∃I,U. T1 = ⓕ{I}T2.U.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=3/ /3 width=4/
-[ #I #L #K #V #i #HLK #H destruct
- lapply (ldrop_pair2_fwd_cw … HLK V) -HLK #H
- elim (lt_refl_false … H)
-| #a #I #L #V #T #H
- elim (discr_lpair_x_xy … H)
-]
-qed-.
-
-lemma csup_inv_all4_refl: ∀L,T1,T2. ⦃L, T1⦄ > ⦃L, T2⦄ →
- ∨∨ ∃∃a,I,U. T1 = ⓑ{a,I}T2.U
- | ∃∃I,W. T1 = ⓕ{I}W.T2
- | ∃∃I,U. T1 = ⓕ{I}T2.U.
-/2 width=4 by csup_inv_all4_refl_aux/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_ldrop.ma".
-include "basic_2/substitution/csup.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* Main forward lemmas ******************************************************)
-
-theorem csup_trans_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ →
- ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ →
- L = L0 ∨ ⦃L, T1⦄ > ⦃L, T3⦄.
-#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/
-[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H
- lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1
- lapply (csup_fwd_cw … H) -H #H2
- lapply (transitive_lt … H1 H2) -H1 -H2 #H
- elim (lt_refl_false … H)
-| #a #I #L0 #V2 #T2 #T3 #HT23
- elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/
- qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/csup_csup.ma".
-include "basic_2/unfold/csups.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Advanced forward lemmas **************************************************)
-
-(*
-lemma csupp_strap2_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ →
- ∀T3. ⦃L0, T2⦄ >+ ⦃L, T3⦄ →
- L = L0 ∨ ⦃L, T1⦄ >+ ⦃L, T3⦄.
-#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/
-[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H
- lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1
- lapply (csupp_fwd_cw … H) -H #H2
- lapply (transitive_lt … H1 H2) -H1 -H2 #H
- elim (lt_refl_false … H)
-| #a #I #L0 #V2 #T2 #T3 #HT23
- /3 width=5/
-
- elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/
- qed-.
-
-
-
-
-
-
-
-
-lemma csups_strap1_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ >* ⦃L0, T2⦄ →
- ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ → L = L0.
-#L #L0 #T1 #T2 #H @(csups_ind_dx … H) -L -T1 //
-#L1 #L #T1 #T #HL1 #_ #IHL0 #T3 #HL0
-lapply (csup_trans_fwd_refl … HL10) … HL0) -T2
-*)
-lemma lift_csups_trans_aux: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀L1,L2,U2. ⦃L1, U1⦄ >* ⦃L2, U2⦄ → L1 = L2 →
- ∃T2. ⇧[d, e] T2 ≡ U2.
-#T1 #U1 #d #e #HTU1 #L1 #L2 #U2 #H @(csups_ind … H) -L2 -U2 /2 width=2/ -T1
-#L #L2 #U #U2 #HL1 #HL2 #IHL1 #H destruct
-
-* -T1 -U1 -d -e
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/delift_lift.ma".
-include "basic_2/unfold/delift_delift.ma".
-include "basic_2/computation/cprs_delift.ma".
-include "basic_2/equivalence/cpcs_cpcs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Properties on inverse basic term relocation ******************************)
-
-lemma cpcs_zeta_delift_comm: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 →
- L ⊢ T2 ⬌* +ⓓV.T1.
-/3 width=1/ qed.
-
-(* Basic_1: was only: pc3_gen_cabbr *)
-lemma thin_cpcs_delift_mono: ∀L,U1,U2. L ⊢ U1 ⬌* U2 →
- ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
- ∀T2. L ⊢ ▼*[d, e] U2 ≡ T2 → K ⊢ T1 ⬌* T2.
-#L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2
-elim (cpcs_inv_cprs … H) -H #U #HU1 #HU2
-elim (thin_cprs_delift_conf … HU1 … HLK … HTU1) -U1 #T #HT1 #HUT
-elim (thin_cprs_delift_conf … HU2 … HLK … HTU2) -U2 -HLK #X #HT2 #H
-lapply (delift_mono … H … HUT) -L #H destruct /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/thin_delift.ma".
-include "basic_2/reducibility/tpr_delift.ma".
-include "basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties on inverse basic term relocation ******************************)
-
-(* Basic_1: was only: pr2_gen_cabbr *)
-lemma thin_cpr_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
- ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
- ∃∃T2. K ⊢ T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
-#L #U1 #U2 * #U #HU1 #HU2 #K #d #e #HLK #T1 #HTU1
-elim (tpr_delift_conf … HU1 … HTU1) -U1 #T #HT1 #HUT
-elim (le_or_ge (|L|) d) #Hd
-[ elim (thin_delift_tpss_conf_le … HU2 … HUT … HLK ?)
-| elim (le_or_ge (|L|) (d+e)) #Hde
- [ elim (thin_delift_tpss_conf_le_up … HU2 … HUT … HLK ? ? ?)
- | elim (thin_delift_tpss_conf_be … HU2 … HUT … HLK ? ?)
- ]
-] -U -HLK // -Hd [2,3: -Hde] #T2 #HT2
-lapply (cpr_intro … HT1 HT2) -T /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_delift.ma".
-include "basic_2/reducibility/cpr_cpr.ma".
-include "basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties on inverse basic term relocation ******************************)
-
-(* Note: this should be stated with tprs *)
-lemma cprs_zeta_delift: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 → L ⊢ +ⓓV.T1 ➡* T2.
-#L #V #T1 #T2 * #T #HT1 #HT2
-@(cprs_strap2 … (+ⓓV.T)) [ /3 width=3/ | @inj /3 width=3/ ] (**) (* explicit constructor, /5 width=3/ is too slow *)
-qed.
-
-(* Basic_1: was only: pr3_gen_cabbr *)
-lemma thin_cprs_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡* U2 →
- ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
- ∃∃T2. K ⊢ T1 ➡* T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
-#L #U1 #U2 #H @(cprs_ind … H) -U2 /2 width=3/
-#U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1
-elim (IHU1 … HLK … HTU1) -U1 #T #HT1 #HUT
-elim (thin_cpr_delift_conf … HU2 … HLK … HUT) -U -HLK /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'TSubst $L $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-definition delift: nat → nat → lenv → relation term ≝
- λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T.
-
-interpretation "inverse basic relocation (term)"
- 'TSubst L T1 d e T2 = (delift d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 →
- ∀L. L ⊢ ▼*[d, e] T2 ≡ T1.
-/2 width=3/ qed.
-
-lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T.
-/2 width=3/ qed.
-
-lemma delift_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2.
-#L1 #T1 #T2 #d #e * /3 width=3/
-qed.
-
-lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k.
-/2 width=3/ qed.
-
-lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i.
-/3 width=3/ qed.
-
-lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e).
-/3 width=3/ qed.
-
-lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p.
-/2 width=3/ qed.
-
-lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e.
- L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 →
- L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
-#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
-lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
-qed.
-
-lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
- L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 →
- L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k.
-#L #U2 #d #e #k * #U #HU
->(tpss_inv_sort1 … HU) -HU #HU2
->(lift_inv_sort2 … HU2) -HU2 //
-qed-.
-
-lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p.
-#L #U #d #e #p * #U #HU
->(tpss_inv_gref1 … HU) -HU #HU2
->(lift_inv_gref2 … HU2) -HU2 //
-qed-.
-
-lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 →
- ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
- L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 &
- U2 = ⓑ{a,I} V2. T2.
-#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
-elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
-elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
-lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
-qed-.
-
-lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 →
- ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
- L ⊢ ▼*[d, e] T1 ≡ T2 &
- U2 = ⓕ{I} V2. T2.
-#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
-elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
-elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
-qed-.
-
-lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2.
-#L #T1 #T2 #d * #T #HT1
->(tpss_inv_refl_O2 … HT1) -HT1 #HT2
->(lift_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e * #T #HT1 #HT2
->(lift_fwd_tw … HT2) -T2 /2 width=4 by tpss_fwd_tw/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'TSubstAlt $L $T1 $d $e $T2 }.
-
-include "basic_2/unfold/delift_lift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* alternative definition of inverse basic term relocation *)
-inductive delifta: nat → nat → lenv → relation term ≝
-| delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k)
-| delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i)
-| delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 →
- ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2
-| delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e))
-| delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p)
-| delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
- delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| delifta_flat : ∀L,I,V1,V2,T1,T2,d,e.
- delifta d e L V1 V2 → delifta d e L T1 T2 →
- delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "inverse basic relocation (term) alternative"
- 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma delifta_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/
-[ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
-#L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
-[ * #i #Hn #T2 #d #e #H destruct
- [ >(delift_inv_sort1 … H) -H //
- | elim (delift_inv_lref1 … H) -H * /2 width=1/
- #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
- lapply (ldrop_pair2_fwd_fw … HLK) #H
- lapply (IH … HV12) // -H /2 width=6/
- | >(delift_inv_gref1 … H) -H //
- ]
-| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
- [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
- lapply (IH … HV12) -HV12 // #HV12
- lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12
- lapply (delifta_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
- | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HV12) -HV12 //
- lapply (IH … HT12) -IH -HT12 // /2 width=1/
- ]
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/
-qed-.
-
-lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
- (∀L,d,e,k. R d e L (⋆k) (⋆k)) →
- (∀L,d,e,i. i < d → R d e L (#i) (#i)) →
- (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
- ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
- ) →
- (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
- (∀L,d,e,p. R d e L (§p) (§p)) →
- (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
- L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 →
- R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
- L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 →
- R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e
-// /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Main properties **********************************************************)
-
-theorem delift_mono: ∀L,T,T1,T2,d,e.
- L ⊢ ▼*[d, e] T ≡ T1 → L ⊢ ▼*[d, e] T ≡ T2 → T1 = T2.
-#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2
-elim (tpss_conf_eq … H1TU1 … H1TU2) -T #U #HU1 #HU2
-lapply (tpss_inv_lift1_eq … HU1 … H2TU1) -HU1 #H destruct
-lapply (tpss_inv_lift1_eq … HU2 … H2TU2) -HU2 #H destruct
-lapply (lift_inj … H2TU1 … H2TU2) //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_lbotr.ma".
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 →
- ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2.
-#L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2
-elim (lift_total V 0 (i+1)) #U #HVU
-lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m /2 width=1/ #HV2U
-lapply (lift_conf_be … HVU2 … HV2U ?) //
->commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
-qed.
-
-lemma lbotr_delift: ∀L,T1,d,e. d + e ≤ |L| → ⊒[d, e] L →
- ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 @(f2_ind … fw … L T1) -L -T1
-#n #IH #L * * /2 width=2/
-[ #i #H #d #e #Hde #HL destruct
- elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
- elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
- lapply (lt_to_le_to_lt … Hide Hde) #Hi
- elim (ldrop_O1_lt … Hi) -Hi #I #K #V1 #HLK
- lapply (lbotr_inv_ldrop … HLK … HL ? ?) // #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK (#i)) #HKL
- lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
- lapply (ldrop_fwd_O1_length … HLK0) #H
- lapply (lbotr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
- [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
- elim (IH … HKL … HK) -IH -HKL -HK
- [2: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
- elim (lift_total V2 0 d) /3 width=7/
-| #a #I #V1 #T1 #H #d #e #Hde #HL destruct
- elim (IH … V1 … Hde HL) // #V2 #HV12
- elim (IH (L.ⓑ{I}V1) T1 … (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12
- lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
-| #I #V1 #T1 #H #d #e #Hde #HL destruct
- elim (IH … V1 … Hde HL) // #V2 #HV12
- elim (IH … T1 … Hde HL) -IH -Hde -HL // /3 width=2/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i.
-#L #U2 #i #d #e * #U #HU #HU2 #Hid
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct >(lift_inv_lref2_lt … HU2) //
-| * #K #V1 #V2 #Hdi
- lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
- elim (lt_refl_false … Hi)
-]
-qed-.
-
-lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 →
- d ≤ i → i < d + e →
- ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
- ⇧[0, d] V2 ≡ U2.
-#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
-| * #K #V1 #V #_ #_ #HLK #HV1 #HVU
- elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
-]
-qed-.
-
-lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
- d + e ≤ i → U2 = #(i - e).
-#L #U2 #i #d #e * #U #HU #HU2 #Hdei
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct >(lift_inv_lref2_ge … HU2) //
-| * #K #V1 #V2 #_ #Hide
- lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
- elim (lt_refl_false … Hi)
-]
-qed-.
-
-lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
- ∨∨ (i < d ∧ U2 = #i)
- | (∃∃K,V1,V2. d ≤ i & i < d + e &
- ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
- ⇧[0, d] V2 ≡ U2
- )
- | (d + e ≤ i ∧ U2 = #(i - e)).
-#L #U2 #i #d #e #H
-elim (lt_or_ge i d) #Hdi
-[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
-| elim (lt_or_ge i (d+e)) #Hide
- [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
- | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
- ]
-]
-qed-.
-
-(* Properties on basic term relocation **************************************)
-
-lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
- ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
- L ⊢ ▼*[dt, et] U1 ≡ U2.
-#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
-elim (lift_total T d e) #U #HTU
-lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
-elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
->(lift_mono … HTU2 … HT2) -T2 /2 width=3/
-qed.
-
-lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
- ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
- ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
- L ⊢ ▼*[dt, et + e] U1 ≡ T2.
-#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
-elim (lift_total T d e) #U #HTU
-lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
-lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
-qed.
-
-lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
- ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ ▼*[dt + e, et] U1 ≡ U2.
-#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
-elim (lift_total T d e) #U #HTU
-lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
-elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
->(lift_mono … HTU2 … HT2) -T2 /2 width=3/
-qed.
-
-lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 →
- ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2.
-#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1
-lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct
-lapply (lift_inj … HTU1 … HTU2) -U2 //
-qed-.
-
-lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T →
- ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e →
- L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide
-lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10
-lapply (lift_trans_be … HT2 … HT0 ? ?) -T //
->commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?);
-<minus_le_minus_minus_comm // <plus_minus_m_m [ /2 width=3/ | /2 width=1/ ]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_alt.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Properties on sn partial unfold on local environments ********************)
-
-lemma delift_ltpss_sn_conf_eq: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 →
- ∀K. L ⊢ ▶* [d, e] K → K ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 #T2 #d #e * #T #HT1 #HT2 #K #HLK
-elim (ltpss_sn_tpss_conf … HT1 … HLK) -HT1 -HLK #T0 #HT10 #HT0
-lapply (tpss_inv_lift1_eq … HT0 … HT2) -HT0 #H destruct /2 width=3/
-qed.
-
-lemma ltpss_sn_delift_trans_eq: ∀L,K,d,e. L ⊢ ▶* [d, e] K →
- ∀T1,T2. K ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #K #d #e #HLK #T1 #T2 * #T #HT1 #HT2
-lapply (ltpss_sn_tpss_trans_eq … HT1 … HLK) -K /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Properties on partial unfold on terms ************************************)
-
-lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-elim (tpss_inv_lift1_le … HXU1 … HLK … HTX1 ?) -X1 -HLK // -H1 /3 width=5/
-qed.
-
-lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
-
-lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2 #H3
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-elim (tpss_inv_lift1_le_up … HXU1 … HLK … HTX1 ? ? ?) -X1 -HLK // -H1 -H2 -H3 /3 width=5/
-qed.
-
-lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=6/ qed.
-
-lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // -H1 -H2 /3 width=5/
-qed.
-
-lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
-
-lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
-#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
-qed.
-
-lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
-/3 width=3/ qed.
-
-lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
-#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
-lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
-qed.
-
-lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'TSubst $T1 $d $e $T2 }.
-
-include "basic_2/unfold/ltpss_sn.ma".
-
-(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
-
-definition thin: nat → nat → relation lenv ≝
- λd,e,L1,L2. ∃∃L. L1 ⊢ ▶* [d, e] L & ⇩[d, e] L ≡ L2.
-
-interpretation "basic thinning (local environment)"
- 'TSubst L1 d e L2 = (thin d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ▼*[d, e] L1 ≡ L2.
-/2 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma thin_inv_thin1: ∀I,K1,V1,L2,e. ▼*[0, e] K1.ⓑ{I} V1 ≡ L2 → 0 < e →
- ▼*[0, e - 1] K1 ≡ L2.
-#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e
-elim (ltpss_sn_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct
-lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/delift_tpss.ma".
-include "basic_2/unfold/delift_ltpss.ma".
-include "basic_2/unfold/thin.ma".
-
-(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
-
-(* Inversion lemmas on inverse basic term relocation ************************)
-
-lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
- ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
- K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
- L2 = K2. ⓑ{I} V2.
-#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
-elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
-elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
-qed-.
-
-(* Properties on inverse basic term relocation ******************************)
-
-lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
- ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
-#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
-elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
-lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
-lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
-qed.
-
-lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
-lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
-elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
-elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
-lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
-lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
-qed.
-
-lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
-
-lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
-lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
-elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
-elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
-lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
-lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
-qed.
-
-lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
-
-lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
-lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
-elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
-elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
-lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
-lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
-qed.
-
-lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/ltpss_sn_ldrop.ma".
-include "basic_2/unfold/thin.ma".
-
-(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
-
-(* Properties on local environment slicing **********************************)
-
-lemma thin_ldrop_conf_ge: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2 - e1] L1 ≡ L2.
-#L0 #L1 #d1 #e1 * /3 width=8 by ltpss_sn_ldrop_conf_ge, ldrop_conf_ge/
-qed.
-
-lemma thin_ldrop_conf_be: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. ▼*[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
-#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #Hd1e2 #He2de1
-elim (ltpss_sn_ldrop_conf_be … HL0 … HL02 ? ?) -L0 // #L0 #HL20 #HL0
-elim (ldrop_conf_be … HL1 … HL0 ? ?) -L // -Hd1e2 -He2de1 /3 width=3/
-qed.
-
-lemma thin_ldrop_conf_le: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. ▼*[d1 - e2, e1] L2 ≡ L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #He2d1
-elim (ltpss_sn_ldrop_conf_le … HL0 … HL02 ?) -L0 // #L0 #HL20 #HL0
-elim (ldrop_conf_le … HL1 … HL0 ?) -L // -He2d1 /3 width=3/
-qed.
-
-lemma thin_ldrop_trans_ge: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #Hd1e2
-lapply (ldrop_trans_ge … HL0 … HL02 ?) -L0 // #HL2
-lapply (ltpss_sn_ldrop_trans_ge … HL1 … HL2 ?) -L // /2 width=1/
-qed.
-
-lemma thin_ldrop_trans_le: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. ▼*[d1 - e2, e1] L ≡ L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #He2d1
-elim (ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL0 #HL02
-elim (ltpss_sn_ldrop_trans_le … HL1 … HL0 He2d1) -L -He2d1 /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/delift.ma".
-include "basic_2/reducibility/ltpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Properties on inverse basic term relocation ******************************)
-
-lemma tpr_delift_conf: ∀U1,U2. U1 ➡ U2 → ∀L,T1,d,e. L ⊢ ▼*[d, e] U1 ≡ T1 →
- ∃∃T2. T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
-#U1 #U2 #HU12 #L #T1 #d #e * #W1 #HUW1 #HTW1
-elim (tpr_tpss_conf … HU12 … HUW1) -U1 #U1 #HWU1 #HU21
-elim (tpr_inv_lift1 … HWU1 … HTW1) -W1 /3 width=5/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-(*
-include "basic_2/reduction/lpr_ldrop.ma".
-
-include "basic_2/unfold/fsups.ma".
-include "basic_2/reducibility/lpr_ldrop.ma".
-
-lamma pippo: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀V1,i. ⇧[i, 1] V1 ≡ T1 → T2 = #i → ⊥.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2
-[ #I #L1 #V #W1 #j #H1 #H2
- elim (lift_inv_lref2 … H1) -H1 * #H1 #H3
-
- HVT2 : ()
- HV2 : (K2⊢V➡V2)
-
-
-thaorem cpr_trans_lpr: ∀L1,T1,T. L1 ⊢ T1 ➡ T → ∀L2. L1 ⊢ ➡ L2 →
- ∀T2. L2 ⊢ T ➡ T2 →
- (⦃L2, T2⦄ ⊃* ⦃L1, T1⦄ → ⊥) ∨ T1 = T.
-#L1 #T1 @(fsupp_wf_ind … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
-[ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct -IH
- elim (cpr_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpr_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct //
- ]
- | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -V0 #HLK2
- elim (cpr_inv_lift1 … HT2 … HLK2 … HVT) -HT2 -HLK2 -HVT #V2 #HVT2 #HV2
- @or_introl #H
-
-
-
-
-
-
- elim (lift_inv_lref2 … HVT2) -HVT2 * #H #_
- [ elim (lt_zero_false … H)
- | >commutative_plus in H; >plus_plus_comm_23 #H
- elim (le_plus_xySz_x_false … H)
- ]
- ]
-| #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpr_inv_bind1 … H1) -H1 *
- [ #V #T #HV1 #HT1 #H destruct
- elim (cpr_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV2 #HT2 #H destruct
- elim (IH … HV1 … HV2) // #HV12 destruct
- [ @or_introl #H destruct /2 width=1/
- | elim (IH … HT1 … HT2) // /2 width=1/ -L1 -L2 #HT12 destruct
- @or_introl #H destruct /2 width=1/
- ]
- | #T2 #HT2 #HXT2 #H1 #H2 destruct
- elim (IH … HT1 … HT2) // /2 width=1/ -L1 -L2 #HT12 destruct
- | elim (term_eq_dec V1 V) #HV1 destruct
-
- ]
- | #Y1 #HTY1 #HXY1 #H11 #H12 destruct
- elim (lift_total (+ⓓV1.T1) 0 1) #Y2 #HXY2
- lapply (cpr_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
- ]
-| #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpr_inv_flat1 … H1) -H1 *
- [ #V #T #HV1 #HT1 #H destruct
- elim (cpr_inv_flat1 … H2) -H2 *
- [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
- | #HX2 #H destruct /3 width=5/
- ]
- | #HX1 #H destruct /3 width=5/
-]
-qed-.
-*)
+++ /dev/null
-lemma ldrop_bor1: ∀L1,L2,s1,s2,d,e. ⇩[s1, d, e] L1 ≡ L2 → ⇩[s1 ∨ s2, d, e] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by ldrop_gen/
-qed.
-
-lemma ldrop_bor2: ∀L1,L2,s1,s2,d,e. ⇩[s2, d, e] L1 ≡ L2 → ⇩[s1 ∨ s2, d, e] L1 ≡ L2.
-#L1 #L2 #s1 #s2 >commutative_orb /2 width=1 by ldrop_bor1/
-qed.
-
-(* Basic_1: was: drop_conf_rev *)
-axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
- ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
+++ /dev/null
-include "basic_2/reduction/lpx_ldrop.ma".
-
-lemma lpx_fpbc_trans: ∀h,g,G1,G2,K1,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ≻[h, g] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #K1 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-[ #G2 #L2 #T2 #H12 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
- /3 width=5 by fpbc_fqu, ex3_2_intro/
-| #T2 #HT12 #H #HKL1
+++ /dev/null
-lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-(* ALTERNATIVE PROOF
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
-[ /2 width=1 by or_introl/
-| #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
- elim (fpb_fpbu … H1) -H1 #H1
- [ /3 width=1 by
-*)
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
-#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
-[ -HT1 elim (fqus_inv_gen … HT2) -HT2
- [ #H elim (fqup_inv_step_sn … H) -H
- /4 width=11 by fpbs_intro_alt, fpbu_fqu, ex2_3_intro, or_intror/
- | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
- [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
- | elim (lpxs_nlleq_inv_step_sn … HL0 H) -HL0 -H
- /5 width=7 by lpxs_lleq_fpbs, fpbu_lpx, lleq_trans, ex2_3_intro, or_intror/
- ]
- ]
-| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
- /5 width=11 by fpbs_intro_alt, fpbu_cpx, ex2_3_intro, or_intror/
-]
-qed-.
-
-(* alternative proof that needs decidability of bteq to go in fpbs.ma
- * or lpx_fpbc_trans to go in fpbs_lift.ma (possibly)
-*)
-
-axiom lpx_bteq_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1⦄ ⊢➡ [h, g] L →
- ⦃G1, L, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄.
-
-lemma fpbs_fwd_fpb_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1 [ /2 width=1 by or_introl/ ] (**) (* auto fails without brackets *)
-#G1 #G #L1 #L #T1 #T *
-[ #G0 #L0 #T0 #H #H02 #IH1 elim (fquq_inv_gen … H) -H
- [ -IH1 /4 width=5 by fpbc_fqu, ex2_3_intro, or_intror/
- | -H02 * #HG #HL #HT destruct /2 width=1 by/
- ]
-| #T0 #HT10 #H02 #IH02 elim (eq_term_dec T1 T0) #H destruct
- [ -H02 /2 width=1 by/
- | -IH02 /5 width=5 by fpbc_cpx, ex2_3_intro, or_intror/
- ]
-| #L0 #HL10 #_ * [ /3 width=3 by or_introl, lpx_bteq_trans/ ]
- * #G3 #L3 #T3 #H13 #H32
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/rajust_5.ma".
-include "basic_2/substitution/drop.ma".
-
-(* AJUSTMENT ****************************************************************)
-
-inductive fpa (s:bool): bi_relation lenv term ≝
-| fpa_fwd: ∀L,K,T,U,d,e. ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U → fpa s K T L U
-| fpa_bwd: ∀L,K,T,U,d,e. ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U → fpa s L U K T
-.
-
-interpretation
- "ajustment (restricted closure)"
- 'RAjust L1 T1 s L2 T2 = (fpa s L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpa_refl: ∀s. bi_reflexive … (fpa s).
-/2 width=4 by fpa_fwd/ qed.
-
-lemma fpa_sym: ∀s. bi_symmetric … (fpa s).
-#s #L1 #L2 #T1 #T2 * /2 width=4 by fpa_fwd, fpa_bwd/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fpa.ma".
-
-(* AJUSTMENT ****************************************************************)
-
-(* Main properties **********************************************************)
-
-theorem fpa_conf: ∀s. bi_confluent … (fpa s).
-/3 width=4 by fpa_sym, ex2_2_intro/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/rajuststar_5.ma".
-include "basic_2/substitution/fpa.ma".
-
-(* MULTIPLE AJUSTMENT *******************************************************)
-
-definition fpas: bool → bi_relation lenv term ≝ λs. bi_TC … (fpa s).
-
-interpretation
- "multiple ajustment (restricted closure)"
- 'RAjustStar L1 T1 s L2 T2 = (fpas s L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fpas_ind: ∀L1,T1,s. ∀R:relation2 …. R L1 T1 →
- (∀L,L2,T,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳[s] ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → R L2 T2.
-#L1 #T1 #s #R #IH1 #IH2 #L2 #T2 #H
-@(bi_TC_star_ind … IH1 IH2 L2 T2 H) //
-qed-.
-
-lemma fpas_ind_dx: ∀L2,T2,s. ∀R:relation2 …. R L2 T2 →
- (∀L1,L,T1,T. ⦃L1, T1⦄ ⇳[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → R L1 T1.
-#L2 #T2 #s #R #IH1 #IH2 #L1 #T1 #H
-@(bi_TC_star_ind_dx … IH1 IH2 L1 T1 H) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fpas_refl: ∀s. bi_reflexive … (fpas s).
-/2 width=1 by bi_inj/ qed.
-
-lemma fpas_sym: ∀s. bi_symmetric … (fpas s).
-/3 width=1 by fpa_sym, bi_TC_symmetric/ qed-.
-
-lemma fpa_fpas: ∀L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L2, T2⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
-/2 width=1 by bi_inj/ qed.
-
-lemma fpas_strap1: ∀L1,L,L2,T1,T,T2,s. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳[s] ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
-/2 width=4 by bi_step/ qed-.
-
-lemma fpas_strap2: ∀L1,L,L2,T1,T,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
-/2 width=4 by bi_TC_strap/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fpa_fpa.ma".
-include "basic_2/multiple/fpas.ma".
-
-(* MULTIPLE AJUSTMENT *******************************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fpas_strip: ∀L0,L1,L2,T0,T1,T2,s. ⦃L0, T0⦄ ⇳[s] ⦃L1, T1⦄ → ⦃L0, T0⦄ ⇳*[s] ⦃L2, T2⦄ →
- ∃∃L,T. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ & ⦃L2, T2⦄ ⇳[s] ⦃L, T⦄.
-/3 width=4 by fpa_conf, bi_TC_strip/ qed-.
-
-(* Main properties **********************************************************)
-
-theorem fpas_conf: ∀s. bi_confluent … (fpas s).
-/3 width=4 by fpa_conf, bi_TC_confluent/ qed-.
-
-theorem fpas_trans: ∀s. bi_transitive … (fpas s).
-/2 width=4 by bi_TC_transitive/ qed-.
-
-theorem fpas_canc_sn: ∀L,L1,L2,T,T1,T2,s.
- ⦃L, T⦄ ⇳*[s] ⦃L1, T1⦄→ ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
-/3 width=4 by fpas_trans, fpas_sym/ qed-.
-
-theorem fpas_canc_dx: ∀L1,L2,L,T1,T2,T,s.
- ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L2, T2⦄ ⇳*[s] ⦃L, T⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
-/3 width=4 by fpas_trans, fpas_sym/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/term_vector.ma".
-include "basic_2/multiple/fpas.ma".
-
-(* MULTIPLE VECTOR AJUSTMENT ************************************************)
-
-inductive fpasv (s:bool): bi_relation lenv (list term) ≝
-| fpasv_nil : ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → fpasv s L1 (◊) L2 (◊)
-| fpasv_cons: ∀L1,L2,T1s,T2s,T1,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ →
- fpasv s L1 T1s L2 T2s →
- fpasv s L1 (T1 @ T1s) L2 (T2 @ T2s)
-.
-
-interpretation
- "multiple vector ajustment (restricted closure)"
- 'RAjustStar L1 T1s s L2 T2s = (fpasv s L1 T1s L2 T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-
-
-(* Basic_1: was just: lifts1_flat (left to right) *)
-lemma fpas_inv_applv1: ∀L1,L2,V1s,T1,X,s. ⦃L1, Ⓐ V1s.T1⦄ ⇳*[s] ⦃L2, X⦄ →
- ∃∃V2s,T2. ⦃L1, V1s⦄ ⇳*[s] ⦃L2, V2s⦄ & ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ &
- X = Ⓐ V2s.T2.
-#L1 #L2 #V1s elim V1s -V1s
-[ #T1 #X #s #H
- @(ex3_2_intro … (◊) X) /2 width=3 by fpasv_nil/ (**) (* explicit constructor *)
-| #V1 #V1s #IHV1s #T1 #X #s #H
- elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
- elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
- @(ex3_2_intro) [4: // |3: /2 width=2 by liftsv_cons/ |1,2: skip | // ] (**) (* explicit constructor *)
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: lifts1_flat (right to left) *)
-lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 →
- ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
-#V1s #V2s #des #H elim H -V1s -V2s /3 width=1 by lifts_flat/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/genv.ma".
-include "basic_2/multiple/fpas.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-definition candidate: Type[0] ≝ relation3 genv lenv term.
-
-definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G,L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L2, T2⦄ →
- NF … (RR G L1) RS T1 → NF … (RR G L2) RS T2.
-
-definition CP0s ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G,L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ →
- NF … (RR G L1) RS T1 → NF … (RR G L2) RS T2.
-
-definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G,L. ∃k. NF … (RR G L) RS (⋆k).
-
-definition CP2 ≝ λRP:candidate.
- ∀G,L,T,k. RP G L (ⓐ⋆k.T) → RP G L T.
-
-(* requirements for generic computation properties *)
-record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
-{ cp0: CP0 RR RS;
- cp1: CP1 RR RS;
- cp2: CP2 RP
-}.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: nf2_lift1 *)
-lemma gcp_fpas: ∀RR,RS. CP0 RR RS → CP0s RR RS.
-#RR #RS #HRR #G #L1 #L2 #T1 #T2 #s #H @(fpas_ind … H) -L2 -T2 /3 width=5 by/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/ineint_5.ma".
-include "basic_2/grammar/aarity.ma".
-include "basic_2/substitution/lift_vector.ma".
-include "basic_2/computation/gcp.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-definition S0 ≝ λC:candidate. ∀G,L1,L2,T1,T2. ⦃L1, T1⦄ ⇳[Ⓕ] ⦃L2, T2⦄ →
- C G L1 T1 → C G L2 T2.
-
-definition S0s ≝ λC:candidate. ∀G,L1,L2,T1,T2. ⦃L1, T1⦄ ⇳*[Ⓕ] ⦃L2, T2⦄ →
- C G L1 T1 → C G L2 T2.
-
-(* Note: this is Girard's CR1 *)
-definition S1 ≝ λRP,C:candidate.
- ∀G,L,T. C G L T → RP G L T.
-
-(* Note: this is Tait's iii, or Girard's CR4 *)
-definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs →
- ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
-
-(* Note: this generalizes Tait's ii *)
-definition S3 ≝ λC:candidate.
- ∀a,G,L,Vs,V,T,W.
- C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
-
-definition S4 ≝ λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
-
-definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
- C G L (ⒶVs.V2) → ⇧[0, i+1] V1 ≡ V2 →
- ⇩[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
-
-definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
-
-definition S7 ≝ λC:candidate.
- ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
-
-(* requirements for the generic reducibility candidate *)
-record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
-{ s0: S0 C;
- s1: S1 RP C;
- s2: S2 RR RS RP C;
- s3: S3 C;
- s4: S4 RP C;
- s5: S5 C;
- s6: S6 RP C;
- s7: S7 C
-}.
-
-(* the functional construction for candidates *)
-definition cfun: candidate → candidate → candidate ≝
- λC1,C2,G,K,T. ∀L,V,U. ⦃K, T⦄ ⇳*[Ⓕ] ⦃L, U⦄ →
- C1 G L V → C2 G L (ⓐV.U).
-
-(* the reducibility candidate associated to an atomic arity *)
-let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
-match A with
-[ AAtom ⇒ RP
-| APair B A ⇒ cfun (acr RP B) (acr RP A)
-].
-
-interpretation
- "candidate of reducibility of an atomic arity (abstract)"
- 'InEInt RP G L T A = (acr RP A G L T).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: sc3_lift1 *)
-lemma gcr_fpas: ∀C. S0 C → S0s C.
-#C #HC #G #L1 #L2 #T1 #T2 #H @(fpas_ind … H) -L2 -T2 /3 width=5 by/
-qed.
-(*
-lemma rp_lifts: ∀RR,RS,RP. gcr RR RS RP RP →
- ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
- RP G L V → RP G L0 V0.
-#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
-@gcr_lifts /width=7 by/
-@(s0 … HRP)
-qed.
-
-(* Basic_1: was only: sns3_lifts1 *)
-lemma rp_liftsv_all: ∀RR,RS,RP. gcr RR RS RP RP →
- ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
- all … (RP G L) Vs → all … (RP G L0) V0s.
-#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
-#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
-qed.
-*)
-(* Basic_1: was:
- sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
-*)
-lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀A. gcr RR RS RP (acr RP A).
-#RR #RS #RP #H1RP #H2RP #A elim A -A //
-#B #A #IHB #IHA @mk_gcr
-[ /3 width=4 by fpas_strap2/
-| #G #L #T #H
- elim (cp1 … H1RP G L) #k #HK
- lapply (H L (⋆k) T ? ?) -H //
- [ @(s2 … IHB … (◊)) //
- | #H @(cp2 … H1RP … k) @(s1 … IHA) //
- ]
-| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
- lapply (s1 … IHB … HB) #HV0
- @(s2 … IHA … (V0 @ V0s))
- /3 width=14 by rp_liftsv_all, gcp_lifts, cp0, lifts_simple_dx, conj/
-| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
- elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
- @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #k #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- >(lifts_inv_sort1 … HY) -Y
- lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0 @ V0s)) /3 width=7 by rp_liftsv_all, conj/
-| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
- elim (drops_drop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
- >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
- elim (drops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
- elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
- elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
- >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
- @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
-| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
- elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
- elim (lift_total V10 0 1) #V20 #HV120
- elim (liftv_total 0 1 V10s) #V20s #HV120s
- @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/
- @(HA … (des + 1)) /2 width=2 by drops_skip/
- [ @lifts_applv //
- elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
- >(liftv_mono … HV12s … HV10s) -V1s //
- | @(s0 … IHB … HB … HV120) /2 width=2 by drop_drop/
- ]
-| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
- @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
-]
-qed.
-
-lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀L0,V0,W0,T0,des. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
- ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
- ) →
- ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
-#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HL0 #H #HB
-lapply (acr_gcr … H1RP H2RP A) #HCA
-lapply (acr_gcr … H1RP H2RP B) #HCB
-elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
-lapply (gcr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
-@(s3 … HCA … (◊))
-@(s6 … HCA … (◊) (◊)) //
-[ @(HA … HL0) //
-| lapply (s1 … HCB) -HCB #HCB
- @(s7 … H2RP … (◊)) /2 width=1 by/
-]
-qed.
-
-(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
-(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⇳ [ break term 46 s ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'RAjust $L1 $T1 $s $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⇳ * [ break term 46 s ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'RAjustStar $L1 $T1 $s $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazybtpredproper_8.ma".
-include "basic_2/multiple/fleq.ma".
-include "basic_2/reduction/fpbu.ma".
-
-(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
-
-definition fpbc: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g,G1,L1,T1,G2,L2,T2.
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄.
-
-interpretation
- "'qrst' proper parallel reduction (closure)"
- 'LazyBTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
-
-(* Baic properties **********************************************************)
-
-lemma fpbu_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by ex2_3_intro/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/multiple/fleq_fleq.ma".
-include "basic_2/reduction/fpbu_fleq.ma".
-include "basic_2/reduction/fpbc.ma".
-
-(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
-
-(* Properties on lazy equivalence on closures *******************************)
-
-lemma fpbc_fleq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
-/3 width=9 by fleq_trans, ex2_3_intro/
-qed-.
-
-lemma fleq_fpbc_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 *
-#G0 #L0 #T0 #H0 #H02 elim (fleq_fpbu_trans … H1 … H0) -G -L -T
-/3 width=9 by fleq_trans, ex2_3_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazybtpredstarproper_8.ma".
-include "basic_2/reduction/fpbc.ma".
-
-(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
-
-definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g. tri_TC … (fpbc h g).
-
-interpretation "'qrst' proper parallel computation (closure)"
- 'LazyBTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed.
-
-lemma fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed.
-
-lemma fpbu_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/3 width=1 by fpbu_fpbc, fpbc_fpbg/ qed.
-
-(* Basic eliminators ********************************************************)
-
-lemma fpbg_ind: ∀h,g,G1,L1,T1. ∀R:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
-qed-.
-
-lemma fpbg_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
-#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/fpbc_fleq.ma".
-include "basic_2/computation/fpbg.ma".
-
-(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
-
-(* Properties on lazy equivalence for closures ******************************)
-
-lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind … H) -G -L -T
-[ /3 width=5 by fpbc_fpbg, fpbc_fleq_trans/
-| /4 width=9 by fpbg_strap1, fpbc_fleq_trans/
-]
-qed-.
-
-lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx … H) -G -L -T
-[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/
-| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fpbs_fpbu.ma".
-include "basic_2/computation/fpbs_fpbc.ma".
-include "basic_2/computation/fpbs_fpbs.ma".
-include "basic_2/computation/fpbg_fpbs.ma".
-
-(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpbg_inv_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx … H) -G1 -L1 -T1
-[ #G1 #L1 #T1 * /3 width=5 by fleq_fpbs, ex2_3_intro/
-| #G1 #G #L1 #L #T1 #T *
- #G0 #L0 #T0 #H10 #H0 #_ *
- /5 width=9 by fpbu_fpbs, fpbs_trans, fleq_fpbs, ex2_3_intro/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind … H) -G2 -L2 -T2
-[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans … IH1) -IH1 ] (**) (* full auto fails *)
-/2 width=1 by fpbc_fpbs/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fpbs_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
-/5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-.
-
-(* Man properties ***********************************************************)
-
-theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
-/2 width=5 by tri_TC_transitive/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/lpxs_lleq.ma".
-include "basic_2/computation/fpbs_lift.ma".
-include "basic_2/computation/fpbg_fleq.ma".
-
-(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
-
-(* Properties on "qrst" parallel reduction on closures **********************)
-
-lemma fpbg_fpb_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu … H2) -H2
-/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/
-qed-.
-
-lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
-/3 width=5 by fleq_fpbg_trans, fpbg_strap2, fpbu_fpbc/
-qed-.
-
-(* Properties on "qrst" parallel compuutation on closures *******************)
-
-lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
-qed-.
-
-(* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/
-qed-.
-
-lemma fpbu_fpbs_fpbg: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed.
-
-(* Note: this is used in the closure proof *)
-lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
-/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/
-qed.
-
-lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
- (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/
-qed.
-
-lemma lstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, l2] T2 → (T1 = T2 → ⊥) →
- ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
-/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
-
-lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
-#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/
-qed.
-
-lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
-[ /2 width=1 by or_introl/
-| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu … H2) -H2 #H2
- [ /3 width=5 by fleq_trans, or_introl/
- | /5 width=5 by fpbc_fpbg, fleq_fpbc_trans, fpbu_fpbc, or_intror/
- | /3 width=5 by fpbg_fleq_trans, or_intror/
- | /4 width=5 by fpbg_strap1, fpbu_fpbc, or_intror/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/fpbu_lift.ma".
-include "basic_2/computation/fpbg.ma".
-
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma sta_fpbg: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
- ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
-/4 width=2 by fpbu_fpbg, sta_fpbu/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/fpbc.ma".
-include "basic_2/computation/fpbs_fleq.ma".
-
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
-
-(* Properties on "qrst" proper parallel reduction for closures **************)
-
-lemma fpbc_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
-/3 width=5 by fpbu_fwd_fpb, fpbs_strap2, fleq_fpbs/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻≡ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'LazyBTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/bteq_6.ma".
-include "basic_2/grammar/lenv_length.ma".
-include "basic_2/grammar/genv.ma".
-
-(* EQUIVALENT "BIG TREE" NORMAL FORMS ***************************************)
-
-definition bteq: tri_relation genv lenv term ≝
- λG1,L1,T1,G2,L2,T2.
- ∧∧ G1 = G2 & |L1| = |L2| & T1 = T2.
-
-interpretation
- "equivalent 'big tree' normal forms (closure)"
- 'BTEq G1 L1 T1 G2 L2 T2 = (bteq G1 L1 T1 G2 L2 T2).
-
-(* Basic_properties *********************************************************)
-
-lemma bteq_refl: tri_reflexive … bteq.
-/2 width=1 by and3_intro/ qed.
-
-lemma bteq_sym: tri_symmetric … bteq.
-#G1 #G2 #L1 #L2 #T1 #T2 * //
-qed-.
-
-lemma bteq_dec: ∀G1,G2,L1,L2,T1,T2. Decidable (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄).
-#G1 #G2 #L1 #L2 #T1 #T2 elim (genv_eq_dec G1 G2)
-#H1G [2: @or_intror * #H2G #H2L #H2T destruct /2 width=1 by/ ]
-elim (eq_nat_dec (|L1|) (|L2|))
-#H1L [2: @or_intror * #H2G #H2L #H2T destruct /2 width=1 by/ ]
-elim (term_eq_dec T1 T2)
-#H1T [2: @or_intror * #H2G #H2L #H2T destruct /2 width=1 by/ ]
-/3 width=1 by and3_intro, or_introl/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⋕ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTEq $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/bteq.ma".
-
-(* EQUIVALENT "BIG TREE" NORMAL FORMS ***************************************)
-
-(* Main properties **********************************************************)
-
-theorem bteq_trans: tri_transitive … bteq.
-#G1 #G #L1 #L #T1 #T * //
-qed-.
-
-theorem bteq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2. ⦃G, L, T⦄ ⋕ ⦃G1, L1, T1⦄ →
- ⦃G, L, T⦄ ⋕ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
-/3 width=5 by bteq_trans, bteq_sym/ qed-.
-
-theorem bteq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T. ⦃G1, L1, T1⦄ ⋕ ⦃G, L, T⦄ →
- ⦃G2, L2, T2⦄ ⋕ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
-/3 width=5 by bteq_trans, bteq_sym/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊢ ⋕ ➡ break [ term 46 h , break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRedSn $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-include "basic_2/reduction/fpn.ma".
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpb_bteq_fwd_fpn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
-[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
- [ #H1 #H2 elim (fqu_fwd_bteq … H1 H2)
- | * #HG #HL #HT #_ destruct //
- ]
-| #T2 #HT12 * //
-]
-qed-.
+++ /dev/null
-lemma fpb_fpbc_or_fpn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ ∨
- ⦃G1, L1, T1⦄ ⊢ ➡[h,g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=1 by and3_intro, or_intror/
-[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H [| * ]
- /3 width=1 by fpbc_fqu, and3_intro, or_introl, or_intror/
-| #T2 #HT12 elim (eq_term_dec T1 T2) #H destruct
- /4 width=1 by and3_intro, or_introl, or_intror, fpbc_cpx/
-]
-qed-.
-
-lemma fpb_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) →
- ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 elim (fpb_fpbc_or_fpn … H) -H //
-#H elim H0 -H0 /2 width=3 by fpn_fwd_bteq/
-qed.
-
-lemma fpbc_fwd_bteq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=8 by fqu_fwd_bteq/
-#T2 #_ #HT12 * /2 width=1 by/
-qed-.
+++ /dev/null
-(*
-inclade "basic_2/computation/fpns.ma".
-inclade "basic_2/computation/fpbs.ma".
-inclade "basic_2/reduction/fpbc.ma".
-
-lemma fpn_dec: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- Decidable (⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄).
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fpb_fpbc_or_fpn … H) -H /2 width=1 by or_introl/
-#H12 @or_intror
-#H @(fpbc_fwd_bteq … H12) -H12 @(fpn_fwd_bteq … H)
-qed-.
-*)
-(*
-lemma fpns_dec: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- Decidable (⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄).
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=1 by or_introl/
-#G #G2 #L #L2 #T #T2 #H #H2 *
-#H1 elim (fpn_dec … H2) -H2 #H2 /3 width=5 by fpns_strap1, or_introl/
-[ @or_intror #H12
-| @or_intror #H12 @H1 -H1
-*)
-(*
-inclade "basic_2/grammar/bteq_bteq.ma".
-inclade "basic_2/computation/fpns.ma".
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpbs_bteq_fwd_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 //
-#G #G2 #L #L2 #T #T2 #H1 #H2 #IH1 #H12 elim (bteq_dec G1 G L1 L T1 T)
-[ -H1 /4 width=10 by fpns_strap1, fpb_bteq_fwd_fpn, bteq_canc_sn/
-| -IH1 #H
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/btpredsn_8.ma".
-include "basic_2/relocation/lleq.ma".
-include "basic_2/reduction/lpx.ma".
-
-(* REDUCTION FOR "BIG TREE" NORMAL FORMS ************************************)
-
-inductive fpn (h) (g) (G) (L1) (T): relation3 genv lenv term ≝
-| fpn_intro: ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → L1 ⋕[T] L2 → fpn h g G L1 T G L2 T
-.
-
-interpretation
- "reduction for 'big tree' normal forms (closure)"
- 'BTPRedSn h g G1 L1 T1 G2 L2 T2 = (fpn h g G1 L1 T1 G2 L2 T2).
-
-(* Basic_properties *********************************************************)
-
-lemma fpn_refl: ∀h,g. tri_reflexive … (fpn h g).
-/2 width=1 by fpn_intro/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fpn_inv_gen: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡[h, g] L2 & L1 ⋕[T1] L2 & T1 = T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and4_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/lpx_lpx.ma".
-include "basic_2/reduction/fpn.ma".
-
-(* REDUCTION FOR "BIG TREE" NORMAL FORMS ************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fpn_fqu_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
-#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
-#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqu_trans … H12 … HKL1 HV1) -L1
-/3 width=5 by fpn_intro, ex2_3_intro/
-qed-.
-
-lemma fpn_fquq_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃⸮ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
-#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
-#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fquq_trans … H12 … HKL1 HV1) -L1
-/3 width=5 by fpn_intro, ex2_3_intro/
-qed-.
-
-lemma fpn_fqup_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
- ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃+ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
-#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
-#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqup_trans … H12 … HKL1 HV1) -L1
-/3 width=5 by fpn_intro, ex2_3_intro/
-qed-.
-
-lemma fpn_fqus_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃* ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
-#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
-#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqus_trans … H12 … HKL1 HV1) -L1
-/3 width=5 by fpn_intro, ex2_3_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/btpredsnstar_8.ma".
-include "basic_2/reduction/fpn.ma".
-
-(* COMPUTATION FOR "BIG TREE" NORMAL FORMS **********************************)
-
-definition fpns: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g. tri_TC … (fpn h g).
-
-interpretation
- "computation for 'big tree' normal forms (closure)"
- 'BTPRedSnStar h g G1 L1 T1 G2 L2 T2 = (fpns h g G1 L1 T1 G2 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fpns_ind: ∀h,g,G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-lapply (tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
-qed-.
-
-lemma fpns_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
-#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
-qed-.
-
-(* Basic_properties *********************************************************)
-
-lemma fpns_refl: ∀h,g. tri_reflexive … (fpns h g).
-/2 width=1 by tri_inj/ qed.
-
-lemma fpn_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fpns_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed-.
-
-lemma fpns_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lleq_lleq.ma".
-include "basic_2/computation/lpxs.ma".
-include "basic_2/computation/fpns.ma".
-
-(* COMPUTATION FOR "BIG TREE" NORMAL FORMS **********************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpns_inv_gen: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 & L1 ⋕[T1] L2 & T1 = T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpns_ind … H) -G2 -L2 -T2 /2 width=1 by and4_intro/
-#G #G2 #L #L2 #T #T2 #_ #H2 * #HG #HL1 #HT1 #HT destruct
-elim (fpn_inv_gen … H2) -H2 #HG #HL2 #HT #HT2 destruct
-/3 width=3 by lpxs_strap1, lleq_trans, and4_intro/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lpxs_lleq_fpns: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → L1 ⋕[T] L2 →
- ⦃G, L1, T⦄ ⊢ ⋕➡*[h, g] ⦃G, L2, T⦄.
-#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2 //
-#L #L2 #HL1 #HL2 #IHL1 #HL12 elim (lleq_dec T L1 L) #HT
-[ -HL1 @fpns_strap1 [4: @IHL1 // |1,2,3: skip ]
- /3 width=3 by fpn_intro, lleq_canc_sn/
-| -IHL1
-
-(* Main properties **********************************************************)
-
-theorem fpns_trans: tri_transitive … fqus.
-/2 width=5 by tri_TC_transitive/ qed-.
-
-
-(*
-lemma fpns_refl: ∀h,g. tri_reflexive … (fpns h g).
-/2 width=1 by and3_intro/ qed.
-
-lemma fpn_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=1 by lpx_lpxs, and3_intro/
-qed.
-
-lemma fpns_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * #H1G #H1L #G1T *
-/3 width=3 by lpxs_strap1, and3_intro/
-qed-.
-
-lemma fpns_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * #H1G #H1L #G1T *
-/3 width=3 by lpxs_strap2, and3_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpns_fwd_bteq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=4 by lpxs_fwd_length, and3_intro/
-qed-.
-*)
+++ /dev/null
-include "basic_2/grammar/bteq.ma".
-
-lemma fqu_fwd_bteq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #V * #_ #H elim (plus_xSy_x_false … H)
-| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_x … H)
-| #a #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y … H)
-| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y … H)
-| #G #L #K #T #U #e #HLK #_ * #_ #H
- lapply (ldrop_fwd_length_lt4 … HLK ?) // >H -L #H
- elim (lt_refl_false … H)
-]
-qed-.
+++ /dev/null
-(* Basic eliminators ********************************************************)
-
-theorem fsb_ind_alt: ∀h,g. ∀R: relation3 …. (
- ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h,g] T1 → (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → R G2 L2 T2
- ) → R G1 L1 T1
- ) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → R G L T.
-#h #g #R #IH #G #L #T #H elim H -G -L -T
-/5 width=1 by fpb_fpbc, fsb_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/btsnalt_5.ma".
-include "basic_2/computation/fpbs_fpbs.ma".
-include "basic_2/computation/fsb.ma".
-
-(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
-
-(* Note: alternative definition of fsb *)
-inductive fsba (h) (g): relation3 genv lenv term ≝
-| fsba_intro: ∀G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
- ) → fsba h g G1 L1 T1.
-
-interpretation
- "'big tree' strong normalization (closure) alternative"
- 'BTSNAlt h g G L T = (fsba h g G L T).
-
-(* Basic eliminators ********************************************************)
-
-theorem fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
- ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h,g] T1 → (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → R G2 L2 T2
- ) → R G1 L1 T1
- ) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → R G L T.
-#h #g #R #IH #G #L #T #H elim H -G -L -T
-/5 width=1 by fsba_intro/
-qed-.
-
-(* Basic_properties *********************************************************)
-
-fact fsba_intro_aux: ∀h,g,G1,L1,T1. (
- ∀G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕ ⦃G, L, T⦄ →
- (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
- ) → fsba h g G1 L1 T1.
-/4 width=5 by fsba_intro/ qed-.
-
-lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1 →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2.
-#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
-#G1 #L1 #T1 #H0 #IH0 #G #L #T #H1 @fsba_intro
-#G2 #L2 #T2 #H2 #_ lapply (fpbs_trans … H1 … H2) -G -L -T
-#H12 elim (bteq_dec G1 G2 L1 L2 T1 T2) /3 width=6 by fpb_fpbs/
--IH0 #H212
-
-
- -H0 -H #H @(IH0 … H) -IH0 -H // @(fpbs_trans … H1 … H2)
-
-lemma fsba_intro_fpb: ∀h,g,G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2
- ) → ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1.
-#h #g #G1 #L1 #T1 #IH1 @fsba_intro_aux
-#G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T
-[ #H1 #H2 -IH1 elim H2 -H2 //
-| #G0 #G #L0 #L #T0 #T #H10 #H12 #IH2 #H210 #H212 elim (bteq_dec G1 G L1 L T1 T)
- [ -IH1 -H210 -H10 -H12 /3 width=1 by/
- | -IH2 -H212 #H21 lapply (IH1 … H21) -IH1 -H21
- [
- | -H10 -H210 #H
-(*
-(* Main inversion lemmas ****************************************************)
-
-theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
-#h #g #G #L #T #H elim H -G -L -T
-/5 width=12 by fsb_intro, fpb_fpbs, fpbc_fwd_fpb, fpbc_fwd_gen/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem fsb_fsba: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥⦥[h, g] T.
-#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
-/4 width=1 by fsba_intro_fpb/
-qed.
-(*
-| fsba_intro: ∀G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → fsb h g G2 L2 T2
- ) → fsb h g G1 L1 T1
-.
-
-
-
-(****************************************************************************)
-
-include "basic_2/substitution/fqup.ma".
-
-lemma fsb_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
-#h #g #G #L #T #H @(csx_ind … H) -T
-*)*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⋕ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'LazyEq $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }.
-
-include "basic_2/reducibility/cpr.ma".
-include "basic_2/reducibility/fpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-definition cfpr: lenv → bi_relation lenv term ≝
- λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2.
-
-interpretation
- "context-sensitive parallel reduction (closure)"
- 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L).
-/2 width=1/ qed.
-
-lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 * /3 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆.
-#L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V2. V1 ➡ V2 &
- ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
- L2 = ⋆.ⓑ{I}V2@@K2.
-#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H
-elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
->shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H
-elim (tpr_inv_bind1 … H) -H *
-[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/
-| #T0 #_ #_ #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_aaa.ma".
-include "basic_2/reducibility/cfpr_cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_fpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #HT1 #L2 #T2 #H
-elim (fpr_inv_all … H) -H
-/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_cpr.ma".
-include "basic_2/reducibility/cfpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Main properties **********************************************************)
-
-theorem cfpr_conf: ∀L. bi_confluent … (cfpr L).
-#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
-elim (cpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
-elim (cpr_fwd_shift1 … H1) #L0 #T0 #HL10 #H destruct /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_tpss.ma".
-include "basic_2/reducibility/cpr_cpr.ma".
-include "basic_2/reducibility/cfpr_ltpss.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 →
- L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
-#L1 #L #H elim H -L1 -L
-[ #L2 #T1 #T2 #HT12 #HL2
- lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct
- lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/
-| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H
- elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
- lapply (cpr_bind_dx false … HV1 HT12) -HV1 -HT12 #HT12
- lapply (cpr_tpss_trans … HT12 (-ⓑ{I}V2.T2) 0 (|L|) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
-]
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cfpr_inv_all: ∀L1,L2,L0,T1,T2. L0 ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃L. L0 @@ L1 ➡ L0 @@ L & L0 @@ L ⊢ T1 ➡ T2 &
- L0 @@ L ⊢ ▶* [0, |L0| + |L|] L0 @@ L2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L2 #L0 #T1 #T2 #H
- elim (cfpr_inv_atom1 … H) -H #HT12 #H destruct /3 width=4/
-| #I #L1 #V1 #IH #X #L0 #T1 #T2 #H
- elim (cfpr_inv_pair1 … H) -H #L2 #V #V2 #HV1 #HV2 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L #HL1 #HT12 #HL2
- elim (ltpr_inv_append1 … HL1) -HL1 #X #Y #HX #HY #H
- lapply (ltpr_fwd_length … HX) -HX #HX
- elim (append_inj_dx … H ?) -H // -HX #_ #H destruct -X
- lapply (ltpss_sn_fwd_length … HL2) >append_length >append_length #H
- lapply (injective_plus_r … H) -H #H
- @(ex3_intro … (⋆.ⓑ{I}V@@Y)) <append_assoc // -HT12
- <append_assoc [ /3 width=1/ ] -HV1 -HY
- >append_length <associative_plus
- @(ltpss_sn_dx_trans_eq … HL2) -HL2 >H -H >commutative_plus /3 width=1/
-]
-qed-.
-
-lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2.
-#L1 #L2 #T1 #T2 #H
-lapply (fpr_cfpr … H) -H #H
-elim (cfpr_inv_all … H) -H /2 width=4/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cpr_lift.ma".
-include "basic_2/reducibility/cpr_ltpss_sn.ma".
-include "basic_2/reducibility/cfpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cfpr_inv_pair1: ∀I,L,K1,L2,V1,T1,T2. L ⊢ ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V,V2. V1 ➡ V & L ⊢ V ▶* [0, |L|] V2 &
- L.ⓑ{I}V ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
- L2 = ⋆.ⓑ{I}V2@@K2.
-* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H
-elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
->shift_append_assoc >shift_append_assoc normalize in ⊢ (??%%→?); #H
-[ elim (cpr_inv_abbr1 … H) -H *
- [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/
- | #T0 #_ #_ #H destruct
- ]
-| elim (cpr_inv_abst1 … H Abst V2) -H
- #V #T * #V0 #HV10 #HV0 #HT1 #H destruct
- lapply (ltpss_sn_cpr_trans (L.ⓛV0) … 0 (|L|+1) … HT1) -HT1 /2 width=1/ #HT12
- /3 width=7/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }.
-
-notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }.
-
-include "basic_2/reducibility/fpr.ma".
-
-(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
-
-definition fpc: bi_relation lenv term ≝
- λL1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ ∨ ⦃L2, T2⦄ ➡ ⦃L1, T1⦄.
-
-interpretation
- "context-free parallel conversion (closure)"
- 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpc_refl: bi_reflexive … fpc.
-/2 width=1/ qed.
-
-lemma fpc_sym: bi_symmetric … fpc.
-#L1 #L2 #T1 #T2 * /2 width=1/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpc_fwd_fpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ →
- ∃∃L,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄.
-#L1 #L2 #T1 #T2 * /2 width=4/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/conversion/fpc.ma".
-
-(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
-
-(* Main properties **********************************************************)
-
-theorem fpc_conf: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
- ∀L2,T2. ⦃L0, T0⦄ ⬌ ⦃L2, T2⦄ →
- ∃∃L,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
-/3 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }.
-
-notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }.
-
-include "basic_2/conversion/fpc.ma".
-
-(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
-
-definition fpcs: bi_relation lenv term ≝ bi_TC … fpc.
-
-interpretation "context-free parallel equivalence (closure)"
- 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fpcs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L2 T2.
-/3 width=7 by bi_TC_star_ind/ qed-.
-
-lemma fpcs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L1 T1.
-/3 width=7 by bi_TC_star_ind_dx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fpcs_refl: bi_reflexive … fpcs.
-/2 width=1/ qed.
-
-lemma fpcs_sym: bi_symmetric … fpcs.
-/3 width=1/ qed.
-
-lemma fpc_fpcs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma fpcs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma fpcs_fpr_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=1/ qed.
-
-lemma fpcs_fpr_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡ ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=1/ qed.
-
-lemma fpcs_fpr_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
- ∀L2,T2. ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4/ qed.
-
-lemma fpcs_fpr_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ →
- ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4/ qed.
-
-lemma fpcs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
- ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4/ qed.
-
-lemma fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4/ qed-.
-
-lemma fpcs_fpr_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡ ⦃L1, T1⦄ →
- ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fprs_aaa.ma".
-include "basic_2/equivalence/fpcs_fpcs.ma".
-
-(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
-
-(* Main properties about atomic arity assignment on terms *******************)
-
-theorem aaa_fpcs_mono: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
- ∀A1. L1 ⊢ T1 ⁝ A1 → ∀A2. L2 ⊢ T2 ⁝ A2 →
- A1 = A2.
-#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2
-elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
-lapply (aaa_fprs_conf … HT1 … H1) -L1 -T1 #HT1
-lapply (aaa_fprs_conf … HT2 … H2) -L2 -T2 #HT2
-lapply (aaa_mono … HT1 … HT2) -L -T //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fprs_cprs.ma".
-include "basic_2/equivalence/cpcs_cpcs.ma".
-include "basic_2/equivalence/fpcs_fpcs.ma".
-
-(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fpcs_flat_dx_tpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⬌* ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #T1 #T2 #HT12
-elim (fpcs_inv_fprs … HT12) -HT12
-/3 width=6 by fprs_flat_dx_tpr, fprs_div/ (**) (* auto too slow without trace *)
-qed.
-
-lemma fpcs_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
- ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄.
-#I #L1 #L2 #V1 #V2 #T1 #T2 #H12
-elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
-elim (fprs_bind2_minus … H1) -H1 #W1 #U1 #HTU1 #H destruct
-elim (fprs_bind2_minus … H2) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpcs_inv_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
- ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
-#I #L1 #L2 #V1 #V2 #T1 #T2 #H12
-elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
-elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
-elim (fprs_inv_pair1 … H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpcs_fwd_bind_minus: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
- ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
-#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b
-elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
-elim (fprs_fwd_bind2_minus … H1 b) -H1 #W1 #U1 #HTU1 #H destruct
-elim (fprs_fwd_bind2_minus … H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
-qed-.
-
-lemma fpcs_fwd_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
- ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
-/3 width=1 by fpcs_inv_shift, fpcs_fwd_bind_minus/ qed-.
-
-lemma fpcs_fwd_abst24: ∀a,L1,L2,V1,V2,T1,T2. ⦃L1, ⓛ{a}V1.T1⦄ ⬌* ⦃L2, ⓛ{a}V2.T2⦄ →
- ∀b,I,W. ⦃L1, ⓑ{b,I}W.T1⦄ ⬌* ⦃L2, ⓑ{b,I}W.T2⦄.
-#a #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b #I #W
-elim (fpcs_inv_fprs … H12) -H12 #L #U #H1 #H2
-elim (fprs_fwd_abst2 … H1 b I W) -H1 #W1 #U1 #HTU1 #H destruct
-elim (fprs_fwd_abst2 … H2 b I W) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
-qed-.
-
-lemma fpcs_fwd_abst13: ∀L1,L2,V1,V2,T1,T2. ⦃L1.ⓛV1, T1⦄ ⬌* ⦃L2.ⓛV2, T2⦄ →
- ∀I,W. ⦃L1.ⓑ{I}W, T1⦄ ⬌* ⦃L2.ⓑ{I}W, T2⦄.
-/4 width=4 by fpcs_fwd_shift, fpcs_fwd_abst24, fpcs_shift/ qed-.
-
-(* Properties on context-sensitive parallel equivalence for terms ***********)
-
-lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
-#L #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
-qed.
-
-(* Inversion lemmas on context-sensitive parallel equivalence for terms *****)
-
-lemma fpcs_inv_cpcs: ∀L,T1,T2. ⦃L, T1⦄ ⬌* ⦃L, T2⦄ → L ⊢ T1 ⬌* T2.
-#L #T1 #T2 #H
-elim (fpcs_inv_fprs … H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fprs_fprs.ma".
-include "basic_2/conversion/fpc_fpc.ma".
-include "basic_2/equivalence/fpcs_fprs.ma".
-
-(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpcs_inv_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
- ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡* ⦃L, T⦄.
-#L1 #L2 #T1 #T2 #H @(fpcs_ind … H) -L2 -T2
-[ /3 width=4/
-| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
- [ elim (fprs_strip … HT2 … HT0) -L -T #L #T #HT2 #HT0
- lapply (fprs_strap1 … HT10 … HT0) -L0 -T0 /2 width=4/
- | lapply (fprs_strap2 … HT2 … HT0) -L -T /2 width=4/
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fpr_fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
-elim (fprs_strip … HT2 … HT1) /2 width=4 by fpr_fprs_div/
-qed-.
-
-lemma fprs_fpr_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L2, T2⦄ ⬌* ⦃L1, T1⦄.
-#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
-elim (fprs_strip … HT2 … HT1) /2 width=4 by fprs_fpr_div/
-qed-.
-
-lemma fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
-elim (fprs_conf … HT1 … HT2) /2 width=4/
-qed-.
-
-lemma fpcs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
- ∀L2,T2. ⦃L0, T0⦄ ⬌* ⦃L2, T2⦄ →
- ∃∃L,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
-/3 width=4/ qed.
-
-(* Main properties **********************************************************)
-
-theorem fpcs_trans: bi_transitive … fpcs.
-/2 width=4/ qed.
-
-theorem fpcs_canc_sn: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ⬌* ⦃L1, T1⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
-
-theorem fpcs_canc_dx: ∀L1,L2,L,T1,T2,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L2, T2⦄ ⬌* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fprs.ma".
-include "basic_2/equivalence/fpcs.ma".
-
-(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
-
-(* Properties on context-free parallel computation for closures *************)
-
-(* Note: was lemma 1000 *)
-lemma fpcs_fprs_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /3 width=4/
-qed.
-
-lemma fpcs_fprs_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡* ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /3 width=4/
-qed.
-
-lemma fpcs_fprs_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /2 width=4/
-qed.
-
-lemma fpcs_fprs_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx … H) -L1 -T1 /width=1/ /2 width=4/
-qed.
-
-lemma fpcs_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /2 width=4/
-qed.
-
-lemma fpcs_fprs_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind … H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *)
-qed.
-
-lemma fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx … H) -L2 -T2 /2 width=1/ /2 width=4/
-qed.
-
-lemma fprs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=7 by bi_step, fprs_div/ qed-.
-
-lemma fpr_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
-/3 width=4 by bi_step, fprs_div/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }.
-
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
-
-definition fpr: bi_relation lenv term ≝
- λL1,T1,L2,T2. |L1| = |L2| ∧ L1 @@ T1 ➡ L2 @@ T2.
-
-interpretation
- "context-free parallel reduction (closure)"
- 'FocalizedPRed L1 T1 L2 T2 = (fpr L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpr_refl: bi_reflexive … fpr.
-/2 width=1/ qed.
-
-lemma fpr_shift: ∀I1,I2,L1,L2,V1,V2,T1,T2.
- ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
- ⦃L1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2.ⓑ{I2}V2, T2⦄.
-#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12
-@conj // normalize // (**) (* explicit constructor *)
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fpr_inv_atom1: ∀L2,T1,T2. ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → T1 ➡ T2 ∧ L2 = ⋆.
-#L2 #T1 #T2 * #H
-lapply (length_inv_zero_sn … H) -H #H destruct /2 width=1/
-qed-.
-
-lemma fpr_inv_atom3: ∀L1,T1,T2. ⦃L1,T1⦄ ➡ ⦃⋆,T2⦄ → T1 ➡ T2 ∧ L1 = ⋆.
-#L1 #T1 #T2 * #H
-lapply (length_inv_zero_dx … H) -H #H destruct /2 width=1/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpr_fwd_pair1: ∀I1,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃I2,K2,V2. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
- L2 = K2.ⓑ{I2}V2.
-#I1 #K1 #L2 #V1 #T1 #T2 * #H
-elim (length_inv_pos_sn … H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/
-qed-.
-
-lemma fpr_fwd_pair3: ∀I2,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I2}V2, T2⦄ →
- ∃∃I1,K1,V1. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
- L1 = K1.ⓑ{I1}V1.
-#I2 #L1 #K2 #V2 #T1 #T2 * #H
-elim (length_inv_pos_dx … H) -H #I1 #K1 #V1 #HK12 #H destruct /3 width=5/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cfpr_cpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma ltpr_tpr_fpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
-/3 width=4/ qed.
-
-lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
-/2 width=4/ qed.
-
-lemma fpr_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ →
- ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
- ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- ∃∃L2. ⦃L1, U1⦄ ➡ ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
-#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
-elim (fpr_inv_all … HT12) -HT12 #K #HK1 #HT12 #HK2
-elim (ldrop_ltpr_trans … HLK1 … HK1) -K1 #L #HL1 #HLK
-lapply (cpr_lift … HLK … HTU1 … HTU2 HT12) -T1 -T2 #HU12
-elim (le_or_ge (|K|) d) #Hd
-[ elim (ldrop_ltpss_sn_trans_ge … HLK … HK2 …)
-| elim (ldrop_ltpss_sn_trans_be … HLK … HK2 …)
-] // -Hd #L2 #HL2 #HLK2
-lapply (ltpss_sn_weak_full … HL2) -K /3 width=4/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fpr_flat_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡ ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #T1 #T2 #HT12
-elim (fpr_inv_all … HT12) -HT12 /4 width=4/
-qed.
-
-lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
- ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
-#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
-elim (fpr_inv_all … H) /3 width=4/
-qed.
-
-lemma fpr_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L2, U2⦄ →
- ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡ ⦃L2.ⓑ{I}V2, T2⦄ &
- U2 = -ⓑ{I}V2.T2.
-#I1 #L1 #L2 #V1 #T1 #U2 #H
-elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
-elim (cpr_fwd_bind1_minus … H false) -H /4 width=4/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
- ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
- T = -ⓑ{I}V2.T2.
-#I #L1 #L #V1 #T1 #T #H1 #b
-elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
-elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
-qed-.
-
-lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
- ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
- ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
-* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
-elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
-[ elim (cpr_inv_abbr1 … H) -H *
- [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
- | #T #_ #_ #H destruct
- ]
-| elim (cpr_inv_abst1 … H Abst V2) -H
- #V #T #HV1 #_ #H destruct /3 width=4/
-]
-qed-.
-
-lemma fpr_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡ ⦃L2, U2⦄ → ∀b,I,W.
- ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡ ⦃L2, ⓑ{b,I}W.T2⦄ &
- U2 = ⓛ{a}V2.T2.
-#a #L1 #L2 #V1 #T1 #U2 #H
-elim (fpr_inv_all … H) #L #HL1 #H #HL2 #b #I #W
-elim (cpr_fwd_abst1 … H b I W) -H /3 width=4/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I1 #K1 #X #V1 #T1 #T2 #H
-elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
-elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
-qed-.
-
-lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
- ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
- L1 = K1.ⓑ{I}V1.
-#I2 #X #K2 #V2 #T1 #T2 #H
-elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
-elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
-qed-.
-
-(* More advanced forward lemmas *********************************************)
-
-lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I #K1 #L2 #V1 #T1 #T2 #H #b
-elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
-elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/tpr_tpr.ma".
-include "basic_2/reducibility/fpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
-
-(* Main properties **********************************************************)
-
-theorem fpr_conf: bi_confluent … fpr.
-#L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
-elim (tpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
-elim (tpr_fwd_shift1 … H1) #L #T #HL1 #H destruct /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }.
-
-notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRedStarAlt $L1 $T1 $L2 $T2 }.
-
-include "basic_2/reducibility/fpr.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
-
-definition fprs: bi_relation lenv term ≝ bi_TC … fpr.
-
-interpretation "context-free parallel computation (closure)"
- 'FocalizedPRedStar L1 T1 L2 T2 = (fprs L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fprs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L2 T2.
-/3 width=7 by bi_TC_star_ind/ qed-.
-
-lemma fprs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L1 T1.
-/3 width=7 by bi_TC_star_ind_dx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fpr_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma fprs_refl: bi_reflexive … fprs.
-/2 width=1/ qed.
-
-lemma fprs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma fprs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
-/2 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/cfpr_aaa.ma".
-include "basic_2/computation/fprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_fprs_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #HT1 #L2 #T2 #HLT12
-@(bi_TC_Conf3 … HT1 ?? HLT12) /2 width=4/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/fpr_cpr.ma".
-include "basic_2/computation/cprs_lfprs.ma".
-include "basic_2/computation/lfprs_ltprs.ma".
-include "basic_2/computation/lfprs_fprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fprs_flat_dx_tpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡* ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #T1 #T2 #HT12 @(fprs_ind … HT12) -L2 -T2 /3 width=1/
-#L #L2 #T #T2 #_ #HT2 #IHT2 #V1 #V2 #HV12 #I
-lapply (IHT2 … HV12 I) -IHT2 -HV12 /3 width=6/
-qed.
-
-lemma fprs_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L2, U2⦄ →
- ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡* ⦃L2.ⓑ{I}V2, T2⦄ &
- U2 = -ⓑ{I}V2.T2.
-#I #L1 #L2 #V1 #T1 #U2 #H @(fprs_ind … H) -L2 -U2 /2 width=4/
-#L #L2 #U #U2 #_ #HU2 * #V #T #HT1 #H destruct
-elim (fpr_bind2_minus … HU2) -HU2 /3 width=4/
-qed-.
-
-lemma fprs_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡* ⦃K2, T2⦄ →
- ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
- ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- ∃∃L2. ⦃L1, U1⦄ ➡* ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
-#K1 #K2 #T1 #T2 #HT12 @(fprs_ind … HT12) -K2 -T2
-[ #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
- >(lift_mono … HTU2 … HTU1) -U2 /2 width=3/
-| #K #K2 #T #T2 #_ #HT2 #IHT1 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
- elim (lift_total T d e) #U #HTU
- elim (IHT1 … HLK1 … HTU1 HTU) -K1 -T1 #L #HU1 #HKL
- elim (fpr_lift … HT2 … HKL … HTU HTU2) -K -T -T2 /3 width=4/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fprs_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
- ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /2 width=5/
-#L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct
-elim (fpr_inv_pair1 … HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/
-qed-.
-
-lemma fprs_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡* ⦃K2.ⓑ{I}V2, T2⦄ →
- ∃∃K1,V1. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
- L1 = K1.ⓑ{I}V1.
-#I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx … H) -L1 -T1 /2 width=5/
-#L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct
-elim (fpr_inv_pair3 … HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fprs_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L, T⦄ → ∀b.
- ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡* ⦃L, ⓑ{b,I}V2.T2⦄ &
- T = -ⓑ{I}V2.T2.
-#I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind … H1) -L -T /2 width=4/
-#L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct
-elim (fpr_fwd_bind2_minus … H0 b) -H0 /3 width=4/
-qed-.
-
-lemma fprs_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
- ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
- ⦃K1, ⓑ{b,I}V1.T1⦄ ➡* ⦃K2, ⓑ{b,I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I #K1 #L2 #V1 #T1 #T2 #H #b
-elim (fprs_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
-elim (fprs_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
-qed-.
-
-lemma fprs_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡* ⦃L2, U2⦄ → ∀b,I,W.
- ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡* ⦃L2, ⓑ{b,I}W.T2⦄ &
- U2 = ⓛ{a}V2.T2.
-#a #L1 #L2 #V1 #T1 #U2 #H #b #I #W @(fprs_ind … H) -L2 -U2 /2 width=4/
-#L #L2 #U #U2 #_ #H2 * #V #T #HT1 #H destruct
-elim (fpr_fwd_abst2 … H2 b I W) -H2 /3 width=4/
-qed-.
-
-(* Properties on context-sensitive parallel computation for terms ***********)
-
-lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄.
-#L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/
-qed.
-
-(* Forward lemmas on context-sensitive parallel computation for terms *******)
-
-lemma fprs_fwd_cprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L1 ⊢ T1 ➡* T2.
-#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 //
-#L #L2 #T #T2 #H1 #H2 #IH1
-elim (fpr_inv_all … H2) -H2 #L0 #HL0 #HT2 #_ -L2
-lapply (lfprs_cpr_trans L1 … HT2) -HT2 /3 width=3/
-qed-.
-(*
-(* Advanced properties ******************************************************)
-
-lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
- ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
-#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
-elim (fpr_inv_all … H) /3 width=4/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
- ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
- ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
-* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
-elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
-[ elim (cpr_inv_abbr1 … H) -H *
- [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
- | #T #_ #_ #H destruct
- ]
-| elim (cpr_inv_abst1 … H Abst V2) -H
- #V #T #HV1 #_ #H destruct /3 width=4/
-]
-qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/fpr_fpr.ma".
-include "basic_2/computation/fprs.ma".
-
-(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fprs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ➡ ⦃L1, T1⦄ →
- ∀L2,T2. ⦃L0, T0⦄ ➡* ⦃L2, T2⦄ →
- ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄.
-#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8
-/2 width=4/ qed.
-
-(* Main propertis ***********************************************************)
-
-theorem fprs_conf: bi_confluent … fprs.
-/2 width=4/ qed.
-
-theorem fprs_trans: bi_transitive … fprs.
-/2 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_length.ma".
-
-(* POINTWISE EXTENSION OF A FOCALIZED REALTION FOR TERMS ********************)
-
-inductive lpx_bi (R:bi_relation lenv term): relation lenv ≝
-| lpx_bi_stom: lpx_bi R (⋆) (⋆)
-| lpx_bi_pair: ∀I,K1,K2,V1,V2.
- lpx_bi R K1 K2 → R K1 V1 K2 V2 →
- lpx_bi R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lpx_bi_inv_atom1_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L1 = ⋆ → L2 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_bi_inv_atom1: ∀R,L2. lpx_bi R (⋆) L2 → L2 = ⋆.
-/2 width=4 by lpx_bi_inv_atom1_aux/ qed-.
-
-fact lpx_bi_inv_pair1_aux: ∀R,L1,L2. lpx_bi R L1 L2 →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. lpx_bi R K1 K2 &
- R K1 V1 K2 V2 & L2 = K2. ⓑ{I} V2.
-#R #L1 #L2 * -L1 -L2
-[ #J #K1 #V1 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
-]
-qed-.
-
-lemma lpx_bi_inv_pair1: ∀R,I,K1,V1,L2. lpx_bi R (K1. ⓑ{I} V1) L2 →
- ∃∃K2,V2. lpx_bi R K1 K2 & R K1 V1 K2 V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_bi_inv_pair1_aux/ qed-.
-
-fact lpx_bi_inv_atom2_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L2 = ⋆ → L1 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_bi_inv_atom2: ∀R,L1. lpx_bi R L1 (⋆) → L1 = ⋆.
-/2 width=4 by lpx_bi_inv_atom2_aux/ qed-.
-
-fact lpx_bi_inv_pair2_aux: ∀R,L1,L2. lpx_bi R L1 L2 →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 &
- L1 = K1. ⓑ{I} V1.
-#R #L1 #L2 * -L1 -L2
-[ #J #K2 #V2 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
-]
-qed-.
-
-lemma lpx_bi_inv_pair2: ∀R,I,L1,K2,V2. lpx_bi R L1 (K2. ⓑ{I} V2) →
- ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_bi_inv_pair2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpx_bi_fwd_length: ∀R,L1,L2. lpx_bi R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L1 -L2 normalize //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_bi_refl: ∀R. bi_reflexive ? ? R → reflexive … (lpx_bi R).
-#R #HR #L elim L -L // /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L1 ⊢ ➡ ➡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PRedSnAlt $L1 $L2 }.
-
-notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRedAlt $L1 $L2 }.
-
-include "basic_2/grammar/lenv_px_bi.ma".
-include "basic_2/reducibility/fpr_cpr.ma".
-include "basic_2/reducibility/lfpr_fpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-
-(* alternative definition *)
-definition lfpra: relation lenv ≝ lpx_bi fpr.
-
-interpretation
- "focalized parallel reduction (environment) alternative"
- 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lfpra_refl: reflexive … lfpra.
-/2 width=1/ qed.
-
-lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
-#L1 elim L1 -L1
-[ #L2 #T1 #T2 #H
- elim (fpr_inv_atom1 … H) -H #_ #H destruct //
-| #L1 #I #V1 #IH #L2 #T1 #T2 #H
- elim (fpr_inv_pair1 … H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆.
-/2 width=2 by lpx_bi_inv_atom1/ qed-.
-
-lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ →
- ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- L2 = K2. ⓑ{I} V2.
-/2 width=1 by lpx_bi_inv_pair1/ qed-.
-
-lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆.
-/2 width=2 by lpx_bi_inv_atom2/ qed-.
-
-lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ →
- ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- L1 = K1. ⓑ{I} V1.
-/2 width=1 by lpx_bi_inv_pair2/ qed-.
-
-lemma lfpra_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ∀T.⦃L1, T⦄ ➡ ⦃L2, T⦄.
-#L1 #L2 * -L1 -L2 // /3 width=1/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|.
-/2 width=2 by lpx_bi_fwd_length/ qed-.
-
-(* Main properties **********************************************************)
-
-theorem lfpr_lfpra: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
-#L1 #L2 #H
-lapply (lfpr_inv_fpr … H (⋆0)) -H /2 width=3/
-qed.
-
-theorem lfpra_lfpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
-#L1 #L2 #H
-lapply (lfpra_inv_fpr … H (⋆0)) -H /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/lfpr.ma".
-include "basic_2/reducibility/cfpr_cpr.ma".
-
-(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Properties on context-free parallel reduction for closures ***************)
-
-lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
-#L1 #L2 #T1 #T2 #H
-elim (fpr_inv_all … H) -H /2 width=3/
-qed.
-
-(* Inversion lemmas on context-free parallel reduction for closures *********)
-
-lemma lfpr_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡ ⦃L2, T⦄.
-#L1 #L2 * /2 width=4/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/lfpr_fpr.ma".
-include "basic_2/computation/fprs_fprs.ma".
-include "basic_2/computation/lfprs.ma".
-
-(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
-
-(* Inversion lemmas on context-free parallel reduction for closures *********)
-
-lemma lfprs_inv_fprs: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡* ⦃L2, T⦄.
-#L1 #L2 #H @(lfprs_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL1 #T
-lapply (lfpr_inv_fpr … HL2 T) -HL2 /3 width=4/
-qed-.
-
-(* Properties on context-free parallel computation for closures *************)
-
-lemma fprs_lfprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1⦄ ➡* ⦃L2⦄.
-#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 // /3 width=5/
-qed.
-
-lemma lfprs_fprs_trans: ∀L1,L,L2,T1,T2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
-#L1 #L #L2 #T1 #T2 #HL1 #HL2
-lapply (lfprs_inv_fprs … HL1 T1) -HL1 /2 width=4/
-qed.
-(*
-lamma lfprs_cprs_conf: ∀L1,L,L2,T1,T2. ⦃L1⦄ ➡* ⦃L2⦄ → L1 ⊢ T1 ➡* T2 → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'RestSupTerm $L1 $T1 $L2 $T2 }.
-
-include "basic_2/grammar/cl_weight.ma".
-include "basic_2/substitution/lift.ma".
-
-(* RESTRICTED SUPCLOSURE ****************************************************)
-
-inductive frsup: bi_relation lenv term ≝
-| frsup_bind_sn: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) L V
-| frsup_bind_dx: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T
-| frsup_flat_sn: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L V
-| frsup_flat_dx: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L T
-.
-
-interpretation
- "restricted structural predecessor (closure)"
- 'RestSupTerm L1 T1 L2 T2 = (frsup L1 T1 L2 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact frsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
- ∀J. T1 = ⓪{J} → ⊥.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #a #I #L #V #T #J #H destruct
-| #a #I #L #V #T #J #H destruct
-| #I #L #V #T #J #H destruct
-| #I #L #V #T #J #H destruct
-]
-qed-.
-
-lemma frsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁ ⦃L2, T2⦄ → ⊥.
-/2 width=7 by frsup_inv_atom1_aux/ qed-.
-
-fact frsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
- ∀b,J,W,U. T1 = ⓑ{b,J}W.U →
- (L2 = L1 ∧ T2 = W) ∨
- (L2 = L1.ⓑ{J}W ∧ T2 = U).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
-| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
-| #I #L #V #T #b #J #W #U #H destruct
-| #I #L #V #T #b #J #W #U #H destruct
-]
-qed-.
-
-lemma frsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁ ⦃L2, T2⦄ →
- (L2 = L1 ∧ T2 = W) ∨
- (L2 = L1.ⓑ{J}W ∧ T2 = U).
-/2 width=4 by frsup_inv_bind1_aux/ qed-.
-
-fact frsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
- ∀J,W,U. T1 = ⓕ{J}W.U →
- L2 = L1 ∧ (T2 = W ∨ T2 = U).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #a #I #L #V #T #J #W #U #H destruct
-| #a #I #L #V #T #J #W #U #H destruct
-| #I #L #V #T #J #W #U #H destruct /3 width=1/
-| #I #L #V #T #J #W #U #H destruct /3 width=1/
-]
-qed-.
-
-lemma frsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁ ⦃L2, T2⦄ →
- L2 = L1 ∧ (T2 = W ∨ T2 = U).
-/2 width=4 by frsup_inv_flat1_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma frsup_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/
-qed-.
-
-lemma frsup_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/
-qed-.
-
-lemma frsup_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{T2} < ♯{T1}.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/
-qed-.
-
-lemma frsup_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #a
-| #a #I #L #V #_ @(ex_intro … (⋆.ⓑ{I}V)) //
-]
-#I #L #V #T @(ex_intro … (⋆)) //
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lift_frsup_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀L,K,U2. ⦃L, U1⦄ ⧁ ⦃L @@ K, U2⦄ →
- ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
-#T1 #U1 #d #e * -T1 -U1 -d -e
-[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HTU1 #L #K #X #H
- elim (frsup_inv_bind1 … H) -H *
- [ -HTU1 #H1 #H2 destruct
- >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/
- | -HVW1 #H1 #H2 destruct
- >(append_inv_pair_dx … H1) -L -K normalize /2 width=2/
- ]
-|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #K #X #H
- elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct
- >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/
-]
-#i #d #e [2,3: #_ ] #L #K #X #H
-elim (frsup_inv_atom1 … H)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'RestSupTermPlus $L1 $T1 $L2 $T2 }.
-
-include "basic_2/substitution/frsup.ma".
-
-(* PLUS-ITERATED RESTRICTED SUPCLOSURE **************************************)
-
-definition frsupp: bi_relation lenv term ≝ bi_TC … frsup.
-
-interpretation "plus-iterated restricted structural predecessor (closure)"
- 'RestSupTermPlus L1 T1 L2 T2 = (frsupp L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma frsupp_ind: ∀L1,T1. ∀R:relation2 lenv term.
- (∀L2,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → R L2 T2) →
- (∀L,T,L2,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → R L2 T2.
-#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_TC_ind … IH1 IH2 ? ? H)
-qed-.
-
-lemma frsupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term.
- (∀L1,T1. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → R L1 T1) →
- (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → R L1 T1.
-#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
-@(bi_TC_ind_dx … IH1 IH2 ? ? H)
-qed-.
-
-(* Baic inversion lemmas ****************************************************)
-
-lemma frsupp_inv_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ ∨
- ∃∃L,T. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ & ⦃L, T⦄ ⧁ ⦃L2, T2⦄.
-/2 width=1 by bi_TC_decomp_r/ qed-.
-
-lemma frsupp_inv_sn: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ ∨
- ∃∃L,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ & ⦃L, T⦄ ⧁+ ⦃L2, T2⦄.
-/2 width=1 by bi_TC_decomp_l/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma frsup_frsupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma frsupp_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma frsupp_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma frsupp_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2
-/3 width=3 by frsup_fwd_fw, transitive_lt/
-qed-.
-
-lemma frsupp_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
-#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2
-/2 width=3 by frsup_fwd_lw/ (**) (* /3 width=5 by frsup_fwd_lw, transitive_le/ is too slow *)
-#L #T #L2 #T2 #_ #HL2 #HL1
-lapply (frsup_fwd_lw … HL2) -HL2 /2 width=3 by transitive_le/
-qed-.
-
-lemma frsupp_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ♯{T2} < ♯{T1}.
-#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2
-/3 width=3 by frsup_fwd_tw, transitive_lt/
-qed-.
-
-lemma frsupp_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
-#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2 /2 width=3 by frsup_fwd_append/
-#L #T #L2 #T2 #_ #HL2 * #K1 #H destruct
-elim (frsup_fwd_append … HL2) -HL2 #K2 #H destruct /2 width=2/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lift_frsupp_trans: ∀L,U1,K,U2. ⦃L, U1⦄ ⧁+ ⦃L @@ K, U2⦄ →
- ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
-#L #U1 @(f2_ind … fw … L U1) -L -U1 #n #IH
-#L #U1 #Hn #K #U2 #H #T1 #d #e #HTU1 destruct
-elim (frsupp_inv_sn … H) -H /2 width=5 by lift_frsup_trans/ *
-#L0 #U0 #HL0 #HL
-elim (frsup_fwd_append … HL0) #K0 #H destruct
-elim (frsupp_fwd_append … HL) #L0 >append_assoc #H
-elim (append_inj_dx … H ?) -H // #_ #H destruct
-<append_assoc in HL; #HL
-elim (lift_frsup_trans … HTU1 … HL0) -T1 #T #HTU
-lapply (frsup_fwd_fw … HL0) -HL0 #HL0
-elim (IH … HL … HTU) -IH -HL -T // -L -U1 -U0 /2 width=2/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/frsupp.ma".
-
-(* PLUS-ITERATED RESTRICTED SUPCLOSURE **************************************)
-
-(* Main propertis ***********************************************************)
-
-theorem frsupp_trans: bi_transitive … frsupp.
-/2 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'RestSupTermStar $L1 $T1 $L2 $T2 }.
-
-include "basic_2/unfold/frsupp.ma".
-
-(* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
-
-definition frsups: bi_relation lenv term ≝ bi_star … frsup.
-
-interpretation "star-iterated restricted structural predecessor (closure)"
- 'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma frsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L2 T2.
-#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_star_ind … IH1 IH2 ? ? H)
-qed-.
-
-lemma frsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L1 T1.
-#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
-@(bi_star_ind_dx … IH1 IH2 ? ? H)
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma frsups_refl: bi_reflexive … frsups.
-/2 width=1/ qed.
-
-lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma frsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma frsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma frsups_frsupp_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ →
- ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma frsupp_frsups_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ →
- ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
- (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 * /2 width=1/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
-/3 width=1 by frsupp_fwd_fw, lt_to_le/
-qed-.
-
-lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
-#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
-/2 width=3 by frsupp_fwd_lw/
-qed-.
-
-lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{T2} ≤ ♯{T1}.
-#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
-/3 width=3 by frsupp_fwd_tw, lt_to_le/
-qed-.
-
-lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
-#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
-[ * #H1 #H2 destruct
- @(ex_intro … (⋆)) //
-| /2 width=3 by frsupp_fwd_append/
-qed-.
-
-(* Advanced forward lemmas ***************************************************)
-
-lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
- ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
-#T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
-[ * #H1 #H2 destruct
- >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
-| /2 width=5 by lift_frsupp_trans/
-]
-qed-.
-
-(* Advanced inversion lemmas for frsupp **************************************)
-
-lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
-#J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
-#L2 #T2 #H elim (frsup_inv_atom1 … H)
-qed-.
-
-lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
- ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
-#b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
- elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
-| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
-]
-qed-.
-
-lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
- ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
-#J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
- elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
-| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/frsups.ma".
-
-(* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
-
-(* Main propertis ***********************************************************)
-
-theorem frsups_trans: bi_transitive … frsups.
-/2 width=4/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/frsups.ma".
-include "basic_2/static/ssta.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
-#h #g #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #l #_ #H
- elim (frsupp_inv_atom1_frsups … H)
-| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
- elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
- elim (lift_inv_lref2_be … H ? ?) -H //
-| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
- elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
- elim (lift_inv_lref2_be … H ? ?) -H //
-| #a #I #L #V #T #U #l #_ #IHTU #H
- elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
- lapply (frsups_fwd_fw … H) -H normalize
- <associative_plus <associative_plus #H
- elim (le_plus_xySz_x_false … H)
-| #L #V #T #U #l #_ #IHTU #H
- elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
- lapply (frsups_fwd_fw … H) -H normalize
- <associative_plus <associative_plus #H
- elim (le_plus_xySz_x_false … H)
-| /3 width=4/
-]
-qed-.
-
-fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → T = U → ⊥.
-#h #g #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #l #_ #H
- lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
- <e1 -e1 #H elim (lt_refl_false … H)
-| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
- elim (lift_inv_lref2_be … HWU ? ?) -HWU //
-| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
- elim (lift_inv_lref2_be … HWU ? ?) -HWU //
-| #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
-| #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
-| #L #W #T #U #l #HTU #_ #H destruct
- elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
-]
-qed-.
-
-lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, T⦄ → ⊥.
-/2 width=8 by ssta_inv_refl_aux/ qed-.
-
-lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
-#h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
-[ * #_ #H destruct /2 width=6 by ssta_inv_refl/
-| /2 width=8 by ssta_inv_frsupp/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* Basic inversion lemmas ***************************************************)
-
-fact fsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀J. T1 = ⓪{J} →
- (∃∃I,K,V. L1 = K.ⓑ{I}V & J = LRef 0 & L2 = K & T2 = V) ∨
- ∃∃I,K,V,i. ⦃K, #i⦄ ⊃ ⦃L2, T2⦄ & L1 = K.ⓑ{I}V & J = LRef (i+1).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #I #L #V #J #H destruct /3 width=6/
-| #I #L #K #V #T #i #HLK #J #H destruct /3 width=7/
-| #a #I #L #V #T #J #H destruct
-| #a #I #L #V #T #J #H destruct
-| #I #L #V #T #J #H destruct
-| #I #L #V #T #J #H destruct
-]
-qed-.
-
-lemma fsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⊃ ⦃L2, T2⦄ →
- (∃∃I,K,V. L1 = K.ⓑ{I}V & J = LRef 0 & L2 = K & T2 = V) ∨
- ∃∃I,K,V,i. ⦃K, #i⦄ ⊃ ⦃L2, T2⦄ & L1 = K.ⓑ{I}V & J = LRef (i+1).
-/2 width=3 by fsup_inv_atom1_aux/ qed-.
-
-fact fsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
- ∀b,J,W,U. T1 = ⓑ{b,J}W.U →
- (L2 = L1 ∧ T2 = W) ∨
- (L2 = L1.ⓑ{J}W ∧ T2 = U).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #I #L #V #b #J #W #U #H destruct
-| #I #L #K #V #T #i #_ #b #J #W #U #H destruct
-| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
-| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
-| #I #L #V #T #b #J #W #U #H destruct
-| #I #L #V #T #b #J #W #U #H destruct
-]
-qed-.
-
-lemma fsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃ ⦃L2, T2⦄ →
- (L2 = L1 ∧ T2 = W) ∨
- (L2 = L1.ⓑ{J}W ∧ T2 = U).
-/2 width=4 by fsup_inv_bind1_aux/ qed-.
-
-fact fsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
- ∀J,W,U. T1 = ⓕ{J}W.U →
- L2 = L1 ∧ (T2 = W ∨ T2 = U).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #I #L #K #J #W #U #H destruct
-| #I #L #K #V #T #i #_ #J #W #U #H destruct
-| #a #I #L #V #T #J #W #U #H destruct
-| #a #I #L #V #T #J #W #U #H destruct
-| #I #L #V #T #J #W #U #H destruct /3 width=1/
-| #I #L #V #T #J #W #U #H destruct /3 width=1/
-]
-qed-.
-
-lemma fsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃ ⦃L2, T2⦄ →
- L2 = L1 ∧ (T2 = W ∨ T2 = U).
-/2 width=4 by fsup_inv_flat1_aux/ qed-.
+++ /dev/null
-(* Advanced inversion lemmas on plus-iterated supclosure ********************)
-
-lamma fsupp_inv_bind1_fsups: ∀b,J,G1,G2,L1,L2,W,U,T2. ⦃G1, L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, W⦄ ⊃* ⦃G2, L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃G2, L2, T2⦄.
-#b #J #G1 #G2 #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H
- elim (fsup_inv_bind1 … H) -H * #H1 #H2 #H3 destruct /2 width=1/
-| #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=4/
-]
-qad-.
-
-lamma fsupp_inv_flat1_fsups: ∀J,G1,G2,L1,L2,W,U,T2. ⦃G1, L1, ⓕ{J}W.U⦄ ⊃+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, W⦄ ⊃* ⦃G2, L2, T2⦄ ∨ ⦃G1, L1, U⦄ ⊃* ⦃G2, L2, T2⦄.
-#J #G1 #G2 #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H
- elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
-| #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=4/
-]
-qad-.
-
-lamma fsupp_fsups: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lamma fsups_lref: ∀I,G,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃* ⦃G, K, V⦄.
-/3 width=5 by _/ qed.
-
-lamma fsups_lref_S_lt: ∀I,G1,G2,L,K,V,T,i.
- 0 < i → ⦃G1, L, #(i-1)⦄ ⊃* ⦃G2, K, T⦄ → ⦃G1, L.ⓑ{I}V, #i⦄ ⊃+ ⦃G2, K, T⦄.
-/3 width=7 by _/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fsup.ma".
-include "basic_2/substitution/ldrop_ldrop.ma".
-
-(* LOCAL ENVIRONMENT SLICING ************************************************)
-
-(* Inversion lemmas on supclosure *******************************************)
-
-lemma fsup_inv_atom1_ldrop: ∀K,V,L,I. ⦃L, ⓪{I}⦄ ⊃ ⦃K, V⦄ →
- ∃∃J,i. ⇩[0, i] L ≡ K.ⓑ{J}V & I = LRef i.
-#K #V #L @(f_ind … length … L) -L #n #IH #L #Hn #I #H
-elim (fsup_inv_atom1 … H) -H *
-[ #J #L0 #V0 #H1 #H2 #H3 #H4 destruct /2 width=4/
-| #J #L0 #V0 #i #HLK #H1 #H2 destruct
- elim (IH … HLK) -IH -HLK [2: normalize // ] #I #j #HLK #H destruct /3 width=4/
-]
-qed-.
-
-(* Advanced eliminators on supclosure ***************************************)
-
-lemma fsup_ind_ldrop: ∀R:bi_relation lenv term.
- (∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → R L (#i) K V) →
- (∀a,I,L,V,T. R L (ⓑ{a,I}V.T) L V) →
- (∀a,I,L,V,T. R L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T) →
- (∀I,L,V,T. R L (ⓕ{I}V.T) L V) →
- (∀I,L,V,T. R L (ⓕ{I}V.T) L T) →
- ∀L1,T1,L2,T2. ⦃L1,T1⦄⊃⦃L2,T2⦄ → R L1 T1 L2 T2.
-#R #H1 #H2 #H3 #H4 #H5 #L1 #T1 #L2 #T2 #H elim H -L1 -T1 -L2 -T2 //
-[ /3 width=2/
-| #I #L #K #V #T #i #H #H1LK
- elim (fsup_inv_atom1_ldrop … H) -H #J #j #H2LK #H destruct /3 width=2/
-]
-qed-.
-
-(* Advanced inversion lemmas on supclosure **********************************)
-
-lemma fsup_inv_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
- ∀J,W,j. ⇩[0, j] L1 ≡ L2.ⓑ{J}W → T1 = #j ∧ T2 = W.
-#L1 #L2 #T1 #T2 #H @(fsup_ind_ldrop … H) -L1 -L2 -T1 -T2
-[ #I #L #K #V #i #HLKV #J #W #j #HLKW
- elim (ldrop_conf_div … HLKV … HLKW) -L /2 width=1/
-| #a
-| #a
-]
-#I #L #V #T #J #W #j #H
-lapply (ldrop_pair2_fwd_fw … H W) -H #H
-[2: lapply (transitive_lt (♯{L,W}) … H) /2 width=1/ -H #H ]
-elim (lt_refl_false … H)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/fsups.ma".
-include "basic_2/substitution/ldrop_fsup.ma".
-
-(* LOCAL ENVIRONMENT SLICING ************************************************)
-
-(* Inversion lemmas on plus-iterated supclosure ****************************)
-
-lemma fsupp_inv_atom1_fsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⊃+ ⦃L2, T2⦄ →
- ∃∃I,K,V,i. ⇩[0, i] L1 ≡ K.ⓑ{I}V &
- ⦃K, V⦄ ⊃* ⦃L2, T2⦄ & J = LRef i.
-#J #L1 #L2 #T2 #H @(fsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
- elim (fsup_inv_atom1_ldrop … H) -H * #i #HL12 #H destruct /2 width=7/
-| #L #T #L2 #T2 #_ #HT2 * #I #K #V #i #HLK #HVT #H destruct /3 width=8/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/multiple/lifts_lifts.ma".
-include "basic_2/multiple/drops_drops.ma".
-include "basic_2/static/aaa_lifts.ma".
-include "basic_2/static/aaa_aaa.ma".
-include "basic_2/computation/lsubc_drops.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: sc3_arity_csubc *)
-theorem acr_aaa_csubc: ∀RR,RS,RP.
- gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
- ∀L2. G ⊢ L2 ⫃[RP] L1 → ⦃G, L2, T⦄ ϵ[RP] 〚A〛.
-#RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
-[ #G #L1 #k #L2 #HL21
- lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
- lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L2 #HL21
- lapply (acr_gcr … H1RP H2RP B) #HB
- elim (lsubc_drop_O1_trans … HL21 … HLK1) -L1 #X #HLK2 #H
- elim (lsubc_inv_pair2 … H) -H *
- [ #K2 #HK21 #H destruct -HKV1B
- lapply (drop_fwd_drop2 … HLK2) #H
- elim (lift_total V1 0 (i +1)) #V #HV1
- lapply (s5 … HB ? G ? ? (◊) … HV1 HLK2) /3 width=7 by s0/
- | #K2 #V2 #A2 #HVA2 #H1V1A2 #H2V1A2 #_ #H1 #H2 destruct -IHB
- lapply (aaa_mono … H2V1A2 … HKV1B) #H destruct -H2V1A2 -HKV1B
- lapply (drop_fwd_drop2 … HLK2) #H
- elim (lift_total V1 0 (i +1)) #V3 #HV13
- elim (lift_total V2 0 (i +1)) #V #HV2
- lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/ -HLK2
- lapply (s7 … HB G L2 (◊)) /3 width=7 by s0/
- ]
-| #a #G #L1 #V #T #B #A #_ #_ #IHB #IHA #L2 #HL21
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (acr_gcr … H1RP H2RP B) #HB
- lapply (s1 … HB) -HB #HB
- lapply (s6 … HA G L2 (◊) (◊)) /4 width=1 by lsubc_pair/
-| #a #G #L1 #W #T #B #A #HLWB #_ #IHB #IHA #L2 #HL21
- @(acr_abst … H1RP H2RP) [ /2 width=5 by/ ]
- #L3 #V3 #W3 #T3 #des3 #HL32 #HW03 #HT03 #H1B #H2B
- @(gcr_lifts … L2.ⓓⓝW.V3,T … HL32)
- elim (drops_lsubc_trans … H1RP H2RP … HL32 … HL21) -L2 #L2 #HL32 #HL21
- lapply (aaa_lifts … L2 W3 … des3 … HLWB) -HLWB /2 width=4 by drops_trans, lifts_trans/ #HLW2B
- @(s0
-
- @(IHA (L2. ⓛW3) … (des3 + 1)) -IHA
- /3 width=5 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
-| #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
- elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
- /3 width=10 by drops_nil, lifts_nil/
-| #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
- elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (s7 … HA G L2 (◊)) /3 width=5 by/
-]
-qed.
-
-(* Basic_1: was: sc3_arity *)
-lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
-/2 width=8 by drops_nil, lifts_nil, acr_aaa_csubc_lifts/ qed.
-
-lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
-#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
-lapply (acr_gcr … H1RP H2RP A) #HA
-@(s1 … HA) /2 width=4 by acr_aaa/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/ineint_5.ma".
-include "basic_2/grammar/aarity.ma".
-include "basic_2/multiple/mr2_mr2.ma".
-include "basic_2/multiple/lifts_lift_vector.ma".
-include "basic_2/multiple/drops_drop.ma".
-include "basic_2/computation/gcp.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-definition S0 ≝ λC:candidate. ∀G,L2,L1,T1,d,e.
- C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
-
-definition S0s ≝ λC:candidate.
- ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
-
-(* Note: this is Girard's CR1 *)
-definition S1 ≝ λRP,C:candidate.
- ∀G,L,T. C G L T → RP G L T.
-
-(* Note: this is Tait's iii, or Girard's CR4 *)
-definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs →
- ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
-
-(* Note: this generalizes Tait's ii *)
-definition S3 ≝ λC:candidate.
- ∀a,G,L,Vs,V,T,W.
- C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
-
-definition S4 ≝ λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
-
-definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[0, i+1] V1 ≡ V2 →
- C G L (ⒶVs.V2) → C G L (ⒶVs.#i).
-
-definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
-
-definition S7 ≝ λC:candidate.
- ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
-
-(* requirements for the generic reducibility candidate *)
-record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
-{ (* s0: S0 C; *)
- s1: S1 RP C;
- s2: S2 RR RS RP C;
- s3: S3 C;
- s4: S4 RP C;
- s5: S5 C;
- s6: S6 RP C;
- s7: S7 C
-}.
-
-(* the functional construction for candidates *)
-definition cfun: candidate → candidate → candidate ≝
- λC1,C2,G,K,T. ∀V. C1 G K V → C2 G K (ⓐV.T).
-
-(* the reducibility candidate associated to an atomic arity *)
-let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
-match A with
-[ AAtom ⇒ RP
-| APair B A ⇒ cfun (acr RP B) (acr RP A)
-].
-
-interpretation
- "candidate of reducibility of an atomic arity (abstract)"
- 'InEInt RP G L T A = (acr RP A G L T).
-
-(* Basic properties *********************************************************)
-(*
-(* Basic_1: was: sc3_lift1 *)
-lemma gcr_lifts: ∀C. S0 C → S0s C.
-#C #HC #G #L1 #L2 #des #H elim H -L1 -L2 -des
-[ #L #T1 #T2 #H #HT1 <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
- elim (lifts_inv_cons … H) -H /3 width=10 by/
-]
-qed.
-*)
-axiom rp_lift: ∀RP. S0 RP.
-
-
-axiom rp_lifts: ∀RR,RS,RP. gcr RR RS RP RP →
- ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
- RP G L V → RP G L0 V0.
-(*
-#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
-@gcr_lifts /width=7 by/
-@(s0 … HRP)
-qed.
-*)
-(* Basic_1: was only: sns3_lifts1 *)
-axiom rp_liftsv_all: ∀RR,RS,RP. gcr RR RS RP RP →
- ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
- all … (RP G L) Vs → all … (RP G L0) V0s.
-(*
-#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
-#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
-qed.
-*)
-
-lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀A. S0 (acr RP A).
-#RR #RS #RP #H1RP #H2RP #A elim A -A /2 width=7 by rp_lift/
-#B #A #IHB #IHA #G #L2 #L1 #T1 #d #e #IH #T2 #HL21 #HT12 #V #HB
-@(IHA … HL21) [3: @(lift_flat … HT12) |1: skip |
-
-(* Basic_1: was:
- sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
-*)
-lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀A. gcr RR RS RP (acr RP A).
-#RR #RS #RP #H1RP #H2RP #A elim A -A //
-#B #A #IHB #IHA @mk_gcr
-[ #G #L #T #H
- elim (cp1 … H1RP G L) #k #HK
- lapply (H (⋆k) ?) -H
- [ lapply (s2 … IHB G L (◊) … HK) //
- | #H @(cp2 … H1RP … k) @(s1 … IHA) //
- ]
-| #G #L #Vs #HVs #T #H1T #H2T #V #HB
- lapply (s1 … IHB … HB) #HV
- @(s2 … IHA … (V @ Vs))
- /3 width=14 by rp_liftsv_all, gcp_lifts, cp0, lifts_simple_dx, conj/
-| #a #G #L #Vs #U #T #W #HA #V #HB
- @(s3 … IHA … (V @ Vs)) /2 width=1 by/
-| #G #L #Vs #HVs #k #V #HB
- lapply (s1 … IHB … HB) #HV
- @(s4 … IHA … (V @ Vs)) /3 width=7 by rp_liftsv_all, conj/
-| #I #G #L #K #Vs #V1 #V2 #i #HLK #HV12 #HA #V #HB
- @(s5 … IHA … (V @ Vs) … HLK HV12) /2 width=1 by/
-| #G #L #V1s #V2s #HV12s #a #W #T #HA #HW #V1 #HB
- elim (lift_total V1 0 1) #V2 #HV12
- @(s6 … IHA … (V1 @ V1s) (V2 @ V2s)) /2 width=1 by liftv_cons/
- @HA @(gcr_lift … H1RP H2RP … HB … HV12) /2 width=2 by drop_drop/
-| #G #L #Vs #T #W #HA #HW #V #HB
- @(s7 … IHA … (V @ Vs)) /2 width=1 by/
-]
-qed.
-
-lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀a,G,L,W,T,B,A. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀V. ⦃G, L, V⦄ ϵ[RP] 〚B〛 → ⦃G, L.ⓓⓝW.V, T⦄ ϵ[RP] 〚A〛
- ) →
- ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
-#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #B #A #HW #HA #L0 #V0 #X #des #HL0 #H #HB
-lapply (acr_gcr … H1RP H2RP A) #HCA
-lapply (acr_gcr … H1RP H2RP B) #HCB
-elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
-lapply (gcr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
-lapply (s3 … HCA … a G L0 (◊)) #H @H -H
-lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
-[ @(HA … HL0) //
-| lapply (s1 … HCB) -HCB #HCB
- lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
-]
-qed.
-
-(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
-(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ≅ break [ term 46 d , break term 46 e ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'CoEq $d $e $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_lt.ma".
-include "basic_2/notation/relations/coeq_4.ma".
-include "basic_2/grammar/lenv_length.ma".
-
-(* COEQUIVALENCE FOR LOCAL ENVIRONMENTS *************************************)
-
-inductive lcoeq: relation4 ynat ynat lenv lenv ≝
-| lcoeq_atom: ∀d,e. lcoeq d e (⋆) (⋆)
-| lcoeq_zero: ∀I,L1,L2,V.
- lcoeq 0 0 L1 L2 → lcoeq 0 0 (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lcoeq_pair: ∀I1,I2,L1,L2,V1,V2,e. lcoeq 0 e L1 L2 →
- lcoeq 0 (⫯e) (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-| lcoeq_succ: ∀I,L1,L2,V,d,e.
- lcoeq d e L1 L2 → lcoeq (⫯d) e (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-.
-
-interpretation
- "coequivalence (local environment)"
- 'CoEq d e L1 L2 = (lcoeq d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lcoeq_pair_lt: ∀I1,I2,L1,L2,V1,V2,e. L1 ≅[0, ⫰e] L2 → 0 < e →
- L1.ⓑ{I1}V1 ≅[0, e] L2.ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lcoeq_pair/
-qed.
-
-lemma lcoeq_succ_lt: ∀I,L1,L2,V,d,e. L1 ≅[⫰d, e] L2 → 0 < d →
- L1.ⓑ{I}V ≅[d, e] L2. ⓑ{I}V.
-#I #L1 #L2 #V #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lcoeq_succ/
-qed.
-
-lemma lcoeq_pair_O_Y: ∀L1,L2. L1 ≅[0, ∞] L2 →
- ∀I1,I2,V1,V2. L1.ⓑ{I1}V1 ≅[0,∞] L2.ⓑ{I2}V2.
-#L1 #L2 #HL12 #I1 #I2 #V1 #V2 lapply (lcoeq_pair I1 I2 … V1 V2 … HL12) -HL12 //
-qed.
-
-lemma lcoeq_refl: ∀L,d,e. L ≅[d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
-#Hd destruct /2 width=1 by lcoeq_succ/
-#e elim (ynat_cases … e) [| * #x ]
-#He destruct /2 width=1 by lcoeq_zero, lcoeq_pair/
-qed.
-
-lemma lcoeq_O_Y: ∀L1,L2. |L1| = |L2| → L1 ≅[0, ∞] L2.
-#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
-* [2,4: #L2 #I2 #V1 ] normalize /3 width=2 by lcoeq_pair_O_Y/
-<plus_n_Sm #H destruct
-qed.
-
-lemma lcoeq_sym: ∀d,e. symmetric … (lcoeq d e).
-#d #e #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by lcoeq_zero, lcoeq_pair, lcoeq_succ/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lcoeq_inv_atom1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #d #e * -L1 -L2 -d -e //
-[ #I #L1 #L2 #V #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #H destruct
-| #I #L1 #L2 #V #d #e #_ #H destruct
-]
-qed-.
-
-lemma lcoeq_inv_atom1: ∀L2,d,e. ⋆ ≅[d, e] L2 → L2 = ⋆.
-/2 width=5 by lcoeq_inv_atom1_aux/ qed-.
-
-fact lcoeq_inv_zero1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
- ∀J,K1,W. L1 = K1.ⓑ{J}W → d = 0 → e = 0 →
- ∃∃K2. K1 ≅[0, 0] K2 & L2 = K2.ⓑ{J}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #J #K1 #W #H destruct
-| #I #L1 #L2 #V #HL12 #J #K1 #W #H destruct /2 width=3 by ex2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #J #K1 #W #_ #_ #H elim (ysucc_inv_O_dx … H)
-| #I #L1 #L2 #V #d #e #_ #J #K1 #W #_ #H elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lcoeq_inv_zero1: ∀I,K1,L2,V. K1.ⓑ{I}V ≅[0, 0] L2 →
- ∃∃K2. K1 ≅[0, 0] K2 & L2 = K2.ⓑ{I}V.
-/2 width=7 by lcoeq_inv_zero1_aux/ qed-.
-
-fact lcoeq_inv_pair1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → 0 < e →
- ∃∃J2,K2,W2. K1 ≅[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #J1 #K1 #W1 #H destruct
-| #I #L1 #L2 #V #_ #J1 #K1 #W1 #_ #_ #H elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #e #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I #L1 #L2 #V #d #e #_ #J1 #K1 #W1 #_ #H elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lcoeq_inv_pair1: ∀I1,K1,L2,V1,e. K1.ⓑ{I1}V1 ≅[0, e] L2 → 0 < e →
- ∃∃I2,K2,V2. K1 ≅[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=7 by lcoeq_inv_pair1_aux/ qed-.
-
-fact lcoeq_inv_succ1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
- ∀J,K1,W. L1 = K1.ⓑ{J}W → 0 < d →
- ∃∃K2. K1 ≅[⫰d, e] K2 & L2 = K2.ⓑ{J}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #J #K1 #W #H destruct
-| #I #L1 #L2 #V #_ #J #K1 #W #_ #H elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #J #K1 #W #_ #H elim (ylt_yle_false … H) //
-| #I #L1 #L2 #V #d #e #HL12 #J #K1 #W #H destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lcoeq_inv_succ1: ∀I,K1,L2,V,d,e. K1.ⓑ{I}V ≅[d, e] L2 → 0 < d →
- ∃∃K2. K1 ≅[⫰d, e] K2 & L2 = K2.ⓑ{I}V.
-/2 width=3 by lcoeq_inv_succ1_aux/ qed-.
-
-lemma lcoeq_inv_atom2: ∀L1,d,e. L1 ≅[d, e] ⋆ → L1 = ⋆.
-/3 width=3 by lcoeq_inv_atom1, lcoeq_sym/ qed-.
-
-lemma lcoeq_inv_zero2: ∀I,K2,L1,V. L1 ≅[0, 0] K2.ⓑ{I}V →
- ∃∃K1. K1 ≅[0, 0] K2 & L1 = K1.ⓑ{I}V.
-#I #K2 #L1 #V #H elim (lcoeq_inv_zero1 … (lcoeq_sym … H)) -H
-/3 width=3 by lcoeq_sym, ex2_intro/
-qed-.
-
-lemma lcoeq_inv_pair2: ∀I2,K2,L1,V2,e. L1 ≅[0, e] K2.ⓑ{I2}V2 → 0 < e →
- ∃∃I1,K1,V1. K1 ≅[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V1.
-#I2 #K2 #L1 #V2 #e #H #He elim (lcoeq_inv_pair1 … (lcoeq_sym … H)) -H
-/3 width=5 by lcoeq_sym, ex2_3_intro/
-qed-.
-
-lemma lcoeq_inv_succ2: ∀I,K2,L1,V,d,e. L1 ≅[d, e] K2.ⓑ{I}V → 0 < d →
- ∃∃K1. K1 ≅[⫰d, e] K2 & L1 = K1.ⓑ{I}V.
-#I #K2 #L1 #V #d #e #H #Hd elim (lcoeq_inv_succ1 … (lcoeq_sym … H)) -H
-/3 width=3 by lcoeq_sym, ex2_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lcoeq_fwd_length: ∀L1,L2,d,e. L1 ≅[d, e] L2 → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
-qed-.
-
-(* Advanced inversionn lemmas ***********************************************)
-
-fact lcoeq_inv_O2_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 → e = 0 → L1 = L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e /3 width=1 by eq_f3/
-#I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #H elim (ysucc_inv_O_dx … H)
-qed-.
-
-lemma lcoeq_inv_O2: ∀L1,L2,d. L1 ≅[d, 0] L2 → L1 = L2.
-/2 width=4 by lcoeq_inv_O2_aux/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lcoeq.ma".
-
-(* COEQUIVALENCE FOR LOCAL ENVIRONMENTS *************************************)
-
-(* Main properties **********************************************************)
-
-theorem lcoeq_trans: ∀d,e. Transitive … (lcoeq d e).
-#d #e #L1 #L2 #H elim H -L1 -L2 -d -e //
-[ #I #L1 #L #V #HL1 #_ #X #H >(lcoeq_inv_O2 … HL1) -HL1 //
-| #I1 #I #L1 #L #V1 #V #e #_ #IHL1 #X #H elim (lcoeq_inv_pair1 … H) -H //
- #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lcoeq_pair/
-| #I #L1 #L #V #d #e #_ #IHL1 #X #H elim (lcoeq_inv_succ1 … H) -H //
- #L2 #HL2 #H destruct /3 width=1 by lcoeq_succ/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_minus.ma".
-include "basic_2/grammar/lcoeq.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-(* Properties on coequivalence **********************************************)
-
-lemma lcoeq_ldrop_trans_lt: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
- ∀I,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W → i < d →
- ∃∃K1. K1 ≅[⫰(d-i), e] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #J #K2 #W #s #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I #L1 #L2 #V #_ #_ #J #K2 #W #s #i #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #J #K2 #W #s #i #_ #H
- elim (ylt_yle_false … H) //
-| #I #L1 #L2 #V #d #e #HL12 #IHL12 #J #K2 #W #s #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct >ypred_succ
- /2 width=3 by ldrop_pair, ex2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ … H) -H
- #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
- >yminus_succ <yminus_inj /3 width=3 by ldrop_drop_lt, ex2_intro/
- ]
-]
-qed-.
-
-lemma lcoeq_ldrop_conf_lt: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
- ∀I,K1,W,s,i. ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W → i < d →
- ∃∃K2. K1 ≅[⫰(d-i), e] K2 & ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W.
-#L1 #L2 #d #e #HL12 #I #K1 #W #s #i #HLK1 #Hid
-elim (lcoeq_ldrop_trans_lt … (lcoeq_sym … HL12) … HLK1) // -L1 -Hid
-/3 width=3 by lcoeq_sym, ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )"
- non associative with precedence 45
- for @{ 'CPConvStar $L1 $L2 }.
-
-include "basic_2/grammar/lenv_px_sn.ma".
-include "basic_2/equivalence/cpcs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
-
-definition lcpcs: relation lenv ≝ lpx_sn … cpcs.
-
-interpretation "context-sensitive parallel equivalence (local environment)"
- 'CPConvStar L1 L2 = (lcpcs L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lcpcs_inv_atom1: ∀L2. ⋆ ⊢ ⬌* L2 → L2 = ⋆.
-/2 width=2 by lpx_sn_inv_atom1/ qed-.
-
-lemma lcpcs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ⬌* L2 →
- ∃∃K2,V2. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L2 = K2. ⓑ{I} V2.
-/2 width=1 by lpx_sn_inv_pair1/ qed-.
-
-lemma lcpcs_inv_atom2: ∀L1. L1 ⊢ ⬌* ⋆ → L1 = ⋆.
-/2 width=2 by lpx_sn_inv_atom2/ qed-.
-
-lemma lcpcs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ⬌* K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L1 = K1. ⓑ{I} V1.
-/2 width=1 by lpx_sn_inv_pair2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lcpcs_fwd_length: ∀L1,L2. L1 ⊢ ⬌* L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/ltpr.ma".
-include "basic_2/equivalence/lcpcs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-lemma ltpr_lcpcs: ∀L1,L2. L1 ➡ L2 → L1 ⊢ ⬌* L2.
-#L1 #L2 #H elim H -L1 -L2 // /4 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⋕ ⋕ break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyEqAlt $T $d $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazyeq_4.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-definition lleq: relation4 ynat term lenv lenv ≝
- λd,T,L1,L2. |L1| = |L2| ∧
- (∀U. ⦃⋆, L1⦄ ⊢ T ▶*[d, ∞] U ↔ ⦃⋆, L2⦄ ⊢ T ▶*[d, ∞] U).
-
-interpretation
- "lazy equivalence (local environment)"
- 'LazyEq T d L1 L2 = (lleq d T L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
-/3 width=1 by conj/ qed.
-
-lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
-#d #T #L1 #L2 * /3 width=1 by iff_sym, conj/
-qed-.
-
-lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ⋕[⋆k, d] L2.
-#L1 #L2 #d #k #HL12 @conj // -HL12
-#U @conj #H >(cpys_inv_sort1 … H) -H //
-qed.
-
-lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ⋕[§p, d] L2.
-#L1 #L2 #d #k #HL12 @conj // -HL12
-#U @conj #H >(cpys_inv_gref1 … H) -H //
-qed.
-
-lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
- L1 ⋕[V, d] L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V →
- L1 ⋕[ⓑ{a,I}V.T, d] L2.
-#a #I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
-#X @conj #H elim (cpys_inv_bind1 … H) -H
-#W #U #HVW #HTU #H destruct
-elim (IHV W) -IHV elim (IHT U) -IHT /3 width=1 by cpys_bind/
-qed.
-
-lemma lleq_flat: ∀I,L1,L2,V,T,d.
- L1 ⋕[V, d] L2 → L1 ⋕[T, d] L2 → L1 ⋕[ⓕ{I}V.T, d] L2.
-#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
-#X @conj #H elim (cpys_inv_flat1 … H) -H
-#W #U #HVW #HTU #H destruct
-elim (IHV W) -IHV elim (IHT U) -IHT
-/3 width=1 by cpys_flat/
-qed.
-
-lemma lleq_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
- d ≤ dt → dt ≤ d + e → L1 ⋕[U, d] L2.
-#L1 #L2 #U #dt * #HL12 #IH #T #d #e #HTU #Hddt #Hdtde @conj // -HL12
-#U0 elim (IH U0) -IH #H12 #H21 @conj
-#HU0 elim (cpys_fwd_up … HU0 … HTU) // -HU0 /4 width=5 by cpys_weak/
-qed-.
-
-lemma lsuby_lleq_trans: ∀L2,L,T,d. L2 ⋕[T, d] L →
- ∀L1. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → L1 ⋕[T, d] L.
-#L2 #L #T #d * #HL2 #IH #L1 #HL12 #H @conj // -HL2
-#U elim (IH U) -IH #Hdx #Hsn @conj #HTU
-[ @Hdx -Hdx -Hsn @(lsuby_cpys_trans … HTU) -HTU
- /2 width=1 by lsuby_sym/ (**) (* full auto does not work *)
-| -H -Hdx /3 width=3 by lsuby_cpys_trans/
-]
-qed-.
-
-lemma lleq_lsuby_trans: ∀L,L1,T,d. L ⋕[T, d] L1 →
- ∀L2. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → L ⋕[T, d] L2.
-/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-.
-
-lemma lleq_lsuby_repl: ∀L1,L2,T,d. L1 ⋕[T, d] L2 →
- ∀K1. K1 ⊑×[d, ∞] L1 → |K1| = |L1| →
- ∀K2. L2 ⊑×[d, ∞] K2 → |L2| = |K2| →
- K1 ⋕[T, d] K2.
-/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → |L1| = |L2|.
-#L1 #L2 #T #d * //
-qed-.
-
-lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ⋕[d, T] L2 → ∀K1,i. ⇩[i] L1 ≡ K1 →
- ∃K2. ⇩[i] L2 ≡ K2.
-#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length … H) -H
-#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
-qed-.
-
-lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ⋕[d, T] L2 → ∀K2,i. ⇩[i] L2 ≡ K2 →
- ∃K1. ⇩[i] L1 ≡ K1.
-/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-.
-
-lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
- L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1 ⋕[V, d] L2.
-#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
-#U elim (H (ⓑ{a,I}U.T)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_bind/ -H
-#H elim (cpys_inv_bind1 … H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,d.
- L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V.
-#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12
-#U elim (H (ⓑ{a,I}V.U)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_bind/ -H
-#H elim (cpys_inv_bind1 … H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
- L1 ⋕[ⓕ{I}V.T, d] L2 → L1 ⋕[V, d] L2.
-#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
-#U elim (H (ⓕ{I}U.T)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_flat/ -H
-#H elim (cpys_inv_flat1 … H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
- L1 ⋕[ⓕ{I}V.T, d] L2 → L1 ⋕[T, d] L2.
-#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
-#U elim (H (ⓕ{I}V.U)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_flat/ -H
-#H elim (cpys_inv_flat1 … H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,d. L1 ⋕[ⓑ{a,I}V.T, d] L2 →
- L1 ⋕[V, d] L2 ∧ L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V.
-/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-.
-
-lemma lleq_inv_flat: ∀I,L1,L2,V,T,d. L1 ⋕[ⓕ{I}V.T, d] L2 →
- L1 ⋕[V, d] L2 ∧ L1 ⋕[T, d] L2.
-/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazyeqalt_4.ma".
-include "basic_2/substitution/lleq_lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Note: alternative definition of lleq *)
-inductive lleqa: relation4 ynat term lenv lenv ≝
-| lleqa_sort: ∀L1,L2,d,k. |L1| = |L2| → lleqa d (⋆k) L1 L2
-| lleqa_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → lleqa d (#i) L1 L2
-| lleqa_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
- lleqa (yinj 0) V K1 K2 → lleqa d (#i) L1 L2
-| lleqa_free: ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → lleqa d (#i) L1 L2
-| lleqa_gref: ∀L1,L2,d,p. |L1| = |L2| → lleqa d (§p) L1 L2
-| lleqa_bind: ∀a,I,L1,L2,V,T,d.
- lleqa d V L1 L2 → lleqa (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- lleqa d (ⓑ{a,I}V.T) L1 L2
-| lleqa_flat: ∀I,L1,L2,V,T,d.
- lleqa d V L1 L2 → lleqa d T L1 L2 → lleqa d (ⓕ{I}V.T) L1 L2
-.
-
-interpretation
- "lazy equivalence (local environment) alternative"
- 'LazyEqAlt T d L1 L2 = (lleqa d T L1 L2).
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lleqa_inv_lleq: ∀L1,L2,T,d. L1 ⋕⋕[T, d] L2 → L1 ⋕[T, d] L2.
-#L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/2 width=8 by lleq_flat, lleq_bind, lleq_gref, lleq_free, lleq_lref, lleq_skip, lleq_sort/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lleq_lleqa: ∀L1,T,L2,d. L1 ⋕[T, d] L2 → L1 ⋕⋕[T, d] L2.
-#L1 #T @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * * /3 width=3 by lleqa_gref, lleqa_sort, lleq_fwd_length/
-[ #i #Hn #L2 #d #H elim (lleq_fwd_lref … H) [ * || * ]
- /4 width=9 by lleqa_free, lleqa_lref, lleqa_skip, lleq_fwd_length, ldrop_fwd_rfw/
-| #a #I #V #T #Hn #L2 #d #H elim (lleq_inv_bind … H) -H /3 width=1 by lleqa_bind/
-| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat … H) -H /3 width=1 by lleqa_flat/
-]
-qed.
-
-(* Advanced eliminators *****************************************************)
-
-lemma lleq_ind_alt: ∀R:relation4 ynat term lenv lenv. (
- ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
- ) → (
- ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
- ) → (
- ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
- K1 ⋕[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
- ) → (
- ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
- ) → (
- ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
- ) → (
- ∀a,I,L1,L2,V,T,d.
- L1 ⋕[V, d]L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V →
- R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
- ) → (
- ∀I,L1,L2,V,T,d.
- L1 ⋕[V, d]L2 → L1 ⋕[T, d] L2 →
- R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
- ) →
- ∀d,T,L1,L2. L1 ⋕[T, d] L2 → R d T L1 L2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa … H) -H
-/3 width=9 by lleqa_inv_lleq/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lleq_alt.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact lleq_inv_S_aux: ∀L1,L2,T,d0. L1 ⋕[T, d0] L2 → ∀d. d0 = d + 1 →
- ∀K1,K2,I,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
-#L1 #L2 #T #d0 #H @(lleq_ind_alt … H) -L1 -L2 -T -d0
-/2 width=1 by lleq_gref, lleq_free, lleq_sort/
-[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V #HLK1 #HLK2 #HV destruct
- elim (yle_split_eq i d) /2 width=1 by lleq_skip, ylt_fwd_succ2/ -HL12 -Hid
- #H destruct /2 width=8 by lleq_lref/
-| #I1 #I2 #L1 #L2 #K11 #K22 #V #d0 #i #Hd0i #HLK11 #HLK22 #HV #_ #d #H #K1 #K2 #J #W #_ #_ #_ destruct
- /3 width=8 by lleq_lref, yle_pred_sn/
-| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /4 width=7 by lleq_bind, ldrop_drop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /3 width=7 by lleq_flat/
-]
-qed-.
-
-lemma lleq_inv_S: ∀T,L1,L2,d. L1 ⋕[T, d+1] L2 →
- ∀K1,K2,I,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
-/2 width=7 by lleq_inv_S_aux/ qed-.
-
-lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
- L1 ⋕[V, 0] L2 ∧ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind … H) -H
-/3 width=7 by ldrop_pair, conj, lleq_inv_S/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
- L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O … H) -H //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_ge: ∀L1,L2,T,d1. L1 ⋕[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ⋕[T, d2] L2.
-#L1 #L2 #T #d1 #H @(lleq_ind_alt … H) -L1 -L2 -T -d1
-/4 width=1 by lleq_sort, lleq_free, lleq_gref, lleq_bind, lleq_flat, yle_succ/
-[ /3 width=3 by lleq_skip, ylt_yle_trans/
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d1 #i #Hi #HLK1 #HLK2 #HV #IHV #d2 #Hd12 elim (ylt_split i d2)
- [ lapply (lleq_fwd_length … HV) #HK12 #Hid2
- lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
- normalize in ⊢ (%→%→?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/
- | /3 width=8 by lleq_lref, yle_trans/
- ]
-]
-qed-.
-
-lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[V, 0] L2 → L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
- L1 ⋕[ⓑ{a,I}V.T, 0] L2.
-/3 width=3 by lleq_ge, lleq_bind/ qed.
-
-lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ⋕[T, 0] L2.ⓑ{I2}V2 →
- ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ⋕[T, 1] L2.ⓑ{J2}W2.
-#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge … HT 1 ?) // -HT
-#HT @(lleq_lsuby_repl … HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *)
-qed-.
-
-lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
- ∀J,W. L1 ⋕[W, 0] L2 → L1.ⓑ{J}W ⋕[T, 0] L2.ⓑ{J}W.
-/3 width=7 by lleq_bind_repl_SO, lleq_inv_S/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ⋕[ⓑ{a,I}V.T, 0] L2 → ⊥) →
- (L1 ⋕[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V → ⊥).
-#a #I #L1 #L2 #V #T #H elim (lleq_dec V L1 L2 0)
-/4 width=1 by lleq_bind_O, or_intror, or_introl/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fqus_alt.ma".
-include "basic_2/substitution/lleq_ext.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Properties on supclosure and derivatives *********************************)
-
-lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃ ⦃G2, K2, U⦄ →
- ∀L1. L1 ⋕[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊃ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
-[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
- #I1 #K1 #H1 #H2 lapply (ldrop_inv_O2 … H1) -H1
- #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
-| * [ #a ] #I #G #L2 #V #T #L1 #H
- [ elim (lleq_inv_bind … H)
- | elim (lleq_inv_flat … H)
- ] -H
- /2 width=3 by fqu_pair_sn, ex2_intro/
-| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
- #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
-| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
- /2 width=3 by fqu_flat_dx, ex2_intro/
-| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
- elim (ldrop_O1_le (e+1) L1)
- [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
- | lapply (ldrop_fwd_length_le2 … HLK2) -K2
- lapply (lleq_fwd_length … HL12) -T -U //
- ]
-]
-qed-.
-
-lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃⸮ ⦃G2, K2, U⦄ →
- ∀L1. L1 ⋕[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊃⸮ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
-[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
-| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃+ ⦃G2, K2, U⦄ →
- ∀L1. L1 ⋕[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊃+ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
-[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
- /3 width=3 by fqu_fqup, ex2_intro/
-| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
- #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
- /3 width=5 by fqup_strap1, ex2_intro/
-]
-qed-.
-
-lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃* ⦃G2, K2, U⦄ →
- ∀L1. L1 ⋕[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊃* ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
-[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
-| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_lift.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ⋕[#i, d] L2.
-#L1 #L2 #d #i #Hid #HL12 @conj // -HL12
-#U @conj #H elim (cpys_inv_lref1 … H) -H // *
-#I #Z #Y #X #H elim (ylt_yle_false … Hid … H)
-qed.
-
-lemma lleq_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[#i, d] L2.
-#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ]
-[ lapply (ldrop_fwd_length … HLK1) -HLK1 #H1
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H2
- >H1 >H2 -H1 -H2 normalize //
-| #U @conj #H elim (cpys_inv_lref1 … H) -H // *
- >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU
- [ letin HLK ≝ HLK1 | letin HLK ≝ HLK2 ]
- lapply (ldrop_mono … H … HLK) -H #H destruct elim (IH W)
- /3 width=7 by cpys_subst_Y2/
-]
-qed.
-
-lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ⋕[#i, d] L2.
-#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12
-#U @conj #H elim (cpys_inv_lref1 … H) -H // *
-#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 … H) -H
-#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
-qed.
-
-(* Properties on relocation *************************************************)
-
-lemma lleq_lift_le: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → dt ≤ d → L1 ⋕[U, dt] L2.
-#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd
-lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
-#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ]
-elim (cpys_inv_lift1_be … HU0 … HLKA … HTU) // -HU0 >yminus_Y_inj #T0 #HT0 #HTU0
-elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=12 by cpys_lift_be/
-qed-.
-
-lemma lleq_lift_ge: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → d ≤ dt → L1 ⋕[U, dt+e] L2.
-#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt
-lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
-#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ]
-elim (cpys_inv_lift1_ge … HU0 … HLKA … HTU) /2 width=1 by monotonic_yle_plus_dx/ -HU0 >yplus_minus_inj #T0 #HT0 #HTU0
-elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=10 by cpys_lift_ge/
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma lleq_inv_lift_le: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → dt ≤ d → K1 ⋕[T, dt] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd
-lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
-lapply (cpys_lift_be … HT0 … HLKA … HTU … HTU0) // -HT0
->yplus_Y1 #HU0 elim (cpys_inv_lift1_be … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdtd
-#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
-qed-.
-
-lemma lleq_inv_lift_ge: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ dt → K1 ⋕[T, dt-e] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt
-lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-elim (yle_inv_plus_inj2 … Hdedt) #Hddt #Hedt
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
-lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0 -Hddt
->ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdedt -Hedt
-#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
-qed-.
-
-lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ⋕[T, d] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde
-lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
-lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0
-#HU0 lapply (cpys_weak … HU0 dt (∞) ? ?) // -HU0
-#HU0 lapply (H0 HU0)
-#HU0 lapply (cpys_weak … HU0 d (∞) ? ?) // -HU0
-#HU0 elim (cpys_inv_lift1_ge_up … HU0 … HLKB … HTU) // -L1 -L2 -U -Hddt -Hdtde
-#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_cpys.ma".
-include "basic_2/substitution/lleq_ldrop.ma".
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_lref: ∀L1,L2. ∀d:ynat. ∀i:nat. L1 ⋕[#i, d] L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I1,I2,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V &
- ⇩[i] L2 ≡ K2.ⓑ{I2}V &
- K1 ⋕[V, yinj 0] K2 & d ≤ yinj i.
-#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
-elim (ylt_split i d) /2 width=1 by or3_intro1/ #Hdi #Hi
-elim (ldrop_O1_lt … Hi) #I1 #K1 #V1 #HLK1
-elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
-lapply (ldrop_fwd_length_minus2 … HLK2) #H
-lapply (ldrop_fwd_length_minus2 … HLK1) >HL12 <H -HL12 -H
-#H lapply (injective_plus_l … H) -H #HK12
-elim (lift_total V1 0 (i+1)) #W1 #HVW1
-elim (lift_total V2 0 (i+1)) #W2 #HVW2
-elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
-elim (cpys_inv_lref1_ldrop … (H1 ?) … HLK2 … HVW1) -H1
-[ elim (cpys_inv_lref1_ldrop … (H2 ?) … HLK1 … HVW2) -H2 ]
-/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
-lapply (cpys_antisym_eq … H12 … H21) -H12 -H21 #H destruct
-@or3_intro2 @(ex4_5_intro … HLK1 HLK2) // @conj // -HK12
-#V elim (lift_total V 0 (i+1))
-#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
-[ elim (cpys_inv_lref1_ldrop … (H12 ?) … HLK2 … HVW) -H12 -H21
-| elim (cpys_inv_lref1_ldrop … (H21 ?) … HLK1 … HVW) -H21 -H12
-] [1,3: >yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/
-qed-.
-
-lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
- ∀I2,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I2}V →
- i < d ∨
- ∃∃I1,K1. ⇩[i] L1 ≡ K1.ⓑ{I1}V & K1 ⋕[V, 0] K2 & d ≤ i.
-#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref … H) -H [ * || * ]
-[ #_ #H elim (lt_refl_false i)
- lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
- /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I1 #I2 #K11 #K22 #V0 #HLK11 #HLK22 #HV0 #Hdi lapply (ldrop_mono … HLK22 … HLK2) -L2
- #H destruct /3 width=5 by ex3_2_intro, or_intror/
-]
-qed-.
-
-lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
- ∀I1,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V →
- i < d ∨
- ∃∃I2,K2. ⇩[i] L2 ≡ K2.ⓑ{I2}V & K1 ⋕[V, 0] K2 & d ≤ i.
-#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 … d … HLK1) -HLK1
-[2: * ] /4 width=6 by lleq_sym, ex3_2_intro, or_introl, or_intror/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lleq_inv_lref_ge_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
- ∀I2,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I2}V →
- ∃∃I1,K1. ⇩[i] L1 ≡ K1.ⓑ{I1}V & K1 ⋕[V, 0] K2.
-#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx … H … HLK2) -L2
-[ #H elim (ylt_yle_false … H Hdi)
-| * /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lleq_inv_lref_ge_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
- ∀I1,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V →
- ∃∃I2,K2. ⇩[i] L2 ≡ K2.ⓑ{I2}V & K1 ⋕[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 … Hdi … HLK1) -Hdi -HLK1
-/3 width=4 by lleq_sym, ex2_2_intro/
-qed-.
-
-lemma lleq_inv_lref_ge_gen: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- V1 = V2 ∧ K1 ⋕[V2, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
-elim (lleq_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
-#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by conj/
-qed-.
-
-lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
- ∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
-elim (lleq_inv_lref_ge_gen … HL12 … HLK1 HLK2) //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ⋕[T, d] L2).
-#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_sort/
-| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
- [ #HL12 #d elim (ylt_split i d) /3 width=1 by lleq_skip, or_introl/
- #Hdi elim (lt_or_ge i (|L1|)) #HiL1
- elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, lleq_free/
- elim (ldrop_O1_lt … HiL2) #I2 #K2 #V2 #HLK2
- elim (ldrop_O1_lt … HiL1) #I1 #K1 #V1 #HLK1
- elim (eq_term_dec V2 V1)
- [ #H3 elim (IH K1 V1 … K2 0) destruct
- /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/
- ]
- -IH #H3 @or_intror
- #H elim (lleq_fwd_lref … H) -H [1,3,4,6: * ]
- [1,3: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
- |5,6: /2 width=4 by ylt_yle_false/
- ]
- #Z1 #Z2 #Y1 #Y2 #X #HLY1 #HLY2 #HX #_
- lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
- lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
- #H2 #H1 destruct /2 width=1 by/
- ]
-| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_gref/
-| #a #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V … L2 d) /2 width=1 by/
- elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/
- #H1 #H2 @or_intror
- #H elim (lleq_inv_bind … H) -H /2 width=1 by/
-| #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V … L2 d) /2 width=1 by/
- elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, lleq_flat/
- #H1 #H2 @or_intror
- #H elim (lleq_inv_flat … H) -H /2 width=1 by/
-]
--n /4 width=3 by lleq_fwd_length, or_intror/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lleq_trans: ∀d,T. Transitive … (lleq d T).
-#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/
-qed-.
-
-theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ⋕[d, T] L1→ L ⋕[d, T] L2 → L1 ⋕[d, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-theorem lleq_canc_dx: ∀L1,L2,L,T,d. L1 ⋕[d, T] L → L2 ⋕[d, T] L → L1 ⋕[d, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,d. (L1 ⋕[ⓑ{a,I}V.T, d] L2 → ⊥) →
- (L1 ⋕[V, d] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V → ⊥).
-#a #I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
-/4 width=1 by lleq_bind, or_intror, or_introl/
-qed-.
-
-lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ⋕[ⓕ{I}V.T, d] L2 → ⊥) →
- (L1 ⋕[V, d] L2 → ⊥) ∨ (L1 ⋕[T, d] L2 → ⊥).
-#I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
-/4 width=1 by lleq_flat, or_intror, or_introl/
-qed-.
-
-(* Note: lleq_nlleq_trans: ∀d,T,L1,L. L1⋕[T, d] L →
- ∀L2. (L ⋕[T, d] L2 → ⊥) → (L1 ⋕[T, d] L2 → ⊥).
-/3 width=3 by lleq_canc_sn/ qed-.
-works with /4 width=8/ so lleq_canc_sn is more convenient
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⧣ break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyNegatedEq $T $d $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⧣ ⧣ break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyNegatedEqAlt $T $d $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazynegatedeq_4.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)
-
-definition llneq: relation4 ynat term lenv lenv ≝
- λd,T,L1,L2. |L1| = |L2| ∧ (L1 ⋕[T, d] L2 → ⊥).
-
-interpretation
- "negated lazy equivalence (local environment)"
- 'LazyNegatedEq T d L1 L2 = (llneq d T L1 L2).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazynegatedeqalt_4.ma".
-include "basic_2/substitution/lleq_lleq.ma".
-include "basic_2/substitution/llneq.ma".
-
-(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)
-
-(* alternative definition of llneq *)
-inductive llneqa: relation4 ynat term lenv lenv ≝
-| llneqa_neq: ∀I1,I2,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i]L1 ≡ K1.ⓑ{I1}V1 → ⇩[i]L2 ≡ K2.ⓑ{I2}V2 →
- |K1| = |K2| → (V1 = V2 → ⊥) → llneqa d (#i) L1 L2
-| llneqa_eq : ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i]L1 ≡ K1.ⓑ{I1}V → ⇩[i]L2 ≡ K2.ⓑ{I2}V →
- llneqa 0 (V) K1 K2 → llneqa d (#i) L1 L2
-| llneqa_bind_sn: ∀a,I,L1,L2,V,T,d.
- llneqa d V L1 L2 → llneqa d (ⓑ{a,I}V.T) L1 L2
-| llneqa_bind_dx: ∀a,I,L1,L2,V,T,d.
- llneqa (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → llneqa d (ⓑ{a,I}V.T) L1 L2
-| llneqa_flat_sn: ∀I,L1,L2,V,T,d.
- llneqa d V L1 L2 → llneqa d (ⓕ{I}V.T) L1 L2
-| llneqa_flat_dx: ∀I,L1,L2,V,T,d.
- llneqa d T L1 L2 → llneqa d (ⓕ{I}V.T) L1 L2
-.
-
-interpretation
- "negated lazy equivalence (local environment) alternative"
- 'LazyNegatedEqAlt T d L1 L2 = (llneqa d T L1 L2).
-
-(* Main properties **********************************************************)
-
-theorem llneq_llneqa: ∀T,L1,L2,d. L1 ⧣[T, d] L2 → L1 ⧣⧣[T, d] L2.
-#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 #d * #HL12 #H elim H /2 width=1 by lleq_sort/
-| #i #Hn #L2 #d * #HL12 #H elim (ylt_split i d) #Hdi
- [ elim H /2 width=1 by lleq_skip/ ]
- elim (lt_or_ge i (|L1|)) #HiL1
- [2: elim H /3 width=3 by lleq_free, le_repl_sn_aux/ ]
- elim (ldrop_O1_lt … HiL1) #I1 #K1 #V1 #HLK1
- elim (ldrop_O1_lt L2 i) /2 width=1 by/ #I2 #K2 #V2 #HLK2
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) normalize
- elim (eq_term_dec V1 V2) #HnV12 destruct
- [2: #H @(llneqa_neq … HLK1 … HLK2) /2 width=1 by/ ] (**) (* explicit constructor *)
- elim (lleq_dec V2 K1 K2 0) #HnV2 [ elim H /2 width=8 by lleq_lref/ ]
- #H @(llneqa_eq … HLK1 … HLK2) /4 width=2 by ldrop_fwd_rfw, conj/ (**) (* explicit constructor *)
-| #p #Hn #L2 #d * #HL12 #H elim H /2 width=1 by lleq_gref/
-| #a #I #V #T #Hn #L2 #d * #HL12 #H destruct elim (nlleq_inv_bind … H) -H
- [ /5 width=1 by llneqa_bind_sn, conj/
- | #H @llneqa_bind_dx @IH // @conj normalize /2 width=1 by/
- ]
-| #I #V #T #Hn #L2 #d * #HL12 #H destruct elim (nlleq_inv_flat … H) -H
- /5 width=1 by llneqa_flat_dx, llneqa_flat_sn, conj/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/llneq_alt.ma".
-
-(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma llneq_inv_atom1: ∀L1,L2,T,d. L1 ⧣⧣[T, d] L2 → |L1| ≤ d → ⊥.
-#L1 #L2 #T #d #H elim H -L1 -L2 -T -d /2 width=1 by/
-[ #I1 #I2 #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #_ #_ #_
- >(ldrop_fwd_length … HLK1) -HLK1 normalize
- #H lapply (yle_trans … H … Hdi) -d
- /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #_ #_ #_
- >(ldrop_fwd_length … HLK1) -HLK1 normalize
- #H lapply (yle_trans … H … Hdi) -d
- /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/
-| #a #I #L1 <yplus_inj /4 width=1 by yle_succ/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/llneq_alt.ma".
-include "basic_2/computation/lpxs_ldrop.ma".
-include "basic_2/computation/lpxs_cpxs.ma".
-
-(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
-
-(* Forward lemmas on negated lazy equivalence for local environments ********)
-
-lemma lpxs_llneq_fwd_cpxs: ∀h,g,G,L1,L2,T,d. L1 ⧣⧣[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- ∃∃T1,T2. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] T1 & ⦃G, L2⦄ ⊢ T ▶*[d, ∞] T2 &
- ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & (T1 = T2 → ⊥).
-#h #g #G #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-[ #I1 #I2 #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #_ #HnV12 #HL12
- elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HY
- elim (lpxs_inv_pair1 … H) -H #Z #X #_ #HV12 #H destruct
- lapply (ldrop_mono … HY … HLK2) -HY #H destruct
- lapply (ldrop_fwd_drop2 … HLK1) #H2LK1
- elim (lift_total V1 0 (i+1)) #T1 #HVT1
- elim (lift_total V2 0 (i+1)) #T2 #HVT2
- @(ex4_2_intro … T1 T2) /3 width=10 by cpxs_lift, cpys_subst, lift_inj/ (**) (* explicit constructor *)
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #_ #IHK12 #HL12
- elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HY
- elim (lpxs_inv_pair1 … H) -H #Z #X #HK12 #_ #H destruct
- lapply (ldrop_mono … HY … HLK2) -HY #H destruct
- elim (IHK12 HK12) -IHK12 -HK12 #V1 #V2 #HV1 #HV2 #HV12 #HnV12
- lapply (ldrop_fwd_drop2 … HLK1) #H2LK1
- elim (lift_total V1 0 (i+1)) #T1 #HVT1
- elim (lift_total V2 0 (i+1)) #T2 #HVT2
- @(ex4_2_intro … T1 T2) /3 width=10 by cpxs_lift, cpys_subst_Y2, lift_inj/ (**) (* explicit constructor *)
-| #a #I #L1 #L2 #V #T #d #_ #IHV #HL12 elim (IHV HL12) -IHV -HL12
- #V1 #V2 #HV1 #HV2 #HV12 #HnV12
- @(ex4_2_intro … (ⓑ{a,I}V1.T) (ⓑ{a,I}V2.T)) /2 width=1 by cpys_bind, cpxs_pair_sn/
- #H destruct /2 width=1 by/
-| #a #I #L1 #L2 #V #T #d #_ #IHT #HL12 elim (IHT ?) /2 width=1 by lpxs_pair_refl/ -IHT -HL12
- #T1 #T2 #HT1 #HT2 #HT12 #HnT12
- @(ex4_2_intro … (ⓑ{a,I}V.T1) (ⓑ{a,I}V.T2)) /2 width=1 by cpys_bind, cpxs_bind_dx/
- #H destruct /2 width=1 by/
-| #I #L1 #L2 #V #T #d #_ #IHV #HL12 elim (IHV HL12) -IHV -HL12
- #V1 #V2 #HV1 #HV2 #HV12 #HnV12
- @(ex4_2_intro … (ⓕ{I}V1.T) (ⓕ{I}V2.T)) /2 width=1 by cpys_flat, cpxs_pair_sn/
- #H destruct /2 width=1 by/
-| #I #L1 #L2 #V #T #d #_ #IHT #HL12 elim (IHT HL12) -IHT -HL12
- #T1 #T2 #HT1 #HT2 #HT12 #HnT12
- @(ex4_2_intro … (ⓕ{I}V.T1) (ⓕ{I}V.T2)) /2 width=1 by cpys_flat, cpxs_flat_dx/
- #H destruct /2 width=1 by/
-]
-qed-.
-
-lemma lpxs_nlleq_fwd_cpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- (L1 ⋕[T, d] L2 → ⊥) →
- ∃∃T1,T2. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] T1 & ⦃G, L2⦄ ⊢ T ▶*[d, ∞] T2 &
- ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & (T1 = T2 → ⊥).
-/5 width=4 by lpxs_llneq_fwd_cpxs, lpxs_fwd_length, llneq_llneqa, conj/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/llprs_cprs.ma".
-include "basic_2/conversion/cpc_cpc.ma".
-include "basic_2/equivalence/cpcs_cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
-#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
-[ /3 width=3 by ex2_intro/
-| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
- [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
- | /3 width=5 by cprs_strap2, ex2_intro/
- ]
-]
-qed-.
-
-(* Basic_1: was: pc3_gen_sort *)
-lemma cpcs_inv_sort: ∀G,L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
-#G #L #k1 #k2 #H elim (cpcs_inv_cprs … H) -H
-#T #H1 >(cprs_inv_sort1 … H1) -T #H2
-lapply (cprs_inv_sort1 … H2) -L #H destruct //
-qed-.
-
-lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
-#a #G #L #W1 #T1 #T #H
-elim (cpcs_inv_cprs … H) -H #X #H1 #H2
-elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
-/3 width=6 by cprs_bind, ex2_2_intro/
-qed-.
-
-lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
-/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
-
-(* Basic_1: was: pc3_gen_sort_abst *)
-lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
-#a #G #L #W #T #k #H
-elim (cpcs_inv_cprs … H) -H #X #H1
->(cprs_inv_sort1 … H1) -X #H2
-elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: pc3_gen_lift *)
-lemma cpcs_inv_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
- ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
-#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
-elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
-elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
-elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
->(lift_inj … HXU … HTU) -X -U -d -e /2 width=3 by cprs_div/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llpr_cpcs_trans: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
- ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
-/4 width=5 by cprs_div, cprs_llpr_trans/
-qed-.
-
-lemma llprs_cpcs_trans: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡*[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡*[T2, 0] L2 →
- ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
-/4 width=5 by cprs_div, llprs_cprs_trans/
-qed-.
-
-lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
-/2 width=3 by cpr_cprs_div/
-qed-.
-
-lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
-#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
-/2 width=3 by cprs_cpr_div/
-qed-.
-
-lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
-/2 width=3 by cprs_div/
-qed-.
-
-(* Basic_1: was: pc3_wcpr0_t *)
-(* Basic_1: note: pc3_wcpr0_t should be renamed *)
-lemma llpr_cprs_conf: ∀G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #T1 #HL12 #T2 #HT12 elim (cprs_llpr_conf_dx … HT12 … HL12) -L1
-/2 width=3 by cprs_div/
-qed-.
-
-(* Basic_1: was only: pc3_pr0_pr2_t *)
-(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
-lemma llpr_cpr_conf: ∀G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-/3 width=5 by llpr_cprs_conf, cpr_cprs/ qed-.
-
-(* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
-#G #L #V1 #V2 #HV12 #T1 #T2 #HT12
-elim (cpcs_inv_cprs … HV12) -HV12
-elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_flat, cprs_div/
-qed.
-
-lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
-/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
-
-lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
-#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_div, cprs_bind/
-qed.
-
-lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
-#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
-/3 width=5 by cprs_div, cprs_bind/
-qed.
-
-lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
- ∀L2. L2 ⊑ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_div, lsubr_cprs_trans/
-qed-.
-
-(* Basic_1: was: pc3_lift *)
-lemma cpcs_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
- ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
-#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
-elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
-elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/
-qed.
-
-lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
- ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
-#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
-
-(* More inversion lemmas ****************************************************)
-
-axiom cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
-(*
-#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
-elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
-lapply (llprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by llprs_pair/ -HT2 #HT2
-lapply (llprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by llprs_pair/ -HT2 #HT2
-/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
-qed-.
-*)
-lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
-#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
-qed-.
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was pc3_t *)
-theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
-
-theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
-
-theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
-
-lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
- ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
-/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
-
-lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
- ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
-/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
-
-(* Basic_1: was: pc3_wcpr0 *)
-lemma llpr_cpcs_conf: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
- ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
-/3 width=5 by cpcs_canc_dx, llpr_cprs_conf/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/llpr_llpr.ma".
-include "basic_2/computation/cprs_lift.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: pr3_t *)
-(* Basic_1: includes: pr1_t *)
-theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
-normalize /2 width=3 by trans_TC/ qed-.
-
-(* Basic_1: was: pr3_confluence *)
-(* Basic_1: includes: pr1_confluence *)
-theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
-normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
-
-theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
-/3 width=5 by cprs_trans, cprs_bind_dx/
-qed.
-
-(* Basic_1: was: pr3_flat *)
-theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
-#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
-/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
-qed.
-
-theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
-#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
-@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
-qed.
-
-theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
-#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
-qed.
-
-theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2
-/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
-qed.
-
-theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
-/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was pr3_gen_appl *)
-lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
- U2 = ⓐV2. T2
- | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
- ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
- | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
- ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
- ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
-#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
-#U #U2 #_ #HU2 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_appl1 … HU2) -HU2 *
- [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
- | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
- lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
- lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
- /5 width=5 by cprs_flat_dx, cpr_cprs, cprs_bind, lsubr_abst, ex2_3_intro, or3_intro1/
- | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
- @or3_intro2 @(ex4_5_intro … HV2 HT10) /3 width=3 by cprs_flat_sn, cprs_strap1, cpr_cprs, cprs_bind/ (**) (* full auto is too slow 11s *)
- ]
-| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
-| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
-]
-qed-.
-
-(* Properties concerning sn parallel reduction on local environments ********)
-
-(* Basic_1: was just: pr3_pr2_pr2_t *)
-(* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma llpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (llpr G 0).
-#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3 by/
-| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (llpr_inv_lref_ge_dx … HL12 … HLK2) -L2
- /5 width=7 by cprs_delta, cprs_strap2, llpr_cpr_conf/
-| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpr_inv_bind_O … HL12) -HL12 /4 width=1 by cprs_bind/
-| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 /3 width=1 by cprs_flat/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_zeta/
-| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) /3 width=1 by cprs_tau/
-| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_beta/
-| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_theta/
-]
-qed-.
-
-lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=9 by llpr_cpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma cprs_llpr_trans: ∀G. s_rs_transitive … (cpr G) (llpr G 0).
-/3 width=6 by llpr_cpr_trans, llpr_cpr_conf, s_r_trans_LTC1/ qed-.
-
-(* Basic_1: was: pr3_strip *)
-(* Basic_1: includes: pr1_strip *)
-lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
-normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
-
-lemma cprs_llpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #H @(cprs_ind_dx … T0 H) -T0 /2 width=3 by ex2_intro/
-#T0 #T #HT0 #_ #IHT1 #L1 #HL01
-elim (IHT1 … L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2
-elim (llpr_cpr_conf_dx … HT0 … HL01) -L0 #T3 #HT03 #HT3
-elim (cprs_strip … HT2 … HT3) -T
-/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
-qed-.
-
-lemma cprs_llpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cprs_llpr_conf_dx … HT01 … HL01)
-/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/
-qed-.
-
-lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/llpr_ldrop.ma".
-include "basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Advanced properties ******************************************************)
-
-(* Note: apparently this was missing in basic_1 *)
-lemma cprs_delta: ∀G,L,K,V,V2,i.
- ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡* V2 →
- ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡* W2.
-#G #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=6 by cpr_cprs, cpr_delta/ ]
-#V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
-lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
-elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/
-qed.
-
-lemma cprs_llpr_conf: ∀G. s_r_confluent1 … (cprs G) (llpr G 0).
-/3 width=5 by llpr_cpr_conf, s_r_conf1_LTC1/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was: pr3_gen_lref *)
-lemma cprs_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡* T2 →
- T2 = #i ∨
- ∃∃K,V1,T1. ⇩[i] L ≡ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡* T1 &
- ⇧[0, i + 1] T1 ≡ T2.
-#G #L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1 by or_introl/
-#T #T2 #_ #HT2 *
-[ #H destruct
- elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
- * /4 width=6 by cpr_cprs, ex3_3_intro, or_intror/
-| * #K #V1 #T1 #HLK #HVT1 #HT1
- lapply (ldrop_fwd_drop2 … HLK) #H0LK
- elim (cpr_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
- /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
-]
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr3_lift *)
-lemma cprs_lift: ∀G. l_liftable (cprs G).
-/3 width=10 by l_liftable_LTC, cpr_lift/ qed.
-
-(* Basic_1: was: pr3_gen_lift *)
-lemma cprs_inv_lift1: ∀G. l_deliftable_sn (cprs G).
-/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡ break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyPRedSn $G $L1 $L2 $T $d }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡* break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyPRedSnStar $G $L1 $L2 $T $d }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazypredsn_5.ma".
-include "basic_2/relocation/llpx_sn.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
-
-definition llpr: genv → relation4 ynat term lenv lenv ≝ λG. llpx_sn (cpr G).
-
-interpretation "lazy parallel reduction (local environment, sn variant)"
- 'LazyPRedSn G L1 L2 T d = (llpr G d T L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma llpr_inv_flat: ∀I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[T, d] L2.
-/2 width=2 by llpx_sn_inv_flat/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpr_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → |L1| = |L2|.
-/2 width=4 by llpx_sn_fwd_length/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpr_lref: ∀I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 → ⦃G, L1⦄ ⊢ ➡[#i, d] L2.
-/2 width=9 by llpx_sn_lref/ qed.
-
-(* Note: lemma 250 *)
-lemma llpr_refl: ∀G,T,d. reflexive … (llpr G d T).
-/2 width=1 by llpx_sn_refl/ qed.
-
-(* Basic_1: removed theorems 5: wcpr0_gen_sort wcpr0_gen_head
- wcpr0_getl wcpr0_getl_back
- pr0_subst1_back
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/fquq_alt.ma".
-include "basic_2/static/ssta_llpx_sn.ma".
-include "basic_2/reduction/cpr_lift.ma".
-include "basic_2/reduction/cpr_llpx_sn.ma".
-include "basic_2/reduction/llpr.ma".
-
-(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma llpr_inv_lref_ge_dx: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
-/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
-
-lemma llpr_inv_lref_ge_sn: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
-/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
-
-lemma llpr_inv_lref_ge_bi: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
-/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
-
-lemma llpr_inv_bind_O: ∀a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [ⓑ{a,I}V.T, 0] L2 →
- ⦃G, L1⦄ ⊢ ➡[V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind_O/ qed-.
-
-lemma llpr_bind_repl_O: ∀I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V2 →
- ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡ W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[T, 0] L2.ⓑ{J}W2.
-/2 width=4 by llpx_sn_bind_repl_O/ qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llpr_ssta_conf: ∀h,g,G. s_r_confluent1 … (ssta h g G) (llpr G 0).
-/3 width=10 by ssta_llpx_sn_conf, cpr_lift/ qed-.
-
-lemma llpr_cpr_conf: ∀G. s_r_confluent1 … (cpr G) (llpr G 0).
-/3 width=10 by cpr_llpx_sn_conf, cpr_inv_lift1, cpr_lift/ qed-.
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=10 by llpr_lref, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
-qed-.
-
-lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=10 by llpr_lref, cpr_pair_sn, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
-qed-.
-
-lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/llpx_sn_llpx_sn.ma".
-include "basic_2/substitution/fqup.ma".
-include "basic_2/reduction/llpr_ldrop.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Main properties on context-sensitive parallel reduction for terms ********)
-
-fact cpr_conf_llpr_atom_atom:
- ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
-/2 width=3 by cpr_atom, ex2_intro/ qed-.
-
-fact cpr_conf_llpr_atom_delta:
- ∀G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
- ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01
-elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
-lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1))
-/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
-qed-.
-
-(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
-fact cpr_conf_llpr_delta_delta:
- ∀G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
- ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
- ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_
-lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1
-elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
-lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_bind_bind:
- ∀a,I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
-#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
-/3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_bind_zeta:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
-#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_zeta_zeta:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
-#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_flat_flat:
- ∀I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
-#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) //
-elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_flat_tau:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
-elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_tau_tau:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
-elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_flat_beta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
-/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
-qed-.
-
-(* Basic-1: includes:
- pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
- pr0_cong_upsilon_cong pr0_cong_upsilon_delta
-*)
-fact cpr_conf_llpr_flat_theta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2
-elim (cpr_inv_abbr1 … H) -H *
-[ #W1 #T1 #HW01 #HT01 #H destruct
- elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
- elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
- /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
-| #T1 #HT01 #HXT1 #H destruct
- elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
- elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
- /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/
-]
-qed-.
-
-fact cpr_conf_llpr_beta_beta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
-/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-(* Basic_1: includes: pr0_upsilon_upsilon *)
-fact cpr_conf_llpr_theta_theta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
- ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
-elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/
-/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-theorem cpr_conf_llpr: ∀G. llpx_sn_confluent2 (cpr G) (cpr G).
-#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
-[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_atom1 … H1) -H1
- elim (cpr_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpr_conf_llpr_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpr_conf_llpr_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpr_conf_llpr_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpr_conf_llpr_delta_delta/
- ]
-| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpr_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpr_conf_llpr_bind_bind/
- | /4 width=11 by ex2_commute, cpr_conf_llpr_bind_zeta/
- | /3 width=11 by cpr_conf_llpr_bind_zeta/
- | /3 width=12 by cpr_conf_llpr_zeta_zeta/
- ]
-| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
- | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
- ]
- elim (cpr_inv_flat1 … H2) -H2 *
- [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
- |2,6,10,14: #HX2 #H2
- |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
- |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
- ] destruct
- [ /3 width=10 by cpr_conf_llpr_flat_flat/
- | /4 width=8 by ex2_commute, cpr_conf_llpr_flat_tau/
- | /4 width=12 by ex2_commute, cpr_conf_llpr_flat_beta/
- | /4 width=14 by ex2_commute, cpr_conf_llpr_flat_theta/
- | /3 width=8 by cpr_conf_llpr_flat_tau/
- | /3 width=8 by cpr_conf_llpr_tau_tau/
- | /3 width=12 by cpr_conf_llpr_flat_beta/
- | /3 width=13 by cpr_conf_llpr_beta_beta/
- | /3 width=14 by cpr_conf_llpr_flat_theta/
- | /3 width=17 by cpr_conf_llpr_theta_theta/
- ]
-]
-qed-.
-
-(* Basic_1: includes: pr0_confluence pr2_confluence *)
-theorem cpr_conf: ∀G,L. confluent … (cpr G L).
-/2 width=6 by cpr_conf_llpr/ qed-.
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma llpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_llpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
-qed-.
-
-lemma llpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_llpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazypredsnstar_5.ma".
-include "basic_2/reduction/llpr.ma".
-
-(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
-
-definition llprs: genv → relation4 ynat term lenv lenv ≝
- λG,d. LTC … (llpr G d).
-
-interpretation "lazy parallel computation (local environment, sn variant)"
- 'LazyPRedSnStar G L1 L2 T d = (llprs G d T L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma llprs_ind: ∀G,L1,T,d. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L2.
-#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma llprs_ind_dx: ∀G,L2,T,d. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L1.
-#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpr_llprs: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
-/2 width=1 by inj/ qed.
-
-lemma llprs_refl: ∀G,L,T,d. ⦃G, L⦄ ⊢ ➡*[T, d] L.
-/2 width=1 by lpr_llprs/ qed.
-
-lemma llprs_strap1: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
-normalize /2 width=3 by step/ qed-.
-
-lemma llprs_strap2: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
-normalize /2 width=3 by TC_strap/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llprs_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → |L1| = |L2|.
-#G #L1 #L2 #T #d #H @(llprs_ind … H) -L2
-/3 width=6 by llpr_fwd_length, trans_eq/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/llpx_sn_tc.ma".
-include "basic_2/computation/cprs_cprs.ma".
-include "basic_2/computation/llprs.ma".
-
-(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Advanced properties ******************************************************)
-
-lemma llprs_pair_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡*[T, 0] L.ⓑ{I}V2.
-/2 width=1 by llpx_sn_TC_pair_dx/ qed.
-
-(* Properties on context-sensitive parallel computation for terms ***********)
-
-lemma llprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (llprs G 0).
-/3 width=5 by cprs_llpr_trans, s_r_trans_LTC2/ qed-.
-
-(* Basic_1: was just: pr3_pr3_pr3_t *)
-lemma llprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (llprs G 0).
-#G @s_r_to_s_rs_trans @s_r_trans_LTC2
-/3 width=5 by cprs_llpr_trans, s_rs_trans_TC1/ (**) (* full auto too slow *)
-qed-.
-
-(* Note: this is an instance of a general theorem *)
-lemma llprs_cprs_conf_dx: ∀G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡* U2 →
- ∀L0. ⦃G2, L0⦄ ⊢ ➡*[T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[U2, O] L2.
-#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx … H) -L0 //
-#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 … IHL2) -IHL2
-lapply (llprs_cprs_trans … HTU2 … HL2) -L2 #HTU2
-/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/
-qed-.
-
-(* Note: this is an instance of a general theorem *)
-lemma llprs_cpr_conf_dx: ∀G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∀L0. ⦃G2, L0⦄ ⊢ ➡*[T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[U2, O] L2.
-#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx … H) -L0 //
-#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 … IHL2) -IHL2
-lapply (llprs_cpr_trans … HTU2 … HL2) -L2 #HTU2
-/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/
-qed-.
-
-lemma llprs_cprs_conf_sn: ∀G,L0,L1,T0. ⦃G, L0⦄ ⊢ ➡*[T0, 0] L1 →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #L1 #T0 #H @(llprs_ind_dx … H) -L0 /2 width=3 by ex2_intro/
-#L0 #L #HL0 #HL1 #IHL1 #T1 #HT01 elim (cprs_llpr_conf_sn … HT01 … HL0)
-#T2 #HT12 #HT02 elim (IHL1 … HT02) -IHL1 -HT02
-lapply (cprs_trans … HT01 … HT12) #HT02
-lapply (cprs_llpr_conf … HT02 … HL0) -HT02 -HL0
-/4 width=5 by cprs_llpr_trans, cprs_trans, ex2_intro/
-qed-.
-
-lemma llprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[T0, 0] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-/3 width=3 by llprs_cprs_conf_sn, cpr_cprs/ qed-.
-
-lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=3 by llprs_cprs_trans, llprs_pair_dx, cprs_bind/ qed-.
-
-(* Inversion lemmas on context-sensitive parallel computation for terms *****)
-
-(* Basic_1: was: pr3_gen_abst *)
-lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓛ{a}W2.T2.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
-#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
-elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (llprs_cpr_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro/
-qed-.
-
-lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
- ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
-#a #G #L #W1 #W2 #T1 #T2 #H
-elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
-qed-.
-
-(* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓓ{a}V2.T2
- ) ∨
- ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
-#U0 #U2 #_ #HU02 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_abbr1 … HU02) -HU02 *
- [ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?)
- /4 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
- | #T2 #HT02 #HUT2
- lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
- /4 width=3 by llprs_pair_dx, cprs_trans, ex3_intro, or_intror/
- ]
-| #U1 #HTU1 #HU01
- elim (lift_total U2 0 1) #U #HU2
- lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
- /4 width=3 by cprs_strap1, ldrop_drop, ex3_intro, or_intror/
-]
-qed-.
-
-(* Note: we loose lprs_cprs_conf_dx and derivatives:
- lprs_cpr_conf_dx lprs_cprs_conf
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/llprs.ma".
-
-(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Main properties **********************************************************)
-
-theorem llprs_trans: ∀G,T,d. Transitive … (llprs G d T).
-normalize /2 width=3 by trans_TC/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-definition llpx_sn_confluent2: relation (lenv→relation term) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. llpx_sn R1 0 T0 L0 L1 → ∀L2. llpx_sn R2 0 T0 L0 L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-(* Note: we miss llpx_sn_conf and derivatives: lpr_conf lprs_conf *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/llpx_ldrop.ma".
-include "basic_2/computation/cpxs_cpxs.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-
-(* Properties on lazy sn reduction on local environments ********************)
-
-lemma cpxs_llpx_conf: ∀h,g,G. s_r_confluent1 … (cpxs h g G) (llpx h g G 0).
-/3 width=5 by llpx_cpx_conf, s_r_conf1_LTC1/ qed-.
-
-lemma llpx_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (llpx h g G 0).
-#h #g #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3 by/
-| /3 width=2 by cpx_cpxs, cpx_sort/
-| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (llpx_inv_lref_ge_dx … HL12 … HLK2) -L2
- /5 width=8 by cpxs_delta, cpxs_strap2, llpx_cpx_conf/
-| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpx_inv_bind_O … HL12) -HL12 /4 width=1 by cpxs_bind/
-| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpx_inv_flat … HL12) -HL12 /3 width=1 by cpxs_flat/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
- elim (llpx_inv_bind_O … HL12) /3 width=3 by cpxs_zeta/
-| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
- elim (llpx_inv_flat … HL12) /3 width=1 by cpxs_tau/
-| #G #L2 #V1 #V2 #T2 #HV12 #IHV12 #L1 #HL12
- elim (llpx_inv_flat … HL12) /3 width=1 by cpxs_ti/
-| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
- elim (llpx_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpx_inv_bind_O … HL12) /3 width=3 by cpxs_beta/
-| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
- elim (llpx_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpx_inv_bind_O … HL12) /3 width=3 by cpxs_theta/
-]
-qed-.
-
-lemma llpx_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (llpx h g G 0).
-#h #g #G @s_r_trans_LTC1 /2 width=3 by llpx_cpx_trans, llpx_cpx_conf/ (**) (* full auto fails here but works in cprs_cprs *)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cpxs_llpx.ma".
-include "basic_2/computation/csx_alt.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* Properties on lazy sn extended reduction for local environments **********)
-
-lemma csx_llpx_conf: ∀h,g,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #T #H @(csx_ind_alt … H) -T
-/5 width=3 by csx_intro_cpxs, llpx_cpxs_trans, cpxs_llpx_conf/ (* 2 cpxs_llpx_trans *)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/csx_llpx.ma".
-include "basic_2/computation/llpxs.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* Properties on lazy sn extended computation for local environments ********)
-
-lemma csx_llpxs_conf: ∀h,g,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #T #HT #L2 #H @(llpxs_ind … H) -L2 /3 by llpxs_strap1, csx_llpx_conf/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cpx_lift.ma".
-include "basic_2/reduction/llpx_lpr.ma".
-include "basic_2/computation/fpbs.ma".
-
-(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
-
-(* Properties on sn parallel reduction for local environments ***************)
-
-(* Note: this is used in the closure proof *)
-(* Note: original proof: /4 width=5 by fpbs_strap1, lpr_fpb, cpr_fpb/ *)
-(* Note: this should be moved *)
-lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
-/5 width=5 by fpbs_strap1, cpr_fpb, fpb_llpx, lpr_llpx/ qed.
-
-(* Note: this is used in the closure proof *)
-(* Note: this should be moved *)
-lemma cpr_lpr_ssta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,l2.
- ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L2⦄ ⊢ T2 ▪[h, g] l2+1 → ⦃G, L2⦄ ⊢ T2 •[h, g] U2 →
- ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
-/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, ssta_cpx, fpb_cpx/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡ break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyPRedSn $G $L1 $L2 $h $g $T $d }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡* break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyPRedSnStar $G $L1 $L2 $h $g $T $d }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( G ⊢ ⋕ ⬊ * break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L )"
- non associative with precedence 45
- for @{ 'LazySN $h $g $T $d $G $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( G ⊢ ⋕ ⬊ ⬊ * break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L )"
- non associative with precedence 45
- for @{ 'LazySNAlt $h $g $T $d $G $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazypredsn_7.ma".
-include "basic_2/relocation/llpx_sn.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
-
-definition llpx: ∀h. sd h → genv → relation4 ynat term lenv lenv ≝
- λh,g,G. llpx_sn (cpx h g G).
-
-interpretation "lazy extended parallel reduction (local environment, sn variant)"
- 'LazyPRedSn G L1 L2 h g T d = (llpx h g G d T L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma llpx_inv_flat: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
-/2 width=2 by llpx_sn_inv_flat/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpx_fwd_length: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → |L1| = |L2|.
-/2 width=4 by llpx_sn_fwd_length/ qed-.
-
-lemma llpx_fwd_flat_dx: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
-/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
-
-lemma llpx_fwd_pair_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ②{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
-/2 width=3 by llpx_sn_fwd_pair_sn/ qed-.
-
-(* Note: this might be removed *)
-lemma llpx_fwd_bind_sn: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
-/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
-
-(* Note: this might be removed *)
-lemma llpx_fwd_bind_dx: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, d] L2 →
- ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, ⫯d] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
-
-(* Note: this might be removed *)
-lemma llpx_fwd_flat_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
-/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpx_lref: ∀h,g,I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2.
-/2 width=9 by llpx_sn_lref/ qed.
-
-lemma llpx_refl: ∀h,g,G,T,d. reflexive … (llpx h g G d T).
-/2 width=1 by llpx_sn_refl/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa_lift.ma".
-include "basic_2/static/lsuba_aaa.ma".
-include "basic_2/reduction/llpx_ldrop.ma".
-
-(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-(* Note: lemma 500 *)
-lemma aaa_cpx_llpx_conf: ∀h,g,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T1, 0] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #g #L1 #k #X #H elim (cpx_inv_sort1 … H) -H // * //
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
- elim (cpx_inv_lref1 … H) -H
- [ #H destruct
- elim (llpx_inv_lref_ge_sn … HL12 … HLK1) -L1 /3 width=6 by aaa_lref/
- | * #J #Y #Z #V2 #H #HV12 #HV2
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- elim (llpx_inv_lref_ge_sn … HL12 … HLK1) -L1 /3 width=8 by aaa_lift, ldrop_fwd_drop2/
- ]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (llpx_inv_bind_O … HL12) -HL12 #HV1 #HT1
- elim (cpx_inv_abbr1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=4 by llpx_bind_repl_O, aaa_abbr/
- | #T2 #HT12 #HT2 #H destruct -IHV1 /3 width=8 by aaa_inv_lift, ldrop_drop/
- ]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (llpx_inv_bind_O … HL12) -HL12 #HV1 #HT1
- elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=4 by llpx_bind_repl_O, aaa_abst/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (llpx_inv_flat … HL12) -HL12 #HV1 #HT1
- elim (cpx_inv_appl1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
- | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
- lapply (IHV1 … HV12 … HV1) -IHV1 -HV12 #HV2
- lapply (IHT1 (ⓛ{b}W2.U2) … HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
- elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
- /5 width=6 by lsuba_aaa_trans, lsuba_abbr, aaa_abbr, aaa_cast/
- | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
- lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=2 by ldrop_drop/ #HV2
- lapply (IHT1 (ⓓ{b}W2.U2) … HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
- elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
- ]
-| #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (llpx_inv_flat … HL12) -HL12 #HV1 #HT1
- elim (cpx_inv_cast1 … H) -H
- [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by aaa_cast/
- | -IHV1 /2 width=1 by/
- | -IHT1 /2 width=1 by/
- ]
-]
-qed-.
-
-lemma aaa_cpx_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
-/2 width=7 by aaa_cpx_llpx_conf/ qed-.
-
-lemma aaa_llpx_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-/2 width=7 by aaa_cpx_llpx_conf/ qed-.
-
-lemma aaa_cpr_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
-/3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-.
-(*
-lamma aaa_llpr_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-/3 width=5 by aaa_llpx_conf, llpr_llpx/ qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cpx_llpx_sn.ma".
-include "basic_2/reduction/cpx_lift.ma".
-include "basic_2/reduction/llpx.ma".
-
-(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma llpx_inv_lref_ge_dx: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
-/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
-
-lemma llpx_inv_lref_ge_sn: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
-/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
-
-lemma llpx_inv_lref_ge_bi: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
-/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
-
-lemma llpx_inv_bind_O: ∀h,g,a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [h, g, ⓑ{a,I}V.T, 0] L2 →
- ⦃G, L1⦄ ⊢ ➡[h, g, V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind_O/ qed-.
-
-lemma llpx_bind_repl_O: ∀h,g,I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V2 →
- ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[h, g, W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{J}W2.
-/2 width=4 by llpx_sn_bind_repl_O/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma llpx_fwd_bind_O_dx: ∀h,g,a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, 0] L2 →
- ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V.
-/2 width=3 by llpx_sn_fwd_bind_O_dx/ qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llpx_cpx_conf: ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx h g G 0).
-/3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/ qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma llpx_ldrop_trans_O: ∀h,g,G,L1,L2,U. ⦃G, L1⦄ ⊢ ➡[h, g, U, 0] L2 →
- ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
- ∃∃K1. ⇩[e] L1 ≡ K1 & ⦃G, K1⦄ ⊢ ➡[h, g, T, 0] K2.
-/2 width=5 by llpx_sn_ldrop_trans_O/ qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma llpx_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=7 by llpx_fwd_bind_O_dx, llpx_fwd_pair_sn,llpx_fwd_flat_dx, ex3_2_intro/
-[ #I #G1 #L1 #V1 #K1 #H elim (llpx_inv_lref_ge_dx … H) -H [5,6: // |2,3,4: skip ]
- #Y1 #W1 #HKL1 #HW1 #HWV1 elim (lift_total V1 0 1)
- /4 width=7 by llpx_cpx_conf, cpx_delta, fqu_drop, ldrop_fwd_drop2, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HU1
- elim (llpx_ldrop_trans_O … HU1 … HLK1) -L1
- /3 width=5 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma llpx_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (llpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cpx_lleq.ma".
-include "basic_2/reduction/llpx.ma".
-
-(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma llpx_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
-/2 width=1 by llpx_sn_lrefl/ qed-.
-
-lemma lleq_llpx_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
- ∀L.⦃G, L2⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L.
-/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_trans/ qed-.
-
-lemma lleq_llpx_conf: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
- ∀L.⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L2⦄ ⊢ ➡[h, g, T, d] L.
-/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_conf/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/llpx_sn_lpx_sn.ma".
-include "basic_2/reduction/lpr.ma".
-include "basic_2/reduction/llpx.ma".
-
-(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
-
-(* Properties on sn parallel reduction **************************************)
-
-(* Note: this should be moved *)
-lemma lpr_llpx: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
-/3 width=4 by cpr_cpx, lpx_sn_llpx_sn, llpx_sn_co/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazypredsnstar_7.ma".
-include "basic_2/reduction/llpx.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
-
-definition llpxs: ∀h. sd h → genv → relation4 ynat term lenv lenv ≝
- λh,g,G,d. LTC … (llpx h g G d).
-
-interpretation "lazy extended parallel computation (local environment, sn variant)"
- 'LazyPRedSnStar G L1 L2 h g T d = (llpxs h g G d T L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma llpxs_ind: ∀h,g,G,L1,T,d. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡[h, g, T, d] L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → R L2.
-#h #g #G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma llpxs_ind_dx: ∀h,g,G,L2,T,d. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → R L1.
-#h #g #G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpx_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-normalize /2 width=1 by inj/ qed.
-
-lemma llpxs_refl: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L.
-/2 width=1 by llpx_llpxs/ qed.
-
-lemma llpxs_strap1: ∀h,g,G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-normalize /2 width=3 by step/ qed.
-
-lemma llpxs_strap2: ∀h,g,G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-normalize /2 width=3 by TC_strap/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpxs_fwd_length: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → |L1| = |L2|.
-#h #g #G #L1 #L2 #T #d #H @(llpxs_ind … H) -L2
-/3 width=8 by llpx_fwd_length, trans_eq/
-qed-.
-
-(* Note: this might be moved *)
-lemma llpxs_fwd_bind_sn: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓑ{a,I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡*[h, g, V, d] L2.
-#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
-/3 width=6 by llpx_fwd_bind_sn, llpxs_strap1/
-qed-.
-
-(* Note: this might be moved *)
-lemma llpxs_fwd_bind_dx: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓑ{a,I}V.T, d] L2 →
- ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, g, T, ⫯d] L2.ⓑ{I}V.
-#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
-/3 width=6 by llpx_fwd_bind_dx, llpxs_strap1/
-qed-.
-
-(* Note: this might be moved *)
-lemma llpxs_fwd_flat_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡*[h, g, V, d] L2.
-#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
-/3 width=6 by llpx_fwd_flat_sn, llpxs_strap1/
-qed-.
-
-(* Note: this might be moved *)
-lemma llpxs_fwd_flat_dx: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
-/3 width=6 by llpx_fwd_flat_dx, llpxs_strap1/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/llpx_aaa.ma".
-include "basic_2/computation/llpxs.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
-
-(* Properties about atomic arity assignment on terms ************************)
-
-lemma aaa_llpxs_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-#h #g #G #L1 #T #A #HT #L2 #HL12
-@(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_llpx_conf/
-qed-.
-(*
-lamma aaa_llprs_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-/3 width=5 by aaa_llpxs_conf, llprs_llpxs/ qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/llpx_sn_tc.ma".
-include "basic_2/computation/cpxs_llpx.ma".
-include "basic_2/computation/llpxs.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
-
-(* Advanced properties ******************************************************)
-
-lemma llpxs_pair_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡*[h, g, T, 0] L.ⓑ{I}V2.
-/2 width=1 by llpx_sn_TC_pair_dx/ qed.
-
-(* Properties on context-sensitive extended parallel computation for terms **)
-
-lemma llpxs_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (llpxs h g G 0).
-/3 width=5 by s_r_trans_LTC2, llpx_cpxs_trans/ qed-.
-
-lemma llpxs_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (llpxs h g G 0).
-#h #g #G @s_r_to_s_rs_trans @s_r_trans_LTC2
-/3 width=5 by llpx_cpxs_trans, s_rs_trans_TC1/ (**) (* full auto too slow *)
-qed-.
-
-(* Note: this is an instance of a general theorem *)
-lemma llpxs_cpxs_conf_dx: ∀h,g,G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
- ∀L0. ⦃G2, L0⦄ ⊢ ➡*[h, g, T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[h, g, U2, O] L2.
-#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx … H) -L0 //
-#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 … IHL2) -IHL2
-lapply (llpxs_cpxs_trans … HTU2 … HL2) -L2 #HTU2
-/3 width=3 by llpx_cpxs_trans, cpxs_llpx_conf/
-qed-.
-
-(* Note: this is an instance of a general theorem *)
-lemma llpxs_cpx_conf_dx: ∀h,g,G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- ∀L0. ⦃G2, L0⦄ ⊢ ➡*[h, g, T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[h, g, U2, O] L2.
-#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx … H) -L0 //
-#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 … IHL2) -IHL2
-lapply (llpxs_cpx_trans … HTU2 … HL2) -L2 #HTU2
-/3 width=3 by llpx_cpxs_trans, cpxs_llpx_conf/
-qed-.
-
-lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-/4 width=5 by llpxs_cpxs_trans, llpxs_pair_dx, cpxs_bind/ qed.
-
-(* Advanced forward lemmas **************************************************)
-
-(* Note: this might be moved *)
-lemma llpxs_fwd_lref_ge_sn: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡*[h, g, #i, d] L2 → d ≤ i →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- ⦃G, K1⦄ ⊢ ➡*[h, g, V2, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2.
-#h #g #G #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 @(llpxs_ind … H) -L2 /2 width=5 by ex3_2_intro/ -HLK1
-#L #L2 #_ #HL2 * #K #V #HLK #HK1 #HV1 elim (llpx_inv_lref_ge_sn … HL2 … HLK) // -HL2 -HLK -Hdi
-#K2 #V2 #HLK2 #HK2 #HV2
-@(ex3_2_intro … HLK2) -HLK2
-[ /3 width=5 by llpxs_cpx_conf_dx, llpxs_strap1, llpx_cpx_conf/
-| /3 width=5 by llpxs_cpx_trans, cpxs_trans/
-]
-qed-.
-
-(* Inversion lemmas on context-sensitive ext parallel computation for terms *)
-
-lemma cpxs_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
- U2 = ⓛ{a}V2.T2.
-#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
-#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
-elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (llpxs_cpx_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro/
-qed-.
-
-lemma cpxs_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
- U2 = ⓓ{a}V2.T2
- ) ∨
- ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
-#U0 #U2 #_ #HU02 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpx_inv_abbr1 … HU02) -HU02 *
- [ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (llpxs_cpx_trans … HT02 (L.ⓓV1) ?)
- /4 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
- | #T2 #HT02 #HUT2
- lapply (llpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
- /4 width=3 by llpxs_pair_dx, cpxs_trans, ex3_intro, or_intror/
- ]
-| #U1 #HTU1 #HU01
- elim (lift_total U2 0 1) #U #HU2
- /6 width=12 by cpxs_strap1, cpx_lift, ldrop_drop, ex3_intro, or_intror/
-]
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma llpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (llpx_fqu_trans … H12 … HKL1) -L1
- /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
-| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
- #L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fqu_trans … H2 … HL0) -L
- #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
- /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
-]
-qed-.
-
-lemma llpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
-#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
-#L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fquq_trans … H2 … HL0) -L
-#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
-/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
-qed-.
-
-lemma llpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx … H) -K1
-[ /2 width=5 by ex3_2_intro/
-| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
- lapply (llpx_cpxs_trans … HT1 … HK1) -HT1 #HT1
- lapply (cpxs_llpx_conf … HT1 … HK1) -HK1 #HK1
- elim (llpx_fquq_trans … HT2 … HK1) -K
- /3 width=7 by llpxs_strap2, cpxs_strap1, ex3_2_intro/
-]
-qed-.
-
-lemma llpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx … H) -K1
-[ /2 width=5 by ex3_2_intro/
-| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
- lapply (llpx_cpxs_trans … HT1 … HK1) -HT1 #HT1
- lapply (cpxs_llpx_conf … HT1 … HK1) -HK1 #HK1
- elim (llpx_fqup_trans … HT2 … HK1) -K
- /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/
-]
-qed-.
-
-lemma llpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g, T1, 0] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g, T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx … H) -K1
-[ /2 width=5 by ex3_2_intro/
-| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
- lapply (llpx_cpxs_trans … HT1 … HK1) -HT1 #HT1
- lapply (cpxs_llpx_conf … HT1 … HK1) -HK1 #HK1
- elim (llpx_fqus_trans … HT2 … HK1) -K
- /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lleq_leq.ma".
-include "basic_2/reduction/llpx_lleq.ma".
-include "basic_2/computation/cpxs_lleq.ma".
-include "basic_2/computation/llpxs_cpxs.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS *************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma llpxs_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-/3 width=1 by llpx_lrefl, llpx_llpxs/ qed-.
-
-lemma lleq_llpxs_trans: ∀h,g,G,L2,L,T,d. ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L →
- ∀L1. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L.
-#h #g #G #L2 #L #T #d #H @(llpxs_ind … H) -L
-/3 width=3 by llpxs_strap1, llpxs_lrefl/
-qed-.
-
-lemma lleq_llpxs_conf: ∀h,g,G,L1,L,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L →
- ∀L2. L1 ⋕[T, d] L2 → ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L.
-/3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-.
-(*
-foct leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
- ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
- ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
- (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
-#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
-[ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
- /3 width=5 by ex3_intro, conj/
-| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
-| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
- elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
- lapply (ysucc_inv_Y_dx … He) -He #He
- elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
- #T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
-| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
- elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
- elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
- #T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
-]
-qed-.
-
-lamma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
- ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
- ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
- (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
-/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/llpxs.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
-
-(* Main properties **********************************************************)
-
-theorem llpxs_trans: ∀h,g,G,T,d. Transitive … (llpxs h g G d T).
-normalize /2 width=3 by trans_TC/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/llpx_lpr.ma".
-include "basic_2/computation/lprs.ma".
-include "basic_2/computation/llpxs.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
-
-(* Properties on sn parallel computation ************************************)
-
-(* Note: this should be moved *)
-lemma lprs_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-normalize /3 width=3 by lpr_llpx, monotonic_TC/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazysn_6.ma".
-include "basic_2/substitution/lleq.ma".
-include "basic_2/reduction/llpx.ma".
-
-(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
-
-definition llsx: ∀h. sd h → relation4 ynat term genv lenv ≝
- λh,g,d,T,G. SN … (llpx h g G d T) (lleq d T).
-
-interpretation
- "lazy extended strong normalization (local environment)"
- 'LazySN h g d T G L = (llsx h g T d G L).
-
-(* Basic eliminators ********************************************************)
-
-lemma llsx_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
- (∀L1. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #H0 #L1 #H elim H -L1
-/5 width=1 by lleq_sym, SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llsx_intro: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊*[h, g, T, d] L1.
-/5 width=1 by lleq_sym, SN_intro/ qed.
-
-lemma llsx_sort: ∀h,g,G,L,d,k. G ⊢ ⋕⬊*[h, g, ⋆k, d] L.
-#h #g #G #L1 #d #k @llsx_intro
-#L2 #HL12 #H elim H -H
-/3 width=6 by llpx_fwd_length, lleq_sort/
-qed.
-
-lemma llsx_gref: ∀h,g,G,L,d,p. G ⊢ ⋕⬊*[h, g, §p, d] L.
-#h #g #G #L1 #d #p @llsx_intro
-#L2 #HL12 #H elim H -H
-/3 width=6 by llpx_fwd_length, lleq_gref/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazysnalt_6.ma".
-include "basic_2/substitution/lleq_lleq.ma".
-include "basic_2/computation/llpxs_lleq.ma".
-include "basic_2/computation/llsx.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* alternative definition of llsx *)
-definition llsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
- λh,g,d,T,G. SN … (llpxs h g G d T) (lleq d T).
-
-interpretation
- "lazy extended strong normalization (local environment) alternative"
- 'LazySNAlt h g d T G L = (llsxa h g T d G L).
-
-(* Basic eliminators ********************************************************)
-
-lemma llsxa_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
- (∀L1. G ⊢ ⋕⬊⬊*[h, g, T, d] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⋕⬊⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #H0 #L1 #H elim H -L1
-/5 width=1 by lleq_sym, SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llsxa_intro: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
-/5 width=1 by lleq_sym, SN_intro/ qed.
-
-fact llsxa_intro_aux: ∀h,g,G,L1,T,d.
- (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → L1 ⋕[T, d] L → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
-/4 width=3 by llsxa_intro/ qed-.
-
-lemma llsxa_llpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊⬊*[h, g, T, d] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → G ⊢ ⋕⬊⬊*[h, g, T, d] L2.
-#h #g #G #L1 #T #d #H @(llsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 @llsxa_intro
-elim (lleq_dec T L1 L2 d) /4 width=4 by lleq_llpxs_trans, lleq_canc_sn/
-qed-.
-
-lemma llsxa_intro_llpx: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
-#h #g #G #L1 #T #d #IH @llsxa_intro_aux
-#L #L2 #H @(llpxs_ind_dx … H) -L
-[ #H destruct #H elim H //
-| #L0 #L elim (lleq_dec T L1 L d)
- /4 width=3 by llsxa_llpxs_trans, lleq_llpx_trans/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem llsx_llsxa: ∀h,g,G,L,T,d. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊⬊*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(llsx_ind … H) -L
-/4 width=1 by llsxa_intro_llpx/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem llsxa_inv_llsx: ∀h,g,G,L,T,d. G ⊢ ⋕⬊⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(llsxa_ind … H) -L
-/4 width=1 by llsx_intro, llpx_llpxs/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llsx_intro_alt: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊*[h, g, T, d] L1.
-/6 width=1 by llsxa_inv_llsx, llsx_llsxa, llsxa_intro/ qed.
-
-lemma llsx_llpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
-/4 width=3 by llsxa_inv_llsx, llsx_llsxa, llsxa_llpxs_trans/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma llsx_ind_alt: ∀h,g,G,T,d. ∀R:predicate lenv.
- (∀L1. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #IH #L #H @(llsxa_ind h g G T d … L)
-/4 width=1 by llsxa_inv_llsx, llsx_llsxa/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/csx_lift.ma".
-include "basic_2/computation/csx_llpxs.ma".
-include "basic_2/computation/llsx_ldrop.ma".
-include "basic_2/computation/llsx_llpx.ma".
-include "basic_2/computation/llsx_llpxs.ma".
-(*
-axiom cpx_llpx_trans: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L2. ⦃G, L1⦄⊢ ➡[h, g, T2, O] L2 →
- ∃∃L. ⦃G, L1⦄⊢ ➡[h, g, T1, O] L & L ⋕[T2, 0] L2.
-(*
-fact llsx_cpx_trans_aux: ∀h,g,G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L1,d. G ⊢ ⋕⬊*[h, g, T1, d] L1 → d = 0 →
- L0 ⋕[T1, d] L1 → ∀L2. L1 ⋕[T2, d] L2 →
- G ⊢ ⋕⬊*[h, g, T2, d] L2.
-#h #g #G #L0 #T1 #T2 #HT12 #L1 #d #H @(llsx_ind … H) -L1
-#L1 #_ #IHL1 #Hd #He011 #L2 #He122 @llsx_intro
-#L3 #Hx223 #Hn223 destruct
-lapply (lleq_cpx_conf_sn … HT12 … He011) #He021
-lapply (lleq_cpx_conf … HT12 … He011) -HT12 #HT12
-lapply (lleq_llpx_trans … He122 … Hx223) -Hx223 #Hx123
-elim (cpx_llpx_trans … HT12 … Hx123) -Hx123 #L4 #Hx114 #He423
-(* lapply (lleq_cpx_conf … Hx114 … He011) #He120 *)
-@(IHL1 … Hx114) // -IHL1
-[ #HL13 @HnL2 -HnL2
-*)
-
-fact llsx_cpx_trans_aux: ∀h,g,G,L1,T1,d. G ⊢ ⋕⬊*[h, g, T1, d] L1 → d = 0 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L2. L1 ⋕[T1, d] L2 → G ⊢ ⋕⬊*[h, g, T2, 0] L2.
-#h #g #G #L1 #T1 #d #H @(llsx_ind … H) -L1
-#L1 #_ #IHL1 #Hd #T2 #HT12 #L2 #He112 @llsx_intro
-#L3 #Hx223 #Hn223 destruct
-lapply (lleq_cpx_conf_sn … HT12 … He112) #He122
-lapply (lleq_cpx_conf … HT12 … He112) -HT12 #HT12
-elim (cpx_llpx_trans … HT12 … Hx223) #L4 #Hx214 #He423
-@(IHL1 … L4) //
-*)
-axiom llsx_cpx_trans_O: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
- G ⊢ ⋕⬊*[h, g, T1, 0] L → G ⊢ ⋕⬊*[h, g, T2, 0] L.
-
-(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
-
-(* Advanced properties ******************************************************)
-
-lemma llsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,d. d ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
- ∀K2. G ⊢ ⋕⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g, V, 0] K2 →
- ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L2.
-#h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt … H) -V
-#V0 #_ #IHV0 #K2 #H @(llsx_ind … H) -K2
-#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @llsx_intro
-#L2 #HL02 #HnL02 elim (llpx_inv_lref_ge_sn … HL02 … HLK0) // -HL02
-#K2 #V2 #HLK2 #HK02 #HV02 elim (eq_term_dec V0 V2)
-#HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
-[ /4 width=7 by llpxs_strap1, lleq_lref/
-| lapply (llpx_cpx_conf … HV02 … HK02) -HK02 #HK02
- @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
- /3 width=3 by llsx_cpx_trans_O, llpxs_cpx_conf_dx, llsx_llpx_trans, llpxs_cpx_trans, llpxs_strap1/
-]
-qed.
-
-lemma llsx_lref_be: ∀h,g,I,G,K,V,i,d. d ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
- G ⊢ ⋕⬊*[h, g, V, 0] K →
- ∀L. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L.
-/2 width=8 by llsx_lref_be_lpxs/ qed.
-
-(* Main properties **********************************************************)
-
-theorem csx_llsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀d. G ⊢ ⋕⬊*[h, g, T, d] L.
-#h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
-#Z #Y #X #IH #G #L * * //
-[ #i #HG #HL #HT #H #d destruct
- elim (lt_or_ge i (|L|)) /2 width=1 by llsx_lref_free/
- elim (ylt_split i d) /2 width=1 by llsx_lref_skip/
- #Hdi #Hi elim (ldrop_O1_lt … Hi) -Hi
- #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
- /4 width=6 by llsx_lref_be, fqup_lref/
-| #a #I #V #T #HG #HL #HT #H #d destruct
- elim (csx_fwd_bind … H) -H /3 width=1 by llsx_bind/
-| #I #V #T #HG #HL #HT #H #d destruct
- elim (csx_fwd_flat … H) -H /3 width=1 by llsx_flat/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lleq_ldrop.ma".
-include "basic_2/computation/llsx.ma".
-
-(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
-
-(* Advanced properties ******************************************************)
-
-lemma llsx_lref_free: ∀h,g,G,L,d,i. |L| ≤ i → G ⊢ ⋕⬊*[h, g, #i, d] L.
-#h #g #G #L1 #d #i #HL1 @llsx_intro
-#L2 #HL12 #H elim H -H
-/4 width=8 by llpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
-qed.
-
-lemma llsx_lref_skip: ∀h,g,G,L,d,i. yinj i < d → G ⊢ ⋕⬊*[h, g, #i, d] L.
-#h #g #G #L1 #d #i #HL1 @llsx_intro
-#L2 #HL12 #H elim H -H
-/3 width=6 by llpx_fwd_length, lleq_skip/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lleq_lleq.ma".
-include "basic_2/reduction/llpx_lleq.ma".
-include "basic_2/computation/llsx.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* Advanced properties ******************************************************)
-
-lemma llsx_llpx_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
-#h #g #G #L1 #T #d #H @(llsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 @llsx_intro
-elim (lleq_dec T L1 L2 d) /4 width=4 by lleq_llpx_trans, lleq_canc_sn/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/llpxs_llpxs.ma".
-include "basic_2/computation/llsx_alt.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* Advanced properties ******************************************************)
-
-fact llsx_bind_llpxs_aux: ∀h,g,a,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
- ∀Y,T. G ⊢ ⋕⬊*[h, g, T, ⫯d] Y →
- ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g, ⓑ{a,I}V.T, d] L2 →
- G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L2.
-#h #g #a #I #G #L1 #V #d #H @(llsx_ind_alt … H) -L1
-#L1 #HL1 #IHL1 #Y #T #H @(llsx_ind_alt … H) -Y
-#Y #HY #IHY #L2 #H #HL12 destruct @llsx_intro_alt
-#L0 #HL20 lapply (llpxs_fwd_bind_dx … HL20)
-lapply (llpxs_trans … HL12 … HL20)
-#HL10 #HT #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 d)
- [ -HL20 #HV @(IHL1 … L0)
- /3 width=4 by llsx_llpxs_trans, llpxs_fwd_bind_sn, lleq_canc_sn/ (**) (* full auto too slow *)
- | -HnV -HL10
- /3 width=4 by llsx_llpxs_trans, llpxs_fwd_bind_sn/
- ]
-| /3 width=4 by/
-]
-qed-.
-
-lemma llsx_bind: ∀h,g,a,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
- ∀T. G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V →
- G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L.
-/2 width=3 by llsx_bind_llpxs_aux/ qed.
-
-lemma llsx_flat_llpxs: ∀h,g,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
- ∀L2,T. G ⊢ ⋕⬊*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, ⓕ{I}V.T, d] L2 →
- G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L2.
-#h #g #I #G #L1 #V #d #H @(llsx_ind_alt … H) -L1
-#L1 #HL1 #IHL1 #L2 #T #H @(llsx_ind_alt … H) -L2
-#L2 #HL2 #IHL2 #HL12 @llsx_intro_alt
-#L0 #HL20 lapply (llpxs_fwd_flat_dx … HL20)
-lapply (llpxs_trans … HL12 … HL20)
-#HL10 #HT #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 d)
- [ #HV @(IHL1 … L0) /3 width=3 by llsx_llpxs_trans, llpxs_fwd_flat_sn, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
- | -HnV -HL10 /3 width=4 by llsx_llpxs_trans, llpxs_fwd_flat_sn/
- ]
-| /3 width=1 by/
-]
-qed-.
-
-lemma llsx_flat: ∀h,g,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
- ∀T. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L.
-/2 width=3 by llsx_flat_llpxs/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_leq.ma".
-include "basic_2/relocation/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-definition TC_llpx_sn_confluent1: relation (relation3 lenv term term) ≝ λS,R.
- ∀Ls,T1,T2. S Ls T1 T2 →
- ∀Ld. TC … (llpx_sn R 0 T1) Ls Ld → TC … (llpx_sn R 0 T2) Ls Ld.
-
-lemma TC_llpx_sn_s_confluent: ∀S,R. (llpx_sn_confluent1 S R) → TC_llpx_sn_confluent1 S R.
-#S #R #HSR #Ls #T1 #T2 #HT12 #Ld #H
-generalize in match HT12; -HT12
-@(TC_ind_dx … Ls H) -Ls
-[ /3 width=3 by inj/
-| #Ls #L #HLs #_ #IHLd #HT12
- @(TC_strap … L) /2 width=3 by/ @IHLd -IHLd
-
-lemma TC_llpx_sn_lref_refl: ∀R. (∀L.reflexive … (R L)) →
- ∀I,L1,K1,K2,V,d,i. d ≤ yinj i → ⇩[i] L1 ≡ K1.ⓑ{I}V →
- TC lenv (llpx_sn R 0 V) K1 K2 →
- ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → TC … (llpx_sn R d (#i)) L1 L2.
-#R #HR #I #L1 #K1 #K2 #V #d #i #Hdi #HLK1 #H @(TC_star_ind … K2 H) -K2
-[ /2 width=1 by llpx_sn_refl/
-| /4 width=9 by llpx_sn_refl, llpx_sn_lref, inj/
-| #K #K2 #_ #HV #IHK1 #L2 #HLK2 lapply (ldrop_fwd_length … HLK2)
- #H elim (ldrop_O1_ex (K.ⓑ{I}V) i L2) [2: normalize in H ⊢ %; >(llpx_sn_fwd_length … HV) ]
- /4 width=11 by llpx_sn_lref, step/
-]
-qed-.
-
-lemma TC_llpx_sn_lref: ∀R. (∀L.reflexive … (R L)) → (llpx_sn_confluent1 R R) →
- ∀I,K1,V1,V2,d,i. d ≤ yinj i → LTC … R K1 V1 V2 →
- ∀K2. TC lenv (llpx_sn R 0 V1) K1 K2 → ∀L1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 → TC … (llpx_sn R d (#i)) L1 L2.
-#R #H1R #H2R #I #K1 #V1 #V2 #d #i #Hdi #H @(TC_star_ind_dx … V1 H) -V1
-[ /2 width=1 by llpx_sn_refl/
-| /2 width=7 by TC_llpx_sn_lref_refl/
-| #V1 #V #HV1 #_ #IHV2 #K2 #HK12 #L1 #HLK1 #L2 #HLK2
- lapply (ldrop_fwd_length … HLK1)
- #H elim (ldrop_O1_ex (K1.ⓑ{I}V) i L1) [2: normalize in H ⊢ %; // ] -H
- #L #_ #HLK @(TC_strap … L)
- [ @(llpx_sn_lref … HLK1 … HLK) /2 width=1 by llpx_sn_refl/
- | @(IHV2 … HLK … HLK2)
- -HLK1 -HLK2 -HLK -IHV2 -Hdi @(TC_llpx_sn_s_confluent R R … HK12) //
- ]
-]
-
-
-lemma llpx_sn_LTC_TC_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2,T,d. llpx_sn (LTC … R) d T L1 L2 →
- TC … (llpx_sn R d T) L1 L2.
-#R #HR #L1 #L2 #T #d #H elim H -L1 -L2
-/3 width=3 by llpx_sn_gref, llpx_sn_free, llpx_sn_skip, llpx_sn_sort, inj/
-[ #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #_ #HV12 #IHV1
-
-(* Properties on transitive_closure *****************************************)
-
-lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … R L1 V1 V2 →
- TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
-#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
-[ /2 width=1 by TC_lpx_sn_pair_refl/
-| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
-]
-qed-.
-
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
- TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
-qed-.
-
-(* Inversion lemmas on transitive closure ***********************************)
-
-lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
-#R #L1 #H @(TC_ind_dx … L1 H) -L1
-[ /2 width=2 by lpx_sn_inv_atom2/
-| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
-[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
-| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
- elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
- lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_ind: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀S:relation lenv.
- S (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
- S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
- elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
- #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
-#R #L2 #H elim H -L2
-[ /2 width=2 by lpx_sn_inv_atom1/
-| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
-]
-qed-.
-
-fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
-[ #J #K #W #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (LTC … R) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
-(* Forward lemmas on transitive closure *************************************)
-
-lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L2
-[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
-]
-qed-.
+++ /dev/null
-include "basic_2/relocation/lleq_alt.ma".
-include "basic_2/reduction/cpx_lleq.ma".
-include "basic_2/reduction/lpx_lleq.ma".
-
-lemma not_ex_to_all_not: ∀A:Type[0]. ∀R:predicate A.
- ((∃a. R a)→⊥) → (∀a. R a → ⊥).
-/3 width=2 by ex_intro/ qed-.
-
-lemma lt_repl_sn_trans_tw: ∀L1,L2,T1,T2. ♯{L1, T1} < ♯{L2, T2} →
- ∀U1. ♯{U1} = ♯{T1} → ♯{L1, U1} < ♯{L2, T2}.
-normalize in ⊢ (?→?→?→?→?%%→?→?→?%%); //
-qed-.
-
-axiom cpx_fwd_lift1: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
- ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → ∃T2. ⇧[d, e] T2 ≡ U2.
-(*
-#h #g #G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #i #G #L #T1 #d #e #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=2 by ex_intro/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=2 by lift_lref_ge_minus, lift_lref_lt, ex_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=2 by ex_intro/
- ]
-| #G #L #k #l #Hkl #T1 #d #e #H
- lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by ex_intro/
-| #I #G #L #K #V1 #V2 #W2 #i #HLK #HV12 #HVW2 #IHV2 #T1 #d #e #H
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
- elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
- elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpx_delta, ex2_intro/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpx_delta, ex2_intro/
- ]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
- elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpx_bind, ldrop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -V1
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpx_flat, lift_flat, ex2_intro/
-| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU1 (K.ⓓW1) s … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
- elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_tau, ex2_intro/
-| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ti, ex2_intro/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HLK … HV01) -V1 #V3 #HV32 #HV03
- elim (IHT12 (K.ⓛW0) s … HT01) -T1 /2 width=1 by ldrop_skip/ #T3 #HT32 #HT03
- elim (IHW12 … HLK … HW01) -W1
- /4 width=7 by cpx_beta, lift_bind, lift_flat, ex2_intro/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
- elim (IHT12 (K.ⓓW0) s … HT01) -T1 /2 width=1 by ldrop_skip/ #T3 #HT32 #HT03
- elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
- elim (lift_trans_le … HV3 … HV2) -V
- /4 width=9 by cpx_theta, lift_bind, lift_flat, ex2_intro/
-]
-qed-.
-*)
-lemma cpx_fwd_nlift2: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 → ∀d,e.
- (∀T2. ⇧[d, e] T2 ≡ U2 → ⊥) → (∀T1. ⇧[d, e] T1 ≡ U1 → ⊥).
-#h #g #G #L #U1 #U2 #HU12 #d #e #HnU2 #T1 #HTU1
-elim (cpx_fwd_lift1 … HU12 … HTU1) -L -U1 /2 width=2 by/
-qed-.
-
-fact lleq_lpx_cpx_trans_aux: ∀h,g,G,T1,L1s,L1d,d. L1s ⋕[T1, d] L1d →
- ∀Y,L2d. ⦃G, Y⦄ ⊢ ➡[h, g] L2d → Y = L1d → d = 0 →
- ∀T2. ⦃G, L1d⦄ ⊢ T1 ➡[h, g] T2 →
- ∃∃L2s. ⦃G, L1s⦄ ⊢ ➡[h, g] L2s & L2s ⋕[T2, d] L2d.
-#h #g #G #T1 #L1s @(f2_ind … rfw … L1s T1) -L1s -T1 #n #IH
-#Ys #U1 #Hn #Yd #d #HU1 elim (lleq_fwd_alt … HU1) #H #IHU1
-#Y #L2d * -Y -L2d
-[ -IH -IHU1 -HU1 #HY #Hd #U2 #HU12 destruct
- >(length_inv_zero_dx … H) -Ys /2 width=3 by ex2_intro/
-| #Id #L1d #L2d #W1d #W2d #HL12d #HW12d #HY #Hd #U2 #HU12 destruct
- elim (length_inv_pos_dx … H) -H #Is #L1s #W1s #_ #H destruct
- elim (is_lift_dec U1 0 1) [ -IHU1 -HW12d | -HU1 ]
- [ * #T1 #HTU1 lapply (lift_fwd_tw … HTU1) #H
- lapply (lleq_inv_lift_le … HU1 L1s L1d … HTU1 ?) -HU1 /2 width=1 by ldrop_drop/
- #HT1 elim (cpx_inv_lift1 … HU12 L1d … HTU1) -HU12 -HTU1 /2 width=4 by ldrop_drop/
- #T2 #HTU2 #HT12 elim (IH … HT1 … HL12d … HT12) /2 width=3 by lt_repl_sn_trans_tw/ -IH -HT1 -HT12 -H
- #L2s #HL12s #HT2 @(ex2_intro … (L2s.ⓑ{Is}W1s))
- /3 width=10 by lleq_lift_le, lpx_pair, ldrop_drop/ (**) (* full auto too slow *)
- | #HnU1 lapply (not_ex_to_all_not … HnU1) -HnU1 #HnU1
- elim (IHU1 … HnU1) [2,3,4: // |5,6,7,8,9,10: skip ] -HnU1 #H1 #H2 #HW1s destruct
- elim (IH … HW1s … HL12d … HW12d) // #L2s #HL12s #HW2d
- @(ex2_intro … (L2s.ⓑ{Id}W2d)) /3 width=3 by lleq_cpx_trans, lpx_pair/
- lapply (lleq_fwd_length … HW2d) #HL2sd -HW12d -HW1s
- @lleq_intro_alt [ normalize // ] -HL2sd
- #I2s #I2d #K2s #K2d #V2s #V2d #i @(nat_ind_plus … i) -i
- [ #_ #_ #HLK2s #HLK2d -IH -IHU1 -HU12 -HL12s -HL12d
- lapply (ldrop_inv_O2 … HLK2s) -HLK2s #H destruct
- lapply (ldrop_inv_O2 … HLK2d) -HLK2d #H destruct /2 width=1 by and3_intro/
- | #i #_ #_ #HnU2 #HLK2s #HLK2d
- lapply (cpx_fwd_nlift2 … HU12 … HnU2) -HU12 -HnU2 #HnU1
- lapply (ldrop_inv_drop1 … HLK2s) -HLK2s #HLK2s
- lapply (ldrop_inv_drop1 … HLK2d) -HLK2d #HLK2d
- elim (lpx_ldrop_trans_O1 … HL12s … HLK2s) -L2s #Y #HLK1s #H
- elim (lpx_inv_pair2 … H) -H #K1s #V1s #HK12s #HV12s #H destruct
- elim (lpx_ldrop_trans_O1 … HL12d … HLK2d) -L2d #Y #HLK1d #H
- elim (lpx_inv_pair2 … H) -H #K1d #V1d #HK12d #HV12d #H destruct
- elim (IHU1 … HnU1) [2,3,4: /2 width=2 by ldrop_drop/ | 5,6,7,8,9,10: skip ] -IHU1 -HnU1 -HLK1d
- #H1 #H2 #HV1d destruct
- lapply (ldrop_fwd_rfw … HLK1s) -HLK1s #H
- elim (IH … HV1d … HK12d … HV12d) // -IH -HV1d -HK12d -HV12d
- [ #Y #H1Y #H2Y
-
-
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPConvAlt $L1 $L2 }.
-
-include "basic_2/reducibility/lfpr.ma".
-
-(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************)
-
-definition lfpc: relation lenv ≝
- λL1,L2. ⦃L1⦄ ➡ ⦃L2⦄ ∨ ⦃L2⦄ ➡ ⦃L1⦄.
-
-interpretation
- "focalized parallel conversion (local environment)"
- 'FocalizedPConv L1 L2 = (lfpc L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lfpc_refl: ∀L. ⦃L⦄ ⬌ ⦃L⦄.
-/2 width=1/ qed.
-
-lemma lfpc_sym: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L2⦄ ⬌ ⦃L1⦄.
-#L1 #L2 * /2 width=1/
-qed.
-
-lemma lfpc_lfpr: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄.
-#L1 #L2 * /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/conversion/lfpc.ma".
-
-(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************)
-
-(* Main properties **********************************************************)
-
-theorem lfpc_conf: ∀L0,L1,L2. ⦃L0⦄ ⬌ ⦃L1⦄ → ⦃L0⦄ ⬌ ⦃L2⦄ →
- ∃∃L. ⦃L1⦄ ⬌ ⦃L⦄ & ⦃L2⦄ ⬌ ⦃L⦄.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPConvStarAlt $L1 $L2 }.
-
-include "basic_2/conversion/lfpc.ma".
-
-(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
-
-definition lfpcs: relation lenv ≝ TC … lfpc.
-
-interpretation "focalized parallel equivalence (local environment)"
- 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lfpcs_ind: ∀L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → R L → R L2) →
- ∀L2. ⦃L1⦄ ⬌* ⦃L2⦄ → R L2.
-#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma lfpcs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → R L → R L1) →
- ∀L1. ⦃L1⦄ ⬌* ⦃L2⦄ → R L1.
-#L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lfpcs_refl: reflexive … lfpcs.
-/2 width=1/ qed.
-
-lemma lfpcs_sym: symmetric … lfpcs.
-/3 width=1/ qed.
-
-lemma lfpc_lfpcs: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/2 width=1/ qed.
-
-lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/2 width=3/ qed.
-
-lemma lfpcs_strap2: ∀L1,L,L2. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/2 width=3/ qed.
-
-lemma lfpcs_lfpr_dx: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=1/ qed.
-
-lemma lfpcs_lfpr_sn: ∀L1,L2. ⦃L2⦄ ➡ ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=1/ qed.
-
-lemma lfpcs_lfpr_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3/ qed.
-
-lemma lfpcs_lfpr_strap2: ∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3/ qed.
-
-lemma lfpcs_lfpr_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡ ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3/ qed.
-
-lemma lfpcs_lfpr_conf: ∀L1,L. ⦃L⦄ ➡ ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/lfprs_aaa.ma".
-include "basic_2/equivalence/lfpcs_lfpcs.ma".
-
-(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
-
-(* Main properties about atomic arity assignment on terms *******************)
-
-theorem aaa_lfpcs_mono: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
- ∀T,A1. L1 ⊢ T ⁝ A1 → ∀A2. L2 ⊢ T ⁝ A2 →
- A1 = A2.
-#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2
-elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2
-lapply (aaa_lfprs_conf … HT1 … HL1) -L1 #HT1
-lapply (aaa_lfprs_conf … HT2 … HL2) -L2 #HT2
-lapply (aaa_mono … HT1 … HT2) -L -T //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/fprs_cprs.ma".
-include "basic_2/computation/lfprs_fprs.ma".
-include "basic_2/equivalence/fpcs_fpcs.ma".
-include "basic_2/equivalence/lfpcs_lfpcs.ma".
-
-(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
-
-(* Inversion lemmas on context-free parallel equivalence for closures *******)
-
-lemma lfpcs_inv_fpcs: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ → ∀T. ⦃L1, T⦄ ⬌* ⦃L2, T⦄.
-#L1 #L2 #HL12 #T
-elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2
-lapply (lfprs_inv_fprs … HL1 T) -HL1
-lapply (lfprs_inv_fprs … HL2 T) -HL2 /2 width=4/
-qed-.
-
-(* Properties on context-free parallel equivalence for closures *************)
-
-lemma fpcs_lfpcs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L2 #T1 #T2 #HT12
-elim (fpcs_inv_fprs … HT12) -HT12 /3 width=5 by fprs_lfprs, lfprs_div/ (**) (* auto too slow without trace *)
-qed.
-
-lemma fpcs_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ⬌* ⦃K2, T2⦄ →
- ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
- ∀d,e. ⇩[d, e] L1 ≡ K1 → ⇩[d, e] L2 ≡ K2 →
- ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#K1 #K2 #T1 #T2 #HT12 #L1 #L2 #HL12 #d #e #HLK1 #HLK2 #U1 #U2 #HTU1 #HTU2
-elim (fpcs_inv_fprs … HT12) -HT12 #K #T #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-elim (fprs_lift … HT1 … HLK1 … HTU1 HTU) -K1 -T1 #K1 #HU1 #_
-elim (fprs_lift … HT2 … HLK2 … HTU2 HTU) -K2 -T2 -T #K2 #HU2 #_ -K -d -e
-lapply (lfpcs_lfprs_conf K1 … HL12) -HL12 /2 width=3/ #H
-lapply (lfpcs_lfprs_strap1 … H K2 ?) -H /2 width=3/ #HK12
-lapply (lfpcs_inv_fpcs … HK12 U) -HK12 #HU
-/3 width=4 by fpcs_fprs_strap2, fpcs_fprs_div/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/lfprs_lfprs.ma".
-include "basic_2/conversion/lfpc_lfpc.ma".
-include "basic_2/equivalence/lfpcs_lfprs.ma".
-
-(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfpcs_inv_lfprs: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
- ∃∃L. ⦃L1⦄ ➡* ⦃L⦄ & ⦃L2⦄ ➡* ⦃L⦄.
-#L1 #L2 #H @(lfpcs_ind … H) -L2
-[ /3 width=3/
-| #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0
- [ elim (lfprs_strip … HL0 … HL2) -L #L #HL0 #HL2
- lapply (lfprs_strap1 … HL10 … HL0) -L0 /2 width=3/
- | lapply (lfprs_strap2 … HL2 … HL0) -L /2 width=3/
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lfpcs_strip: ∀L,L1. ⦃L⦄ ⬌* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌ ⦃L2⦄ →
- ∃∃L0. ⦃L1⦄ ⬌ ⦃L0⦄ & ⦃L2⦄ ⬌* ⦃L0⦄.
-/3 width=3/ qed.
-
-(* Main properties **********************************************************)
-
-theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/2 width=3/ qed.
-
-theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
-
-theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/lfprs.ma".
-include "basic_2/equivalence/lfpcs.ma".
-
-(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
-
-(* Properties on focalized computation for local environments ***************)
-
-lemma lfpcs_lfprs_dx: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /3 width=3/
-qed.
-
-lemma lfpcs_lfprs_sn: ∀L1,L2. ⦃L2⦄ ➡* ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /3 width=3/
-qed.
-
-lemma lfpcs_lfprs_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L #HL1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /2 width=3/
-qed.
-
-lemma lfpcs_lfprs_strap2: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L #H #L2 #HL2 @(lfprs_ind_dx … H) -L1 /width=1/ /2 width=3/
-qed.
-
-lemma lfpcs_lfprs_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /2 width=3/
-qed.
-
-lemma lfpcs_lfprs_conf: ∀L1,L. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L #H #L2 #HL2 @(lfprs_ind … H) -L1 /width=1/ /2 width=3/
-qed.
-
-lemma lfprs_div: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /2 width=1/ /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/llpx_sn_ldrop.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
-
-(* Properties on lazy sn pointwise extensions *******************************)
-
-lemma cpr_llpx_sn_conf: ∀R. (∀I,L.reflexive … (R I L)) →
- (∀I.l_liftable (R I)) →
- (∀I.l_deliftable_sn (R I)) →
- ∀G. s_r_confluent1 … (cpr G) (llpx_sn R 0).
-#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
-[ //
-| #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
- lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
- lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
-| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
- /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
-| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
- /3 width=10 by llpx_sn_inv_lift_le, ldrop_drop/
-| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
-| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- #HV1 #H elim (llpx_sn_inv_bind_O … H) -H
- /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/
-| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- #HV1 #H elim (llpx_sn_inv_bind_O … H) -H //
- #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *)
- [ /3 width=10 by llpx_sn_lift_le, ldrop_drop/
- | /3 width=4 by llpx_sn_bind_repl_O/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/llpx_sn_ldrop.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-(* Properties on lazy sn pointwise extensions *******************************)
-
-(* Note: lemma 1000 *)
-lemma cpx_llpx_sn_conf: ∀R. (∀I,L.reflexive … (R I L)) →
- (∀I.l_liftable (R I)) →
- (∀I.l_deliftable_sn (R I)) →
- ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx_sn R 0).
-#R #H1R #H2R #H3R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
-[ //
-| /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
-| #I #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
- lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
- lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
-| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
- /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
-| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
- /3 width=10 by llpx_sn_inv_lift_le, ldrop_drop/
-| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
-| #G #Ls #V1 #V2 #T #_ #IHV12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
-| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- #HV1 #H elim (llpx_sn_inv_bind_O … H) -H
- /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/
-| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- #HV1 #H elim (llpx_sn_inv_bind_O … H) -H //
- #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *)
- [ /3 width=10 by llpx_sn_lift_le, ldrop_drop/
- | /3 width=4 by llpx_sn_bind_repl_O/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazyeq_4.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-definition ceq: relation4 bind2 lenv term term ≝ λI,L,T1,T2. T1 = T2.
-
-definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
-
-interpretation
- "lazy equivalence (local environment)"
- 'LazyEq T d L1 L2 = (lleq d T L1 L2).
-
-definition lleq_transitive: predicate (relation4 bind2 lenv term term) ≝
- λR. ∀I,L2,T1,T2. R I L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R I L1 T1 T2.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
- ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
- ) → (
- ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
- ) → (
- ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
- ) → (
- ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
- ) → (
- ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
- ) → (
- ∀a,I,L1,L2,V,T,d.
- L1 ≡[V, d]L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
- R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
- ) → (
- ∀I,L1,L2,V,T,d.
- L1 ≡[V, d]L2 → L1 ≡[T, d] L2 →
- R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
- ) →
- ∀d,T,L1,L2. L1 ≡[T, d] L2 → R d T L1 L2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim H -L1 -L2 -T -d /2 width=8 by/
-qed-.
-
-lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,d. L1 ≡[ⓑ{a,I}V.T, d] L2 →
- L1 ≡[V, d] L2 ∧ L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind/ qed-.
-
-lemma lleq_inv_flat: ∀I,L1,L2,V,T,d. L1 ≡[ⓕ{I}V.T, d] L2 →
- L1 ≡[V, d] L2 ∧ L1 ≡[T, d] L2.
-/2 width=2 by llpx_sn_inv_flat/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ≡[T, d] L2 → |L1| = |L2|.
-/2 width=4 by llpx_sn_fwd_length/ qed-.
-
-lemma lleq_fwd_lref: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V &
- ⇩[i] L2 ≡ K2.ⓑ{I}V &
- K1 ≡[V, yinj 0] K2 & d ≤ yinj i.
-#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref … H) /2 width=1/
-* /3 width=7 by or3_intro2, ex4_4_intro/
-qed-.
-
-lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K1,i. ⇩[i] L1 ≡ K1 →
- ∃K2. ⇩[i] L2 ≡ K2.
-/2 width=7 by llpx_sn_fwd_ldrop_sn/ qed-.
-
-lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K2,i. ⇩[i] L2 ≡ K2 →
- ∃K1. ⇩[i] L1 ≡ K1.
-/2 width=7 by llpx_sn_fwd_ldrop_dx/ qed-.
-
-lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
- L1 ≡[ⓑ{a,I}V.T, d] L2 → L1 ≡[V, d] L2.
-/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
-
-lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,d.
- L1 ≡[ⓑ{a,I}V.T, d] L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
-
-lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
- L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[V, d] L2.
-/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
-
-lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
- L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[T, d] L2.
-/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ≡[⋆k, d] L2.
-/2 width=1 by llpx_sn_sort/ qed.
-
-lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ≡[#i, d] L2.
-/2 width=1 by llpx_sn_skip/ qed.
-
-lemma lleq_lref: ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 ≡[V, 0] K2 → L1 ≡[#i, d] L2.
-/2 width=9 by llpx_sn_lref/ qed.
-
-lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, d] L2.
-/2 width=1 by llpx_sn_free/ qed.
-
-lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ≡[§p, d] L2.
-/2 width=1 by llpx_sn_gref/ qed.
-
-lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
- L1 ≡[V, d] L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
- L1 ≡[ⓑ{a,I}V.T, d] L2.
-/2 width=1 by llpx_sn_bind/ qed.
-
-lemma lleq_flat: ∀I,L1,L2,V,T,d.
- L1 ≡[V, d] L2 → L1 ≡[T, d] L2 → L1 ≡[ⓕ{I}V.T, d] L2.
-/2 width=1 by llpx_sn_flat/ qed.
-
-lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
-/2 width=1 by llpx_sn_refl/ qed.
-
-lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
-/2 width=1 by llpx_sn_Y/ qed.
-
-lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
-#d #T #L1 #L2 #H @(lleq_ind … H) -d -T -L1 -L2
-/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
-qed-.
-
-lemma lleq_ge_up: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
- ∀T,d,e. ⇧[d, e] T ≡ U →
- dt ≤ d + e → L1 ≡[U, d] L2.
-/2 width=6 by llpx_sn_ge_up/ qed-.
-
-lemma lleq_ge: ∀L1,L2,T,d1. L1 ≡[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ≡[T, d2] L2.
-/2 width=3 by llpx_sn_ge/ qed-.
-
-lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
- L1 ≡[ⓑ{a,I}V.T, 0] L2.
-/2 width=1 by llpx_sn_bind_O/ qed-.
-
-(* Advancded properties on lazy pointwise exyensions ************************)
-
-lemma llpx_sn_lrefl: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀L1,L2,T,d. L1 ≡[T, d] L2 → llpx_sn R d T L1 L2.
-/2 width=3 by llpx_sn_co/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-inductive llpx_sn (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
-| llpx_sn_sort: ∀L1,L2,d,k. |L1| = |L2| → llpx_sn R d (⋆k) L1 L2
-| llpx_sn_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn R d (#i) L1 L2
-| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn R (yinj 0) V1 K1 K2 → R I K1 V1 V2 → llpx_sn R d (#i) L1 L2
-| llpx_sn_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R d (#i) L1 L2
-| llpx_sn_gref: ∀L1,L2,d,p. |L1| = |L2| → llpx_sn R d (§p) L1 L2
-| llpx_sn_bind: ∀a,I,L1,L2,V,T,d.
- llpx_sn R d V L1 L2 → llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn R d (ⓑ{a,I}V.T) L1 L2
-| llpx_sn_flat: ∀I,L1,L2,V,T,d.
- llpx_sn R d V L1 L2 → llpx_sn R d T L1 L2 → llpx_sn R d (ⓕ{I}V.T) L1 L2
-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
- ∀a,I,V,T. X = ⓑ{a,I}V.T →
- llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #b #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #p #_ #b #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #d #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
-| #I #L1 #L2 #V #T #d #_ #_ #b #J #W #U #H destruct
-]
-qed-.
-
-lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
-
-fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
- ∀I,V,T. X = ⓕ{I}V.T →
- llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #d #p #_ #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #d #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #V #T #d #HV #HT #J #W #U #H destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
-/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpx_sn_fwd_length: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d //
-#I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #HLK1 #HLK2 #_ #_ #HK12
-lapply (ldrop_fwd_length … HLK1) -HLK1
-lapply (ldrop_fwd_length … HLK2) -HLK2
-normalize //
-qed-.
-
-lemma llpx_sn_fwd_ldrop_sn: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
- ∀K1,i. ⇩[i] L1 ≡ K1 → ∃K2. ⇩[i] L2 ≡ K2.
-#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
-#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
-qed-.
-
-lemma llpx_sn_fwd_ldrop_dx: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
- ∀K2,i. ⇩[i] L2 ≡ K2 → ∃K1. ⇩[i] L1 ≡ K1.
-#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H
-#HL12 lapply (ldrop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by ldrop_O1_le/
-qed-.
-
-fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 → ∀i. X = #i →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R (yinj 0) V1 K1 K2 &
- R I K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #j #H destruct
-| #L1 #L2 #d #i #_ #Hid #j #H destruct /2 width=1 by or3_intro1/
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
- /3 width=9 by or3_intro2, ex5_5_intro/
-| #L1 #L2 #d #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
-| #L1 #L2 #d #p #_ #j #H destruct
-| #a #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
-| #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
-]
-qed-.
-
-lemma llpx_sn_fwd_lref: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R (yinj 0) V1 K1 K2 &
- R I K1 V1 V2 & d ≤ yinj i.
-/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
-
-lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R d V L1 L2.
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn R d V L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (②{I}V.T) L1 L2 →
- llpx_sn R d V L1 L2.
-#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
-qed-.
-
-(* Basic_properties *********************************************************)
-
-lemma llpx_sn_refl: ∀R. (∀I,L. reflexive … (R I L)) → ∀T,L,d. llpx_sn R d T L L.
-#R #HR #T #L @(f2_ind … rfw … L T) -L -T
-#n #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
-#i #Hn elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
-#HiL #d elim (ylt_split i d) /2 width=1 by llpx_sn_skip/
-elim (ldrop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
-qed-.
-
-lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2.
-#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/
-#a #I #V1 #T1 #Hn #L2 #HL12
-@llpx_sn_bind /2 width=1/ (**) (* explicit constructor *)
-@IH -IH // normalize /2 width=1 by eq_f2/
-qed-.
-
-lemma llpx_sn_ge_up: ∀R,L1,L2,U,dt. llpx_sn R dt U L1 L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
- dt ≤ d + e → llpx_sn R d U L1 L2.
-#R #L1 #L2 #U #dt #H elim H -L1 -L2 -U -dt
-[ #L1 #L2 #dt #k #HL12 #X #d #e #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
- elim (ylt_yle_false … Hidt) -Hidt
- @(yle_trans … Hdtde) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
-| #I #L1 #L2 #K1 #K2 #W1 #W2 #dt #i #Hdti #HLK1 #HLK2 #HW1 #HW12 #_ #X #d #e #H #_
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12
- lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
- normalize in ⊢ (%→%→?); -I -W1 -W2 -dt /3 width=1 by llpx_sn_skip, ylt_inj/
- | /4 width=9 by llpx_sn_lref, yle_inj, le_plus_b/
- ]
-| /2 width=1 by llpx_sn_free/
-| #L1 #L2 #dt #p #HL12 #X #d #e #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
- elim (lift_inv_bind2 … H) -H #V #T #HVW >commutative_plus #HTU #H destruct
- @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
- @(IHT … HTU) /2 width=1 by yle_succ/
-| #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
- elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct
- /3 width=4 by llpx_sn_flat/
-]
-qed-.
-
-(**) (* the minor premise comes first *)
-lemma llpx_sn_ge: ∀R,L1,L2,T,d1,d2. d1 ≤ d2 →
- llpx_sn R d1 T L1 L2 → llpx_sn R d2 T L1 L2.
-#R #L1 #L2 #T #d1 #d2 * -d1 -d2 (**) (* destructed yle *)
-/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
-qed-.
-
-lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 →
- llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2.
-/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
-
-lemma llpx_sn_co: ∀R1,R2. (∀I,L,T1,T2. R1 I L T1 T2 → R2 I L T1 T2) →
- ∀L1,L2,T,d. llpx_sn R1 d T L1 L2 → llpx_sn R2 d T L1 L2.
-#R1 #R2 #HR12 #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cofrees_alt.ma".
-include "basic_2/substitution/llpx_sn_alt_rec.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* alternative definition of llpx_sn (not recursive) *)
-definition llpx_sn_alt: relation4 bind2 lenv term term → relation4 ynat term lenv lenv ≝
- λR,d,T,L1,L2. |L1| = |L2| ∧
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R I1 K1 V1 V2
- ).
-
-(* Main properties **********************************************************)
-
-theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt R d T L1 L2.
-#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r … H) -H
-#HL12 #IHU @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv_ge … H) -H //
-[ -n #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
-| * #J1 #K10 #W10 #j #Hdj #Hji #HLK10 #HnW10 #HnU destruct
- lapply (ldrop_fwd_drop2 … HLK10) #H
- lapply (ldrop_conf_ge … H … HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
- elim (ldrop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by ldrop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
- lapply (ldrop_fwd_drop2 … HLK20) #H
- lapply (ldrop_conf_ge … H … HLK2 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK20
- elim (IHn K10 W10 … K20 0) -IHn -HL12 /3 width=6 by ldrop_fwd_rfw/
- elim (IHU … HnU HLK10 HLK20) -IHU -HnU -HLK10 -HLK20 //
-]
-qed.
-
-theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx_sn R d T L1 L2.
-#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
-elim (IHU … HLK1 HLK2) /3 width=6 by nlift_frees/
-#H #HV12 @and3_intro // @IHn -IHn /3 width=6 by ldrop_fwd_rfw/
-lapply (ldrop_fwd_drop2 … HLK1) #H1
-lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #H2
-@conj [ @(ldrop_fwd_length_eq1 … H1 H2) // ] -HL12
-#Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #_
->(minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus <minus_plus
-#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *)
-lapply (ldrop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1
-lapply (ldrop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2
-/4 width=14 by frees_be, yle_plus_dx2_trans, yle_succ_dx/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lift_neg.ma".
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* alternative definition of llpx_sn (recursive) *)
-inductive llpx_sn_alt_r (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
-| llpx_sn_alt_r_intro: ∀L1,L2,T,d.
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2
- ) →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
- ) → |L1| = |L2| → llpx_sn_alt_r R d T L1 L2
-.
-
-(* Compact definition of llpx_sn_alt_r **************************************)
-
-lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
- ) → llpx_sn_alt_r R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
-qed.
-
-lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
- (∀L1,L2,T,d. |L1| = |L2| → (
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
- ) → S d T L1 L2) →
- ∀L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → S d T L1 L2.
-#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
-qed-.
-
-lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
-/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
-qed-.
-
-lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
-[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
- /3 width=9 by nlift_bind_sn, and3_intro/
-| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
- lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
- lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
- elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
- @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H //
-qed-.
-
-lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
- R I K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
-#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
-#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
-/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpx_sn_alt_r_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt_r R d (⋆k) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
-qed.
-
-lemma llpx_sn_alt_r_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt_r R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
-qed.
-
-lemma llpx_sn_alt_r_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt_r R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
-/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
-qed.
-
-lemma llpx_sn_alt_r_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
- llpx_sn_alt_r R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
-lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
-/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
-qed.
-
-lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn_alt_r R 0 V1 K1 K2 → R I K1 V1 V2 →
- llpx_sn_alt_r R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
-[ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
- @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
-| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
- elim (lt_or_eq_or_gt i j) #Hij destruct
- [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
- | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
- lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
- | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
- ]
-]
-qed.
-
-lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,d.
- llpx_sn_alt_r R d V L1 L2 → llpx_sn_alt_r R d T L1 L2 →
- llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #T #d #HV #HT
-elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
-elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
-@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_flat … HnVT) -HnVT #H
-[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
-| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
-]
-qed.
-
-lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,d.
- llpx_sn_alt_r R d V L1 L2 →
- llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #T #d #HV #HT
-elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
-elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
-@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_bind … HnVT) -HnVT #H
-[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
-| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt_r R d T L1 L2.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 → llpx_sn R d T L1 L2.
-#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
-[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
-| #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length … H)
- #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
- [ * /2 width=1 by llpx_sn_free/
- | /2 width=1 by llpx_sn_skip/
- | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
- ]
-| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind … H) -H
- /3 width=1 by llpx_sn_bind/
-| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-]
-qed-.
-
-(* Alternative definition of llpx_sn (recursive) ****************************)
-
-lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
- ) → llpx_sn R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
-@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
-qed.
-
-lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
- (∀L1,L2,T,d. |L1| = |L2| → (
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
- ) → S d T L1 L2) →
- ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
-#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
-#H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
-qed-.
-
-lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
-#H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/llpx_sn_leq.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Advanced forward lemmas **************************************************)
-
-lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- i < d ∨
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
- R I K1 V1 V2 & d ≤ i.
-#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
-[ #_ #H elim (lt_refl_false i)
- lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
- /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
- lapply (ldrop_mono … HLK22 … HLK2) -L2 #H destruct
- /3 width=5 by ex4_2_intro, or_intror/
-]
-qed-.
-
-lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- i < d ∨
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
- R I K1 V1 V2 & d ≤ i.
-#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
-[ #H #_ elim (lt_refl_false i)
- lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
- /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
- lapply (ldrop_mono … HLK11 … HLK1) -L1 #H destruct
- /3 width=5 by ex4_2_intro, or_intror/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- llpx_sn R 0 V1 K1 K2 & R I K1 V1 V2.
-#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
-[ #H elim (ylt_yle_false … H Hdi)
-| * /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R 0 V1 K1 K2 & R I K1 V1 V2.
-#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
-[ #H elim (ylt_yle_false … H Hdi)
-| * /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R I1 K1 V1 V2.
-#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
-elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
-#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
-qed-.
-
-fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 →
- ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn R 0 V1 K1 K2 → R I K1 V1 V2 → llpx_sn R d T L1 L2.
-#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
-/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
-[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
- elim (yle_split_eq i d) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hid
- #H destruct /2 width=9 by llpx_sn_lref/
-| #I #L1 #L2 #K11 #K22 #V1 #V2 #d0 #i #Hd0i #HLK11 #HLK22 #HK12 #HV12 #_ #d #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
- /3 width=9 by llpx_sn_lref, yle_pred_sn/
-| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
- /4 width=9 by llpx_sn_bind, ldrop_drop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
- /3 width=9 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 →
- ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn R 0 V1 K1 K2 → R I K1 V1 V2 → llpx_sn R d T L1 L2.
-/2 width=9 by llpx_sn_inv_S_aux/ qed-.
-
-lemma llpx_sn_inv_bind_O: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
-/3 width=9 by ldrop_pair, conj, llpx_sn_inv_S/
-qed-.
-
-(* More advanced forward lemmas *********************************************)
-
-lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
- ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R J L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
-/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
-
-lemma llpx_sn_dec: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
- ∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
-#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
-| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
- [ #HL12 #d elim (ylt_split i d) /3 width=1 by llpx_sn_skip, or_introl/
- #Hdi elim (lt_or_ge i (|L1|)) #HiL1
- elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
- elim (ldrop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
- elim (ldrop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
- elim (eq_bind2_dec I2 I1)
- [ #H2 elim (HR I1 K1 V1 V2) -HR
- [ #H3 elim (IH K1 V1 … K2 0) destruct
- /3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
- ]
- ]
- -IH #H3 @or_intror
- #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ]
- [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
- |7,8,9: /2 width=4 by ylt_yle_false/
- ]
- #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
- lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
- lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
- #H #H0 destruct /2 width=1 by/
- ]
-| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
-| #a #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V … L2 d) /2 width=1 by/
- elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯d)) -IH /3 width=1 by or_introl, llpx_sn_bind/
- #H1 #H2 @or_intror
- #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
-| #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V … L2 d) /2 width=1 by/
- elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, llpx_sn_flat/
- #H1 #H2 @or_intror
- #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
-]
--n /4 width=4 by llpx_sn_fwd_length, or_intror/
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma llpx_sn_lift_le: ∀R. (∀I. l_liftable (R I)) →
- ∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2.
-#R #HR #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
-[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_sort/
-| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hdi #H destruct
- [ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_skip/
- | elim (ylt_yle_false … Hid0) -L1 -L2 -K1 -K2 -e -Hid0
- /3 width=3 by yle_trans, yle_inj/
- ]
-| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hdi #H destruct [ -HK12 | -IHK12 ]
- [ elim (ldrop_trans_lt … HLK1 … HK11) // -K1
- elim (ldrop_trans_lt … HLK2 … HK22) // -Hdi -K2
- /3 width=18 by llpx_sn_lref/
- | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
- lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hdi -Hd0 -K2
- /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
- ]
-| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hid #H destruct
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
- [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
- lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
- ]
-| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d -e
- /2 width=1 by llpx_sn_gref/
-| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
- #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
-| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
- #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_lift_ge: ∀R,K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → d ≤ d0 → llpx_sn R (d0+e) U L1 L2.
-#R #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
-[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_sort/
-| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
- [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
- | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
- ]
-| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hid #H destruct
- [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -e -Hid0
- /3 width=3 by ylt_yle_trans, ylt_inj/
- | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
- lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hid -Hd0 -K2
- /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
- ]
-| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hid #H destruct
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
- [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
- lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
- ]
-| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_gref/
-| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
- #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, ldrop_skip, yle_succ/
-| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
- #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma llpx_sn_inv_lift_le: ∀R. (∀I. l_deliftable_sn (R I)) →
- ∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2.
-#R #HR #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
-[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- [ /2 width=1 by llpx_sn_skip/
- | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct [ -HK12 | -IHK12 ]
- [ elim (ldrop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
- elim (ldrop_conf_lt … HLK2 … HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
- elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
- lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
- /3 width=10 by llpx_sn_lref/
- | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0
- elim (le_inv_plus_l … Hid) -Hid /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
- ]
-| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
- lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
-| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d ≤ d0 → d0 ≤ yinj d + e → llpx_sn R d T K1 K2.
-#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
-[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct
- [ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/
- | elim (ylt_yle_false … Hid0) -L1 -L2 -Hd0 -Hid0
- /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct
- [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0
- /3 width=3 by ylt_yle_trans, ylt_inj/
- | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e
- elim (le_inv_plus_l … Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/
- ]
-| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
- lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_bind2 … H) -H
- >commutative_plus #V #T #HVW #HTU #H destruct
- @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
- @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
-| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ d0 → llpx_sn R (d0-e) T K1 K2.
-#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
-[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct [ -Hid0 | -Hded0 ]
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- [ /4 width=3 by llpx_sn_skip, yle_plus_to_minus_inj2, ylt_yle_trans, ylt_inj/
- | elim (le_inv_plus_l … Hid) -Hid #_
- /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct
- [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hid0
- /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
- | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hded0 -Hid
- /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
- ]
-| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
- lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct
- @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
- <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
- @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
-| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K1,K2,e. ⇩[e] L1 ≡ K1 → ⇩[e] L2 ≡ K2 →
- ∀T. ⇧[0, e] T ≡ U → llpx_sn R 0 T K1 K2.
-/2 width=11 by llpx_sn_inv_lift_be/ qed-.
-
-lemma llpx_sn_ldrop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K1,e. ⇩[e] L1 ≡ K1 → ∀T. ⇧[0, e] T ≡ U →
- ∃∃K2. ⇩[e] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K1 #e #HLK1 #T #HTU elim (llpx_sn_fwd_ldrop_sn … HU … HLK1)
-/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
-qed-.
-
-lemma llpx_sn_ldrop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
- ∃∃K1. ⇩[e] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K2 #e #HLK2 #T #HTU elim (llpx_sn_fwd_ldrop_dx … HU … HLK2)
-/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
-qed-.
-
-(* Inversion lemmas on negated lazy pointwise extension *********************)
-
-lemma nllpx_sn_inv_bind: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
- ∀a,I,L1,L2,V,T,d. (llpx_sn R d (ⓑ{a,I}V.T) L1 L2 → ⊥) →
- (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
-#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
-/4 width=1 by llpx_sn_bind, or_intror, or_introl/
-qed-.
-
-lemma nllpx_sn_inv_flat: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
- ∀I,L1,L2,V,T,d. (llpx_sn R d (ⓕ{I}V.T) L1 L2 → ⊥) →
- (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R d T L1 L2 → ⊥).
-#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
-/4 width=1 by llpx_sn_flat, or_intror, or_introl/
-qed-.
-
-lemma nllpx_sn_inv_bind_O: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
- ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
- (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
-/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lpx_sn_ldrop.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Properties on pointwise extensions ***************************************)
-
-lemma lpx_sn_llpx_sn: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀T,L1,L2,d. lpx_sn R L1 L2 → llpx_sn R d T L1 L2.
-#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
-| -HR #i elim (lt_or_ge i (|L1|))
- [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
- #Hi #Hn #L2 #d elim (ylt_split i d)
- [ -n /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
- #Hdi #HL12 elim (ldrop_O1_lt (Ⓕ) L1 i) //
- #I #K1 #V1 #HLK1 elim (lpx_sn_ldrop_conf … HL12 … HLK1) -HL12
- /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
-| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
-| /4 width=1 by llpx_sn_bind, lpx_sn_pair/
-| -HR /3 width=1 by llpx_sn_flat/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/llpx_sn_ldrop.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Properties about transitive closure **************************************)
-
-lemma llpx_sn_TC_pair_dx: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀I,L,V1,V2,T. LTC … (R I) L V1 V2 →
- LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
-#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2
-/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/predsn_3.ma".
-include "basic_2/relocation/lpx_sn.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (λ_.cpr G).
-
-interpretation "parallel reduction (local environment, sn variant)"
- 'PRedSn G L1 L2 = (lpr G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: includes: wcpr0_gen_sort *)
-lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-(* Basic_1: includes: wcpr0_gen_head *)
-lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Note: lemma 250 *)
-lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
- ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
-/2 width=1 by lpx_sn_pair/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
- pr0_subst1_back
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/predsn_5.ma".
-include "basic_2/reduction/lpr.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
-
-definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
- λh,g,G. lpx_sn (λ_.cpx h g G).
-
-interpretation "extended parallel reduction (local environment, sn variant)"
- 'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpx_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-lemma lpx_inv_pair1: ∀h,g,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpx_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpx_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
-/2 width=1 by lpx_sn_inv_pair/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡[h, g] L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
- ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
-/2 width=1 by lpx_sn_pair/ qed.
-
-lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_length.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-inductive lpx_sn (R:relation4 bind2 lenv term term): relation lenv ≝
-| lpx_sn_atom: lpx_sn R (⋆) (⋆)
-| lpx_sn_pair: ∀I,K1,K2,V1,V2.
- lpx_sn R K1 K2 → R I K1 V1 V2 →
- lpx_sn R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_sn_refl: ∀R. (∀I,L. reflexive ? (R I L)) → reflexive … (lpx_sn R).
-#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. lpx_sn R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-#R #L1 #L2 * -L1 -L2
-[ #J #K1 #V1 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. lpx_sn R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
- ∃∃K1,V1. lpx_sn R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #L1 #L2 * -L1 -L2
-[ #J #K2 #V2 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. lpx_sn R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
- lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
- ∧∧ lpx_sn R L1 L2 & R I1 L1 V1 V2 & I1 = I2.
-#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L1 -L2 normalize //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop.ma".
-include "basic_2/relocation/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* alternative definition of lpx_sn *)
-definition lpx_sn_alt: relation4 bind2 lenv term term → relation lenv ≝
- λR,L1,L2. |L1| = |L2| ∧
- (∀I1,I2,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R I1 K1 V1 V2
- ).
-
-(* Basic forward lemmas ******************************************************)
-
-lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
-#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
-normalize /2 width=1 by length_inv_zero_sn/
-qed-.
-
-lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. lpx_sn_alt R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-#R #I1 #L2 #K1 #V1 #H elim H -H
-#H #IH elim (length_inv_pos_sn … H) -H
-#I2 #K2 #V2 #HK12 #H destruct
-elim (IH I1 I2 K1 K2 V1 V2 0) //
-#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
-@conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
-/2 width=1 by ldrop_drop, conj/
-qed-.
-
-lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
-#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
-normalize /2 width=1 by length_inv_zero_dx/
-qed-.
-
-lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. lpx_sn_alt R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #I2 #L1 #K2 #V2 #H elim H -H
-#H #IH elim (length_inv_pos_dx … H) -H
-#I1 #K1 #V1 #HK12 #H destruct
-elim (IH I1 I2 K1 K2 V1 V2 0) //
-#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
-@conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
-/2 width=1 by ldrop_drop, conj/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
-#R @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
-#H destruct
-qed.
-
-lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
- lpx_sn_alt R L1 L2 → R I L1 V1 V2 →
- lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
-#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
-#HL12 #IH @conj normalize //
-#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
-[ #HLK1 #HLK2
- lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
- lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
- /2 width=1 by conj/
-| -HL12 -HV12 /3 width=6 by ldrop_inv_drop1/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
-#R #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
-#R #L1 elim L1 -L1
-[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
-| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
-]
-qed-.
-
-(* alternative definition of lpx_sn *****************************************)
-
-lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R I1 K1 V1 V2
- ) → lpx_sn R L1 L2.
-/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
-
-lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R I1 K1 V1 V2.
-#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
-#H elim H -H /3 width=4 by conj/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_leq.ma".
-include "basic_2/relocation/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* Properies on dropping ****************************************************)
-
-lemma lpx_sn_ldrop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
- ∀I,K1,V1,i. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R I K1 V1 V2.
-#R #L1 #L2 #H elim H -L1 -L2
-[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
- [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
- | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
- /3 width=5 by ldrop_drop_lt, ex3_2_intro/
- ]
-]
-qed-.
-
-lemma lpx_sn_ldrop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
- ∀I,K2,V2,i. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R I K1 V1 V2.
-#R #L1 #L2 #H elim H -L1 -L2
-[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
- [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
- | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
- /3 width=5 by ldrop_drop_lt, ex3_2_intro/
- ]
-]
-qed-.
-
-lemma lpx_sn_deliftable_dropable: ∀R. (∀I. l_deliftable_sn (R I)) → dropable_sn (lpx_sn R).
-#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
- /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
-| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct
- /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 /3 width=3 by ldrop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (HR … HV12 … HLK1 … HWV1) -V1
- elim (IHLK1 … HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/
-]
-qed-.
-
-lemma lpx_sn_liftable_dedropable: ∀R. (∀I,L. reflexive ? (R I L)) →
- (∀I. l_liftable (R I)) → dedropable_sn (lpx_sn R).
-#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
- /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/
-| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #K2 #V2 #HK12 #HV12 #H destruct
- lapply (lpx_sn_fwd_length … HK12)
- #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
- /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
- @leq_O2 normalize //
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
- /3 width=5 by ldrop_drop, leq_pair, lpx_sn_pair, ex3_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
- elim (IHLK1 … HK12) -K1
- /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/
-]
-qed-.
-
-fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
- d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
- /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
-| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
- #K1 #V1 #HK12 #HV12 #H destruct
- /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
-| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
- #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
- <plus_n_Sm #H destruct
-]
-qed-.
-
-lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
-/2 width=5 by lpx_sn_dropable_aux/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. lpx_sn (λ_.R1) L0 L1 → ∀L2. lpx_sn (λ_.R2) L0 L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn (λ_.R1) L1 L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn (λ_.R)).
-#R #HR #L1 #L #H elim H -L1 -L //
-#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
-elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
-qed-.
-
-theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
- confluent2 … (lpx_sn (λ_.R1)) (lpx_sn (λ_.R2)).
-#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
-[ #_ #X1 #H1 #X2 #H2 -n
- >(lpx_sn_inv_atom1 … H1) -X1
- >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
-| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
- elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
- elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* Properties on transitive_closure *****************************************)
-
-lemma TC_lpx_sn_pair_refl: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,V. TC … (lpx_sn R) (L1.ⓑ{I}V) (L2. ⓑ{I}V).
-#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
-[ /2 width=1 by lpx_sn_refl/
-| /3 width=1 by TC_reflexive, lpx_sn_refl/
-| /3 width=5 by lpx_sn_pair, step/
-]
-qed-.
-
-lemma TC_lpx_sn_pair: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … (R I) L1 V1 V2 →
- TC … (lpx_sn R) (L1.ⓑ{I}V1) (L2. ⓑ{I}V2).
-#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
-[ /2 width=1 by TC_lpx_sn_pair_refl/
-| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
-]
-qed-.
-
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀I,L. reflexive … (R I L)) →
- ∀L1,L2. lpx_sn (λI.LTC … (R I)) L1 L2 →
- TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
-qed-.
-
-(* Inversion lemmas on transitive closure ***********************************)
-
-lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
-#R #L1 #H @(TC_ind_dx … L1 H) -L1
-[ /2 width=2 by lpx_sn_inv_atom2/
-| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair2: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
- ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … (R I) K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
-[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
-| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
- elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
- lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_ind: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
- ∀S:relation lenv.
- S (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … (R I) K1 V1 V2 →
- S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
- elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
- #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
-#R #L2 #H elim H -L2
-[ /2 width=2 by lpx_sn_inv_atom1/
-| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
-]
-qed-.
-
-fact TC_lpx_sn_inv_pair1_aux: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … (R I) K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
-[ #J #K #W #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair1: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
- ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … (R I) K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (λI.LTC … (R I)) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
-(* Forward lemmas on transitive closure *************************************)
-
-lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L2
-[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/llpx_sn_ldrop.ma".
-include "basic_2/static/ssta.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ******************************)
-
-(* Properties on lazy sn pointwise extensions *******************************)
-
-lemma ssta_llpx_sn_conf: ∀R. (∀I,L.reflexive … (R I L)) →
- (∀I.l_liftable (R I)) →
- ∀h,g,G. s_r_confluent1 … (ssta h g G) (llpx_sn R 0).
-#R #H1R #H2R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
-[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
-| #G #Ls #Ks #V1s #W2s #V2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
- lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
- lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
-| #G #Ls #Ks #V1s #W1s #l #i #HLKs #Hl #HVW1s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
- lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
- lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW1s) -HLKs -HLKd -HVW1s /2 width=1 by/ (**) (* full auto too slow *)
-| #a #I #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
- /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
-| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( h ⊢ break term 46 L1 : ⊑ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'CrSubEqN $h $L1 $L2 }.
-
-notation "hvbox( h ⊢ break term 46 L1 : : ⊑ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'CrSubEqNAlt $h $L1 $L2 }.
-
-include "basic_2/dynamic/nta.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
-
-(* Note: may not be transitive *)
-inductive lsubn (h:sh): relation lenv ≝
-| lsubn_atom: lsubn h (⋆) (⋆)
-| lsubn_pair: ∀I,L1,L2,W. lsubn h L1 L2 → lsubn h (L1. ⓑ{I} W) (L2. ⓑ{I} W)
-| lsubn_abbr: ∀L1,L2,V,W. ⦃h, L1⦄ ⊢ V : W → ⦃h, L2⦄ ⊢ V : W →
- lsubn h L1 L2 → lsubn h (L1. ⓓV) (L2. ⓛW)
-.
-
-interpretation
- "local environment refinement (native type assigment)"
- 'CrSubEqN h L1 L2 = (lsubn h L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#h #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑ L2 → L2 = ⋆.
-/2 width=4/ qed-.
-
-fact lsubn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
- h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr.
-#h #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
-]
-qed.
-
-lemma lsubn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑ L2 →
- (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
- h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
-
-fact lsubn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L2 = ⋆ → L1 = ⋆.
-#h #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubc_inv_atom2: ∀h,L1. h ⊢ L1 :⊑ ⋆ → L1 = ⋆.
-/2 width=4/ qed-.
-
-fact lsubn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
- h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst.
-#h #L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
-]
-qed.
-
-(* Basic_1: was: csubt_gen_bind *)
-lemma lsubn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑ K2. ⓑ{I} W →
- (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
- h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
-
-(* Basic_forward lemmas *****************************************************)
-
-lemma lsubn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L1|] L2.
-#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L2|] L2.
-#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: csubt_refl *)
-lemma lsubn_refl: ∀h,L. h ⊢ L :⊑ L.
-#h #L elim L -L // /2 width=1/
-qed.
-
-(* Basic_1: removed theorems 6:
- csubt_gen_flat csubt_drop_flat csubt_clear_conf
- csubt_getl_abbr csubt_getl_abst csubt_ty3_ld
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/equivalence/cpcs_cpcs.ma".
-include "basic_2/dynamic/lsubn.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
-
-(* Properties on context-sensitive parallel equivalence for terms ***********)
-
-(* Basic_1: was: csubt_pr2 *)
-lemma cpr_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 →
- ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
-/3 width=4 by lsubn_fwd_lsubs2, cpr_lsubs_trans/ qed.
-
-lemma cprs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 →
- ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-/3 width=4 by lsubn_fwd_lsubs2, cprs_lsubs_trans/ qed.
-
-(* Basic_1: was: csubt_pc3 *)
-lemma cpcs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 →
- ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
-/3 width=4 by lsubn_fwd_lsubs2, cpcs_lsubs_trans/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/lsubn.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
-
-(* Properties concerning basic local environment slicing ********************)
-
-(* Note: the constant 0 cannot be generalized *)
-lemma lsubn_ldrop_O1_conf: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. h ⊢ K1 :⊑ K2 & ⇩[0, e] L2 ≡ K2.
-#h #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
- [ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
- ]
-| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
- [ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
- ]
-]
-qed.
-
-(* Note: the constant 0 cannot be generalized *)
-(* Basic_1: was only: csubt_drop_abbr csubt_drop_abst *)
-lemma lsubn_ldrop_O1_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. h ⊢ K1 :⊑ K2 & ⇩[0, e] L1 ≡ K1.
-#h #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
- [ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
- ]
-| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
- [ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/lsubn_nta.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
-
-(* Main properties **********************************************************)
-
-(* Note: new property *)
-theorem lsubn_trans: ∀h,L1,L. h ⊢ L1 :⊑ L → ∀L2. h ⊢ L :⊑ L2 → h ⊢ L1 :⊑ L2.
-#h #L1 #L #H elim H -L1 -L
-[ #X #H >(lsubn_inv_atom1 … H) -H //
-| #I #L1 #L #V #HL1 #H1W #IHL1 #X #H
- elim (lsubn_inv_pair1 … H) -H * #L2
- [ #HL2 #H #H2W destruct /4 width=1/
- | #W #H1VW #H2VW #HL2 #H1 #H2 destruct /3 width=3/
- ]
-| #L1 #L #V1 #W1 #H1VW1 #H2VW1 #HL1 #IHL1 #X #H
- elim (lsubn_inv_pair1 … H) -H * #L2
- [ #HL2 #H #HW destruct /3 width=1/
- | #V #_ #_ #_ #_ #H destruct
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/nta_nta.ma".
-include "basic_2/dynamic/lsubn_ldrop.ma".
-include "basic_2/dynamic/lsubn_cpcs.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
-
-(* Properties concerning atomic arity assignment ****************************)
-
-(* Note: the corresponding confluence property does not hold *)
-(* Basic_1: was: csubt_ty3 *)
-lemma lsubn_nta_trans: ∀h,L2,T,U. ⦃h, L2⦄ ⊢ T : U → ∀L1. h ⊢ L1 :⊑ L2 →
- ⦃h, L1⦄ ⊢ T : U.
-#h #L2 #T #U #H elim H -L2 -T -U
-[ //
-| #L2 #K2 #V2 #W2 #U2 #i #HLK2 #_ #WU2 #IHVW2 #L1 #HL12
- elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsubn_inv_pair2 … H) -H * #K1
- [ #HK12 #H destruct /3 width=6/
- | #V1 #_ #_ #_ #_ #H destruct
- ]
-| #L2 #K2 #W2 #V2 #U2 #i #HLK2 #_ #HWU2 #IHWV2 #L1 #HL12
- elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsubn_inv_pair2 … H) -H * #K1 [ | -IHWV2 ]
- [ #HK12 #H destruct /3 width=6/
- | #V1 #H1V1W2 #_ #_ #H #_ destruct /2 width=6/
- ]
-| /4 width=2/
-| /3 width=1/
-| /3 width=2/
-| /3 width=1/
-| /4 width=6/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lpss_ldrop.ma".
-include "basic_2/static/aaa_lift.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about sn parallel substitution ********************************)
-
-(* Note: lemma 500 *)
-lemma aaa_cpss_lpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. L1 ⊢ T1 ▶* T2 →
- ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #H elim H -L1 -T1 -A
-[ #L1 #k #X #H
- >(cpss_inv_sort1 … H) -H //
-| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
- elim (cpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
- | * #Y #Z #V2 #H #HV12 #HV2
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
- ]
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
-| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
-| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
-]
-qed-.
-
-lemma aaa_cpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ▶* T2 → L ⊢ T2 ⁝ A.
-/2 width=5 by aaa_cpss_lpss_conf/ qed-.
-
-lemma aaa_lpss_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T ⁝ A.
-/2 width=5 by aaa_cpss_lpss_conf/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cprs_lpss.ma".
-include "basic_2/equivalence/cpcs_cpcs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-(* Properties on sn parallel substitution for local environments ************)
-
-lemma cpcs_lpss_conf: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
- ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T1 ⬌* T2.
-#L1 #T1 #T2 #H #L2 #HL12
-elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
-elim (cprs_lpss_conf_dx … HT1 … HL12) -HT1 #U1 #H1 #HTU1
-elim (cprs_lpss_conf_dx … HT2 … HL12) -L1 #U2 #H2 #HTU2
-elim (cpss_conf … H1 … H2) -T #U #HU1 #HU2
-lapply (cprs_cpss_trans … HTU1 … HU1) -U1
-lapply (cprs_cpss_trans … HTU2 … HU2) -U2 /2 width=3/
-qed-.
-
-lemma cpcs_cpss_lpss_conf: ∀L1,T,T2. L1 ⊢ T ⬌* T2 → ∀T1. L1 ⊢ T ▶* T1 →
- ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T1 ⬌* T2.
-#L1 #T #T2 #HT2 #T1 #HT1 #L2 #HL12
-lapply (cpcs_lpss_conf … HT2 … HL12) -HT2 #HT2
-elim (lpss_cpss_conf_dx … HT1 … HL12) -L1 #T0 #HT0 #HT10
-lapply (cpcs_cpss_conf … HT0 … HT2) -T #HT02
-lapply (cpcs_cpss_strap2 … HT10 … HT02) -T0 //
-qed-.
-
-lemma cpcs_cpss2_lpss_conf: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
- ∀T3. L1 ⊢ T1 ▶* T3 → ∀T4. L1 ⊢ T2 ▶* T4 →
- ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T3 ⬌* T4.
-#L1 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 #L2 #HL12
-lapply (cpcs_cpss_lpss_conf … HT12 … HT13 … HL12) -T1 #HT32
-elim (lpss_cpss_conf_dx … HT24 … HL12) -L1 #T #HT2 #HT4
-lapply (cpcs_cpss_strap1 … HT32 … HT2) -T2 #HT3
-lapply (cpcs_cpss_div … HT3 … HT4) -T //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 ➤ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRestStar $L $T1 $T2 }.
-
-include "basic_2/substitution/cpss.ma".
-
-(* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
-
-inductive cpqs: lenv → relation term ≝
-| cpqs_atom : ∀I,L. cpqs L (⓪{I}) (⓪{I})
-| cpqs_delta: ∀L,K,V,V2,W2,i.
- ⇩[0, i] L ≡ K. ⓓV → cpqs K V V2 →
- ⇧[0, i + 1] V2 ≡ W2 → cpqs L (#i) W2
-| cpqs_bind : ∀a,I,L,V1,V2,T1,T2.
- cpqs L V1 V2 → cpqs (L. ⓑ{I} V1) T1 T2 →
- cpqs L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| cpqs_flat : ∀I,L,V1,V2,T1,T2.
- cpqs L V1 V2 → cpqs L T1 T2 →
- cpqs L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| cpqs_zeta : ∀L,V,T1,T,T2. cpqs (L.ⓓV) T1 T →
- ⇧[0, 1] T2 ≡ T → cpqs L (+ⓓV. T1) T2
-| cpqs_tau : ∀L,V,T1,T2. cpqs L T1 T2 → cpqs L (ⓝV. T1) T2
-.
-
-interpretation "context-sensitive restricted parallel computation (term)"
- 'PRestStar L T1 T2 = (cpqs L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpqs_lsubr_trans: lsub_trans … cpqs lsubr.
-#L1 #T1 #T2 #H elim H -L1 -T1 -T2
-[ //
-| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
-| /4 width=1/
-|4,6: /3 width=1/
-| /4 width=3/
-]
-qed-.
-
-lemma cpss_cpqs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➤* T2.
-#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
-qed.
-
-lemma cpqs_refl: ∀T,L. L ⊢ T ➤* T.
-/2 width=1/ qed.
-
-lemma cpqs_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
- ∃∃T2,T. L ⊢ T1 ➤* T2 & ⇧[d, 1] T ≡ T2.
-#L #K #V #T1 #d #HLK
-elim (cpss_delift … T1 … HLK) -HLK /3 width=4/
-qed-.
-
-lemma cpqs_append: l_appendable_sn … cpqs.
-#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
-#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpqs_delta … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpqs_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➤* T2 → ∀I. T1 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ➤* V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-#L #T1 #T2 * -L -T1 -T2
-[ #I #L #J #H destruct /2 width=1/
-| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #L #V #T1 #T #T2 #_ #_ #J #H destruct
-| #L #V #T1 #T2 #_ #J #H destruct
-]
-qed-.
-
-lemma cpqs_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➤* T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ➤* V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-/2 width=3 by cpqs_inv_atom1_aux/ qed-.
-
-lemma cpqs_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➤* T2 → T2 = ⋆k.
-#L #T2 #k #H
-elim (cpqs_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-lemma cpqs_inv_lref1: ∀L,T2,i. L ⊢ #i ➤* T2 →
- T2 = #i ∨
- ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ➤* V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #H
-elim (cpqs_inv_atom1 … H) -H /2 width=1/
-* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
-qed-.
-
-lemma cpqs_inv_gref1: ∀L,T2,p. L ⊢ §p ➤* T2 → T2 = §p.
-#L #T2 #p #H
-elim (cpqs_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-fact cpqs_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #b #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
-| #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
-]
-qed-.
-
-lemma cpqs_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-/2 width=3 by cpqs_inv_bind1_aux/ qed-.
-
-lemma cpqs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓓ V1 ⊢ T1 ➤* T2 &
- U2 = ⓓ{a} V2. T2
- ) ∨
- ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true.
-#a #L #V1 #T1 #U2 #H
-elim (cpqs_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
-qed-.
-
-lemma cpqs_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➤* U2 →
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓛ V1 ⊢ T1 ➤* T2 &
- U2 = ⓛ{a} V2. T2.
-#a #L #V1 #T1 #U2 #H
-elim (cpqs_inv_bind1 … H) -H *
-[ /3 width=5/
-| #T #_ #_ #_ #H destruct
-]
-qed-.
-
-fact cpqs_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓕ{I} V2. T2
- ) ∨
- (L ⊢ T1 ➤* U2 ∧ I = Cast).
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
-| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
-| #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
-| #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
-]
-qed-.
-
-lemma cpqs_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓕ{I} V2. T2
- ) ∨
- (L ⊢ T1 ➤* U2 ∧ I = Cast).
-/2 width=3 by cpqs_inv_flat1_aux/ qed-.
-
-lemma cpqs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐ V1. T1 ➤* U2 →
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓐ V2. T2.
-#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H *
-[ /3 width=5/
-| #_ #H destruct
-]
-qed-.
-
-lemma cpqs_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝ V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓝ V2. T2
- ) ∨
- L ⊢ T1 ➤* U2.
-#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H * /2 width=1/ /3 width=5/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpqs_inv_bind1 … H) -H *
- [ #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/unfold/cpqs.ma".
-
-(* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
-
-(* Relocation properties ****************************************************)
-
-lemma cpqs_lift: l_liftable cpqs.
-#K #T1 #T2 #H elim H -K -T1 -T2
-[ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
- elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=8/
- | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=6/
- ]
-| #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
-| #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-| #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
- elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5/
-| #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
-]
-qed.
-
-lemma cpqs_inv_lift1: l_deliftable_sn cpqs.
-#L #U1 #U2 #H elim H -L -U1 -U2
-[ * #L #i #K #d #e #_ #T1 #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
- elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
- elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=8/
- ]
-| #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
- elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
-| #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -V1
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
-| #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU1 (K.ⓓW1) … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
- elim (lift_div_le … HU2 … HTU) -U // /3 width=5/
-| #L #V #U1 #U2 #_ #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lpss_lpss.ma".
-include "basic_2/reduction/lpr_lpss.ma".
-include "basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties on parallel substitution for terms ****************************)
-
-(* Basic_1: was: pr3_subst1 *)
-lemma cprs_cpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 → ∀T2. L ⊢ T0 ▶* T2 →
- ∃∃T. L ⊢ T1 ▶* T & L ⊢ T2 ➡* T.
-#L @TC_strip1 /2 width=3 by cpr_cpss_conf/ qed-. (**) (* auto /3 width=3/ fails because a δ-expansion gets in the way *)
-
-(* Properties on sn parallel substitution for local environments ************)
-
-lemma cprs_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ▶* L1 →
- ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡* T.
-#L0 #T0 #T1 #H elim H -T1
-[ #T1 #HT01 #L1 #HL01
- elim (cpr_lpss_conf_dx … HT01 … HL01) -L0 /3 width=3/
-| #T #T1 #_ #HT1 #IHT0 #L1 #HL01
- elim (IHT0 … HL01) #T2 #HT2 #HT02
- elim (cpr_lpss_conf_dx … HT1 … HL01) -L0 #T3 #HT13 #HT3
- elim (cpr_cpss_conf … HT3 … HT2) -T #T #HT3 #HT2
- lapply (cpss_trans … HT13 … HT3) -T3
- lapply (cprs_strap1 … HT02 … HT2) -T2 /2 width=3/
-]
-qed-.
-
-lemma cprs_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ▶* L1 →
- ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cprs_lpss_conf_dx … HT01 … HL01) -HT01 #T #HT1
-lapply (lpss_cpss_trans … HL01 … HT1) -HT1 /2 width=3/
-qed-.
-
-lemma cprs_cpss_lpss_conf_sn: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
- ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. L2 ⊢ T2 ➡* U2 & L1 ⊢ U1 ▶* U2.
-#L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
-elim (cprs_cpss_conf … HTU1 … HT12) -T1 #U #HU1 #HT2U
-elim (cprs_lpss_conf_sn … HT2U … HL12) -HT2U -HL12 #U2 #HU2 #HTU2
-lapply (cpss_trans … HU1 … HU2) -U /2 width=3/
-qed-.
-
-lemma cprs_cpss_lpss_conf_dx: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
- ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. L2 ⊢ T2 ➡* U2 & L2 ⊢ U1 ▶* U2.
-#L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
-elim (cprs_lpss_conf_dx … HTU1 … HL12) -HTU1 #U2 #HU12 #HT1U2
-elim (lpss_cpss_conf_dx … HT12 … HL12) -L1 #T #HT1 #HT2
-elim (cprs_cpss_conf … HT1U2 … HT1) -T1 #U #HU2 #HTU
-lapply (cpss_trans … HU12 … HU2) -U2
-lapply (cpss_cprs_trans … HT2 … HTU) -T /2 width=3/
-qed-.
-
-
-lemma cprs_cpss2_lpss_conf_sn: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 → ∀T2. L1 ⊢ T1 ▶* T2 →
- ∀U2. L1 ⊢ U1 ▶* U2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U. L2 ⊢ T2 ➡* U & L1 ⊢ U2 ▶* U.
-#L1 #T1 #U1 #HTU1 #T2 #HT12 #U2 #HU12 #L2 #HL12
-elim (cprs_cpss_lpss_conf_sn … HTU1 … HT12 … HL12) -T1 #T1 #HT21 #HUT1
-elim (cpss_conf … HU12 … HUT1) -U1 #U1 #HU21 #HTU1
-elim (lpss_cpss_conf_sn … HTU1 … HL12) -HTU1 -HL12 #U2 #HT1U2 #HU12
-lapply (cpss_trans … HU21 … HU12) -U1
-lapply (cprs_cpss_trans … HT21 … HT1U2) -T1 /2 width=3/
-qed-.
-
-lemma cprs_cpss2_lpss_conf_dx: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 → ∀T2. L1 ⊢ T1 ▶* T2 →
- ∀U2. L1 ⊢ U1 ▶* U2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U. L2 ⊢ T2 ➡* U & L2 ⊢ U2 ▶* U.
-#L1 #T1 #U1 #HTU1 #T2 #HT12 #U2 #HU12 #L2 #HL12
-elim (cprs_cpss_lpss_conf_dx … HTU1 … HT12 … HL12) -T1 #T1 #HT21 #HUT1
-elim (lpss_cpss_conf_dx … HU12 … HL12) -L1 #U #HU1 #HU2
-elim (cpss_conf … HU1 … HUT1) -U1 #U1 #HU1 #HTU1
-lapply (cpss_trans … HU2 … HU1) -U
-lapply (cprs_cpss_trans … HT21 … HTU1) -T1 /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 ▶* break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $L $T1 $T2 }.
-
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
-include "basic_2/substitution/lsubr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
-
-inductive cpss: lenv → relation term ≝
-| cpss_atom : ∀I,L. cpss L (⓪{I}) (⓪{I})
-| cpss_delta: ∀L,K,V,V2,W2,i.
- ⇩[0, i] L ≡ K. ⓓV → cpss K V V2 →
- ⇧[0, i + 1] V2 ≡ W2 → cpss L (#i) W2
-| cpss_bind : ∀a,I,L,V1,V2,T1,T2.
- cpss L V1 V2 → cpss (L. ⓑ{I} V1) T1 T2 →
- cpss L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| cpss_flat : ∀I,L,V1,V2,T1,T2.
- cpss L V1 V2 → cpss L T1 T2 →
- cpss L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "context-sensitive parallel substitution (term)"
- 'PSubstStar L T1 T2 = (cpss L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpss_lsubr_trans: lsub_trans … cpss lsubr.
-#L1 #T1 #T2 #H elim H -L1 -T1 -T2
-[ //
-| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
-| /4 width=1/
-| /3 width=1/
-]
-qed-.
-
-(* Basic_1: was by definition: subst1_refl *)
-lemma cpss_refl: ∀T,L. L ⊢ T ▶* T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic_1: was only: subst1_ex *)
-lemma cpss_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
- ∃∃T2,T. L ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
-#K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK /2 width=4/
- elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i) // /3 width=6/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
- ]
-]
-qed-.
-
-lemma cpss_append: l_appendable_sn … cpss.
-#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/
-#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpss_delta … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpss_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ∀I. T1 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-#L #T1 #T2 * -L -T1 -T2
-[ #I #L #J #H destruct /2 width=1/
-| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #I #H destruct /3 width=8/
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-]
-qed-.
-
-lemma cpss_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ▶* T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-/2 width=3 by cpss_inv_atom1_aux/ qed-.
-
-(* Basic_1: was only: subst1_gen_sort *)
-lemma cpss_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ▶* T2 → T2 = ⋆k.
-#L #T2 #k #H
-elim (cpss_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was only: subst1_gen_lref *)
-lemma cpss_inv_lref1: ∀L,T2,i. L ⊢ #i ▶* T2 →
- T2 = #i ∨
- ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #H
-elim (cpss_inv_atom1 … H) -H /2 width=1/
-* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
-qed-.
-
-lemma cpss_inv_gref1: ∀L,T2,p. L ⊢ §p ▶* T2 → T2 = §p.
-#L #T2 #p #H
-elim (cpss_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-fact cpss_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 ▶* V2 &
- L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
- U2 = ⓑ{a,I} V2. T2.
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #b #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5/
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-]
-qed-.
-
-lemma cpss_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* U2 →
- ∃∃V2,T2. L ⊢ V1 ▶* V2 &
- L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
- U2 = ⓑ{a,I} V2. T2.
-/2 width=3 by cpss_inv_bind1_aux/ qed-.
-
-fact cpss_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
- U2 = ⓕ{I} V2. T2.
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
-| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5/
-]
-qed-.
-
-lemma cpss_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* U2 →
- ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
- U2 = ⓕ{I} V2. T2.
-/2 width=3 by cpss_inv_flat1_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpss_fwd_tw: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #H elim H -L -T1 -T2 normalize
-/3 width=1 by monotonic_le_plus_l, le_plus/ (**) (* auto is too slow without trace *)
-qed-.
-
-lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpss_inv_bind1 … H) -H
- #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
-]
-qed-.
-
-(* Basic_1: removed theorems 27:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
- subst1_gen_lift_eq subst1_confluence_neq
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/cpss.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was only: subst1_lift_lt subst1_lift_ge *)
-lemma cpss_lift: l_liftable cpss.
-#K #T1 #T2 #H elim H -K -T1 -T2
-[ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
- elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=8/
- | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=6/
- ]
-| #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
-| #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-]
-qed.
-
-(* Basic_1: was only: subst1_gen_lift_lt subst1_gen_lift_ge *)
-lemma cpss_inv_lift1: l_deliftable_sn cpss.
-#L #U1 #U2 #H elim H -L -U1 -U2
-[ * #L #i #K #d #e #_ #T1 #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
- elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
- elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=8/
- ]
-| #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
- elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
-| #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -V1
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/sstas_lpss.ma".
-include "basic_2/computation/cprs_lpss.ma".
-include "basic_2/computation/dxprs.ma".
-
-(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
-
-(* Properties about sn parallel substitution for local environments *********)
-
-lemma dxprs_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L1 ⊢ U1 ▶* U2.
-#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #HL12
-elim (sstas_lpss_conf … HTU … HL12) -HTU #U0 #HTU0 #HU0
-elim (cprs_cpss_conf … HU1 … HU0) -U #U #HU1 #HU0
-elim (cprs_lpss_conf_sn … HU0 … HL12) -HU0 -HL12 #U2 #HU2 #HU02
-lapply (cpss_trans … HU1 … HU2) -U /3 width=3/
-qed-.
-
-lemma dxprs_cpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*➡*[g] U1 → ∀T2. L ⊢ T1 ▶* T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*➡*[g] U2 & L ⊢ U1 ▶* U2.
-#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12
-elim (sstas_cpss_conf … HTW1 … HT12) -T1 #W2 #HTW2 #HW12
-elim (cprs_cpss_conf … HWU1 … HW12) -W1 /3 width=3/
-qed-.
-
-lemma dxprs_cpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
- ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L1 ⊢ U1 ▶* U2.
-#h #g #L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
-elim (dxprs_cpss_conf … HTU1 … HT12) -T1 #U2 #HTU2 #HU12
-elim (dxprs_lpss_conf … HTU2 … HL12) -HTU2 -HL12 #U #HT2U #HU2
-lapply (cpss_trans … HU12 … HU2) -U2 /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 ⊢ ➤ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRestStarSn $T1 $T2 }.
-
-include "basic_2/substitution/lpss.ma".
-include "basic_2/unfold/cpqs.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
-
-definition lpqs: relation lenv ≝ lpx_sn cpqs.
-
-interpretation "restricted parallel computation (local environment, sn variant)"
- 'PRestStarSn L1 L2 = (lpqs L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpqs_inv_atom1: ∀L2. ⋆ ⊢ ➤* L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-lemma lpqs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ➤* L2 →
- ∃∃K2,V2. K1 ⊢ ➤* K2 & K1 ⊢ V1 ➤* V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpqs_inv_atom2: ∀L1. L1 ⊢ ➤* ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpqs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➤* K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ➤* K2 & K1 ⊢ V1 ➤* V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpqs_refl: ∀L. L ⊢ ➤* L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpqs_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ➤* K2 → K1 ⊢ V1 ➤* V2 →
- K1.ⓑ{I}V1 ⊢ ➤* K2.ⓑ{I}V2.
-/2 width=1/ qed.
-
-lemma lpqs_append: ∀K1,K2. K1 ⊢ ➤* K2 → ∀L1,L2. L1 ⊢ ➤* L2 →
- L1 @@ K1 ⊢ ➤* L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpqs_append/ qed.
-
-lemma lpss_lpqs: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➤* L2.
-#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpqs_fwd_length: ∀L1,L2. L1 ⊢ ➤* L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpqs_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➤* L →
- ∃∃K2,L2. K1 ⊢ ➤* K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpqs_fwd_append2: ∀L,K2,L2. L ⊢ ➤* K2 @@ L2 →
- ∃∃K1,L1. K1 ⊢ ➤* K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fsupp.ma".
-include "basic_2/unfold/lpqs_ldrop.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
-
-(* Main properties on context-sensitive rest parallel computation for terms *)
-
-theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs.
-#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L0 #T0 #IH #L1 * [|*]
-[ #I #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
- elim (cpqs_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpqs_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
- elim (lpqs_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpqs_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (fsupp_lref … HLK1) /3 width=9/
- ]
- | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- elim (lpqs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpqs_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
- elim (cpqs_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (fsupp_lref … HLK1) /3 width=9/
- ]
-| #a #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpqs_inv_bind1 … H1) -H1 *
- [ #V #T #HV1 #HT1 #H destruct
- elim (cpqs_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
- | #T2 #HT2 #HXT2 #H1 #H2 destruct /4 width=5/
- ]
- | #Y1 #HTY1 #HXY1 #H11 #H12 destruct
- elim (lift_total X2 0 1) #Y2 #HXY2
- lapply (cpqs_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
- ]
-| #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpqs_inv_flat1 … H1) -H1 *
- [ #V #T #HV1 #HT1 #H destruct
- elim (cpqs_inv_flat1 … H2) -H2 *
- [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
- | #HX2 #H destruct /3 width=5/
- ]
- | #HX1 #H destruct /3 width=5/
-]
-qed-.
-
-theorem cpqs_trans: ∀L. Transitive … (cpqs L).
-/2 width=5 by cpqs_trans_lpqs/ qed-.
-
-(* Properties on context-sensitive rest. parallel computation for terms *****)
-
-lemma lpqs_cpqs_trans: ∀L1,L2. L1 ⊢ ➤* L2 →
- ∀T1,T2. L2 ⊢ T1 ➤* T2 → L1 ⊢ T1 ➤* T2.
-/2 width=5 by cpqs_trans_lpqs/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/fsup.ma".
-include "basic_2/relocation/ldrop_lpx_sn.ma".
-include "basic_2/unfold/cpqs_lift.ma".
-include "basic_2/unfold/lpqs.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma lpqs_ldrop_conf: dropable_sn lpqs.
-/3 width=5 by lpx_sn_deliftable_dropable, cpqs_inv_lift1/ qed-.
-
-lemma ldrop_lpqs_trans: dedropable_sn lpqs.
-/3 width=9 by lpx_sn_liftable_dedropable, cpqs_lift/ qed-.
-
-lemma lpqs_ldrop_trans_O1: dropable_dx lpqs.
-/2 width=3 by lpx_sn_dropable/ qed-.
-
-(* Properties on context-sensitive rest. parallel computation for terms *****)
-
-lemma fsup_cpqs_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➤* U2 →
- ∃∃L,U1. L1 ⊢ ➤* L & L ⊢ T1 ➤* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [2: * ] [1,2,3,4,5: /3 width=5/ ]
-[ #L #K #U #T #d #e #HLK #HUT #He #U2 #HU2
- elim (lift_total U2 d e) #T2 #HUT2
- lapply (cpqs_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9/
-| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
- elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
- elim (lift_total T d e) #U #HTU
- elim (ldrop_lpqs_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (cpqs_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/lpqs_cpqs.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *****************)
-
-(* Main properties on context-sensitive rest parallel computation for terms *)
-
-fact cpqs_conf_lpqs_atom_atom:
- ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T.
-/2 width=3/ qed-.
-
-fact cpqs_conf_lpqs_atom_delta:
- ∀L0,i. (
- ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
-qed-.
-
-fact cpqs_conf_lpqs_delta_delta:
- ∀L0,i. (
- ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
- ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
-lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_bind_bind:
- ∀a,I,L0,V0,T0. (
- ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 →
- ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T.
-#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
-qed-.
-
-fact cpqs_conf_lpqs_bind_zeta:
- ∀L0,V0,T0. (
- ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 →
- ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T.
-#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_zeta_zeta:
- ∀L0,V0,T0. (
- ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T.
-#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
-elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
-elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_flat_flat:
- ∀I,L0,V0,T0. (
- ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 →
- ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T.
-#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
-qed-.
-
-fact cpqs_conf_lpqs_flat_tau:
- ∀L0,V0,T0. (
- ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_tau_tau:
- ∀L0,V0,T0. (
- ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
-qed-.
-
-theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs.
-#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
-[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_atom1 … H1) -H1
- elim (cpqs_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpqs_conf_lpqs_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpqs_conf_lpqs_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpqs_conf_lpqs_delta_delta/
- ]
-| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpqs_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpqs_conf_lpqs_bind_bind/
- | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/
- | /3 width=11 by cpqs_conf_lpqs_bind_zeta/
- | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/
- ]
-| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- ]
- elim (cpqs_inv_flat1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #HX2 #H2
- ] destruct
- [ /3 width=10 by cpqs_conf_lpqs_flat_flat/
- | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/
- | /3 width=8 by cpqs_conf_lpqs_flat_tau/
- | /3 width=7 by cpqs_conf_lpqs_tau_tau/
- ]
-]
-qed-.
-
-theorem cpqs_conf: ∀L. confluent … (cpqs L).
-/2 width=6 by cpqs_conf_lpqs/ qed-.
-
-(* Properties on context-sensitive rest. parallel computation for terms *****)
-
-lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
- ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
-qed-.
-
-lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
- ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lpqs_conf: confluent … lpqs.
-/3 width=6 by lpx_sn_conf, cpqs_conf_lpqs/
-qed-.
-
-theorem lpqs_trans: Transitive … lpqs.
-/3 width=5 by lpx_sn_trans, cpqs_trans_lpqs/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
- ∃∃L2,T2. L @@ L1 ⊢ ➤* L @@ L2 & L @@ L1 ⊢ T1 ➤* T2 &
- T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #HT1
- @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
-| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
- elim (cpqs_inv_bind1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
- lapply (lpqs_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
- @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fsupp.ma".
-include "basic_2/substitution/lpss_ldrop.ma".
-include "basic_2/reduction/lpr_ldrop.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Properties on context-sensitive parallel substitution for terms **********)
-
-fact cpr_cpss_conf_lpr_lpss_atom_atom:
- ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T.
-/2 width=3/ qed-.
-
-fact cpr_cpss_conf_lpr_lpss_atom_delta:
- ∀L0,i. (
- ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T.
-#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_delta_atom:
- ∀L0,i. (
- ∀L,T.⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02
-elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_delta_delta:
- ∀L0,i. (
- ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
- ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
-lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_bind_bind:
- ∀a,I,L0,V0,T0. (
- ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
-#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_bind_zeta:
- ∀L0,V0,T0. (
- ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T.
-#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_flat_flat:
- ∀I,L0,V0,T0. (
- ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
-#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_flat_tau:
- ∀L0,V0,T0. (
- ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T.
-#L0 #V0 #T0 #IH #T1 #HT01
-#V2 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_flat_beta:
- ∀a,L0,V0,W0,T0. (
- ∀L,T. ⦃L0,ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
-#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
-elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/
-qed-.
-
-fact cpr_cpss_conf_lpr_lpss_flat_theta:
- ∀a,L0,V0,W0,T0. (
- ∀L,T. ⦃L0,ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
- ) →
- ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
- ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 →
- ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
-#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (lift_total V 0 1) #U #HVU
-lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1
-elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
-elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
-elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
-/4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *)
-qed-.
-
-lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss.
-#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
-[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_atom1 … H1) -H1
- elim (cpss_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/
- ]
-| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
- elim (cpr_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ] destruct
- [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/
- | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/
- ]
-| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
- elim (cpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
- | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
- ] destruct
- [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/
- | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/
- | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/
- | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/
- ]
-]
-qed-.
-
-(* Basic_1: includes: pr0_subst1 *)
-(* Basic_1: was: pr2_subst1 *)
-lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L).
-/2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-.
-
-lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
- ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/
-qed-.
-
-lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
- ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
-qed-.
-
-(* Basic_1: includes: pr0_subst1_fwd *)
-lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 →
- ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/
-qed-.
-
-(* Properties on sn parallel substitution on local environments *************)
-
-lemma lpr_lpss_conf: confluent2 … lpr lpss.
-/3 width=6 by lpx_sn_conf, cpr_cpss_conf_lpr_lpss/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/lpr_lpss.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-
-(* Properties on sn parallel substitution on local environments *************)
-
-lemma lprs_lpss_conf: confluent2 … lprs lpss.
-/3 width=3 by TC_strip1, lpr_lpss_conf/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 ⊢ ▶ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarSn $T1 $T2 }.
-
-include "basic_2/grammar/lpx_sn.ma".
-include "basic_2/substitution/cpss.ma".
-
-(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
-
-(* Basic_1: includes: csubst1_bind *)
-definition lpss: relation lenv ≝ lpx_sn cpss.
-
-interpretation "parallel substitution (local environment, sn variant)"
- 'PSubstStarSn L1 L2 = (lpss L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpss_inv_atom1: ∀L2. ⋆ ⊢ ▶* L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-lemma lpss_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* L2 →
- ∃∃K2,V2. K1 ⊢ ▶* K2 & K1 ⊢ V1 ▶* V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpss_inv_atom2: ∀L1. L1 ⊢ ▶* ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpss_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ▶* K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ▶* K2 & K1 ⊢ V1 ▶* V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was by definition: csubst1_refl *)
-lemma lpss_refl: ∀L. L ⊢ ▶* L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpss_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ▶* K2 → K1 ⊢ V1 ▶* V2 →
- K1.ⓑ{I}V1 ⊢ ▶* K2.ⓑ{I}V2.
-/2 width=1/ qed.
-
-lemma lpss_append: ∀K1,K2. K1 ⊢ ▶* K2 → ∀L1,L2. L1 ⊢ ▶* L2 →
- L1 @@ K1 ⊢ ▶* L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpss_append/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpss_fwd_length: ∀L1,L2. L1 ⊢ ▶* L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpss_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ▶* L →
- ∃∃K2,L2. K1 ⊢ ▶* K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpss_fwd_append2: ∀L,K2,L2. L ⊢ ▶* K2 @@ L2 →
- ∃∃K1,L1. K1 ⊢ ▶* K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
-
-(* Basic_1: removed theorems 28:
- csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
- csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
- csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
- csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
- csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
- csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
- csubst1_head csubst1_flat csubst1_gen_head
- csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
- fsubst0_gen_base
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fsupp.ma".
-include "basic_2/substitution/lpss_ldrop.ma".
-
-(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
-
-(* Main properties on context-sensitive parallel substitution for terms *****)
-
-theorem cpss_trans_lpss: lpx_sn_transitive cpss cpss.
-#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L0 #T0 #IH #L1 * [|*]
-[ #I #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
- elim (cpss_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpss_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
- elim (lpss_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpss_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (fsupp_lref … HLK1) /3 width=9/
- ]
- | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
- elim (cpss_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (fsupp_lref … HLK1) /3 width=9/
- ]
-| #a #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpss_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
-| #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpss_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
-]
-qed-.
-
-(* Basic_1: was only: subst1_trans *)
-theorem cpss_trans: ∀L. Transitive … (cpss L).
-/2 width=5 by cpss_trans_lpss/ qed-.
-
-(* Properties on context-sensitive parallel substitution for terms **********)
-
-(* Basic_1: was only: subst1_subst1 *)
-lemma lpss_cpss_trans: ∀L1,L2. L1 ⊢ ▶* L2 →
- ∀T1,T2. L2 ⊢ T1 ▶* T2 → L1 ⊢ T1 ▶* T2.
-/2 width=5 by cpss_trans_lpss/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/fsup.ma".
-include "basic_2/relocation/ldrop_lpx_sn.ma".
-include "basic_2/substitution/cpss_lift.ma".
-include "basic_2/substitution/lpss.ma".
-
-(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma lpss_ldrop_conf: dropable_sn lpss.
-/3 width=5 by lpx_sn_deliftable_dropable, cpss_inv_lift1/ qed-.
-
-lemma ldrop_lpss_trans: dedropable_sn lpss.
-/3 width=9 by lpx_sn_liftable_dedropable, cpss_lift/ qed-.
-
-lemma lpss_ldrop_trans_O1: dropable_dx lpss.
-/2 width=3 by lpx_sn_dropable/ qed-.
-
-(* Properties on context-sensitive parallel substitution for terms **********)
-
-lemma fsup_cpss_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶* U2 →
- ∃∃L,U1. L1 ⊢ ▶* L & L ⊢ T1 ▶* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [2: * ] [1,2,3,4,5: /3 width=5/ ]
-[ #L #K #U #T #d #e #HLK #HUT #He #U2 #HU2
- elim (lift_total U2 d e) #T2 #HUT2
- lapply (cpss_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9/
-| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
- elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
- elim (lift_total T d e) #U #HTU
- elim (ldrop_lpss_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (cpss_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lpss_cpss.ma".
-
-(* SN PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ***************************)
-
-(* Main properties on context-sensitive parallel substitution for terms *****)
-
-fact cpss_conf_lpss_atom_atom:
- ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ▶* T.
-/2 width=3/ qed-.
-
-fact cpss_conf_lpss_atom_delta:
- ∀L0,i. (
- ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ▶* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpss_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
-qed-.
-
-fact cpss_conf_lpss_delta_delta:
- ∀L0,i. (
- ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
- ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V1. K0 ⊢ V0 ▶* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
- ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ▶* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpss_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (fsupp_lref … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
-lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
-qed-.
-
-fact cpss_conf_lpss_bind_bind:
- ∀a,I,L0,V0,T0. (
- ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
- ) →
- ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ▶* T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
- ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ▶* T.
-#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
-qed-.
-
-fact cpss_conf_lpss_flat_flat:
- ∀I,L0,V0,T0. (
- ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
- ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
- ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
- ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
- ) →
- ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0 ⊢ T0 ▶* T1 →
- ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
- ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
- ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ▶* T.
-#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
-qed-.
-
-theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss.
-#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
-[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpss_inv_atom1 … H1) -H1
- elim (cpss_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpss_conf_lpss_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpss_conf_lpss_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpss_conf_lpss_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpss_conf_lpss_delta_delta/
- ]
-| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpss_inv_bind1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct
- elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
- /3 width=10 by cpss_conf_lpss_bind_bind/
-| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpss_inv_flat1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct
- elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
- /3 width=10 by cpss_conf_lpss_flat_flat/
-]
-qed-.
-
-(* Basic_1: was only: subst1_confluence_eq *)
-theorem cpss_conf: ∀L. confluent … (cpss L).
-/2 width=6 by cpss_conf_lpss/ qed-.
-
-(* Properties on context-sensitive parallel substitution for terms **********)
-
-(* Basic_1: was only: subst1_subst1_back *)
-lemma lpss_cpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 →
- ∃∃T. L1 ⊢ T0 ▶* T & L1 ⊢ T1 ▶* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpss_conf_lpss … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
-qed-.
-
-lemma lpss_cpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 →
- ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ▶* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpss_conf_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lpss_conf: confluent … lpss.
-/3 width=6 by lpx_sn_conf, cpss_conf_lpss/
-qed-.
-
-theorem lpss_trans: Transitive … lpss.
-/3 width=5 by lpx_sn_trans, cpss_trans_lpss/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
- ∃∃L2,T2. L @@ L1 ⊢ ▶* L @@ L2 & L @@ L1 ⊢ T1 ▶* T2 &
- T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #HT1
- @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
-| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
- elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
- lapply (lpss_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
- @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L1 ⓝ ⊑ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEqT $L1 $L2 }.
-
-include "basic_2/relocation/ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-inductive lsubr: relation lenv ≝
-| lsubr_sort: ∀L. lsubr L (⋆)
-| lsubr_abbr: ∀L1,L2,V. lsubr L1 L2 → lsubr (L1. ⓓV) (L2.ⓓV)
-| lsubr_abst: ∀I,L1,L2,V1,V2. lsubr L1 L2 → lsubr (L1. ⓑ{I}V1) (L2. ⓛV2)
-.
-
-interpretation
- "local environment refinement (substitution)"
- 'CrSubEq L1 L2 = (lsubr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsubr_bind: ∀I,L1,L2,V. L1 ⊑ L2 → L1. ⓑ{I} V ⊑ L2.ⓑ{I} V.
-* /2 width=1/ qed.
-
-lemma lsubr_abbr: ∀I,L1,L2,V. L1 ⊑ L2 → L1. ⓓV ⊑ L2. ⓑ{I}V.
-* /2 width=1/ qed.
-
-lemma lsubr_refl: ∀L. L ⊑ L.
-#L elim L -L // /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #L1 #L2 #V #_ #H destruct
-| #I #L1 #L2 #V1 #V2 #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_atom1: ∀L2. ⋆ ⊑ L2 → L2 = ⋆.
-/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-
-fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
- ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
-#L1 #L2 * -L1 -L2
-[ #L #K2 #W #H destruct
-| #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3/
-| #I #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⊑ K2.ⓓW →
- ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
-/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
-
-fact lsubr_inv_abst2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W2. L2 = K2.ⓛW2 →
- ∃∃I,K1,W1. K1 ⊑ K2 & L1 = K1.ⓑ{I}W1.
-#L1 #L2 * -L1 -L2
-[ #L #K2 #W2 #H destruct
-| #L1 #L2 #V #_ #K2 #W2 #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #K2 #W2 #H destruct /2 width=5/
-]
-qed-.
-
-lemma lsubr_inv_abst2: ∀L1,K2,W2. L1 ⊑ K2.ⓛW2 →
- ∃∃I,K1,W1. K1 ⊑ K2 & L1 = K1.ⓑ{I}W1.
-/2 width=4 by lsubr_inv_abst2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubr_fwd_length: ∀L1,L2. L1 ⊑ L2 → |L2| ≤ |L1|.
-#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubr_fwd_ldrop2_abbr: ∀L1,L2. L1 ⊑ L2 →
- ∀K2,W,i. ⇩[0, i] L2 ≡ K2. ⓓW →
- ∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1. ⓓW.
-#L1 #L2 #H elim H -L1 -L2
-[ #L #K2 #W #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #L1 #L2 #V #HL12 #IHL12 #K2 #W #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /2 width=3/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 /3 width=3/
- ]
-| #I #L1 #L2 #V1 #V2 #_ #IHL12 #K2 #W #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct
- elim (IHL12 … HLK2) -IHL12 -HLK2 /3 width=3/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-lemma lsubr_fwd_lsubr: ∀L1,L2. L1 ⓝ⊑ L2 → L1 ⊑ L2.
-#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma cpqs_cpr: ∀L,T1,T2. L ⊢ T1 ➤* T2 → L ⊢ T1 ➡ T2.
-#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
-qed.
-
-lemma cpss_cpr: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡ T2.
-/3 width=1/ qed.
-
-lemma lpqs_lpr: ∀L1,L2. L1 ⊢ ➤* L2 → L1 ⊢ ➡ L2.
-#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
-qed.
-
-lemma lpss_lpr: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➡ L2.
-/3 width=1/ qed.
-
-lemma cpss_fwd_cir_eq: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ 𝐈⦃T1⦄ → T2 = T1.
-/3 width=3 by cpr_fwd_cir, cpss_cpr/ qed-.
-
-lemma cpss_cprs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡* T2.
-/3 width=1/ qed.
-
-lemma cprs_cpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ▶* T2 → L ⊢ T1 ➡* T2.
-/3 width=3/ qed-.
-
-lemma cpss_cprs_trans: ∀L,T1,T. L ⊢ T1 ▶* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
-/3 width=3/ qed-.
-
-lemma cpcs_cpss_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ▶* T2 → L ⊢ T1 ⬌* T2.
-#L #T1 #T #HT1 #T2 #HT2
-@(cpcs_cpr_strap1 … HT1) -T1 /2 width=3/
-qed-.
-
-lemma cpcs_cpss_strap2: ∀L,T1,T. L ⊢ T1 ▶* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
-#L #T1 #T #HT1 #T2 #HT2
-@(cpcs_cpr_strap2 … HT2) -T2 /2 width=3/
-qed-.
-
-lemma cpcs_cpss_conf: ∀L,T,T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
-#L #T #T1 #HT1 #T2 #HT2
-@(cpcs_cpr_conf … HT2) -T2 /2 width=3/
-qed-.
-
-lemma cpcs_cpss_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ▶* T → L ⊢ T1 ⬌* T2.
-#L #T1 #T #HT1 #T2 #HT2
-@(cpcs_cpr_div … HT1) -T1 /2 width=3/
-qed-.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lsubr.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-(* Auxiliary inversion lemmas ***********************************************)
-
-fact lsubr_inv_abbr1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓓW →
- ∨∨ L2 = ⋆
- | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
- | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
-#L1 #L2 * -L1 -L2
-[ #L #K1 #W #H destruct /2 width=1/
-| #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
-| #I #L1 #L2 #V1 #V2 #HL12 #K1 #W #H destruct /3 width=4/
-]
-qed-.
-
-lemma lsubr_inv_abbr1: ∀K1,L2,W. K1.ⓓW ⊑ L2 →
- ∨∨ L2 = ⋆
- | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
- | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
-/2 width=3 by lsubr_inv_abbr1_aux/ qed-.
-
-fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W1. L1 = K1.ⓛW1 →
- L2 = ⋆ ∨
- ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
-#L1 #L2 * -L1 -L2
-[ #L #K1 #W1 #H destruct /2 width=1/
-| #L1 #L2 #V #_ #K1 #W1 #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #K1 #W1 #H destruct /3 width=4/
-]
-qed-.
-
-lemma lsubr_inv_abst1: ∀K1,L2,W1. K1.ⓛW1 ⊑ L2 →
- L2 = ⋆ ∨
- ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
-/2 width=4 by lsubr_inv_abst1_aux/ qed-.
-
-(* Main properties **********************************************************)
-
-theorem lsubr_trans: Transitive … lsubr.
-#L1 #L #H elim H -L1 -L
-[ #L1 #X #H
- lapply (lsubr_inv_atom1 … H) -H //
-| #L1 #L #V #_ #IHL1 #X #H
- elim (lsubr_inv_abbr1 … H) -H // *
- #L2 [2: #V2 ] #HL2 #H destruct /3 width=1/
-| #I #L1 #L #V1 #V #_ #IHL1 #X #H
- elim (lsubr_inv_abst1 … H) -H // *
- #L2 #V2 #HL2 #H destruct /3 width=1/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/dxprs_lpss.ma".
-include "basic_2/equivalence/cpcs_lpss.ma".
-include "basic_2/dynamic/snv_lift.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties about sn parallel substitution for local environments *********)
-
-lemma snv_cpss_lpss_conf: ∀h,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] → ∀T2. L1 ⊢ T1 ▶* T2 →
- ∀L2. L1 ⊢ ▶* L2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
-#h #g #L1 #T1 #H elim H -L1 -T1
-[ #L1 #k #X #H #L2 #_
- >(cpss_inv_sort1 … H) -X //
-| #I #L1 #K1 #V1 #i #HLK1 #_ #IHV1 #W2 #H #L2 #HL12
- elim (cpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (IHV1 … HV12 … HK12) -IHV1 -HV12 -HK12 /2 width=5/
- | * #K0 #V0 #V2 #HLK0 #HV12 #HVW2
- lapply (ldrop_mono … HLK0 … HLK1) -HLK0 #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #V #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -V #HLK2
- lapply (IHV1 … HV12 … HK12) -IHV1 -HV12 -HK12 /2 width=7/
- ]
-| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IHV1 … HV12 … HL12) -IHV1 #HV2
- lapply (IHT1 … HT12 (L2.ⓑ{I}V2) ?) -IHT1 -HT12 /2 width=1/
-| #a #L1 #V1 #W1 #W0 #T1 #U1 #l #_ #_ #HVW1 #HW10 #HTU1 #IHV1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IHV1 … HV12 … HL12) -IHV1 #HV2
- lapply (IHT1 … HT12 … HL12) -IHT1 #HT2
- elim (ssta_cpss_lpss_conf … HVW1 … HV12 … HL12) -V1 #W2 #HVW2 #HW12
- elim (dxprs_cpss_lpss_conf … HTU1 … HT12 … HL12) -T1 #X #HTU2 #H
- elim (cpss_inv_bind1 … H) -H #W #U2 #HW0 #_ #H -U1 destruct
- elim (cprs_cpss2_lpss_conf_dx … HW10 … HW12 … HW0 … HL12) -L1 -W1 -W0 #W0 #HW20 #HW0
- lapply (dxprs_strap1 … (ⓛ{a}W0.U2) HTU2 ?) -HTU2 /3 width=3/ -HW0 /2 width=8/
-| #L1 #W1 #T1 #U1 #l #_ #_ #HTU1 #HUW1 #IHW1 #IHT1 #X #H #L2 #HL12
- elim (cpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
- lapply (IHW1 … HW12 … HL12) -IHW1 #HW2
- lapply (IHT1 … HT12 … HL12) -IHT1 #HT2
- elim (ssta_cpss_lpss_conf … HTU1 … HT12 … HL12) -T1 #U2 #HTU2 #HU12
- lapply (cpcs_cpss2_lpss_conf … HUW1 … HU12 … HW12 … HL12) -L1 -W1 -U1 /2 width=4/
-]
-qed-.
-
-lemma snv_lpss_conf: ∀h,g,L1,T. ⦃h, L1⦄ ⊢ T ¡[g] →
- ∀L2. L1 ⊢ ▶* L2 → ⦃h, L2⦄ ⊢ T ¡[g].
-#h #g #L1 #T #HT #L2 #HL12
-@(snv_cpss_lpss_conf … HT … HL12) //
-qed-.
-
-lemma snv_cpss_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊢ T1 ¡[g] →
- ∀T2. L ⊢ T1 ▶* T2 → ⦃h, L⦄ ⊢ T2 ¡[g].
-#h #g #L #T1 #HT1 #T2 #HT12
-@(snv_cpss_lpss_conf … HT1 … HT12) //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lpss_ldrop.ma".
-include "basic_2/static/ssta_lift.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about sn parallel substitution ********************************)
-
-(* Note: apparently this was missing in basic_1 *)
-lemma ssta_cpss_lpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
-#h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
-[ #L1 #k1 #l1 #Hkl1 #X #H
- >(cpss_inv_sort1 … H) -H /3 width=3/
-| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #_ #HWU1 #IHVW1 #X #H #L2 #HL12
- elim (cpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
- elim (IHVW1 … HV12 … HK12) -IHVW1 -HV12 -HK12 #W2 #HVW2 #HW12
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
- | * #Y #Z #V2 #H #HV12 #HV2
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #Z #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -V0 #HLK2
- lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
- elim (IHVW1 … HV12 … HK12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (ssta_lift … HVW2 … HLK2 … HV2 … HWU2) -HVW2 -HLK2 -HV2
- lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 -HWU2 /3 width=3/
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #_ #HWU1 #IHWV1 #X #H #L2 #HL12
- elim (cpss_inv_lref1 … H) -H [ | -IHWV1 -HWU1 -HL12 ]
- [ #H destruct
- elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
- elim (IHWV1 … HW12 … HK12) -IHWV1 -HK12 #V2 #HWV2 #_
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
- | * #K2 #V2 #W2 #HLK2 #_ #_
- lapply (ldrop_mono … HLK2 … HLK1) -HLK1 -HLK2 #H destruct
- ]
-| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
- elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12 (L2.ⓑ{I}V2)) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
-| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
- elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=5/
-| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
- elim (cpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
- elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=3/
-]
-qed-.
-
-lemma ssta_cpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2. L ⊢ T1 ▶* T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* U2.
-/2 width=3 by ssta_cpss_lpss_conf/ qed-.
-
-lemma ssta_lpss_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
- ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
-/2 width=3 by ssta_cpss_lpss_conf/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_lpss.ma".
-include "basic_2/unfold/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Properties about sn parallel substitution for local environments *********)
-
-lemma sstas_cpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ▶* U2.
-#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
-#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12
-elim (ssta_cpss_lpss_conf … HTU0 … HT0 … HL12) -HTU0 -HT0 #U #HTU #HU0
-elim (IHU01 … HU0 … HL12) -IHU01 -U0 -HL12 /3 width=4/
-qed-.
-
-lemma sstas_cpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2. L ⊢ T1 ▶* T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* U2.
-/2 width=3 by sstas_cpss_lpss_conf/ qed-.
-
-lemma sstas_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ▶* U2.
-/2 width=3 by sstas_cpss_lpss_conf/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( h ⊢ break term 46 L1 : ⊑ [ ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'StratifiedCrSubEqN $h $L1 $L2 }.
-
-include "basic_2/dynamic/snta.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
-
-(* Note: may not be transitive *)
-inductive lsubsn (h:sh): relation lenv ≝
-| lsubsn_atom: lsubsn h (⋆) (⋆)
-| lsubsn_pair: ∀I,L1,L2,W. lsubsn h L1 L2 →
- lsubsn h (L1. ⓑ{I} W) (L2. ⓑ{I} W)
-| lsubsn_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V :[l+1] W → ⦃h, L2⦄ ⊢ V :[l+1] W →
- lsubsn h L1 L2 → lsubsn h (L1. ⓓV) (L2. ⓛW)
-.
-
-interpretation
- "local environment refinement (stratified native type assigment)"
- 'StratifiedCrSubEqN h L1 L2 = (lsubsn h L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubsn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L1 = ⋆ → L2 = ⋆.
-#h #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubsn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑[] L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact lsubsn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. h ⊢ K1 :⊑[] K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
- h ⊢ K1 :⊑[] K2 & L2 = K2. ⓛW & I = Abbr.
-#h #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
-]
-qed.
-
-lemma lsubsn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑[] L2 →
- (∃∃K2. h ⊢ K1 :⊑[] K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
- h ⊢ K1 :⊑[] K2 & L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
-
-fact lsubsn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L2 = ⋆ → L1 = ⋆.
-#h #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubsn_inv_atom2: ∀h,L1. h ⊢ L1 :⊑[] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
-
-fact lsubsn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. h ⊢ K1 :⊑[] K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
- h ⊢ K1 :⊑[] K2 & L1 = K1. ⓓV & I = Abst.
-#h #L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
-]
-qed.
-
-lemma lsubsn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑[] K2. ⓑ{I} W →
- (∃∃K1. h ⊢ K1 :⊑[] K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
- h ⊢ K1 :⊑[] K2 & L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
-
-(* Basic_forward lemmas *****************************************************)
-
-lemma lsubsn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L1 ≼[0, |L1|] L2.
-#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubsn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L1 ≼[0, |L2|] L2.
-#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsubsn_refl: ∀h,L. h ⊢ L :⊑[] L.
-#h #L elim L -L // /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/equivalence/cpcs_cpcs.ma".
-include "basic_2/dynamic/lsubsn.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
-
-(* Properties on context-sensitive parallel equivalence for terms ***********)
-
-lemma cpr_lsubsn_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
-/3 width=5 by lsubsn_fwd_lsubs2, cpr_lsubs_trans/ qed-.
-
-lemma cprs_lsubsn_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-/3 width=5 by lsubsn_fwd_lsubs2, cprs_lsubs_trans/ qed-.
-
-lemma cpcs_lsubsn_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
-/3 width=5 by lsubsn_fwd_lsubs2, cpcs_lsubs_trans/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/lsubsn.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
-
-(* Properties concerning basic local environment slicing ********************)
-
-(* Note: the constant 0 cannot be generalized *)
-lemma lsubsn_ldrop_O1_conf: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. h ⊢ K1 :⊑[] K2 & ⇩[0, e] L2 ≡ K2.
-#h #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
- [ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
- ]
-| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
- [ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
- ]
-]
-qed.
-
-(* Note: the constant 0 cannot be generalized *)
-lemma lsubsn_ldrop_O1_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
- ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. h ⊢ K1 :⊑[] K2 & ⇩[0, e] L1 ≡ K1.
-#h #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
- [ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
- ]
-| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
- [ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snta_snta.ma".
-include "basic_2/dynamic/lsubsn_ldrop.ma".
-include "basic_2/dynamic/lsubsn_cpcs.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
-
-(* Properties concerning stratified native type assignment ******************)
-
-(* Note: the corresponding confluence property does not hold *)
-lemma lsubsn_snta_trans: ∀h,L2,T,U,l. ⦃h, L2⦄ ⊢ T :[l] U →
- ∀L1. h ⊢ L1 :⊑[] L2 → ⦃h, L1⦄ ⊢ T :[l] U.
-#h #L2 #T #U #l #H elim H -L2 -T -U -l
-[ //
-| #L2 #K2 #V2 #W2 #U2 #i #l #HLK2 #_ #WU2 #IHVW2 #L1 #HL12
- elim (lsubsn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsubsn_inv_pair2 … H) -H * #K1
- [ #HK12 #H destruct /3 width=6/
- | #V1 #l0 #_ #_ #_ #_ #H destruct
- ]
-| #L2 #K2 #W2 #V2 #U2 #i #l #HLK2 #HWV2 #HWU2 #IHWV2 #L1 #HL12
- elim (lsubsn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsubsn_inv_pair2 … H) -H * #K1 [ -HWV2 | -IHWV2 ]
- [ #HK12 #H destruct /3 width=6/
- | #V1 #l0 #H1 #H2 #_ #H #_ destruct
- elim (snta_fwd_correct … H2) -H2 #V #H
- elim (snta_mono … HWV2 … H) -HWV2 -H /2 width=6/
- ]
-| /4 width=3/
-| /3 width=2/
-| /3 width=2/
-| /3 width=1/
-| #L2 #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #IHUV2 #L1 #HL12
- lapply (cpcs_lsubsn_trans … HL12 … HU12) -HU12 /3 width=3/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 : * break [ l ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'NativeTypeStar $h $l $L $T1 $T2 }.
-
-notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 : break [ l ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StratifiedNativeType $h $l $L $T1 $T2 }.
-
-include "basic_2/static/sh.ma".
-include "basic_2/equivalence/cpcs.ma".
-
-(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
-
-inductive snta (h:sh): nat → lenv → relation term ≝
-| snta_sort: ∀L,k. snta h 0 L (⋆k) (⋆(next h k))
-| snta_ldef: ∀L,K,V,W,U,i,l. ⇩[0, i] L ≡ K. ⓓV → snta h l K V W →
- ⇧[0, i + 1] W ≡ U → snta h l L (#i) U
-| snta_ldec: ∀L,K,W,V,U,i,l. ⇩[0, i] L ≡ K. ⓛW → snta h l K W V →
- ⇧[0, i + 1] W ≡ U → snta h (l+1) L (#i) U
-| snta_bind: ∀I,L,V,W,T,U,l1,l2. snta h l1 L V W → snta h l2 (L. ⓑ{I} V) T U →
- snta h l2 L (ⓑ{I}V.T) (ⓑ{I}V.U)
-| snta_appl: ∀L,V,W1,W2,T,U,l1,l2. snta h (l1+1) L V W2 →
- snta h l2 L (ⓛW1.T) (ⓛW2.U) →
- snta h l2 L (ⓐV.ⓛW1.T) (ⓐV.ⓛW2.U)
-| snta_pure: ∀L,V,T,U,W,l. snta h (l+1) L T U → snta h l L (ⓐV.U) W →
- snta h (l+1) L (ⓐV.T) (ⓐV.U)
-| snta_cast: ∀L,T,U,W,l1,l2. snta h l2 L T U → snta h l1 L U W →
- snta h l2 L (ⓝU.T) U
-| snta_conv: ∀L,T,U1,U2,V2,l. snta h l L T U1 → L ⊢ U1 ⬌* U2 →
- snta h (l-1) L U2 V2 → snta h l L T U2
-.
-
-interpretation "stratified native type assignment (term)"
- 'StratifiedNativeType h l L T U = (snta h l L T U).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/equivalence/cpcs_cpcs.ma".
-include "basic_2/dynamic/snta.ma".
-
-(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact snta_inv_sort1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀k0. T = ⋆k0 →
- l = 0 ∧ L ⊢ ⋆(next h k0) ⬌* U.
-#h #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #k0 #H destruct /2 width=1/
-| #L #K #V #W #U #i #l #_ #_ #_ #_ #k0 #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #_ #k0 #H destruct
-| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #k0 #H destruct
-| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #k0 #H destruct
-| #L #V #T #U #W #l #_ #_ #_ #_ #k0 #H destruct
-| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #k0 #H destruct
-| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #k0 #H destruct
- elim (IHTU1 ??) -IHTU1 [3: // |2: skip ] #H #Hk0
- lapply (cpcs_trans … Hk0 … HU12) -U1 /2 width=1/
-]
-qed.
-
-lemma snta_inv_sort1: ∀h,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k :[l] U →
- l = 0 ∧ L ⊢ ⋆(next h k) ⬌* U.
-/2 width=4/ qed-.
-
-fact snta_inv_lref1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀j. T = #j →
- (∃∃K,V,W,U0. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V :[l] W &
- ⇧[0, j + 1] W ≡ U0 & L ⊢ U0 ⬌* U
- ) ∨
- (∃∃K,W,V,U0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W :[l-1] V &
- ⇧[0, j + 1] W ≡ U0 & l > 0 & L ⊢ U0 ⬌* U
- ).
-#h #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #j #H destruct
-| #L #K #V #W #U #i #l #HLK #HVW #HWU #_ #j #H destruct /3 width=8/
-| #L #K #W #V #U #i #l #HLK #HWV #HWU #_ #j #H destruct /3 width=8/
-| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct
-| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct
-| #L #V #T #U #W #l #_ #_ #_ #_ #j #H destruct
-| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #j #H destruct
-| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #j #H destruct
- elim (IHTU1 ??) -IHTU1 [4: // |2: skip ] * #K #V #W #U0 #HLK #HVW #HWU0 [2: #H ] #HU01
- lapply (cpcs_trans … HU01 … HU12) -U1 /3 width=8/
-]
-qed.
-
-lemma snta_inv_lref1: ∀h,L,U,i,l. ⦃h, L⦄ ⊢ #i :[l] U →
- (∃∃K,V,W,U0. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V :[l] W &
- ⇧[0, i + 1] W ≡ U0 & L ⊢ U0 ⬌* U
- ) ∨
- (∃∃K,W,V,U0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W :[l-1] V &
- ⇧[0, i + 1] W ≡ U0 & l > 0 & L ⊢ U0 ⬌* U
- ).
-/2 width=3/ qed-.
-
-fact snta_inv_bind1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀J,X,Y. T = ⓑ{J}Y.X →
- ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 &
- ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 &
- L ⊢ ⓑ{J}Y.Z2 ⬌* U.
-#h #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #J #X #Y #H destruct
-| #L #K #V #W #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct
-| #I #L #V #W #T #U #l1 #l2 #HVW #HTU #_ #_ #J #X #Y #H destruct /2 width=3/
-| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct
-| #L #V #T #U #W #l #_ #_ #_ #_ #J #X #Y #H destruct
-| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct
-| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #J #X #Y #H destruct
- elim (IHTU1 ????) -IHTU1 [5: // |2,3,4: skip ] #Z1 #Z2 #l0 #HZ1 #HZ2 #HU1
- lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/
-]
-qed.
-
-lemma snta_inv_bind1: ∀h,J,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{J}Y.X :[l] U →
- ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 &
- L ⊢ ⓑ{J}Y.Z2 ⬌* U.
-/2 width=3/ qed-.
-
-fact snta_inv_cast1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀X,Y. T = ⓝY.X →
- ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U.
-#h #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #X #Y #H destruct
-| #L #K #V #W #U #i #l #_ #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #_ #X #Y #H destruct
-| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
-| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
-| #L #V #T #U #W #l #_ #_ #_ #_ #X #Y #H destruct
-| #L #T #U #W #l1 #l2 #HTU #_ #_ #_ #X #Y #H destruct /2 width=1/
-| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
- elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #HXY #HU1
- lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=1/
-]
-qed.
-
-lemma snta_inv_cast1: ∀h,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X :[l] U →
- ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U.
-/2 width=3/ qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma snta_lift: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 →
- ∀L2,d,e. ⇩[d, e] L2 ≡ L1 →
- ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 →
- ⦃h, L2⦄ ⊢ T2 :[l] U2.
-#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
-[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2
- >(lift_inv_sort1 … H1) -X1
- >(lift_inv_sort1 … H2) -X2 //
-| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2
- elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=8/
- | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
- lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
- ]
-| #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // <minus_plus #W #HW1 #HWU2
- elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
- lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
- elim (lift_total V1 (d-i-1) e) /3 width=8/
- | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
- lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
- ]
-| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
- elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
- elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
- lapply (lift_mono … H1 … HV12) -H1 #H destruct
- elim (lift_total W1 d e) /4 width=6/
-| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
- elim (lift_inv_flat1 … H1) -H1 #V2 #X #HV12 #H1 #H destruct
- elim (lift_inv_bind1 … H1) -H1 #W21 #T2 #HW121 #HT12 #H destruct
- elim (lift_inv_flat1 … H2) -H2 #Y2 #X #HY #H2 #H destruct
- elim (lift_inv_bind1 … H2) -H2 #W22 #U2 #HW122 #HU12 #H destruct
- lapply (lift_mono … HY … HV12) -HY #H destruct /4 width=6/
-| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
- elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
- elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
- lapply (lift_mono … H1 … HV12) -H1 #H destruct
- elim (lift_total W1 d e) #W2 #HW12 /4 width=6/
-| #L1 #T1 #U1 #W1 #l1 #l2 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL21 #X #H #U2 #HU12
- elim (lift_inv_flat1 … H) -H #X2 #T2 #HUX2 #HT12 #H destruct
- lapply (lift_mono … HUX2 … HU12) -HUX2 #H destruct
- elim (lift_total W1 d e) /3 width=6/
-| #L1 #T1 #U11 #U12 #V12 #l #_ #HU112 #_ #IHTU11 #IHUV12 #L2 #d #e #HL21 #U1 #HTU1 #U2 #HU12
- elim (lift_total U11 d e) #U #HU11
- elim (lift_total V12 d e) #V22 #HV122
- lapply (cpcs_lift … HL21 … HU11 … HU12 HU112) -HU112 /3 width=6/
-]
-qed.
-
-(* Advanced forvard lemmas **************************************************)
-
-fact snta_fwd_pure1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀X,Y. T = ⓐY.X →
- ∃∃V,W,l0. ⦃h, L⦄ ⊢ Y :[l0+1] W & ⦃h, L⦄ ⊢ X :[l] V &
- L ⊢ ⓐY.V ⬌* U.
-#h #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #X #Y #H destruct
-| #L #K #V #W #U #i #l #_ #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #_ #X #Y #H destruct
-| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
-| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #HTU #_ #_ #X #Y #H destruct /2 width=3/
-| #L #V #T #U #W #l #HTU #_ #_ #IHU #X #Y #H destruct
- elim (IHU U Y ?) -IHU // /3 width=3/
-| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
-| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
- elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #V #W #l0 #HYW #HXV #HU1
- lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/
-]
-qed.
-
-lemma snta_fwd_pure1: ∀h,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓐY.X :[l] U →
- ∃∃V,W,l0. ⦃h, L⦄ ⊢ Y :[l0+1] W & ⦃h, L⦄ ⊢ X :[l] V &
- L ⊢ ⓐY.V ⬌* U.
-/2 width=3/ qed-.
-
-lemma snta_fwd_correct: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U →
- ∃T0. ⦃h, L⦄ ⊢ U :[l-1] T0.
-#h #L #T #U #l #H elim H -L -T -U -l
-[ /2 width=2/
-| #L #K #V #W #W0 #i #l #HLK #_ #HW0 * #V0 #HWV0
- lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- elim (lift_total V0 0 (i+1)) /3 width=10/
-| #L #K #W #V #V0 #i #l #HLK #HWV #HWV0 #_
- lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- elim (lift_total V 0 (i+1)) /3 width=10/
-| #I #L #V #W #T #U #l1 #l2 #HVW #_ #_ * /3 width=3/
-| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #_ #_ * #X #H
- elim (snta_inv_bind1 … H) -H /4 width=5/
-| /3 width=2/
-| /2 width=2/
-| /2 width=2/
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma snta_cast_short: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ⦃h, L⦄ ⊢ ⓝU.T :[l] U.
-#h #L #T #U #l #HTU
-elim (snta_fwd_correct … HTU) /2 width=3/
-qed.
-
-lemma snta_typecheck: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U →
- ∃T0. ⦃h, L⦄ ⊢ ⓝU.T :[l] T0.
-/3 width=2/ qed.
-
-lemma snta_cast_old: ∀h,L,W,T,U,l.
- ⦃h, L⦄ ⊢ T :[l] U → ⦃h, L⦄ ⊢ U :[l-1] W → ⦃h, L⦄ ⊢ ⓝU.T :[l] ⓝW.U.
-#h #L #W #T #U #l #HTU #HUW
-@(snta_conv … U) /2 width=2/ /3 width=1/ (**) (* /4 width=3/ is a bit slow *)
-qed.
-
-lemma snta_appl_old: ∀h,L,V,W,T,U,l1,l2.
- ⦃h, L⦄ ⊢ V :[l1+1] W → ⦃h, L⦄ ⊢ T :[l2+1] ⓛW.U →
- ⦃h, L⦄ ⊢ ⓐV.T :[l2+1] ⓐV.ⓛW.U.
-#h #L #V #W #T #U #l1 #l2 #HVW #HTU
-elim (snta_fwd_correct … HTU) #X #H
-elim (snta_inv_bind1 … H) -H /4 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snta_ltpss.ma".
-include "basic_2/dynamic/snta_thin.ma".
-include "basic_2/dynamic/lsubsn_snta.ma".
-
-(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
-(*
-lemma snta_fwd_abst: ∀h,L,W1,W2,T,U,l2. ⦃h, L⦄ ⊢ ⓛW1.T :[l2] ⓛW2.U →
- ∃∃V1,V2,l1. ⦃h, L⦄ ⊢ W1 :[l1] V1 & ⦃h, L⦄ ⊢ W2 :[l1] V2 &
- L ⊢ W1 ⬌* W2.
-#h #L #W1 #W2 #T #U #l2 #HTU
-elim (snta_fwd_correct … HTU) #X #H
-elim (snta_inv_bind1 … H) -H #W #T0 #l #HW2 #_ #_ -X
-elim (snta_inv_bind1 … HTU) -HTU #V1 #U0 #l0 #HWV1 #_ #H
-elim (cpcs_inv_abst … H Abst W1) -H
-#HW12 #_ -U0
-@(ex3_3_intro … HWV1 … HW12)
-[3: @(snta_conv … HTU0 HU0)
-
- /3 width=3/
-
-*)
-(*
-#h #L #V #T #U #l2 #HTU
-elim (snta_fwd_correct … HTU) #X #H
-elim (snta_inv_bind1 … H) -H #W #T0 #l1 #HVW #HUT0 #_ -X
-elim (snta_inv_bind1 … HTU) -HTU #W0 #U0 #l0 #_ #HTU0 #H -l0
-elim (cpcs_inv_abst … H Abst V) -H /3 width=3/
-qed-.
-*)
-(*
-lemma snta_fwd_appl1_sound_aux: ∀h,l0. (∀L1,L2,T1,T2,U,l.
- l < l0 → ⦃h, L1⦄ ⊢ T1 :[l] U →
- L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
- ) →
- ∀L,T,U,l2. ⦃h, L⦄ ⊢ T :[l2] U →
- ∀Z,Y,X1. T = ⓐZ.ⓛY.X1 → l0 = l2 →
- ∃l1. ⦃h, L⦄ ⊢ Z :[l1+1] Y.
-#h #l0 #IH #L #T #U #l2 #H elim H -L -T -U -l2
-[
-|
-|
-|
-| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #HTU #_ #_ #Z #Y #X1 #H1 #H2 destruct -IH
- elim (snta_fwd_abst … HTU) -X1 -U -l2 #Y0 #W0 #l0 #HY0 #H1 #HYW2
- elim (snta_fwd_correct … HVW2) #W #H2
- elim (snta_mono … H1 … H2) -H1 -H2 #H #_ destruct -W0 -W /4 width=6/
-| #L #V #T #U #W #l #HTU #HUW #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
- elim (snta_inv_abst_sn … HTU) -HTU #Y0 #l0 #HY0 #HX12
-|
-| #L #T #U1 #U2 #V2 #l #HTU1 #HU12 #HUV2 #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-lemma snta_inv_appl_aux: ∀h,l0. (∀L1,L2,T1,T2,U,l.
- l < l0 + 1 → ⦃h, L1⦄ ⊢ T1 :[l] U →
- L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
- ) →
- ∀L,T,U,l2. ⦃h, L⦄ ⊢ T :[l2] U →
- ∀Z,Y,X1,X2. T = ⓐZ.ⓛY.X1 → U = ⓐZ.ⓛY.X2 → l0 = l2 →
- ∃∃l1. ⦃h, L⦄ ⊢ Z :[l1+1] Y & ⦃h, L.ⓛY⦄ ⊢ X1 :[l2] X2.
-#h #l0 #IH #L #T #U #l2 * -L -T -U -l2
-[
-|
-|
-|
-| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #HTU #Z #Y #X1 #X2 #H1 #H2 #H3 destruct -IH
- elim (snta_inv_abst … HTU) -HTU /2 width=2/
-| #L #V #T #U #W #l #HTU #HUW #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
- elim (snta_inv_abst … HTU) -HTU #Y0 #l0 #HY0 #HX12
-|
-| #L #T #U1 #U2 #V2 #l #HTU1 #HU12 #HUV2 #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
-
- /2 width=2/
-
-
-axiom pippo: ∀h,l0. (∀L1,L2,T1,T2,U,l.
- l < l0 + 1 → ⦃h, L1⦄ ⊢ T1 :[l] U →
- L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
- ) →
- ∀L,T1,U1,l. ⦃h, L⦄ ⊢ T1 :[l] U1 →
- ∀V2,W2,T2. L ⊢ T1 ➡* ⓐV2.ⓛW2.T2 → l0 = l →
- ∃l0. ⦃h, L2⦄ ⊢ V2 :[l0+1] W2.
-(*
-#h #l #IH #L1 #T1 #U1 #l1 * -L1 -T1 -U1 -l1
-[
-|
-|
-|
-| #L1 #V1 #W1 #T1 #U1 #l1 #HVW1 #HTU1 #Y1 #X1 #H1 #L2 #Y2 #HL12 #HY12 #Z2 #X2 #HX12 #H2 destruct
- elim (IH ??? Y2 … HVW1 HL12 ?) -HVW1 // [2: /3 width=1/ ] -HY12 #l21 #HY2W1 #H1l21 #H2l21
- elim (IH … HTU1 HL12 HX12) -IH -HTU1 -HL12 -HX12 // #l22 #H #_ #H2l22
- elim (snta_inv_bind1 … H) -H #Z #X #HZ2 #_ #H
- elim (cpcs_inv_abst … H Abst W1) -H #H #_
- lapply (transitive_le … (l21+l22) … H1l21 ?) -H1l21 // #Hl21
- @(ex3_1_intro … Hl21) [2: /3 width=1/ ]
- @(snta_conv … W1) /2 width=2/ (**) (* explicit constructors *)
-| #L1 #V1 #T1 #U1 #W1 #l1 #HTU1 #HUW1 #Y1 #X1 #H1 #L2 #Y2 #HL12 #HY12 #Z2 #X2 #HX12 #H2 destruct
-
-*)
-(* Properties on context-free parallel reduction for local environments *****)
-*)
-fact snta_ltpr_tpr_conf_aux: ∀h,l0. (∀L1,L2,T1,T2,U,l.
- l < l0 → ⦃h, L1⦄ ⊢ T1 :[l] U →
- L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
- ) →
- ∀L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U → ∀L2. L1 ➡ L2 →
- ∀T2. T1 ➡ T2 → l0 = l → ⦃h, L2⦄ ⊢ T2 :[l] U.
-#h #l0 #IH #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
-[ #L1 #k1 #L2 #_ #T2 #H #_ -l0
- >(tpr_inv_atom1 … H) -H //
-| #L1 #K1 #V1 #W #U #i1 #l #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H #Hl -IH
- >(tpr_inv_atom1 … H) -T2
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
-| #L1 #K1 #W1 #V1 #U1 #i1 #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #HL12 #T2 #H #Hl -IH
-(*
- >(tpr_inv_atom1 … H) -T2
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H
- elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total V1 0 (i+1)) #W #HW
- lapply (snta_lift h … HLK … HWU1 … HW) /2 width=1/ -HLK -HW
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpr_lift … HW12 … HWU1 … HWU2) -HWU1 #HU12
- @(snta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /3 width=6/ is too slow *)
-*)
-| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH
-(*
- elim (tpr_inv_bind1 … H) -H *
- [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
- lapply (IHVW1 … HL12 … HV12) #HV2W1
- lapply (IHVW1 L2 … V1 ?) // -IHVW1 #HWV1
- lapply (IHTU1 (L2.ⓑ{I}V2) … HT1) -HT1 /2 width=1/ #HTU1
- lapply (IHTU1 (L2.ⓑ{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H
- lapply (tps_lsubs_trans … HT2 (L2.ⓑ{I}V2) ?) -HT2 /2 width=1/ #HT2
- lapply (snta_tps_conf … HTU1 … HT2) -T #HT2U1
- elim (snta_fwd_correct … H) -H #U2 #HU12
- @(snta_conv … (ⓑ{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *)
- | #T #HT1 #HTX #H destruct
- lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HVW1
- lapply (IHTU1 (L2.ⓓV1) … HT1) -T1 /2 width=1/ -L1 #H
- elim (snta_fwd_correct … H) #T1 #HUT1
- elim (snta_ldrop_conf … H L2 0 1 ? ?) -H // /2 width=1/ #T0 #U0 #HTU0 #H #HU10
- lapply (delift_inv_lift1_eq … H L2 … HTX) -H -HTX /2 width=1/ #H destruct
- @(snta_conv … HTU0) /2 width=2/
- ]
-*)
-| #L1 #V1 #W11 #W2 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH
-(*
- elim (tpr_inv_appl1 … H) -H *
- [ #V2 #Y #HV12 #HY #H destruct
- elim (tpr_inv_abst1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct
- lapply (IHTU1 L2 ? (ⓛW1.T1) ?) // #H
- elim (snta_fwd_correct … H) -H #X #H
- elim (snta_inv_bind1 … H) -H #W #U #HW #HU #_
- @(snta_conv … (ⓐV2.ⓛW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *)
- | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
- lapply (IHVW1 … HL12 … HV12) #HVW2
- lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HV1W2
- lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1
- elim (snta_fwd_correct … H1) #T #H2
- elim (snta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H
- elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21
- elim (snta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0
- lapply (lsubsn_snta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2
- @(snta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *)
- | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct
- ]
-*)
-| #L1 #V1 #T1 #U1 #W1 #l #_ #HUW1 #IHTU1 #_ #L2 #HL12 #X #H #Hl
- elim (tpr_inv_appl1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (cpr_tpr … HV12 L2) #HV
- elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=6/ ] #U
- @(snta_conv … (ⓐV2.U1)) /2 width=1/ -HV12 /4 width=8 by snta_pure, cprs_flat_dx/ (**) (* explicit constructor, /4 width=8/ is too slow without trace *)
- | #V2 #W0 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
- lapply (IHTU1 … HL12 (ⓛW0.T2) ? ?) -IHTU1 // /2 width=1/ -T0 #H1
- lapply (IH … (ⓐV2.U1) … HUW1 HL12 ?) // /3 width=1/ #H2
- lapply (snta_pure … H1 H2) -H2 #H
- elim (snta_inv_bind1 … H1) -H1 #V0 #U2 #l1 #HWV0 #HTU2 #HU21
- @(snta_conv … (ⓓV2.U2)) (**) (* explicit constructor *)
- [2:
-(*
- @snta_bind /3 width=2/ /3 width=6/ (**) (* /4 width=6/ is a bit slow *)
-*)
- |3: @(cpcs_cpr_conf … (ⓐV1.ⓛW0.U2)) /2 width=1/
- |4: /2 width=5/
- | skip
- ]
-(*
- elim (snta_fwd_pure1 … H) -H #T1 #W2 #HVW2 #HUT1 #HTW1
-
- elim (cpcs_inv_abst1 … HU21) #W3 #U3 #HU13 #H
- elim (cprs_inv_abst … H Abst W0) -H #HW03 #_
- elim (pippo … IH … HUW1 ? V2 W3 U3 HL12 ? ?) -IH -HUW1 -HL12 // /3 width=1/ -HU13 #l2 #HV2W3
- lapply (snta_conv h L2 V2 W3 W0 V0 (l1+1) ? ? ?) /2 width=1/ -HV2W3 -HW03 -HWV0 #HV2W0
-*)
-(* SEGMENT 1.5
- lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
- lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
- lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/
-
-axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y →
- ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W.
-
-*)
-(* SEGMENT 2
-| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct
- lapply (cpr_tpss … HU12) /4 width=4/
-| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12
- @(snta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *)
-]
-qed.
-*)
-
-(* SEGMENT 3
-fact snta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U.
-
-
- | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
- elim (snta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0
- lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
- lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0
- lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2
- @(snta_abbr … HW2) -HW2
- @(snta_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
- ]
-| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct
- elim (tpr_inv_cast1 … H) -H
- [ * #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
- lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
- | -HV1 #HT1X
- lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
- ]
-]
-qed.
-
-lemma snta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 →
- ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U.
-
-/2 width=9/ qed.
-
-axiom snta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A.
-/2 width=5/ qed.
-
-axiom snta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A.
-/2 width=5/ qed.
-*)
-*)*)
\ No newline at end of file
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/equivalence/cpcs_ltpss.ma".
-include "basic_2/dynamic/snta_snta.ma".
-
-(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about parallel unfold *****************************************)
-
-lemma snta_ltpss_tpss_conf: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U.
-#h #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
-[ #L1 #k #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H //
-| #L1 #K1 #V1 #W #U #i #l #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lt_or_ge i d) #Hdi
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=7/
- | elim (lt_or_ge i (d + e)) #Hide [ | -Hdi ]
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=7/
- | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=7/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=9/
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
- [ #H destruct
- elim (lift_total V1 0 (i+1)) #W #HW
- elim (lt_or_ge i d) #Hdi [ -HWV1 ]
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12
- lapply (cpr_tpss … HU12) -HU12 #HU12
- @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *)
- | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -HW -Hdi ]
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12
- lapply (cpr_tpss … HU12) -HU12 #HU12
- @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *)
- | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
- ]
-| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (cpr_tpss … HV12) #HV
- lapply (IHTU1 (L2.ⓑ{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H
- elim (snta_fwd_correct … H) -H #U2 #HU12
- @(snta_conv … (ⓑ{I}V2.U1)) /3 width=2/ /3 width=4/ /4 width=4/ (**) (* explicit constructor, /5 width=6/ is too slow *)
-| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
- elim (tpss_inv_bind1 … HY) -HY #W21 #T2 #HW121 #HT12 #H destruct
- lapply (cpr_tpss … HV12) #HVV12
- lapply (IHTU1 L2 d e ? (ⓛW21.T2) ?) -IHTU1 // /2 width=1/ -HW121 -HT12 #H0
- elim (snta_fwd_correct … H0) #X #H
- elim (snta_inv_bind1 … H) -H #W #U #l0 #HW #HU #_
- @(snta_conv … (ⓐV2.ⓛW12.U1)) /3 width=2/ /3 width=4/ /3 width=5/ (**) (* explicit constructor, /4 width=5/ is too slow *)
-| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (cpr_tpss … HV12) #HV
- elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=4/ ] #U
- @(snta_conv … (ⓐV2.U1)) /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *)
-| #L1 #T1 #U1 #W1 #l1 #l2 #HTU1 #HUW1 #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
- elim (snta_fwd_correct … HTU1) -HTU1 #U #H
- elim (snta_mono … HUW1 … H) -HUW1 -H #H #_ -U destruct
- elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct
- lapply (cpr_tpss … HU12) #HU /4 width=4/
-| #L1 #T1 #U11 #U12 #U #l #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12
- @(snta_conv … U11) /2 width=5/ (**) (* explicit constructor, /3 width=7/ is too slow *)
-]
-qed.
-
-lemma snta_ltpss_tps_conf: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U.
-/3 width=7/ qed.
-
-lemma snta_ltpss_conf: ∀h,L1,T,U,l. ⦃h, L1⦄ ⊢ T :[l] U →
- ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T :[l] U.
-/2 width=7/ qed.
-
-lemma snta_tpss_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U.
-/2 width=7/ qed.
-
-lemma snta_tps_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U.
-/2 width=7/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snta_lift.ma".
-
-(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Main properties **********************************************************)
-
-theorem snta_mono: ∀h,L,T,U1,l1. ⦃h, L⦄ ⊢ T :[l1] U1 →
- ∀U2,l2. ⦃h, L⦄ ⊢ T :[l2] U2 → l1 = l2 ∧ L ⊢ U1 ⬌* U2.
-#h #L #T #U1 #l1 #H elim H -L -T -U1 -l1
-[ #L #k #X #l2 #H
- lapply (snta_inv_sort1 … H) -H * /2 width=1/
-| #L #K #V #W11 #W12 #i #l1 #HLK #_ #HW112 #IHVW11 #X #l2 #H
- elim (snta_inv_lref1 … H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX
- lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- elim (IHVW11 … HVW21) -IHVW11 -HVW21 #Hl12 #HW121
- lapply (cpcs_lift … HLK … HW112 … HW212 ?) // -K -W11 -W21 /3 width=3/
-| #L #K #W #V1 #V #i #l1 #HLK #_ #HWV #IHWV1 #X #l2 #H
- elim (snta_inv_lref1 … H) -H * #K0 #W0 #V2 #V0 #HLK0 #HW0V2 #HWV0 [2: #HL2 ] #HX
- lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H destruct
- lapply (lift_mono … HWV0 … HWV) -HWV0 -HWV #H destruct
- elim (IHWV1 … HW0V2) -IHWV1 -HW0V2 /3 width=1/
-| #I #L #V #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H
- elim (snta_inv_bind1 … H) -H #W2 #U2 #l20 #_ #HTU2 #H
- elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12
- lapply (cpcs_trans … (ⓑ{I}V.U1) … H) -H /2 width=1/
-| #L #V #W #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H
- elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H
- elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12
- lapply (cpcs_trans … (ⓐV.ⓛW1.U1) … H) -H /2 width=1/
-| #L #V #T #U1 #W1 #l1 #_ #_ #IHTU1 #_ #X #l2 #H
- elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H
- elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12
- lapply (cpcs_trans … (ⓐV.U1) … H) -H /2 width=1/
-| #L #T #U1 #W1 #l10 #l1 #_ #_ #IHTU1 #_ #X #l2 #H
- elim (snta_inv_cast1 … H) -H #HTU1
- elim (IHTU1 … HTU1) -IHTU1 -HTU1 /2 width=1/
-| #L #T #U11 #U12 #V12 #l1 #_ #HU112 #_ #IHTU11 #_ #U2 #l2 #HTU2
- elim (IHTU11 … HTU2) -IHTU11 -HTU2 #Hl12 #H
- lapply (cpcs_canc_sn … HU112 … H) -U11 /2 width=1/
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma snta_cast_alt: ∀h,L,T,W,U,l. ⦃h, L⦄ ⊢ T :[l] W → ⦃h, L⦄ ⊢ T :[l] U →
- ⦃h, L⦄ ⊢ ⓝW.T :[l] U.
-#h #L #T #W #U #l #HTW #HTU
-elim (snta_mono … HTW … HTU) #_ #HWU
-elim (snta_fwd_correct … HTU) -HTU /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/thin_ldrop.ma".
-include "basic_2/equivalence/cpcs_delift.ma".
-include "basic_2/dynamic/snta_lift.ma".
-
-(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties on basic local environment thinning ***************************)
-
-(* Note: this is known as the substitution lemma *)
-lemma snta_thin_conf: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 →
- ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 →
- ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 :[l] U2 &
- L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2.
-#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
-[ /2 width=5/
-| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12
- elim (lt_or_ge i d) #Hdi [ -HVW1 ]
- [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H
- lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1
- elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2
- elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12
- lapply (delift_mono … H … HV12) -H -HV12 #H destruct
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1
- lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 //
- >minus_plus <plus_minus_m_m // /3 width=6/
- | elim (lt_or_ge i (d+e)) #Hide [ -HVW1 | -Hdi -IHVW1 -HL1 ]
- [ lapply (sfr_ldrop_trans_be_up … HLK1 … HL1 ? ?) -HL1 // /2 width=2/ <minus_n_O #H
- elim (sfr_inv_bind … H ?) -H /2 width=1/ #HK1 #_
- elim (thin_ldrop_conf_be … HL12 … HLK1 ? ?) -HL12 /2 width=2/ #K2 #H #HLK2
- lapply (thin_inv_thin1 … H ?) -H /2 width=1/ #HK12
- elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #V2 #W2 #HVW2 #HV12 #HW12
- elim (lift_total V2 0 d) #T2 #HVT2
- elim (lift_total W2 0 d) #U2 #HWU2
- elim (lift_total W2 0 (i+1)) #U #HW2U
- lapply (snta_lift … HVW2 … HLK2 … HVT2 … HWU2) -HVW2 -HLK2 #HTU2
- lapply (ldrop_fwd_ldrop2 … HLK1) #HLK0
- lapply (delift_lift_ge … HW12 … HLK0 HWU1 … HW2U) -HW12 -HLK0 -HWU1 // >minus_plus #HU1
- lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2
- lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/
- | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei
- lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1
- elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W
- <plus_minus in ⊢ (??%??→?); /2 width=2/ #HW1
- <minus_minus // /2 width=2/ -Hdei >commutative_plus <minus_n_n /3 width=6/
- ]
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL1 #HL12
- elim (lt_or_ge i d) #Hdi [ -HWV1 | -IHWV1 ]
- [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H
- lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1
- elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2
- elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK1 HK12) -IHWV1 -HK1 -HK12 #X2 #V2 #HWV2 #H #_
- lapply (delift_mono … H … HW12) -H #H destruct
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
- lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 //
- >minus_plus <plus_minus_m_m // /3 width=6/
- | elim (lt_or_ge i (d+e)) #Hide [ -HWV1 -HWU1 -HL12 | -Hdi -HL1 ]
- [ lapply (sfr_inv_ldrop … HLK1 … HL1 ? ?) -HLK1 -HL1 // -Hdi -Hide #H destruct
- | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei
- lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1
- elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W
- <plus_minus in ⊢ (??%??→?); /2 width=2/ #HW1
- <minus_minus // /2 width=2/ -Hdei >commutative_plus <minus_n_n /3 width=6/
- ]
- ]
-| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL1 #HL12
- elim (IHVW1 … HL1 HL12) -IHVW1 #V2 #W2 #HVW2 #HV12 #_
- elim (IHTU1 (L2.ⓑ{I}V2) (d+1) e ? ?) -IHTU1 /2 width=1/ -HL1 -HL12 #T2 #U2 #HTU2 #HT12 #HU12
- lapply (delift_lsubs_trans … HT12 (L1.ⓑ{I}V2) ?) -HT12 /2 width=1/
- lapply (delift_lsubs_trans … HU12 (L1.ⓑ{I}V2) ?) -HU12 /2 width=1/ /3 width=7/
-| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL1 #HL12
- elim (IHVW1 … HL1 HL12) -IHVW1 #V2 #W22 #HVW2 #HV12 #HW122
- elim (IHTU1 … HL1 HL12) -IHTU1 -HL1 -HL12 #X2 #Y2 #HXY2 #HX2 #HY2
- elim (delift_inv_bind1 … HX2) -HX2 #W21 #T2 #W121 #HT12 #H destruct
- elim (delift_inv_bind1 … HY2) -HY2 #X #U2 #HX #HU12 #H destruct
- lapply (delift_mono … HX … HW122) -HX #H destruct
- @(ex3_2_intro … (ⓐV2.ⓛW21.T2) (ⓐV2.ⓛW22.U2)) [ /2 width=2/ | 2,3: /3 width=1/ ] (**) (* explict constructor, /4 depth=?/ is too slow *)
-| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL1 #HL12
- elim (IHTU1 … HL1 HL12) -IHTU1 #T2 #U2 #HTU2 #HT12 #HU12
- elim (IHUW1 … HL1 HL12) -IHUW1 -HL1 -HL12 #X2 #W2 #HXW2 #H #HW12
- elim (delift_inv_flat1 … H) -H #V2 #Y2 #HV12 #HY2 #H destruct
- lapply (delift_mono … HY2 … HU12) -HY2 #H destruct /3 width=7/
-| #L1 #T1 #U1 #W1 #l1 #l2 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL1 #HL12
- elim (IHTU1 … HL1 HL12) -IHTU1 #T2 #U2 #HTU2 #HT12 #HU12
- elim (IHUW1 … HL1 HL12) -IHUW1 -HL1 -HL12 #Y2 #W2 #HUW2 #HY2 #HW12
- lapply (delift_mono … HY2 … HU12) -HY2 #H destruct /3 width=5/
-| #L1 #T1 #U11 #U12 #V1 #l #_ #HU112 #_ #IHT1 #IHU12 #L2 #d #e #HL1 #HL12
- elim (IHT1 … HL1 HL12) -IHT1 #T2 #U21 #HT2 #HT12 #HU121
- elim (IHU12 … HL1 HL12) -IHU12 -HL1 #U22 #V2 #HU22 #HU122 #_
- lapply (thin_cpcs_delift_mono … HU112 … HL12 … HU121 … HU122) -HU112 -HL12 -HU121 /3 width=5/
-]
-qed.
-
-lemma snta_ldrop_conf: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 →
- ∀L2,d,e. ≽ [d, e] L1 → ⇩[d, e] L1 ≡ L2 →
- ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 :[l] U2 &
- L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2.
-/3 width=1/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/lstas_lstas.ma".
-include "basic_2/computation/fpbs_lift.ma".
-include "basic_2/computation/fpbg_fleq.ma".
-include "basic_2/equivalence/cpes_cpds.ma".
-include "basic_2/dynamic/snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Inductive premises for the preservation results **************************)
-
-definition IH_snv_cpx_lpx: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
-
-definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
-
-definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
- λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-
-(* Properties for the preservation results **********************************)
-
-fact snv_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
-/3 width=6 by lpr_lpx, cpr_cpx/ qed-.
-
-fact snv_sta_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- ∀G,L,T. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
- ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
- ∀U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ U ¡[h, g].
-/3 width=6 by sta_cpx/ qed-.
-
-fact snv_cpxs_lpx_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
-#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
-@(cpxs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cpxs_fpbs/
-qed-.
-
-fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
-/3 width=10 by snv_cpxs_lpx_aux, cprs_cpxs, lpr_lpx/ qed-.
-
-fact snv_lstas_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 → ⦃G, L1⦄ ⊢ U1 ¡[h, g].
-/3 width=12 by snv_cpxs_lpx_aux, lstas_cpxs/ qed-.
-
-fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H
-@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
-qed-.
-
-fact da_cpcs_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] →
- ∀l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ∀l2. ⦃G, L⦄ ⊢ T2 ▪[h, g] l2 →
- ⦃G, L⦄ ⊢ T1 ⬌* T2 → l1 = l2.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #l1 #Hl1 #l2 #Hl2 #H
-elim (cpcs_inv_cprs … H) -H /4 width=18 by da_cprs_lpr_aux, da_mono/
-qed-.
-
-fact sta_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l+1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •[h] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #H01 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12
-elim (IH … H01 … 1 … Hl U1 … HT12 … HL12) -H01 -Hl -HT12 -HL12
-/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/
-qed-.
-
-fact lstas_cprs_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H
-@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
-#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
-elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (IH1 … Hl21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
-[2: /3 width=12 by da_cprs_lpr_aux/
-|3: /3 width=10 by snv_cprs_lpr_aux/
-|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
-] -G0 -L0 -T0 -T1 -T -l1
-/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
-qed-.
-
-fact lstas_cpcs_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l,l1. l ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l] U1 →
- ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
- ∀l2. l ≤ l2 → ⦃G, L1⦄ ⊢ T2 ▪[h, g] l2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, l] U2 →
- ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l #l1 #Hl1 #HTl1 #U1 #HTU1 #T2 #H02 #HT2 #l2 #Hl2 #HTl2 #U2 #HTU2 #H #L2 #HL12
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (lstas_cprs_lpr_aux … H01 HT1 … Hl1 HTl1 … HTU1 … H1 … HL12) -T1 /2 width=1 by/ #W1 #H1 #HUW1
-elim (lstas_cprs_lpr_aux … H02 HT2 … Hl2 HTl2 … HTU2 … H2 … HL12) -T2 /2 width=1 by/ #W2 #H2 #HUW2 -L0 -T0
-lapply (lstas_mono … H1 … H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/
-qed-.
-
-fact lstas_cpds_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
- ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
- ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
- ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
-lapply (da_mono … H … Hl1) -H #H destruct
-lapply (lstas_da_conf … HTU1 … Hl1) #Hl12
-elim (le_or_ge l2 l) #Hl2
-[ lapply (lstas_conf_le … HTU1 … HT1T) -HT1T
- /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
-| lapply (lstas_da_conf … HT1T … Hl1) #Hl1l
- lapply (lstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
- elim (lstas_cprs_lpr_aux … IH3 IH2 IH1 … Hl1l … HTU1 … HTT2 L) -IH2 -IH1 -Hl1l -HTU1 -HTT2
- /3 width=12 by snv_lstas_aux, cpcs_cpes, fpbg_fpbs_trans, lstas_fpbs, monotonic_le_minus_l, ex3_2_intro/
-]
-qed-.
-
-fact cpds_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 * #W1 #l1 #l2 #Hl21 #Hl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
-elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
-lapply (IH2 … H01 … Hl1 … HT12 … HL12) -L0 -T0 // -T1
-lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
-lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=7 by ex4_3_intro, ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lsubd_da.ma".
-include "basic_2/computation/cpds_cpds.ma".
-include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_cpcs.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on degree assignment for terms ********************************)
-
-fact da_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h g G1 L1 T1.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #_ #_ #_ #_ #l #H2 #X3 #H3 #L2 #_ -IH2 -IH1
- lapply (da_inv_sort … H2) -H2
- lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1 by da_sort/
-| #i #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
- elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #H #HX0
- elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #l1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- elim (cpr_inv_lref1 … H3) -H3
- [1,3: #H destruct
- lapply (fqup_lref … G1 … HLK1)
- elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- /4 width=10 by da_ldef, da_ldec, fqup_fpbg/
- |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (fqup_lref … G1 … HLK1)
- elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_drop2 … HLK2) -V2
- /4 width=8 by da_lift, fqup_fpbg/
- ]
-| #p #_ #_ #HT0 #H1 destruct -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
- elim (snv_inv_bind … H1) -H1 #_ #HT1
- lapply (da_inv_bind … H2) -H2
- elim (cpr_inv_bind1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=9 by da_bind, fqup_fpbg, lpr_pair/
- | #T2 #HT12 #HT2 #H1 #H2 destruct
- /4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, ldrop_drop/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #b0 #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10
- lapply (da_inv_flat … H2) -H2 #Hl
- elim (cpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH2 /4 width=7 by da_flat, fqup_fpbg/
- | #b #V2 #W #W2 #U1 #U2 #HV12 #HW2 #HU12 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW #HU1
- lapply (da_inv_bind … Hl) -Hl #Hl
- elim (cpds_inv_abst1 … HT10) -HT10 #W3 #U3 #HW3 #_ #H destruct -U3
- lapply (cprs_div … HW3 … HW10) -W3 #HWW1
- lapply (da_sta_conf … HVW1 … Hl0) <minus_plus_m_m #H
- elim (snv_fwd_da … HW) #l1 #Hl1
- lapply (snv_lstas_aux … IH2 … HV1 … 1 … Hl0 W1 ?) /2 width=2 by fqup_fpbg, sta_lstas/ #HW1
- lapply (da_cpcs_aux … IH2 IH1 … Hl1 … H … HWW1) -H
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, sta_fpbs/ #H destruct
- lapply (IH1 … HV1 … Hl0 … HV12 … HL12) -HV1 -Hl0 -HV12 [ /2 by fqup_fpbg/ ] #Hl0
- lapply (IH1 … Hl1 … HW2 … HL12) -Hl1 // /2 width=1 by fqup_fpbg/ -HW
- lapply (IH1 … HU1 … Hl … HU12 (L2.ⓛW2) ?) -IH1 -HU1 -Hl -HU12 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
- /4 width=6 by da_bind, lsubd_da_trans, lsubd_abbr/
- | #b #V #V2 #W #W2 #U1 #U2 #HV1 #HV2 #HW2 #HU12 #H1 #H2 destruct -IH2 -V -W0 -T0 -l0 -HV1 -HVW1
- elim (snv_inv_bind … HT1) -HT1 #_
- lapply (da_inv_bind … Hl) -Hl
- /5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
- elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
- lapply (da_inv_flat … H2) -H2 #Hl
- elim (cpr_inv_cast1 … H3) -H3
- [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
- | /3 width=7 by fqup_fpbg/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snv_lift.ma".
-include "basic_2/dynamic/snv_cpcs.ma".
-include "basic_2/dynamic/lsubsv_snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-fact snv_cpx_lpx_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpx_lpx h g G1 L1 T1.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH3 -IH2 -IH1 -H1
- elim (cpx_inv_sort1 … H2) -H2 // * //
-| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
- elim (lpx_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (fqup_lref … G1 … HLK1) #HKL
- elim (cpx_inv_lref1 … H2) -H2
- [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/
- | * #I0 #K0 #V0 #W0 #H #HVW0 #W0 -HV12
- lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
- lapply (ldrop_fwd_drop2 … HLK2) -HLK2 /4 width=8 by fqup_fpbg, snv_lift/
- ]
-| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #HV1 #HT1
- elim (cpx_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbg, snv_bind, lpx_pair/
- | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
- /4 width=10 by fqup_fpbg, snv_inv_lift, lpx_pair, ldrop_drop/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1
- elim (cpx_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
-(* lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #H2l0
- elim (sta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fqup_fpbg/ -HV1 #W2 #HVW2 #HW12
- elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -HT12 -HTU1 #X #HTU2 #H
- elim (cprs_inv_abst1 … H) -H #W20 #U2 #HW120 #_ #H destruct
- lapply (lpr_cprs_conf … HL12 … HW10) -L1 #HW10
- lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
- lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
- elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
- lapply (cpds_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?)
- /2 width=7 by snv_appl, cprs_bind/ *)
- | #b #V2 #W20 #W2 #T20 #T2 #HV12 #HW202 #HT202 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
- elim (cpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
- lapply (cprs_div … HW10 … HW230) -W30 #HW120
- lapply (snv_sta_aux … IH1 … Hl0 … HVW1) /2 width=1 by fqup_fpbg/ #HW10
- lapply (da_sta_conf … HVW1 … Hl0) <minus_plus_m_m #HlW10
- elim (snv_fwd_da … HW20) #l #Hl
- lapply (da_cpcs_aux … IH1 IH2 … HlW10 … Hl … HW120) // -HlW10
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, sta_fpbs/ #H destruct
-(* lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HlV2
- lapply (IH2 … Hl … HW202 … HL12) /2 width=1 by fqup_fpbg/ #HlW2
- elim (sta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #W3 #HV2W3 #HW103
- lapply (da_sta_conf … HV2W3 … HlV2) <minus_plus_m_m #HlW3
- lapply (cpcs_cpr_strap1 … HW120 … HW202) -HW120 #HW102
- lapply (lpr_cpcs_conf … HL12 … HW102) -HW102 #HW102
- lapply (cpcs_canc_sn … HW103 … HW102) -W10 #HW32
- lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1 #HV2
- lapply (IH1 … HW202 … HL12) /2 width=1 by fqup_fpbg/ -HW20 #HW2
- lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT20 #HT2
- lapply (snv_sta_aux … IH4 … HlV2 … HV2W3)
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/ #HW3
- lapply (lsubsv_snv_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /3 width=3 by snv_bind, snv_cast/
- @(lsubsv_abbr … l) /3 width=7 by fqup_fpbg/ #W #W0 #l0 #Hl0 #HV2W #HW20
- lapply (lstas_sta_conf_pos … HV2W3 … HV2W) -HV2W #HW3W
- @(lstas_cpcs_lpr_aux … IH1 IH2 IH3 … HlW3 … HW3W … HlW2 … HW20 … HW32) //
- [ /3 width=9 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_sta_fpbs/
- | /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/
- ] *)
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
- elim (cpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
- elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
-(* lapply (lpr_cpr_conf … HL12 … HW10) -HW10 #HW10
- elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbg, lpr_pair/ -HTU0 #X #HTU2 #H
- elim (cprs_inv_abst1 … H) -H #W #U2 #HW1 #_ #H destruct -U3
- elim (sta_cpr_lpr_aux … IH3 … HVW1 … HV10 … HL12) /2 width=2 by fqup_fpbg/ -IH3 -HVW1 #X #H1 #H2
- lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
- elim (lift_total X 0 1) #W20 #H3
- lapply (sta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=2 by ldrop_drop/ -H1 #HVW20
- lapply (cpcs_lift … (L2.ⓓW2) … H3 … HW13 H2) /2 width=2 by ldrop_drop/ -HW13 -H3 -H2 #HW320
- lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
- elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
- lapply (cpds_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) /2 width=1 by cprs_bind/ -HW3 -HTU2 #HTU2
- lapply (IH2 … Hl0 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -IH2 -Hl0 #Hl0
- lapply (da_lift … Hl0 (L2.ⓓW2) … HV02) /2 width=2 by ldrop_drop/ -Hl0 #Hl0
- lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ -HW0 #HW2
- lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV10 #HV0
- lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -L1 #HT2
- lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /3 width=7 by snv_bind, snv_appl, ldrop_drop/ *)
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
- elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
- elim (cpx_inv_cast1 … H2) -H2
- [ * #W2 #T2 #HW12 #HT12 #H destruct
-(* lapply (cpcs_cprs_strap1 … HUW1 W2 ?) /2 width=1 by cpr_cprs/ -HUW1 #H1
-*) lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/ -HW1 -HW12 #HW2
- lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -IH1 #HT2
-(* elim (sta_cpr_lpr_aux … IH3 … Hl0 … HTU1 … HT12 … HL12) /2 width=2 by fqup_fpbg/ -IH3 -HTU1 #U2 #HTU2 #HU12
- lapply (IH2 … Hl0 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -IH2 -HT1 -HT12 -Hl0 #Hl0
- /4 width=7 by snv_cast, lpr_cpcs_conf, cpcs_canc_sn/ *)
- | #H -IH3 -IH2 -HW1 -HTU1 -HUW1
- lapply (IH1 … H … HL12) -IH1 -H -HL12 /2 width=1 by fqup_fpbg/
- | #H -IH3 -IH2 -HTU1 -HUW1
- lapply (IH1 … H … HL12) -IH1 -H -HL12 /2 width=1 by fqup_fpbg/
- ]
-]
-[4: @snv_cast //
-qed-.
-snv_lstas, snv_cpr_lpr
\ No newline at end of file
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cpds_cpds.ma".
-include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_cpcs.ma".
-include "basic_2/dynamic/lsubsv_lstas.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on sn parallel reduction for local environments ***************)
-
-fact lstas_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_lstas_cpr_lpr h g G1 L1 T1.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #_ #_ #_ #_ #l1 #l2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
- >(lstas_inv_sort1 … H2) -X2
- >(cpr_inv_sort1 … H3) -X3 /2 width=3 by ex2_intro/
-| #i #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus … l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3
- [ lapply (lstas_inv_O … H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ]
- elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HK0 #HX0
- elim (da_inv_lref … Hl1) -Hl1 * #K1 [ #V1 | #W1 #l0 ] #HLK1 [ #HVl1 | #HWl1 #H destruct ]
- lapply (ldrop_mono … HK0 … HLK1) -HK0 #H destruct
- elim (lstas_inv_lref1 … H2) -H2 * #K0 #V0 #W0 [2,4: #X0 ] #HK0 [1,2: #_ -X0 ] #HVW0 #HX2
- lapply (ldrop_mono … HK0 … HLK1) -HK0 #H destruct
- [ lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21 ]
- lapply (fqup_lref … G1 … HLK1) #HKV1
- elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #V2 ] #HK12 [ #HW12 | #HV12 ] #H destruct
- lapply (ldrop_fwd_drop2 … HLK2) #H2
- elim (cpr_inv_lref1 … H3) -H3
- [1,3: #H destruct -HLK1
- |2,4: * #K0 #V0 #X0 #H #HVX0 #HX0
- lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
- ]
- [ lapply (IH2 … HWl1 … HW12 … HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
- elim (da_inv_sta … H) -H
- elim (IH1 … HWl1 … HVW0 … HW12 … HK12) -IH1 -HVW0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
- elim (lift_total V2 0 (i+1))
- /3 width=12 by cpcs_lift, lstas_ldec, ex2_intro/
- | elim (IH1 … HVl1 … HVW0 … HV12 … HK12) -IH1 -HVl1 -HVW0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
- elim (lift_total W2 0 (i+1))
- /4 width=12 by cpcs_lift, lstas_ldef, ex2_intro/
- | elim (IH1 … HVl1 … HVW0 … HVX0 … HK12) -IH1 -HVl1 -HVW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -l1 #W2 #HXW2 #HW02
- elim (lift_total W2 0 (i+1))
- /3 width=12 by cpcs_lift, lstas_lift, ex2_intro/
- ]
-| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #_ #HT1
- lapply (da_inv_bind … Hl1) -Hl1 #Hl1
- elim (lstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
- elim (cpr_inv_bind1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
- /4 width=5 by cpcs_bind2, lpr_cpr_conf, lstas_bind, ex2_intro/
- | #T3 #HT13 #HXT3 #H1 #H2 destruct
- elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
- elim (lstas_inv_lift1 … HTU3 L2 … HXT3) -T3
- /5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, ldrop_drop, ex2_intro/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU10
- lapply (da_inv_flat … Hl1) -Hl1 #Hl1
- elim (lstas_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
- elim (cpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
- elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1
- /4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lstas_appl, ex2_intro/
- | #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
- lapply (da_inv_bind … Hl1) -Hl1 #Hl1
- elim (lstas_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
- elim (cpds_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW20 #_ #H destruct
- lapply (cprs_div … HW10 … HW20) -W0 #HW12
- lapply (da_sta_conf … HVW1 … Hl0) <minus_plus_m_m #H
- elim (snv_fwd_da … HW2) #l #Hl
- lapply (snv_lstas_aux … IH3 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fqup_fpbg, sta_lstas/ #HW1
- lapply (da_cpcs_aux … IH3 IH2 … H … Hl … HW12) // -H
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, sta_fpbs/ #H destruct
- lapply (snv_cpr_lpr_aux … IH3 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2l
- elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fqup_fpbg, sta_lstas/ -HVW1 #W4 #H #HW14
- lapply (lstas_inv_SO … H) #HV2W4
- lapply (da_sta_conf … HV2W4 … HV2l) <minus_plus_m_m #HW4l
- lapply (snv_lstas_aux … IH3 … HV2 … HV2l … H) -H /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/ #HW4
- lapply (snv_cpr_lpr_aux … IH3 … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3
- lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3l
- elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
- lapply (cpcs_cpr_strap1 … HW12 … HW23) #H
- lapply (lpr_cpcs_conf … HL12 … H) -H #H
- lapply (cpcs_canc_sn … HW14 H) -H #HW43
- elim (lsubsv_lstas_trans … g … HTU3 … Hl21 … (L2.ⓓⓝW3.V2)) -HTU3
- [ #U4 #HT3U4 #HU43 -HW12 -HW3 -HW3l -W4 -IH3 -IH2
- @(ex2_intro … (ⓓ{b}ⓝW3.V2.U4)) /2 width=1 by lstas_bind/ -HT3U4
- @(cpcs_canc_dx … (ⓓ{b}ⓝW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
- @(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
- /4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
- | -U3
- @(lsubsv_abbr … l) /3 width=7 by fqup_fpbg/
- #W #W0 #l0 #Hl0 #HV2W #HW30
- lapply (lstas_sta_conf_pos … HV2W4 … HV2W) -HV2W #HW4W
- @(lstas_cpcs_lpr_aux … IH3 IH2 IH1 … Hl0 … HW4W … Hl0 … HW30 … HW43) //
- [ /3 width=9 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_sta_fpbs/
- | /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/
- ]
- | -IH3 -IH1 /3 width=9 by fqup_fpbg, lpr_pair/
- ]
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
- elim (snv_inv_bind … HT1) -HT1 #_ #HT0
- lapply (da_inv_bind … Hl1) -Hl1 #Hl1
- elim (lstas_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
- elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
- lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
- lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
- lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
- lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
- lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?)
- /4 width=3 by lstas_appl, lstas_bind, cpr_theta, ex2_intro/
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus … l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
- [ lapply (lstas_inv_O … H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ]
- elim (snv_inv_cast … H1) -H1 #U1 #l #_ #HT1 #_ #_ #_ -U1 -l
- lapply (da_inv_flat … Hl1) -Hl1 #Hl1
- lapply (lstas_inv_cast1 … H2) -H2 #HTU1
- elim (cpr_inv_cast1 … H3) -H3
- [ * #U2 #T2 #_ #HT12 #H destruct
- elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1 -HL12
- /3 width=3 by fqup_fpbg, lstas_cast, ex2_intro/
- | #HT1X3
- elim (IH1 … Hl1 … HTU1 … HT1X3 … HL12) -IH1 -Hl1 -HTU1 -HL12
- /2 width=3 by fqup_fpbg, ex2_intro/
- ]
-]
-qed-.
+++ /dev/null
-lemma da_cpcs: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G, L⦄ ⊢ T2 ¡[h, g] →
- ∀l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ∀l2. ⦃G, L⦄ ⊢ T2 ▪[h, g] l2 →
- ⦃G, L⦄ ⊢ T1 ⬌* T2 → l1 = l2.
-#h #g #G #L #T1 #HT1 #T2 #HT2 #l1 #Hl1 #l2 #Hl2 #H
-elim (cpcs_inv_cprs … H) -H /3 width=12 by da_cprs_lpr, da_mono/
-qed-.
-
-lemma sta_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l+1 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •[h] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-#h #g #G #L1 #T1 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12
-elim (lstas_cpr_lpr … 1 … Hl U1 … HT12 … HL12) -Hl -HT12 -HL12
-/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/
-qed-.
-
-lemma snv_sta: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
- ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
- ∀U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ U ¡[h, g].
-/3 width=7 by lstas_inv_SO, sta_lstas, snv_lstas/ qed-.
-
-lemma lstas_cpds: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ T1 ¡[h, g] →
- ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
- ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
- ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
-#h #g #G #L #T1 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
-lapply (da_mono … H … Hl1) -H #H destruct
-lapply (lstas_da_conf … HTU1 … Hl1) #Hl12
-elim (le_or_ge l2 l) #Hl2
-[ lapply (lstas_conf_le … HTU1 … HT1T) -HT1T //
- /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
-| lapply (lstas_da_conf … HT1T … Hl1) #Hl1l
- lapply (lstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
- elim (lstas_cprs_lpr … Hl1l … HTU1 … HTT2 L) -Hl1l -HTU1 -HTT2
- /3 width=7 by snv_lstas, cpcs_cpes, monotonic_le_minus_l, ex3_2_intro/
-]
-qed-.
-
-lemma cpds_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
-#h #g #G #L1 #T1 #HT1 #U1 * #W1 #l1 #l2 #Hl21 #Hl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
-elim (lstas_cpr_lpr … Hl1 … HTW1 … HT12 … HL12) // #W2 #HTW2 #HW12
-lapply (da_cpr_lpr … Hl1 … HT12 … HL12) // -T1
-lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
-lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=7 by ex4_3_intro, ex2_intro/
-qed-.
-
-(* Note: missing da_scpds_lpr, da_scpes *)
-
-lemma scpds_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1,l. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, l] U1 →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g, l] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
-#h #g #G #L1 #T1 #HT1 #U1 #l2 * #W1 #l1 #Hl21 #HTl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
-elim (lstas_cpr_lpr … HTl1 … HTW1 … HT12 … HL12) // #W2 #HTW2 #HW12
-lapply (da_cpr_lpr … HTl1 … HT12 … HL12) // -T1
-lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
-lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=6 by ex4_2_intro, ex2_intro/
-qed-.
-
-lemma scpes_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
- ∀l1,l2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2 →
- ∀U1. ⦃G, L1⦄ ⊢ T1 ➡ U1 → ∀U2. ⦃G, L1⦄ ⊢ T2 ➡ U2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L2⦄ ⊢ U1 •*⬌*[h, g, l1, l2] U2.
-#h #g #G #L1 #T1 #HT1 #T2 #HT2 #l1 #l2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
-elim (scpds_cpr_lpr … HT10 … HTU1 … HL12) -HT10 -HTU1 // #X1 #HUX1 #H1
-elim (scpds_cpr_lpr … HT20 … HTU2 … HL12) -HT20 -HTU2 // #X2 #HUX2 #H2
-elim (cprs_conf … H1 … H2) -T0 /3 width=5 by scpds_div, scpds_cprs_trans/
-qed-.
-
-(* Note: missing lstas_scpds, scpes_le *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/da_sta.ma".
-include "basic_2/static/sta_aaa.ma".
-
-(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma aaa_da: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
-#h #g #G #L #T #A #H elim (aaa_sta h … H) -A /2 width=2 by sta_da/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/sta.ma".
-include "basic_2/static/da_da.ma".
-
-(* Properties on static type assignment for terms ***************************)
-
-lemma da_sta_conf: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
- ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ U ▪[h, g] l-1.
-#h #g #G #L #T #U #H elim H -G -L -T -U
-[ #G #L #k #l #H
- lapply (da_inv_sort … H) -H /3 width=1 by da_sort, deg_next/
-| #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #l #H
- elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0
- lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
- lapply (drop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
-| #G #L #K #W #V #U #i #HLK #_ #HWU #IHWV #l #H
- elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0 [| #H0 ]
- lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
- lapply (drop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
-| #a #I #G #L #V #T #U #_ #IHTU #l #H
- lapply (da_inv_bind … H) -H /3 width=1 by da_bind/
-| #G #L #V #T #U #_ #IHTU #l #H
- lapply (da_inv_flat … H) -H /3 width=1 by da_flat/
-| #G #L #W #T #U #_ #IHTU #l #H
- lapply (da_inv_flat … H) -H /2 width=1 by/
-]
-qed-.
-
-lemma sta_da: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
- ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
-#h #g #G #L #T #U #H elim H -G -L -T -U
-[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
-| #G #L #K #V #W #W0 #i #HLK #_ #_ * /3 width=5 by da_ldef, ex_intro/
-| #G #L #K #W #V #W0 #i #HLK #_ #_ * /3 width=5 by da_ldec, ex_intro/
-| #a #I #G #L #V #T #U #_ * /3 width=2 by da_bind, ex_intro/
-| #G #L #V #T #U #_ * /3 width=2 by da_flat, ex_intro/
-| #G #L #W #T #U #_ * /3 width=2 by da_flat, ex_intro/
-]
-qed-.
-
-lemma sta_da_ge: ∀h,G,L,T,U,l0. ⦃G, L⦄ ⊢ T •[h] U →
- ∃∃g,l. ⦃G, L⦄ ⊢ T ▪[h, g] l & l0 ≤ l.
-#h #G #L #T #U #l0 #H elim H -G -L -T -U
-[ /3 width=4 by da_sort, ex2_2_intro/
-| #G #L #K #V #W #W0 #i #HLK #_ #_ * /3 width=5 by da_ldef, ex2_2_intro/
-| #G #L #K #W #V #W0 #i #HLK #_ #_ * /4 width=5 by da_ldec, lt_to_le, le_S_S, ex2_2_intro/
-| #a #I #G #L #V #T #U #_ * /3 width=4 by da_bind, ex2_2_intro/
-| #G #L #V #T #U #_ * /3 width=4 by da_flat, ex2_2_intro/
-| #G #L #W #T #U #_ * /3 width=4 by da_flat, ex2_2_intro/
-]
-qed-.
-
-(* Inversion lrmmas on static type assignment for terms *********************)
-
-lemma da_inv_sta: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l →
- ∃U. ⦃G, L⦄ ⊢ T •[h] U.
-#h #g #G #L #T #l #H elim H -G -L -T -l
-[ /2 width=2/
-| #G #L #K #V #i #l #HLK #_ * #W #HVW
- elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldef, ex_intro/
-| #G #L #K #W #i #l #HLK #_ * #V #HWV
- elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldec, ex_intro/
-| #a #I #G #L #V #T #l #_ * /3 width=2 by sta_bind, ex_intro/
-| * #G #L #V #T #l #_ * /3 width=2 by sta_appl, sta_cast, ex_intro/
-]
-qed-.
-
-lemma sta_inv_refl_pos: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h] T → ⊥.
-#h #g #G #L #T #l #H1T #HTT
-lapply (da_sta_conf … HTT … H1T) -HTT <minus_plus_m_m #H2T
-lapply (da_mono … H2T … H1T) -h -G -L -T #H
-elim (plus_xySz_x_false 0 l 0) //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/statictypestar_6.ma".
-include "basic_2/static/sta.ma".
-
-(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
-
-definition lstas: ∀h. genv → lenv → nat → relation term ≝
- λh,G,L. lstar … (sta h G L).
-
-interpretation "nat-iterated static type assignment (term)"
- 'StaticTypeStar h G L l T U = (lstas h G L l T U).
-
-(* Basic eliminators ********************************************************)
-
-lemma lstas_ind_sn: ∀h,G,L,U2. ∀R:relation2 nat term.
- R 0 U2 → (
- ∀l,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 → ⦃G, L⦄ ⊢ U1 •* [h, l] U2 →
- R l U1 → R (l+1) T
- ) →
- ∀l,T. ⦃G, L⦄ ⊢ T •*[h, l] U2 → R l T.
-/3 width=5 by lstar_ind_l/ qed-.
-
-lemma lstas_ind_dx: ∀h,G,L,T. ∀R:relation2 nat term.
- R 0 T → (
- ∀l,U1,U2. ⦃G, L⦄ ⊢ T •* [h, l] U1 → ⦃G, L⦄ ⊢ U1 •[h] U2 →
- R l U1 → R (l+1) U2
- ) →
- ∀l,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R l U.
-/3 width=5 by lstar_ind_r/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lstas_inv_O: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 0] U → T = U.
-/2 width=4 by lstar_inv_O/ qed-.
-
-lemma lstas_inv_SO: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 1] U → ⦃G, L⦄ ⊢ T •[h] U.
-/2 width=1 by lstar_inv_step/ qed-.
-
-lemma lstas_inv_step_sn: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 •[h] T & ⦃G, L⦄ ⊢ T •*[h, l] T2.
-/2 width=3 by lstar_inv_S/ qed-.
-
-lemma lstas_inv_step_dx: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l] T & ⦃G, L⦄ ⊢ T •[h] T2.
-/2 width=3 by lstar_inv_S_dx/ qed-.
-
-lemma lstas_inv_sort1: ∀h,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, l] X → X = ⋆((next h)^l k).
-#h #G #L #X #k #l #H @(lstas_ind_dx … H) -X -l //
-#l #X #X0 #_ #H #IHX destruct
-lapply (sta_inv_sort1 … H) -H #H destruct
->iter_SO //
-qed-.
-
-lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l+1] X → ⊥.
-#h #G #L #X #p #l #H elim (lstas_inv_step_sn … H) -H
-#U #H #HUX elim (sta_inv_gref1 … H)
-qed-.
-
-lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
- ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
-#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
-#l #X #X0 #_ #HX0 * #U #HTU #H destruct
-elim (sta_inv_bind1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
-qed-.
-
-lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
- ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
-#h #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
-#l #X #X0 #_ #HX0 * #U #HTU #H destruct
-elim (sta_inv_appl1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
-qed-.
-
-lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U → ⦃G, L⦄ ⊢ T •*[h, l+1] U.
-#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn … H) -H
-#U #H #HUX lapply (sta_inv_cast1 … H) -H /2 width=3 by lstar_S/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lstas_refl: ∀h,G,L. reflexive … (lstas h G L 0).
-// qed.
-
-lemma sta_lstas: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T •*[h, 1] U.
-/2 width=1 by lstar_step/ qed.
-
-lemma lstas_step_sn: ∀h,G,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h] U1 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2 →
- ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
-/2 width=3 by lstar_S/ qed.
-
-lemma lstas_step_dx: ∀h,G,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T2 •[h] U2 →
- ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
-/2 width=3 by lstar_dx/ qed.
-
-lemma lstas_split: ∀h,G,L. inv_ltransitive … (lstas h G L).
-/2 width=1 by lstar_inv_ltransitive/ qed-.
-
-lemma lstas_sort: ∀h,G,L,l,k. ⦃G, L⦄ ⊢ ⋆k •*[h, l] ⋆((next h)^l k).
-#h #G #L #l @(nat_ind_plus … l) -l //
-#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/
-qed.
-
-lemma lstas_bind: ∀h,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] ⓑ{a,I}V.U.
-#h #I #G #L #V #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/
-qed.
-
-lemma lstas_appl: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
- ∀V.⦃G, L⦄ ⊢ ⓐV.T •*[h, l] ⓐV.U.
-#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/
-qed.
-
-lemma lstas_cast: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U →
- ∀W. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U.
-#h #G #L #T #U #l #H elim (lstas_inv_step_sn … H) -H /3 width=3 by sta_cast, lstas_step_sn/
-qed.
-
-(* Basic_1: removed theorems 7:
- sty1_abbr sty1_appl sty1_bind sty1_cast2
- sty1_correct sty1_lift sty1_trans
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/sta_aaa.ma".
-include "basic_2/unfold/lstas.ma".
-
-(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma lstas_aaa_conf: ∀h,G,L,l. Conf3 … (aaa G L) (lstas h G L l).
-/3 width=6 by sta_aaa_conf, lstar_Conf3/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/statictypestaralt_6.ma".
-include "basic_2/unfold/lstas_lift.ma".
-
-(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
-
-(* alternative definition of lstas *)
-inductive lstasa (h): genv → relation4 lenv nat term term ≝
-| lstasa_O : ∀G,L,T. lstasa h G L 0 T T
-| lstasa_sort: ∀G,L,l,k. lstasa h G L l (⋆k) (⋆((next h)^l k))
-| lstasa_ldef: ∀G,L,K,V,W,U,i,l. ⇩[i] L ≡ K.ⓓV → lstasa h G K (l+1) V W →
- ⇧[0, i+1] W ≡ U → lstasa h G L (l+1) (#i) U
-| lstasa_ldec: ∀G,L,K,W,V,V0,U,i,l. ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W •[h] V0 →
- lstasa h G K l W V → ⇧[0, i+1] V ≡ U → lstasa h G L (l+1) (#i) U
-| lstasa_bind: ∀a,I,G,L,V,T,U,l. lstasa h G (L.ⓑ{I}V) l T U →
- lstasa h G L l (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
-| lstasa_appl: ∀G,L,V,T,U,l. lstasa h G L l T U → lstasa h G L l (ⓐV.T) (ⓐV.U)
-| lstasa_cast: ∀G,L,W,T,U,l. lstasa h G L (l+1) T U → lstasa h G L (l+1) (ⓝW.T) U
-.
-
-interpretation "nat-iterated static type assignment (term) alternative"
- 'StaticTypeStarAlt h G L l T U = (lstasa h G L l T U).
-
-(* Base properties **********************************************************)
-
-lemma sta_lstasa: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T ••*[h, 1] U.
-#h #G #L #T #U #H elim H -G -L -T -U
-/2 width=8 by lstasa_O, lstasa_sort, lstasa_ldef, lstasa_ldec, lstasa_bind, lstasa_appl, lstasa_cast/
-qed.
-
-lemma lstasa_step_dx: ∀h,G,L,T1,T,l. ⦃G, L⦄ ⊢ T1 ••*[h, l] T →
- ∀T2. ⦃G, L⦄ ⊢ T •[h] T2 → ⦃G, L⦄ ⊢ T1 ••*[h, l+1] T2.
-#h #G #L #T1 #T #l #H elim H -G -L -T1 -T -l
-[ /2 width=1 by sta_lstasa/
-| #G #L #l #k #X #H >(sta_inv_sort1 … H) -X >commutative_plus //
-| #G #L #K #V #W #U #i #l #HLK #_ #HWU #IHVW #U2 #HU2
- lapply (drop_fwd_drop2 … HLK) #H
- elim (sta_inv_lift1 … HU2 … H … HWU) -H -U /3 width=6 by lstasa_ldef/
-| #G #L #K #W #V #V0 #U #i #l #HLK #HWl0 #_ #HVU #IHWV #U2 #HU2
- lapply (drop_fwd_drop2 … HLK) #H
- elim (sta_inv_lift1 … HU2 … H … HVU) -H -U /3 width=8 by lstasa_ldec/
-| #a #I #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H
- elim (sta_inv_bind1 … H) -H #U #HU1 #H destruct /3 width=1 by lstasa_bind/
-| #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H
- elim (sta_inv_appl1 … H) -H #U #HU1 #H destruct /3 width=1 by lstasa_appl/
-| /3 width=1 by lstasa_cast/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem lstas_lstasa: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ⦃G, L⦄ ⊢ T ••*[h, l] U.
-#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /2 width=3 by lstasa_step_dx, lstasa_O/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lstasa_inv_lstas: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T ••*[h, l] U → ⦃G, L⦄ ⊢ T •*[h, l] U.
-#h #G #L #T #U #l #H elim H -G -L -T -U -l
-/2 width=8 by lstas_inv_SO, lstas_ldec, lstas_ldef, lstas_cast, lstas_appl, lstas_bind/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma lstas_ind_alt: ∀h. ∀R:genv→relation4 lenv nat term term.
- (∀G,L,T. R G L O T T) →
- (∀G,L,l,k. R G L l (⋆k) (⋆((next h)^l k))) → (
- ∀G,L,K,V,W,U,i,l.
- ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V •*[h, l+1] W → ⇧[O, i+1] W ≡ U →
- R G K (l+1) V W → R G L (l+1) (#i) U
- ) → (
- ∀G,L,K,W,V,V0,U,i,l.
- ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W •[h] V0 →
- ⦃G, K⦄ ⊢ W •*[h, l]V → ⇧[O, i+1] V ≡ U →
- R G K l W V → R G L (l+1) (#i) U
- ) → (
- ∀a,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
- R G (L.ⓑ{I}V) l T U → R G L l (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
- ) → (
- ∀G,L,V,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
- R G L l T U → R G L l (ⓐV.T) (ⓐV.U)
- ) → (
- ∀G,L,W,T,U,l. ⦃G, L⦄⊢ T •*[h, l+1] U →
- R G L (l+1) T U → R G L (l+1) (ⓝW.T) U
- ) →
- ∀G,L,l,T,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R G L l T U.
-#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #G #L #l #T #U #H
-elim (lstas_lstasa … H) /3 width=10 by lstasa_inv_lstas/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/lstas.ma".
-include "basic_2/static/da_sta.ma".
-
-(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
-
-(* Properties on degree assignment for terms ********************************)
-
-lemma lstas_da_conf: ∀h,g,G,L,T,U,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U →
- ∀l2. ⦃G, L⦄ ⊢ T ▪[h, g] l2 → ⦃G, L⦄ ⊢ U ▪[h, g] l2-l1.
-#h #g #G #L #T #U #l1 #H @(lstas_ind_dx … H) -U -l1 //
-#l1 #U #U0 #_ #HU0 #IHTU #l2 #HT
-<minus_plus /3 width=3 by da_sta_conf/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lstas_inv_refl_pos: ∀h,G,L,T,l. ⦃G, L⦄ ⊢ T •*[h, l+1] T → ⊥.
-#h #G #L #T #l #H elim (lstas_inv_step_sn … H)
-#U #HTU #_ elim (sta_da_ge … (l+1) HTU) -U
-#g #l0 #HT #Hl0 lapply (lstas_da_conf … H … HT) -H
-#H0T lapply (da_mono … HT … H0T) -h -G -L -T
-#H elim (discr_x_minus_xy … H) -H
-[ #H destruct /2 width=3 by le_plus_xSy_O_false/
-| -Hl0 <plus_n_Sm #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/sta_lift.ma".
-include "basic_2/unfold/lstas.ma".
-
-(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
-
-(* Properties on relocation *************************************************)
-
-lemma lstas_lift: ∀h,G,l. l_liftable (llstar … (sta h G) l).
-/3 width=10 by l_liftable_llstar, sta_lift/ qed.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (llstar … (sta h G) l).
-/3 width=6 by l_deliftable_sn_llstar, sta_inv_lift1/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lstas_inv_lref1: ∀h,G,L,U,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] U →
- (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
- ⇧[0, i+1] W ≡ U
- ) ∨
- (∃∃K,W,V,V0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V0 &
- ⦃G, K⦄ ⊢ W •*[h, l] V & ⇧[0, i+1] V ≡ U
- ).
-#h #G #L #U #i #l #H elim (lstas_inv_step_sn … H) -H
-#X #H #HXU elim (sta_inv_lref1 … H) -H
-* #K #V #W #HLK #HVW #HWX
-lapply (drop_fwd_drop2 … HLK) #H0LK
-elim (lstas_inv_lift1 … HXU … H0LK … HWX) -H0LK -X
-/4 width=8 by lstas_step_sn, ex4_4_intro, ex3_3_intro, or_introl, or_intror/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lstas_fwd_correct: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
- ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
- ∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2.
-#h #G #L #T1 #U1 #HTU1 #T2 #l #H @(lstas_ind_dx … H) -l -T2 /2 width=3 by ex_intro/ -HTU1
-#l #T #T2 #_ #HT2 #_ -T1 -U1 -l
-elim (sta_fwd_correct … HT2) -T /2 width=2 by ex_intro/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lstas_total: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
- ∀l. ∃U0. ⦃G, L⦄ ⊢ T •*[h, l] U0.
-#h #G #L #T #U #HTU #l @(nat_ind_plus … l) -l /2 width=2 by ex_intro/
-#l * #U0 #HTU0 elim (lstas_fwd_correct … HTU … HTU0) -U
-/3 width=4 by lstas_step_dx, ex_intro/
-qed-.
-
-lemma lstas_ldef: ∀h,G,L,K,V,i. ⇩[i] L ≡ K.ⓓV →
- ∀W,l. ⦃G, K⦄ ⊢ V •*[h, l+1] W →
- ∀U. ⇧[0, i+1] W ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
-#h #G #L #K #V #i #HLK #W #l #HVW #U #HWU
-lapply (drop_fwd_drop2 … HLK)
-elim (lstas_inv_step_sn … HVW) -HVW #W0
-elim (lift_total W0 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldef, lstas_lift/
-qed.
-
-lemma lstas_ldec: ∀h,G,L,K,W,i. ⇩[i] L ≡ K.ⓛW → ∀V0. ⦃G, K⦄ ⊢ W •[h] V0 →
- ∀V,l. ⦃G, K⦄ ⊢ W •*[h, l] V →
- ∀U. ⇧[0, i+1] V ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
-#h #G #L #K #W #i #HLK #V0 #HWV0 #V #l #HWV #U #HVU
-lapply (drop_fwd_drop2 … HLK) #H
-elim (lift_total W 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldec, lstas_lift/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/sta_sta.ma".
-include "basic_2/unfold/lstas_lift.ma".
-
-(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
-
-(* Main properties **********************************************************)
-
-theorem lstas_trans: ∀h,G,L. ltransitive … (lstas h G L).
-/2 width=3 by lstar_ltransitive/ qed-.
-
-theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h G L l).
-/3 width=7 by sta_mono, lstar_singlevalued/ qed-.
-
-theorem lstas_conf_le: ∀h,G,L,T,U1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U1 →
- ∀U2,l2. l1 ≤ l2 → ⦃G, L⦄ ⊢ T •*[h, l2] U2 →
- ⦃G, L⦄ ⊢ U1 •*[h, l2-l1] U2.
-#h #G #L #T #U1 #l1 #HTU1 #U2 #l2 #Hl12
->(plus_minus_m_m … Hl12) in ⊢ (%→?); -Hl12 >commutative_plus #H
-elim (lstas_split … H) -H #U #HTU
->(lstas_mono … HTU … HTU1) -T //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lstas_sta_conf_pos: ∀h,G,L,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 →
- ∀U2,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U2 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2.
-#h #G #L #T #U1 #HTU1 #U2 #l #HTU2
-lapply (lstas_conf_le … T U1 1 … HTU2) -HTU2 /2 width=1 by sta_lstas/
-qed-.
-
-lemma lstas_strip_pos: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
- ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
- ∃∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2 & ⦃G, L⦄ ⊢ U1 •*[h, l+1] U2.
-#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12
-elim (lstas_fwd_correct … HTU1 … HT12)
-lapply (lstas_sta_conf_pos … HTU1 … HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/statictype_5.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/drop.ma".
-include "basic_2/static/sh.ma".
-
-(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
-
-(* activate genv *)
-inductive sta (h:sh): relation4 genv lenv term term ≝
-| sta_sort: ∀G,L,k. sta h G L (⋆k) (⋆(next h k))
-| sta_ldef: ∀G,L,K,V,W,U,i. ⇩[i] L ≡ K.ⓓV → sta h G K V W →
- ⇧[0, i + 1] W ≡ U → sta h G L (#i) U
-| sta_ldec: ∀G,L,K,W,V,U,i. ⇩[i] L ≡ K.ⓛW → sta h G K W V →
- ⇧[0, i + 1] W ≡ U → sta h G L (#i) U
-| sta_bind: ∀a,I,G,L,V,T,U. sta h G (L.ⓑ{I}V) T U →
- sta h G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
-| sta_appl: ∀G,L,V,T,U. sta h G L T U → sta h G L (ⓐV.T) (ⓐV.U)
-| sta_cast: ∀G,L,W,T,U. sta h G L T U → sta h G L (ⓝW.T) U
-.
-
-interpretation "static type assignment (term)"
- 'StaticType h G L T U = (sta h G L T U).
-
-(* Basic inversion lemmas ************************************************)
-
-fact sta_inv_sort1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀k0. T = ⋆k0 →
- U = ⋆(next h k0).
-#h #G #L #T #U * -G -L -T -U
-[ #G #L #k #k0 #H destruct //
-| #G #L #K #V #W #U #i #_ #_ #_ #k0 #H destruct
-| #G #L #K #W #V #U #i #_ #_ #_ #k0 #H destruct
-| #a #I #G #L #V #T #U #_ #k0 #H destruct
-| #G #L #V #T #U #_ #k0 #H destruct
-| #G #L #W #T #U #_ #k0 #H destruct
-qed-.
-
-(* Basic_1: was: sty0_gen_sort *)
-lemma sta_inv_sort1: ∀h,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k •[h] U → U = ⋆(next h k).
-/2 width=5 by sta_inv_sort1_aux/ qed-.
-
-fact sta_inv_lref1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀j. T = #j →
- (∃∃K,V,W. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h] W &
- ⇧[0, j+1] W ≡ U
- ) ∨
- (∃∃K,W,V. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V &
- ⇧[0, j+1] W ≡ U
- ).
-#h #G #L #T #U * -G -L -T -U
-[ #G #L #k #j #H destruct
-| #G #L #K #V #W #U #i #HLK #HVW #HWU #j #H destruct /3 width=6 by or_introl, ex3_3_intro/
-| #G #L #K #W #V #U #i #HLK #HWV #HWU #j #H destruct /3 width=6 by or_intror, ex3_3_intro/
-| #a #I #G #L #V #T #U #_ #j #H destruct
-| #G #L #V #T #U #_ #j #H destruct
-| #G #L #W #T #U #_ #j #H destruct
-]
-qed-.
-
-(* Basic_1: was sty0_gen_lref *)
-lemma sta_inv_lref1: ∀h,G,L,U,i. ⦃G, L⦄ ⊢ #i •[h] U →
- (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h] W &
- ⇧[0, i+1] W ≡ U
- ) ∨
- (∃∃K,W,V. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V &
- ⇧[0, i+1] W ≡ U
- ).
-/2 width=3 by sta_inv_lref1_aux/ qed-.
-
-fact sta_inv_gref1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀p0. T = §p0 → ⊥.
-#h #G #L #T #U * -G -L -T -U
-[ #G #L #k #p0 #H destruct
-| #G #L #K #V #W #U #i #_ #_ #_ #p0 #H destruct
-| #G #L #K #W #V #U #i #_ #_ #_ #p0 #H destruct
-| #a #I #G #L #V #T #U #_ #p0 #H destruct
-| #G #L #V #T #U #_ #p0 #H destruct
-| #G #L #W #T #U #_ #p0 #H destruct
-qed-.
-
-lemma sta_inv_gref1: ∀h,G,L,U,p. ⦃G, L⦄ ⊢ §p •[h] U → ⊥.
-/2 width=8 by sta_inv_gref1_aux/ qed-.
-
-fact sta_inv_bind1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
- ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •[h] Z & U = ⓑ{b,J}Y.Z.
-#h #G #L #T #U * -G -L -T -U
-[ #G #L #k #b #J #X #Y #H destruct
-| #G #L #K #V #W #U #i #_ #_ #_ #b #J #X #Y #H destruct
-| #G #L #K #W #V #U #i #_ #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
-| #G #L #V #T #U #_ #b #J #X #Y #H destruct
-| #G #L #W #T #U #_ #b #J #X #Y #H destruct
-]
-qed-.
-
-(* Basic_1: was: sty0_gen_bind *)
-lemma sta_inv_bind1: ∀h,b,J,G,L,Y,X,U. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X •[h] U →
- ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •[h] Z & U = ⓑ{b,J}Y.Z.
-/2 width=3 by sta_inv_bind1_aux/ qed-.
-
-fact sta_inv_appl1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀X,Y. T = ⓐY.X →
- ∃∃Z. ⦃G, L⦄ ⊢ X •[h] Z & U = ⓐY.Z.
-#h #G #L #T #U * -G -L -T -U
-[ #G #L #k #X #Y #H destruct
-| #G #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct
-| #G #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct
-| #a #I #G #L #V #T #U #_ #X #Y #H destruct
-| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
-| #G #L #W #T #U #_ #X #Y #H destruct
-]
-qed-.
-
-(* Basic_1: was: sty0_gen_appl *)
-lemma sta_inv_appl1: ∀h,G,L,Y,X,U. ⦃G, L⦄ ⊢ ⓐY.X •[h] U →
- ∃∃Z. ⦃G, L⦄ ⊢ X •[h] Z & U = ⓐY.Z.
-/2 width=3 by sta_inv_appl1_aux/ qed-.
-
-fact sta_inv_cast1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀X,Y. T = ⓝY.X →
- ⦃G, L⦄ ⊢ X •[h] U.
-#h #G #L #T #U * -G -L -T -U
-[ #G #L #k #X #Y #H destruct
-| #G #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct
-| #G #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct
-| #a #I #G #L #V #T #U #_ #X #Y #H destruct
-| #G #L #V #T #U #_ #X #Y #H destruct
-| #G #L #W #T #U #HTU #X #Y #H destruct //
-]
-qed-.
-
-(* Basic_1: was: sty0_gen_cast *)
-lemma sta_inv_cast1: ∀h,G,L,X,Y,U. ⦃G, L⦄ ⊢ ⓝY.X •[h] U → ⦃G, L⦄ ⊢ X •[h] U.
-/2 width=4 by sta_inv_cast1_aux/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/sta.ma".
-include "basic_2/static/aaa_lift.ma".
-
-(* STATIC TYPE ASSIGNMENT FOR TERMS *****************************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma aaa_sta: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G, L⦄ ⊢ T •[h] U.
-#h #G #L #T #A #H elim H -G -L -T -A
-[ /2 width=2 by sta_sort, ex_intro/
-| * #G #L #K [ #V | #W ] #B #i #HLK #_ * [ #W | #V ] #HVW
- elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldef, sta_ldec, ex_intro/
-| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_bind, ex_intro/
-| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_bind, ex_intro/
-| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_appl, ex_intro/
-| #G #L #W #T #A #_ #_ #_ * /3 width=2 by sta_cast, ex_intro/
-]
-qed-.
-
-lemma sta_aaa_conf: ∀h,G,L. Conf3 … (aaa G L) (sta h G L).
-#h #G #L #T #A #H elim H -G -L -T -A
-[ #G #L #k #U #H
- lapply (sta_inv_sort1 … H) -H #H destruct //
-| #I #G #L #K #V #B #i #HLK #HV #IHV #U #H
- elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HU
- lapply (drop_mono … HLK0 … HLK) -HLK0 #H0 destruct
- lapply (drop_fwd_drop2 … HLK) -HLK #HLK
- @(aaa_lift … HLK … HU) -HU -L /2 width=2 by/
-| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
- elim (sta_inv_bind1 … H) -H #U #HTU #H destruct /3 width=2 by aaa_abbr/
-| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
- elim (sta_inv_bind1 … H) -H #U #HTU #H destruct /3 width=2 by aaa_abst/
-| #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
- elim (sta_inv_appl1 … H) -H #U #HTU #H destruct /3 width=3 by aaa_appl/
-| #G #L #V #T #A #_ #_ #IHV #IHT #X #H
- lapply (sta_inv_cast1 … H) -H /2 width=2 by/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/drop_drop.ma".
-include "basic_2/static/sta.ma".
-
-(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
-
-(* Properties on relocation *************************************************)
-
-(* Basic_1: was: sty0_lift *)
-lemma sta_lift: ∀h,G. l_liftable (sta h G).
-#h #G #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1
-[ #G #L1 #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
- >(lift_inv_sort1 … H1) -X1
- >(lift_inv_sort1 … H2) -X2 //
-| #G #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
- elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=9 by sta_ldef/
- | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
- lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by sta_ldef, drop_inv_gen/
- ]
-| #G #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
- elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
- lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
- elim (lift_total V1 (d-i-1) e) /3 width=9 by sta_ldec/
- | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
- lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by sta_ldec, drop_inv_gen/
- ]
-| #a #I #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
- elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
- elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
- lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by sta_bind, drop_skip/
-| #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
- elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
- elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
- lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by sta_appl/
-| #G #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
- elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by sta_cast/
-]
-qed.
-
-(* Note: apparently this was missing in basic_1 *)
-lemma sta_inv_lift1: ∀h,G. l_deliftable_sn (sta h G).
-#h #G #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2
-[ #G #L2 #k #L1 #s #d #e #_ #X #H
- >(lift_inv_sort2 … H) -X /2 width=3 by sta_sort, lift_sort, ex2_intro/
-| #G #L2 #K2 #V2 #W2 #W #i #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
- [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
- elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
- elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by sta_ldef, ex2_intro/
- | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
- elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
- elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
- #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by sta_ldef, le_S, ex2_intro/
- ]
-| #G #L2 #K2 #W2 #V2 #W #i #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
- [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
- elim (IHWV2 … HK21 … HW12) -K2 #V1 #_ #HWV1
- elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by sta_ldec, ex2_intro/
- | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
- elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
- elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
- #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by sta_ldec, le_S, ex2_intro/
- ]
-| #a #I #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by sta_bind, drop_skip, lift_bind, ex2_intro/
-| #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by sta_appl, lift_flat, ex2_intro/
-| #G #L2 #W2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
- elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by sta_cast, ex2_intro/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-(* Basic_1: was: sty0_correct *)
-lemma sta_fwd_correct: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∃T0. ⦃G, L⦄ ⊢ U •[h] T0.
-#h #G #L #T #U #H elim H -G -L -T -U
-[ /2 width=2/
-| #G #L #K #V #W #W0 #i #HLK #_ #HW0 * #V0 #HWV0
- lapply (drop_fwd_drop2 … HLK) -HLK #HLK
- elim (lift_total V0 0 (i+1)) /3 width=11 by ex_intro, sta_lift/
-| #G #L #K #W #V #V0 #i #HLK #HWV #HWV0 #_
- lapply (drop_fwd_drop2 … HLK) -HLK #HLK
- elim (lift_total V 0 (i+1)) /3 width=11 by ex_intro, sta_lift/
-| #a #I #G #L #V #T #U #_ * /3 width=2 by sta_bind, ex_intro/
-| #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2 by sta_appl, ex_intro/
-| #G #L #W #T #U #_ * /2 width=2 by ex_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/multiple/llpx_sn_drop.ma".
-include "basic_2/static/sta.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ******************************)
-
-(* Properties on lazy sn pointwise extensions *******************************)
-
-lemma sta_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R →
- ∀h,G. s_r_confluent1 … (sta h G) (llpx_sn R 0).
-#R #H1R #H2R #h #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
-[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
-| #G #Ls #Ks #V1s #W2s #V2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
- lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
- lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
-| #G #Ls #Ks #V1s #W1s #V2s #i #HLKs #_ #HV12s #IHVW1s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
- lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
- lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HV12s) -HLKs -HLKd -HV12s /2 width=1 by/ (**) (* full auto too slow *)
-| #a #I #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
- /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
-| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/drop_drop.ma".
-include "basic_2/static/sta.ma".
-
-(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
-
-(* Main properties **********************************************************)
-
-(* Note: apparently this was missing in basic_1 *)
-theorem sta_mono: ∀h,G,L. singlevalued … (sta h G L).
-#h #G #L #T #U1 #H elim H -G -L -T -U1
-[ #G #L #k #X #H >(sta_inv_sort1 … H) -X //
-| #G #L #K #V #W #U1 #i #HLK #_ #HWU1 #IHVW #U2 #H
- elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HW0U2
- lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
- lapply (IHVW … HVW0) -IHVW -HVW0 #H destruct
- >(lift_mono … HWU1 … HW0U2) -W0 -U1 //
-| #G #L #K #W #V #U1 #i #HLK #_ #HWU1 #IHWV #U2 #H
- elim (sta_inv_lref1 … H) -H * #K0 #W0 #V0 #HLK0 #HWV0 #HV0U2
- lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
- lapply (IHWV … HWV0) -IHWV -HWV0 #H destruct
- >(lift_mono … HWU1 … HV0U2) -W -U1 //
-| #a #I #G #L #V #T #U1 #_ #IHTU1 #X #H
- elim (sta_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
-| #G #L #V #T #U1 #_ #IHTU1 #X #H
- elim (sta_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
-| #G #L #W #T #U1 #_ #IHTU1 #U2 #H
- lapply (sta_inv_cast1 … H) -H /2 width=1 by/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • break [ term 46 h ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StaticType $h $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • • * break [ term 46 h , break term 46 l ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StaticTypeStarAlt $h $G $L $l $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break 𝐇𝐍 ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'HdNormal $L $T }.
-
-include "basic_2/reduction/cpr_tshf.ma".
-
-(* CONTEXT-SENSITIVE WEAK HEAD NORMAL TERMS *********************************)
-
-definition chnf: lenv → predicate term ≝ λL. NF … (cpr L) tshf.
-
-interpretation
- "context-sensitive head normality (term)"
- 'HdNormal L T = (chnf L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma chnf_inv_tshf: ∀L,T. L ⊢ 𝐇𝐍⦃T⦄ → T ≈ T.
-normalize /2 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tshf_thnf: ∀T. T ≈ T → ⋆ ⊢ 𝐇𝐍⦃T⦄.
-#T #HT #T2 #H elim (cpr_fwd_tshf1 … H) -H //
-#H elim H //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/tshf.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
-
-(* Forward lemmas on same head forms for terms ******************************)
-
-lemma cpr_fwd_tshf1: ∀L,T1,T2. L ⊢ T1 ➡ T2 → T1 ≈ T1 →
- T2 ≈ T1 ∨ (L = ⋆ → ⊥).
-#L #T1 #T2 #H elim H -L -T1 -T2
-[ /2 width=1/
-| #L #K #V1 #V2 #W2 #i #HLK #_ #_ #_ #_
- @or_intror #H destruct
- lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H
- elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct /2 width=1/
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
- lapply (IHT12 HT1U2) -IHT12 -HT1U2 * #HUT2 /3 width=1/
- lapply (simple_tshf_repl_sn … HUT2 HT1) /3 width=1/
-| #L #V #T #T1 #T2 #_ #_ #_ #H
- elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct
-| #L #V #T1 #T2 #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
-| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H
- elim (simple_inv_bind … H)
-| #a #L #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H
- elim (simple_inv_bind … H)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 ≈ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'Hom $L $T1 $T2 }.
-
-notation "hvbox( L ⊢ break 𝐇𝐑 ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'HdReducible $L $T }.
-
-notation "hvbox( L ⊢ break 𝐇𝐈 ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'NotHdReducible $L $T }.
-
-include "basic_2/grammar/term_simple.ma".
-
-(* SAME HEAD TERM FORMS *****************************************************)
-
-inductive tshf: relation term ≝
- | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I})
- | tshf_abbr: ∀V1,V2,T1,T2. tshf (-ⓓV1. T1) (-ⓓV2. T2)
- | tshf_abst: ∀a,V1,V2,T1,T2. tshf (ⓛ{a}V1. T1) (ⓛ{a}V2. T2)
- | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄ →
- tshf (ⓐV1. T1) (ⓐV2. T2)
-.
-
-interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1.
-#T1 #T2 #H elim H -T1 -T2 /2 width=1/
-qed.
-
-lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2.
-#T1 #T2 #H elim H -T1 -T2 // /2 width=1/
-qed.
-
-lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1.
-/3 width=2/ qed.
-
-lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#T1 #T2 #H elim H -T1 -T2 //
-[ #V1 #V2 #T1 #T2 #H
- elim (simple_inv_bind … H)
-| #a #V1 #V2 #T1 #T2 #H
- elim (simple_inv_bind … H)
-]
-qed. (**) (* remove from index *)
-
-lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-/3 width=3/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀a,I,W1,U1. T1 = ⓑ{a,I}W1.U1 →
- ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
- (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
-#T1 #T2 * -T1 -T2
-[ #J #a #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
-| #b #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
-| #V1 #V2 #T1 #T2 #_ #_ #_ #a #I #W1 #U1 #H destruct
-]
-qed.
-
-lemma tshf_inv_bind1: ∀a,I,W1,U1,T2. ⓑ{a,I}W1.U1 ≈ T2 →
- ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
- (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
-/2 width=5/ qed-.
-
-fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 →
- ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
- I = Appl & T2 = ⓐW2. U2.
-#T1 #T2 * -T1 -T2
-[ #J #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
-| #a #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/
-]
-qed.
-
-lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 →
- ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
- I = Appl & T2 = ⓐW2. U2.
-/2 width=4/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_aaa.ma".
-include "basic_2/reducibility/cpr_aaa.ma".
-include "basic_2/reducibility/xpr.ma".
-
-(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma xpr_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •➡[g] U → L ⊢ U ⁝ A.
-#h #g #L #T #A #HT #U * /2 width=3/ /2 width=6/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_lift.ma".
-include "basic_2/reducibility/cpr_lift.ma".
-include "basic_2/reducibility/xpr.ma".
-
-(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma xpr_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 •➡[g] U2 →
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & ⦃h, L. ⓛV1⦄ ⊢ T1 •➡[g] T2 &
- U2 = ⓛ{a}V2. T2.
-#h #g #a #L #V1 #T1 #U2 *
-[ #H elim (cpr_inv_abst1 … H Abst V1) /3 width=5/
-| #l #H elim (ssta_inv_bind1 … H) /3 width=5/
-]
-qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma xpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
- ∀h,g. ⦃h, K⦄ ⊢ T1 •➡[g] T2 → ⦃h, L⦄ ⊢ U1 •➡[g] U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #h #g *
-/3 width=9/ /3 width=10/
-qed.
-
-lemma xpr_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •➡[g] U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •➡[g] T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 * [ #HU12 | #l #HU12 ]
-[ elim (cpr_inv_lift1 … HLK … HTU1 … HU12) -L -U1 /3 width=3/
-| elim (ssta_inv_lift1 … HU12 … HLK … HTU1) -L -U1 /3 width=4/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lsubss_ssta.ma".
-include "basic_2/reducibility/xpr.ma".
-
-(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
-
-(* Properties on lenv ref for stratified type assignment ********************)
-
-lemma lsubss_xpr_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
- ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡[g] T2.
-#h #g #L1 #L2 #HL12 #T1 #T2 * [ | #l ] #HT12
-[ lapply (lsubss_fwd_lsubs2 … HL12) -HL12 /3 width=3/
-| /3 width=4/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • ➡ * break [ term 46 g ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'XPRedStar $h $g $L $T1 $T2 }.
-
-notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • ⬊ * break [ term 46 g ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'XSN $h $g $L $T }.
-
-include "basic_2/static/lsubss.ma".
-include "basic_2/reducibility/xpr.ma".
-(*
-include "basic_2/reducibility/cnf.ma".
-*)
-(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
-
-definition xprs: ∀h. sd h → lenv → relation term ≝
- λh,g,L. TC … (xpr h g L).
-
-interpretation "extended parallel computation (term)"
- 'XPRedStar h g L T1 T2 = (xprs h g L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma xprs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → R T → R T2) →
- ∀T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T2.
-#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-lemma xprs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → R T → R T1) →
- ∀T1. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T1.
-#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma xprs_refl: ∀h,g,L. reflexive … (xprs h g L).
-/2 width=1/ qed.
-
-lemma xprs_strap1: ∀h,g,L,T1,T,T2.
- ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2.
-/2 width=3/ qed.
-
-lemma xprs_strap2: ∀h,g,L,T1,T,T2.
- ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2.
-/2 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-(*
-axiom xprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U.
-#L #T #U #H @(xprs_ind_dx … H) -T //
-#T0 #T #H1T0 #_ #IHT #H2T0
-lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
-qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/xpr_aaa.ma".
-include "basic_2/computation/xprs.ma".
-
-(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma xprs_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •➡*[g] U → L ⊢ U ⁝ A.
-#h #g #L #T #A #HT #U #H @(xprs_ind … H) -U // /2 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cprs.ma".
-include "basic_2/computation/xprs.ma".
-
-(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
-
-(* properties on context sensitive parallel computation for terms ***********)
-
-lemma cprs_xprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2.
-#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/xpr_lift.ma".
-include "basic_2/computation/cprs.ma".
-include "basic_2/computation/xprs.ma".
-
-(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
-
-(* Advanced forward lemmas **************************************************)
-
-lemma xprs_fwd_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •➡*[g] U2 →
- ∃∃V2,T2. L ⊢ V1 ➡* V2 & U2 = ⓛ{a}V2. T2.
-#h #g #a #L #V1 #T1 #U2 #H @(xprs_ind … H) -U2 /2 width=4/
-#U #U2 #_ #HU2 * #V #T #HV1 #H destruct
-elim (xpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #_ #H destruct /3 width=4/
-qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma xprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 →
- ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •➡*[g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- ⦃h, L⦄ ⊢ U1 •➡*[g] U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 #HT12 @(xprs_ind … HT12) -T2
-[ -HLK #T2 #HT12
- <(lift_mono … HTU1 … HT12) -T1 //
-| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (xpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/
-]
-qed.
-
-lemma xprs_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •➡*[g] U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •➡*[g] T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 #HU12 @(xprs_ind … HU12) -U2 /2 width=3/
--HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1
-elim (xpr_inv_lift1 … HLK … HTU … HU2) -U -HLK /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reducibility/xpr_lsubss.ma".
-include "basic_2/computation/xprs.ma".
-
-(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
-
-(* Properties on lenv ref for stratified type assignment ********************)
-
-lemma lsubss_xprs_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
- ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2.
-#h #g #L1 #L2 #HL12 #T1 #T2 #H @(xprs_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1
-lapply (lsubss_xpr_trans … HL12 … HT2) -L2 /2 width=3/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/xprs.ma".
-
-(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
-
-theorem xprs_trans: ∀h,g,L. transitive … (xprs h g L).
-/2 width=3/ qed-.
</body>
<table name="basic_2_sum"/>
- <subsection name="B">Stage "B"</subsection>
- <news class="gamma" date="2019 November 19.">
+ <news class="gamma" date="2020-02-27.">
+ λδ-2B is repackaged without λδ-ground-2.
+ </news>
+ <news class="gamma" date="2019-11-19.">
λδ-2B is released.
</news>
- <news class="beta" date="2019 September 3.">
+ <news class="beta" date="2019-09-03.">
Applicability condition is now parametrized
with a generic subset of numbers.
</news>
- <news class="beta" date="2019 June 2.">
+ <news class="beta" date="2019-06-02.">
Applicability condition parametrized
with an initial interval of numbers
allows λδ-2B to generalize both λδ-2A and λδ-1B.
</news>
- <news class="beta" date="2019 April 16.">
+ <news class="beta" date="2019-04-16.">
Extended (λδ-2A) and restricted (λδ-1B) validity is decidable
(anniversary milestone).
</news>
- <news class="beta" date="2019 March 25.">
+ <news class="beta" date="2019-03-25.">
Preservation of validity for rt-computation
does not need the sort degree parameter
(i.e. no induction on the degree).
</news>
- <news class="beta" date="2018 November 1.">
+ <news class="beta" date="2018-11-01.">
Extended (λδ-2A) and restricted (λδ-1B) validity rules justified.
</news>
- <news class="alpha" date="2018 September 21.">
+ <news class="alpha" date="2018-09-21.">
λδ-2A completed with
confluence of rt-computation and
preservation of validity for rt-computation.
</news>
- <news class="alpha" date="2018 June 8.">
+ <news class="alpha" date="2018-06-08.">
Behavioral component rt_computation completed.
</news>
- <news class="alpha" date="2018 April 16.">
+ <news class="alpha" date="2018-04-16.">
"Big tree" theorem
(anniversary milestone).
</news>
- <news class="alpha" date="2018 March 9.">
+ <news class="alpha" date="2018-03-09.">
Support for rt-computation completed.
</news>
- <news class="alpha" date="2017 October 17.">
+ <news class="alpha" date="2017-10-17.">
Exclusion binder in local environments.
Syntactic component updated:
syntax, relocation, s_transition, s_computation, static, i_static.
</news>
- <news class="alpha" date="2017 April 16.">
+ <news class="alpha" date="2017-04-16.">
Strong rt-normalization
for simply typed terms
(anniversary milestone).
</news>
- <news class="alpha" date="2017 March 16.">
+ <news class="alpha" date="2017-03-16.">
Behavioral component rt_transition completed.
</news>
- <news class="alpha" date="2017 February 19.">
+ <news class="alpha" date="2017-02-19.">
Generic candidates of reducibility.
</news>
- <news class="alpha" date="2017 January 17.">
+ <news class="alpha" date="2017-01-17.">
Confluence for parallel r-transition on referred entries of local environments.
</news>
- <news class="alpha" date="2016 September 15.">
+ <news class="alpha" date="2016-09-15.">
Confluence for context-sensitive parallel r-transition on terms.
</news>
- <news class="alpha" date="2016 April 16.">
+ <news class="alpha" date="2016-04-16.">
Syntactic component completed:
syntax, relocation, s_transition, s_computation, static
(anniversary milestone).
</news>
- <news class="alpha" date="2016 March 25.">
+ <news class="alpha" date="2016-03-25.">
Relocation with reference transforming maps (rtmap).
</news>
- <news class="alpha" date="2015 October 9.">
+ <news class="alpha" date="2015-10-09.">
λδ-2B is started.
</news>
- <subsection name="A">Stage "A" </subsection>
- <news class="delta" date="2019 November 20.">
- λδ-2A is repackaged (was λδ-2A1).
- </news>
- <news class="delta" date="2015 August 27.">
- λδ-2A appears too complex and is dismissed.
- </news>
- <news class="gamma" date="2014 October 28.">
- λδ-2A is released.
- </news>
- <news class="beta" date="2014 September 9.">
- Iterated static type assignment defined (more elegantly)
- as a primitive notion.
- </news>
- <news class="beta" date="2014 June 18.">
- Preservation of stratified native validity
- for context-sensitive computation on terms.
- </news>
- <news class="alpha" date="2014 June 9.">
- Strong qrst-normalization
- for simply typed terms.
- </news>
- <news class="alpha" date="2014 April 16.">
- Lazy equivalence on local environments
- added as q-step to rst-computation on closures
- (anniversary milestone).
- </news>
- <news class="alpha" date="2014 January 20.">
- Parametrized slicing on local environments
- comprises both versions of this operation
- (one from basic_1, the other used in basic_2 till now).
- </news>
- <news class="alpha" date="2013 August 7.">
- Passive support for global environments.
- </news>
- <news class="alpha" date="2013 July 27.">
- Reaxiomatized β-reductum as in rt-reduction.
- </news>
- <news class="alpha" date="2013 July 20.">
- Context-sensitive strong rt-normalization
- for simply typed terms.
- </news>
- <news class="alpha" date="2013 April 16.">
- Reaxiomatized substitution and reduction
- commute with respect to subclosure
- (anniversary milestone).
- </news>
- <news class="alpha" date="2013 March 16.">
- Mutual recursive preservation of stratified native validity
- for rst-computation on closures.
- </news>
- <news class="alpha" date="2012 October 16.">
- Confluence for context-free parallel reduction on closures.
- </news>
- <news class="alpha" date="2012 July 26.">
- Term binders polarized to control ζ-reduction (not released).
- </news>
- <news class="alpha" date="2012 April 16.">
- Context-sensitive subject equivalence
- for atomic arity assignment
- (anniversary milestone).
- </news>
- <news class="alpha" date="2012 March 15.">
- Context-sensitive strong normalization
- for simply typed terms.
- </news>
- <news class="alpha" date="2012 January 27.">
- Generic candidates of reducibility.
- </news>
- <news class="alpha" date="2011 September 21.">
- Confluence for context-sensitive parallel reduction on terms.
- </news>
- <news class="alpha" date="2011 September 6.">
- Confluence for context-free parallel reduction on terms.
- </news>
- <news class="alpha" date="2011 April 17.">
- λδ-2A is started.
- </news>
-
<section4 name="structure">Logical Structure of the Specification</section4>
<body>This table reports the specification's components and their planes.
</body>
(* Basic_1: includes: pr1_pr0 *)
definition cprs: relation4 genv lenv term term ≝
- λG. LTC … (cpr G).
+ λG. CTC … (cpr G).
interpretation "context-sensitive parallel computation (term)"
'PRedStar G L T1 T2 = (cprs G L T1 T2).
normalize /2 width=3 by TC_strap/ qed-.
lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr.
-/3 width=5 by lsubr_cpr_trans, LTC_lsub_trans/
+/3 width=5 by lsubr_cpr_trans, CTC_lsub_trans/
qed-.
(* Basic_1: was: pr3_pr1 *)
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_5.ma".
include "basic_2A/reduction/lpr_lpr.ma".
include "basic_2A/computation/cprs_lift.ma".
(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
lemma lpr_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lpr G).
-#G @s_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
+#G @s_r_trans_CTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
qed-.
(* Basic_1: was: pr3_strip *)
(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝
- λh,g,G. LTC … (cpx h g G).
+ λh,g,G. CTC … (cpx h g G).
interpretation "extended context-sensitive parallel computation (term)"
'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2).
normalize /2 width=3 by TC_strap/ qed.
lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
-/3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
+/3 width=5 by lsubr_cpx_trans, CTC_lsub_trans/
qed-.
lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
elim (cpx_inv_sort1 … HU2) -HU2
[ #H destruct /2 width=4 by ex2_2_intro/
| * #d0 #Hkd0 #H destruct -d
- @(ex2_2_intro … (n+1) d0) /2 width=1 by deg_inv_prec/ >iter_SO //
+ @(ex2_2_intro … (n+1) d0)
+ [ <plus_plus_comm_23 /2 width=1 by deg_inv_prec/
+ | >iter_SO //
+ ]
]
]
qed-.
(* Advanced properties ******************************************************)
lemma lpx_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpx h g G).
-#h #g #G @s_r_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
+#h #g #G @s_r_trans_CTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
qed-.
lemma cpxs_bind2_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
(* Properties on equivalence for local environments *************************)
lemma lreq_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) (lreq 0 (∞)).
-/3 width=5 by lreq_cpx_trans, LTC_lsub_trans/
+/3 width=5 by lreq_cpx_trans, CTC_lsub_trans/
qed-.
@csx_intro #T #HLT2 #HT2
elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1 by/
+>(lift_mono … HT0 … HT12) in HT2; -T0 /2 width=1 by/
qed.
(* Basic_1: was just: sn3_gen_lift *)
elim (lift_total T l m) #T0 #HT0
lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1 by/
+>(lift_inj … HT0 … HT21) in HT2; -T0 /2 width=1 by/
qed.
(* Advanced inversion lemmas ************************************************)
#X #H #H0
elim (cpxs_fwd_sort_vector … H) -H #H
[ elim H0 -H0 //
-| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h k))) /2 width=1 by cpxs_flat_dx/
+| -H0 @(csx_cpxs_trans … (Ⓐ(V⨮Vs).⋆(next h k))) /2 width=1 by cpxs_flat_dx/
]
qed.
lapply (csx_fwd_pair_sn … H) #HW1
lapply (csx_fwd_flat_dx … H) #H1
@csx_appl_simple_tsts /2 width=3 by simple_flat/ -IHV12s -HW1 -H1 #X #H1 #H2
-elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1 by liftv_cons/ -HV12s -HV12
+elim (cpxs_fwd_theta_vector … (V2⨮V2s) … H1) -H1 /2 width=1 by liftv_cons/ -HV12s -HV12
[ -H #H elim H2 -H2 //
| -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
]
(* Basic inversion lemmas ***************************************************)
-lemma csxv_inv_cons: ∀h,g,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T @ Ts →
+lemma csxv_inv_cons: ∀h,g,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T ⨮ Ts →
⦃G, L⦄ ⊢ ⬊*[h, g] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] Ts.
normalize // qed-.
[ #G #L #k #L0 #cs #HL0 #X #H #L2 #HL20
>(lifts_inv_sort1 … H) -H
lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
- lapply (s4 â\80¦ HAtom G L2 (â\97\8a)) /2 width=1 by/
+ lapply (s4 â\80¦ HAtom G L2 (â\92º)) /2 width=1 by/
| #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #cs #HL01 #X #H #L2 #HL20
lapply (acr_gcr … H1RP H2RP B) #HB
elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
[ #K2 #HK20 #H destruct
elim (lift_total V0 0 (i0 +1)) #V #HV0
elim (lifts_lift_trans … Hi0 … Hcs0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
- lapply (s5 â\80¦ HB ? G ? ? (â\97\8a) … HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
+ lapply (s5 â\80¦ HB ? G ? ? (â\92º) … HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
| -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hcs0
#K2 #V2 #A2 #HKV2A #H1KV0A #H2KV0A #_ #H1 #H2 destruct
lapply (drop_fwd_drop2 … HLK2) #HLK2b
lapply (aaa_mono … H2KV0A … HKV0B) #H destruct -H2KV0A -HKV0B
elim (lift_total V0 0 (i0 +1)) #V3 #HV03
elim (lift_total V2 0 (i0 +1)) #V #HV2
- lapply (s5 â\80¦ HB ? G ? ? (â\97\8a) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/
- lapply (s7 â\80¦ HB G L2 (â\97\8a)) /3 width=7 by gcr_lift/
+ lapply (s5 â\80¦ HB ? G ? ? (â\92º) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/
+ lapply (s7 â\80¦ HB G L2 (â\92º)) /3 width=7 by gcr_lift/
]
| #a #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL20
elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
lapply (acr_gcr … H1RP H2RP A) #HA
lapply (acr_gcr … H1RP H2RP B) #HB
lapply (s1 … HB) -HB #HB
- lapply (s6 â\80¦ HA G L2 (â\97\8a) (â\97\8a)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/
+ lapply (s6 â\80¦ HA G L2 (â\92º) (â\92º)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/
| #a #G #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL02
elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
@(acr_abst … H1RP H2RP) /2 width=5 by/
| #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #cs #HL0 #X #H #L2 #HL20
elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (s7 â\80¦ HA G L2 (â\97\8a)) /3 width=5 by/
+ lapply (s7 â\80¦ HA G L2 (â\92º)) /3 width=5 by/
]
qed.
include "basic_2A/notation/relations/ineint_5.ma".
include "basic_2A/grammar/aarity.ma".
-include "basic_2A/multiple/mr2_mr2.ma".
include "basic_2A/multiple/lifts_lift_vector.ma".
include "basic_2A/multiple/drops_drop.ma".
include "basic_2A/computation/gcp.ma".
[ #G #L #T #H
elim (cp1 … H1RP G L) #k #HK
lapply (H L (⋆k) T (◊) ? ? ?) -H //
- [ lapply (s2 â\80¦ IHB G L (â\97\8a) … HK) //
+ [ lapply (s2 â\80¦ IHB G L (â\92º) … HK) //
| /3 width=6 by s1, cp3/
]
| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
lapply (s1 … IHB … HB) #HV0
- @(s2 … IHA … (V0 @ V0s))
+ @(s2 … IHA … (V0 ⨮ V0s))
/3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/
| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
- @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+ @(s3 … IHA … (V0 ⨮ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
| #G #L #Vs #HVs #k #L0 #V0 #X #cs #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
>(lifts_inv_sort1 … HY) -Y
lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0 @ V0s)) /3 width=7 by gcp2_lifts_all, conj/
+ @(s4 … IHA … (V0 ⨮ V0s)) /3 width=7 by gcp2_lifts_all, conj/
| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
elim (lifts_lift_trans … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
>(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
- @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
+ @(s5 … IHA … (V0 ⨮ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
elim (lift_total V10 0 1) #V20 #HV120
elim (liftv_total 0 1 V10s) #V20s #HV120s
- @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/
+ @(s6 … IHA … (V10 ⨮ V10s) (V20 ⨮ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/
@(HA … (cs + 1)) /2 width=2 by drops_skip/
[ @lifts_applv //
elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
- @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
+ @(s7 … IHA … (V0 ⨮ V0s)) /3 width=5 by lifts_applv/
]
qed.
lapply (acr_gcr … H1RP H2RP B) #HCB
elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0
-lapply (s3 â\80¦ HCA â\80¦ a G L0 (â\97\8a)) #H @H -H
-lapply (s6 â\80¦ HCA G L0 (â\97\8a) (â\97\8a) ?) // #H @H -H
+lapply (s3 â\80¦ HCA â\80¦ a G L0 (â\92º)) #H @H -H
+lapply (s6 â\80¦ HCA G L0 (â\92º) (â\92º) ?) // #H @H -H
[ @(HA … HL0) //
| lapply (s1 … HCB) -HCB #HCB
- lapply (s7 â\80¦ H2RP G L0 (â\97\8a)) /3 width=1 by/
+ lapply (s7 â\80¦ H2RP G L0 (â\92º)) /3 width=1 by/
]
qed.
(* *)
(**************************************************************************)
+include "ground_2/ynat/ynat_minus_sn.ma".
include "basic_2A/notation/relations/cosn_5.ma".
include "basic_2A/computation/lsx.ma".
| lcosx_sort: ∀l. lcosx h g G l (⋆)
| lcosx_skip: ∀I,L,T. lcosx h g G 0 L → lcosx h g G 0 (L.ⓑ{I}T)
| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, g, T, l] L →
- lcosx h g G l L â\86\92 lcosx h g G (⫯l) (L.ⓑ{I}T)
+ lcosx h g G l L â\86\92 lcosx h g G (â\86\91l) (L.ⓑ{I}T)
.
interpretation
lemma lcosx_drop_trans_lt: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, l] L →
∀I,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → i < l →
- G â\8a¢ ~â¬\8a*[h, g, â«°(l-i)] K â\88§ G â\8a¢ â¬\8a*[h, g, V, â«°(l-i)] K.
+ G â\8a¢ ~â¬\8a*[h, g, â\86\93(l-i)] K â\88§ G â\8a¢ â¬\8a*[h, g, V, â\86\93(l-i)] K.
#h #g #G #L #l #H elim H -L -l
[ #l #J #K #V #i #H elim (drop_inv_atom1 … H) -H #H destruct
| #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false … H) -H //
elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK destruct
[ >ypred_succ /2 width=1 by conj/
| lapply (ylt_pred … Hil ?) -Hil /2 width=1 by ylt_inj/ >ypred_succ #Hil
- elim (IHL … HLK ?) -IHL -HLK <yminus_inj >yminus_SO2 //
+ elim (IHL … HLK ?) -IHL -HLK >minus_SO_dx //
<(ypred_succ l) in ⊢ (%→%→?); >yminus_pred /2 width=1 by ylt_inj, conj/
]
]
(* Basic inversion lemmas ***************************************************)
-fact lcosx_inv_succ_aux: â\88\80h,g,G,L,x. G â\8a¢ ~â¬\8a*[h, g, x] L â\86\92 â\88\80l. x = ⫯l →
+fact lcosx_inv_succ_aux: â\88\80h,g,G,L,x. G â\8a¢ ~â¬\8a*[h, g, x] L â\86\92 â\88\80l. x = â\86\91l →
L = ⋆ ∨
∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
G ⊢ ⬊*[h, g, V, l] K.
#h #g #G #L #l * -L -l /2 width=1 by or_introl/
[ #I #L #T #_ #x #H elim (ysucc_inv_O_sn … H)
-| #I #L #T #l #HT #HL #x #H <(ysucc_inj … H) -x
+| #I #L #T #l #HT #HL #x #H <(ysucc_inv_inj … H) -x
/3 width=6 by ex3_3_intro, or_intror/
]
qed-.
-lemma lcosx_inv_succ: â\88\80h,g,G,L,l. G â\8a¢ ~â¬\8a*[h, g, ⫯l] L → L = ⋆ ∨
+lemma lcosx_inv_succ: â\88\80h,g,G,L,l. G â\8a¢ ~â¬\8a*[h, g, â\86\91l] L → L = ⋆ ∨
∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
G ⊢ ⬊*[h, g, V, l] K.
/2 width=3 by lcosx_inv_succ_aux/ qed-.
-lemma lcosx_inv_pair: â\88\80h,g,I,G,L,T,l. G â\8a¢ ~â¬\8a*[h, g, ⫯l] L.ⓑ{I}T →
+lemma lcosx_inv_pair: â\88\80h,g,I,G,L,T,l. G â\8a¢ ~â¬\8a*[h, g, â\86\91l] L.ⓑ{I}T →
G ⊢ ~⬊*[h, g, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
#h #g #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
[ #H destruct
(* *)
(**************************************************************************)
-include "ground_2A/ynat/ynat_max.ma".
include "basic_2A/computation/lsx_drop.ma".
include "basic_2A/computation/lsx_lpx.ma".
include "basic_2A/computation/lsx_lpxs.ma".
#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #l #HL #H
elim (ylt_split i l) #Hli [ -H | -HL ]
- [ <(ymax_pre_sn l (⫯i)) /2 width=1 by ylt_fwd_le_succ/
- elim (lcosx_drop_trans_lt … HL … HLK) // -HL -Hli
- lapply (drop_fwd_drop2 … HLK) -HLK /3 width=7 by lsx_lift_ge/
+ [ elim (lcosx_drop_trans_lt … HL … HLK) // -HL
+ <yminus_succ2 #H1K #H2K
+ lapply (ylt_fwd_le_succ1 … Hli) -Hli #Hli
+ >(yminus_plus l (↑i))
+ /4 width=7 by lsx_lift_ge, drop_fwd_drop2/
| lapply (lsx_fwd_lref_be … H … HLK) // -H -Hli
lapply (drop_fwd_drop2 … HLK) -HLK
/4 width=10 by lsx_ge, lsx_lift_le/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #l #HL #H
elim (lsx_inv_flat … H) -H /3 width=1 by lsx_flat/
| #G #L #V #U1 #U2 #T2 #_ #HTU2 #IHU12 #l #HL #H
- elim (lsx_inv_bind … H) -H
+ elim (lsx_inv_bind … H) -H #H1L #H2L <(yplus_minus l 1)
/4 width=9 by lcosx_pair, lsx_inv_lift_ge, drop_drop/
| #G #L #V #T1 #T2 #_ #IHT12 #l #HL #H
elim (lsx_inv_flat … H) -H /2 width=1 by/
elim (lsx_inv_flat … H) -H #HV1 #H
elim (lsx_inv_bind … H) -H #HW1 #HT1
@lsx_bind /2 width=1 by/ (**) (* explicit constructor *)
- @lsx_flat [ /3 width=7 by lsx_lift_ge, drop_drop/ ]
+ @lsx_flat [ <(yplus_SO2 l) /3 width=7 by lsx_lift_ge, drop_drop/ ]
@(lsx_lreq_conf … (L.ⓓW1)) /3 width=1 by lcosx_pair, lreq_succ/
]
qed-.
(* Properties on context-sensitive parallel computation for terms ***********)
lemma lprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lprs G).
-/3 width=5 by s_r_trans_LTC2, lpr_cprs_trans/ qed-.
+/3 width=5 by s_r_trans_CTC2, lpr_cprs_trans/ qed-.
(* Basic_1: was just: pr3_pr3_pr3_t *)
(* Note: alternative proof /3 width=5 by s_r_trans_LTC1, lprs_cpr_trans/ *)
lemma lprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lprs G).
-#G @s_r_to_s_rs_trans @s_r_trans_LTC2
+#G @s_r_to_s_rs_trans @s_r_trans_CTC2
@s_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
qed-.
(* Properties on context-sensitive extended parallel computation for terms **)
lemma lpxs_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpxs h g G).
-/3 width=5 by s_r_trans_LTC2, lpx_cpxs_trans/ qed-.
+/3 width=5 by s_r_trans_CTC2, lpx_cpxs_trans/ qed-.
(* Note: alternative proof: /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ *)
lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpxs h g G).
-#h #g #G @s_r_to_s_rs_trans @s_r_trans_LTC2
+#h #g #G @s_r_to_s_rs_trans @s_r_trans_CTC2
@s_rs_trans_TC1 /2 width=3 by lpx_cpxs_trans/ (**) (* full auto too slow *)
qed-.
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_6_3.ma".
+include "ground_2/xoa/ex_7_4.ma".
include "basic_2A/notation/relations/lrsubeqc_4.ma".
include "basic_2A/static/lsubr.ma".
include "basic_2A/static/aaa.ma".
(* Advanced forward lemmas **************************************************)
lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
- G â\8a¢ â¬\8a*[h, g, T, ⫯l] L.ⓑ{I}V.
+ G â\8a¢ â¬\8a*[h, g, T, â\86\91l] L.ⓑ{I}V.
#h #g #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L
#L1 #_ #IHL1 @lsx_intro
#Y #H #HT elim (lpx_inv_pair1 … H) -H
(* Advanced inversion lemmas ************************************************)
lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a, I}V.T, l] L →
- G â\8a¢ â¬\8a*[h, g, V, l] L â\88§ G â\8a¢ â¬\8a*[h, g, T, ⫯l] L.ⓑ{I}V.
+ G â\8a¢ â¬\8a*[h, g, V, l] L â\88§ G â\8a¢ â¬\8a*[h, g, T, â\86\91l] L.ⓑ{I}V.
/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.
(* Advanced properties ******************************************************)
fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
- â\88\80Y,T. G â\8a¢ â¬\8a*[h, g, T, ⫯l] Y →
+ â\88\80Y,T. G â\8a¢ â¬\8a*[h, g, T, â\86\91l] Y →
∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L2.
#h #g #a #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
qed-.
lemma lsx_bind: ∀h,g,a,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
- â\88\80T. G â\8a¢ â¬\8a*[h, g, T, ⫯l] L.ⓑ{I}V →
+ â\88\80T. G â\8a¢ â¬\8a*[h, g, T, â\86\91l] L.ⓑ{I}V →
G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L.
/2 width=3 by lsx_bind_lpxs_aux/ qed.
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_2.ma".
include "basic_2A/notation/relations/dpredstar_7.ma".
include "basic_2A/static/da.ma".
include "basic_2A/computation/cprs.ma".
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_7_3.ma".
+include "ground_2/xoa/ex_8_4.ma".
include "basic_2A/notation/relations/lrsubeqv_5.ma".
include "basic_2A/dynamic/shnv.ma".
(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
definition cpcs: relation4 genv lenv term term ≝
- λG. LTC … (cpc G).
+ λG. CTC … (cpc G).
interpretation "context-sensitive parallel equivalence (term)"
'PConvStar G L T1 T2 = (cpcs G L T1 T2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/crr_append.ma".
+include "basic_2/reduction/cir.ma".
+
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************)
+
+(* Advanved properties ******************************************************)
+
+lemma cir_labst_last: ∀G,L,T,W. ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, ⋆.ⓛW @@ L⦄ ⊢ ➡ 𝐈⦃T⦄.
+/3 width=2 by crr_inv_labst_last/ qed.
+
+lemma cir_tif: ∀G,T,W. ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐈⦃T⦄.
+/3 width=2 by crr_inv_trr/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cir_inv_append_sn: ∀G,L,K,T. ⦃G, K @@ L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
+/3 width=1/ qed-.
+
+lemma cir_inv_tir: ∀G,T,W. ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃T⦄.
+/3 width=1/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/crx_append.ma".
+include "basic_2/reduction/cix.ma".
+
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cix_inv_append_sn: ∀h,g,G,L,K,T. ⦃G, K @@ L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+/3 width=1 by crx_append_sn/ qed-.
+
+lemma cix_inv_tix: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+/3 width=1 by trx_crx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_append.ma".
+
+(* SHIFT OF A CLOSURE *******************************************************)
+
+let rec shift L T on L ≝ match L with
+[ LAtom ⇒ T
+| LPair L I V ⇒ shift L (-ⓑ{I} V. T)
+].
+
+interpretation "shift (closure)" 'Append L T = (shift L T).
+
+(* Basic properties *********************************************************)
+
+lemma shift_append_assoc: ∀L,K. ∀T:term. (L @@ K) @@ T = L @@ K @@ T.
+#L #K elim K -K // normalize //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma shift_inj: ∀L1,L2. ∀T1,T2:term. L1 @@ T1 = L2 @@ T2 → |L1| = |L2| →
+ L1 = L2 ∧ T1 = T2.
+#L1 elim L1 -L1
+[ * normalize /2 width=1/
+ #L2 #I2 #V2 #T1 #T2 #_ <plus_n_Sm #H destruct
+| #L1 #H1 #V1 #IH * normalize
+ [ #T1 #T2 #_ <plus_n_Sm #H destruct
+ | #L2 #I2 #V2 #T1 #T2 #H1 #H2
+ elim (IH … H1) -IH -H1 /2 width=1/ -H2 #H1 #H2 destruct /2 width=1/
+ ]
+]
+qed-.
--- /dev/null
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+
+lemma cpr_append: ∀G. l_appendable_sn … (cpr G).
+#G #K #T1 #T2 #H elim H -G -K -T1 -T2
+/2 width=3 by cpr_bind, cpr_flat, cpr_zeta, cpr_tau, cpr_beta, cpr_theta/
+#G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpr_delta … (L@@K0) V1 … HVW2) //
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
+qed.
+
+lemma cpr_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X
+ >shift_append_assoc normalize #H
+ elim (cpr_inv_bind1 … H) -H *
+ [ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
+
--- /dev/null
+lemma cpx_append: ∀h,g,G. l_appendable_sn … (cpx h g G).
+#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
+/2 width=3 by cpx_sort, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_ti, cpx_beta, cpx_theta/
+#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpx_delta … I … (L@@K0) V1 … HVW2) //
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
+qed.
+
+lemma cpx_fwd_shift1: ∀h,g,G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#h #g #G #L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X
+ >shift_append_assoc normalize #H
+ elim (cpx_inv_bind1 … H) -H *
+ [ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
--- /dev/null
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+
+lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
+#G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
+/2 width=1 by cpy_atom, cpy_bind, cpy_flat/
+#I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
+qed-.
+
+lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #d #e #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X #d #e
+ >shift_append_assoc normalize #H
+ elim (cpy_inv_bind1 … H) -H
+ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
+ /2 width=3 by trans_eq/
+]
+qed-.
--- /dev/null
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+
+lemma cpys_append: ∀G. l_appendable_sn … (cpys G).
+#G #K #T1 #T2 #H elim H -G -K -T1 -T2
+/2 width=3 by cpys_bind, cpys_flat/
+#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpys_delta … I … (L@@K0) V1 … HVW2) //
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
+qed.
+
+lemma cpys_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X >shift_append_assoc normalize
+ #H elim (cpys_inv_bind1 … H) -H
+ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
+]
+qed-.
--- /dev/null
+lemma cpys_append: ∀G,d,e. l_appendable_sn … (cpys d e G).
+#G #d #e #K #T1 #T2 #H @(cpys_ind … H) -T2
+/3 width=3 by cpys_strap1, cpy_append/
+qed-.
+
+lemma cpys_fwd_shift1: ∀G,L,L1,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#G #L #L1 #T1 #T #d #e #H @(cpys_ind … H) -T
+[ /2 width=4 by ex2_2_intro/
+| #T #X #_ #HX * #L0 #T0 #HL10 #H destruct
+ elim (cpy_fwd_shift1 … HX) -HX #L2 #T2 #HL02 #H destruct
+ /2 width=4 by ex2_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/reduction/crr.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
+
+(* Advanved properties ******************************************************)
+
+lemma crr_append_sn: ∀G,L,K,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, K @@ L⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L #K0 #T #H elim H -L -T /2 width=1/
+#L #K #V #i #HLK
+lapply (ldrop_fwd_length_lt2 … HLK) #Hi
+lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=2/ -Hi /2 width=3/
+qed.
+
+lemma trr_crr: ∀G,L,T. ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L #T #H lapply (crr_append_sn … H) //
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact crr_inv_labst_last_aux: ∀G,L1,T,W. ⦃G, L1⦄ ⊢ ➡ 𝐑⦃T⦄ →
+ ∀L2. L1 = ⋆.ⓛW @@ L2 → ⦃G, L2⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L1 #T #W #H elim H -L1 -T /2 width=1/ /3 width=1/
+[ #L1 #K1 #V1 #i #HLK1 #L2 #H destruct
+ lapply (ldrop_fwd_length_lt2 … HLK1)
+ >append_length >commutative_plus normalize in ⊢ (??% → ?); #H
+ elim (le_to_or_lt_eq i (|L2|)) /2 width=1/ -H #Hi destruct
+ [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2
+ lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi
+ normalize #H destruct /2 width=3/
+ | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // <minus_n_n #H
+ lapply (ldrop_inv_O2 … H) -H #H destruct
+ ]
+| #a #I #L1 #V #T #HI #_ #IHT #L2 #H destruct /3 width=1/
+]
+qed.
+
+lemma crr_inv_labst_last: ∀G,L,T,W. ⦃G, ⋆.ⓛW @@ L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄.
+/2 width=4/ qed-.
+
+lemma crr_inv_trr: ∀G,T,W. ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃T⦄.
+/2 width=4/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/reduction/crx.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************)
+
+(* Advanved properties ******************************************************)
+
+lemma crx_append_sn: ∀h,g,G,L,K,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⦃G, K @@ L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+#h #g #G #L #K0 #T #H elim H -L -T
+/2 width=2 by crx_sort, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/
+#I #L #K #V #i #HLK
+lapply (ldrop_fwd_length_lt2 … HLK) #Hi
+lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=4 by crx_delta, lt_to_le/
+qed.
+
+lemma trx_crx: ∀h,g,G,L,T. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+#h #g #G #L #T #H lapply (crx_append_sn … H) //
+qed.
--- /dev/null
+lemma lpr_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ➡ K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L1 @@ K1⦄ ⊢ ➡ L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpr_append/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpr_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡ L →
+ ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡ K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpr_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ➡ K2 @@ L2 →
+ ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡ K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
--- /dev/null
+lemma lpx_append: ∀h,g,G,K1,K2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ⦃G, L1 @@ K1⦄ ⊢ ➡[h, g] L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpx_append/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpx_fwd_append1: ∀h,g,G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡[h, g] L →
+ ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpx_fwd_append2: ∀h,g,G,L,K2,L2. ⦃G, L⦄ ⊢ ➡[h, g] K2 @@ L2 →
+ ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
--- /dev/null
+lemma lpx_sn_append: ∀R. l_appendable_sn R →
+ ∀K1,K2. lpx_sn R K1 K2 → ∀L1,L2. lpx_sn R L1 L2 →
+ lpx_sn R (L1 @@ K1) (L2 @@ K2).
+#R #HR #K1 #K2 #H elim H -K1 -K2 /3 width=1 by lpx_sn_pair/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpx_sn_fwd_append1: ∀R,L1,K1,L. lpx_sn R (K1 @@ L1) L →
+ ∃∃K2,L2. lpx_sn R K1 K2 & L = K2 @@ L2.
+#R #L1 elim L1 -L1
+[ #K1 #K2 #HK12
+ @(ex2_2_intro … K2 (⋆)) // (* explicit constructor, /2 width=4/ does not work *)
+| #L1 #I #V1 #IH #K1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #H destruct
+ @(ex2_2_intro … (L2.ⓑ{I}V2) HK12) // (* explicit constructor, /2 width=4/ does not work *)
+]
+qed-.
+
+lemma lpx_sn_fwd_append2: ∀R,L2,K2,L. lpx_sn R L (K2 @@ L2) →
+ ∃∃K1,L1. lpx_sn R K1 K2 & L = K1 @@ L1.
+#R #L2 elim L2 -L2
+[ #K2 #K1 #HK12
+ @(ex2_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=4/ does not work *)
+| #L2 #I #V2 #IH #K2 #X #H
+ elim (lpx_sn_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #H destruct
+ @(ex2_2_intro … (L1.ⓑ{I}V1) HK12) // (* explicit constructor, /2 width=4/ does not work *)
+]
+qed-.
--- /dev/null
+lemma lpys_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ▶*× K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 →
+ ⦃G, L1 @@ K1⦄ ⊢ ▶*× L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpys_append/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpys_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ▶*× L →
+ ∃∃K2,L2. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpys_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ▶*× K2 @@ L2 →
+ ∃∃K1,L1. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpys_fwd_shift1_ext: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
+ ∃∃L2,T2. ⦃G, L @@ L1⦄ ⊢ ▶*× L @@ L2 & ⦃G, L @@ L1⦄ ⊢ T1 ▶*× T2 &
+ T = L2 @@ T2.
+#G #L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #HT1 @ex3_2_intro
+ [3: // |4,5: // |1,2: skip ] (**) (* auto does not work *)
+| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
+ elim (cpys_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ lapply (lpys_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1 by lpys_append, lpys_pair/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3 by trans_eq/ | skip ] <append_assoc // (**) (* explicit constructor *)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubstnormal_5.ma".
+include "basic_2/relocation/cpy.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION *****************)
+
+definition cny: ∀d,e. relation3 genv lenv term ≝
+ λd,e,G,L. NF … (cpy d e G L) (eq …).
+
+interpretation
+ "normality for context-sensitive extended substitution (term)"
+ 'PSubstNormal G L T d e = (cny d e G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cny_inv_lref: ∀G,L,d,e,i. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄ →
+ ∨∨ yinj i < d | d + e ≤ yinj i | |L| ≤ i.
+#G #L #d #e #i #H elim (ylt_split i d) /2 width=1 by or3_intro0/
+#Hdi elim (ylt_split i (d+e)) /2 width=1 by or3_intro1/
+#Hide elim (lt_or_ge i (|L|)) /2 width=1 by or3_intro2/
+#Hi elim (ldrop_O1_lt L i) //
+#I #K #V #HLK elim (lift_total V 0 (i+1))
+#W #HVW lapply (H W ?) -H /2 width=5 by cpy_subst/ -HLK
+#H destruct elim (lift_inv_lref2_be … HVW) -L -d -e //
+qed-.
+
+lemma cny_inv_bind: ∀a,I,G,L,V,T,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓑ{a,I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ▶[⫯d, e] 𝐍⦃T⦄.
+#a #I #G #L #V1 #T1 #d #e #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓑ{a,I}V2.T1) ?) -HVT1
+| #T2 #HT2 lapply (HVT1 (ⓑ{a,I}V1.T2) ?) -HVT1
+]
+/2 width=1 by cpy_bind/ #H destruct //
+qed-.
+
+lemma cny_inv_flat: ∀I,G,L,V,T,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓕ{I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ ∧ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T⦄.
+#I #G #L #V1 #T1 #d #e #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓕ{I}V2.T1) ?) -HVT1
+| #T2 #HT2 lapply (HVT1 (ⓕ{I}V1.T2) ?) -HVT1
+]
+/2 width=1 by cpy_flat/ #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cny_conf: ∀G,d,e.
+ ∀L1,T. ⦃G, L1⦄ ⊢ ▶[d, e] 𝐍⦃T⦄ →
+ ∀L2. L1 ⊑×[d, e] L2 → ⦃G, L2⦄ ⊢ ▶[d, e] 𝐍⦃T⦄.
+#G #d #e #L1 #T1 #HT1 #L2 #HL12 #T2 #HT12
+@HT1 /3 width=3 by lsuby_cpy_trans/
+qed-.
+
+lemma cny_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃⋆k⦄.
+#G #L #d #e #k #X #H elim (cpy_inv_sort1 … H) -H //
+qed.
+
+lemma cny_lref_free: ∀G,L,d,e,i. |L| ≤ i → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
+#G #L #d #e #i #Hi #X #H elim (cpy_inv_lref1 … H) -H // *
+#I #K #V #_ #_ #HLK #_ lapply (ldrop_fwd_length_lt2 … HLK) -HLK
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed.
+
+lemma cny_lref_atom: ∀G,L,d,e,i. ⇩[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
+#G #L #d #e #i #HL @cny_lref_free >(ldrop_fwd_length … HL) -HL //
+qed.
+
+lemma cny_lref_top: ∀G,L,d,e,i. d+e ≤ yinj i → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
+#G #L #d #e #i #Hdei #X #H elim (cpy_inv_lref1 … H) -H // *
+#I #K #V #_ #H elim (ylt_yle_false … H) //
+qed.
+
+lemma cny_lref_skip: ∀G,L,d,e,i. yinj i < d → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃#i⦄.
+#G #L #d #e #i #Hid #X #H elim (cpy_inv_lref1 … H) -H // *
+#I #K #V #H elim (ylt_yle_false … H) //
+qed.
+
+lemma cny_gref: ∀G,L,d,e,p. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃§p⦄.
+#G #L #d #e #p #X #H elim (cpy_inv_gref1 … H) -H //
+qed.
+
+lemma cny_bind: ∀G,L,V,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ →
+ ∀I,T. ⦃G, L.ⓑ{I}V⦄ ⊢ ▶[⫯d, e] 𝐍⦃T⦄ →
+ ∀a. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓑ{a,I}V.T⦄.
+#G #L #V1 #d #e #HV1 #I #T1 #HT1 #a #X #H
+elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+>(HV1 … HV12) -V2 >(HT1 … HT12) -T2 //
+qed.
+
+lemma cny_flat: ∀G,L,V,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃V⦄ →
+ ∀T. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T⦄ →
+ ∀I. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃ⓕ{I}V.T⦄.
+#G #L #V1 #d #e #HV1 #T1 #HT1 #I #X #H
+elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+>(HV1 … HV12) -V2 >(HT1 … HT12) -T2 //
+qed.
+
+lemma cny_narrow: ∀G,L,T,d1,e1. ⦃G, L⦄ ⊢ ▶[d1, e1] 𝐍⦃T⦄ →
+ ∀d2,e2. d1 ≤ d2 → d2 + e2 ≤ d1 + e1 → ⦃G, L⦄ ⊢ ▶[d2, e2] 𝐍⦃T⦄.
+#G #L #T1 #d1 #e1 #HT1 #d2 #e2 #Hd12 #Hde21 #T2 #HT12
+@HT1 /2 width=5 by cpy_weak/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cpy_lift.ma".
+include "basic_2/relocation/cny.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION *****************)
+
+(* Properties on relocation *************************************************)
+
+lemma cny_lift_le: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → dt + et ≤ d → ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdetd #U2 #HU12
+elim (cpy_inv_lift1_le … HU12 … HLK … HTU1) // -L -Hdetd #T2 #HT12
+>(HT1 … HT12) -K /2 width=5 by lift_mono/
+qed-.
+
+lemma cny_lift_be: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → dt ≤ d → yinj d ≤ dt + et → ⦃G, L⦄ ⊢ ▶[dt, et+e] 𝐍⦃U⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdtd #Hddet #U2 #HU12
+elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hdtd -Hddet #T2
+>yplus_minus_inj #HT12 >(HT1 … HT12) -K /2 width=5 by lift_mono/
+qed-.
+
+lemma cny_lift_ge: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → d ≤ dt → ⦃G, L⦄ ⊢ ▶[dt+e, et] 𝐍⦃U⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #U2 #HU12
+elim (cpy_inv_lift1_ge … HU12 … HLK … HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hddt #T2
+>yplus_minus_inj #HT12 >(HT1 … HT12) -K /2 width=5 by lift_mono/
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma cny_inv_lift_le: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → dt + et ≤ d → ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdetd #T2 #HT12
+elim (lift_total T2 d e) #U2 #HTU2
+lapply (cpy_lift_le … HT12 … HLK … HTU1 … HTU2 ?) // -K -Hdetd #HU12
+lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
+qed-.
+
+lemma cny_inv_lift_be: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → dt ≤ d → yinj d + e ≤ dt + et → ⦃G, K⦄ ⊢ ▶[dt, et-e] 𝐍⦃T⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdtd #Hdedet #T2 #HT12
+lapply (yle_fwd_plus_ge_inj … Hdedet) // #Heet
+elim (yle_inv_plus_inj2 … Hdedet) -Hdedet #Hddete #Hedet
+elim (lift_total T2 d e) #U2 #HTU2
+lapply (cpy_lift_be … HT12 … HLK … HTU1 … HTU2 ? ?) // [ >yplus_minus_assoc_inj // ] -K -Hdtd -Hddete
+>ymax_pre_sn // -Heet #HU12
+lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
+qed-.
+
+lemma cny_inv_lift_ge: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → yinj d + e ≤ dt → ⦃G, K⦄ ⊢ ▶[dt-e, et] 𝐍⦃T⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdedt #T2 #HT12
+elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #Hddte #Hedt
+elim (lift_total T2 d e) #U2 #HTU2
+lapply (cpy_lift_ge … HT12 … HLK … HTU1 … HTU2 ?) // -K -Hddte
+>ymax_pre_sn // -Hedt #HU12
+lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma cny_inv_lift_ge_up: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ⦃G, K⦄ ⊢ ▶[d, dt + et - (yinj d + e)] 𝐍⦃T⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hddt #Hdtde #Hdedet
+lapply (cny_narrow … HU1 (d+e) (dt+et-(d+e)) ? ?) -HU1 [ >ymax_pre_sn_comm ] // #HU1
+lapply (cny_inv_lift_ge … HU1 … HLK … HTU1 ?) // -L -U1
+>yplus_minus_inj //
+qed-.
+
+lemma cny_inv_lift_subst: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i+1] L ≡ K → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
+#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW
+lapply (cny_inv_lift_ge_up … HW … HLK … HVW ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+| >yminus_succ2 //
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Note: this should be applicable in a forward manner *)
+lemma cny_lift_ge_up: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[yinj d, dt + et - (yinj d + yinj e)] 𝐍⦃T⦄ →
+ ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U →
+ yinj d ≤ dt → dt ≤ yinj d + yinj e → yinj d + yinj e ≤ dt + et →
+ ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #Hdtde #Hdedet
+lapply (cny_lift_be … HT1 … HLK … HTU1 ? ?) // -K -T1
+#HU1 @(cny_narrow … HU1) -HU1 // (**) (* auto fails *)
+qed-.
+
+lemma cny_lift_subst: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i+1] L ≡ K → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
+#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW
+@(cny_lift_ge_up … HLK … HVW) // >yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 //
+| /2 width=3 by yle_trans/
+| /2 width=1 by ylt_fwd_le_succ1/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/delta_equivalence/cpzs.ma".
+include "basic_2/reduction/cpx.ma".
+
+fact destruct_tsort_tsort: ∀k1,k2. ⋆k1 = ⋆k2 → k1 = k2.
+#k1 #k2 #H destruct //
+qed-.
+
+axiom cpzs_inv_subst: ∀I,G,L,K,V1,V2,W2,i.
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ W2 →
+ ⦃G, L⦄ ⊢ #i ◆*[O, ∞] W2 → ⦃G, K⦄⊢ V1 ◆*[O, ∞] V2.
+
+axiom cpzs_subst: ∀I,G,L,K,V1,V2,W2,i.
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ W2 →
+ ⦃G, K⦄⊢ V1 ◆*[O, ∞] V2 → ⦃G, L⦄ ⊢ #i ◆*[O, ∞] W2.
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Forward lemmas on delta-equivalence for terms ****************************)
+
+lemma cpx_fwd_cpys_cpzs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀d,e. ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2 ↔ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2
+[ /2 width=1 by conj/
+| #G #L #k #l #_ #d #e @conj #H lapply (next_lt h k)
+ [ <(cpzs_inv_sort … H)
+ | lapply (cpys_inv_sort1 … H) -H #H >(destruct_tsort_tsort … H)
+ ] -H #H elim (lt_refl_false … H)
+| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #d #e @conj #H
+(*
+ [ @(cpys_subst … HLK … HVW2) // >yminus_Y_inj /3 width=7 by cpzs_inv_subst/
+ | elim (cpys_inv_lref1_ldrop … H … HLK … HVW2) -H /3 width=7 by cpzs_subst/
+ ]
+*)
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e
+ elim (IHV12 d e) -IHV12 elim (IHT12 (⫯d) e) -IHT12
+ #IHTdx #IHTsn #IHVdx #IHVsn @conj #H
+ [ elim (cpzs_inv_bind … H) -H /3 width=1 by cpys_bind/
+ | elim (cpys_inv_bind1 … H) -H #X1 #X2 #H1 #H2 #H destruct /3 width=1 by cpzs_bind/
+ ]
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e
+ elim (IHV12 d e) -IHV12 elim (IHT12 (d) e) -IHT12
+ #IHTdx #IHTsn #IHVdx #IHVsn @conj #H
+ [ elim (cpzs_inv_flat … H) -H /3 width=1 by cpys_flat/
+ | elim (cpys_inv_flat1 … H) -H #X1 #X2 #H1 #H2 #H destruct /3 width=1 by cpzs_flat/
+ ]
+| #G #L #V #U1 #U2 #T2 #_ #HTU2 #_ #d #e @conj #H
+| #G #L #V1 #T1 #T2 #_ #_ #d #e @conj #H
+| #G #L #V1 #V2 #T1 #HV12 #_ #d #e @conj #H
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #d #e @conj #H
+ [ elim (cpzs_inv_flat_bind … H)
+ | elim (cpys_inv_flat1 … H) -H #X1 #X2 #H1 #H2 #H destruct
+ ]
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #d #e @conj #H
+ [ elim (cpzs_inv_flat_bind … H)
+ | elim (cpys_inv_flat1 … H) -H #X1 #X2 #H1 #H2 #H destruct
+ ]
+]
+
+
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubsteval_6.ma".
+include "basic_2/relocation/cny.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+definition cpye: ynat → ynat → relation4 genv lenv term term ≝
+ λd,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 ∧ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive extended substitution (term)"
+ 'PSubstEval G L T1 T2 d e = (cpye d e G L T1 T2).
+
+(* Basic_properties *********************************************************)
+
+(* Note: this should go in subconversion *)
+lemma leqy_cpye_trans: ∀G,L2,T1,T2,d,e. ⦃G, L2⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀L1. L1 ⊑×[d, e] L2 → L2 ⊑×[d, e] L1 → ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+#G #L2 #T1 #T2 #d #e *
+/4 width=8 by lsuby_cpys_trans, lsuby_cny_conf, conj/
+qed-.
+
+lemma cpye_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃⋆k⦄.
+/3 width=5 by cny_sort, conj/ qed.
+
+lemma cpye_free: ∀G,L,d,e,i. |L| ≤ i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
+/3 width=6 by cny_lref_free, conj/ qed.
+
+lemma cpye_top: ∀G,L,d,e,i. d + e ≤ yinj i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
+/3 width=6 by cny_lref_top, conj/ qed.
+
+lemma cpye_skip: ∀G,L,d,e,i. yinj i < d → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
+/3 width=6 by cny_lref_skip, conj/ qed.
+
+lemma cpye_gref: ∀G,L,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃§p⦄.
+/3 width=5 by cny_gref, conj/ qed.
+
+lemma cpye_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃ⓑ{a,I}V2.T2⦄.
+#G #L #V1 #V2 #d #e * #HV12 #HV2 #I #T1 #T2 *
+/5 width=8 by cpys_bind, cny_bind, lsuby_cny_conf, lsuby_succ, conj/
+qed.
+
+lemma cpye_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃ⓕ{I}V2.T2⦄.
+#G #L #V1 #V2 #d #e * #HV12 #HV2 #T1 #T2 *
+/3 width=7 by cpys_flat, cny_flat, conj/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpye_inv_sort1: ∀G,L,X,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃X⦄ → X = ⋆k.
+#G #L #X #d #e #k * /2 width=5 by cpys_inv_sort1/
+qed-.
+
+lemma cpye_inv_gref1: ∀G,L,X,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃X⦄ → X = §p.
+#G #L #X #d #e #p * /2 width=5 by cpys_inv_gref1/
+qed-.
+
+lemma cpye_inv_bind1: ∀a,I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ &
+ X = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_bind1 … H1) -H1
+#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_bind … H2) -H2
+/5 width=8 by lsuby_cny_conf, lsuby_succ, ex3_2_intro, conj/
+qed-.
+
+lemma cpye_inv_flat1: ∀I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ &
+ X = ⓕ{I}V2.T2.
+#I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_flat1 … H1) -H1
+#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_flat … H2) -H2
+/3 width=5 by ex3_2_intro, conj/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubstevalalt_6.ma".
+include "basic_2/substitution/cpye_lift.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+(* Note: alternative definition of cpye *)
+inductive cpyea: ynat → ynat → relation4 genv lenv term term ≝
+| cpyea_sort : ∀G,L,d,e,k. cpyea d e G L (⋆k) (⋆k)
+| cpyea_free : ∀G,L,d,e,i. |L| ≤ i → cpyea d e G L (#i) (#i)
+| cpyea_top : ∀G,L,d,e,i. d + e ≤ yinj i → cpyea d e G L (#i) (#i)
+| cpyea_skip : ∀G,L,d,e,i. yinj i < d → cpyea d e G L (#i) (#i)
+| cpyea_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d+e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → cpyea (yinj 0) (⫰(d+e-yinj i)) G K V1 V2 →
+ ⇧[0, i+1] V2 ≡ W2 → cpyea d e G L (#i) W2
+| cpyea_gref : ∀G,L,d,e,p. cpyea d e G L (§p) (§p)
+| cpyea_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
+ cpyea d e G L V1 V2 → cpyea (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
+ cpyea d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpyea_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
+ cpyea d e G L V1 V2 → cpyea d e G L T1 T2 →
+ cpyea d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+ "evaluation for context-sensitive extended substitution (term) alternative"
+ 'PSubstEvalAlt G L T1 T2 d e = (cpyea d e G L T1 T2).
+
+(* Main properties **********************************************************)
+
+theorem cpye_cpyea: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
+#Z #Y #X #IH #G #L * *
+[ #k #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_sort1 … H) -H //
+| #i #HG #HL #HT #T2 #d #e #H destruct
+ elim (cpye_inv_lref1 … H) -H *
+ /4 width=7 by cpyea_subst, cpyea_free, cpyea_top, cpyea_skip, fqup_lref/
+| #p #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_gref1 … H) -H //
+| #a #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
+ elim (cpye_inv_bind1 … H) -H /3 width=1 by cpyea_bind/
+| #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
+ elim (cpye_inv_flat1 … H) -H /3 width=1 by cpyea_flat/
+]
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem cpyea_inv_cpye: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+/2 width=7 by cpye_subst, cpye_flat, cpye_bind, cpye_skip, cpye_top, cpye_free/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpye_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
+ (∀G,L,d,e,k. R d e G L (⋆k) (⋆k)) →
+ (∀G,L,d,e,i. |L| ≤ i → R d e G L (#i) (#i)) →
+ (∀G,L,d,e,i. d + e ≤ yinj i → R d e G L (#i) (#i)) →
+ (∀G,L,d,e,i. yinj i < d → R d e G L (#i) (#i)) →
+ (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[yinj O, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ →
+ ⇧[O, i+1] V2 ≡ W2 → R (yinj O) (⫰(d+e-yinj i)) G K V1 V2 → R d e G L (#i) W2
+ ) →
+ (∀G,L,d,e,p. R d e G L (§p) (§p)) →
+ (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
+ R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
+ R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L T1 T2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #d #e #G #L #T1 #T2 #H elim (cpye_cpyea … H) -G -L -T1 -T2 -d -e
+/3 width=8 by cpyea_inv_cpye/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_cny.ma".
+include "basic_2/substitution/cpys_cpys.ma".
+include "basic_2/substitution/cpye.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+(* Advanced properties ******************************************************)
+
+lemma cpye_cpys_conf: ∀G,L,T,T2,d,e. ⦃G, L⦄ ⊢ T ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ▶*[d, e] T1 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+#G #L #T #T2 #d #e * #H2 #HT2 #T1 #H1 elim (cpys_conf_eq … H1 … H2) -T
+#T0 #HT10 #HT20 >(cpys_inv_cny1 … HT2 … HT20) -T2 //
+qed-.
+
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cny_lift.ma".
+include "basic_2/substitution/fqup.ma".
+include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/cpye.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+(* Advanced properties ******************************************************)
+
+lemma cpye_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ →
+ ⇧[O, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃W2⦄.
+#I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK *
+/4 width=13 by cpys_subst, cny_lift_subst, ldrop_fwd_drop2, conj/
+qed.
+
+lemma cpye_total: ∀G,L,T1,d,e. ∃T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
+#Z #Y #X #IH #G #L * *
+[ #k #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
+| #i #HG #HL #HT #d #e destruct
+ elim (ylt_split i d) /3 width=2 by cpye_skip, ex_intro/
+ elim (ylt_split i (d+e)) /3 width=2 by cpye_top, ex_intro/
+ elim (lt_or_ge i (|L|)) /3 width=2 by cpye_free, ex_intro/
+ #Hi #Hide #Hdi elim (ldrop_O1_lt L i) // -Hi
+ #I #K #V1 #HLK elim (IH G K V1 … 0 (⫰(d+e-i))) -IH /2 width=2 by fqup_lref/
+ #V2 elim (lift_total V2 0 (i+1)) /3 width=8 by ex_intro, cpye_subst/
+| #p #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
+| #a #I #V1 #T1 #HG #HL #HT #d #e destruct
+ elim (IH G L V1 … d e) // elim (IH G (L.ⓑ{I}V1) T1 … (⫯d) e) //
+ /3 width=2 by cpye_bind, ex_intro/
+| #I #V1 #T1 #HG #HL #HT #d #e destruct
+ elim (IH G L V1 … d e) // elim (IH G L T1 … d e) //
+ /3 width=2 by cpye_flat, ex_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpye_inv_lref1: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∨∨ |L| ≤ i ∧ T2 = #i
+ | d + e ≤ yinj i ∧ T2 = #i
+ | yinj i < d ∧ T2 = #i
+ | ∃∃I,K,V1,V2. d ≤ yinj i & yinj i < d + e &
+ ⇩[i] L ≡ K.ⓑ{I}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #d #e * #H1 #H2 elim (cpys_inv_lref1 … H1) -H1
+[ #H destruct elim (cny_inv_lref … H2) -H2
+ /3 width=1 by or4_intro0, or4_intro1, or4_intro2, conj/
+| * #I #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
+ @or4_intro3 @(ex5_4_intro … HLK … HVT2) (**) (* explicit constructor *)
+ /4 width=13 by cny_inv_lift_subst, ldrop_fwd_drop2, conj/
+]
+qed-.
+
+lemma cpye_inv_lref1_free: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ (∨∨ |L| ≤ i | d + e ≤ yinj i | yinj i < d) → T2 = #i.
+#G #L #T2 #d #e #i #H * elim (cpye_inv_lref1 … H) -H * //
+#I #K #V1 #V2 #Hdi #Hide #HLK #_ #_ #H
+[ elim (lt_refl_false i) -d
+ @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/ (**) (* full auto slow: 19s *)
+]
+elim (ylt_yle_false … H) //
+qed-.
+
+lemma cpye_inv_lref1_lget: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 →
+ ∨∨ d + e ≤ yinj i ∧ T2 = #i
+ | yinj i < d ∧ T2 = #i
+ | ∃∃V2. d ≤ yinj i & yinj i < d + e &
+ ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #d #e #i #H #I #K #V1 #HLK elim (cpye_inv_lref1 … H) -H *
+[ #H elim (lt_refl_false i) -T2 -d
+ @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/
+| /3 width=1 by or3_intro0, conj/
+| /3 width=1 by or3_intro1, conj/
+| #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
+ lapply (ldrop_mono … HLY … HLK) -HLY -HLK #H destruct
+ /3 width=3 by or3_intro2, ex4_intro/
+]
+qed-.
+
+lemma cpye_inv_lref1_subst_ex: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I,K,V1. d ≤ yinj i → yinj i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #d #e #i #H #I #K #V1 #Hdi #Hide #HLK
+elim (cpye_inv_lref1_lget … H … HLK) -H * /2 width=3 by ex2_intro/
+#H elim (ylt_yle_false … H) //
+qed-.
+
+lemma cpye_inv_lref1_subst: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I,K,V1,V2. d ≤ yinj i → yinj i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ T2 →
+ ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄.
+#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2
+elim (cpye_inv_lref1_subst_ex … H … HLK) -H -HLK //
+#X2 #H0 #HXT2 lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma cpye_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdetd
+elim (cpys_inv_lift1_le … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_le … HU2 … HLK … HTU2 ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet
+elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_be … HU2 … HLK … HTU2 ? ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ yinj d + e ≤ dt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdedt
+elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_ge … HU2 … HLK … HTU2 ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] 𝐍⦃T2⦄ &
+ ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_ge_up … HU2 … HLK … HTU2 ? ? ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] 𝐍⦃W2⦄ →
+ ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
+ d ≤ yinj i → i < d + e →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ & ⇧[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #d #e * #HW12 #HW2 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_subst … HW12 … HLK … HVW1) -W1 // #V2 #HV12 #HVW2
+lapply (cny_inv_lift_subst … HLK HW2 HVW2) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cny.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Inversion lemmas on normality for extended substitution ******************)
+
+lemma cpys_inv_cny1: ∀G,L,T1,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → T1 = T2.
+#G #L #T1 #d #e #HT1 #T2 #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct <(HT1 … HT2) -T //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/pdeltaconvstar_6.ma".
+include "basic_2/substitution/cpye_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED DELTA-EQUIVALENCE FOR TERMS *******************)
+
+definition cpzs: ynat → ynat → relation4 genv lenv term term ≝
+ λd,e,G,L,T1,T2.
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T⦄ & ⦃G, L⦄ ⊢ T2 ▶*[d, e] 𝐍⦃T⦄.
+
+interpretation "context-sensitive extended delta-equivalence (term)"
+ 'PDeltaConvStar G L T1 d e T2 = (cpzs d e G L T1 T2).
+
+(* Basic properties **********************************************************)
+
+lemma cpye_div: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T2 ▶*[d, e] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2.
+/2 width=3 by ex2_intro/ qed.
+
+lemma cpzs_refl: ∀G,L,d,e. reflexive … (cpzs d e G L).
+#G #L #d #e #T elim (cpye_total G L T d e) /2 width=3 by cpye_div/
+qed.
+
+lemma cpzs_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ◆*[⫯d, e] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ◆*[d, e] ⓑ{a,I}V2.T2.
+#G #L #V1 #V2 #d #e * #V #HV1 #HV2 #I #T1 #T2 *
+/5 width=10 by cpye_div, cpye_bind, leqy_cpye_trans, cny_bind, lsuby_succ/
+qed.
+
+lemma cpzs_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 →
+ ∀T1,T2. ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ◆*[d, e] ⓕ{I}V2.T2.
+#G #L #V1 #V2 #d #e * #V #HV1 #HV2 #T1 #T2 *
+/3 width=5 by cpye_div, cpye_flat, cny_flat/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpzs_inv_sort: ∀G,L,d,e,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ◆*[d, e] ⋆k2 → k1 = k2.
+#G #L #d #e #k1 #k2 * #X #H1 #H2
+lapply (cpye_inv_sort1 … H1) -H1 #H1
+lapply (cpye_inv_sort1 … H2) -H2 #H2
+destruct //
+qed-.
+
+lemma cpzs_inv_bind: ∀a1,a2,I1,I2,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ ⓑ{a1,I1}V1.T1 ◆*[d, e] ⓑ{a2,I2}V2.T2 →
+ ∧∧ a1 = a2 & I1 = I2
+ & ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 & ⦃G, L.ⓑ{I1}V1⦄ ⊢ T1 ◆*[⫯d, e] T2.
+#a1 #a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2
+elim (cpye_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1
+elim (cpye_inv_bind1 … H2) -H2 #W2 #U2 #HW12 #HU12 #H2
+destruct /5 width=8 by cpye_div, leqy_cpye_trans, lsuby_succ, and4_intro/
+qed-.
+
+lemma cpzs_inv_flat: ∀I1,I2,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ ⓕ{I1}V1.T1 ◆*[d, e] ⓕ{I2}V2.T2 →
+ ∧∧ I1 = I2
+ & ⦃G, L⦄ ⊢ V1 ◆*[d, e] V2 & ⦃G, L⦄ ⊢ T1 ◆*[d, e] T2.
+#I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2
+elim (cpye_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1
+elim (cpye_inv_flat1 … H2) -H2 #W2 #U2 #HW12 #HU12 #H2
+destruct /3 width=3 by cpye_div, and3_intro/
+qed-.
+
+lemma cpzs_inv_flat_bind: ∀a2,I1,I2,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ ⓕ{I1}V1.T1 ◆*[d, e] ⓑ{a2,I2}V2.T2 → ⊥.
+#a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2
+elim (cpye_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1
+elim (cpye_inv_bind1 … H2) -H2 #W2 #U2 #HW12 #HU12 #H2
+destruct
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpye_lift.ma".
+include "basic_2/substitution/lleq_alt.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma lleq_fwd_cpye: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → ∀G,T1. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G, L2⦄ ⊢ T ▶*[d, ∞] 𝐍⦃T2⦄ → T1 = T2.
+#L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d
+[ #L1 #L2 #d #k #_ #G #T1 #H1 #T2 #H2
+ >(cpye_inv_sort1 … H1) -H1 >(cpye_inv_sort1 … H2) -H2 //
+| #L1 #L2 #d #i #_ #Hid #G #T1 #H1 #T2 #H2
+ >(cpye_inv_lref1_free … H1) -H1 [ >(cpye_inv_lref1_free … H2) -H2 ]
+ /2 width=1 by or3_intro2/
+| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #_ #IHV #G #T1 #H1 #T2 #H2
+ elim (cpye_inv_lref1_subst_ex … H1 … HLK1) -H1 -HLK1 //
+ elim (cpye_inv_lref1_subst_ex … H2 … HLK2) -H2 -HLK2 //
+ >yminus_Y_inj #V2 #HV2 #HVT2 #V1 #HV1 #HVT1
+ lapply (IHV … HV1 … HV2) -IHV -HV1 -HV2 #H destruct /2 width=5 by lift_mono/
+| #L1 #L2 #d #i #_ #HL1 #HL2 #G #T1 #H1 #T2 #H2
+ >(cpye_inv_lref1_free … H1) -H1 [ >(cpye_inv_lref1_free … H2) -H2 ]
+ /2 width=1 by or3_intro0/
+| #L1 #L2 #d #p #_ #G #T1 #H1 #T2 #H2
+ >(cpye_inv_gref1 … H1) -H1 >(cpye_inv_gref1 … H2) -H2 //
+| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2
+ elim (cpye_inv_bind1 … H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct
+ elim (cpye_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ /3 width=3 by eq_f2/
+| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2
+ elim (cpye_inv_flat1 … H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct
+ elim (cpye_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ /3 width=3 by eq_f2/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpye_cpye.ma".
+include "basic_2/reduction/lpx_cpys.ma".
+
+axiom cpx_cpys_conf_lpx: ∀h,g,G,d,e.
+ ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h, g] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, g] L1 →
+ ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*[d, e] T2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡[h, g] T.
+
+(* SN EXTENDED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *********************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma cpx_cpys_cpye_fwd_lpx: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] U1 →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2.
+#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e #HTU1
+elim (cpx_cpys_conf_lpx … HT12 … HL12 … HTU1) -T1
+/3 width=9 by cpx_cpys_trans_lpx, cpye_cpys_conf/
+qed-.
+
+lemma cpx_cpye_fwd_lpx: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2.
+#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e *
+/2 width=9 by cpx_cpys_cpye_fwd_lpx/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpye_lift.ma".
+include "basic_2/reduction/lpx_cpye.ma".
+include "basic_2/computation/cpxs_cpxs.ma".
+include "basic_2/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma cpx_cpye_fwd_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡*[h, g] U2.
+#h #g #G #L1 #L2 #H @(lpxs_ind_dx … H) -L1
+[ /3 width=9 by cpx_cpxs, cpx_cpye_fwd_lpx/
+| #L1 #L #HL1 #_ #IHL2 #T1 #T2 #HT12 #U1 #d #e #HTU1 #U2 #HTU2
+ elim (cpye_total G L T2 d e) #X2 #HTX2
+ lapply (cpx_cpye_fwd_lpx … HT12 … HL1 … HTU1 … HTX2) -T1
+ /4 width=9 by lpx_cpxs_trans, cpxs_strap2/ (**) (* full auto too long: 41s *)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ◆ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PDeltaConvStar $G $L $T1 $d $e $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PSubstEval $G $L $T1 $T2 $d $e }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PSubstEvalAlt $G $L $T1 $T2 $d $e }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ▶ break [ term 46 d , break term 46 e ] 𝐍 break ⦃ term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'PSubstNormal $G $L $T $d $e }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/cofreestar_4.ma".
+include "basic_2/relocation/lift_neg.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+definition cofrees: relation4 ynat nat lenv term ≝
+ λd,i,L,U1. ∀U2. ⦃⋆, L⦄ ⊢ U1 ▶*[d, ∞] U2 → ∃T2. ⇧[i, 1] T2 ≡ U2.
+
+interpretation
+ "context-sensitive exclusion from free variables (term)"
+ 'CoFreeStar L i d T = (cofrees d i L T).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cofrees_fwd_lift: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
+/2 width=1 by/ qed-.
+
+lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
+#a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
+#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_bind_dx: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
+#a #I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓑ{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1
+#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_flat_sn: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
+#I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓕ{I}W2.U)) /2 width=1 by cpys_flat/ -W1
+#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_flat_dx: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓕ{I}W.U2)) /2 width=1 by cpys_flat/ -U1
+#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cofrees_inv_gen: ∀L,U,U0,d,i. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 → (∀T. ⇧[i, 1] T ≡ U0 → ⊥) →
+ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥.
+#L #U #U0 #d #i #HU0 #HnU0 #HU elim (HU … HU0) -L -U -d /2 width=2 by/
+qed-.
+
+lemma cofrees_inv_lref_eq: ∀L,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃#i⦄ → ⊥.
+#L #d #i #H elim (H (#i)) -H //
+#X #H elim (lift_inv_lref2_be … H) -H //
+qed-.
+
+lemma cofrees_inv_bind: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
+/3 width=8 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-.
+
+lemma cofrees_inv_flat: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+/3 width=7 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-.
+
+(* Basic Properties *********************************************************)
+
+lemma cofrees_lsuby_conf: ∀L1,U,d,i. L1 ⊢ i ~ϵ 𝐅*[d]⦃U⦄ →
+ ∀L2. L1 ⊆[d, ∞] L2 → L2 ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+/3 width=3 by lsuby_cpys_trans/ qed-.
+
+lemma cofrees_sort: ∀L,d,i,k. L ⊢ i ~ϵ 𝐅*[d]⦃⋆k⦄.
+#L #d #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_gref: ∀L,d,i,p. L ⊢ i ~ϵ 𝐅*[d]⦃§p⦄.
+#L #d #i #p #X #H >(cpys_inv_gref1 … H) -X /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_bind: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d] ⦃V⦄ →
+ ∀I,T. L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ →
+ ∀a. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄.
+#L #W1 #d #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 … H) -H
+#W2 #U2 #HW12 #HU12 #H destruct
+elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/
+qed.
+
+lemma cofrees_flat: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ∀T. L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ →
+ ∀I. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄.
+#L #W1 #d #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 … H) -H
+#W2 #U2 #HW12 #HU12 #H destruct
+elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/
+qed.
+
+lemma cofrees_cpy_trans: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
+ ∀i. L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄.
+/3 width=3 by cpys_strap2/ qed-.
+
+axiom cofrees_dec: ∀L,T,d,i. Decidable (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄).
+
+(* Basic negated properties *************************************************)
+
+lemma frees_cpy_div: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
+ ∀i. (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → ⊥).
+/3 width=7 by cofrees_cpy_trans/ qed-.
+
+(* Basic negated inversion lemmas *******************************************)
+
+lemma frees_inv_bind: ∀a,I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄ → ⊥) →
+ (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ → ⊥).
+#a #I #L #W #U #d #i #H elim (cofrees_dec L W d i)
+/4 width=9 by cofrees_bind, or_intror, or_introl/
+qed-.
+
+lemma frees_inv_flat: ∀I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄ → ⊥) →
+ (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥).
+#I #L #W #U #d #H elim (cofrees_dec L W d)
+/4 width=8 by cofrees_flat, or_intror, or_introl/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cpy_nlift.ma".
+include "basic_2/substitution/cofrees_lift.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+(* Alternative definition of frees_ge ***************************************)
+
+lemma nlift_frees: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
+#L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
+qed-.
+
+lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
+ (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
+ ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
+ (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
+#L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
+#U1 #U2 #HU12 #HU2 *
+[ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
+ * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
+| * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
+ * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
+ lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
+ #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
+ #HK12
+ @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
+ @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
+ >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
+]
+qed-.
+
+lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
+ (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
+ (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
+ ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
+#R #IH1 #IH2 #d #i #L #U
+generalize in match d; -d generalize in match i; -i
+@(f2_ind … rfw … L U) -L -U
+#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
+-IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/cofrees.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cofrees_inv_lref_be: ∀L,d,i,j. L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → d ≤ yinj j → j < i →
+ ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #d #i #j #Hj #Hdj #Hji #I #K #W1 #HLK #W2 #HW12 elim (lift_total W2 0 (j+1))
+#X2 #HWX2 elim (Hj X2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
+#Z2 #HZX2 elim (lift_div_le … HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/
+>minus_plus <plus_minus_m_m //
+qed-.
+
+lemma cofrees_inv_be: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∀j. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
+ ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → d ≤ yinj j → j < i → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH #k #_ #d #i #_ #j #H elim (H (⋆k)) -H //
+| -IH #j #_ #d #i #Hi0 #j0 #H <(nlift_inv_lref_be_SO … H) -j0
+ /2 width=9 by cofrees_inv_lref_be/
+| -IH #p #_ #d #i #_ #j #H elim (H (§p)) -H //
+| #a #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
+ elim (cofrees_inv_bind … H1) -H1 #HW #HU
+ elim (nlift_inv_bind … H2) -H2 [ -HU /3 width=9 by/ ]
+ -HW #HnU lapply (IH … HU … HnU I K V ? ? ?)
+ /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/ -a -I -J -L -W -U -d
+ >minus_plus_plus_l //
+| #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
+ elim (cofrees_inv_flat … H1) -H1 #HW #HU
+ elim (nlift_inv_flat … H2) -H2 [ /3 width=9 by/ ]
+ #HnU @(IH … HU … HnU … HLK) // (**) (* full auto fails *)
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cofrees_lref_skip: ∀L,d,i,j. j < i → yinj j < d → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I #K #W1 #W2 #Hdj elim (ylt_yle_false … Hdj) -i -I -L -K -W1 -W2 -X //
+]
+qed.
+
+lemma cofrees_lref_lt: ∀L,d,i,j. i < j → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
+| * #I #K #V1 #V2 #_ #_ #_ #H -I -L -K -V1 -d
+ elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
+]
+qed.
+
+lemma cofrees_lref_gt: ∀I,L,K,W,d,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
+ K ⊢ (i-j-1) ~ϵ 𝐅*[O]⦃W⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
+ #H destruct elim (HW1 … HW12) -I -K -W1 -d
+ #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
+ >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
+]
+qed.
+
+lemma cofrees_lref_free: ∀L,d,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#L #d #i #j #Hj #Hji #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I #K #W1 #W2 #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
+ #H elim (lt_refl_false j) -d -i -K -W1 -W2 -X /2 width=3 by lt_to_le_to_lt/
+]
+qed.
+
+(* Advanced negated inversion lemmas ****************************************)
+
+lemma frees_inv_lref_gt: ∀L,d,i,j. j < i → (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) →
+ ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*[0]⦃W⦄ → ⊥) & d ≤ yinj j.
+#L #d #i #j #Hji #H elim (ylt_split j d) #Hjd
+[ elim H -H /2 width=6 by cofrees_lref_skip/
+| elim (lt_or_ge j (|L|)) #Hj
+ [ elim (ldrop_O1_lt … Hj) -Hj /4 width=10 by cofrees_lref_gt, ex3_3_intro/
+ | elim H -H /2 width=6 by cofrees_lref_free/
+ ]
+]
+qed-.
+
+lemma frees_inv_lref_free: ∀L,d,i,j. (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) → |L| ≤ j → j = i.
+#L #d #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
+#Hij elim H -H /2 width=6 by cofrees_lref_lt, cofrees_lref_free/
+qed-.
+
+lemma frees_inv_gen: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
+ ∃∃U0. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH #k #_ #d #i #H elim H -H //
+| #j #Hn #d #i #H elim (lt_or_eq_or_gt i j)
+ [ -n #Hij elim H -H /2 width=5 by cofrees_lref_lt/
+ | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
+ | #Hji elim (frees_inv_lref_gt … H) // -H
+ #I #K #W1 #HLK #H #Hdj elim (IH … H) /2 width=3 by ldrop_fwd_rfw/ -H -n
+ #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
+ #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
+ #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
+ >minus_plus <plus_minus_m_m //
+ ]
+| -IH #p #_ #d #i #H elim H -H //
+| #a #I #W #U #Hn #d #i #H elim (frees_inv_bind … H) -H
+ #H elim (IH … H) // -H -n
+ /4 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn, ex2_intro/
+| #I #W #U #Hn #d #i #H elim (frees_inv_flat … H) -H
+ #H elim (IH … H) // -H -n
+ /4 width=9 by cpys_flat, nlift_flat_dx, nlift_flat_sn, ex2_intro/
+]
+qed-.
+
+lemma frees_ind: ∀L,d,i. ∀R:predicate term.
+ (∀U1. (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) → R U1) →
+ (∀U1,U2. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 → (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → R U2 → R U1) →
+ ∀U. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U.
+#L #d #i #R #IH1 #IH2 #U1 #H elim (frees_inv_gen … H) -H
+#U2 #H #HnU2 @(cpys_ind_dx … H) -U1 /4 width=8 by cofrees_inv_gen/
+qed-.
+
+(* Advanced negated properties **********************************************)
+
+lemma frees_be: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
+ ∀i. j < i → (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) →
+ ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
+ ∀d. d ≤ yinj j → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
+/4 width=11 by cofrees_inv_be/ qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma cofrees_lift_be: ∀d0,e0,i. d0 ≤ i → i ≤ d0 + e0 →
+ ∀L,K,s. ⇩[s, d0, e0+1] L ≡ K → ∀T,U. ⇧[d0, e0+1] T ≡ U →
+ ∀d. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#d0 #e0 #i #Hd0i #Hide0 #L #K #s #HLK #T1 #U1 #HTU1 #d #U2 #HU12
+elim (yle_split d0 d) #H1
+[ elim (yle_split d (d0+e0+1)) #H2
+ [ letin cpys_inv ≝ cpys_inv_lift1_ge_up
+ | letin cpys_inv ≝ cpys_inv_lift1_ge
+ ]
+| letin cpys_inv ≝ cpys_inv_lift1_be
+]
+elim (cpys_inv … HU12 … HLK … HTU1) // #T2 #_ #HTU2 -s -L -K -U1 -T1 -d
+elim (lift_split … HTU2 i e0) /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_lift_ge: ∀d0,e0,i. d0 + e0 ≤ i →
+ ∀L,K,s. ⇩[s, d0, e0] L ≡ K → ∀T,U. ⇧[d0, e0] T ≡ U →
+ ∀d. K ⊢ i-e0 ~ϵ 𝐅*[d-yinj e0]⦃T⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#d0 #e0 #i #Hde0i #L #K #s #HLK #T1 #U1 #HTU1 #d #HT1 #U2 #HU12
+elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i
+elim (yle_split d0 d) #H1
+[ elim (yle_split d (d0+e0)) #H2
+ [ elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) // >yplus_inj >yminus_Y_inj #T2 #HT12
+ lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12
+ | elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) // #T2
+ ]
+| elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) // >yminus_Y_inj #T2 #HT12
+ lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) // -HT12
+]
+-s -L #HT12 #HTU2
+elim (HT1 … HT12) -T1 #V2 #HVT2
+elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_intro/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L ⊢ break term 46 i ~ ϵ 𝐅 * [ break term 46 d ] ⦃ break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'CoFreeStar $L $i $d $T }.
--- /dev/null
+lemma cpcs_beta_dx: ∀a,L,V1,V2,W1,W2,T1,T2.
+ L ⊢ V1 ⬌* V2 → L ⊢ W1 ⬌* W2 → L.ⓛW2 ⊢ T1 ⬌* T2 →
+ L ⊢ ⓐV1.ⓛ{a}W1.T1 ⬌* ⓓ{a}ⓝW2.V2.T2.
+#a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12
+@(cpcs_cpr_strap1 … (ⓐV2.ⓛ{a}W2.T2)) /2 width=1/ /3 width=1/
+qed.
+
+lemma cpcs_beta_sn: ∀a,L,V1,V2,W1,W2,T1,T2.
+ L ⊢ V1 ⬌* V2 → L ⊢ W1 ⬌* W2 → L.ⓛW1 ⊢ T1 ⬌* T2 →
+ L ⊢ ⓐV1.ⓛ{a}W1.T1 ⬌* ⓓ{a}ⓝW2.V2.T2.
+#a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12
+lapply (lsubr_cpcs_trans … HT12 (L.ⓓⓝW1.V1) ?) /2 width=1/ #H2T12
+@(cpcs_cpr_strap2 … (ⓓ{a}ⓝW1.V1.T1)) /2 width=1/ -HT12 /3 width=1/
+qed.
--- /dev/null
+(* Advanced properties ******************************************************)
+
+lemma cpds_cpes_dx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T2.
+#h #g #G #L #T1 #T2 #l * /3 width=6 by cpcs_cprs_dx, ex4_2_intro/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpes_inv_abst2: ∀h,g,a,G,L,W1,T1,T,l. ⦃G, L⦄ ⊢ T •*⬌*[h, g, l] ⓛ{a}W1.T1 →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ T •*➡*[h, g, l] ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+#h #g #a #G #L #W1 #T1 #T #l2 * #T0 #l1 #Hl21 #HT #HT0 #H
+elim (cpcs_inv_abst2 … H) -H /3 width=6 by ex4_2_intro, ex2_2_intro/
+qed-.
+
+(****************************************************************************)
+
+lemma sta_cpcs_cpes: ∀h,g,G,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h] T →
+ ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, 1] T2.
+/3 width=6 by sta_lstas, ex4_2_intro/ qed.
+
+lemma lstas_cpes: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T2.
+/2 width=6 by ex4_2_intro/ qed.
+
+lemma cpes_strap1: ∀h,g,G,L,T1,T,T2,l.
+ ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l] T2.
+#h #g #G #L #T1 #T #T2 #l * /3 width=8 by cpcs_strap1, ex4_2_intro/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_ldrop.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ | elim (lt_or_ge i (d + e)) #Hide
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ | -Hdi
+ lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+]
+qed.
+
+lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=7/ qed.
+
+lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
+/2 width=7/ qed.
+
+lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A.
+/2 width=7/ qed.
+
+lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A.
+/2 width=7/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/static/aaa_ltpss_dx.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma aaa_ltpss_sn_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
+#L1 #T #A #HT #L2 #d #e #HL12
+lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
+@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/
+qed.
+
+lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=5/ qed.
+
+lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cif.ma".
+include "basic_2/reducibility/cnf_lift.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
+ elim (cif_inv_delta … HLK ?) //
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
+ elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+ lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
+ elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
+]
+qed.
+
+lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 #HT1
+lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
+qed.
+
+lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#T1 #T2 #H elim H -T1 -T2
+[ //
+| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
+ [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
+ >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+ | elim (cif_inv_ri2 … H) /2 width=1/
+ ]
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
+ elim (cif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
+ [ lapply (tps_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+ elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+ lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+ lapply (IHT1 … HT1) -IHT1 #H destruct
+ lapply (tps_cif_eq … HT2 ?) -HT2 //
+ | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
+ elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
+ ]
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
+ elim (cif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #V1 #T1 #T #T2 #_ #_ #_ #L #H
+ elim (cif_inv_ri2 … H) /2 width=1/
+| #V1 #T1 #T2 #_ #_ #L #H
+ elim (cif_inv_ri2 … H) /2 width=1/
+]
+qed.
+
+lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 * #T0 #HT10 #HT02 #HT1
+lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
+qed.
+
+theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+/3 width=3/ qed.
+
+(* Note: this property is unusual *)
+lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
+#L #T #H elim H -L -T
+[ #L #K #V #i #HLK #H
+ elim (cnf_inv_delta … HLK H)
+| #L #V #T #_ #IHV #H
+ elim (cnf_inv_appl … H) -H /2 width=1/
+| #L #V #T #_ #IHT #H
+ elim (cnf_inv_appl … H) -H /2 width=1/
+| #I #L #V #T * #H1 #H2 destruct
+ [ elim (cnf_inv_zeta … H2)
+ | elim (cnf_inv_tau … H2)
+ ]
+|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
+ [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
+ |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
+ ]
+| #a #L #V #W #T #H
+ elim (cnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a #L #V #W #T #H
+ elim (cnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+]
+qed.
+
+theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
+/2 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/cprs_ltpss_dx.ma".
+include "basic_2/equivalence/cpcs_cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties concerning dx partial unfold on local environments ************)
+
+lemma cpcs_ltpss_dx_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #d #e #HL12 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
+elim (cprs_ltpss_dx_conf … HT1 … HL12) -HT1 #U1 #HTU1 #H1
+elim (cprs_ltpss_dx_conf … HT2 … HL12) -L1 #U2 #HTU2 #H2
+elim (tpss_conf_eq … H1 … H2) -T #U #HU1 #HU2
+lapply (cprs_tpss_trans … HTU1 … HU1) -U1
+lapply (cprs_tpss_trans … HTU2 … HU2) -U2 /2 width=3/
+qed-.
+
+lemma cpcs_ltpss_dx_tpss_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T,T2. L1 ⊢ T ⬌* T2 →
+ ∀T1. L2 ⊢ T ▶* [d, e] T1 →
+ L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #d #e #HL12 #T #T2 #HT2 #T1 #HT1
+lapply (cpcs_ltpss_dx_conf … HL12 … HT2) -L1 #HT2
+lapply (cpcs_tpss_conf … HT1 … HT2) -T //
+qed-.
+
+lemma cpcs_ltpss_dx_tpss2_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T1,T2. L1 ⊢ T1 ⬌* T2 →
+ ∀T3. L2 ⊢ T1 ▶* [d, e] T3 →
+ ∀T4. L2 ⊢ T2 ▶* [d, e] T4 →
+ L2 ⊢ T3 ⬌* T4.
+#L1 #L2 #d #e #HL12 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24
+lapply (cpcs_ltpss_dx_tpss_conf … HL12 … HT12 … HT13) -L1 -T1 #HT32
+lapply (cpcs_tpss_strap1 … HT32 … HT24) -T2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/equivalence/cpcs_ltpss_dx.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties concerning sn partial unfold on local environments ************)
+
+lemma cpcs_ltpss_sn_conf: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 #T1 #U1 #HTU1
+lapply (IHL1 … HTU1) -IHL1 -HTU1 #HTU1
+lapply (cpcs_ltpss_dx_conf … HL2 … HTU1) -HTU1 -HL2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss.ma".
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Basic_1: includes: pr2_delta1 *)
+definition cpr: lenv → relation term ≝
+ λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2.
+
+interpretation
+ "context-sensitive parallel reduction (term)"
+ 'PRed L T1 T2 = (cpr L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+/3 width=5/ qed-.
+
+(* Basic_1: was by definition: pr2_free *)
+lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
+/2 width=3/ qed.
+
+lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+/3 width=5/ qed.
+
+lemma cpr_refl: ∀L,T. L ⊢ T ➡ T.
+/2 width=1/ qed.
+
+(* Note: new property *)
+(* Basic_1: was only: pr2_thin_dx *)
+lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
+ L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+lemma cpr_cast: ∀L,V,T1,T2.
+ L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2.
+#L #V #T1 #T2 * /3 width=3/
+qed.
+
+(* Note: it does not hold replacing |L1| with |L2| *)
+(* Basic_1: was only: pr2_change *)
+lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
+#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
+lapply (tpss_lsubr_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: pr2_gen_csort *)
+lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2.
+#T1 #T2 * #T #HT normalize #HT2
+<(tpss_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+(* Basic_1: was: pr2_gen_sort *)
+lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
+#L #T2 #k * #X #H
+>(tpr_inv_atom1 … H) -H #H
+>(tpss_inv_sort1 … H) -H //
+qed-.
+
+(* Basic_1: was: pr2_gen_cast *)
+lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U2 = ⓝV2. T2
+ ) ∨ L ⊢ T1 ➡ U2.
+#L #V1 #T1 #U2 * #X #H #HU2
+elim (tpr_inv_cast1 … H) -H /3 width=3/
+* #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #L #V1 #T1 #T * #X #H1 #H2 #b
+elim (tpr_fwd_bind1_minus … H1 b) -H1 #V0 #T0 #HT10 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+qed-.
+
+lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L #L1 #T1 #T * #X #H1 #H2
+elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct
+elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/
+qed-.
+
+(* Basic_1: removed theorems 6:
+ pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
+ pr2_gen_ctail pr2_ctail
+ Basic_1: removed local theorems 3:
+ pr2_free_free pr2_free_delta pr2_delta_delta
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/aaa_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_aaa.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_cpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A.
+#L #T1 #A #HT1 #T2 * /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/tpr_tpr.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpr_bind_sn: ∀a,I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
+@ex2_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *)
+qed.
+
+(* Basic_1: was only: pr2_gen_cbind *)
+lemma cpr_bind_dx: ∀a,I,L,V1,V2,T1,T2. V1 ➡ V2 → L. ⓑ{I} V2 ⊢ T1 ➡ T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
+elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
+lapply (tpss_lsubr_trans … HT0 (⋆. ⓑ{I} V2) ?) -HT0 /2 width=1/ #HT0
+lapply (tpss_inv_SO2 … HT0) -HT0 #HT0
+@ex2_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
+qed.
+
+(* Basic_1: was only: pr2_head_1 *)
+lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
+ L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
+* /2 width=1/ /3 width=1/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @@ T1 ➡ L @@ T2.
+#L elim L -L
+[ #T1 #T2 #HT12 @(cpr_inv_atom … HT12)
+| normalize /3 width=1/
+].
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: pr2_confluence *)
+theorem cpr_conf: ∀L,U0,T1,T2. L ⊢ U0 ➡ T1 → L ⊢ U0 ➡ T2 →
+ ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
+#L #U0 #T1 #T2 * #U1 #HU01 #HUT1 * #U2 #HU02 #HUT2
+elim (tpr_conf … HU01 HU02) -U0 #U #HU1 #HU2
+elim (ltpr_tpr_tpss_conf ? L … HU1 … HUT1) -U1 // #U1 #HTU1 #HU1
+elim (ltpr_tpr_tpss_conf ? L … HU2 … HUT2) -U2 // #U2 #HTU2 #HU2
+elim (tpss_conf_eq … HU1 … HU2) -U /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/reducibility/tpr_lift.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
+ ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
+ ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
+#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
+lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+@ex2_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
+qed.
+
+lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
+ ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
+#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 #a #I
+lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
+lapply (tpss_lsubr_trans … HT02 (L.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
+@(ex2_intro … (ⓑ{a,I}V0.T0)) /2 width=1/ (* explicit constructors *)
+qed.
+
+lemma cpr_beta: ∀a,L,V1,V2,W,T1,T2.
+ L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
+#a #L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
+lapply (tpss_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+@(ex2_intro … (ⓓ{a}V.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
+qed.
+
+lemma cpr_beta_dx: ∀a,L,V1,V2,W,T1,T2.
+ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
+/3 width=1/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr2_gen_lref *)
+lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
+ T2 = #i ∨
+ ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
+ ⇧[0, i + 1] T1 ≡ T2 &
+ i < |L|.
+#L #T2 #i * #X #H
+>(tpr_inv_atom1 … H) -H #H
+elim (tpss_inv_lref1 … H) -H /2 width=1/
+* /3 width=6/
+qed-.
+
+(* Basic_1: was pr2_gen_abbr *)
+lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡ U2 →
+ (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 &
+ L. ⓓV ⊢ T1 ➡ T2 &
+ U2 = ⓓ{a}V2. T2
+ ) ∨
+ ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2 & a = true.
+#a #L #V1 #T1 #Y * #X #H1 #H2
+elim (tpr_inv_abbr1 … H1) -H1 *
+[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
+ lapply (tps_weak_full … HT0) -HT0 #HT0
+ lapply (tpss_lsubr_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02
+ lapply (tpss_weak_full … HT02) -HT02 #HT02
+ lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/
+| #T2 #HT12 #HXT2 #H destruct
+ elim (lift_total Y 0 1) #Z #HYZ
+ lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H
+ lapply (cpr_intro … HT12 … H) -T2 /3 width=3/
+]
+qed-.
+
+(* Basic_1: was: pr2_gen_abst *)
+lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡ U2 → ∀I,W.
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
+#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W
+elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+lapply (tpss_lsubr_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
+qed-.
+
+(* Basic_1: was pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2
+ | ∃∃a,V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2.
+#L #V1 #U0 #Y * #X #H1 #H2
+elim (tpr_inv_appl1 … H1) -H1 *
+[ #V #U #HV1 #HU0 #H destruct
+ elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
+| #a #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=9/
+| #a #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
+ elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
+ elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=13/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U = ⓐV2. T2.
+#L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
+ ∃∃V2,T2. L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
+ U2 = ⓛ{a}V2.T2.
+#a #L #V1 #T1 #U2 * #U #H #HU2 #b #I #W
+elim (tpr_fwd_abst1 … H b I W) -H #V #T #HT1 #H destruct
+elim (tpss_inv_bind1 … HU2) -HU2 #V2 #T2 #_ #HT2
+lapply (tpss_lsubr_trans … HT2 (L.ⓑ{I}W) ?) -HT2 /2 width=1/ /4 width=5/
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr2_lift *)
+lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
+elim (lt_or_ge (|K|) d) #HKd
+[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
+| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
+]
+qed.
+
+(* Basic_1: was: pr2_gen_lift *)
+lemma cpr_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
+elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #T #HTU #T1
+elim (lt_or_ge (|L|) d) #HLd
+[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK /2 width=2/
+ /3 width=7 by ex2_intro, cpr_intro/
+| elim (lt_or_ge (|L|) (d + e)) #HLde
+ [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // /2 width=2/
+ /3 width=7 by ex2_intro, cpr_intro/
+ | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK //
+ /3 width=7 by ex2_intro, cpr_intro/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_tpss.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning parallel unfold on terms ***************************)
+
+(* Note: we could invoke tpss_weak_full instead of ltpr_fwd_length *)
+(* Basic_1: was only: pr2_subst1 *)
+lemma cpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#L1 #L2 #HL12 #T1 #T2 * #T #HT1 #HT2 #d #e #U1 #HTU1
+elim (ltpr_tpr_tpss_conf … HL12 … HT1 … HTU1) -L1 -HT1 #U #HU1 #HTU
+elim (tpss_conf_eq … HT2 … HTU) -T /3 width=3/
+qed.
+
+lemma cpr_ltpr_conf_eq: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 ➡ T & T2 ➡ T.
+#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
+>(ltpr_fwd_length … HL12) in HT2; #HT2
+elim (ltpr_tpr_tpss_conf … HL12 … HT2) -L1 /3 width=3/
+qed.
+
+lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2.
+#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1
+elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2
+elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2
+lapply (tpss_weak_full … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *)
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning dx partial unfold on local environments ************)
+
+lemma ltpss_dx_cpr_conf: ∀L1,T,U1. L1 ⊢ T ➡ U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. L2 ⊢ T ➡ U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
+lapply (ltpss_dx_fwd_length … HL12) #H >H in HU1; -H #HU1
+elim (ltpss_dx_tpss_conf … HU1 … HL12) -L1 /3 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning sn partial unfold on local environments ************)
+
+lemma ltpss_sn_cpr_trans: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
+#L1 #L2 #d #e #HL12 #T1 #T2 *
+lapply (ltpss_sn_weak_full … HL12)
+<(ltpss_sn_fwd_length … HL12) -HL12 /3 width=5/
+qed.
+
+lemma ltpss_sn_cpr_conf: ∀L1,T,U1. L1 ⊢ T ➡ U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∃∃U2. L2 ⊢ T ➡ U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
+lapply (ltpss_sn_fwd_length … HL12) #H >H in HU1; -H #HU1
+elim (ltpss_sn_tpss_conf … HU1 … HL12) -HU1 -HL12 /3 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_tpss.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties on partial unfold for terms ***********************************)
+
+lemma cpr_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡ T →
+ ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2
+lapply (tpss_weak_full … HT2) -HT2 #HT2
+lapply (tpss_trans_eq … HT0 HT2) -T /2 width=3/
+qed.
+
+lemma cpr_tps_trans: ∀L,T1,T. L ⊢ T1 ➡ T →
+ ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡ T2.
+/3 width=5/ qed.
+
+lemma cpr_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡ T1 →
+ ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 →
+ ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡ T.
+#L #T0 #T1 * #U0 #HTU0 #HU0T1 #T2 #d #e #HT02
+elim (tpr_tpss_conf … HTU0 … HT02) -T0 #T0 #HT20 #HUT0
+elim (tpss_conf_eq … HU0T1 … HUT0) -U0 /3 width=5/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_ltpr.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties concerning parallel unfold on terms ***************************)
+
+(* Basic_1: was only: pr3_subst1 *)
+lemma cprs_tpss_ltpr: ∀L1,T1,U1,d,e. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
+ ∃∃U2. L2 ⊢ U1 ➡* U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2
+[ #T2 #HT12
+ elim (cpr_tpss_ltpr … HL12 … HT12 … HTU1) -L1 -T1 /3 width=3/
+| #T #T2 #_ #HT2 * #U #HU1 #HTU
+ elim (cpr_tpss_ltpr … HT2 … HTU) -L1 -T // /3 width=3/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_ltpss_dx.ma".
+include "basic_2/computation/cprs_tpss.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties concerning dx partial unfold on local environments ************)
+
+lemma cprs_ltpss_dx_conf: ∀L1,T,U1. L1 ⊢ T ➡* U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. L2 ⊢ T ➡* U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#L1 #T #U1 #H @(cprs_ind … H) -U1 /2 width=3/
+#T1 #U1 #_ #HTU1 #IHT1 #L2 #d #e #HL12
+elim (IHT1 … HL12) -IHT1 #U #HTU #HT1U
+elim (ltpss_dx_cpr_conf … HTU1 … HL12) -L1 #U0 #HT1U0 #HU10
+elim (cpr_tpss_conf … HT1U0 … HT1U) -T1 #U2 #HU02 #HU2
+lapply (tpss_trans_eq … HU10 HU02) -U0 /3 width=3/
+qed-.
+
+lemma cprs_ltpss_dx_tpss_conf: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. L2 ⊢ T2 ➡* U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (cprs_ltpss_dx_conf … HTU1 … HL12) -L1 #U #HT1U #HU1
+elim (cprs_tpss_conf … HT1U … HT12) -T1 #T #HUT #HT2
+lapply (tpss_trans_eq … HU1 HUT) -U /2 width=3/
+qed-.
+
+lemma cprs_ltpss_dx_tpss2_conf: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∀U2. L2 ⊢ U1 ▶* [d, e] U2 →
+ ∃∃U. L2 ⊢ T2 ➡* U & L2 ⊢ U2 ▶* [d, e] U.
+#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 #U2 #HU12
+elim (cprs_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -L1 -T1 #U #HT2U #HU1
+elim (tpss_conf_eq … HU12 … HU1) -U1 #U0 #HU20 #HU0
+lapply (cprs_tpss_trans … HT2U … HU0) -U /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/computation/cprs_ltpss_dx.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties concerning sn partial unfold on local environments ************)
+
+lemma cprs_ltpss_sn_conf: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T,U1. L1 ⊢ T ➡* U1 →
+ ∃∃U2. L2 ⊢ T ➡* U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
+#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
+elim (IHL1 … HTU1) -IHL1 -HTU1 #U #HTU #HU1
+elim (cprs_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_tpss.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties on partial unfold for terms ***********************************)
+
+lemma cprs_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T →
+ ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡* T2.
+#L #T1 #T #H @(cprs_ind … H) -T /2 width=3/ /3 width=5/
+qed.
+
+lemma cprs_tps_trans: ∀L,T1,T. L ⊢ T1 ➡* T →
+ ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡* T2.
+/3 width=5 by inj, cprs_tpss_trans/ qed. (**) (* auto too slow without trace *)
+
+lemma cprs_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 →
+ ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 →
+ ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡* T.
+#L #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3/
+#T #T1 #_ #HT1 #IHT0 #T2 #d #e #HT02
+elim (IHT0 … HT02) -T0 #T0 #HT0 #HT20
+elim (cpr_tpss_conf … HT1 … HT0) -T /3 width=5/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unwind/sstas_ltpss_dx.ma".
+include "basic_2/computation/cprs_ltpss_dx.ma".
+include "basic_2/computation/dxprs.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma dxprs_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12
+elim (sstas_ltpss_dx_conf … HTU … HL12) -HTU #U0 #HTU0 #HU0
+elim (cprs_ltpss_dx_conf … HU1 … HL12) -L1 #U2 #HU2 #HU12
+elim (cprs_tpss_conf … HU2 … HU0) -U #U #HU2 #HU0
+lapply (tpss_trans_eq … HU12 HU2) -U2 /3 width=3/
+qed-.
+
+lemma dxprs_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*➡*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #d #e #HT12
+elim (sstas_tpss_conf … HTW1 … HT12) -T1 #W2 #HTW2 #HW12
+elim (cprs_tpss_conf … HWU1 … HW12) -W1 /3 width=3/
+qed-.
+
+lemma dxprs_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (dxprs_ltpss_dx_conf … HTU1 … HL12) -L1 #U #HT1U #HU1
+elim (dxprs_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/computation/dxprs_ltpss_dx.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma dxprs_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 [ /2 width=3/ ]
+#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
+elim (IHL1 … HTU1) -IHL1 -HTU1 #U #HTU #HU1
+elim (dxprs_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lsubr_lbotr.ma".
+include "basic_2/relocation/ldrop_ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Inversion lemmas about local env. full refinement for substitution *******)
+
+(* Note: ldrop_ldrop not needed *)
+lemma lbotr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ⊒[d, e] L →
+ d ≤ i → i < d + e → I = Abbr.
+#I #L elim L -L
+[ #K #V #i #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L #J #W #IHL #K #V #i #H
+ elim (ldrop_inv_O1 … H) -H *
+ [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
+ lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
+ lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
+ elim (lsubr_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
+ | #Hi #HLK #d @(nat_ind_plus … d) -d
+ [ #e #H #_ #Hide
+ elim (lbotr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
+ @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/
+ | #d #_ #e #H #Hdi #Hide
+ lapply (lbotr_inv_skip … H ?) -H // #HL
+ @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/
+ ]
+ ]
+]
+qed-.
+
+(* Properties about local env. full refinement for substitution *************)
+
+(* Note: ldrop_ldrop not needed *)
+lemma lbotr_ldrop: ∀L,d,e.
+ (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
+ ⊒[d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d @(nat_ind_plus … d) -d
+[ #e @(nat_ind_plus … e) -e //
+ #e #_ #H0
+ >(H0 I L V 0 ? ? ?) //
+ /5 width=6 by lbotr_abbr, ldrop_ldrop, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
+| #d #_ #e #H0
+ /5 width=6 by lbotr_skip, ldrop_ldrop, le_S_S, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
+]
+qed.
+
+lemma lbotr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
+ dd + ee ≤ d → ⊒[dd, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
+elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
+elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
+@(lbotr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
+qed.
+
+lemma lbotr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
+ ∀dd,ee. ⊒[dd, ee] L1 →
+ dd ≤ d + e → d + e ≤ dd + ee →
+ ⊒[d, dd + ee - d - e] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
+lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
+>commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi #HL1K2
+@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
+qed.
+
+lemma lbotr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
+ d + e ≤ dd → ⊒[dd - e, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
+>plus_minus in Hiddee; // #Hiddee
+lapply (transitive_le … Hdde Hddi) -Hdde #Hid
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
+@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_px.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Properties on pointwise extension ****************************************)
+
+lemma lpx_deliftable_dropable: ∀R. t_deliftable_sn R → dropable_sn (lpx R).
+#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 /3 width=3/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (HR … HV12 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5/
+]
+qed.
+
+lemma lpx_liftable_dedropable: ∀R. reflexive ? R →
+ t_liftable R → dedropable_sn (lpx R).
+#R #H1R #H2R #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (H2R … HW12 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+]
+qed.
+
+fact lpx_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx R K1 K2.
+#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #X #H >(lpx_inv_atom2 … H) -H /2 width=3/
+| #K2 #I #V2 #X #H
+ elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
+| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
+ elim (lpx_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
+| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ >commutative_plus normalize #H destruct
+]
+qed-.
+
+lemma lpx_dropable: ∀R. dropable_dx (lpx R).
+/2 width=5 by lpx_dropable_aux/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_append.ma".
+
+(* POINTWISE EXTENSION OF A CONTEXT-FREE REALTION FOR TERMS *****************)
+
+inductive lpx (R:relation term): relation lenv ≝
+| lpx_stom: lpx R (⋆) (⋆)
+| lpx_pair: ∀I,K1,K2,V1,V2.
+ lpx R K1 K2 → R V1 V2 → lpx R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_inv_atom1_aux: ∀R,L1,L2. lpx R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_inv_atom1: ∀R,L2. lpx R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_inv_atom1_aux/ qed-.
+
+fact lpx_inv_pair1_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #L1 #L2 * -L1 -L2
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
+]
+qed-.
+
+lemma lpx_inv_pair1: ∀R,I,K1,V1,L2. lpx R (K1. ⓑ{I} V1) L2 →
+ ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_inv_pair1_aux/ qed-.
+
+fact lpx_inv_atom2_aux: ∀R,L1,L2. lpx R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_inv_atom2: ∀R,L1. lpx R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_inv_atom2_aux/ qed-.
+
+fact lpx_inv_pair2_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #L1 #L2 * -L1 -L2
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
+]
+qed-.
+
+lemma lpx_inv_pair2: ∀R,I,L1,K2,V2. lpx R L1 (K2. ⓑ{I} V2) →
+ ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_fwd_length: ∀R,L1,L2. lpx R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lpx_inv_append1: ∀R,L1,K1,L. lpx R (K1 @@ L1) L →
+ ∃∃K2,L2. lpx R K1 K2 & lpx R L1 L2 & L = K2 @@ L2.
+#R #L1 elim L1 -L1 normalize
+[ #K1 #K2 #HK12
+ @(ex3_2_intro … K2 (⋆)) // (**) (* explicit constructor, /2 width=5/ does not work *)
+| #L1 #I #V1 #IH #K1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #HL12 #H destruct
+ @(ex3_2_intro … HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *)
+]
+qed-.
+
+lemma lpx_inv_append2: ∀R,L2,K2,L. lpx R L (K2 @@ L2) →
+ ∃∃K1,L1. lpx R K1 K2 & lpx R L1 L2 & L = K1 @@ L1.
+#R #L2 elim L2 -L2 normalize
+[ #K2 #K1 #HK12
+ @(ex3_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=5/ does not work *)
+| #L2 #I #V2 #IH #K2 #X #H
+ elim (lpx_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #HL12 #H destruct
+ @(ex3_2_intro … HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R).
+#R #HR #L elim L -L // /2 width=1/
+qed.
+
+lemma lpx_sym: ∀R. symmetric ? R → symmetric … (lpx R).
+#R #HR #L1 #L2 #H elim H -H // /3 width=1/
+qed.
+
+lemma lpx_trans: ∀R. Transitive ? R → Transitive … (lpx R).
+#R #HR #L1 #L #H elim H -L //
+#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
+elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
+qed.
+
+lemma lpx_conf: ∀R. confluent ? R → confluent … (lpx R).
+#R #HR #L0 #L1 #H elim H -L1
+[ #X #H >(lpx_inv_atom1 … H) -X /2 width=3/
+| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
+ elim (IHK01 … HK02) -K0 #K #HK1 #HK2
+ elim (HR … HV01 … HV02) -HR -V0 /3 width=5/
+]
+qed.
+
+lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+qed.
+
+lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L →
+ ∀L2. lpx R L L2 → lpx (TC … R) L1 L2.
+#R #L1 #L #H elim H -L /2 width=1/
+#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
+elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
+qed.
+
+lemma TC_lpx_pair_dx: ∀R. reflexive ? R →
+ ∀I,K,V1,V2. TC … R V1 V2 →
+ TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2).
+#R #HR #I #K #V1 #V2 #H elim H -V2
+/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
+qed.
+
+lemma TC_lpx_pair_sn: ∀R. reflexive ? R →
+ ∀I,V,K1,K2. TC … (lpx R) K1 K2 →
+ TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V).
+#R #HR #I #V #K1 #K2 #H elim H -K2
+/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
+qed.
+
+lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2.
+#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/
+qed.
+
+lemma lpx_inv_TC: ∀R. reflexive ? R →
+ ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/
+qed.
+
+lemma lpx_append: ∀R,K1,K2. lpx R K1 K2 → ∀L1,L2. lpx R L1 L2 →
+ lpx R (L1 @@ K1) (L2 @@ K2).
+#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 𝟙 break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'RTop $T1 $T2 }.
+
+include "basic_2/grammar/lenv_px.ma".
+
+(* POINTWISE EXTENSION OF TOP RELATION FOR TERMS ****************************)
+
+definition ttop: relation term ≝ λT1,T2. True.
+
+definition ltop: relation lenv ≝ lpx ttop.
+
+interpretation
+ "top reduction (environment)"
+ 'RTop L1 L2 = (ltop L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ltop_refl: reflexive … ltop.
+/2 width=1/ qed.
+
+lemma ltop_sym: symmetric … ltop.
+/2 width=1/ qed.
+
+lemma ltop_trans: transitive … ltop.
+/2 width=3/ qed.
+
+lemma ltop_append: ∀K1,K2. K1 𝟙 K2 → ∀L1,L2. L1 𝟙 L2 → L1 @@ K1 𝟙 L2 @@ K2.
+/2 width=1/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ltop_inv_atom1: ∀L2. ⋆ 𝟙 L2 → L2 = ⋆.
+/2 width=2 by lpx_inv_atom1/ qed-.
+
+lemma ltop_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 𝟙 L2 →
+ ∃∃K2,V2. K1 𝟙 K2 & L2 = K2. ⓑ{I} V2.
+#K1 #I #V1 #L2 #H
+elim (lpx_inv_pair1 … H) -H /2 width=4/
+qed-.
+
+lemma ltop_inv_atom2: ∀L1. L1 𝟙 ⋆ → L1 = ⋆.
+/2 width=2 by lpx_inv_atom2/ qed-.
+
+lemma ltop_inv_pair2: ∀L1,K2,I,V2. L1 𝟙 K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 𝟙 K2 & L1 = K1. ⓑ{I} V1.
+#L1 #K2 #I #V2 #H
+elim (lpx_inv_pair2 … H) -H /2 width=4/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltop_fwd_length: ∀L1,L2. L1 𝟙 L2 → |L1| = |L2|.
+/2 width=2 by lpx_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 ⦄ ➡ break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRed $L1 $L2 }.
+
+include "basic_2/unfold/ltpss_sn.ma".
+include "basic_2/reducibility/ltpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
+
+definition lfpr: relation lenv ≝
+ λL1,L2. ∃∃L. L1 ➡ L & L ⊢ ▶* [0, |L|] L2
+.
+
+interpretation
+ "focalized parallel reduction (environment)"
+ 'FocalizedPRed L1 L2 = (lfpr L1 L2).
+
+(* Basic properties *********************************************************)
+
+(* Note: lemma 250 *)
+lemma lfpr_refl: ∀L. ⦃L⦄ ➡ ⦃L⦄.
+/2 width=3/ qed.
+
+lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄.
+/3 width=5/ qed.
+
+lemma ltpr_lfpr: ∀L1,L2. L1 ➡ L2 → ⦃L1⦄ ➡ ⦃L2⦄.
+/3 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆.
+#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_sn_inv_atom1 … HL2) -HL2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/aaa_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_aaa.ma".
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_lfpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → L2 ⊢ T ⁝ A.
+#L1 #T #A #HT #L2 * /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_ldrop.ma".
+include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
+ ∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄.
+#L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
+<(ltpss_sn_fwd_length … HL2) #V #HV1 #HV2 #I
+lapply (ltpss_sn_tpss_trans_eq … HV2 … HL2) -HV2 #V2
+@(ex2_intro … (L.ⓑ{I}V)) /2 width=1/ (**) (* explicit constructor *)
+qed.
+
+(* Properties on supclosure *************************************************)
+(*
+lamma fsub_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃L1⦄ ➡ ⦃L⦄ & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #HT12 #U2 * #T #H1 #H2
+elim (fsub_tpr_trans … HT12 … H1) -T2 #L #U #HL1 #HT1U #HUT
+elim (fsup_tpss_trans_full … HUT … H2) -T -HUT -H2 #L #U #HL1 #HT1U #HUT
+
+
+
+
+
+
+ #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
+elim (lift_total T d e) #U #HTU
+elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
+lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+qed-.
+*)
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_ltpss_sn.ma".
+include "basic_2/reducibility/ltpr_ltpr.ma".
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Main properties **********************************************************)
+
+theorem lfpr_conf: ∀L0,L1,L2. ⦃L0⦄ ➡ ⦃L1⦄ → ⦃L0⦄ ➡ ⦃L2⦄ →
+ ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄.
+#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2
+lapply (ltpr_fwd_length … HK01) #H
+>(ltpr_fwd_length … HK02) in H; #H
+elim (ltpr_conf … HK01 … HK02) -K0 #K #HK1 #HK2
+lapply (ltpss_sn_fwd_length … HKL1) #H1
+lapply (ltpss_sn_fwd_length … HKL2) #H2
+>H1 in HKL1 H; -H1 #HKL1
+>H2 in HKL2; -H2 #HKL2 #H
+elim (ltpr_ltpss_sn_conf … HKL1 … HK1) -K1 #K1 #HK1 #HLK1
+elim (ltpr_ltpss_sn_conf … HKL2 … HK2) -K2 #K2 #HK2 #HLK2
+elim (ltpss_sn_conf … HK1 … HK2) -K #K #HK1 #HK2
+lapply (ltpr_fwd_length … HLK1) #H1
+lapply (ltpr_fwd_length … HLK2) #H2
+/3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/ltprs.ma".
+include "basic_2/computation/lfprs.ma".
+
+(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
+
+(* Properties on context-free parallel computation for local environments ***)
+
+lemma ltprs_lfprs: ∀L1,L2. L1 ➡* L2 → ⦃L1⦄ ➡* ⦃L2⦄.
+/3 width=3/ qed.
--- /dev/null
+definition t_liftable: relation term → Prop ≝
+ λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2.
+
+definition t_deliftable_sn: relation term → Prop ≝
+ λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2.
+
+lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R).
+#R #HR #T1 #T2 #H elim H -T2
+[ /3 width=7/
+| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /3 width=9/
+]
+qed.
+
+lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R).
+#R #HR #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #T1 #d #e #HTU1
+ elim (HR … HU12 … HTU1) -U1 /3 width=3/
+| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1
+ elim (IHU1 … HTU1) -U1 #T #HTU #HT1
+ elim (HR … HU2 … HTU) -U /3 width=5/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L1 break ⊑ [ term 46 d , break term 46 e ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'SubEq $L1 $d $e $L2 }.
+
+include "basic_2/grammar/lenv_length.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
+
+inductive lsubr: nat → nat → relation lenv ≝
+| lsubr_sort: ∀d,e. lsubr d e (⋆) (⋆)
+| lsubr_OO: ∀L1,L2. lsubr 0 0 L1 L2
+| lsubr_abbr: ∀L1,L2,V,e. lsubr 0 e L1 L2 →
+ lsubr 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
+| lsubr_abst: ∀L1,L2,I,V1,V2,e. lsubr 0 e L1 L2 →
+ lsubr 0 (e + 1) (L1. ⓑ{I}V1) (L2. ⓛV2)
+| lsubr_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
+ lsubr d e L1 L2 → lsubr (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
+.
+
+interpretation
+ "local environment refinement (substitution)"
+ 'SubEq L1 d e L2 = (lsubr d e L1 L2).
+
+definition lsubr_trans: ∀S. (lenv → relation S) → Prop ≝ λS,R.
+ ∀L2,s1,s2. R L2 s1 s2 →
+ ∀L1,d,e. L1 ⊑ [d, e] L2 → R L1 s1 s2.
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_bind_eq: ∀L1,L2,e. L1 ⊑ [0, e] L2 → ∀I,V.
+ L1. ⓑ{I} V ⊑ [0, e + 1] L2.ⓑ{I} V.
+#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
+qed.
+
+lemma lsubr_abbr_lt: ∀L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
+ L1. ⓓV ⊑ [0, e] L2.ⓓV.
+#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma lsubr_abst_lt: ∀L1,L2,I,V1,V2,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
+ L1. ⓑ{I}V1 ⊑ [0, e] L2. ⓛV2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma lsubr_skip_lt: ∀L1,L2,d,e. L1 ⊑ [d - 1, e] L2 → 0 < d →
+ ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 ⊑ [d, e] L2. ⓑ{I2} V2.
+#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
+qed.
+
+lemma lsubr_bind_lt: ∀I,L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
+ L1. ⓓV ⊑ [0, e] L2. ⓑ{I}V.
+* /2 width=1/ qed.
+
+lemma lsubr_refl: ∀d,e,L. L ⊑ [d, e] L.
+#d elim d -d
+[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
+| #d #IHd #e #L elim L -L // /2 width=1/
+]
+qed.
+
+lemma TC_lsubr_trans: ∀S,R. lsubr_trans S R → lsubr_trans S (λL. (TC … (R L))).
+#S #R #HR #L1 #s1 #s2 #H elim H -s2
+[ /3 width=5/
+| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
+ lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubr_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L1 = ⋆ →
+ L2 = ⋆ ∨ (d = 0 ∧ e = 0).
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ /2 width=1/
+| /3 width=1/
+| #L1 #L2 #W #e #_ #H destruct
+| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
+| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
+]
+qed.
+
+lemma lsubr_inv_atom1: ∀L2,d,e. ⋆ ⊑ [d, e] L2 →
+ L2 = ⋆ ∨ (d = 0 ∧ e = 0).
+/2 width=3/ qed-.
+
+fact lsubr_inv_skip1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ ∀I1,K1,V1. L1 = K1.ⓑ{I1}V1 → 0 < d →
+ ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #I1 #K1 #V1 #H destruct
+| #L1 #L2 #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
+]
+qed.
+
+lemma lsubr_inv_skip1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑ [d, e] L2 → 0 < d →
+ ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5/ qed-.
+
+fact lsubr_inv_atom2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L2 = ⋆ →
+ L1 = ⋆ ∨ (d = 0 ∧ e = 0).
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ /2 width=1/
+| /3 width=1/
+| #L1 #L2 #W #e #_ #H destruct
+| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
+| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
+]
+qed.
+
+lemma lsubr_inv_atom2: ∀L1,d,e. L1 ⊑ [d, e] ⋆ →
+ L1 = ⋆ ∨ (d = 0 ∧ e = 0).
+/2 width=3/ qed-.
+
+fact lsubr_inv_abbr2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ ∀K2,V. L2 = K2.ⓓV → d = 0 → 0 < e →
+ ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #K1 #V #H destruct
+| #L1 #L2 #K1 #V #_ #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #W #e #HL12 #K1 #V #H #_ #_ destruct /2 width=3/
+| #L1 #L2 #I #W1 #W2 #e #_ #K1 #V #H destruct
+| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #K1 #V #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,V,e. L1 ⊑ [0, e] K2.ⓓV → 0 < e →
+ ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
+/2 width=5/ qed-.
+
+fact lsubr_inv_skip2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ ∀I2,K2,V2. L2 = K2.ⓑ{I2}V2 → 0 < d →
+ ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #I1 #K1 #V1 #H destruct
+| #L1 #L2 #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
+]
+qed.
+
+lemma lsubr_inv_skip2: ∀I2,L1,K2,V2,d,e. L1 ⊑ [d, e] K2.ⓑ{I2}V2 → 0 < d →
+ ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=5/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+fact lsubr_fwd_length_full1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ d = 0 → e = |L1| → |L1| ≤ |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
+[ //
+| /2 width=1/
+| /3 width=1/
+| /3 width=1/
+| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubr_fwd_length_full1: ∀L1,L2. L1 ⊑ [0, |L1|] L2 → |L1| ≤ |L2|.
+/2 width=5/ qed-.
+
+fact lsubr_fwd_length_full2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ d = 0 → e = |L2| → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
+[ //
+| /2 width=1/
+| /3 width=1/
+| /3 width=1/
+| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubr_fwd_length_full2: ∀L1,L2. L1 ⊑ [0, |L2|] L2 → |L2| ≤ |L1|.
+/2 width=5/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⊒ [ term 46 d , break term 46 e ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'SubEqBottom $d $e $L2 }.
+
+include "basic_2/relocation/lsubr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
+
+(* bottom element of the refinement *)
+definition lbotr: nat → nat → predicate lenv ≝
+ λd,e. NF_sn … (lsubr d e) (lsubr d e …).
+
+interpretation
+ "local environment full refinement (substitution)"
+ 'SubEqBottom d e L = (lbotr d e L).
+
+(* Basic properties *********************************************************)
+
+lemma lbotr_atom: ∀d,e. ⊒[d, e] ⋆.
+#d #e #L #H
+elim (lsubr_inv_atom2 … H) -H
+[ #H destruct //
+| * #H1 #H2 destruct //
+]
+qed.
+
+lemma lbotr_OO: ∀L. ⊒[0, 0] L.
+// qed.
+
+lemma lbotr_abbr: ∀L,V,e. ⊒[0, e] L → ⊒[0, e + 1] L.ⓓV.
+#L #V #e #HL #K #H
+elim (lsubr_inv_abbr2 … H ?) -H // <minus_plus_m_m #X #HLX #H destruct
+lapply (HL … HLX) -HL -HLX /2 width=1/
+qed.
+
+lemma lbotr_abbr_O: ∀L,V. ⊒[0,1] L.ⓓV.
+#L #V
+@(lbotr_abbr … 0) //
+qed.
+
+lemma lbotr_skip: ∀I,L,V,d,e. ⊒[d, e] L → ⊒[d + 1, e] L.ⓑ{I}V.
+#I #L #V #d #e #HL #K #H
+elim (lsubr_inv_skip2 … H ?) -H // <minus_plus_m_m #J #X #W #HLX #H destruct
+lapply (HL … HLX) -HL -HLX /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lbotr_inv_bind: ∀I,L,V,e. ⊒[0, e] L.ⓑ{I}V → 0 < e →
+ ⊒[0, e - 1] L ∧ I = Abbr.
+#I #L #V #e #HL #He
+lapply (HL (L.ⓓV) ?) /2 width=1/ #H
+elim (lsubr_inv_abbr2 … H ?) -H // #K #_ #H destruct
+@conj // #L #HKL
+lapply (HL (L.ⓓV) ?) -HL /2 width=1/ -HKL #H
+elim (lsubr_inv_abbr2 … H ?) -H // -He #X #HLX #H destruct //
+qed-.
+
+lemma lbotr_inv_skip: ∀I,L,V,d,e. ⊒[d, e] L.ⓑ{I}V → 0 < d → ⊒[d - 1, e] L.
+#I #L #V #d #e #HL #Hd #K #HLK
+lapply (HL (K.ⓑ{I}V) ?) -HL /2 width=1/ -HLK #H
+elim (lsubr_inv_skip2 … H ?) -H // -Hd #J #X #W #HKX #H destruct //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_px.ma".
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+definition ltpr: relation lenv ≝ lpx tpr.
+
+interpretation
+ "context-free parallel reduction (environment)"
+ 'PRed L1 L2 = (ltpr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ltpr_refl: reflexive … ltpr.
+/2 width=1/ qed.
+
+lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2.
+/2 width=1/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: wcpr0_gen_sort *)
+lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
+/2 width=2 by lpx_inv_atom1/ qed-.
+
+(* Basic_1: was: wcpr0_gen_head *)
+lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
+ ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
+/2 width=1 by lpx_inv_pair1/ qed-.
+
+lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
+/2 width=2 by lpx_inv_atom2/ qed-.
+
+lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=1 by lpx_inv_pair2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
+/2 width=2 by lpx_fwd_length/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L →
+ ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2.
+/2 width=1 by lpx_inv_append1/ qed-.
+
+lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 →
+ ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1.
+/2 width=1 by lpx_inv_append2/ qed-.
+
+(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/aaa_ltpss_dx.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 →
+ ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * *
+[1,2,3:
+ #i #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
+ >(tpr_inv_atom1 … H2) -Y
+|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
+]
+[ >(aaa_inv_sort … H1) -X //
+| elim (aaa_inv_lref … H1) #I #K1 #V1 #HLK1 #HA
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #Y #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IH … HKV1 … HA … HK12 … HV12) -L1 -K1 -V1 /2 width=5/
+| elim (aaa_inv_gref … H1)
+| elim (aaa_inv_abbr … H1) -H1 #B #HB #HA
+ elim (tpr_inv_abbr1 … H2) -H2 *
+ [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
+ lapply (tps_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+ lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
+ lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/
+ | -B #T #HT1 #HXT #H destruct
+ lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA
+ @(aaa_inv_lift … HA … HXT) /2 width=1/
+ ]
+| elim (aaa_inv_abst … H1) -H1 #B #A #HB #HA #H destruct
+ elim (tpr_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
+ lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/
+| elim (aaa_inv_appl … H1) -H1 #B #HB #HA
+ elim (tpr_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
+ lapply (IH … HA … HL12 … HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/
+ | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
+ elim (aaa_inv_abst … HA) -HA #B0 #A0 #HB0 #HA0 #H destruct
+ lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
+ lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
+ lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 // /2 width=1/ -T0 -L1 -V1 /4 width=7/
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
+ elim (aaa_inv_abbr … HA) -HA #B0 #HW0 #HT0
+ lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
+ lapply (IH … HB … HL12 … HV10) -HB -HV10 /width=5/ #HV0
+ lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2
+ @(aaa_abbr … HW2) -HW2
+ @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
+ ]
+| elim (aaa_inv_cast … H1) -H1 #HV1 #HT1
+ elim (tpr_inv_cast1 … H2) -H2
+ [ * #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
+ lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
+ | -HV1 #HT1X
+ lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
+ ]
+]
+qed.
+
+lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A.
+/2 width=5/ qed.
+
+lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A.
+/2 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/ldrop_lpx.ma".
+include "basic_2/substitution/fsup.ma".
+include "basic_2/reducibility/tpr_lift.ma".
+include "basic_2/reducibility/ltpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properies on local environment slicing ***********************************)
+
+(* Basic_1: was: wcpr0_drop *)
+lemma ltpr_ldrop_conf: dropable_sn ltpr.
+/3 width=3 by lpx_deliftable_dropable, tpr_inv_lift1/ qed.
+
+(* Basic_1: was: wcpr0_drop_back *)
+lemma ldrop_ltpr_trans: dedropable_sn ltpr.
+/2 width=3/ qed.
+
+lemma ltpr_ldrop_trans_O1: dropable_dx ltpr.
+/2 width=3/ qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma fsub_tpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. T2 ➡ U2 →
+ ∃∃L,U1. L1 ➡ L & T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
+elim (lift_total T d e) #U #HTU
+elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
+lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/tpr_tpr.ma".
+include "basic_2/reducibility/ltpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Main properties **********************************************************)
+
+theorem ltpr_conf: ∀L0:lenv. ∀L1. L0 ➡ L1 → ∀L2. L0 ➡ L2 →
+ ∃∃L. L1 ➡ L & L2 ➡ L.
+#L0 #L1 #H elim H -L0 -L1 /2 width=3/
+#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H
+elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
+elim (IHK01 … HK02) -K0 #K #HK1 #HK2
+elim (tpr_conf … HV01 HV02) -V0 /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties concerning dx parallel unfold on local environments ***********)
+
+lemma ltpr_ltpss_dx_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2.
+#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ /2 width=3/
+| #L1 #I #V1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
+ elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
+ elim (ltpr_tpr_tpss_conf … HK12 … HV12 … HVW1) -V1 /3 width=5/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/reducibility/ltpr_ltpss_dx.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties on sn parallel unfold on local environments *******************)
+
+(* Note: this can also be proved like ltpr_ltpss_dx_conf *)
+lemma ltpr_ltpss_sn_conf: ∀L1,K1,d,e. L1 ⊢ ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 ⊢ ▶* [d, e] K2 & K1 ➡ K2.
+#L1 #K1 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H
+@(ltpssa_ind … H) -K1 /2 width=3/
+#K #K1 #_ #HK1 #IHK #L2 #HL12
+elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
+elim (ltpr_ltpss_dx_conf … HK1 … HK2) -K /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties concerning parallel substitution on terms *********************)
+
+(* Basic_1: was: pr0_subst1_fwd *)
+lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ /2 width=3/
+| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
+ elim (lift_total V2 0 (i+1)) #W2 #HVW2
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
+| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
+| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L1 /3 width=5/
+]
+qed-.
+
+(* Basic_1: was: pr0_subst1_back *)
+lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 →
+ ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2.
+#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
+[ /2 width=3/
+| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
+ elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
+ elim (lift_total V1 0 (i+1)) #W1 #HVW1
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
+| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
+| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L2 /3 width=5/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
+include "basic_2/reducibility/ltpr_tps.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties on partial unfold for terms ***********************************)
+
+(* Basic_1: was: pr0_subst1 *)
+lemma ltpr_tpr_tps_conf: ∀T1,T2. T1 ➡ T2 →
+ ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 →
+ ∀L2. L1 ➡ L2 →
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#T1 #T2 #H elim H -T1 -T2
+[ #I #L1 #d #e #U1 #H #L2 #HL12
+ elim (ltpr_tps_conf … H … HL12) -L1 /3 width=3/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV12 … HVW1 … HL12) -V1
+ elim (IHT12 … HTU1 … HL12) -T1 -HL12 /3 width=5/
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
+ elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ lapply (tpss_lsubr_trans … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/
+| #a #I #V1 #V2 #T1 #T #T2 #HV12 #_ #HT2 #IHV12 #IHT1 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV12 … HVW1 … HL12) -V1 #W2 #HW12 #HVW2
+ elim (IHT1 … HTU1 (L2. ⓑ{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
+ elim (tpss_strip_neq … HTU … HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2
+ lapply (tps_lsubr_trans … HU2 (L2. ⓑ{I} V2) ?) -HU2 /2 width=1/ #HU2
+ elim (ltpss_dx_tps_conf … HU2 (L2. ⓑ{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23
+ lapply (tps_lsubr_trans … HU3 (⋆. ⓑ{I} W2) ?) -HU3 /2 width=1/ #HU3
+ lapply (tpss_lsubr_trans … HU23 (L2. ⓑ{I} W2) ?) -HU23 /2 width=1/ #HU23
+ lapply (tpss_trans_eq … HTU2 … HU23) -U2 /3 width=5/
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
+ elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
+ elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ elim (lift_total VV2 0 1) #VV #H2VV
+ lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV
+ @ex2_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_bind1 … H) -H #W #U1 #_ #HTU1 #H destruct -V
+ elim (IHT1 … HTU1 (L2.ⓓW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU
+ elim (tpss_inv_lift1_ge … HTU L2 … HT2 ?) -T <minus_plus_m_m /3 width=3/
+| #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+ elim (IHT12 … HTT1 … HL12) -T1 -HL12 /3 width=3/
+]
+qed-.
+
+lemma tpr_tps_conf_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
+ ⋆. ⓑ{I} V1 ⊢ T1 ▶ [0, 1] U1 →
+ ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 ▶ [0, 1] U2.
+#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
+elim (ltpr_tpr_tps_conf … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ -V1 #U2 #HU12 #HTU2
+lapply (tpss_inv_SO2 … HTU2) -HTU2 /2 width=3/
+qed-.
+
+lemma ltpr_tpr_tpss_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
+#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
+[ /2 width=3/
+| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
+ elim (ltpr_tpr_tps_conf … HUT … HU1 … HL12) -U -HL12 #U2 #HU12 #HTU2
+ lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
+]
+qed-.
+
+lemma tpr_tpss_conf: ∀T1,T2. T1 ➡ T2 →
+ ∀L,U1,d,e. L ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. U1 ➡ U2 & L ⊢ T2 ▶* [d, e] U2.
+/2 width=5 by ltpr_tpr_tpss_conf/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr.ma".
+include "basic_2/computation/tprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
+
+definition ltprs: relation lenv ≝ TC … ltpr.
+
+interpretation
+ "context-free parallel computation (environment)"
+ 'PRedStar L1 L2 = (ltprs L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma ltprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. L1 ➡* L → L ➡ L2 → R L → R L2) →
+ ∀L2. L1 ➡* L2 → R L2.
+#L1 #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma ltprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. L1 ➡ L → L ➡* L2 → R L → R L1) →
+ ∀L1. L1 ➡* L2 → R L1.
+#L2 #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltprs_refl: reflexive … ltprs.
+/2 width=1/ qed.
+
+lemma ltpr_ltprs: ∀L1,L2. L1 ➡ L2 → L1 ➡* L2.
+/2 width=1/ qed.
+
+lemma ltprs_strap1: ∀L1,L,L2. L1 ➡* L → L ➡ L2 → L1 ➡* L2.
+/2 width=3/ qed.
+
+lemma ltprs_strap2: ∀L1,L,L2. L1 ➡ L → L ➡* L2 → L1 ➡* L2.
+/2 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ltprs_inv_atom1: ∀L2. ⋆ ➡* L2 → L2 = ⋆.
+#L2 #H @(ltprs_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 destruct
+>(ltpr_inv_atom1 … HL2) -L2 //
+qed-.
+
+lemma ltprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ➡* L2 →
+ ∃∃K2,V2. K1 ➡* K2 & V1 ➡* V2 & L2 = K2. ⓑ{I} V2.
+#I #K1 #L2 #V1 #H @(ltprs_ind … H) -L2 /2 width=5/
+#L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
+elim (ltpr_inv_pair1 … HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
+qed-.
+
+lemma ltprs_inv_atom2: ∀L1. L1 ➡* ⋆ → L1 = ⋆.
+#L1 #H @(ltprs_ind_dx … H) -L1 //
+#L1 #L #HL1 #_ #IHL2 destruct
+>(ltpr_inv_atom2 … HL1) -L1 //
+qed-.
+
+lemma ltprs_inv_pair2: ∀I,L1,K2,V2. L1 ➡* K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ➡* K2 & V1 ➡* V2 & L1 = K1. ⓑ{I} V1.
+#I #L1 #K2 #V2 #H @(ltprs_ind_dx … H) -L1 /2 width=5/
+#L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+elim (ltpr_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltprs_fwd_length: ∀L1,L2. L1 ➡* L2 → |L1| = |L2|.
+#L1 #L2 #H @(ltprs_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1
+>IHL1 -L1 >(ltpr_fwd_length … HL2) -HL2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 ➡ ➡ * break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRedStarAlt $T1 $T2 }.
+
+include "basic_2/computation/ltprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
+
+(* alternative definition of ltprs *)
+definition ltprsa: relation lenv ≝ lpx tprs.
+
+interpretation
+ "context-free parallel computation (environment) alternative"
+ 'PRedStarAlt L1 L2 = (ltprsa L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ltprs_ltprsa: ∀L1,L2. L1 ➡* L2 → L1 ➡➡* L2.
+/2 width=1/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ltprsa_ltprs: ∀L1,L2. L1 ➡➡* L2 → L1 ➡* L2.
+/2 width=1/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_ldrop.ma".
+include "basic_2/computation/ltprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
+
+lemma ltprs_ldrop_conf: dropable_sn ltprs.
+/2 width=3/ qed.
+
+lemma ldrop_ltprs_trans: dedropable_sn ltprs.
+/2 width=3/ qed.
+
+lemma ltprs_ldrop_trans_O1: dropable_dx ltprs.
+/2 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_ltpr.ma".
+include "basic_2/computation/ltprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
+
+(* Advanced properties ******************************************************)
+
+lemma ltprs_strip: ∀L1. ∀L:lenv. L ➡* L1 → ∀L2. L ➡ L2 →
+ ∃∃L0. L1 ➡ L0 & L2 ➡* L0.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem ltprs_conf: confluent … ltprs.
+/3 width=3/ qed.
+
+theorem ltprs_trans: Transitive … ltprs.
+/2 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Basic_1: includes: csubst1_bind *)
+inductive ltpss_dx: nat → nat → relation lenv ≝
+| ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆)
+| ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e.
+ ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
+ ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+| ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e.
+ ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
+ ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "parallel unfold (local environment, dx variant)"
+ 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
+[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
+ >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
+]
+qed.
+
+lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
+/2 width=4/ qed-.
+
+fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2.
+ L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K1 #V1 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
+
+fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2.
+ L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K2 #V2 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
+ L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
+ L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) /2 width=1/
+qed.
+
+lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) /2 width=1/
+qed.
+
+lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+(* Basic_1: was by definition: csubst1_refl *)
+lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L.
+#L elim L -L //
+#L #I #V #IHL * /2 width=1/ * /2 width=1/
+qed.
+
+lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
+ lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
+ >plus_plus_comm_23 in Hde12; #Hde12
+ elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
+ [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
+ lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
+ | -Hd21 normalize in Hde12;
+ lapply (lt_to_le_to_lt 0 … Hde12) // #He2
+ lapply (le_plus_to_minus_r … Hde12) -Hde12
+ /3 width=5 by ltpss_dx_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
+ ]
+]
+qed.
+
+lemma ltpss_dx_weak_full: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+// /3 width=2/ /3 width=3/
+qed.
+
+fact ltpss_dx_append_le_aux: ∀K1,K2,d,x. K1 ▶* [d, x] K2 → x = |K1| - d →
+ ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ▶* [d, x + e] L2 @@ K2.
+#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
+[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct //
+| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
+ lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
+ /4 width=5 by ltpss_dx_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_plus_to_le_r … H2) -H2 #Hd
+ /4 width=5 by ltpss_dx_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+]
+qed-.
+
+lemma ltpss_dx_append_le: ∀K1,K2,d. K1 ▶* [d, |K1| - d] K2 →
+ ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ▶* [d, |K1| - d + e] L2 @@ K2.
+/2 width=1 by ltpss_dx_append_le_aux/ qed.
+
+lemma ltpss_dx_append_zero: ∀K1,K2. K1 ▶* [0, |K1|] K2 →
+ ∀L1,L2,e. L1 ▶* [0, e] L2 →
+ L1 @@ K1 ▶* [0, |K1| + e] L2 @@ K2.
+/2 width=1/ qed.
+
+lemma ltpss_dx_append_ge: ∀K1,K2,d,e. K1 ▶* [d, e] K2 →
+ ∀L1,L2. L1 ▶* [d - |K1|, e] L2 → |K1| ≤ d →
+ L1 @@ K1 ▶* [d, e] L2 @@ K2.
+#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
+[ #d #e #L1 #L2 <minus_n_O //
+| #K #I #V #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
+ normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
+ lapply (le_plus_to_le_r … H) -H /3 width=1/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpss_dx_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+normalize //
+qed-.
+
+(* Basic_1: removed theorems 28:
+ csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
+ csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
+ csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
+ csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
+ csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
+ csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
+ csubst1_head csubst1_flat csubst1_gen_head
+ csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
+ fsubst0_gen_base
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fsup.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/ltpss_dx.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma ltpss_dx_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ldrop_ltpss_dx_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d1 ≤ d2 →
+ ∃∃L2. L1 ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
+ elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
+ elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
+ elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_ge … HW12 … HLK2 HWV1 … HWV2) -W1 // -Hd12
+ <le_plus_minus_comm // /4 width=5/
+]
+qed-.
+
+lemma ldrop_ltpss_dx_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ▶* [d2, e2] K2 →
+ d2 ≤ d1 → d1 ≤ d2 + e2 →
+ ∃∃L2. L1 ▶* [d2, e1 + e2] L2 &
+ ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
+ >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
+ lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
+ elim (IHLK1 … HK12 H1 H2) -K1 -H2
+ lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
+ elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
+ [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
+ | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 // /2 width=1/
+ >plus_minus // >commutative_plus /4 width=5/
+ | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // ] /2 width=1/
+ >commutative_plus /3 width=5/
+ ]
+]
+qed-.
+
+lemma ldrop_ltpss_dx_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
+ ∃∃L2. L1 ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
+ elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
+ elim (IHLK1 … HK12 H) -K1
+ elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
+ >(ltpss_dx_inv_refl_O2 … HL12) -L1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
+ elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
+ [ -IHLK1 -Hd21 <(ltpss_dx_inv_refl_O2 … H) -X /3 width=5/
+ | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 /2 width=1/ /3 width=5/
+ | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
+ ]
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fsup_tps_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶[0,|L2|] U2 →
+ ∃∃L,U1. L1 ▶*[0,|L|] L & L ⊢ T1 ▶[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+elim (le_or_ge d (|K|)) #Hd
+[ elim (ldrop_ltpss_dx_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tps_lift_be … HT1 … HL2K … HTU1 HTU … Hd) // -HT1 -HTU1 #HU1
+| elim (ldrop_ltpss_dx_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tps_lift_le … HT1 … HL2K … HTU1 HTU Hd) -HT1 -HTU1 #HU1
+]
+lapply (ltpss_dx_weak_full … HL12) -HL12 #HL12
+lapply (tps_weak_full … HU1) -HU1 #HU1
+@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_tpss.ma".
+
+(* DX PARTIAL UNFOLD ON LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_dx_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
+ L1 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind … H) -U2 /2 width=3/
+#U #U2 #_ #HU2 * #X2 #HTX2 #HUX2
+elim (ltpss_dx_tps_conf … HU2 … HL01) -L0 #X1 #HUX1 #HU2X1
+elim (tpss_strip_eq … HUX2 … HUX1) -U #X #HX2 #HX1
+lapply (tpss_trans_eq … HU2X1 … HX1) -X1 /3 width=3/
+qed.
+
+lemma ltpss_dx_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
+ L1 ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L0 ⊢ T ▶* [d1, e1] U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
+[ /2 width=3/
+| #U #U2 #_ #HU2 * #T #HT2 #HTU
+ elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
+ elim (ltpss_dx_tps_trans … HTU … HL10) -HTU -HL10 #X #HTX #HXU
+ lapply (tpss_trans_eq … HXU HU2) -U /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
+ ∀L0. L0 ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
+#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
+[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
+ elim (tpss_inv_atom1 … H) -H // *
+ #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
+ lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
+ elim (ltpss_dx_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
+ elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
+ lapply (tpss_fwd_tw … HV01) #H2
+ lapply (transitive_le (♯{K1} + ♯{V0}) … H1) -H1 /2 width=1/ -H2 #H
+ lapply (tpss_trans_eq … HV01 HV12) -V1 #HV02
+ lapply (IH … HV02 … HK01) -IH -HV02 -HK01
+ [ normalize /2 width=1/ | /2 width=6/ ]
+| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
+ [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
+ lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
+ lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
+ lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
+ | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (IH … HVW2 … HL0) -HVW2 //
+ lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ▶* [d, e] L1 →
+ L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem ltpss_dx_conf: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
+ ∃∃L. L1 ▶* [d2, e2] L & L2 ▶* [d1, e1] L.
+#L0 @(f_ind … lw … L0) -L0 #n #IH *
+[ #_ #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 -n
+ >(ltpss_dx_inv_atom1 … H1) -L1
+ >(ltpss_dx_inv_atom1 … H2) -L2 /2 width=3/
+| #K0 #I0 #V0 #Hn #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 destruct
+ elim (eq_or_gt d1) #Hd1 [ elim (eq_or_gt e1) #He1 ] destruct
+ [ lapply (ltpss_dx_inv_refl_O2 … H1) -H1 #H1
+ | elim (ltpss_dx_inv_tpss21 … H1 … He1) -H1 #K1 #V1 #HK01 #HV01 #H1
+ | elim (ltpss_dx_inv_tpss11 … H1 … Hd1) -H1 #K1 #V1 #HK01 #HV01 #H1
+ ] destruct
+ elim (eq_or_gt d2) #Hd2 [1,3,5: elim (eq_or_gt e2) #He2 ] destruct
+ [1,3,5: lapply (ltpss_dx_inv_refl_O2 … H2) -H2 #H2
+ |2,4,6: elim (ltpss_dx_inv_tpss21 … H2 … He2) -H2 #K2 #V2 #HK02 #HV02 #H2
+ |7,8,9: elim (ltpss_dx_inv_tpss11 … H2 … Hd2) -H2 #K2 #V2 #HK02 #HV02 #H2
+ ] destruct
+ [1: -IH /2 width=3/
+ |2,3,4,7: -IH /3 width=5/
+ |5,6,8,9:
+ elim (IH … HK01 … HK02) // -K0 #K #HK1 #HK2
+ elim (ltpss_dx_tpss_conf … HV01 … HK1) -HV01 #W1 #HW1 #HVW1
+ elim (ltpss_dx_tpss_conf … HV02 … HK2) -HV02 #W2 #HW2 #HVW2
+ elim (tpss_conf_eq … HW1 … HW2) -V0 #V #HW1 #HW2
+ lapply (tpss_trans_eq … HVW1 HW1) -W1
+ lapply (tpss_trans_eq … HVW2 HW2) -W2 /3 width=5/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_dx_ldrop.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial substitution on terms **********************)
+
+lemma ltpss_dx_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
+
+lemma ltpss_dx_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_dx_tps.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial unfold on terms ****************************)
+
+lemma ltpss_dx_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_dx_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+(* Basic_1: was: subst1_subst1_back *)
+lemma ltpss_dx_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L1 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_dx_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_dx_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2
+ elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
+]
+qed.
+
+lemma ltpss_dx_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_dx_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+(* Basic_1: was: subst1_subst1 *)
+lemma ltpss_dx_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L0 ⊢ T ▶* [d1, e1] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_dx_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_dx_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2
+ elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 break ⊢ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarSn $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+inductive ltpss_sn: nat → nat → relation lenv ≝
+| ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆)
+| ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e.
+ ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 →
+ ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+| ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e.
+ ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 →
+ ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "parallel unfold (local environment, sn variant)"
+ 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
+[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
+ >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
+]
+qed.
+
+lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2.
+/2 width=4/ qed-.
+
+fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2.
+ L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e →
+ ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K1 #V1 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d →
+ ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
+
+fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2.
+ L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K2 #V2 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e.
+ L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 →
+ L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e.
+ L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 →
+ L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) /2 width=1/
+qed.
+
+lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) /2 width=1/
+qed.
+
+lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L.
+#L elim L -L //
+#L #I #V #IHL * /2 width=1/ * /2 width=1/
+qed.
+
+lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
+ lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
+ >plus_plus_comm_23 in Hde12; #Hde12
+ elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
+ [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
+ lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
+ | -Hd21 normalize in Hde12;
+ lapply (lt_to_le_to_lt 0 … Hde12) // #He2
+ lapply (le_plus_to_minus_r … Hde12) -Hde12
+ /3 width=5 by ltpss_sn_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
+ ]
+]
+qed.
+
+lemma ltpss_sn_weak_full: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [0, |L1|] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+// /3 width=2/ /3 width=3/
+qed.
+
+fact ltpss_sn_append_le_aux: ∀K1,K2,d,x. K1 ⊢ ▶* [d, x] K2 → x = |K1| - d →
+ ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ⊢ ▶* [d, x + e] L2 @@ K2.
+#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
+[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct //
+| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
+ lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
+ /4 width=5 by ltpss_sn_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_plus_to_le_r … H2) -H2 #Hd
+ /4 width=5 by ltpss_sn_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+]
+qed-.
+
+lemma ltpss_sn_append_le: ∀K1,K2,d. K1 ⊢ ▶* [d, |K1| - d] K2 →
+ ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ⊢ ▶* [d, |K1| - d + e] L2 @@ K2.
+/2 width=1 by ltpss_sn_append_le_aux/ qed.
+
+lemma ltpss_sn_append_ge: ∀K1,K2,d,e. K1 ⊢ ▶* [d, e] K2 →
+ ∀L1,L2. L1 ⊢ ▶* [d - |K1|, e] L2 → |K1| ≤ d →
+ L1 @@ K1 ⊢ ▶* [d, e] L2 @@ K2.
+#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
+[ #d #e #L1 #L2 <minus_n_O //
+| #K #I #V #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
+ normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
+ lapply (le_plus_to_le_r … H) -H /3 width=1/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpss_sn_fwd_length: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+normalize //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }.
+
+include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
+include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* alternative definition of ltpss_sn *)
+definition ltpssa: nat → nat → relation lenv ≝
+ λd,e. TC … (ltpss_dx d e).
+
+interpretation "parallel unfold (local environment, sn variant) alternative"
+ 'PSubstStarSnAlt L1 d e L2 = (ltpssa d e L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma ltpssa_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. L1 ⊢ ▶▶* [d, e] L → L ▶* [d, e] L2 → R L → R L2) →
+ ∀L2. L1 ⊢ ▶▶* [d, e] L2 → R L2.
+#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma ltpssa_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. L1 ▶* [d, e] L → L ⊢ ▶▶* [d, e] L2 → R L → R L1) →
+ ∀L1. L1 ⊢ ▶▶* [d, e] L2 → R L1.
+#d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltpssa_refl: ∀L,d,e. L ⊢ ▶▶* [d, e] L.
+/2 width=1/ qed.
+
+lemma ltpssa_tpss2: ∀I,L1,V1,V2,e. L1 ⊢ V1 ▶*[0, e] V2 →
+ ∀L2. L1 ⊢ ▶▶* [0, e] L2 →
+ L1.ⓑ{I}V1 ⊢ ▶▶* [O, e + 1] L2.ⓑ{I}V2.
+#I #L1 #V1 #V2 #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
+[ /3 width=1/ | /3 width=5/ ]
+qed.
+
+lemma ltpssa_tpss1: ∀I,L1,V1,V2,d,e. L1 ⊢ V1 ▶*[d, e] V2 →
+ ∀L2. L1 ⊢ ▶▶* [d, e] L2 →
+ L1.ⓑ{I}V1 ⊢ ▶▶* [d + 1, e] L2.ⓑ{I}V2.
+#I #L1 #V1 #V2 #d #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
+[ /3 width=1/ | /3 width=5/ ]
+qed.
+
+lemma ltpss_sn_ltpssa: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶▶* [d, e] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // /2 width=1/
+qed.
+
+lemma ltpss_sn_dx_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
+ ∀L2. L ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+#L1 #L #d #e #H elim H -L1 -L -d -e
+[ #d #e #X #H
+ lapply (ltpss_dx_inv_atom1 … H) -H #H destruct //
+| #L #I #V #X #H
+ lapply (ltpss_dx_inv_refl_O2 … H) -H #H destruct //
+| #L1 #L #I #V1 #V #e #_ #HV1 #IHL1 #X #H
+ elim (ltpss_dx_inv_tpss21 … H ?) -H // <minus_plus_m_m
+ #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (IHL1 … HL2) -L #HL12
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
+ lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
+| #L1 #L #I #V1 #V #d #e #_ #HV1 #IHL1 #X #H
+ elim (ltpss_dx_inv_tpss11 … H ?) -H // <minus_plus_m_m
+ #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (IHL1 … HL2) -L #HL12
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
+ lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_sn_trans_eq: ∀L1,L,d,e. L1 ▶* [d, e] L →
+ ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+/3 width=3/ qed.
+
+lemma ltpssa_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
+ ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
+/3 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ltpssa_ltpss_sn: ∀L1,L2,d,e. L1 ⊢ ▶▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+#L1 #L2 #d #e #H @(ltpssa_ind … H) -L2 // /2 width=3/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_sn_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
+ ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
+#L0 #L1 #d1 #e1 #H #L2 #d2 #e2 #HL02
+lapply (ltpss_sn_ltpssa … H) -H #HL01
+elim (ltpssa_strip … HL01 … HL02) -L0
+/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
+qed.
+
+(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
+lemma ltpss_sn_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
+ L0 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #HTU2 #L1 #d1 #e1 #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L1 /2 width=3/ -HTU2
+#L #L1 #H #HL1 * #T #HT2 #HU2T
+lapply (ltpssa_ltpss_sn … H) -H #HL0
+lapply (ltpss_sn_dx_trans_eq … HL0 … HL1) -HL0 #HL01
+elim (ltpss_dx_tpss_conf … HT2 … HL1) -HT2 -HL1 #T0 #HT20 #HT0
+lapply (ltpss_sn_tpss_trans_eq … HT0 … HL01) -HT0 -HL01 #HT0
+lapply (tpss_trans_eq … HU2T HT0) -T /2 width=3/
+qed.
+
+(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
+lemma ltpss_sn_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
+ L1 ⊢ ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L1 ⊢ T ▶* [d1, e1] U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #H #HTU2
+lapply (ltpss_sn_ltpssa … H) -H #HL10
+@(ltpssa_ind_dx … HL10) -L1 /2 width=3/ -HTU2
+#L1 #L #HL1 #_ * #T #HT2 #HTU2
+elim (ltpss_dx_tpss_trans_down … HL1 HT2) -HT2 // #T0 #HT20 #HT0 -Hde2d1
+lapply (tpss_trans_eq … HT0 HTU2) -T #HT0U2
+lapply (ltpss_dx_tpss_trans_eq … HT0U2 … HL1) -HT0U2 -HL1 /2 width=3/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem ltpssa_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ⊢ ▶▶* [d2, e2] L2 →
+ ∃∃L. L1 ⊢ ▶▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
+/3 width=3/ qed.
+
+(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
+theorem ltpss_sn_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ⊢ ▶* [d2, e2] L2 →
+ ∃∃L. L1 ⊢ ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
+#L0 #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2
+lapply (ltpss_sn_ltpssa … H1) -H1 #HL01
+lapply (ltpss_sn_ltpssa … H2) -H2 #HL02
+elim (ltpssa_conf … HL01 … HL02) -L0
+/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fsup.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/ltpss_sn.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma ltpss_sn_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+]
+qed.
+
+lemma ltpss_sn_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
+]
+qed.
+
+lemma ltpss_sn_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L2 ⊢ ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_sn_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L ⊢ ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_sn_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 ⊢ ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ltpss_sn_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L ⊢ ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ldrop_ltpss_sn_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d1 ≤ d2 →
+ ∃∃L2. L1 ⊢ ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
+ elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
+ elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
+ elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_ge … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // -Hd12
+ <le_plus_minus_comm // /4 width=5/
+]
+qed-.
+
+lemma ldrop_ltpss_sn_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 →
+ d2 ≤ d1 → d1 ≤ d2 + e2 →
+ ∃∃L2. L1 ⊢ ▶* [d2, e1 + e2] L2 &
+ ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
+ >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
+ lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
+ elim (IHLK1 … HK12 H1 H2) -K1 -H2
+ lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
+ elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
+ [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
+ | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // /2 width=1/
+ >plus_minus // >commutative_plus /4 width=5/
+ | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // ] /2 width=1/
+ >commutative_plus /3 width=5/
+ ]
+]
+qed-.
+
+lemma ldrop_ltpss_sn_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
+ ∃∃L2. L1 ⊢ ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
+ elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
+ elim (IHLK1 … HK12 H) -K1
+ elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
+ >(ltpss_sn_inv_refl_O2 … HL12) -L1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
+ elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
+ [ -IHLK1 -Hd21 <(ltpss_sn_inv_refl_O2 … H) -X /3 width=5/
+ | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 /2 width=1/ /3 width=5/
+ | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
+ ]
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fsup_tpss_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶*[0,|L2|] U2 →
+ ∃∃L,U1. L1 ⊢ ▶*[0,|L1|] L & L ⊢ T1 ▶*[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (ltpss_sn_fwd_length … HK1) #H >H in HK1; -H #HK1
+elim (le_or_ge d (|K|)) #Hd
+[ elim (ldrop_ltpss_sn_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tpss_lift_be … HT1 … Hd HL2K HTU1 … HTU) // -HT1 -HTU1 #HU1
+| elim (ldrop_ltpss_sn_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tpss_lift_le … HT1 … Hd HL2K HTU1 … HTU) -HT1 -HTU1 #HU1
+]
+lapply (ltpss_sn_weak_full … HL12) -HL12 #HL12
+lapply (tpss_weak_full … HU1) -HU1 #HU1
+@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_sn_tpss.ma".
+
+(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_sn_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
+ ∀L0. L0 ⊢ ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
+#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
+[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
+ elim (tpss_inv_atom1 … H) -H // *
+ #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
+ lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
+ elim (ltpss_sn_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
+ elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
+ lapply (IH … HV12 … HK01) -IH -HV12 -HK01 [ normalize /2 width=1/ ] #HV12
+ lapply (tpss_trans_eq … HV01 HV12) -V1 /2 width=6/
+| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
+ [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
+ lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
+ lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
+ lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
+ | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (IH … HVW2 … HL0) -HVW2 //
+ lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
+]
+qed.
+
+lemma ltpss_sn_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ⊢ ▶* [d, e] L1 →
+ L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem ltpss_sn_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
+ ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+#L1 #L #d #e #H elim H -L1 -L -d -e //
+[ #L1 #L #I #V1 #V #e #HL1 #HV1 #IHL1 #X #H
+ elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
+ lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
+| #L1 #L #I #V1 #V #d #e #HL1 #HV1 #IHL1 #X #H
+ elim (ltpss_sn_inv_tpss11 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
+ lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
+]
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶ [d, e] T →
+ ∃∃L2,T2. L @@ L1 ⊢ ▶* [d + |L1|, e] L @@ L2 &
+ L @@ L2 ⊢ T1 ▶ [d + |L1|, e] T2 &
+ T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #d #e #HT1
+ @ex3_2_intro [3: // |5: // |4: normalize /2 width=1/ |1,2: skip ] (**) (* /2 width=4/ does not work *)
+| #I #L1 #V1 #IH #L #T1 #T #d #e >shift_append_assoc #H
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ <append_assoc >append_length <associative_plus
+ lapply (ltpss_sn_trans_eq (L.ⓑ{I}V1@@L1) … HL12) -HL12 /3 width=1/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_ldrop.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial substitution on terms **********************)
+
+lemma ltpss_sn_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
+
+lemma ltpss_sn_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_tps.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial unfold on terms ****************************)
+
+lemma ltpss_sn_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_sn_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+lemma ltpss_sn_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L0 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_sn_ldrop_conf_le … HL01 … HLK0 ?) /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // >minus_plus <plus_minus_m_m // /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_sn_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H
+ lapply (tpss_lsubr_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2
+ elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
+]
+qed.
+
+lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L1 ⊢ T ▶* [d1, e1] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_sn_ldrop_trans_be … HL10 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H
+ lapply (tpss_lsubr_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2
+ elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/dxprs_ltpss_dx.ma".
+include "basic_2/equivalence/cpcs_ltpss_dx.ma".
+include "basic_2/dynamic/snv_lift.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma snv_ltpss_dx_tpss_conf: ∀h,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+#h #g #L1 #T1 #H elim H -L1 -T1
+[ #L1 #k #L2 #d #e #_ #X #H
+ >(tpss_inv_sort1 … H) -X //
+| #I #L1 #K1 #V1 #i #HLK1 #HV1 #IHV1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ | elim (lt_or_ge i (d + e)) #Hide ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 …) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H …) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IHV1 … HK12 … HV12) -IHV1 -HK12 -HV12 /2 width=5/
+ | elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 …) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H …) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IHV1 … HK12 … HV12) -IHV1 -HK12 -HV12 /2 width=5/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=5/
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 …) -L1 // /2 width=2/ #X #H1 #H2
+ elim (ltpss_dx_inv_tpss21 … H1 …) -H1 /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … H2 … HLK2) -H2 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
+ lapply (IHV1 … HK12 … HV1W2) -IHV1 -HK12 -HV1W2 /2 width=7/
+ ]
+| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IHV1 … HL12 … HV12) -IHV1 #HV2
+ lapply (IHT1 … HT12) -IHT1 -HT12 /2 width=1/
+| #a #L1 #V1 #W1 #W0 #T1 #U1 #l #_ #_ #HVW1 #HW10 #HTU1 #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IHV1 … HL12 … HV12) -IHV1 #HV2
+ lapply (IHT1 … HT12) -IHT1 /2 width=1/ #HT2
+ elim (ssta_ltpss_dx_tpss_conf … HVW1 … HL12 … HV12) -V1 #W2 #HVW2 #HW12
+ elim (dxprs_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -T1 #X #HTU2 #H
+ elim (tpss_inv_bind1 … H) -H #W #U2 #HW0 #_ #H -U1 destruct
+ elim (cprs_ltpss_dx_tpss2_conf … HW10 … HL12 … HW12 … HW0) -L1 -W1 -W0 #W0 #HW20 #HW0
+ lapply (dxprs_strap1 … (ⓛ{a}W0.U2) HTU2 ?) -HTU2 /3 width=3/ -HW0 /2 width=8/
+| #L1 #W1 #T1 #U1 #l #_ #_ #HTU1 #HUW1 #IHW1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (IHW1 … HL12 … HW12) -IHW1 #HW2
+ lapply (IHT1 … HL12 … HT12) -IHT1 #HT2
+ elim (ssta_ltpss_dx_tpss_conf … HTU1 … HL12 … HT12) -T1 #U2 #HTU2 #HU12
+ lapply (cpcs_ltpss_dx_tpss2_conf … HL12 … HUW1 … HU12 … HW12) -L1 -W1 -U1 /2 width=4/
+]
+qed-.
+
+lemma snv_ltpss_dx_conf: ∀h,g,L1,T. ⦃h, L1⦄ ⊢ T ¡[g] →
+ ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T ¡[g].
+#h #g #L1 #T #HT #L2 #d #e #HL12
+@(snv_ltpss_dx_tpss_conf … HT … HL12) //
+qed-.
+
+lemma snv_tpss_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊢ T1 ¡[g] →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 ¡[g].
+#h #g #L #T1 #HT1 #T2 #d #e #HT12
+@(snv_ltpss_dx_tpss_conf … HT1 … HT12) //
+qed-.
+
+lemma snv_tps_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊢ T1 ¡[g] →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 ¡[g].
+#h #g #L #T1 #HT1 #T2 #d #e #HT12
+@(snv_tpss_conf … HT1 T2) /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/snv_ltpss_dx.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma snv_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T. ⦃h, L1⦄ ⊢ T ¡[g] → ⦃h, L2⦄ ⊢ T ¡[g].
+#h #g #L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 #T1 #HT1
+lapply (IHL1 … HT1) -IHL1 -HT1 #HT1
+lapply (snv_ltpss_dx_conf … HT1 … HL2) -HT1 -HL2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_tpss.ma".
+include "basic_2/static/ssta_lift.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma ssta_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
+[ #L1 #k1 #l1 #Hkl1 #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H /3 width=3/
+| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
+ elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (ssta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
+ lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HWV1 ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
+ ]
+| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
+| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
+| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
+]
+qed.
+
+lemma ssta_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=5/ qed.
+
+lemma ssta_ltpss_dx_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ssta_tpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ssta_tps_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/static/ssta_ltpss_dx.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma ssta_ltpss_sn_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T #U1 #l #HTU1 #L2 #d #e #HL12
+lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
+@(ltpssa_ind … HL12) -L2 [ /2 width=3/ ] -HTU1
+#L #L2 #HL1 #HL2 * #U #HTU #HU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+elim (ssta_ltpss_dx_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
+lapply (ltpss_dx_tpss_trans_eq … HU2 … HL2) -HU2 -HL2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL1) -HU2 -HL1 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma ssta_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #l #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (ssta_ltpss_sn_conf … HTU1 … HL12) -HTU1 #U #HT1U #HU1
+elim (ssta_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma ssta_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L1 ⊢ U1 ▶* [d, e] U2.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta_ltpss.ma".
+include "basic_2/unwind/sstas.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************)
+
+(* Properties about parallel unfold *****************************************)
+
+lemma sstas_ltpss_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #H @(sstas_ind_alt … H) -T1
+[ #T1 #HUT1 #L2 #d #e #HL12 #U2 #HU12
+ elim (ssta_ltpss_tpss_conf … HUT1 … HL12 … HU12) -HUT1 -HL12 /3 width=3/
+| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
+ elim (ssta_ltpss_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
+ elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
+]
+qed.
+
+lemma sstas_ltpss_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=5/ qed.
+
+lemma sstas_ltpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta_ltpss_dx.ma".
+include "basic_2/unwind/sstas.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma sstas_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
+#T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
+elim (ssta_ltpss_dx_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
+elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
+qed.
+
+lemma sstas_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=7 by step, sstas_ltpss_dx_tpss_conf/ qed. (**) (* auto fails without trace *)
+
+lemma sstas_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/unwind/sstas_ltpss_dx.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ************************)
+
+(* Properties about sn parallel unfold *****************************************)
+
+lemma sstas_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∃∃U2. L1 ⊢ U1 ▶* [d, e] U2 & ⦃h, L2⦄ ⊢ T •*[g] U2.
+#h #g #L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
+#L #L2 #HL1 #HL2 #IHL1 #T #U1 #H1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
+elim (IHL1 … H1) -IHL1 -H1 #U #HU1 #HTU
+elim (sstas_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #H2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma sstas_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (sstas_ltpss_sn_conf … HL12 … HTU1) -HTU1 #U #HU1 #HT1U
+elim (sstas_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma sstas_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L1 ⊢ U1 ▶* [d, e] U2.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_tpss.ma".
+include "basic_2/static/sta_lift.ma".
+
+(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
+
+(* Properties about parallel unfold *****************************************)
+
+lemma sta_ltpss_tpss_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 • U2 & L2 ⊢ U1 ▶* [d, e] U2.
+#h #L1 #T1 #U #H elim H -L1 -T1 -U
+[ #L1 #k1 #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H /2 width=3/
+| #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
+ [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
+ [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
+ elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (sta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
+ lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HWV1 ]
+ [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
+ [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
+ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
+ ]
+| #I #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
+| #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
+| #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
+]
+qed.
+
+lemma sta_ltpss_tps_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 • U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=5/ qed.
+
+lemma sta_ltpss_conf: ∀h,L1,T,U1. ⦃h, L1⦄ ⊢ T • U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T • U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sta_tpss_conf: ∀h,L,T1,U1. ⦃h, L⦄ ⊢ T1 • U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 • U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sta_tps_conf: ∀h,L,T1,U1. ⦃h, L⦄ ⊢ T1 • U1 →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 • U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 ➡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRed $T1 $T2 }.
+
+include "basic_2/substitution/tps.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Basic_1: includes: pr0_delta1 *)
+inductive tpr: relation term ≝
+| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I})
+| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
+ tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+| tpr_beta : ∀a,V1,V2,W,T1,T2.
+ tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
+| tpr_delta: ∀a,I,V1,V2,T1,T,T2.
+ tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 →
+ tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2.
+ tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
+ tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2)
+| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2
+| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2
+.
+
+interpretation
+ "context-free parallel reduction (term)"
+ 'PRed T1 T2 = (tpr T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
+/2 width=3/ qed.
+
+(* Basic_1: was by definition: pr0_refl *)
+lemma tpr_refl: reflexive … tpr.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}.
+#U1 #U2 * -U1 -U2
+[ //
+| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
+| #V #T1 #T #T2 #_ #_ #k #H destruct
+| #V #T1 #T2 #_ #k #H destruct
+]
+qed.
+
+(* Basic_1: was: pr0_gen_sort pr0_gen_lref *)
+lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}.
+/2 width=3/ qed-.
+
+fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+ (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
+ ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+#U1 #U2 * -U1 -U2
+[ #J #a #I #V #T #H destruct
+| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct
+| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct
+| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/
+| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct
+| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/
+| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct
+]
+qed.
+
+lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 →
+ (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
+ ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr.
+/2 width=3/ qed-.
+
+(* Basic_1: was pr0_gen_abbr *)
+lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 →
+ (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
+ ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 &
+ U2 = ⓓ{a}V2. T2
+ ) ∨
+ ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
+#a #V1 #T1 #U2 #H
+elim (tpr_inv_bind1 … H) -H * /3 width=7/
+qed-.
+
+fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2 &
+ I = Appl
+ | (U0 ➡ U2 ∧ I = Cast).
+#U1 #U2 * -U1 -U2
+[ #I #J #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/
+| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/
+| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct
+| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
+]
+qed.
+
+lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2 &
+ I = Appl
+ | (U0 ➡ U2 ∧ I = Cast).
+/2 width=3/ qed-.
+
+(* Basic_1: was pr0_gen_appl *)
+lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2
+ | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2.
+#V1 #U0 #U2 #H
+elim (tpr_inv_flat1 … H) -H *
+/3 width=5/ /3 width=9/ /3 width=13/
+#_ #H destruct
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U = ⓐV2. T2.
+#V1 #T1 #U #H #HT1
+elim (tpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Basic_1: was: pr0_gen_cast *)
+lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 →
+ (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2)
+ ∨ T1 ➡ U2.
+#V1 #T1 #U2 #H
+elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2
+[ #_ #_ #_ #_ #H destruct
+| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
+ | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
+#T1 #T2 * -T1 -T2
+[ #I #i #H destruct /2 width=1/
+| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
+| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct
+ lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/
+| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/
+]
+qed.
+
+lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
+ | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
+/2 width=3/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tpr_fwd_bind1_minus: ∀I,V1,T1,T. -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #V1 #T1 #T #H #b elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct /3 width=4/
+| #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #T1 #X
+ >shift_append_assoc normalize #H
+ elim (tpr_inv_bind1 … H) -H *
+ [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct
+ elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct
+ elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct
+ >append_length >HL1 >HL2 -L1 -L
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
+
+(* Basic_1: removed theorems 3:
+ pr0_subst0_back pr0_subst0_fwd pr0_subst0
+*)
+(* Basic_1: removed local theorems: 1: pr0_delta_tau *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_lift.ma".
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr0_lift *)
+lemma tpr_lift: t_liftable tpr.
+#T1 #T2 #H elim H -T1 -T2
+[ * #i #U1 #d #e #HU1 #U2 #HU2
+ lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
+ [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
+ | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
+ | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
+ ]
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
+ elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
+ elim (lift_total T (d + 1) e) #U #HTU
+ @tpr_delta
+ [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *)
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
+ elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
+ elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2
+ elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V
+ elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/
+| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/
+]
+qed.
+
+(* Basic_1: was: pr0_gen_lift *)
+lemma tpr_inv_lift1: t_deliftable_sn tpr.
+#T1 #T2 #H elim H -T1 -T2
+[ * #i #X #d #e #HX
+ [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/
+ | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/
+ ]
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
+ elim (IHV12 … HV01) -V1
+ elim (IHT12 … HT01) -T1 /3 width=5/
+| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HV01) -V1
+ elim (IHT12 … HT01) -T1 /3 width=5/
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX
+ elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct
+ elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
+ elim (IHT1 … HUT1) -T1 #U #HUT #HU1
+ elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2
+ @ex2_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *)
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
+ elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
+ elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
+ elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
+ @ex2_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX
+ elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct
+ elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31
+ elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/
+| #V #T1 #T2 #_ #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
+ elim (IHT12 … HT01) -T1 /3 width=3/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr0_gen_abst *)
+lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
+#a #V1 #T1 #U2 #H elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T #T2 #HV12 #HT1 #HT2
+ lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /2 width=5/
+| #T2 #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma tpr_fwd_abst1: ∀a,V1,T1,U2. ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
+ ∃∃V2,T2. ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
+ U2 = ⓛ{a}V2.T2.
+#a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T #T2 #HV12 #HT1 #HT2
+ lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /3 width=4/
+| #T2 #_ #_ #_ #H destruct
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Confluence lemmas ********************************************************)
+
+fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
+/2 width=3/ qed.
+
+fact tpr_conf_flat_flat:
+ ∀I,V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0.
+#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/
+qed.
+
+fact tpr_conf_flat_beta:
+ ∀a,V0,V1,T1,V2,W0,U0,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 →
+ U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X.
+#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
+elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/
+qed.
+
+(* Basic-1: was:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact tpr_conf_flat_theta:
+ ∀a,V0,V1,T1,V2,V,W0,W2,U0,U2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V →
+ W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X.
+#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2
+elim (lift_total VV 0 1) #VVV #HVV
+lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
+elim (tpr_inv_abbr1 … H) -H *
+(* case 1: delta *)
+[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct
+ elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
+ elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
+ elim (tpr_tps_conf_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1
+ @ex2_intro
+ [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
+ |1:skip
+ |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
+ ] (**) (* /5 width=14/ is too slow *)
+(* case 3: zeta *)
+| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1
+ elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2
+ elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct
+ @(ex2_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *)
+]
+qed.
+
+fact tpr_conf_flat_cast:
+ ∀X2,V0,V1,T0,T1. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 →
+ ∃∃X. ⓝV1. T1 ➡ X & X2 ➡ X.
+#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/
+qed.
+
+fact tpr_conf_beta_beta:
+ ∀a. ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X.
+#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/
+qed.
+
+(* Basic_1: was: pr0_cong_delta pr0_delta_delta *)
+fact tpr_conf_delta_delta:
+ ∀a,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
+ ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 →
+ ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 →
+ ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X.
+#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
+elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2
+elim (tpr_tps_conf_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1
+elim (tpr_tps_conf_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
+elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2
+@ex2_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
+qed.
+
+fact tpr_conf_delta_zeta:
+ ∀X2,V0,V1,T0,T1,TT1,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 →
+ T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 →
+ ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X.
+#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2
+elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2
+elim (tpr_tps_conf_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT
+elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2
+lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /3 width=3/
+qed.
+
+(* Basic_1: was: pr0_upsilon_upsilon *)
+fact tpr_conf_theta_theta:
+ ∀a,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 →
+ ⇧[O, 1] V1 ≡ VV1 → ⇧[O, 1] V2 ≡ VV2 →
+ ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X.
+#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
+elim (lift_total V 0 1) #VV #HVV
+lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
+lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2
+@ex2_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
+qed.
+
+fact tpr_conf_zeta_zeta:
+ ∀V0:term. ∀X2,TT0,T0,T1,TT2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{TT0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 →
+ TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 →
+ ∃∃X. T1 ➡ X & X2 ➡ X.
+#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2
+elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2
+elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10
+elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2
+lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/
+qed.
+
+fact tpr_conf_tau_tau:
+ ∀V0,T0:term. ∀X2,T1. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ T0 ➡ T1 → T0 ➡ X2 →
+ ∃∃X. T1 ➡ X & X2 ➡ X.
+#V0 #T0 #X2 #T1 #IH #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/
+qed.
+
+(* Confluence ***************************************************************)
+
+(* Basic_1: was: pr0_confluence *)
+theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃T. T1 ➡ T & T2 ➡ T.
+#T0 @(f_ind … tw … T0) -T0 #n #IH *
+[ #I #_ #X1 #X2 #H1 #H2 -n
+ >(tpr_inv_atom1 … H1) -X1
+ >(tpr_inv_atom1 … H2) -X2
+(* case 1: atom, atom *)
+ //
+| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2
+ [ elim (tpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (tpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #U2 #HV02 #HT02 #HTU2 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+(* case 2: delta, delta *)
+ [ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
+(* case 3: zeta, delta (repeated) *)
+ | @ex2_commute /3 width=10 by tpr_conf_delta_zeta/
+(* case 4: delta, zeta *)
+ | /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
+(* case 5: zeta, zeta *)
+ | /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
+ ]
+ | elim (tpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #b1 #V1 #W1 #U1 #T1 #HV01 #HUT1 #H11 #H12 #H13
+ | #b1 #V1 #Y1 #W1 #Z1 #U1 #T1 #HV01 #HWZ1 #HUT1 #HVY1 #H11 #H12 #H13
+ | #HX1 #H1
+ ]
+ elim (tpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #b2 #V2 #W2 #U2 #T2 #HV02 #HUT2 #H21 #H22 #H23
+ |3,7,11,15: #b2 #V2 #Y2 #W2 #Z2 #U2 #T2 #HV02 #HWZ2 #HUT2 #HVY2 #H21 #H22 #H23
+ |4,8,12,16: #HX2 #H2
+ ] destruct
+(* case 6: flat, flat *)
+ [ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
+(* case 7: beta, flat (repeated) *)
+ | @ex2_commute /3 width=8 by tpr_conf_flat_beta/
+(* case 8: theta, flat (repeated) *)
+ | @ex2_commute /3 width=11 by tpr_conf_flat_theta/
+(* case 9: tau, flat (repeated) *)
+ | @ex2_commute /3 width=6 by tpr_conf_flat_cast/
+(* case 10: flat, beta *)
+ | /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 11: beta, beta *)
+ | /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 12: flat, theta *)
+ | /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
+(* case 13: theta, theta *)
+ | /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
+(* case 14: flat, tau *)
+ | /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
+(* case 15: tau, tau *)
+ | /3 width=5 by tpr_conf_tau_tau/
+ ]
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 ➡ * break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRedStar $T1 $T2 }.
+
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
+
+(* Basic_1: includes: pr1_pr0 *)
+definition tprs: relation term ≝ TC … tpr.
+
+interpretation "context-free parallel computation (term)"
+ 'PRedStar T1 T2 = (tprs T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma tprs_ind: ∀T1. ∀R:predicate term. R T1 →
+ (∀T,T2. T1 ➡* T → T ➡ T2 → R T → R T2) →
+ ∀T2. T1 ➡* T2 → R T2.
+#T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma tprs_ind_dx: ∀T2. ∀R:predicate term. R T2 →
+ (∀T1,T. T1 ➡ T → T ➡* T2 → R T → R T1) →
+ ∀T1. T1 ➡* T2 → R T1.
+#T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tprs_refl: reflexive … tprs.
+/2 width=1/ qed.
+
+lemma tpr_tprs: ∀T1,T2. T1 ➡ T2 → T2 ➡* T2.
+/2 width=1/ qed.
+
+lemma tprs_strap1: ∀T1,T,T2. T1 ➡* T → T ➡ T2 → T1 ➡* T2.
+/2 width=3/ qed.
+
+lemma tprs_strap2: ∀T1,T,T2. T1 ➡ T → T ➡* T2 → T1 ➡* T2.
+/2 width=3/ qed.
+
+(* Basic_1: was only: pr1_head_1 *)
+lemma tprs_pair_sn: ∀I,T1,T2. T1 ➡ T2 → ∀V1,V2. V1 ➡* V2 →
+ ②{I} V1. T1 ➡* ②{I} V2. T2.
+* [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind … H) -V2
+[1,3: /3 width=1/
+|2,4: #V #V2 #_ #HV2 #IHV1
+ @(tprs_strap1 … IHV1) -IHV1 /2 width=1/
+]
+qed.
+
+(* Basic_1: was only: pr1_head_2 *)
+lemma tprs_pair_dx: ∀I,V1,V2. V1 ➡ V2 → ∀T1,T2. T1 ➡* T2 →
+ ②{I} V1. T1 ➡* ②{I} V2. T2.
+* [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind … H) -T2
+[1,3: /3 width=1/
+|2,4: #T #T2 #_ #HT2 #IHT1
+ @(tprs_strap1 … IHT1) -IHT1 /2 width=1/
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tprs_inv_atom1: ∀U2,k. ⋆k ➡* U2 → U2 = ⋆k.
+#U2 #k #H @(tprs_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU1 destruct
+>(tpr_inv_atom1 … HU2) -HU2 //
+qed-.
+
+lemma tprs_inv_cast1: ∀W1,T1,U2. ⓝW1.T1 ➡* U2 → T1 ➡* U2 ∨
+ ∃∃W2,T2. W1 ➡* W2 & T1 ➡* T2 & U2 = ⓝW2.T2.
+#W1 #T1 #U2 #H @(tprs_ind … H) -U2 /3 width=5/
+#U #U2 #_ #HU2 * /3 width=3/ *
+#W #T #HW1 #HT1 #H destruct
+elim (tpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
+#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/tpr_lift.ma".
+include "basic_2/computation/tprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tprs_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡* U2 →
+ ∃∃V2,T2. V1 ➡* V2 & T1 ➡* T2 & U2 = ⓛ{a}V2. T2.
+#a #V1 #T1 #U2 #H @(tprs_ind … H) -U2 /2 width=5/
+#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+elim (tpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
+qed-.
+
+lemma tprs_inv_abst: ∀a,V1,V2,T1,T2. ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 →
+ V1 ➡* V2 ∧ T1 ➡* T2.
+#a #V1 #V2 #T1 #T2 #H
+elim (tprs_inv_abst1 … H) -H #V #T #HV1 #HT1 #H destruct /2 width=1/
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Note: this was missing in basic_1 *)
+lemma tprs_lift: t_liftable tprs.
+/3 width=7/ qed.
+
+(* Note: this was missing in basic_1 *)
+lemma tprs_inv_lift1: t_deliftable_sn tprs.
+/3 width=3 by tpr_inv_lift1, t_deliftable_sn_TC/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/tpr_tpr.ma".
+include "basic_2/computation/tprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was: pr1_strip *)
+lemma tprs_strip: ∀T1,T. T ➡* T1 → ∀T2. T ➡ T2 →
+ ∃∃T0. T1 ➡ T0 & T2 ➡* T0.
+/3 width=3 by TC_strip1, tpr_conf/ qed.
+
+(* Main propertis ***********************************************************)
+
+(* Basic_1: was: pr1_confluence *)
+theorem tprs_conf: confluent … tprs.
+/3 width=3/ qed.
+
+(* Basic_1: was: pr1_t *)
+theorem tprs_trans: Transitive … tprs.
+/2 width=3/ qed.
+
+(* Basic_1: was: pr1_comp *)
+lemma tprs_pair: ∀I,V1,V2. V1 ➡* V2 → ∀T1,T2. T1 ➡* T2 →
+ ②{I} V1. T1 ➡* ②{I} V2. T2.
+#I #V1 #V2 #H @(tprs_ind … H) -V2 /2 width=1/
+#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12
+@(tprs_trans … (②{I}V.T2)) /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubst $L $T1 $d $e $T2 }.
+
+include "basic_2/substitution/ldrop_append.ma".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+inductive tps: nat → nat → lenv → relation term ≝
+| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
+| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV → ⇧[0, i + 1] V ≡ W → tps d e L (#i) W
+| tps_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
+ tps d e L V1 V2 → tps (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+ tps d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ tps d e L V1 V2 → tps d e L T1 T2 →
+ tps d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "parallel substritution (term)"
+ 'PSubst L T1 d e T2 = (tps d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tps_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶ [d, e] T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ //
+| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tps_refl: ∀T,L,d,e. L ⊢ T ▶ [d, e] T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
+ ∃∃T2,T. L ⊢ T1 ▶ [d, 1] T2 & ⇧[d, 1] T ≡ T2.
+#K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK /2 width=4/
+ elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i ? ? ?) // /3 width=4/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+ ]
+]
+qed.
+
+lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶ [d1, e1] T2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
+ L ⊢ T1 ▶ [d2, e2] T2.
+#L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1
+[ //
+| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
+ lapply (transitive_le … Hd12 … Hid1) -Hd12 -Hid1 #Hid2
+ lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2 width=4/
+| /4 width=3/
+| /4 width=1/
+]
+qed.
+
+lemma tps_weak_top: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [d, |L| - d] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
+ lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+ lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
+| normalize /2 width=1/
+| /2 width=1/
+]
+qed.
+
+lemma tps_weak_full: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [0, |L|] T2.
+#L #T1 #T2 #d #e #HT12
+lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
+lapply (tps_weak_top … HT12) //
+qed.
+
+lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 ▶ [d, i - d] T & L ⊢ T ▶ [i, d + e - i] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ /2 width=3/
+| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+ elim (lt_or_ge i j)
+ [ -Hide -Hjde
+ >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/
+ | -Hdi -Hdj #Hij
+ lapply (plus_minus_m_m … Hjde) -Hjde /3 width=8/
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi -Hide /3 width=5/
+]
+qed.
+
+lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 ▶ [i, d + e - i] T &
+ L ⊢ T ▶ [d, i - d] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ /2 width=3/
+| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+ elim (lt_or_ge i j)
+ [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=8/
+ | -Hdi -Hdj
+ >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi -Hide /3 width=5/
+]
+qed.
+
+lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 →
+ ∀L. L @@ K ⊢ T1 ▶ [d, e] T2.
+#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
+#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
+lapply (ldrop_fwd_ldrop2_length … HK0) #H
+@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
+[ #L #I #d #e #J #H destruct /2 width=1/
+| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+]
+qed.
+
+lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶ [d, e] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+/2 width=3/ qed-.
+
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶ [d, e] T2 → T2 = ⋆k.
+#L #T2 #k #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶ [d, e] T2 →
+ T2 = #i ∨
+ ∃∃K,V. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2.
+#L #T2 #i #d #e #H
+elim (tps_inv_atom1 … H) -H /2 width=1/
+* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+qed-.
+
+lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶ [d, e] T2 → T2 = §p.
+#L #T2 #p #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
+ L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
+[ #L #k #d #e #a #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
+| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
+]
+qed.
+
+lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶ [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
+ L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+/2 width=3/ qed-.
+
+fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
+ ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶ [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+/2 width=3/ qed-.
+
+fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → e = 0 → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+ lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
+ elim (lt_refl_false … Hdd)
+| /3 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 0] T2 → T1 = T2.
+/2 width=6/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
+/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+qed-.
+
+lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #d #e #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X #d #e
+ >shift_append_assoc normalize #H
+ elim (tps_inv_bind1 … H) -H
+ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic_1: removed theorems 25:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/substitution/tps.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
+#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
+[ //
+| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
+ elim (lt_or_ge i (d+1)) #HiSd
+ [ -Hide1 -HV0
+ lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
+ lapply (ldrop_mono … HLK0 … HLK) #H destruct
+ | -V -Hdi /2 width=4/
+ ]
+| /4 width=3/
+| /3 width=3/
+]
+qed.
+
+lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
+/2 width=3/ qed-.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #HT12 #K #V #HLK
+lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
+lapply (tps_inv_refl_O2 … HT12) -HT12 //
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt + et ≤ d →
+ L ⊢ U1 ▶ [dt, et] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+]
+qed.
+
+lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt ≤ d → d ≤ dt + et →
+ L ⊢ U1 ▶ [dt, et + e] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+ | -Hdti
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+ ]
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
+ ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+]
+qed.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ d ≤ dt →
+ L ⊢ U1 ▶ [dt + e, et] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
+ lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
+ lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ -Hdtd -Hidet
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+ | -Hdti -Hdedet
+ lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+ elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+ ]
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d + e ≤ dt →
+ ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
+ lapply (transitive_le … Hdedt … Hdti) #Hdei
+ elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+ elim (le_inv_plus_l … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (le_inv_plus_l … Hdetd) #_ #Hedt
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
+ <plus_minus // /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
+ L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
+ elim (lift_inv_lref2 … H) -H * #H
+ [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
+ elim (lt_refl_false … H)
+ | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
+ elim (lt_refl_false … H)
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >IHV12 // >IHT12 //
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >IHV12 // >IHT12 //
+]
+qed.
+(*
+ Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le (plus d h) i) ->
+ (EX u1 | (subst0 (minus i h) v u1 u2) &
+ t1 = (lift h d u1)
+ ).
+
+
+ Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le d i) -> (lt i (plus d h)) ->
+ (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
+*)
+lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
+qed.
+
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
+qed.
+
+lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
+elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
+lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
+lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_lift.ma".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶ [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶ [d1, e1] T.
+#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+ elim (tps_inv_lref1 … H) -H
+ [ #HX destruct /3 width=6/
+ | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
+ ]
+| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV01 … HV02) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
+| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T.
+#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+ elim (tps_inv_lref1 … H1) -H1
+ [ #H destruct /3 width=6/
+ | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
+ [ -Hd1 -Hde2
+ lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
+ lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
+ elim (lt_refl_false … H)
+ | -Hd2 -Hde1
+ lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
+ lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
+ elim (lt_refl_false … H)
+ ]
+ ]
+| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02 ?) -T0
+ [ -H #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
+ | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H
+ [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *)
+ ]
+| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5/
+]
+qed.
+
+(* Note: the constant 1 comes from tps_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 →
+ ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e →
+ L ⊢ T1 ▶ [d, e] T2.
+#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e
+[ #L #I #d #e #T2 #H #He
+ elim (tps_inv_atom1 … H) -H
+ [ #H destruct //
+ | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
+ ]
+| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+ lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
+ <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
+| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ lapply (IHT10 … HT02 He) -T0 #HT12
+ lapply (tps_lsubr_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
+| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
+]
+qed.
+
+theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
+#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
+[ /2 width=3/
+| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+ lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
+ lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
+ <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=8/
+| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02 ?) -V0 //
+ elim (IHT10 … HT02 ?) -T0 // /3 width=6/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $L $T1 $d $e $T2 }.
+
+include "basic_2/substitution/tps.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+definition tpss: nat → nat → lenv → relation term ≝
+ λd,e,L. TC … (tps d e L).
+
+interpretation "partial unfold (term)"
+ 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
+ ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
+#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
+ ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
+#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=1/ qed.
+
+lemma tpss_strap1: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+lemma tpss_strap2: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+lemma tpss_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
+/3 width=3/ qed.
+
+lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
+/2 width=1/ qed.
+
+lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
+#L #V1 #V2 #d #e #HV12 elim HV12 -V2
+[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
+ [ /3 width=5/
+ | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+ ]
+| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
+ lapply (tpss_lsubr_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
+ lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+]
+qed.
+
+lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
+ L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 →
+ L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2.
+#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
+[ #V2 #HV12 #HT12 elim HT12 -T2
+ [ /3 width=1/
+ | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+ ]
+| #V #V2 #_ #HV12 #IHV #HT12
+ lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+]
+qed.
+
+lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
+ L ⊢ T1 ▶* [d2, e2] T2.
+#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT12 #IHT
+ lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
+]
+qed.
+
+lemma tpss_weak_top: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT12 #IHT
+ lapply (tps_weak_top … HT12) -HT12 /2 width=3/
+]
+qed.
+
+lemma tpss_weak_full: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2.
+#L #T1 #T2 #d #e #HT12
+lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
+lapply (tpss_weak_top … HT12) //
+qed.
+
+lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
+ ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
+#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Note: this can be derived from tpss_inv_atom1 *)
+lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k.
+#L #T2 #k #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT destruct
+ >(tps_inv_sort1 … HT2) -HT2 //
+]
+qed-.
+
+(* Note: this can be derived from tpss_inv_atom1 *)
+lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
+#L #T2 #p #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT destruct
+ >(tps_inv_gref1 … HT2) -HT2 //
+]
+qed-.
+
+lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 &
+ L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+[ /2 width=5/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+ lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+]
+qed-.
+
+lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+[ /2 width=5/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/
+]
+qed-.
+
+lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2.
+#L #T1 #T2 #d #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (tps_fwd_tw … HT2) -HT2 #HT2
+@(transitive_le … IHT1) //
+qed-.
+
+lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
+[ /2 width=4/
+| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
+ elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarAlt $L $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss_lift.ma".
+
+(* PARALLEL UNFOLD ON TERMS *************************************************)
+
+(* alternative definition of tpss *)
+inductive tpssa: nat → nat → lenv → relation term ≝
+| tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I})
+| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 →
+ ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2
+| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
+ tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+ tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ tpssa d e L V1 V2 → tpssa d e L T1 T2 →
+ tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "parallel unfold (term) alternative"
+ 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tpssa_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ //
+| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T →
+ ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
+#L #T1 #T #d #e #H elim H -L -T1 -T -d -e
+[ #L #I #d #e #X #H
+ elim (tps_inv_atom1 … H) -H // * /2 width=6/
+| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
+ lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
+ lapply (tps_weak … H 0 (d+e) ? ?) -H // #H
+ elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/
+| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tps_lsubr_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
+ lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
+ lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
+ lapply (tpssa_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
+| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/
+]
+qed.
+
+lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/
+qed-.
+
+lemma tpss_ind_alt: ∀R:nat→nat→lenv→relation term.
+ (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) →
+ (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 →
+ ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
+ ) →
+ (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 →
+ R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 →
+ R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2.
+#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e
+// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_lift.ma".
+include "basic_2/unfold/tpss.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma tpss_subst: ∀L,K,V,U1,i,d,e.
+ d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 →
+ ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2.
+#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1
+[ /3 width=4/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+ elim (lift_total U 0 (i+1)) #U0 #HU0
+ lapply (IHU … HU0) -IHU #H
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+ lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
+ lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
+]
+qed.
+
+(* Advanced inverion lemmas *************************************************)
+
+lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V1,V2,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+#L #T2 #I #d #e #H @(tpss_ind … H) -T2
+[ /2 width=1/
+| #T #T2 #_ #HT2 *
+ [ #H destruct
+ elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
+ | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
+ lapply (ldrop_fwd_ldrop2 … HLK) #H
+ elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
+ @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
+ ]
+]
+qed-.
+
+lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶* [d, e] T2 →
+ T2 = #i ∨
+ ∃∃K,V1,V2. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#L #T2 #i #d #e #H
+elim (tpss_inv_atom1 … H) -H /2 width=1/
+* #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
+qed-.
+
+lemma tpss_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e + 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶* [d + 1, e] T2.
+#L #T1 #T2 #d #e #H #K #V #HLK @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT
+lapply (tps_inv_S2 … HT2 … HLK) -HT2 -HLK /2 width=3/
+qed-.
+
+lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
+ ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 ▶* [dt, et] U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
+ ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 ▶* [dt + e, et] U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d + e ≤ dt →
+ ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e.
+ L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU destruct
+<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
+qed.
+
+lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_tps.ma".
+include "basic_2/unfold/tpss_lift.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tpss_inv_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2.
+#L #T1 #T2 #d #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (tps_trans_ge … IHT1 … HT2 ?) //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
+/3 width=3/ qed.
+
+lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
+/3 width=3/ qed.
+
+lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2.
+#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2
+[ /2 width=3/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
+ elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ]
+ /3 width=7 by ex2_intro, step/ (**) (* just /3 width=7/ is too slow *)
+]
+qed.
+
+lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶* [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶* [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶* [d, e] T2 →
+ L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubststar_4.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/substitution/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* avtivate genv *)
+inductive cpys: relation4 genv lenv term term ≝
+| cpys_atom : ∀I,G,L. cpys G L (⓪{I}) (⓪{I})
+| cpys_delta: ∀I,G,L,K,V,V2,W2,i.
+ ⇩[i] L ≡ K.ⓑ{I}V → cpys G K V V2 →
+ ⇧[0, i + 1] V2 ≡ W2 → cpys G L (#i) W2
+| cpys_bind : ∀a,I,G,L,V1,V2,T1,T2.
+ cpys G L V1 V2 → cpys G (L.ⓑ{I}V1) T1 T2 →
+ cpys G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpys_flat : ∀I,G,L,V1,V2,T1,T2.
+ cpys G L V1 V2 → cpys G L T1 T2 →
+ cpys G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+ "context-sensitive extended multiple substitution (term)"
+ 'PSubstStar G L T1 T2 = (cpys G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpys_trans: ∀G. lsub_trans … (cpys G) lsuby.
+#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsuby_ldrop_trans … HL12 … HLK1) -HL12 -HLK1 *
+ /3 width=7 by cpys_delta/
+| /4 width=1 by lsuby_pair, cpys_bind/
+| /3 width=1 by cpys_flat/
+]
+qed-.
+
+(* Note: this is "∀L. reflexive … (cpys L)" *)
+lemma cpys_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ▶* T.
+#G #T elim T -T // * /2 width=1 by cpys_bind, cpys_flat/
+qed.
+
+lemma cpys_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶* V2 →
+ ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ▶* ②{I}V2.T.
+* /2 width=1 by cpys_bind, cpys_flat/
+qed.
+
+lemma cpys_bind_ext: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶* V2 →
+ ∀J,T1,T2. ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 →
+ ∀a,I. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* ⓑ{a,I}V2.T2.
+/4 width=4 by lsuby_cpys_trans, cpys_bind, lsuby_pair/ qed.
+
+lemma cpys_delift: ∀I,G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓑ{I}V) →
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
+#I #G #K #V #T1 elim T1 -T1
+[ * /2 width=4 by cpys_atom, lift_sort, lift_gref, ex2_2_intro/
+ #i #L #d elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i) /3 width=7 by cpys_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by cpys_bind, ldrop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpys_flat, lift_flat, ex2_2_intro/
+ ]
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpys_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶* T2 → ∀J. T1 = ⓪{J} →
+ T2 = ⓪{J} ∨
+ ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
+#G #L #T1 #T2 * -L -T1 -T2
+[ #I #G #L #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by ex4_5_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpys_inv_atom1: ∀J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ▶* T2 →
+ T2 = ⓪{J} ∨
+ ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
+/2 width=3 by cpys_inv_atom1_aux/ qed-.
+
+lemma cpys_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ▶* T2 → T2 = ⋆k.
+#G #L #T2 #k #H elim (cpys_inv_atom1 … H) -H // *
+#I #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+lemma cpys_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶* T2 →
+ T2 = #i ∨
+ ∃∃I,K,V,V2. ⇩[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#G #L #T2 #i #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+#I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
+qed-.
+
+lemma cpys_inv_lref1_ge: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶* T2 → |L| ≤ i → T2 = #i.
+#G #L #T2 #i #H elim (cpys_inv_lref1 … H) -H // *
+#I #K #V1 #V2 #HLK #_ #_ #HL -V2 lapply (ldrop_fwd_length_lt2 … HLK) -K -I -V1
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma cpys_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ▶* T2 → T2 = §p.
+#G #L #T2 #p #H elim (cpys_inv_atom1 … H) -H // *
+#I #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+fact cpys_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶* U2 →
+ ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 &
+ U2 = ⓑ{a,J}V2.T2.
+#G #L #U1 #U2 * -L -U1 -U2
+[ #I #G #L #b #J #W #U1 #H destruct
+| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
+]
+qed-.
+
+lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶* T2 &
+ U2 = ⓑ{a,I}V2.T2.
+/2 width=3 by cpys_inv_bind1_aux/ qed-.
+
+lemma cpys_inv_bind1_ext: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* U2 → ∀J.
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 &
+ U2 = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #U2 #H #J elim (cpys_inv_bind1 … H) -H
+#V2 #T2 #HV12 #HT12 #H destruct
+/4 width=5 by lsuby_cpys_trans, lsuby_pair, ex3_2_intro/
+qed-.
+
+fact cpys_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ▶* U2 →
+ ∀J,V1,U1. U = ⓕ{J}V1.U1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L⦄ ⊢ U1 ▶* T2 &
+ U2 = ⓕ{J}V2.T2.
+#G #L #U #U2 * -L -U -U2
+[ #I #G #L #J #W #U1 #H destruct
+| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Note: lemma 1250 *)
+lemma cpys_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ▶* U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L⦄ ⊢ U1 ▶* T2 &
+ U2 = ⓕ{I}V2.T2.
+/2 width=3 by cpys_inv_flat1_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpys_fwd_bind1: ∀a,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* T → ∀b,J.
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,J}V1.T1 ▶* ⓑ{b,J}V2.T2 &
+ T = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #T #H #b #J elim (cpys_inv_bind1_ext … H J) -H
+#V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpys_bind, ex2_2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lpx_sn_lpx_sn.ma".
+include "basic_2/substitution/fqup.ma".
+include "basic_2/substitution/lpys_ldrop.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Main properties **********************************************************)
+
+theorem cpys_antisym: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶* T2 → ⦃G, L⦄ ⊢ T2 ▶* T1 → T1 = T2.
+#G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
+[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #_ #HW2 lapply (ldrop_fwd_drop2 … HLK) -I -V1
+ #HLK elim (cpys_inv_lift1 … HW2 … HLK … HVW2) -L -HVW2
+ #X #H #_ elim (lift_inv_lref2_be … H) -G -K -V2 -W2 -X //
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_bind1 … H) -H
+ #V #T #HV2 #HT2 #H destruct
+ lapply (IHV12 HV2) #H destruct -IHV12 -HV2 /3 width=1 by eq_f2/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_flat1 … H) -H
+ #V #T #HV2 #HT2 #H destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+theorem cpys_trans_lpys: ∀G. lpx_sn_transitive (cpys G) (cpys G).
+#G0 #L0 #T0 @(fqup_wf_ind_eq … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #G1 #L1 * [|*]
+[ #I #HG #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
+ elim (cpys_inv_atom1 … H1) -H1
+ [ #H destruct
+ elim (cpys_inv_atom1 … HT2) -HT2
+ [ #H destruct //
+ | * #I2 #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
+ elim (lpys_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpys_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
+ lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
+ ]
+ | * #I1 #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
+ elim (lpys_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpys_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+ elim (cpys_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
+ lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
+ ]
+| #a #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpys_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+ elim (cpys_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ /4 width=5 by cpys_bind, lpys_pair/
+| #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpys_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+ elim (cpys_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ /3 width=5 by cpys_flat/
+]
+qed-.
+
+theorem cpys_trans: ∀G,L. Transitive … (cpys G L).
+/2 width=5 by cpys_trans_lpys/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lpys_cpys_trans: ∀G. lsub_trans … (cpys G) (lpys G).
+/2 width=5 by cpys_trans_lpys/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/fqus_alt.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Relocation properties ****************************************************)
+
+lemma cpys_lift: ∀G. l_liftable (cpys G).
+#G #K #T1 #T2 #H elim H -G -K -T1 -T2
+[ #I #G #K #L #s #d #e #_ #U1 #H1 #U2 #H2
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
+ elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #HKV #HVY #H destruct /3 width=10 by cpys_delta/
+ | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpys_delta, ldrop_inv_gen/
+ ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpys_bind, ldrop_skip/
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpys_flat/
+]
+qed.
+
+lemma cpys_inv_lift1: ∀G. l_deliftable_sn (cpys G).
+#G #L #U1 #U2 #H elim H -G -L -U1 -U2
+[ * #G #L #i #K #s #d #e #_ #T1 #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpys_atom, lift_sort, ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpys_atom, lift_gref, ex2_intro/
+ ]
+| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+ elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+ elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpys_delta, ex2_intro/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
+ elim (lift_split … HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpys_delta, ex2_intro/
+ ]
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpys_bind, ldrop_skip, lift_bind, ex2_intro/
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -V1
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpys_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpys_pair_sn, cpys_bind, cpys_flat, ex2_intro/
+[ #I #G #L #V2 #U2 #HVU2
+ elim (lift_total U2 0 1)
+ /4 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_drop, ex2_intro/
+| #G #L #K #T1 #U1 #e #HLK1 #HTU1 #T2 #HTU2
+ elim (lift_total T2 0 (e+1))
+ /3 width=11 by cpys_lift, fqu_drop, ex2_intro/
+]
+qed-.
+
+lemma fquq_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpys_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fqup_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpys_trans … H12 … HTU2) -T2
+ /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
+ elim (fqu_cpys_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
+ elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma fqus_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
+[ #HT12 elim (fqup_cpys_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fqu_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
+ #U2 #HVU2 @(ex3_intro … U2)
+ [1,3: /3 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_drop/
+ | #H destruct /2 width=7 by lift_inv_lref2_be/
+ ]
+| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
+ [1,3: /2 width=4 by fqu_pair_sn, cpys_pair_sn/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
+ [1,3: /2 width=4 by fqu_bind_dx, cpys_bind/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
+ [1,3: /2 width=4 by fqu_flat_dx, cpys_flat/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
+ #U2 #HTU2 @(ex3_intro … U2)
+ [1,3: /2 width=10 by cpys_lift, fqu_drop/
+ | #H0 destruct /3 width=5 by lift_inj/
+]
+qed-.
+
+lemma fquq_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+[ #H12 elim (fqu_cpys_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
+
+lemma fqup_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpys_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
+ #U1 #HTU1 #H #H12 elim (fqu_cpys_trans_neq … H1 … HTU1 H) -T1
+ /3 width=8 by fqup_strap2, ex3_intro/
+]
+qed-.
+
+lemma fqus_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+[ #H12 elim (fqup_cpys_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubstsnstar_3.ma".
+include "basic_2/relocation/lpx_sn.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
+
+definition lpys: relation3 genv lenv lenv ≝ λG. lpx_sn (cpys G).
+
+interpretation
+ "extended multiple substitution (local environment, sn variant)"
+ 'PSubstSnStar G L1 L2 = (lpys G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpys_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ▶* L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+lemma lpys_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpys_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ▶* ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpys_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ▶* K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpys_inv_pair: ∀I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ▶* L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ▶* L2 & ⦃G, L1⦄ ⊢ V1 ▶* V2 & I1 = I2.
+/2 width=1 by lpx_sn_inv_pair/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpys_refl: ∀G,L. ⦃G, L⦄ ⊢ ▶* L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpys_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ▶* K2 → ⦃G, K1⦄ ⊢ V1 ▶* V2 →
+ ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpys_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ▶* L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lpx_sn_ldrop.ma".
+include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/lpys.ma".
+
+(* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma lpys_ldrop_conf: ∀G. dropable_sn (lpys G).
+/3 width=6 by lpx_sn_deliftable_dropable, cpys_inv_lift1/ qed-.
+
+lemma ldrop_lpys_trans: ∀G. dedropable_sn (lpys G).
+/3 width=10 by lpx_sn_liftable_dedropable, cpys_lift/ qed-.
+
+lemma lpys_ldrop_trans_O1: ∀G. dropable_dx (lpys G).
+/2 width=3 by lpx_sn_dropable/ qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpys_pair, ex3_2_intro/
+[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpys_inv_pair1 … H) -H
+ #K2 #W2 #HLK2 #HVW2 #H destruct
+ /3 width=5 by fqu_fquq, cpys_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
+ elim (ldrop_lpys_trans … HLK1 … HK12) -HK12
+ /3 width=7 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_lpys_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpys_fqu_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpys_pair, ex3_2_intro/
+[ #I #G1 #L1 #V1 #X #H elim (lpys_inv_pair2 … H) -H
+ #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
+ /4 width=7 by cpys_delta, fqu_drop, ldrop_drop, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HL01
+ elim (lpys_ldrop_trans_O1 … HL01 … HLK1) -L1
+ /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma lpys_fquq_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (lpys_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_cpys.ma".
+
+(* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
+
+(* Main properties **********************************************************)
+
+theorem lpys_trans: ∀G. Transitive … (lpys G).
+/3 width=5 by lpx_sn_trans, cpys_trans_lpys/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 break ⊆ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEq $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lrsubeq_2.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+inductive lsuby: relation lenv ≝
+| lsuby_atom: ∀L. lsuby L (⋆)
+| lsuby_pair: ∀I1,I2,L1,L2,V. lsuby L1 L2 → lsuby (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+.
+
+interpretation
+ "local environment refinement (extended substitution)"
+ 'LRSubEq L1 L2 = (lsuby L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_refl: ∀L. L ⊆ L.
+#L elim L -L /2 width=1 by lsuby_pair/
+qed.
+
+lemma lsuby_sym: ∀L1,L2. L1 ⊆ L2 → |L1| = |L2| → L2 ⊆ L1.
+#L1 #L2 #H elim H -L1 -L2
+[ #L1 #H >(length_inv_zero_dx … H) -L1 //
+| #I1 #I2 #L1 #L2 #V #_ #IHL12 #H lapply (injective_plus_l … H) -H
+ /3 width=1 by lsuby_pair/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊆ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2 //
+#I1 #I2 #L1 #L2 #V #_ #H destruct
+qed-.
+
+lemma lsuby_inv_atom1: ∀L2. ⋆ ⊆ L2 → L2 = ⋆.
+/2 width=3 by lsuby_inv_atom1_aux/ qed-.
+
+fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊆ L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
+ L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
+#L1 #L2 * -L1 -L2
+[ #L #J1 #K1 #W #H destruct /2 width=1 by or_introl/
+| #I1 #I2 #L1 #L2 #V #HL12 #J1 #K1 #W #H destruct /3 width=4 by ex2_2_intro, or_intror/
+]
+qed-.
+
+lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊆ L2 →
+ L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
+/2 width=4 by lsuby_inv_pair1_aux/ qed-.
+
+fact lsuby_inv_pair2_aux: ∀L1,L2. L1 ⊆ L2 → ∀J2,K2,W. L2 = K2.ⓑ{J2}W →
+ ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
+#L1 #L2 * -L1 -L2
+[ #L #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V #HL12 #J2 #K2 #W #H destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lsuby_inv_pair2: ∀I2,L1,K2,W. L1 ⊆ K2.ⓑ{I2}W →
+ ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
+/2 width=4 by lsuby_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuby_fwd_length: ∀L1,L2. L1 ⊆ L2 → |L2| ≤ |L1|.
+#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
+qed-.
+
+lemma lsuby_ldrop_trans: ∀L1,L2. L1 ⊆ L2 →
+ ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+ ∃∃I1,K1. K1 ⊆ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
+#L1 #L2 #H elim H -L1 -L2
+[ #L #J2 #K2 #W #s #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #s #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+ [ /3 width=4 by ldrop_pair, ex2_2_intro/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_drop_lt, ex2_2_intro/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: Transitive … lsuby.
+#L1 #L #H elim H -L1 -L
+[ #L1 #X #H lapply (lsuby_inv_atom1 … H) -H //
+| #I1 #I #L1 #L #V #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H // *
+ #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ▶ * break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PSubstSnStar $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_alt.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on local environment refinement for extended substitution *****)
+
+lemma lsuby_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) (lsuby 0 (∞)).
+#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| /2 width=2 by cpx_sort/
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsuby_ldrop_trans_be … HL12 … HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lsuby_pair_O_Y/
+|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsuby_pair_O_Y/
+]
+qed-.
+
+(* Properties on context-sensitive extended multiple substitution for terms *)
+
+lemma cpys_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L -T1 -T2 -d -e
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
+qed.
+
+lemma cpy_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+/3 width=3 by cpy_cpys, cpys_cpx/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_alt.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on context-sensitive extended multiple substitution for terms *)
+
+lemma cpys_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L -T1 -T2 -d -e
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
+qed.
+
+lemma cpy_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+/3 width=3 by cpy_cpys, cpys_cpx/ qed.
+
+lemma cpx_cpy_trans: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+ ∀T2,d,e. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T #H elim H -G -L -T1 -T
+[ #I #G #L #X #d #e #H elim (cpy_inv_atom1 … H) //
+ * /2 width=3 by cpy_cpx/
+| #G #L #k #l #Hkl #X #d #e #H >(cpy_inv_sort1 … H) -X /2 width=2 by cpx_sort/
+| #I #G #L #K #V1 #V #W #i #HLK #_ #HVW #IHV1 #X #d #e #H
+ lapply (ldrop_fwd_drop2 … HLK) #H0
+ lapply (cpy_weak … H 0 (∞) ? ?) -H // #H
+ elim (cpy_inv_lift1_be … H … H0 … HVW) -H -H0 -HVW
+ /3 width=7 by cpx_delta/
+| #a #I #G #L #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #X #d #e #H elim (cpy_inv_bind1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct
+ /5 width=7 by cpx_bind, lsuby_cpy_trans, lsuby_succ/
+| #I #G #L #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #X #d #e #H elim (cpy_inv_flat1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /3 width=3 by cpx_flat/
+| #G #L #V1 #U1 #U #T #_ #HTU #IHU1 #T2 #d #e #HT2
+ lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
+ elim (lift_total T2 0 1) #U2 #HTU2
+ lapply (cpy_lift_be … HT2 (L.ⓓV1) … (Ⓕ) … HTU … HTU2 ? ?) -T
+ /3 width=3 by cpx_zeta, ldrop_drop/
+| /3 width=3 by cpx_tau/
+| /3 width=3 by cpx_ti/
+| #a #G #L #V1 #V #W1 #W #T1 #T #_ #_ #_ #IHV1 #IHW1 #IHT1 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #Y #T2 #HY #HT2 #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
+ /5 width=7 by cpx_beta, lsuby_cpy_trans, lsuby_succ/
+| #a #G #L #V1 #V #U #W1 #W #T1 #T #_ #HVU #_ #_ #IHV1 #IHW1 #IHT1 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #W2 #Y #HW2 #HY #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
+ lapply (cpy_weak … HU2 0 (∞) ? ?) -HU2 // #HU2
+ elim (cpy_inv_lift1_be … HU2 L … HVU) -U
+ /5 width=7 by cpx_theta, lsuby_cpy_trans, lsuby_succ, ldrop_drop/
+]
+qed-.
+
+lemma cpx_cpys_trans: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+ ∀T2,d,e. ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T #HT1 #T2 #d #e #H @(cpys_ind … H) -T2
+/2 width=5 by cpx_cpy_trans/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cpx_cpys.ma".
+include "basic_2/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on local environment refinement for extended substitution *****)
+
+lemma lsuby_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) (lsuby 0 (∞)).
+/3 width=5 by lsuby_cpx_trans, LTC_lsub_trans/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cpy_lift.ma".
+include "basic_2/substitution/cpys.ma".
+include "basic_2/reduction/lpx_ldrop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on context-sensitive extended substitution for terms **********)
+
+lemma cpx_cpy_trans_lpx: ∀h,g,G,L1,T1,T. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀T2,d,e. ⦃G, L2⦄ ⊢ T ▶[d, e] T2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L1 #T1 #T #H elim H -G -L1 -T1 -T
+[ #J #G #L1 #L2 #HL12 #T2 #d #e #H elim (cpy_inv_atom1 … H) -H //
+ * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct
+ elim (lpx_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpx_inv_pair2 … H) -H #K1 #V1 #_ #HV12 #H destruct
+ /2 width=7 by cpx_delta/
+| #G #L1 #k #l #Hkl #L2 #_ #X #d #e #H >(cpy_inv_sort1 … H) -X /2 width=2 by cpx_sort/
+| #I #G #L1 #K1 #V1 #V #T #i #HLK1 #_ #HVT #IHV1 #L2 #HL12 #T2 #d #e #HT2
+ elim (lpx_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_drop2 … HLK2) -V0 #HLK2
+ lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
+ elim (cpy_inv_lift1_be … HT2 … HLK2 … HVT) -HT2 -HLK2 -HVT
+ /3 width=7 by cpx_delta/
+| #a #I #G #L1 #V1 #V #T1 #T #HV1 #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_bind1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5 by lpx_pair, cpx_bind/
+| #I #G #L1 #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_flat1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5 by cpx_flat/
+| #G #L1 #V1 #U1 #U #T #_ #HTU #IHU1 #L2 #HL12 #T2 #d #e #HT2
+ lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
+ elim (lift_total T2 0 1) #U2 #HTU2
+ lapply (cpy_lift_be … HT2 (L2.ⓓV1) … (Ⓕ) … HTU … HTU2 ? ?) -T
+ /4 width=5 by cpx_zeta, lpx_pair, ldrop_drop/
+| /3 width=5 by cpx_tau/
+| /3 width=5 by cpx_ti/
+| #a #G #L1 #V1 #V #W1 #W #T1 #T #HV1 #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #Y #T2 #HY #HT2 #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
+ /5 width=11 by lpx_pair, cpx_beta, lsuby_cpy_trans, lsuby_succ/
+| #a #G #L1 #V1 #V #U #W1 #W #T1 #T #_ #HVU #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #W2 #Y #HW2 #HY #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
+ lapply (cpy_weak … HU2 0 (∞) ? ?) -HU2 // #HU2
+ elim (cpy_inv_lift1_be … HU2 L2 … HVU) -U
+ /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/
+]
+qed-.
+
+lemma cpx_cpys_trans_lpx: ∀h,g,G,L1,T1,T. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀T2,d,e. ⦃G, L2⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind … H) -T2
+/2 width=7 by cpx_cpy_trans_lpx/
+qed-.
--- /dev/null
+(*
+lemma lsuby_weak: ∀L1,L2,d1,e1. L1 ⊑×[d1, e1] L2 →
+ ∀d2,e2. d1 ≤ d2 → e2 ≤ e1 → L1 ⊑×[d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #_ #d2 #e2 #_ #He21
+ >(yle_inv_O2 … He21) -He21
+ /4 width=3 by lsuby_fwd_length, lsuby_O1, monotonic_le_plus_l/
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #d2 #e2 #_ #He21
+ elim (ynat_cases e2) /4 width=3 by lsuby_fwd_length, lsuby_O1, monotonic_le_plus_l/
+ * #e0 #H destruct lapply (yle_inv_succ … He21) -He21 #He21
+ elim (ynat_cases d2) /3 width=1 by lsuby_pair/
+ * #d0 #H destruct @lsuby_succ @IHL12 //
+ [ destruct
+
+*)
--- /dev/null
+(* Basic forward lemmas *****************************************************)
+
+lemma csup_fwd_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ →
+ ∃i. ⇩[0, i] L1 ≡ L2 ∨ ⇩[0, i] L2 ≡ L1.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=2/ /4 width=2/
+#I #L1 #K1 #V1 #i #HLK1
+lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /3 width=2/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lift_csup_trans_eq: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀L,U2. ⦃L, U1⦄ > ⦃L, U2⦄ →
+ ∃T2. ⇧[d, e] T2 ≡ U2.
+#T1 #U1 #d #e * -T1 -U1 -d -e
+[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #_ #L #X #H
+ elim (csup_inv_bind1 … H) -H *
+ [ #_ #H destruct /2 width=2/
+ | #H elim (discr_lpair_x_xy … H)
+ ]
+|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H
+ elim (csup_inv_flat1 … H) -H #_ * #H destruct /2 width=2/
+]
+#i #d #e [2,3: #_ ] #L #X #H
+elim (csup_inv_atom1 … H) -H #I #j #HL #H destruct
+lapply (ldrop_pair2_fwd_cw … HL X) -HL #H
+elim (lt_refl_false … H)
+qed-.
+(*
+lemma lift_csup_trans_gt: ∀L1,L2,U1,U2. ⦃L1, U1⦄ > ⦃L2, U2⦄ →
+ ⇩[0, 1] L2 ≡ L1 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∃T2. ⇧[d + 1, e] T2 ≡ U2.
+#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2
+[ #I #L1 #K1 #V #i #HLK1 #HKL1
+ lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_lw … HKL1) -HKL1 #HKL1
+ lapply (transitive_le … HLK1 HKL1) -L1 normalize #H
+
+
+| #a
+| #a
+]
+#I #L1 #W1 #U1 #HL1
+
+
+
+ #X #d #e #H
+ lapply (ldrop_inv_refl … HL1) -HL1
+| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H
+ lapply (ldrop_inv_ldrop1 … HL1)
+
+ elim (lift_inv_bind2 … H) -H #W2 #U2 #HW21 #HU21 #H destruct
+
+
+ /3 width=2/ /4 width=2/
+
+*)
+
+
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma csup_inv_lref2_be: ∀L,U,i. ⦃L, U⦄ > ⦃L, #i⦄ →
+ ∀T,d,e. ⇧[d, e] T ≡ U → d ≤ i → i < d + e → ⊥.
+#L #U #i #H #T #d #e #HTU #Hdi #Hide
+elim (lift_csup_trans_eq … HTU … H) -H -T #T #H
+elim (lift_inv_lref2_be … H ? ?) //
+qed-.
+
+
+fact csup_inv_all4_refl_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → L1 = L2 →
+ ∨∨ ∃∃a,I,U. T1 = ⓑ{a,I}T2.U
+ | ∃∃I,W. T1 = ⓕ{I}W.T2
+ | ∃∃I,U. T1 = ⓕ{I}T2.U.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=3/ /3 width=4/
+[ #I #L #K #V #i #HLK #H destruct
+ lapply (ldrop_pair2_fwd_cw … HLK V) -HLK #H
+ elim (lt_refl_false … H)
+| #a #I #L #V #T #H
+ elim (discr_lpair_x_xy … H)
+]
+qed-.
+
+lemma csup_inv_all4_refl: ∀L,T1,T2. ⦃L, T1⦄ > ⦃L, T2⦄ →
+ ∨∨ ∃∃a,I,U. T1 = ⓑ{a,I}T2.U
+ | ∃∃I,W. T1 = ⓕ{I}W.T2
+ | ∃∃I,U. T1 = ⓕ{I}T2.U.
+/2 width=4 by csup_inv_all4_refl_aux/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/substitution/csup.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Main forward lemmas ******************************************************)
+
+theorem csup_trans_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ →
+ ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ →
+ L = L0 ∨ ⦃L, T1⦄ > ⦃L, T3⦄.
+#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/
+[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H
+ lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1
+ lapply (csup_fwd_cw … H) -H #H2
+ lapply (transitive_lt … H1 H2) -H1 -H2 #H
+ elim (lt_refl_false … H)
+| #a #I #L0 #V2 #T2 #T3 #HT23
+ elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/
+ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/csup_csup.ma".
+include "basic_2/unfold/csups.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Advanced forward lemmas **************************************************)
+
+(*
+lemma csupp_strap2_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ →
+ ∀T3. ⦃L0, T2⦄ >+ ⦃L, T3⦄ →
+ L = L0 ∨ ⦃L, T1⦄ >+ ⦃L, T3⦄.
+#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/
+[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H
+ lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1
+ lapply (csupp_fwd_cw … H) -H #H2
+ lapply (transitive_lt … H1 H2) -H1 -H2 #H
+ elim (lt_refl_false … H)
+| #a #I #L0 #V2 #T2 #T3 #HT23
+ /3 width=5/
+
+ elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/
+ qed-.
+
+
+
+
+
+
+
+
+lemma csups_strap1_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ >* ⦃L0, T2⦄ →
+ ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ → L = L0.
+#L #L0 #T1 #T2 #H @(csups_ind_dx … H) -L -T1 //
+#L1 #L #T1 #T #HL1 #_ #IHL0 #T3 #HL0
+lapply (csup_trans_fwd_refl … HL10) … HL0) -T2
+*)
+lemma lift_csups_trans_aux: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀L1,L2,U2. ⦃L1, U1⦄ >* ⦃L2, U2⦄ → L1 = L2 →
+ ∃T2. ⇧[d, e] T2 ≡ U2.
+#T1 #U1 #d #e #HTU1 #L1 #L2 #U2 #H @(csups_ind … H) -L2 -U2 /2 width=2/ -T1
+#L #L2 #U #U2 #HL1 #HL2 #IHL1 #H destruct
+
+* -T1 -U1 -d -e
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/delift_lift.ma".
+include "basic_2/unfold/delift_delift.ma".
+include "basic_2/computation/cprs_delift.ma".
+include "basic_2/equivalence/cpcs_cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma cpcs_zeta_delift_comm: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 →
+ L ⊢ T2 ⬌* +ⓓV.T1.
+/3 width=1/ qed.
+
+(* Basic_1: was only: pc3_gen_cabbr *)
+lemma thin_cpcs_delift_mono: ∀L,U1,U2. L ⊢ U1 ⬌* U2 →
+ ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
+ ∀T2. L ⊢ ▼*[d, e] U2 ≡ T2 → K ⊢ T1 ⬌* T2.
+#L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2
+elim (cpcs_inv_cprs … H) -H #U #HU1 #HU2
+elim (thin_cprs_delift_conf … HU1 … HLK … HTU1) -U1 #T #HT1 #HUT
+elim (thin_cprs_delift_conf … HU2 … HLK … HTU2) -U2 -HLK #X #HT2 #H
+lapply (delift_mono … H … HUT) -L #H destruct /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/thin_delift.ma".
+include "basic_2/reducibility/tpr_delift.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties on inverse basic term relocation ******************************)
+
+(* Basic_1: was only: pr2_gen_cabbr *)
+lemma thin_cpr_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
+ ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
+ ∃∃T2. K ⊢ T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
+#L #U1 #U2 * #U #HU1 #HU2 #K #d #e #HLK #T1 #HTU1
+elim (tpr_delift_conf … HU1 … HTU1) -U1 #T #HT1 #HUT
+elim (le_or_ge (|L|) d) #Hd
+[ elim (thin_delift_tpss_conf_le … HU2 … HUT … HLK ?)
+| elim (le_or_ge (|L|) (d+e)) #Hde
+ [ elim (thin_delift_tpss_conf_le_up … HU2 … HUT … HLK ? ? ?)
+ | elim (thin_delift_tpss_conf_be … HU2 … HUT … HLK ? ?)
+ ]
+] -U -HLK // -Hd [2,3: -Hde] #T2 #HT2
+lapply (cpr_intro … HT1 HT2) -T /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_delift.ma".
+include "basic_2/reducibility/cpr_cpr.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties on inverse basic term relocation ******************************)
+
+(* Note: this should be stated with tprs *)
+lemma cprs_zeta_delift: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 → L ⊢ +ⓓV.T1 ➡* T2.
+#L #V #T1 #T2 * #T #HT1 #HT2
+@(cprs_strap2 … (+ⓓV.T)) [ /3 width=3/ | @inj /3 width=3/ ] (**) (* explicit constructor, /5 width=3/ is too slow *)
+qed.
+
+(* Basic_1: was only: pr3_gen_cabbr *)
+lemma thin_cprs_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡* U2 →
+ ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
+ ∃∃T2. K ⊢ T1 ➡* T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
+#L #U1 #U2 #H @(cprs_ind … H) -U2 /2 width=3/
+#U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1
+elim (IHU1 … HLK … HTU1) -U1 #T #HT1 #HUT
+elim (thin_cpr_delift_conf … HU2 … HLK … HUT) -U -HLK /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $L $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+definition delift: nat → nat → lenv → relation term ≝
+ λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T.
+
+interpretation "inverse basic relocation (term)"
+ 'TSubst L T1 d e T2 = (delift d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 →
+ ∀L. L ⊢ ▼*[d, e] T2 ≡ T1.
+/2 width=3/ qed.
+
+lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T.
+/2 width=3/ qed.
+
+lemma delift_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2.
+#L1 #T1 #T2 #d #e * /3 width=3/
+qed.
+
+lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k.
+/2 width=3/ qed.
+
+lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i.
+/3 width=3/ qed.
+
+lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e).
+/3 width=3/ qed.
+
+lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p.
+/2 width=3/ qed.
+
+lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e.
+ L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 →
+ L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
+qed.
+
+lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
+ L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 →
+ L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k.
+#L #U2 #d #e #k * #U #HU
+>(tpss_inv_sort1 … HU) -HU #HU2
+>(lift_inv_sort2 … HU2) -HU2 //
+qed-.
+
+lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p.
+#L #U #d #e #p * #U #HU
+>(tpss_inv_gref1 … HU) -HU #HU2
+>(lift_inv_gref2 … HU2) -HU2 //
+qed-.
+
+lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 →
+ ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
+ L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
+lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 →
+ ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
+ L ⊢ ▼*[d, e] T1 ≡ T2 &
+ U2 = ⓕ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
+qed-.
+
+lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2.
+#L #T1 #T2 #d * #T #HT1
+>(tpss_inv_refl_O2 … HT1) -HT1 #HT2
+>(lift_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e * #T #HT1 #HT2
+>(lift_fwd_tw … HT2) -T2 /2 width=4 by tpss_fwd_tw/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubstAlt $L $T1 $d $e $T2 }.
+
+include "basic_2/unfold/delift_lift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* alternative definition of inverse basic term relocation *)
+inductive delifta: nat → nat → lenv → relation term ≝
+| delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k)
+| delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i)
+| delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 →
+ ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2
+| delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e))
+| delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p)
+| delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
+ delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+ delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| delifta_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ delifta d e L V1 V2 → delifta d e L T1 T2 →
+ delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "inverse basic relocation (term) alternative"
+ 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma delifta_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/
+[ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
+#L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
+[ * #i #Hn #T2 #d #e #H destruct
+ [ >(delift_inv_sort1 … H) -H //
+ | elim (delift_inv_lref1 … H) -H * /2 width=1/
+ #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
+ lapply (ldrop_pair2_fwd_fw … HLK) #H
+ lapply (IH … HV12) // -H /2 width=6/
+ | >(delift_inv_gref1 … H) -H //
+ ]
+| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
+ [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
+ lapply (IH … HV12) -HV12 // #HV12
+ lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12
+ lapply (delifta_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
+ | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HV12) -HV12 //
+ lapply (IH … HT12) -IH -HT12 // /2 width=1/
+ ]
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/
+qed-.
+
+lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
+ (∀L,d,e,k. R d e L (⋆k) (⋆k)) →
+ (∀L,d,e,i. i < d → R d e L (#i) (#i)) →
+ (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
+ ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
+ ) →
+ (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
+ (∀L,d,e,p. R d e L (§p) (§p)) →
+ (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
+ L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 →
+ R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
+ L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 →
+ R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e
+// /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem delift_mono: ∀L,T,T1,T2,d,e.
+ L ⊢ ▼*[d, e] T ≡ T1 → L ⊢ ▼*[d, e] T ≡ T2 → T1 = T2.
+#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2
+elim (tpss_conf_eq … H1TU1 … H1TU2) -T #U #HU1 #HU2
+lapply (tpss_inv_lift1_eq … HU1 … H2TU1) -HU1 #H destruct
+lapply (tpss_inv_lift1_eq … HU2 … H2TU2) -HU2 #H destruct
+lapply (lift_inj … H2TU1 … H2TU2) //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/ldrop_lbotr.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 →
+ ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2.
+#L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2
+elim (lift_total V 0 (i+1)) #U #HVU
+lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m /2 width=1/ #HV2U
+lapply (lift_conf_be … HVU2 … HV2U ?) //
+>commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
+qed.
+
+lemma lbotr_delift: ∀L,T1,d,e. d + e ≤ |L| → ⊒[d, e] L →
+ ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 @(f2_ind … fw … L T1) -L -T1
+#n #IH #L * * /2 width=2/
+[ #i #H #d #e #Hde #HL destruct
+ elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
+ elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
+ lapply (lt_to_le_to_lt … Hide Hde) #Hi
+ elim (ldrop_O1_lt … Hi) -Hi #I #K #V1 #HLK
+ lapply (lbotr_inv_ldrop … HLK … HL ? ?) // #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK (#i)) #HKL
+ lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
+ lapply (ldrop_fwd_O1_length … HLK0) #H
+ lapply (lbotr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
+ [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
+ elim (IH … HKL … HK) -IH -HKL -HK
+ [2: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
+ elim (lift_total V2 0 d) /3 width=7/
+| #a #I #V1 #T1 #H #d #e #Hde #HL destruct
+ elim (IH … V1 … Hde HL) // #V2 #HV12
+ elim (IH (L.ⓑ{I}V1) T1 … (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12
+ lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
+| #I #V1 #T1 #H #d #e #Hde #HL destruct
+ elim (IH … V1 … Hde HL) // #V2 #HV12
+ elim (IH … T1 … Hde HL) -IH -Hde -HL // /3 width=2/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i.
+#L #U2 #i #d #e * #U #HU #HU2 #Hid
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct >(lift_inv_lref2_lt … HU2) //
+| * #K #V1 #V2 #Hdi
+ lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
+ elim (lt_refl_false … Hi)
+]
+qed-.
+
+lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 →
+ d ≤ i → i < d + e →
+ ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
+ ⇧[0, d] V2 ≡ U2.
+#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
+| * #K #V1 #V #_ #_ #HLK #HV1 #HVU
+ elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
+]
+qed-.
+
+lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
+ d + e ≤ i → U2 = #(i - e).
+#L #U2 #i #d #e * #U #HU #HU2 #Hdei
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct >(lift_inv_lref2_ge … HU2) //
+| * #K #V1 #V2 #_ #Hide
+ lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
+ elim (lt_refl_false … Hi)
+]
+qed-.
+
+lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
+ ∨∨ (i < d ∧ U2 = #i)
+ | (∃∃K,V1,V2. d ≤ i & i < d + e &
+ ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
+ ⇧[0, d] V2 ≡ U2
+ )
+ | (d + e ≤ i ∧ U2 = #(i - e)).
+#L #U2 #i #d #e #H
+elim (lt_or_ge i d) #Hdi
+[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
+| elim (lt_or_ge i (d+e)) #Hide
+ [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
+ | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
+ ]
+]
+qed-.
+
+(* Properties on basic term relocation **************************************)
+
+lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
+ L ⊢ ▼*[dt, et] U1 ≡ U2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
+>(lift_mono … HTU2 … HT2) -T2 /2 width=3/
+qed.
+
+lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ L ⊢ ▼*[dt, et + e] U1 ≡ T2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
+qed.
+
+lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ ▼*[dt + e, et] U1 ≡ U2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
+>(lift_mono … HTU2 … HT2) -T2 /2 width=3/
+qed.
+
+lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 →
+ ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2.
+#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1
+lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct
+lapply (lift_inj … HTU1 … HTU2) -U2 //
+qed-.
+
+lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T →
+ ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e →
+ L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide
+lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10
+lapply (lift_trans_be … HT2 … HT0 ? ?) -T //
+>commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?);
+<minus_le_minus_minus_comm // <plus_minus_m_m [ /2 width=3/ | /2 width=1/ ]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_alt.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Properties on sn partial unfold on local environments ********************)
+
+lemma delift_ltpss_sn_conf_eq: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 →
+ ∀K. L ⊢ ▶* [d, e] K → K ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T2 #d #e * #T #HT1 #HT2 #K #HLK
+elim (ltpss_sn_tpss_conf … HT1 … HLK) -HT1 -HLK #T0 #HT10 #HT0
+lapply (tpss_inv_lift1_eq … HT0 … HT2) -HT0 #H destruct /2 width=3/
+qed.
+
+lemma ltpss_sn_delift_trans_eq: ∀L,K,d,e. L ⊢ ▶* [d, e] K →
+ ∀T1,T2. K ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #K #d #e #HLK #T1 #T2 * #T #HT1 #HT2
+lapply (ltpss_sn_tpss_trans_eq … HT1 … HLK) -K /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Properties on partial unfold on terms ************************************)
+
+lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_le … HXU1 … HLK … HTX1 ?) -X1 -HLK // -H1 /3 width=5/
+qed.
+
+lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2 #H3
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_le_up … HXU1 … HLK … HTX1 ? ? ?) -X1 -HLK // -H1 -H2 -H3 /3 width=5/
+qed.
+
+lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=6/ qed.
+
+lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // -H1 -H2 /3 width=5/
+qed.
+
+lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
+#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
+qed.
+
+lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
+/3 width=3/ qed.
+
+lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
+#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
+lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
+qed.
+
+lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $T1 $d $e $T2 }.
+
+include "basic_2/unfold/ltpss_sn.ma".
+
+(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
+
+definition thin: nat → nat → relation lenv ≝
+ λd,e,L1,L2. ∃∃L. L1 ⊢ ▶* [d, e] L & ⇩[d, e] L ≡ L2.
+
+interpretation "basic thinning (local environment)"
+ 'TSubst L1 d e L2 = (thin d e L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ▼*[d, e] L1 ≡ L2.
+/2 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma thin_inv_thin1: ∀I,K1,V1,L2,e. ▼*[0, e] K1.ⓑ{I} V1 ≡ L2 → 0 < e →
+ ▼*[0, e - 1] K1 ≡ L2.
+#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e
+elim (ltpss_sn_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct
+lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/delift_tpss.ma".
+include "basic_2/unfold/delift_ltpss.ma".
+include "basic_2/unfold/thin.ma".
+
+(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
+
+(* Inversion lemmas on inverse basic term relocation ************************)
+
+lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
+ K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
+ L2 = K2. ⓑ{I} V2.
+#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
+elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
+elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
+qed-.
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
+ ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
+elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
+lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
+lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
+qed.
+
+lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
+
+lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/ltpss_sn_ldrop.ma".
+include "basic_2/unfold/thin.ma".
+
+(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
+
+(* Properties on local environment slicing **********************************)
+
+lemma thin_ldrop_conf_ge: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2 - e1] L1 ≡ L2.
+#L0 #L1 #d1 #e1 * /3 width=8 by ltpss_sn_ldrop_conf_ge, ldrop_conf_ge/
+qed.
+
+lemma thin_ldrop_conf_be: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. ▼*[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
+#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #Hd1e2 #He2de1
+elim (ltpss_sn_ldrop_conf_be … HL0 … HL02 ? ?) -L0 // #L0 #HL20 #HL0
+elim (ldrop_conf_be … HL1 … HL0 ? ?) -L // -Hd1e2 -He2de1 /3 width=3/
+qed.
+
+lemma thin_ldrop_conf_le: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ▼*[d1 - e2, e1] L2 ≡ L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #He2d1
+elim (ltpss_sn_ldrop_conf_le … HL0 … HL02 ?) -L0 // #L0 #HL20 #HL0
+elim (ldrop_conf_le … HL1 … HL0 ?) -L // -He2d1 /3 width=3/
+qed.
+
+lemma thin_ldrop_trans_ge: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #Hd1e2
+lapply (ldrop_trans_ge … HL0 … HL02 ?) -L0 // #HL2
+lapply (ltpss_sn_ldrop_trans_ge … HL1 … HL2 ?) -L // /2 width=1/
+qed.
+
+lemma thin_ldrop_trans_le: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ▼*[d1 - e2, e1] L ≡ L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #He2d1
+elim (ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL0 #HL02
+elim (ltpss_sn_ldrop_trans_le … HL1 … HL0 He2d1) -L -He2d1 /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/delift.ma".
+include "basic_2/reducibility/ltpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma tpr_delift_conf: ∀U1,U2. U1 ➡ U2 → ∀L,T1,d,e. L ⊢ ▼*[d, e] U1 ≡ T1 →
+ ∃∃T2. T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2.
+#U1 #U2 #HU12 #L #T1 #d #e * #W1 #HUW1 #HTW1
+elim (tpr_tpss_conf … HU12 … HUW1) -U1 #U1 #HWU1 #HU21
+elim (tpr_inv_lift1 … HWU1 … HTW1) -W1 /3 width=5/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+(*
+include "basic_2/reduction/lpr_ldrop.ma".
+
+include "basic_2/unfold/fsups.ma".
+include "basic_2/reducibility/lpr_ldrop.ma".
+
+lamma pippo: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀V1,i. ⇧[i, 1] V1 ≡ T1 → T2 = #i → ⊥.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2
+[ #I #L1 #V #W1 #j #H1 #H2
+ elim (lift_inv_lref2 … H1) -H1 * #H1 #H3
+
+ HVT2 : ()
+ HV2 : (K2⊢V➡V2)
+
+
+thaorem cpr_trans_lpr: ∀L1,T1,T. L1 ⊢ T1 ➡ T → ∀L2. L1 ⊢ ➡ L2 →
+ ∀T2. L2 ⊢ T ➡ T2 →
+ (⦃L2, T2⦄ ⊃* ⦃L1, T1⦄ → ⊥) ∨ T1 = T.
+#L1 #T1 @(fsupp_wf_ind … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
+[ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct -IH
+ elim (cpr_inv_atom1 … H1) -H1
+ [ #H destruct
+ elim (cpr_inv_atom1 … HT2) -HT2
+ [ #H destruct //
+ | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct //
+ ]
+ | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V0 #HLK2
+ elim (cpr_inv_lift1 … HT2 … HLK2 … HVT) -HT2 -HLK2 -HVT #V2 #HVT2 #HV2
+ @or_introl #H
+
+
+
+
+
+
+ elim (lift_inv_lref2 … HVT2) -HVT2 * #H #_
+ [ elim (lt_zero_false … H)
+ | >commutative_plus in H; >plus_plus_comm_23 #H
+ elim (le_plus_xySz_x_false … H)
+ ]
+ ]
+| #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V #T #HV1 #HT1 #H destruct
+ elim (cpr_inv_bind1 … H2) -H2 *
+ [ #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IH … HV1 … HV2) // #HV12 destruct
+ [ @or_introl #H destruct /2 width=1/
+ | elim (IH … HT1 … HT2) // /2 width=1/ -L1 -L2 #HT12 destruct
+ @or_introl #H destruct /2 width=1/
+ ]
+ | #T2 #HT2 #HXT2 #H1 #H2 destruct
+ elim (IH … HT1 … HT2) // /2 width=1/ -L1 -L2 #HT12 destruct
+ | elim (term_eq_dec V1 V) #HV1 destruct
+
+ ]
+ | #Y1 #HTY1 #HXY1 #H11 #H12 destruct
+ elim (lift_total (+ⓓV1.T1) 0 1) #Y2 #HXY2
+ lapply (cpr_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
+ ]
+| #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V #T #HV1 #HT1 #H destruct
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
+ | #HX2 #H destruct /3 width=5/
+ ]
+ | #HX1 #H destruct /3 width=5/
+]
+qed-.
+*)
--- /dev/null
+lemma ldrop_bor1: ∀L1,L2,s1,s2,d,e. ⇩[s1, d, e] L1 ≡ L2 → ⇩[s1 ∨ s2, d, e] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by ldrop_gen/
+qed.
+
+lemma ldrop_bor2: ∀L1,L2,s1,s2,d,e. ⇩[s2, d, e] L1 ≡ L2 → ⇩[s1 ∨ s2, d, e] L1 ≡ L2.
+#L1 #L2 #s1 #s2 >commutative_orb /2 width=1 by ldrop_bor1/
+qed.
+
+(* Basic_1: was: drop_conf_rev *)
+axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
+ ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
--- /dev/null
+include "basic_2/reduction/lpx_ldrop.ma".
+
+lemma lpx_fpbc_trans: ∀h,g,G1,G2,K1,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ≻[h, g] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #K1 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
+ /3 width=5 by fpbc_fqu, ex3_2_intro/
+| #T2 #HT12 #H #HKL1
--- /dev/null
+lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+(* ALTERNATIVE PROOF
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+[ /2 width=1 by or_introl/
+| #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
+ elim (fpb_fpbu … H1) -H1 #H1
+ [ /3 width=1 by
+*)
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
+#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
+[ -HT1 elim (fqus_inv_gen … HT2) -HT2
+ [ #H elim (fqup_inv_step_sn … H) -H
+ /4 width=11 by fpbs_intro_alt, fpbu_fqu, ex2_3_intro, or_intror/
+ | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
+ [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
+ | elim (lpxs_nlleq_inv_step_sn … HL0 H) -HL0 -H
+ /5 width=7 by lpxs_lleq_fpbs, fpbu_lpx, lleq_trans, ex2_3_intro, or_intror/
+ ]
+ ]
+| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
+ /5 width=11 by fpbs_intro_alt, fpbu_cpx, ex2_3_intro, or_intror/
+]
+qed-.
+
+(* alternative proof that needs decidability of bteq to go in fpbs.ma
+ * or lpx_fpbc_trans to go in fpbs_lift.ma (possibly)
+*)
+
+axiom lpx_bteq_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1⦄ ⊢➡ [h, g] L →
+ ⦃G1, L, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄.
+
+lemma fpbs_fwd_fpb_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ ∨
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1 [ /2 width=1 by or_introl/ ] (**) (* auto fails without brackets *)
+#G1 #G #L1 #L #T1 #T *
+[ #G0 #L0 #T0 #H #H02 #IH1 elim (fquq_inv_gen … H) -H
+ [ -IH1 /4 width=5 by fpbc_fqu, ex2_3_intro, or_intror/
+ | -H02 * #HG #HL #HT destruct /2 width=1 by/
+ ]
+| #T0 #HT10 #H02 #IH02 elim (eq_term_dec T1 T0) #H destruct
+ [ -H02 /2 width=1 by/
+ | -IH02 /5 width=5 by fpbc_cpx, ex2_3_intro, or_intror/
+ ]
+| #L0 #HL10 #_ * [ /3 width=3 by or_introl, lpx_bteq_trans/ ]
+ * #G3 #L3 #T3 #H13 #H32
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/rajust_5.ma".
+include "basic_2/substitution/drop.ma".
+
+(* AJUSTMENT ****************************************************************)
+
+inductive fpa (s:bool): bi_relation lenv term ≝
+| fpa_fwd: ∀L,K,T,U,d,e. ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U → fpa s K T L U
+| fpa_bwd: ∀L,K,T,U,d,e. ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U → fpa s L U K T
+.
+
+interpretation
+ "ajustment (restricted closure)"
+ 'RAjust L1 T1 s L2 T2 = (fpa s L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpa_refl: ∀s. bi_reflexive … (fpa s).
+/2 width=4 by fpa_fwd/ qed.
+
+lemma fpa_sym: ∀s. bi_symmetric … (fpa s).
+#s #L1 #L2 #T1 #T2 * /2 width=4 by fpa_fwd, fpa_bwd/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fpa.ma".
+
+(* AJUSTMENT ****************************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fpa_conf: ∀s. bi_confluent … (fpa s).
+/3 width=4 by fpa_sym, ex2_2_intro/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/rajuststar_5.ma".
+include "basic_2/substitution/fpa.ma".
+
+(* MULTIPLE AJUSTMENT *******************************************************)
+
+definition fpas: bool → bi_relation lenv term ≝ λs. bi_TC … (fpa s).
+
+interpretation
+ "multiple ajustment (restricted closure)"
+ 'RAjustStar L1 T1 s L2 T2 = (fpas s L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fpas_ind: ∀L1,T1,s. ∀R:relation2 …. R L1 T1 →
+ (∀L,L2,T,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳[s] ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → R L2 T2.
+#L1 #T1 #s #R #IH1 #IH2 #L2 #T2 #H
+@(bi_TC_star_ind … IH1 IH2 L2 T2 H) //
+qed-.
+
+lemma fpas_ind_dx: ∀L2,T2,s. ∀R:relation2 …. R L2 T2 →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ⇳[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → R L1 T1.
+#L2 #T2 #s #R #IH1 #IH2 #L1 #T1 #H
+@(bi_TC_star_ind_dx … IH1 IH2 L1 T1 H) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpas_refl: ∀s. bi_reflexive … (fpas s).
+/2 width=1 by bi_inj/ qed.
+
+lemma fpas_sym: ∀s. bi_symmetric … (fpas s).
+/3 width=1 by fpa_sym, bi_TC_symmetric/ qed-.
+
+lemma fpa_fpas: ∀L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L2, T2⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
+/2 width=1 by bi_inj/ qed.
+
+lemma fpas_strap1: ∀L1,L,L2,T1,T,T2,s. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳[s] ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
+/2 width=4 by bi_step/ qed-.
+
+lemma fpas_strap2: ∀L1,L,L2,T1,T,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L, T⦄ → ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
+/2 width=4 by bi_TC_strap/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fpa_fpa.ma".
+include "basic_2/multiple/fpas.ma".
+
+(* MULTIPLE AJUSTMENT *******************************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fpas_strip: ∀L0,L1,L2,T0,T1,T2,s. ⦃L0, T0⦄ ⇳[s] ⦃L1, T1⦄ → ⦃L0, T0⦄ ⇳*[s] ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ & ⦃L2, T2⦄ ⇳[s] ⦃L, T⦄.
+/3 width=4 by fpa_conf, bi_TC_strip/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem fpas_conf: ∀s. bi_confluent … (fpas s).
+/3 width=4 by fpa_conf, bi_TC_confluent/ qed-.
+
+theorem fpas_trans: ∀s. bi_transitive … (fpas s).
+/2 width=4 by bi_TC_transitive/ qed-.
+
+theorem fpas_canc_sn: ∀L,L1,L2,T,T1,T2,s.
+ ⦃L, T⦄ ⇳*[s] ⦃L1, T1⦄→ ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
+/3 width=4 by fpas_trans, fpas_sym/ qed-.
+
+theorem fpas_canc_dx: ∀L1,L2,L,T1,T2,T,s.
+ ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L2, T2⦄ ⇳*[s] ⦃L, T⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
+/3 width=4 by fpas_trans, fpas_sym/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/term_vector.ma".
+include "basic_2/multiple/fpas.ma".
+
+(* MULTIPLE VECTOR AJUSTMENT ************************************************)
+
+inductive fpasv (s:bool): bi_relation lenv (list term) ≝
+| fpasv_nil : ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ → fpasv s L1 (◊) L2 (◊)
+| fpasv_cons: ∀L1,L2,T1s,T2s,T1,T2. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ →
+ fpasv s L1 T1s L2 T2s →
+ fpasv s L1 (T1 @ T1s) L2 (T2 @ T2s)
+.
+
+interpretation
+ "multiple vector ajustment (restricted closure)"
+ 'RAjustStar L1 T1s s L2 T2s = (fpasv s L1 T1s L2 T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+
+
+(* Basic_1: was just: lifts1_flat (left to right) *)
+lemma fpas_inv_applv1: ∀L1,L2,V1s,T1,X,s. ⦃L1, Ⓐ V1s.T1⦄ ⇳*[s] ⦃L2, X⦄ →
+ ∃∃V2s,T2. ⦃L1, V1s⦄ ⇳*[s] ⦃L2, V2s⦄ & ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ &
+ X = Ⓐ V2s.T2.
+#L1 #L2 #V1s elim V1s -V1s
+[ #T1 #X #s #H
+ @(ex3_2_intro … (◊) X) /2 width=3 by fpasv_nil/ (**) (* explicit constructor *)
+| #V1 #V1s #IHV1s #T1 #X #s #H
+ elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
+ elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
+ @(ex3_2_intro) [4: // |3: /2 width=2 by liftsv_cons/ |1,2: skip | // ] (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: lifts1_flat (right to left) *)
+lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 →
+ ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
+#V1s #V2s #des #H elim H -V1s -V2s /3 width=1 by lifts_flat/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/genv.ma".
+include "basic_2/multiple/fpas.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+definition candidate: Type[0] ≝ relation3 genv lenv term.
+
+definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G,L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳[s] ⦃L2, T2⦄ →
+ NF … (RR G L1) RS T1 → NF … (RR G L2) RS T2.
+
+definition CP0s ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G,L1,L2,T1,T2,s. ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄ →
+ NF … (RR G L1) RS T1 → NF … (RR G L2) RS T2.
+
+definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G,L. ∃k. NF … (RR G L) RS (⋆k).
+
+definition CP2 ≝ λRP:candidate.
+ ∀G,L,T,k. RP G L (ⓐ⋆k.T) → RP G L T.
+
+(* requirements for generic computation properties *)
+record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
+{ cp0: CP0 RR RS;
+ cp1: CP1 RR RS;
+ cp2: CP2 RP
+}.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: nf2_lift1 *)
+lemma gcp_fpas: ∀RR,RS. CP0 RR RS → CP0s RR RS.
+#RR #RS #HRR #G #L1 #L2 #T1 #T2 #s #H @(fpas_ind … H) -L2 -T2 /3 width=5 by/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/ineint_5.ma".
+include "basic_2/grammar/aarity.ma".
+include "basic_2/substitution/lift_vector.ma".
+include "basic_2/computation/gcp.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+definition S0 ≝ λC:candidate. ∀G,L1,L2,T1,T2. ⦃L1, T1⦄ ⇳[Ⓕ] ⦃L2, T2⦄ →
+ C G L1 T1 → C G L2 T2.
+
+definition S0s ≝ λC:candidate. ∀G,L1,L2,T1,T2. ⦃L1, T1⦄ ⇳*[Ⓕ] ⦃L2, T2⦄ →
+ C G L1 T1 → C G L2 T2.
+
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:candidate.
+ ∀G,L,T. C G L T → RP G L T.
+
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs →
+ ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
+
+(* Note: this generalizes Tait's ii *)
+definition S3 ≝ λC:candidate.
+ ∀a,G,L,Vs,V,T,W.
+ C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
+
+definition S4 ≝ λRP,C:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
+
+definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
+ C G L (ⒶVs.V2) → ⇧[0, i+1] V1 ≡ V2 →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
+
+definition S6 ≝ λRP,C:candidate.
+ ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
+
+definition S7 ≝ λC:candidate.
+ ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
+
+(* requirements for the generic reducibility candidate *)
+record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
+{ s0: S0 C;
+ s1: S1 RP C;
+ s2: S2 RR RS RP C;
+ s3: S3 C;
+ s4: S4 RP C;
+ s5: S5 C;
+ s6: S6 RP C;
+ s7: S7 C
+}.
+
+(* the functional construction for candidates *)
+definition cfun: candidate → candidate → candidate ≝
+ λC1,C2,G,K,T. ∀L,V,U. ⦃K, T⦄ ⇳*[Ⓕ] ⦃L, U⦄ →
+ C1 G L V → C2 G L (ⓐV.U).
+
+(* the reducibility candidate associated to an atomic arity *)
+let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
+match A with
+[ AAtom ⇒ RP
+| APair B A ⇒ cfun (acr RP B) (acr RP A)
+].
+
+interpretation
+ "candidate of reducibility of an atomic arity (abstract)"
+ 'InEInt RP G L T A = (acr RP A G L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: sc3_lift1 *)
+lemma gcr_fpas: ∀C. S0 C → S0s C.
+#C #HC #G #L1 #L2 #T1 #T2 #H @(fpas_ind … H) -L2 -T2 /3 width=5 by/
+qed.
+(*
+lemma rp_lifts: ∀RR,RS,RP. gcr RR RS RP RP →
+ ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
+ RP G L V → RP G L0 V0.
+#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
+@gcr_lifts /width=7 by/
+@(s0 … HRP)
+qed.
+
+(* Basic_1: was only: sns3_lifts1 *)
+lemma rp_liftsv_all: ∀RR,RS,RP. gcr RR RS RP RP →
+ ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
+ all … (RP G L) Vs → all … (RP G L0) V0s.
+#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
+qed.
+*)
+(* Basic_1: was:
+ sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
+*)
+lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀A. gcr RR RS RP (acr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A //
+#B #A #IHB #IHA @mk_gcr
+[ /3 width=4 by fpas_strap2/
+| #G #L #T #H
+ elim (cp1 … H1RP G L) #k #HK
+ lapply (H L (⋆k) T ? ?) -H //
+ [ @(s2 … IHB … (◊)) //
+ | #H @(cp2 … H1RP … k) @(s1 … IHA) //
+ ]
+| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
+ lapply (s1 … IHB … HB) #HV0
+ @(s2 … IHA … (V0 @ V0s))
+ /3 width=14 by rp_liftsv_all, gcp_lifts, cp0, lifts_simple_dx, conj/
+| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #des #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
+ elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
+ @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+| #G #L #Vs #HVs #k #L0 #V0 #X #des #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ >(lifts_inv_sort1 … HY) -Y
+ lapply (s1 … IHB … HB) #HV0
+ @(s4 … IHA … (V0 @ V0s)) /3 width=7 by rp_liftsv_all, conj/
+| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
+ elim (drops_drop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
+ >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
+ elim (drops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
+ elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
+ elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
+ >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
+ @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
+| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
+ elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
+ elim (lift_total V10 0 1) #V20 #HV120
+ elim (liftv_total 0 1 V10s) #V20s #HV120s
+ @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/
+ @(HA … (des + 1)) /2 width=2 by drops_skip/
+ [ @lifts_applv //
+ elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
+ >(liftv_mono … HV12s … HV10s) -V1s //
+ | @(s0 … IHB … HB … HV120) /2 width=2 by drop_drop/
+ ]
+| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
+ @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
+]
+qed.
+
+lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
+ ∀L0,V0,W0,T0,des. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
+ ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
+ ) →
+ ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
+#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HL0 #H #HB
+lapply (acr_gcr … H1RP H2RP A) #HCA
+lapply (acr_gcr … H1RP H2RP B) #HCB
+elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+lapply (gcr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
+@(s3 … HCA … (◊))
+@(s6 … HCA … (◊) (◊)) //
+[ @(HA … HL0) //
+| lapply (s1 … HCB) -HCB #HCB
+ @(s7 … H2RP … (◊)) /2 width=1 by/
+]
+qed.
+
+(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
+(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⇳ [ break term 46 s ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'RAjust $L1 $T1 $s $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⇳ * [ break term 46 s ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'RAjustStar $L1 $T1 $s $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazybtpredproper_8.ma".
+include "basic_2/multiple/fleq.ma".
+include "basic_2/reduction/fpbu.ma".
+
+(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
+
+definition fpbc: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,g,G1,L1,T1,G2,L2,T2.
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄.
+
+interpretation
+ "'qrst' proper parallel reduction (closure)"
+ 'LazyBTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
+
+(* Baic properties **********************************************************)
+
+lemma fpbu_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by ex2_3_intro/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/multiple/fleq_fleq.ma".
+include "basic_2/reduction/fpbu_fleq.ma".
+include "basic_2/reduction/fpbc.ma".
+
+(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
+
+(* Properties on lazy equivalence on closures *******************************)
+
+lemma fpbc_fleq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+/3 width=9 by fleq_trans, ex2_3_intro/
+qed-.
+
+lemma fleq_fpbc_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 *
+#G0 #L0 #T0 #H0 #H02 elim (fleq_fpbu_trans … H1 … H0) -G -L -T
+/3 width=9 by fleq_trans, ex2_3_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazybtpredstarproper_8.ma".
+include "basic_2/reduction/fpbc.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,g. tri_TC … (fpbc h g).
+
+interpretation "'qrst' proper parallel computation (closure)"
+ 'LazyBTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed.
+
+lemma fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed.
+
+lemma fpbu_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=1 by fpbu_fpbc, fpbc_fpbg/ qed.
+
+(* Basic eliminators ********************************************************)
+
+lemma fpbg_ind: ∀h,g,G1,L1,T1. ∀R:relation3 ….
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
+qed-.
+
+lemma fpbg_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 ….
+ (∀G1,L1,T1. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/fpbc_fleq.ma".
+include "basic_2/computation/fpbg.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on lazy equivalence for closures ******************************)
+
+lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind … H) -G -L -T
+[ /3 width=5 by fpbc_fpbg, fpbc_fleq_trans/
+| /4 width=9 by fpbg_strap1, fpbc_fleq_trans/
+]
+qed-.
+
+lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx … H) -G -L -T
+[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/
+| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fpbs_fpbu.ma".
+include "basic_2/computation/fpbs_fpbc.ma".
+include "basic_2/computation/fpbs_fpbs.ma".
+include "basic_2/computation/fpbg_fpbs.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpbg_inv_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 * /3 width=5 by fleq_fpbs, ex2_3_intro/
+| #G1 #G #L1 #L #T1 #T *
+ #G0 #L0 #T0 #H10 #H0 #_ *
+ /5 width=9 by fpbu_fpbs, fpbs_trans, fleq_fpbs, ex2_3_intro/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind … H) -G2 -L2 -T2
+[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans … IH1) -IH1 ] (**) (* full auto fails *)
+/2 width=1 by fpbc_fpbs/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpbs_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+/5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-.
+
+(* Man properties ***********************************************************)
+
+theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
+/2 width=5 by tri_TC_transitive/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/lpxs_lleq.ma".
+include "basic_2/computation/fpbs_lift.ma".
+include "basic_2/computation/fpbg_fleq.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on "qrst" parallel reduction on closures **********************)
+
+lemma fpbg_fpb_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu … H2) -H2
+/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/
+qed-.
+
+lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
+/3 width=5 by fleq_fpbg_trans, fpbg_strap2, fpbu_fpbc/
+qed-.
+
+(* Properties on "qrst" parallel compuutation on closures *******************)
+
+lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/
+qed-.
+
+lemma fpbu_fpbs_fpbg: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
+/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/
+qed.
+
+lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+ (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/
+qed.
+
+lemma lstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, l2] T2 → (T1 = T2 → ⊥) →
+ ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
+
+lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/
+qed.
+
+lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+[ /2 width=1 by or_introl/
+| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu … H2) -H2 #H2
+ [ /3 width=5 by fleq_trans, or_introl/
+ | /5 width=5 by fpbc_fpbg, fleq_fpbc_trans, fpbu_fpbc, or_intror/
+ | /3 width=5 by fpbg_fleq_trans, or_intror/
+ | /4 width=5 by fpbg_strap1, fpbu_fpbc, or_intror/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/fpbu_lift.ma".
+include "basic_2/computation/fpbg.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma sta_fpbg: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
+ ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/4 width=2 by fpbu_fpbg, sta_fpbu/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/fpbc.ma".
+include "basic_2/computation/fpbs_fleq.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Properties on "qrst" proper parallel reduction for closures **************)
+
+lemma fpbc_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+/3 width=5 by fpbu_fwd_fpb, fpbs_strap2, fleq_fpbs/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻≡ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'LazyBTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/bteq_6.ma".
+include "basic_2/grammar/lenv_length.ma".
+include "basic_2/grammar/genv.ma".
+
+(* EQUIVALENT "BIG TREE" NORMAL FORMS ***************************************)
+
+definition bteq: tri_relation genv lenv term ≝
+ λG1,L1,T1,G2,L2,T2.
+ ∧∧ G1 = G2 & |L1| = |L2| & T1 = T2.
+
+interpretation
+ "equivalent 'big tree' normal forms (closure)"
+ 'BTEq G1 L1 T1 G2 L2 T2 = (bteq G1 L1 T1 G2 L2 T2).
+
+(* Basic_properties *********************************************************)
+
+lemma bteq_refl: tri_reflexive … bteq.
+/2 width=1 by and3_intro/ qed.
+
+lemma bteq_sym: tri_symmetric … bteq.
+#G1 #G2 #L1 #L2 #T1 #T2 * //
+qed-.
+
+lemma bteq_dec: ∀G1,G2,L1,L2,T1,T2. Decidable (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄).
+#G1 #G2 #L1 #L2 #T1 #T2 elim (genv_eq_dec G1 G2)
+#H1G [2: @or_intror * #H2G #H2L #H2T destruct /2 width=1 by/ ]
+elim (eq_nat_dec (|L1|) (|L2|))
+#H1L [2: @or_intror * #H2G #H2L #H2T destruct /2 width=1 by/ ]
+elim (term_eq_dec T1 T2)
+#H1T [2: @or_intror * #H2G #H2L #H2T destruct /2 width=1 by/ ]
+/3 width=1 by and3_intro, or_introl/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⋕ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTEq $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/bteq.ma".
+
+(* EQUIVALENT "BIG TREE" NORMAL FORMS ***************************************)
+
+(* Main properties **********************************************************)
+
+theorem bteq_trans: tri_transitive … bteq.
+#G1 #G #L1 #L #T1 #T * //
+qed-.
+
+theorem bteq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2. ⦃G, L, T⦄ ⋕ ⦃G1, L1, T1⦄ →
+ ⦃G, L, T⦄ ⋕ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
+/3 width=5 by bteq_trans, bteq_sym/ qed-.
+
+theorem bteq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T. ⦃G1, L1, T1⦄ ⋕ ⦃G, L, T⦄ →
+ ⦃G2, L2, T2⦄ ⋕ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
+/3 width=5 by bteq_trans, bteq_sym/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊢ ⋕ ➡ break [ term 46 h , break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTPRedSn $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+include "basic_2/reduction/fpn.ma".
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpb_bteq_fwd_fpn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
+[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
+ [ #H1 #H2 elim (fqu_fwd_bteq … H1 H2)
+ | * #HG #HL #HT #_ destruct //
+ ]
+| #T2 #HT12 * //
+]
+qed-.
--- /dev/null
+lemma fpb_fpbc_or_fpn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ ∨
+ ⦃G1, L1, T1⦄ ⊢ ➡[h,g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=1 by and3_intro, or_intror/
+[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H [| * ]
+ /3 width=1 by fpbc_fqu, and3_intro, or_introl, or_intror/
+| #T2 #HT12 elim (eq_term_dec T1 T2) #H destruct
+ /4 width=1 by and3_intro, or_introl, or_intror, fpbc_cpx/
+]
+qed-.
+
+lemma fpb_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) →
+ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 elim (fpb_fpbc_or_fpn … H) -H //
+#H elim H0 -H0 /2 width=3 by fpn_fwd_bteq/
+qed.
+
+lemma fpbc_fwd_bteq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=8 by fqu_fwd_bteq/
+#T2 #_ #HT12 * /2 width=1 by/
+qed-.
--- /dev/null
+(*
+inclade "basic_2/computation/fpns.ma".
+inclade "basic_2/computation/fpbs.ma".
+inclade "basic_2/reduction/fpbc.ma".
+
+lemma fpn_dec: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ Decidable (⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄).
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fpb_fpbc_or_fpn … H) -H /2 width=1 by or_introl/
+#H12 @or_intror
+#H @(fpbc_fwd_bteq … H12) -H12 @(fpn_fwd_bteq … H)
+qed-.
+*)
+(*
+lemma fpns_dec: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ Decidable (⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄).
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=1 by or_introl/
+#G #G2 #L #L2 #T #T2 #H #H2 *
+#H1 elim (fpn_dec … H2) -H2 #H2 /3 width=5 by fpns_strap1, or_introl/
+[ @or_intror #H12
+| @or_intror #H12 @H1 -H1
+*)
+(*
+inclade "basic_2/grammar/bteq_bteq.ma".
+inclade "basic_2/computation/fpns.ma".
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fpbs_bteq_fwd_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 //
+#G #G2 #L #L2 #T #T2 #H1 #H2 #IH1 #H12 elim (bteq_dec G1 G L1 L T1 T)
+[ -H1 /4 width=10 by fpns_strap1, fpb_bteq_fwd_fpn, bteq_canc_sn/
+| -IH1 #H
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/btpredsn_8.ma".
+include "basic_2/relocation/lleq.ma".
+include "basic_2/reduction/lpx.ma".
+
+(* REDUCTION FOR "BIG TREE" NORMAL FORMS ************************************)
+
+inductive fpn (h) (g) (G) (L1) (T): relation3 genv lenv term ≝
+| fpn_intro: ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → L1 ⋕[T] L2 → fpn h g G L1 T G L2 T
+.
+
+interpretation
+ "reduction for 'big tree' normal forms (closure)"
+ 'BTPRedSn h g G1 L1 T1 G2 L2 T2 = (fpn h g G1 L1 T1 G2 L2 T2).
+
+(* Basic_properties *********************************************************)
+
+lemma fpn_refl: ∀h,g. tri_reflexive … (fpn h g).
+/2 width=1 by fpn_intro/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fpn_inv_gen: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡[h, g] L2 & L1 ⋕[T1] L2 & T1 = T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and4_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/lpx_lpx.ma".
+include "basic_2/reduction/fpn.ma".
+
+(* REDUCTION FOR "BIG TREE" NORMAL FORMS ************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fpn_fqu_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
+#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
+#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqu_trans … H12 … HKL1 HV1) -L1
+/3 width=5 by fpn_intro, ex2_3_intro/
+qed-.
+
+lemma fpn_fquq_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃⸮ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
+#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
+#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fquq_trans … H12 … HKL1 HV1) -L1
+/3 width=5 by fpn_intro, ex2_3_intro/
+qed-.
+
+lemma fpn_fqup_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃+ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
+#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
+#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqup_trans … H12 … HKL1 HV1) -L1
+/3 width=5 by fpn_intro, ex2_3_intro/
+qed-.
+
+lemma fpn_fqus_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃* ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
+#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
+#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqus_trans … H12 … HKL1 HV1) -L1
+/3 width=5 by fpn_intro, ex2_3_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/btpredsnstar_8.ma".
+include "basic_2/reduction/fpn.ma".
+
+(* COMPUTATION FOR "BIG TREE" NORMAL FORMS **********************************)
+
+definition fpns: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,g. tri_TC … (fpn h g).
+
+interpretation
+ "computation for 'big tree' normal forms (closure)"
+ 'BTPRedSnStar h g G1 L1 T1 G2 L2 T2 = (fpns h g G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fpns_ind: ∀h,g,G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+lapply (tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
+qed-.
+
+lemma fpns_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
+qed-.
+
+(* Basic_properties *********************************************************)
+
+lemma fpns_refl: ∀h,g. tri_reflexive … (fpns h g).
+/2 width=1 by tri_inj/ qed.
+
+lemma fpn_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fpns_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed-.
+
+lemma fpns_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lleq_lleq.ma".
+include "basic_2/computation/lpxs.ma".
+include "basic_2/computation/fpns.ma".
+
+(* COMPUTATION FOR "BIG TREE" NORMAL FORMS **********************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpns_inv_gen: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 & L1 ⋕[T1] L2 & T1 = T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpns_ind … H) -G2 -L2 -T2 /2 width=1 by and4_intro/
+#G #G2 #L #L2 #T #T2 #_ #H2 * #HG #HL1 #HT1 #HT destruct
+elim (fpn_inv_gen … H2) -H2 #HG #HL2 #HT #HT2 destruct
+/3 width=3 by lpxs_strap1, lleq_trans, and4_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lpxs_lleq_fpns: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → L1 ⋕[T] L2 →
+ ⦃G, L1, T⦄ ⊢ ⋕➡*[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2 //
+#L #L2 #HL1 #HL2 #IHL1 #HL12 elim (lleq_dec T L1 L) #HT
+[ -HL1 @fpns_strap1 [4: @IHL1 // |1,2,3: skip ]
+ /3 width=3 by fpn_intro, lleq_canc_sn/
+| -IHL1
+
+(* Main properties **********************************************************)
+
+theorem fpns_trans: tri_transitive … fqus.
+/2 width=5 by tri_TC_transitive/ qed-.
+
+
+(*
+lemma fpns_refl: ∀h,g. tri_reflexive … (fpns h g).
+/2 width=1 by and3_intro/ qed.
+
+lemma fpn_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=1 by lpx_lpxs, and3_intro/
+qed.
+
+lemma fpns_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * #H1G #H1L #G1T *
+/3 width=3 by lpxs_strap1, and3_intro/
+qed-.
+
+lemma fpns_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * #H1G #H1L #G1T *
+/3 width=3 by lpxs_strap2, and3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpns_fwd_bteq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=4 by lpxs_fwd_length, and3_intro/
+qed-.
+*)
--- /dev/null
+include "basic_2/grammar/bteq.ma".
+
+lemma fqu_fwd_bteq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V * #_ #H elim (plus_xSy_x_false … H)
+| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_x … H)
+| #a #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y … H)
+| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y … H)
+| #G #L #K #T #U #e #HLK #_ * #_ #H
+ lapply (ldrop_fwd_length_lt4 … HLK ?) // >H -L #H
+ elim (lt_refl_false … H)
+]
+qed-.
--- /dev/null
+(* Basic eliminators ********************************************************)
+
+theorem fsb_ind_alt: ∀h,g. ∀R: relation3 …. (
+ ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h,g] T1 → (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → R G2 L2 T2
+ ) → R G1 L1 T1
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → R G L T.
+#h #g #R #IH #G #L #T #H elim H -G -L -T
+/5 width=1 by fpb_fpbc, fsb_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/btsnalt_5.ma".
+include "basic_2/computation/fpbs_fpbs.ma".
+include "basic_2/computation/fsb.ma".
+
+(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
+
+(* Note: alternative definition of fsb *)
+inductive fsba (h) (g): relation3 genv lenv term ≝
+| fsba_intro: ∀G1,L1,T1. (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
+ ) → fsba h g G1 L1 T1.
+
+interpretation
+ "'big tree' strong normalization (closure) alternative"
+ 'BTSNAlt h g G L T = (fsba h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+theorem fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
+ ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h,g] T1 → (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → R G2 L2 T2
+ ) → R G1 L1 T1
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → R G L T.
+#h #g #R #IH #G #L #T #H elim H -G -L -T
+/5 width=1 by fsba_intro/
+qed-.
+
+(* Basic_properties *********************************************************)
+
+fact fsba_intro_aux: ∀h,g,G1,L1,T1. (
+ ∀G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⋕ ⦃G, L, T⦄ →
+ (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
+ ) → fsba h g G1 L1 T1.
+/4 width=5 by fsba_intro/ qed-.
+
+lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1 →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2.
+#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
+#G1 #L1 #T1 #H0 #IH0 #G #L #T #H1 @fsba_intro
+#G2 #L2 #T2 #H2 #_ lapply (fpbs_trans … H1 … H2) -G -L -T
+#H12 elim (bteq_dec G1 G2 L1 L2 T1 T2) /3 width=6 by fpb_fpbs/
+-IH0 #H212
+
+
+ -H0 -H #H @(IH0 … H) -IH0 -H // @(fpbs_trans … H1 … H2)
+
+lemma fsba_intro_fpb: ∀h,g,G1,L1,T1. (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2
+ ) → ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1.
+#h #g #G1 #L1 #T1 #IH1 @fsba_intro_aux
+#G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T
+[ #H1 #H2 -IH1 elim H2 -H2 //
+| #G0 #G #L0 #L #T0 #T #H10 #H12 #IH2 #H210 #H212 elim (bteq_dec G1 G L1 L T1 T)
+ [ -IH1 -H210 -H10 -H12 /3 width=1 by/
+ | -IH2 -H212 #H21 lapply (IH1 … H21) -IH1 -H21
+ [
+ | -H10 -H210 #H
+(*
+(* Main inversion lemmas ****************************************************)
+
+theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
+#h #g #G #L #T #H elim H -G -L -T
+/5 width=12 by fsb_intro, fpb_fpbs, fpbc_fwd_fpb, fpbc_fwd_gen/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fsb_fsba: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥⦥[h, g] T.
+#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
+/4 width=1 by fsba_intro_fpb/
+qed.
+(*
+| fsba_intro: ∀G1,L1,T1. (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → fsb h g G2 L2 T2
+ ) → fsb h g G1 L1 T1
+.
+
+
+
+(****************************************************************************)
+
+include "basic_2/substitution/fqup.ma".
+
+lemma fsb_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
+#h #g #G #L #T #H @(csx_ind … H) -T
+*)*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⋕ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'LazyEq $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }.
+
+include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/fpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+definition cfpr: lenv → bi_relation lenv term ≝
+ λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2.
+
+interpretation
+ "context-sensitive parallel reduction (closure)"
+ 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L).
+/2 width=1/ qed.
+
+lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 * /3 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆.
+#L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V2. V1 ➡ V2 &
+ ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
+ L2 = ⋆.ⓑ{I}V2@@K2.
+#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H
+elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
+>shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H
+elim (tpr_inv_bind1 … H) -H *
+[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/
+| #T0 #_ #_ #H destruct
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_aaa.ma".
+include "basic_2/reducibility/cfpr_cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_fpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #HT1 #L2 #T2 #H
+elim (fpr_inv_all … H) -H
+/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_cpr.ma".
+include "basic_2/reducibility/cfpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Main properties **********************************************************)
+
+theorem cfpr_conf: ∀L. bi_confluent … (cfpr L).
+#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
+elim (cpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
+elim (cpr_fwd_shift1 … H1) #L0 #T0 #HL10 #H destruct /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_tpss.ma".
+include "basic_2/reducibility/cpr_cpr.ma".
+include "basic_2/reducibility/cfpr_ltpss.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 →
+ L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+#L1 #L #H elim H -L1 -L
+[ #L2 #T1 #T2 #HT12 #HL2
+ lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct
+ lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/
+| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H
+ elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (cpr_bind_dx false … HV1 HT12) -HV1 -HT12 #HT12
+ lapply (cpr_tpss_trans … HT12 (-ⓑ{I}V2.T2) 0 (|L|) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cfpr_inv_all: ∀L1,L2,L0,T1,T2. L0 ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃L. L0 @@ L1 ➡ L0 @@ L & L0 @@ L ⊢ T1 ➡ T2 &
+ L0 @@ L ⊢ ▶* [0, |L0| + |L|] L0 @@ L2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L2 #L0 #T1 #T2 #H
+ elim (cfpr_inv_atom1 … H) -H #HT12 #H destruct /3 width=4/
+| #I #L1 #V1 #IH #X #L0 #T1 #T2 #H
+ elim (cfpr_inv_pair1 … H) -H #L2 #V #V2 #HV1 #HV2 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L #HL1 #HT12 #HL2
+ elim (ltpr_inv_append1 … HL1) -HL1 #X #Y #HX #HY #H
+ lapply (ltpr_fwd_length … HX) -HX #HX
+ elim (append_inj_dx … H ?) -H // -HX #_ #H destruct -X
+ lapply (ltpss_sn_fwd_length … HL2) >append_length >append_length #H
+ lapply (injective_plus_r … H) -H #H
+ @(ex3_intro … (⋆.ⓑ{I}V@@Y)) <append_assoc // -HT12
+ <append_assoc [ /3 width=1/ ] -HV1 -HY
+ >append_length <associative_plus
+ @(ltpss_sn_dx_trans_eq … HL2) -HL2 >H -H >commutative_plus /3 width=1/
+]
+qed-.
+
+lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2.
+#L1 #L2 #T1 #T2 #H
+lapply (fpr_cfpr … H) -H #H
+elim (cfpr_inv_all … H) -H /2 width=4/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cpr_lift.ma".
+include "basic_2/reducibility/cpr_ltpss_sn.ma".
+include "basic_2/reducibility/cfpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cfpr_inv_pair1: ∀I,L,K1,L2,V1,T1,T2. L ⊢ ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V,V2. V1 ➡ V & L ⊢ V ▶* [0, |L|] V2 &
+ L.ⓑ{I}V ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ &
+ L2 = ⋆.ⓑ{I}V2@@K2.
+* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H
+elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
+>shift_append_assoc >shift_append_assoc normalize in ⊢ (??%%→?); #H
+[ elim (cpr_inv_abbr1 … H) -H *
+ [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/
+ | #T0 #_ #_ #H destruct
+ ]
+| elim (cpr_inv_abst1 … H Abst V2) -H
+ #V #T * #V0 #HV10 #HV0 #HT1 #H destruct
+ lapply (ltpss_sn_cpr_trans (L.ⓛV0) … 0 (|L|+1) … HT1) -HT1 /2 width=1/ #HT12
+ /3 width=7/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }.
+
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }.
+
+include "basic_2/reducibility/fpr.ma".
+
+(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
+
+definition fpc: bi_relation lenv term ≝
+ λL1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ ∨ ⦃L2, T2⦄ ➡ ⦃L1, T1⦄.
+
+interpretation
+ "context-free parallel conversion (closure)"
+ 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpc_refl: bi_reflexive … fpc.
+/2 width=1/ qed.
+
+lemma fpc_sym: bi_symmetric … fpc.
+#L1 #L2 #T1 #T2 * /2 width=1/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpc_fwd_fpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄.
+#L1 #L2 #T1 #T2 * /2 width=4/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/conversion/fpc.ma".
+
+(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
+
+(* Main properties **********************************************************)
+
+theorem fpc_conf: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L0, T0⦄ ⬌ ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
+/3 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }.
+
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }.
+
+include "basic_2/conversion/fpc.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+definition fpcs: bi_relation lenv term ≝ bi_TC … fpc.
+
+interpretation "context-free parallel equivalence (closure)"
+ 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fpcs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
+ (∀L,L2,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L2 T2.
+/3 width=7 by bi_TC_star_ind/ qed-.
+
+lemma fpcs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L1 T1.
+/3 width=7 by bi_TC_star_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpcs_refl: bi_reflexive … fpcs.
+/2 width=1/ qed.
+
+lemma fpcs_sym: bi_symmetric … fpcs.
+/3 width=1/ qed.
+
+lemma fpc_fpcs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fpcs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fpcs_fpr_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=1/ qed.
+
+lemma fpcs_fpr_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡ ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=1/ qed.
+
+lemma fpcs_fpr_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
+ ∀L2,T2. ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma fpcs_fpr_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ →
+ ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma fpcs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
+ ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed-.
+
+lemma fpcs_fpr_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fprs_aaa.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Main properties about atomic arity assignment on terms *******************)
+
+theorem aaa_fpcs_mono: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
+ ∀A1. L1 ⊢ T1 ⁝ A1 → ∀A2. L2 ⊢ T2 ⁝ A2 →
+ A1 = A2.
+#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+lapply (aaa_fprs_conf … HT1 … H1) -L1 -T1 #HT1
+lapply (aaa_fprs_conf … HT2 … H2) -L2 -T2 #HT2
+lapply (aaa_mono … HT1 … HT2) -L -T //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fprs_cprs.ma".
+include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fpcs_flat_dx_tpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
+ ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⬌* ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 #HT12
+elim (fpcs_inv_fprs … HT12) -HT12
+/3 width=6 by fprs_flat_dx_tpr, fprs_div/ (**) (* auto too slow without trace *)
+qed.
+
+lemma fpcs_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
+ ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄.
+#I #L1 #L2 #V1 #V2 #T1 #T2 #H12
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+elim (fprs_bind2_minus … H1) -H1 #W1 #U1 #HTU1 #H destruct
+elim (fprs_bind2_minus … H2) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpcs_inv_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
+ ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
+#I #L1 #L2 #V1 #V2 #T1 #T2 #H12
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
+elim (fprs_inv_pair1 … H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fpcs_fwd_bind_minus: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
+ ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
+#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+elim (fprs_fwd_bind2_minus … H1 b) -H1 #W1 #U1 #HTU1 #H destruct
+elim (fprs_fwd_bind2_minus … H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
+qed-.
+
+lemma fpcs_fwd_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
+ ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
+/3 width=1 by fpcs_inv_shift, fpcs_fwd_bind_minus/ qed-.
+
+lemma fpcs_fwd_abst24: ∀a,L1,L2,V1,V2,T1,T2. ⦃L1, ⓛ{a}V1.T1⦄ ⬌* ⦃L2, ⓛ{a}V2.T2⦄ →
+ ∀b,I,W. ⦃L1, ⓑ{b,I}W.T1⦄ ⬌* ⦃L2, ⓑ{b,I}W.T2⦄.
+#a #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b #I #W
+elim (fpcs_inv_fprs … H12) -H12 #L #U #H1 #H2
+elim (fprs_fwd_abst2 … H1 b I W) -H1 #W1 #U1 #HTU1 #H destruct
+elim (fprs_fwd_abst2 … H2 b I W) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
+qed-.
+
+lemma fpcs_fwd_abst13: ∀L1,L2,V1,V2,T1,T2. ⦃L1.ⓛV1, T1⦄ ⬌* ⦃L2.ⓛV2, T2⦄ →
+ ∀I,W. ⦃L1.ⓑ{I}W, T1⦄ ⬌* ⦃L2.ⓑ{I}W, T2⦄.
+/4 width=4 by fpcs_fwd_shift, fpcs_fwd_abst24, fpcs_shift/ qed-.
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
+#L #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
+qed.
+
+(* Inversion lemmas on context-sensitive parallel equivalence for terms *****)
+
+lemma fpcs_inv_cpcs: ∀L,T1,T2. ⦃L, T1⦄ ⬌* ⦃L, T2⦄ → L ⊢ T1 ⬌* T2.
+#L #T1 #T2 #H
+elim (fpcs_inv_fprs … H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fprs_fprs.ma".
+include "basic_2/conversion/fpc_fpc.ma".
+include "basic_2/equivalence/fpcs_fprs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpcs_inv_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡* ⦃L, T⦄.
+#L1 #L2 #T1 #T2 #H @(fpcs_ind … H) -L2 -T2
+[ /3 width=4/
+| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
+ [ elim (fprs_strip … HT2 … HT0) -L -T #L #T #HT2 #HT0
+ lapply (fprs_strap1 … HT10 … HT0) -L0 -T0 /2 width=4/
+ | lapply (fprs_strap2 … HT2 … HT0) -L -T /2 width=4/
+ ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
+elim (fprs_strip … HT2 … HT1) /2 width=4 by fpr_fprs_div/
+qed-.
+
+lemma fprs_fpr_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L2, T2⦄ ⬌* ⦃L1, T1⦄.
+#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
+elim (fprs_strip … HT2 … HT1) /2 width=4 by fprs_fpr_div/
+qed-.
+
+lemma fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
+elim (fprs_conf … HT1 … HT2) /2 width=4/
+qed-.
+
+lemma fpcs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L0, T0⦄ ⬌* ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
+/3 width=4/ qed.
+
+(* Main properties **********************************************************)
+
+theorem fpcs_trans: bi_transitive … fpcs.
+/2 width=4/ qed.
+
+theorem fpcs_canc_sn: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ⬌* ⦃L1, T1⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
+
+theorem fpcs_canc_dx: ∀L1,L2,L,T1,T2,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L2, T2⦄ ⬌* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fprs.ma".
+include "basic_2/equivalence/fpcs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Properties on context-free parallel computation for closures *************)
+
+(* Note: was lemma 1000 *)
+lemma fpcs_fprs_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /3 width=4/
+qed.
+
+lemma fpcs_fprs_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡* ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /3 width=4/
+qed.
+
+lemma fpcs_fprs_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /2 width=4/
+qed.
+
+lemma fpcs_fprs_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx … H) -L1 -T1 /width=1/ /2 width=4/
+qed.
+
+lemma fpcs_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /2 width=4/
+qed.
+
+lemma fpcs_fprs_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind … H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *)
+qed.
+
+lemma fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx … H) -L2 -T2 /2 width=1/ /2 width=4/
+qed.
+
+lemma fprs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=7 by bi_step, fprs_div/ qed-.
+
+lemma fpr_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4 by bi_step, fprs_div/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }.
+
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
+
+definition fpr: bi_relation lenv term ≝
+ λL1,T1,L2,T2. |L1| = |L2| ∧ L1 @@ T1 ➡ L2 @@ T2.
+
+interpretation
+ "context-free parallel reduction (closure)"
+ 'FocalizedPRed L1 T1 L2 T2 = (fpr L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpr_refl: bi_reflexive … fpr.
+/2 width=1/ qed.
+
+lemma fpr_shift: ∀I1,I2,L1,L2,V1,V2,T1,T2.
+ ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
+ ⦃L1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2.ⓑ{I2}V2, T2⦄.
+#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12
+@conj // normalize // (**) (* explicit constructor *)
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fpr_inv_atom1: ∀L2,T1,T2. ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → T1 ➡ T2 ∧ L2 = ⋆.
+#L2 #T1 #T2 * #H
+lapply (length_inv_zero_sn … H) -H #H destruct /2 width=1/
+qed-.
+
+lemma fpr_inv_atom3: ∀L1,T1,T2. ⦃L1,T1⦄ ➡ ⦃⋆,T2⦄ → T1 ➡ T2 ∧ L1 = ⋆.
+#L1 #T1 #T2 * #H
+lapply (length_inv_zero_dx … H) -H #H destruct /2 width=1/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpr_fwd_pair1: ∀I1,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃I2,K2,V2. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
+ L2 = K2.ⓑ{I2}V2.
+#I1 #K1 #L2 #V1 #T1 #T2 * #H
+elim (length_inv_pos_sn … H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/
+qed-.
+
+lemma fpr_fwd_pair3: ∀I2,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I2}V2, T2⦄ →
+ ∃∃I1,K1,V1. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄ &
+ L1 = K1.ⓑ{I1}V1.
+#I2 #L1 #K2 #V2 #T1 #T2 * #H
+elim (length_inv_pos_dx … H) -H #I1 #K1 #V1 #HK12 #H destruct /3 width=5/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cfpr_cpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma ltpr_tpr_fpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
+/2 width=4/ qed.
+
+lemma fpr_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ →
+ ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
+ ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ ∃∃L2. ⦃L1, U1⦄ ➡ ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
+#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
+elim (fpr_inv_all … HT12) -HT12 #K #HK1 #HT12 #HK2
+elim (ldrop_ltpr_trans … HLK1 … HK1) -K1 #L #HL1 #HLK
+lapply (cpr_lift … HLK … HTU1 … HTU2 HT12) -T1 -T2 #HU12
+elim (le_or_ge (|K|) d) #Hd
+[ elim (ldrop_ltpss_sn_trans_ge … HLK … HK2 …)
+| elim (ldrop_ltpss_sn_trans_be … HLK … HK2 …)
+] // -Hd #L2 #HL2 #HLK2
+lapply (ltpss_sn_weak_full … HL2) -K /3 width=4/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_flat_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
+ ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 #HT12
+elim (fpr_inv_all … HT12) -HT12 /4 width=4/
+qed.
+
+lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
+ ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
+#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
+elim (fpr_inv_all … H) /3 width=4/
+qed.
+
+lemma fpr_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L2, U2⦄ →
+ ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡ ⦃L2.ⓑ{I}V2, T2⦄ &
+ U2 = -ⓑ{I}V2.T2.
+#I1 #L1 #L2 #V1 #T1 #U2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+elim (cpr_fwd_bind1_minus … H false) -H /4 width=4/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
+ T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b
+elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
+elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
+qed-.
+
+lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
+ ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
+ ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
+* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+[ elim (cpr_inv_abbr1 … H) -H *
+ [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
+ | #T #_ #_ #H destruct
+ ]
+| elim (cpr_inv_abst1 … H Abst V2) -H
+ #V #T #HV1 #_ #H destruct /3 width=4/
+]
+qed-.
+
+lemma fpr_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡ ⦃L2, U2⦄ → ∀b,I,W.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡ ⦃L2, ⓑ{b,I}W.T2⦄ &
+ U2 = ⓛ{a}V2.T2.
+#a #L1 #L2 #V1 #T1 #U2 #H
+elim (fpr_inv_all … H) #L #HL1 #H #HL2 #b #I #W
+elim (cpr_fwd_abst1 … H b I W) -H /3 width=4/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I1 #K1 #X #V1 #T1 #T2 #H
+elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
+elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
+qed-.
+
+lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
+ ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L1 = K1.ⓑ{I}V1.
+#I2 #X #K2 #V2 #T1 #T2 #H
+elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
+elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
+qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/tpr_tpr.ma".
+include "basic_2/reducibility/fpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
+
+(* Main properties **********************************************************)
+
+theorem fpr_conf: bi_confluent … fpr.
+#L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02
+elim (tpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2
+elim (tpr_fwd_shift1 … H1) #L #T #HL1 #H destruct /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }.
+
+notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRedStarAlt $L1 $T1 $L2 $T2 }.
+
+include "basic_2/reducibility/fpr.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
+
+definition fprs: bi_relation lenv term ≝ bi_TC … fpr.
+
+interpretation "context-free parallel computation (closure)"
+ 'FocalizedPRedStar L1 T1 L2 T2 = (fprs L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fprs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
+ (∀L,L2,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L2 T2.
+/3 width=7 by bi_TC_star_ind/ qed-.
+
+lemma fprs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L1 T1.
+/3 width=7 by bi_TC_star_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpr_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma fprs_refl: bi_reflexive … fprs.
+/2 width=1/ qed.
+
+lemma fprs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fprs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+/2 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/cfpr_aaa.ma".
+include "basic_2/computation/fprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_fprs_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #HT1 #L2 #T2 #HLT12
+@(bi_TC_Conf3 … HT1 ?? HLT12) /2 width=4/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/fpr_cpr.ma".
+include "basic_2/computation/cprs_lfprs.ma".
+include "basic_2/computation/lfprs_ltprs.ma".
+include "basic_2/computation/lfprs_fprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fprs_flat_dx_tpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
+ ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡* ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 #HT12 @(fprs_ind … HT12) -L2 -T2 /3 width=1/
+#L #L2 #T #T2 #_ #HT2 #IHT2 #V1 #V2 #HV12 #I
+lapply (IHT2 … HV12 I) -IHT2 -HV12 /3 width=6/
+qed.
+
+lemma fprs_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L2, U2⦄ →
+ ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡* ⦃L2.ⓑ{I}V2, T2⦄ &
+ U2 = -ⓑ{I}V2.T2.
+#I #L1 #L2 #V1 #T1 #U2 #H @(fprs_ind … H) -L2 -U2 /2 width=4/
+#L #L2 #U #U2 #_ #HU2 * #V #T #HT1 #H destruct
+elim (fpr_bind2_minus … HU2) -HU2 /3 width=4/
+qed-.
+
+lemma fprs_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡* ⦃K2, T2⦄ →
+ ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
+ ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ ∃∃L2. ⦃L1, U1⦄ ➡* ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
+#K1 #K2 #T1 #T2 #HT12 @(fprs_ind … HT12) -K2 -T2
+[ #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
+ >(lift_mono … HTU2 … HTU1) -U2 /2 width=3/
+| #K #K2 #T #T2 #_ #HT2 #IHT1 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
+ elim (lift_total T d e) #U #HTU
+ elim (IHT1 … HLK1 … HTU1 HTU) -K1 -T1 #L #HU1 #HKL
+ elim (fpr_lift … HT2 … HKL … HTU HTU2) -K -T -T2 /3 width=4/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fprs_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
+ ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /2 width=5/
+#L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct
+elim (fpr_inv_pair1 … HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/
+qed-.
+
+lemma fprs_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡* ⦃K2.ⓑ{I}V2, T2⦄ →
+ ∃∃K1,V1. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L1 = K1.ⓑ{I}V1.
+#I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx … H) -L1 -T1 /2 width=5/
+#L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct
+elim (fpr_inv_pair3 … HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fprs_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L, T⦄ → ∀b.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡* ⦃L, ⓑ{b,I}V2.T2⦄ &
+ T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind … H1) -L -T /2 width=4/
+#L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct
+elim (fpr_fwd_bind2_minus … H0 b) -H0 /3 width=4/
+qed-.
+
+lemma fprs_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
+ ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, ⓑ{b,I}V1.T1⦄ ➡* ⦃K2, ⓑ{b,I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fprs_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fprs_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
+qed-.
+
+lemma fprs_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡* ⦃L2, U2⦄ → ∀b,I,W.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡* ⦃L2, ⓑ{b,I}W.T2⦄ &
+ U2 = ⓛ{a}V2.T2.
+#a #L1 #L2 #V1 #T1 #U2 #H #b #I #W @(fprs_ind … H) -L2 -U2 /2 width=4/
+#L #L2 #U #U2 #_ #H2 * #V #T #HT1 #H destruct
+elim (fpr_fwd_abst2 … H2 b I W) -H2 /3 width=4/
+qed-.
+
+(* Properties on context-sensitive parallel computation for terms ***********)
+
+lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄.
+#L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/
+qed.
+
+(* Forward lemmas on context-sensitive parallel computation for terms *******)
+
+lemma fprs_fwd_cprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 //
+#L #L2 #T #T2 #H1 #H2 #IH1
+elim (fpr_inv_all … H2) -H2 #L0 #HL0 #HT2 #_ -L2
+lapply (lfprs_cpr_trans L1 … HT2) -HT2 /3 width=3/
+qed-.
+(*
+(* Advanced properties ******************************************************)
+
+lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
+ ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
+#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
+elim (fpr_inv_all … H) /3 width=4/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
+ ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
+ ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
+* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+[ elim (cpr_inv_abbr1 … H) -H *
+ [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
+ | #T #_ #_ #H destruct
+ ]
+| elim (cpr_inv_abst1 … H Abst V2) -H
+ #V #T #HV1 #_ #H destruct /3 width=4/
+]
+qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/fpr_fpr.ma".
+include "basic_2/computation/fprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fprs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ➡ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L0, T0⦄ ➡* ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄.
+#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8
+/2 width=4/ qed.
+
+(* Main propertis ***********************************************************)
+
+theorem fprs_conf: bi_confluent … fprs.
+/2 width=4/ qed.
+
+theorem fprs_trans: bi_transitive … fprs.
+/2 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_length.ma".
+
+(* POINTWISE EXTENSION OF A FOCALIZED REALTION FOR TERMS ********************)
+
+inductive lpx_bi (R:bi_relation lenv term): relation lenv ≝
+| lpx_bi_stom: lpx_bi R (⋆) (⋆)
+| lpx_bi_pair: ∀I,K1,K2,V1,V2.
+ lpx_bi R K1 K2 → R K1 V1 K2 V2 →
+ lpx_bi R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_bi_inv_atom1_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_bi_inv_atom1: ∀R,L2. lpx_bi R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_bi_inv_atom1_aux/ qed-.
+
+fact lpx_bi_inv_pair1_aux: ∀R,L1,L2. lpx_bi R L1 L2 →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. lpx_bi R K1 K2 &
+ R K1 V1 K2 V2 & L2 = K2. ⓑ{I} V2.
+#R #L1 #L2 * -L1 -L2
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
+]
+qed-.
+
+lemma lpx_bi_inv_pair1: ∀R,I,K1,V1,L2. lpx_bi R (K1. ⓑ{I} V1) L2 →
+ ∃∃K2,V2. lpx_bi R K1 K2 & R K1 V1 K2 V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_bi_inv_pair1_aux/ qed-.
+
+fact lpx_bi_inv_atom2_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_bi_inv_atom2: ∀R,L1. lpx_bi R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_bi_inv_atom2_aux/ qed-.
+
+fact lpx_bi_inv_pair2_aux: ∀R,L1,L2. lpx_bi R L1 L2 →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 &
+ L1 = K1. ⓑ{I} V1.
+#R #L1 #L2 * -L1 -L2
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
+]
+qed-.
+
+lemma lpx_bi_inv_pair2: ∀R,I,L1,K2,V2. lpx_bi R L1 (K2. ⓑ{I} V2) →
+ ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_bi_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_bi_fwd_length: ∀R,L1,L2. lpx_bi R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_bi_refl: ∀R. bi_reflexive ? ? R → reflexive … (lpx_bi R).
+#R #HR #L elim L -L // /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L1 ⊢ ➡ ➡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSnAlt $L1 $L2 }.
+
+notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPRedAlt $L1 $L2 }.
+
+include "basic_2/grammar/lenv_px_bi.ma".
+include "basic_2/reducibility/fpr_cpr.ma".
+include "basic_2/reducibility/lfpr_fpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
+
+(* alternative definition *)
+definition lfpra: relation lenv ≝ lpx_bi fpr.
+
+interpretation
+ "focalized parallel reduction (environment) alternative"
+ 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lfpra_refl: reflexive … lfpra.
+/2 width=1/ qed.
+
+lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
+#L1 elim L1 -L1
+[ #L2 #T1 #T2 #H
+ elim (fpr_inv_atom1 … H) -H #_ #H destruct //
+| #L1 #I #V1 #IH #L2 #T1 #T2 #H
+ elim (fpr_inv_pair1 … H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆.
+/2 width=2 by lpx_bi_inv_atom1/ qed-.
+
+lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ →
+ ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ L2 = K2. ⓑ{I} V2.
+/2 width=1 by lpx_bi_inv_pair1/ qed-.
+
+lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆.
+/2 width=2 by lpx_bi_inv_atom2/ qed-.
+
+lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ →
+ ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ L1 = K1. ⓑ{I} V1.
+/2 width=1 by lpx_bi_inv_pair2/ qed-.
+
+lemma lfpra_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ∀T.⦃L1, T⦄ ➡ ⦃L2, T⦄.
+#L1 #L2 * -L1 -L2 // /3 width=1/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|.
+/2 width=2 by lpx_bi_fwd_length/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lfpr_lfpra: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
+#L1 #L2 #H
+lapply (lfpr_inv_fpr … H (⋆0)) -H /2 width=3/
+qed.
+
+theorem lfpra_lfpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
+#L1 #L2 #H
+lapply (lfpra_inv_fpr … H (⋆0)) -H /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/lfpr.ma".
+include "basic_2/reducibility/cfpr_cpr.ma".
+
+(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Properties on context-free parallel reduction for closures ***************)
+
+lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
+#L1 #L2 #T1 #T2 #H
+elim (fpr_inv_all … H) -H /2 width=3/
+qed.
+
+(* Inversion lemmas on context-free parallel reduction for closures *********)
+
+lemma lfpr_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡ ⦃L2, T⦄.
+#L1 #L2 * /2 width=4/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/lfpr_fpr.ma".
+include "basic_2/computation/fprs_fprs.ma".
+include "basic_2/computation/lfprs.ma".
+
+(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
+
+(* Inversion lemmas on context-free parallel reduction for closures *********)
+
+lemma lfprs_inv_fprs: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡* ⦃L2, T⦄.
+#L1 #L2 #H @(lfprs_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 #T
+lapply (lfpr_inv_fpr … HL2 T) -HL2 /3 width=4/
+qed-.
+
+(* Properties on context-free parallel computation for closures *************)
+
+lemma fprs_lfprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1⦄ ➡* ⦃L2⦄.
+#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 // /3 width=5/
+qed.
+
+lemma lfprs_fprs_trans: ∀L1,L,L2,T1,T2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+#L1 #L #L2 #T1 #T2 #HL1 #HL2
+lapply (lfprs_inv_fprs … HL1 T1) -HL1 /2 width=4/
+qed.
+(*
+lamma lfprs_cprs_conf: ∀L1,L,L2,T1,T2. ⦃L1⦄ ➡* ⦃L2⦄ → L1 ⊢ T1 ➡* T2 → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'RestSupTerm $L1 $T1 $L2 $T2 }.
+
+include "basic_2/grammar/cl_weight.ma".
+include "basic_2/substitution/lift.ma".
+
+(* RESTRICTED SUPCLOSURE ****************************************************)
+
+inductive frsup: bi_relation lenv term ≝
+| frsup_bind_sn: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) L V
+| frsup_bind_dx: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T
+| frsup_flat_sn: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L V
+| frsup_flat_dx: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L T
+.
+
+interpretation
+ "restricted structural predecessor (closure)"
+ 'RestSupTerm L1 T1 L2 T2 = (frsup L1 T1 L2 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact frsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
+ ∀J. T1 = ⓪{J} → ⊥.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a #I #L #V #T #J #H destruct
+| #a #I #L #V #T #J #H destruct
+| #I #L #V #T #J #H destruct
+| #I #L #V #T #J #H destruct
+]
+qed-.
+
+lemma frsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁ ⦃L2, T2⦄ → ⊥.
+/2 width=7 by frsup_inv_atom1_aux/ qed-.
+
+fact frsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
+ ∀b,J,W,U. T1 = ⓑ{b,J}W.U →
+ (L2 = L1 ∧ T2 = W) ∨
+ (L2 = L1.ⓑ{J}W ∧ T2 = U).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
+| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
+| #I #L #V #T #b #J #W #U #H destruct
+| #I #L #V #T #b #J #W #U #H destruct
+]
+qed-.
+
+lemma frsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁ ⦃L2, T2⦄ →
+ (L2 = L1 ∧ T2 = W) ∨
+ (L2 = L1.ⓑ{J}W ∧ T2 = U).
+/2 width=4 by frsup_inv_bind1_aux/ qed-.
+
+fact frsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
+ ∀J,W,U. T1 = ⓕ{J}W.U →
+ L2 = L1 ∧ (T2 = W ∨ T2 = U).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a #I #L #V #T #J #W #U #H destruct
+| #a #I #L #V #T #J #W #U #H destruct
+| #I #L #V #T #J #W #U #H destruct /3 width=1/
+| #I #L #V #T #J #W #U #H destruct /3 width=1/
+]
+qed-.
+
+lemma frsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁ ⦃L2, T2⦄ →
+ L2 = L1 ∧ (T2 = W ∨ T2 = U).
+/2 width=4 by frsup_inv_flat1_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma frsup_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/
+qed-.
+
+lemma frsup_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/
+qed-.
+
+lemma frsup_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{T2} < ♯{T1}.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/
+qed-.
+
+lemma frsup_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a
+| #a #I #L #V #_ @(ex_intro … (⋆.ⓑ{I}V)) //
+]
+#I #L #V #T @(ex_intro … (⋆)) //
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lift_frsup_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀L,K,U2. ⦃L, U1⦄ ⧁ ⦃L @@ K, U2⦄ →
+ ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
+#T1 #U1 #d #e * -T1 -U1 -d -e
+[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HTU1 #L #K #X #H
+ elim (frsup_inv_bind1 … H) -H *
+ [ -HTU1 #H1 #H2 destruct
+ >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/
+ | -HVW1 #H1 #H2 destruct
+ >(append_inv_pair_dx … H1) -L -K normalize /2 width=2/
+ ]
+|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #K #X #H
+ elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct
+ >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/
+]
+#i #d #e [2,3: #_ ] #L #K #X #H
+elim (frsup_inv_atom1 … H)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'RestSupTermPlus $L1 $T1 $L2 $T2 }.
+
+include "basic_2/substitution/frsup.ma".
+
+(* PLUS-ITERATED RESTRICTED SUPCLOSURE **************************************)
+
+definition frsupp: bi_relation lenv term ≝ bi_TC … frsup.
+
+interpretation "plus-iterated restricted structural predecessor (closure)"
+ 'RestSupTermPlus L1 T1 L2 T2 = (frsupp L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma frsupp_ind: ∀L1,T1. ∀R:relation2 lenv term.
+ (∀L2,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → R L2 T2) →
+ (∀L,T,L2,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → R L2 T2.
+#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
+@(bi_TC_ind … IH1 IH2 ? ? H)
+qed-.
+
+lemma frsupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term.
+ (∀L1,T1. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → R L1 T1) →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → R L1 T1.
+#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
+@(bi_TC_ind_dx … IH1 IH2 ? ? H)
+qed-.
+
+(* Baic inversion lemmas ****************************************************)
+
+lemma frsupp_inv_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ ∨
+ ∃∃L,T. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ & ⦃L, T⦄ ⧁ ⦃L2, T2⦄.
+/2 width=1 by bi_TC_decomp_r/ qed-.
+
+lemma frsupp_inv_sn: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ ∨
+ ∃∃L,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ & ⦃L, T⦄ ⧁+ ⦃L2, T2⦄.
+/2 width=1 by bi_TC_decomp_l/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma frsup_frsupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma frsupp_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma frsupp_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma frsupp_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
+#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2
+/3 width=3 by frsup_fwd_fw, transitive_lt/
+qed-.
+
+lemma frsupp_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
+#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2
+/2 width=3 by frsup_fwd_lw/ (**) (* /3 width=5 by frsup_fwd_lw, transitive_le/ is too slow *)
+#L #T #L2 #T2 #_ #HL2 #HL1
+lapply (frsup_fwd_lw … HL2) -HL2 /2 width=3 by transitive_le/
+qed-.
+
+lemma frsupp_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ♯{T2} < ♯{T1}.
+#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2
+/3 width=3 by frsup_fwd_tw, transitive_lt/
+qed-.
+
+lemma frsupp_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
+#L1 #L2 #T1 #T2 #H @(frsupp_ind … H) -L2 -T2 /2 width=3 by frsup_fwd_append/
+#L #T #L2 #T2 #_ #HL2 * #K1 #H destruct
+elim (frsup_fwd_append … HL2) -HL2 #K2 #H destruct /2 width=2/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lift_frsupp_trans: ∀L,U1,K,U2. ⦃L, U1⦄ ⧁+ ⦃L @@ K, U2⦄ →
+ ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
+#L #U1 @(f2_ind … fw … L U1) -L -U1 #n #IH
+#L #U1 #Hn #K #U2 #H #T1 #d #e #HTU1 destruct
+elim (frsupp_inv_sn … H) -H /2 width=5 by lift_frsup_trans/ *
+#L0 #U0 #HL0 #HL
+elim (frsup_fwd_append … HL0) #K0 #H destruct
+elim (frsupp_fwd_append … HL) #L0 >append_assoc #H
+elim (append_inj_dx … H ?) -H // #_ #H destruct
+<append_assoc in HL; #HL
+elim (lift_frsup_trans … HTU1 … HL0) -T1 #T #HTU
+lapply (frsup_fwd_fw … HL0) -HL0 #HL0
+elim (IH … HL … HTU) -IH -HL -T // -L -U1 -U0 /2 width=2/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/frsupp.ma".
+
+(* PLUS-ITERATED RESTRICTED SUPCLOSURE **************************************)
+
+(* Main propertis ***********************************************************)
+
+theorem frsupp_trans: bi_transitive … frsupp.
+/2 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'RestSupTermStar $L1 $T1 $L2 $T2 }.
+
+include "basic_2/unfold/frsupp.ma".
+
+(* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
+
+definition frsups: bi_relation lenv term ≝ bi_star … frsup.
+
+interpretation "star-iterated restricted structural predecessor (closure)"
+ 'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma frsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
+ (∀L,L2,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L2 T2.
+#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
+@(bi_star_ind … IH1 IH2 ? ? H)
+qed-.
+
+lemma frsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L1 T1.
+#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
+@(bi_star_ind_dx … IH1 IH2 ? ? H)
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma frsups_refl: bi_reflexive … frsups.
+/2 width=1/ qed.
+
+lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma frsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma frsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma frsups_frsupp_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ →
+ ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma frsupp_frsups_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ →
+ ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
+ (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 * /2 width=1/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
+#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
+/3 width=1 by frsupp_fwd_fw, lt_to_le/
+qed-.
+
+lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
+#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
+/2 width=3 by frsupp_fwd_lw/
+qed-.
+
+lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{T2} ≤ ♯{T1}.
+#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
+/3 width=3 by frsupp_fwd_tw, lt_to_le/
+qed-.
+
+lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
+#L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
+[ * #H1 #H2 destruct
+ @(ex_intro … (⋆)) //
+| /2 width=3 by frsupp_fwd_append/
+qed-.
+
+(* Advanced forward lemmas ***************************************************)
+
+lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
+ ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
+#T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
+[ * #H1 #H2 destruct
+ >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
+| /2 width=5 by lift_frsupp_trans/
+]
+qed-.
+
+(* Advanced inversion lemmas for frsupp **************************************)
+
+lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
+#J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
+#L2 #T2 #H elim (frsup_inv_atom1 … H)
+qed-.
+
+lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
+ ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
+#b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
+[ #L2 #T2 #H
+ elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
+| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
+]
+qed-.
+
+lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
+ ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
+#J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
+[ #L2 #T2 #H
+ elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
+| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/frsups.ma".
+
+(* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
+
+(* Main propertis ***********************************************************)
+
+theorem frsups_trans: bi_transitive … frsups.
+/2 width=4/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/frsups.ma".
+include "basic_2/static/ssta.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
+#h #g #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #l #_ #H
+ elim (frsupp_inv_atom1_frsups … H)
+| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
+ elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
+ elim (lift_inv_lref2_be … H ? ?) -H //
+| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
+ elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
+ elim (lift_inv_lref2_be … H ? ?) -H //
+| #a #I #L #V #T #U #l #_ #IHTU #H
+ elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
+ lapply (frsups_fwd_fw … H) -H normalize
+ <associative_plus <associative_plus #H
+ elim (le_plus_xySz_x_false … H)
+| #L #V #T #U #l #_ #IHTU #H
+ elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
+ lapply (frsups_fwd_fw … H) -H normalize
+ <associative_plus <associative_plus #H
+ elim (le_plus_xySz_x_false … H)
+| /3 width=4/
+]
+qed-.
+
+fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → T = U → ⊥.
+#h #g #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #l #_ #H
+ lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
+ <e1 -e1 #H elim (lt_refl_false … H)
+| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
+ elim (lift_inv_lref2_be … HWU ? ?) -HWU //
+| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
+ elim (lift_inv_lref2_be … HWU ? ?) -HWU //
+| #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
+| #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
+| #L #W #T #U #l #HTU #_ #H destruct
+ elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
+]
+qed-.
+
+lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, T⦄ → ⊥.
+/2 width=8 by ssta_inv_refl_aux/ qed-.
+
+lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
+#h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
+[ * #_ #H destruct /2 width=6 by ssta_inv_refl/
+| /2 width=8 by ssta_inv_frsupp/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/ldrop.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Basic inversion lemmas ***************************************************)
+
+fact fsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀J. T1 = ⓪{J} →
+ (∃∃I,K,V. L1 = K.ⓑ{I}V & J = LRef 0 & L2 = K & T2 = V) ∨
+ ∃∃I,K,V,i. ⦃K, #i⦄ ⊃ ⦃L2, T2⦄ & L1 = K.ⓑ{I}V & J = LRef (i+1).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #I #L #V #J #H destruct /3 width=6/
+| #I #L #K #V #T #i #HLK #J #H destruct /3 width=7/
+| #a #I #L #V #T #J #H destruct
+| #a #I #L #V #T #J #H destruct
+| #I #L #V #T #J #H destruct
+| #I #L #V #T #J #H destruct
+]
+qed-.
+
+lemma fsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⊃ ⦃L2, T2⦄ →
+ (∃∃I,K,V. L1 = K.ⓑ{I}V & J = LRef 0 & L2 = K & T2 = V) ∨
+ ∃∃I,K,V,i. ⦃K, #i⦄ ⊃ ⦃L2, T2⦄ & L1 = K.ⓑ{I}V & J = LRef (i+1).
+/2 width=3 by fsup_inv_atom1_aux/ qed-.
+
+fact fsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
+ ∀b,J,W,U. T1 = ⓑ{b,J}W.U →
+ (L2 = L1 ∧ T2 = W) ∨
+ (L2 = L1.ⓑ{J}W ∧ T2 = U).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #I #L #V #b #J #W #U #H destruct
+| #I #L #K #V #T #i #_ #b #J #W #U #H destruct
+| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
+| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
+| #I #L #V #T #b #J #W #U #H destruct
+| #I #L #V #T #b #J #W #U #H destruct
+]
+qed-.
+
+lemma fsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃ ⦃L2, T2⦄ →
+ (L2 = L1 ∧ T2 = W) ∨
+ (L2 = L1.ⓑ{J}W ∧ T2 = U).
+/2 width=4 by fsup_inv_bind1_aux/ qed-.
+
+fact fsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
+ ∀J,W,U. T1 = ⓕ{J}W.U →
+ L2 = L1 ∧ (T2 = W ∨ T2 = U).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #I #L #K #J #W #U #H destruct
+| #I #L #K #V #T #i #_ #J #W #U #H destruct
+| #a #I #L #V #T #J #W #U #H destruct
+| #a #I #L #V #T #J #W #U #H destruct
+| #I #L #V #T #J #W #U #H destruct /3 width=1/
+| #I #L #V #T #J #W #U #H destruct /3 width=1/
+]
+qed-.
+
+lemma fsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃ ⦃L2, T2⦄ →
+ L2 = L1 ∧ (T2 = W ∨ T2 = U).
+/2 width=4 by fsup_inv_flat1_aux/ qed-.
--- /dev/null
+(* Advanced inversion lemmas on plus-iterated supclosure ********************)
+
+lamma fsupp_inv_bind1_fsups: ∀b,J,G1,G2,L1,L2,W,U,T2. ⦃G1, L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, W⦄ ⊃* ⦃G2, L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃G2, L2, T2⦄.
+#b #J #G1 #G2 #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H
+ elim (fsup_inv_bind1 … H) -H * #H1 #H2 #H3 destruct /2 width=1/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=4/
+]
+qad-.
+
+lamma fsupp_inv_flat1_fsups: ∀J,G1,G2,L1,L2,W,U,T2. ⦃G1, L1, ⓕ{J}W.U⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, W⦄ ⊃* ⦃G2, L2, T2⦄ ∨ ⦃G1, L1, U⦄ ⊃* ⦃G2, L2, T2⦄.
+#J #G1 #G2 #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H
+ elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=4/
+]
+qad-.
+
+lamma fsupp_fsups: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lamma fsups_lref: ∀I,G,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃* ⦃G, K, V⦄.
+/3 width=5 by _/ qed.
+
+lamma fsups_lref_S_lt: ∀I,G1,G2,L,K,V,T,i.
+ 0 < i → ⦃G1, L, #(i-1)⦄ ⊃* ⦃G2, K, T⦄ → ⦃G1, L.ⓑ{I}V, #i⦄ ⊃+ ⦃G2, K, T⦄.
+/3 width=7 by _/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fsup.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
+
+(* LOCAL ENVIRONMENT SLICING ************************************************)
+
+(* Inversion lemmas on supclosure *******************************************)
+
+lemma fsup_inv_atom1_ldrop: ∀K,V,L,I. ⦃L, ⓪{I}⦄ ⊃ ⦃K, V⦄ →
+ ∃∃J,i. ⇩[0, i] L ≡ K.ⓑ{J}V & I = LRef i.
+#K #V #L @(f_ind … length … L) -L #n #IH #L #Hn #I #H
+elim (fsup_inv_atom1 … H) -H *
+[ #J #L0 #V0 #H1 #H2 #H3 #H4 destruct /2 width=4/
+| #J #L0 #V0 #i #HLK #H1 #H2 destruct
+ elim (IH … HLK) -IH -HLK [2: normalize // ] #I #j #HLK #H destruct /3 width=4/
+]
+qed-.
+
+(* Advanced eliminators on supclosure ***************************************)
+
+lemma fsup_ind_ldrop: ∀R:bi_relation lenv term.
+ (∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → R L (#i) K V) →
+ (∀a,I,L,V,T. R L (ⓑ{a,I}V.T) L V) →
+ (∀a,I,L,V,T. R L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T) →
+ (∀I,L,V,T. R L (ⓕ{I}V.T) L V) →
+ (∀I,L,V,T. R L (ⓕ{I}V.T) L T) →
+ ∀L1,T1,L2,T2. ⦃L1,T1⦄⊃⦃L2,T2⦄ → R L1 T1 L2 T2.
+#R #H1 #H2 #H3 #H4 #H5 #L1 #T1 #L2 #T2 #H elim H -L1 -T1 -L2 -T2 //
+[ /3 width=2/
+| #I #L #K #V #T #i #H #H1LK
+ elim (fsup_inv_atom1_ldrop … H) -H #J #j #H2LK #H destruct /3 width=2/
+]
+qed-.
+
+(* Advanced inversion lemmas on supclosure **********************************)
+
+lemma fsup_inv_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
+ ∀J,W,j. ⇩[0, j] L1 ≡ L2.ⓑ{J}W → T1 = #j ∧ T2 = W.
+#L1 #L2 #T1 #T2 #H @(fsup_ind_ldrop … H) -L1 -L2 -T1 -T2
+[ #I #L #K #V #i #HLKV #J #W #j #HLKW
+ elim (ldrop_conf_div … HLKV … HLKW) -L /2 width=1/
+| #a
+| #a
+]
+#I #L #V #T #J #W #j #H
+lapply (ldrop_pair2_fwd_fw … H W) -H #H
+[2: lapply (transitive_lt (♯{L,W}) … H) /2 width=1/ -H #H ]
+elim (lt_refl_false … H)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/fsups.ma".
+include "basic_2/substitution/ldrop_fsup.ma".
+
+(* LOCAL ENVIRONMENT SLICING ************************************************)
+
+(* Inversion lemmas on plus-iterated supclosure ****************************)
+
+lemma fsupp_inv_atom1_fsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⊃+ ⦃L2, T2⦄ →
+ ∃∃I,K,V,i. ⇩[0, i] L1 ≡ K.ⓑ{I}V &
+ ⦃K, V⦄ ⊃* ⦃L2, T2⦄ & J = LRef i.
+#J #L1 #L2 #T2 #H @(fsupp_ind … H) -L2 -T2
+[ #L2 #T2 #H
+ elim (fsup_inv_atom1_ldrop … H) -H * #i #HL12 #H destruct /2 width=7/
+| #L #T #L2 #T2 #_ #HT2 * #I #K #V #i #HLK #HVT #H destruct /3 width=8/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/multiple/lifts_lifts.ma".
+include "basic_2/multiple/drops_drops.ma".
+include "basic_2/static/aaa_lifts.ma".
+include "basic_2/static/aaa_aaa.ma".
+include "basic_2/computation/lsubc_drops.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: sc3_arity_csubc *)
+theorem acr_aaa_csubc: ∀RR,RS,RP.
+ gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+ ∀L2. G ⊢ L2 ⫃[RP] L1 → ⦃G, L2, T⦄ ϵ[RP] 〚A〛.
+#RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
+[ #G #L1 #k #L2 #HL21
+ lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
+ lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L2 #HL21
+ lapply (acr_gcr … H1RP H2RP B) #HB
+ elim (lsubc_drop_O1_trans … HL21 … HLK1) -L1 #X #HLK2 #H
+ elim (lsubc_inv_pair2 … H) -H *
+ [ #K2 #HK21 #H destruct -HKV1B
+ lapply (drop_fwd_drop2 … HLK2) #H
+ elim (lift_total V1 0 (i +1)) #V #HV1
+ lapply (s5 … HB ? G ? ? (◊) … HV1 HLK2) /3 width=7 by s0/
+ | #K2 #V2 #A2 #HVA2 #H1V1A2 #H2V1A2 #_ #H1 #H2 destruct -IHB
+ lapply (aaa_mono … H2V1A2 … HKV1B) #H destruct -H2V1A2 -HKV1B
+ lapply (drop_fwd_drop2 … HLK2) #H
+ elim (lift_total V1 0 (i +1)) #V3 #HV13
+ elim (lift_total V2 0 (i +1)) #V #HV2
+ lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/ -HLK2
+ lapply (s7 … HB G L2 (◊)) /3 width=7 by s0/
+ ]
+| #a #G #L1 #V #T #B #A #_ #_ #IHB #IHA #L2 #HL21
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (acr_gcr … H1RP H2RP B) #HB
+ lapply (s1 … HB) -HB #HB
+ lapply (s6 … HA G L2 (◊) (◊)) /4 width=1 by lsubc_pair/
+| #a #G #L1 #W #T #B #A #HLWB #_ #IHB #IHA #L2 #HL21
+ @(acr_abst … H1RP H2RP) [ /2 width=5 by/ ]
+ #L3 #V3 #W3 #T3 #des3 #HL32 #HW03 #HT03 #H1B #H2B
+ @(gcr_lifts … L2.ⓓⓝW.V3,T … HL32)
+ elim (drops_lsubc_trans … H1RP H2RP … HL32 … HL21) -L2 #L2 #HL32 #HL21
+ lapply (aaa_lifts … L2 W3 … des3 … HLWB) -HLWB /2 width=4 by drops_trans, lifts_trans/ #HLW2B
+ @(s0
+
+ @(IHA (L2. ⓛW3) … (des3 + 1)) -IHA
+ /3 width=5 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
+| #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ /3 width=10 by drops_nil, lifts_nil/
+| #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (s7 … HA G L2 (◊)) /3 width=5 by/
+]
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+/2 width=8 by drops_nil, lifts_nil, acr_aaa_csubc_lifts/ qed.
+
+lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
+lapply (acr_gcr … H1RP H2RP A) #HA
+@(s1 … HA) /2 width=4 by acr_aaa/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/ineint_5.ma".
+include "basic_2/grammar/aarity.ma".
+include "basic_2/multiple/mr2_mr2.ma".
+include "basic_2/multiple/lifts_lift_vector.ma".
+include "basic_2/multiple/drops_drop.ma".
+include "basic_2/computation/gcp.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+definition S0 ≝ λC:candidate. ∀G,L2,L1,T1,d,e.
+ C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
+
+definition S0s ≝ λC:candidate.
+ ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
+
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:candidate.
+ ∀G,L,T. C G L T → RP G L T.
+
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs →
+ ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
+
+(* Note: this generalizes Tait's ii *)
+definition S3 ≝ λC:candidate.
+ ∀a,G,L,Vs,V,T,W.
+ C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
+
+definition S4 ≝ λRP,C:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
+
+definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[0, i+1] V1 ≡ V2 →
+ C G L (ⒶVs.V2) → C G L (ⒶVs.#i).
+
+definition S6 ≝ λRP,C:candidate.
+ ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
+
+definition S7 ≝ λC:candidate.
+ ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
+
+(* requirements for the generic reducibility candidate *)
+record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
+{ (* s0: S0 C; *)
+ s1: S1 RP C;
+ s2: S2 RR RS RP C;
+ s3: S3 C;
+ s4: S4 RP C;
+ s5: S5 C;
+ s6: S6 RP C;
+ s7: S7 C
+}.
+
+(* the functional construction for candidates *)
+definition cfun: candidate → candidate → candidate ≝
+ λC1,C2,G,K,T. ∀V. C1 G K V → C2 G K (ⓐV.T).
+
+(* the reducibility candidate associated to an atomic arity *)
+let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
+match A with
+[ AAtom ⇒ RP
+| APair B A ⇒ cfun (acr RP B) (acr RP A)
+].
+
+interpretation
+ "candidate of reducibility of an atomic arity (abstract)"
+ 'InEInt RP G L T A = (acr RP A G L T).
+
+(* Basic properties *********************************************************)
+(*
+(* Basic_1: was: sc3_lift1 *)
+lemma gcr_lifts: ∀C. S0 C → S0s C.
+#C #HC #G #L1 #L2 #des #H elim H -L1 -L2 -des
+[ #L #T1 #T2 #H #HT1 <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
+ elim (lifts_inv_cons … H) -H /3 width=10 by/
+]
+qed.
+*)
+axiom rp_lift: ∀RP. S0 RP.
+
+
+axiom rp_lifts: ∀RR,RS,RP. gcr RR RS RP RP →
+ ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
+ RP G L V → RP G L0 V0.
+(*
+#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
+@gcr_lifts /width=7 by/
+@(s0 … HRP)
+qed.
+*)
+(* Basic_1: was only: sns3_lifts1 *)
+axiom rp_liftsv_all: ∀RR,RS,RP. gcr RR RS RP RP →
+ ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
+ all … (RP G L) Vs → all … (RP G L0) V0s.
+(*
+#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
+qed.
+*)
+
+lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀A. S0 (acr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A /2 width=7 by rp_lift/
+#B #A #IHB #IHA #G #L2 #L1 #T1 #d #e #IH #T2 #HL21 #HT12 #V #HB
+@(IHA … HL21) [3: @(lift_flat … HT12) |1: skip |
+
+(* Basic_1: was:
+ sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
+*)
+lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀A. gcr RR RS RP (acr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A //
+#B #A #IHB #IHA @mk_gcr
+[ #G #L #T #H
+ elim (cp1 … H1RP G L) #k #HK
+ lapply (H (⋆k) ?) -H
+ [ lapply (s2 … IHB G L (◊) … HK) //
+ | #H @(cp2 … H1RP … k) @(s1 … IHA) //
+ ]
+| #G #L #Vs #HVs #T #H1T #H2T #V #HB
+ lapply (s1 … IHB … HB) #HV
+ @(s2 … IHA … (V @ Vs))
+ /3 width=14 by rp_liftsv_all, gcp_lifts, cp0, lifts_simple_dx, conj/
+| #a #G #L #Vs #U #T #W #HA #V #HB
+ @(s3 … IHA … (V @ Vs)) /2 width=1 by/
+| #G #L #Vs #HVs #k #V #HB
+ lapply (s1 … IHB … HB) #HV
+ @(s4 … IHA … (V @ Vs)) /3 width=7 by rp_liftsv_all, conj/
+| #I #G #L #K #Vs #V1 #V2 #i #HLK #HV12 #HA #V #HB
+ @(s5 … IHA … (V @ Vs) … HLK HV12) /2 width=1 by/
+| #G #L #V1s #V2s #HV12s #a #W #T #HA #HW #V1 #HB
+ elim (lift_total V1 0 1) #V2 #HV12
+ @(s6 … IHA … (V1 @ V1s) (V2 @ V2s)) /2 width=1 by liftv_cons/
+ @HA @(gcr_lift … H1RP H2RP … HB … HV12) /2 width=2 by drop_drop/
+| #G #L #Vs #T #W #HA #HW #V #HB
+ @(s7 … IHA … (V @ Vs)) /2 width=1 by/
+]
+qed.
+
+lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀a,G,L,W,T,B,A. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
+ ∀V. ⦃G, L, V⦄ ϵ[RP] 〚B〛 → ⦃G, L.ⓓⓝW.V, T⦄ ϵ[RP] 〚A〛
+ ) →
+ ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
+#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #B #A #HW #HA #L0 #V0 #X #des #HL0 #H #HB
+lapply (acr_gcr … H1RP H2RP A) #HCA
+lapply (acr_gcr … H1RP H2RP B) #HCB
+elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+lapply (gcr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
+lapply (s3 … HCA … a G L0 (◊)) #H @H -H
+lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
+[ @(HA … HL0) //
+| lapply (s1 … HCB) -HCB #HCB
+ lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
+]
+qed.
+
+(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
+(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ≅ break [ term 46 d , break term 46 e ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'CoEq $d $e $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_lt.ma".
+include "basic_2/notation/relations/coeq_4.ma".
+include "basic_2/grammar/lenv_length.ma".
+
+(* COEQUIVALENCE FOR LOCAL ENVIRONMENTS *************************************)
+
+inductive lcoeq: relation4 ynat ynat lenv lenv ≝
+| lcoeq_atom: ∀d,e. lcoeq d e (⋆) (⋆)
+| lcoeq_zero: ∀I,L1,L2,V.
+ lcoeq 0 0 L1 L2 → lcoeq 0 0 (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lcoeq_pair: ∀I1,I2,L1,L2,V1,V2,e. lcoeq 0 e L1 L2 →
+ lcoeq 0 (⫯e) (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| lcoeq_succ: ∀I,L1,L2,V,d,e.
+ lcoeq d e L1 L2 → lcoeq (⫯d) e (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+.
+
+interpretation
+ "coequivalence (local environment)"
+ 'CoEq d e L1 L2 = (lcoeq d e L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lcoeq_pair_lt: ∀I1,I2,L1,L2,V1,V2,e. L1 ≅[0, ⫰e] L2 → 0 < e →
+ L1.ⓑ{I1}V1 ≅[0, e] L2.ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lcoeq_pair/
+qed.
+
+lemma lcoeq_succ_lt: ∀I,L1,L2,V,d,e. L1 ≅[⫰d, e] L2 → 0 < d →
+ L1.ⓑ{I}V ≅[d, e] L2. ⓑ{I}V.
+#I #L1 #L2 #V #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lcoeq_succ/
+qed.
+
+lemma lcoeq_pair_O_Y: ∀L1,L2. L1 ≅[0, ∞] L2 →
+ ∀I1,I2,V1,V2. L1.ⓑ{I1}V1 ≅[0,∞] L2.ⓑ{I2}V2.
+#L1 #L2 #HL12 #I1 #I2 #V1 #V2 lapply (lcoeq_pair I1 I2 … V1 V2 … HL12) -HL12 //
+qed.
+
+lemma lcoeq_refl: ∀L,d,e. L ≅[d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
+#Hd destruct /2 width=1 by lcoeq_succ/
+#e elim (ynat_cases … e) [| * #x ]
+#He destruct /2 width=1 by lcoeq_zero, lcoeq_pair/
+qed.
+
+lemma lcoeq_O_Y: ∀L1,L2. |L1| = |L2| → L1 ≅[0, ∞] L2.
+#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
+* [2,4: #L2 #I2 #V1 ] normalize /3 width=2 by lcoeq_pair_O_Y/
+<plus_n_Sm #H destruct
+qed.
+
+lemma lcoeq_sym: ∀d,e. symmetric … (lcoeq d e).
+#d #e #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lcoeq_zero, lcoeq_pair, lcoeq_succ/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lcoeq_inv_atom1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #d #e * -L1 -L2 -d -e //
+[ #I #L1 #L2 #V #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #H destruct
+| #I #L1 #L2 #V #d #e #_ #H destruct
+]
+qed-.
+
+lemma lcoeq_inv_atom1: ∀L2,d,e. ⋆ ≅[d, e] L2 → L2 = ⋆.
+/2 width=5 by lcoeq_inv_atom1_aux/ qed-.
+
+fact lcoeq_inv_zero1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
+ ∀J,K1,W. L1 = K1.ⓑ{J}W → d = 0 → e = 0 →
+ ∃∃K2. K1 ≅[0, 0] K2 & L2 = K2.ⓑ{J}W.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #J #K1 #W #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #W #H destruct /2 width=3 by ex2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #J #K1 #W #_ #_ #H elim (ysucc_inv_O_dx … H)
+| #I #L1 #L2 #V #d #e #_ #J #K1 #W #_ #H elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lcoeq_inv_zero1: ∀I,K1,L2,V. K1.ⓑ{I}V ≅[0, 0] L2 →
+ ∃∃K2. K1 ≅[0, 0] K2 & L2 = K2.ⓑ{I}V.
+/2 width=7 by lcoeq_inv_zero1_aux/ qed-.
+
+fact lcoeq_inv_pair1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → 0 < e →
+ ∃∃J2,K2,W2. K1 ≅[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #J1 #K1 #W1 #H destruct
+| #I #L1 #L2 #V #_ #J1 #K1 #W1 #_ #_ #H elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #e #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+ /2 width=5 by ex2_3_intro/
+| #I #L1 #L2 #V #d #e #_ #J1 #K1 #W1 #_ #H elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lcoeq_inv_pair1: ∀I1,K1,L2,V1,e. K1.ⓑ{I1}V1 ≅[0, e] L2 → 0 < e →
+ ∃∃I2,K2,V2. K1 ≅[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=7 by lcoeq_inv_pair1_aux/ qed-.
+
+fact lcoeq_inv_succ1_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
+ ∀J,K1,W. L1 = K1.ⓑ{J}W → 0 < d →
+ ∃∃K2. K1 ≅[⫰d, e] K2 & L2 = K2.ⓑ{J}W.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #J #K1 #W #H destruct
+| #I #L1 #L2 #V #_ #J #K1 #W #_ #H elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #J #K1 #W #_ #H elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #d #e #HL12 #J #K1 #W #H destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lcoeq_inv_succ1: ∀I,K1,L2,V,d,e. K1.ⓑ{I}V ≅[d, e] L2 → 0 < d →
+ ∃∃K2. K1 ≅[⫰d, e] K2 & L2 = K2.ⓑ{I}V.
+/2 width=3 by lcoeq_inv_succ1_aux/ qed-.
+
+lemma lcoeq_inv_atom2: ∀L1,d,e. L1 ≅[d, e] ⋆ → L1 = ⋆.
+/3 width=3 by lcoeq_inv_atom1, lcoeq_sym/ qed-.
+
+lemma lcoeq_inv_zero2: ∀I,K2,L1,V. L1 ≅[0, 0] K2.ⓑ{I}V →
+ ∃∃K1. K1 ≅[0, 0] K2 & L1 = K1.ⓑ{I}V.
+#I #K2 #L1 #V #H elim (lcoeq_inv_zero1 … (lcoeq_sym … H)) -H
+/3 width=3 by lcoeq_sym, ex2_intro/
+qed-.
+
+lemma lcoeq_inv_pair2: ∀I2,K2,L1,V2,e. L1 ≅[0, e] K2.ⓑ{I2}V2 → 0 < e →
+ ∃∃I1,K1,V1. K1 ≅[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V1.
+#I2 #K2 #L1 #V2 #e #H #He elim (lcoeq_inv_pair1 … (lcoeq_sym … H)) -H
+/3 width=5 by lcoeq_sym, ex2_3_intro/
+qed-.
+
+lemma lcoeq_inv_succ2: ∀I,K2,L1,V,d,e. L1 ≅[d, e] K2.ⓑ{I}V → 0 < d →
+ ∃∃K1. K1 ≅[⫰d, e] K2 & L1 = K1.ⓑ{I}V.
+#I #K2 #L1 #V #d #e #H #Hd elim (lcoeq_inv_succ1 … (lcoeq_sym … H)) -H
+/3 width=3 by lcoeq_sym, ex2_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lcoeq_fwd_length: ∀L1,L2,d,e. L1 ≅[d, e] L2 → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
+qed-.
+
+(* Advanced inversionn lemmas ***********************************************)
+
+fact lcoeq_inv_O2_aux: ∀L1,L2,d,e. L1 ≅[d, e] L2 → e = 0 → L1 = L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e /3 width=1 by eq_f3/
+#I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #H elim (ysucc_inv_O_dx … H)
+qed-.
+
+lemma lcoeq_inv_O2: ∀L1,L2,d. L1 ≅[d, 0] L2 → L1 = L2.
+/2 width=4 by lcoeq_inv_O2_aux/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lcoeq.ma".
+
+(* COEQUIVALENCE FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Main properties **********************************************************)
+
+theorem lcoeq_trans: ∀d,e. Transitive … (lcoeq d e).
+#d #e #L1 #L2 #H elim H -L1 -L2 -d -e //
+[ #I #L1 #L #V #HL1 #_ #X #H >(lcoeq_inv_O2 … HL1) -HL1 //
+| #I1 #I #L1 #L #V1 #V #e #_ #IHL1 #X #H elim (lcoeq_inv_pair1 … H) -H //
+ #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lcoeq_pair/
+| #I #L1 #L #V #d #e #_ #IHL1 #X #H elim (lcoeq_inv_succ1 … H) -H //
+ #L2 #HL2 #H destruct /3 width=1 by lcoeq_succ/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_minus.ma".
+include "basic_2/grammar/lcoeq.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Properties on coequivalence **********************************************)
+
+lemma lcoeq_ldrop_trans_lt: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
+ ∀I,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W → i < d →
+ ∃∃K1. K1 ≅[⫰(d-i), e] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #J #K2 #W #s #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #L1 #L2 #V #_ #_ #J #K2 #W #s #i #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #J #K2 #W #s #i #_ #H
+ elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #d #e #HL12 #IHL12 #J #K2 #W #s #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct >ypred_succ
+ /2 width=3 by ldrop_pair, ex2_intro/
+ | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+ #H <H -H #H lapply (ylt_inv_succ … H) -H
+ #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
+ >yminus_succ <yminus_inj /3 width=3 by ldrop_drop_lt, ex2_intro/
+ ]
+]
+qed-.
+
+lemma lcoeq_ldrop_conf_lt: ∀L1,L2,d,e. L1 ≅[d, e] L2 →
+ ∀I,K1,W,s,i. ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W → i < d →
+ ∃∃K2. K1 ≅[⫰(d-i), e] K2 & ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W.
+#L1 #L2 #d #e #HL12 #I #K1 #W #s #i #HLK1 #Hid
+elim (lcoeq_ldrop_trans_lt … (lcoeq_sym … HL12) … HLK1) // -L1 -Hid
+/3 width=3 by lcoeq_sym, ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'CPConvStar $L1 $L2 }.
+
+include "basic_2/grammar/lenv_px_sn.ma".
+include "basic_2/equivalence/cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
+
+definition lcpcs: relation lenv ≝ lpx_sn … cpcs.
+
+interpretation "context-sensitive parallel equivalence (local environment)"
+ 'CPConvStar L1 L2 = (lcpcs L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lcpcs_inv_atom1: ∀L2. ⋆ ⊢ ⬌* L2 → L2 = ⋆.
+/2 width=2 by lpx_sn_inv_atom1/ qed-.
+
+lemma lcpcs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ⬌* L2 →
+ ∃∃K2,V2. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L2 = K2. ⓑ{I} V2.
+/2 width=1 by lpx_sn_inv_pair1/ qed-.
+
+lemma lcpcs_inv_atom2: ∀L1. L1 ⊢ ⬌* ⋆ → L1 = ⋆.
+/2 width=2 by lpx_sn_inv_atom2/ qed-.
+
+lemma lcpcs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ⬌* K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L1 = K1. ⓑ{I} V1.
+/2 width=1 by lpx_sn_inv_pair2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lcpcs_fwd_length: ∀L1,L2. L1 ⊢ ⬌* L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/ltpr.ma".
+include "basic_2/equivalence/lcpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+lemma ltpr_lcpcs: ∀L1,L2. L1 ➡ L2 → L1 ⊢ ⬌* L2.
+#L1 #L2 #H elim H -L1 -L2 // /4 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⋕ ⋕ break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyEqAlt $T $d $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazyeq_4.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+definition lleq: relation4 ynat term lenv lenv ≝
+ λd,T,L1,L2. |L1| = |L2| ∧
+ (∀U. ⦃⋆, L1⦄ ⊢ T ▶*[d, ∞] U ↔ ⦃⋆, L2⦄ ⊢ T ▶*[d, ∞] U).
+
+interpretation
+ "lazy equivalence (local environment)"
+ 'LazyEq T d L1 L2 = (lleq d T L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
+/3 width=1 by conj/ qed.
+
+lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
+#d #T #L1 #L2 * /3 width=1 by iff_sym, conj/
+qed-.
+
+lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ⋕[⋆k, d] L2.
+#L1 #L2 #d #k #HL12 @conj // -HL12
+#U @conj #H >(cpys_inv_sort1 … H) -H //
+qed.
+
+lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ⋕[§p, d] L2.
+#L1 #L2 #d #k #HL12 @conj // -HL12
+#U @conj #H >(cpys_inv_gref1 … H) -H //
+qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[V, d] L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V →
+ L1 ⋕[ⓑ{a,I}V.T, d] L2.
+#a #I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
+#X @conj #H elim (cpys_inv_bind1 … H) -H
+#W #U #HVW #HTU #H destruct
+elim (IHV W) -IHV elim (IHT U) -IHT /3 width=1 by cpys_bind/
+qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,d.
+ L1 ⋕[V, d] L2 → L1 ⋕[T, d] L2 → L1 ⋕[ⓕ{I}V.T, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
+#X @conj #H elim (cpys_inv_flat1 … H) -H
+#W #U #HVW #HTU #H destruct
+elim (IHV W) -IHV elim (IHT U) -IHT
+/3 width=1 by cpys_flat/
+qed.
+
+lemma lleq_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
+ d ≤ dt → dt ≤ d + e → L1 ⋕[U, d] L2.
+#L1 #L2 #U #dt * #HL12 #IH #T #d #e #HTU #Hddt #Hdtde @conj // -HL12
+#U0 elim (IH U0) -IH #H12 #H21 @conj
+#HU0 elim (cpys_fwd_up … HU0 … HTU) // -HU0 /4 width=5 by cpys_weak/
+qed-.
+
+lemma lsuby_lleq_trans: ∀L2,L,T,d. L2 ⋕[T, d] L →
+ ∀L1. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → L1 ⋕[T, d] L.
+#L2 #L #T #d * #HL2 #IH #L1 #HL12 #H @conj // -HL2
+#U elim (IH U) -IH #Hdx #Hsn @conj #HTU
+[ @Hdx -Hdx -Hsn @(lsuby_cpys_trans … HTU) -HTU
+ /2 width=1 by lsuby_sym/ (**) (* full auto does not work *)
+| -H -Hdx /3 width=3 by lsuby_cpys_trans/
+]
+qed-.
+
+lemma lleq_lsuby_trans: ∀L,L1,T,d. L ⋕[T, d] L1 →
+ ∀L2. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → L ⋕[T, d] L2.
+/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-.
+
+lemma lleq_lsuby_repl: ∀L1,L2,T,d. L1 ⋕[T, d] L2 →
+ ∀K1. K1 ⊑×[d, ∞] L1 → |K1| = |L1| →
+ ∀K2. L2 ⊑×[d, ∞] K2 → |L2| = |K2| →
+ K1 ⋕[T, d] K2.
+/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → |L1| = |L2|.
+#L1 #L2 #T #d * //
+qed-.
+
+lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ⋕[d, T] L2 → ∀K1,i. ⇩[i] L1 ≡ K1 →
+ ∃K2. ⇩[i] L2 ≡ K2.
+#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length … H) -H
+#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
+qed-.
+
+lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ⋕[d, T] L2 → ∀K2,i. ⇩[i] L2 ≡ K2 →
+ ∃K1. ⇩[i] L1 ≡ K1.
+/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-.
+
+lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1 ⋕[V, d] L2.
+#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (ⓑ{a,I}U.T)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_bind/ -H
+#H elim (cpys_inv_bind1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V.
+#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12
+#U elim (H (ⓑ{a,I}V.U)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_bind/ -H
+#H elim (cpys_inv_bind1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
+ L1 ⋕[ⓕ{I}V.T, d] L2 → L1 ⋕[V, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (ⓕ{I}U.T)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_flat/ -H
+#H elim (cpys_inv_flat1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
+ L1 ⋕[ⓕ{I}V.T, d] L2 → L1 ⋕[T, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (ⓕ{I}V.U)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_flat/ -H
+#H elim (cpys_inv_flat1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,d. L1 ⋕[ⓑ{a,I}V.T, d] L2 →
+ L1 ⋕[V, d] L2 ∧ L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V.
+/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-.
+
+lemma lleq_inv_flat: ∀I,L1,L2,V,T,d. L1 ⋕[ⓕ{I}V.T, d] L2 →
+ L1 ⋕[V, d] L2 ∧ L1 ⋕[T, d] L2.
+/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazyeqalt_4.ma".
+include "basic_2/substitution/lleq_lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Note: alternative definition of lleq *)
+inductive lleqa: relation4 ynat term lenv lenv ≝
+| lleqa_sort: ∀L1,L2,d,k. |L1| = |L2| → lleqa d (⋆k) L1 L2
+| lleqa_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → lleqa d (#i) L1 L2
+| lleqa_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
+ lleqa (yinj 0) V K1 K2 → lleqa d (#i) L1 L2
+| lleqa_free: ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → lleqa d (#i) L1 L2
+| lleqa_gref: ∀L1,L2,d,p. |L1| = |L2| → lleqa d (§p) L1 L2
+| lleqa_bind: ∀a,I,L1,L2,V,T,d.
+ lleqa d V L1 L2 → lleqa (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ lleqa d (ⓑ{a,I}V.T) L1 L2
+| lleqa_flat: ∀I,L1,L2,V,T,d.
+ lleqa d V L1 L2 → lleqa d T L1 L2 → lleqa d (ⓕ{I}V.T) L1 L2
+.
+
+interpretation
+ "lazy equivalence (local environment) alternative"
+ 'LazyEqAlt T d L1 L2 = (lleqa d T L1 L2).
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lleqa_inv_lleq: ∀L1,L2,T,d. L1 ⋕⋕[T, d] L2 → L1 ⋕[T, d] L2.
+#L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/2 width=8 by lleq_flat, lleq_bind, lleq_gref, lleq_free, lleq_lref, lleq_skip, lleq_sort/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lleq_lleqa: ∀L1,T,L2,d. L1 ⋕[T, d] L2 → L1 ⋕⋕[T, d] L2.
+#L1 #T @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * * /3 width=3 by lleqa_gref, lleqa_sort, lleq_fwd_length/
+[ #i #Hn #L2 #d #H elim (lleq_fwd_lref … H) [ * || * ]
+ /4 width=9 by lleqa_free, lleqa_lref, lleqa_skip, lleq_fwd_length, ldrop_fwd_rfw/
+| #a #I #V #T #Hn #L2 #d #H elim (lleq_inv_bind … H) -H /3 width=1 by lleqa_bind/
+| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat … H) -H /3 width=1 by lleqa_flat/
+]
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lleq_ind_alt: ∀R:relation4 ynat term lenv lenv. (
+ ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
+ ) → (
+ ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
+ ) → (
+ ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
+ K1 ⋕[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
+ ) → (
+ ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
+ ) → (
+ ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
+ ) → (
+ ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[V, d]L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V →
+ R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
+ ) → (
+ ∀I,L1,L2,V,T,d.
+ L1 ⋕[V, d]L2 → L1 ⋕[T, d] L2 →
+ R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
+ ) →
+ ∀d,T,L1,L2. L1 ⋕[T, d] L2 → R d T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa … H) -H
+/3 width=9 by lleqa_inv_lleq/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lleq_alt.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact lleq_inv_S_aux: ∀L1,L2,T,d0. L1 ⋕[T, d0] L2 → ∀d. d0 = d + 1 →
+ ∀K1,K2,I,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
+ K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
+#L1 #L2 #T #d0 #H @(lleq_ind_alt … H) -L1 -L2 -T -d0
+/2 width=1 by lleq_gref, lleq_free, lleq_sort/
+[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V #HLK1 #HLK2 #HV destruct
+ elim (yle_split_eq i d) /2 width=1 by lleq_skip, ylt_fwd_succ2/ -HL12 -Hid
+ #H destruct /2 width=8 by lleq_lref/
+| #I1 #I2 #L1 #L2 #K11 #K22 #V #d0 #i #Hd0i #HLK11 #HLK22 #HV #_ #d #H #K1 #K2 #J #W #_ #_ #_ destruct
+ /3 width=8 by lleq_lref, yle_pred_sn/
+| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
+ /4 width=7 by lleq_bind, ldrop_drop/
+| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
+ /3 width=7 by lleq_flat/
+]
+qed-.
+
+lemma lleq_inv_S: ∀T,L1,L2,d. L1 ⋕[T, d+1] L2 →
+ ∀K1,K2,I,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
+ K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
+/2 width=7 by lleq_inv_S_aux/ qed-.
+
+lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
+ L1 ⋕[V, 0] L2 ∧ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
+#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind … H) -H
+/3 width=7 by ldrop_pair, conj, lleq_inv_S/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
+ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
+#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O … H) -H //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_ge: ∀L1,L2,T,d1. L1 ⋕[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ⋕[T, d2] L2.
+#L1 #L2 #T #d1 #H @(lleq_ind_alt … H) -L1 -L2 -T -d1
+/4 width=1 by lleq_sort, lleq_free, lleq_gref, lleq_bind, lleq_flat, yle_succ/
+[ /3 width=3 by lleq_skip, ylt_yle_trans/
+| #I1 #I2 #L1 #L2 #K1 #K2 #V #d1 #i #Hi #HLK1 #HLK2 #HV #IHV #d2 #Hd12 elim (ylt_split i d2)
+ [ lapply (lleq_fwd_length … HV) #HK12 #Hid2
+ lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
+ normalize in ⊢ (%→%→?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/
+ | /3 width=8 by lleq_lref, yle_trans/
+ ]
+]
+qed-.
+
+lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[V, 0] L2 → L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
+ L1 ⋕[ⓑ{a,I}V.T, 0] L2.
+/3 width=3 by lleq_ge, lleq_bind/ qed.
+
+lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ⋕[T, 0] L2.ⓑ{I2}V2 →
+ ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ⋕[T, 1] L2.ⓑ{J2}W2.
+#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge … HT 1 ?) // -HT
+#HT @(lleq_lsuby_repl … HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *)
+qed-.
+
+lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
+ ∀J,W. L1 ⋕[W, 0] L2 → L1.ⓑ{J}W ⋕[T, 0] L2.ⓑ{J}W.
+/3 width=7 by lleq_bind_repl_SO, lleq_inv_S/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ⋕[ⓑ{a,I}V.T, 0] L2 → ⊥) →
+ (L1 ⋕[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V → ⊥).
+#a #I #L1 #L2 #V #T #H elim (lleq_dec V L1 L2 0)
+/4 width=1 by lleq_bind_O, or_intror, or_introl/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fqus_alt.ma".
+include "basic_2/substitution/lleq_ext.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on supclosure and derivatives *********************************)
+
+lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃ ⦃G2, K2, U⦄ →
+ ∀L1. L1 ⋕[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊃ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
+ #I1 #K1 #H1 #H2 lapply (ldrop_inv_O2 … H1) -H1
+ #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
+| * [ #a ] #I #G #L2 #V #T #L1 #H
+ [ elim (lleq_inv_bind … H)
+ | elim (lleq_inv_flat … H)
+ ] -H
+ /2 width=3 by fqu_pair_sn, ex2_intro/
+| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
+ #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
+| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
+ /2 width=3 by fqu_flat_dx, ex2_intro/
+| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
+ elim (ldrop_O1_le (e+1) L1)
+ [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
+ | lapply (ldrop_fwd_length_le2 … HLK2) -K2
+ lapply (lleq_fwd_length … HL12) -T -U //
+ ]
+]
+qed-.
+
+lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃⸮ ⦃G2, K2, U⦄ →
+ ∀L1. L1 ⋕[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊃⸮ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
+[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃+ ⦃G2, K2, U⦄ →
+ ∀L1. L1 ⋕[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊃+ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
+ /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
+ #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
+ /3 width=5 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃* ⦃G2, K2, U⦄ →
+ ∀L1. L1 ⋕[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊃* ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
+[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ⋕[#i, d] L2.
+#L1 #L2 #d #i #Hid #HL12 @conj // -HL12
+#U @conj #H elim (cpys_inv_lref1 … H) -H // *
+#I #Z #Y #X #H elim (ylt_yle_false … Hid … H)
+qed.
+
+lemma lleq_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
+ K1 ⋕[V, 0] K2 → L1 ⋕[#i, d] L2.
+#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ]
+[ lapply (ldrop_fwd_length … HLK1) -HLK1 #H1
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H2
+ >H1 >H2 -H1 -H2 normalize //
+| #U @conj #H elim (cpys_inv_lref1 … H) -H // *
+ >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU
+ [ letin HLK ≝ HLK1 | letin HLK ≝ HLK2 ]
+ lapply (ldrop_mono … H … HLK) -H #H destruct elim (IH W)
+ /3 width=7 by cpys_subst_Y2/
+]
+qed.
+
+lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ⋕[#i, d] L2.
+#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12
+#U @conj #H elim (cpys_inv_lref1 … H) -H // *
+#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 … H) -H
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed.
+
+(* Properties on relocation *************************************************)
+
+lemma lleq_lift_le: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → dt ≤ d → L1 ⋕[U, dt] L2.
+#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd
+lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
+#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
+[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ]
+elim (cpys_inv_lift1_be … HU0 … HLKA … HTU) // -HU0 >yminus_Y_inj #T0 #HT0 #HTU0
+elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=12 by cpys_lift_be/
+qed-.
+
+lemma lleq_lift_ge: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → d ≤ dt → L1 ⋕[U, dt+e] L2.
+#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt
+lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
+#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
+[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ]
+elim (cpys_inv_lift1_ge … HU0 … HLKA … HTU) /2 width=1 by monotonic_yle_plus_dx/ -HU0 >yplus_minus_inj #T0 #HT0 #HTU0
+elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=10 by cpys_lift_ge/
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lleq_inv_lift_le: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → dt ≤ d → K1 ⋕[T, dt] K2.
+#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd
+lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
+#H2 #H1 @conj // -HL12 -H1 -H2
+#T0 elim (lift_total T0 d e)
+#U0 #HTU0 elim (IH U0) -IH
+#H12 #H21 @conj #HT0
+[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
+lapply (cpys_lift_be … HT0 … HLKA … HTU … HTU0) // -HT0
+>yplus_Y1 #HU0 elim (cpys_inv_lift1_be … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdtd
+#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
+qed-.
+
+lemma lleq_inv_lift_ge: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ dt → K1 ⋕[T, dt-e] K2.
+#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt
+lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
+#H2 #H1 @conj // -HL12 -H1 -H2
+elim (yle_inv_plus_inj2 … Hdedt) #Hddt #Hedt
+#T0 elim (lift_total T0 d e)
+#U0 #HTU0 elim (IH U0) -IH
+#H12 #H21 @conj #HT0
+[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
+lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0 -Hddt
+>ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdedt -Hedt
+#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
+qed-.
+
+lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ⋕[T, d] K2.
+#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde
+lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
+#H2 #H1 @conj // -HL12 -H1 -H2
+#T0 elim (lift_total T0 d e)
+#U0 #HTU0 elim (IH U0) -IH
+#H12 #H21 @conj #HT0
+[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
+lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0
+#HU0 lapply (cpys_weak … HU0 dt (∞) ? ?) // -HU0
+#HU0 lapply (H0 HU0)
+#HU0 lapply (cpys_weak … HU0 d (∞) ? ?) // -HU0
+#HU0 elim (cpys_inv_lift1_ge_up … HU0 … HLKB … HTU) // -L1 -L2 -U -Hddt -Hdtde
+#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_cpys.ma".
+include "basic_2/substitution/lleq_ldrop.ma".
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref: ∀L1,L2. ∀d:ynat. ∀i:nat. L1 ⋕[#i, d] L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I1,I2,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V &
+ ⇩[i] L2 ≡ K2.ⓑ{I2}V &
+ K1 ⋕[V, yinj 0] K2 & d ≤ yinj i.
+#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
+elim (ylt_split i d) /2 width=1 by or3_intro1/ #Hdi #Hi
+elim (ldrop_O1_lt … Hi) #I1 #K1 #V1 #HLK1
+elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
+lapply (ldrop_fwd_length_minus2 … HLK2) #H
+lapply (ldrop_fwd_length_minus2 … HLK1) >HL12 <H -HL12 -H
+#H lapply (injective_plus_l … H) -H #HK12
+elim (lift_total V1 0 (i+1)) #W1 #HVW1
+elim (lift_total V2 0 (i+1)) #W2 #HVW2
+elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
+elim (cpys_inv_lref1_ldrop … (H1 ?) … HLK2 … HVW1) -H1
+[ elim (cpys_inv_lref1_ldrop … (H2 ?) … HLK1 … HVW2) -H2 ]
+/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
+lapply (cpys_antisym_eq … H12 … H21) -H12 -H21 #H destruct
+@or3_intro2 @(ex4_5_intro … HLK1 HLK2) // @conj // -HK12
+#V elim (lift_total V 0 (i+1))
+#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
+[ elim (cpys_inv_lref1_ldrop … (H12 ?) … HLK2 … HVW) -H12 -H21
+| elim (cpys_inv_lref1_ldrop … (H21 ?) … HLK1 … HVW) -H21 -H12
+] [1,3: >yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/
+qed-.
+
+lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
+ ∀I2,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I2}V →
+ i < d ∨
+ ∃∃I1,K1. ⇩[i] L1 ≡ K1.ⓑ{I1}V & K1 ⋕[V, 0] K2 & d ≤ i.
+#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref … H) -H [ * || * ]
+[ #_ #H elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I1 #I2 #K11 #K22 #V0 #HLK11 #HLK22 #HV0 #Hdi lapply (ldrop_mono … HLK22 … HLK2) -L2
+ #H destruct /3 width=5 by ex3_2_intro, or_intror/
+]
+qed-.
+
+lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
+ ∀I1,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V →
+ i < d ∨
+ ∃∃I2,K2. ⇩[i] L2 ≡ K2.ⓑ{I2}V & K1 ⋕[V, 0] K2 & d ≤ i.
+#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 … d … HLK1) -HLK1
+[2: * ] /4 width=6 by lleq_sym, ex3_2_intro, or_introl, or_intror/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lleq_inv_lref_ge_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I2,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I2}V →
+ ∃∃I1,K1. ⇩[i] L1 ≡ K1.ⓑ{I1}V & K1 ⋕[V, 0] K2.
+#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx … H … HLK2) -L2
+[ #H elim (ylt_yle_false … H Hdi)
+| * /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lleq_inv_lref_ge_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I1,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V →
+ ∃∃I2,K2. ⇩[i] L2 ≡ K2.ⓑ{I2}V & K1 ⋕[V, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 … Hdi … HLK1) -Hdi -HLK1
+/3 width=4 by lleq_sym, ex2_2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_gen: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ V1 = V2 ∧ K1 ⋕[V2, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (lleq_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
+#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by conj/
+qed-.
+
+lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ⋕[V, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
+elim (lleq_inv_lref_ge_gen … HL12 … HLK1 HLK2) //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ⋕[T, d] L2).
+#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_sort/
+| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
+ [ #HL12 #d elim (ylt_split i d) /3 width=1 by lleq_skip, or_introl/
+ #Hdi elim (lt_or_ge i (|L1|)) #HiL1
+ elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, lleq_free/
+ elim (ldrop_O1_lt … HiL2) #I2 #K2 #V2 #HLK2
+ elim (ldrop_O1_lt … HiL1) #I1 #K1 #V1 #HLK1
+ elim (eq_term_dec V2 V1)
+ [ #H3 elim (IH K1 V1 … K2 0) destruct
+ /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/
+ ]
+ -IH #H3 @or_intror
+ #H elim (lleq_fwd_lref … H) -H [1,3,4,6: * ]
+ [1,3: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
+ |5,6: /2 width=4 by ylt_yle_false/
+ ]
+ #Z1 #Z2 #Y1 #Y2 #X #HLY1 #HLY2 #HX #_
+ lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
+ lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
+ #H2 #H1 destruct /2 width=1 by/
+ ]
+| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_gref/
+| #a #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V … L2 d) /2 width=1 by/
+ elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/
+ #H1 #H2 @or_intror
+ #H elim (lleq_inv_bind … H) -H /2 width=1 by/
+| #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V … L2 d) /2 width=1 by/
+ elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, lleq_flat/
+ #H1 #H2 @or_intror
+ #H elim (lleq_inv_flat … H) -H /2 width=1 by/
+]
+-n /4 width=3 by lleq_fwd_length, or_intror/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lleq_trans: ∀d,T. Transitive … (lleq d T).
+#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/
+qed-.
+
+theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ⋕[d, T] L1→ L ⋕[d, T] L2 → L1 ⋕[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+theorem lleq_canc_dx: ∀L1,L2,L,T,d. L1 ⋕[d, T] L → L2 ⋕[d, T] L → L1 ⋕[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,d. (L1 ⋕[ⓑ{a,I}V.T, d] L2 → ⊥) →
+ (L1 ⋕[V, d] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V → ⊥).
+#a #I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
+/4 width=1 by lleq_bind, or_intror, or_introl/
+qed-.
+
+lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ⋕[ⓕ{I}V.T, d] L2 → ⊥) →
+ (L1 ⋕[V, d] L2 → ⊥) ∨ (L1 ⋕[T, d] L2 → ⊥).
+#I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
+/4 width=1 by lleq_flat, or_intror, or_introl/
+qed-.
+
+(* Note: lleq_nlleq_trans: ∀d,T,L1,L. L1⋕[T, d] L →
+ ∀L2. (L ⋕[T, d] L2 → ⊥) → (L1 ⋕[T, d] L2 → ⊥).
+/3 width=3 by lleq_canc_sn/ qed-.
+works with /4 width=8/ so lleq_canc_sn is more convenient
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⧣ break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyNegatedEq $T $d $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⧣ ⧣ break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyNegatedEqAlt $T $d $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazynegatedeq_4.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)
+
+definition llneq: relation4 ynat term lenv lenv ≝
+ λd,T,L1,L2. |L1| = |L2| ∧ (L1 ⋕[T, d] L2 → ⊥).
+
+interpretation
+ "negated lazy equivalence (local environment)"
+ 'LazyNegatedEq T d L1 L2 = (llneq d T L1 L2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazynegatedeqalt_4.ma".
+include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/substitution/llneq.ma".
+
+(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)
+
+(* alternative definition of llneq *)
+inductive llneqa: relation4 ynat term lenv lenv ≝
+| llneqa_neq: ∀I1,I2,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i]L1 ≡ K1.ⓑ{I1}V1 → ⇩[i]L2 ≡ K2.ⓑ{I2}V2 →
+ |K1| = |K2| → (V1 = V2 → ⊥) → llneqa d (#i) L1 L2
+| llneqa_eq : ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i]L1 ≡ K1.ⓑ{I1}V → ⇩[i]L2 ≡ K2.ⓑ{I2}V →
+ llneqa 0 (V) K1 K2 → llneqa d (#i) L1 L2
+| llneqa_bind_sn: ∀a,I,L1,L2,V,T,d.
+ llneqa d V L1 L2 → llneqa d (ⓑ{a,I}V.T) L1 L2
+| llneqa_bind_dx: ∀a,I,L1,L2,V,T,d.
+ llneqa (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → llneqa d (ⓑ{a,I}V.T) L1 L2
+| llneqa_flat_sn: ∀I,L1,L2,V,T,d.
+ llneqa d V L1 L2 → llneqa d (ⓕ{I}V.T) L1 L2
+| llneqa_flat_dx: ∀I,L1,L2,V,T,d.
+ llneqa d T L1 L2 → llneqa d (ⓕ{I}V.T) L1 L2
+.
+
+interpretation
+ "negated lazy equivalence (local environment) alternative"
+ 'LazyNegatedEqAlt T d L1 L2 = (llneqa d T L1 L2).
+
+(* Main properties **********************************************************)
+
+theorem llneq_llneqa: ∀T,L1,L2,d. L1 ⧣[T, d] L2 → L1 ⧣⧣[T, d] L2.
+#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ #k #Hn #L2 #d * #HL12 #H elim H /2 width=1 by lleq_sort/
+| #i #Hn #L2 #d * #HL12 #H elim (ylt_split i d) #Hdi
+ [ elim H /2 width=1 by lleq_skip/ ]
+ elim (lt_or_ge i (|L1|)) #HiL1
+ [2: elim H /3 width=3 by lleq_free, le_repl_sn_aux/ ]
+ elim (ldrop_O1_lt … HiL1) #I1 #K1 #V1 #HLK1
+ elim (ldrop_O1_lt L2 i) /2 width=1 by/ #I2 #K2 #V2 #HLK2
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) normalize
+ elim (eq_term_dec V1 V2) #HnV12 destruct
+ [2: #H @(llneqa_neq … HLK1 … HLK2) /2 width=1 by/ ] (**) (* explicit constructor *)
+ elim (lleq_dec V2 K1 K2 0) #HnV2 [ elim H /2 width=8 by lleq_lref/ ]
+ #H @(llneqa_eq … HLK1 … HLK2) /4 width=2 by ldrop_fwd_rfw, conj/ (**) (* explicit constructor *)
+| #p #Hn #L2 #d * #HL12 #H elim H /2 width=1 by lleq_gref/
+| #a #I #V #T #Hn #L2 #d * #HL12 #H destruct elim (nlleq_inv_bind … H) -H
+ [ /5 width=1 by llneqa_bind_sn, conj/
+ | #H @llneqa_bind_dx @IH // @conj normalize /2 width=1 by/
+ ]
+| #I #V #T #Hn #L2 #d * #HL12 #H destruct elim (nlleq_inv_flat … H) -H
+ /5 width=1 by llneqa_flat_dx, llneqa_flat_sn, conj/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/llneq_alt.ma".
+
+(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llneq_inv_atom1: ∀L1,L2,T,d. L1 ⧣⧣[T, d] L2 → |L1| ≤ d → ⊥.
+#L1 #L2 #T #d #H elim H -L1 -L2 -T -d /2 width=1 by/
+[ #I1 #I2 #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #_ #_ #_
+ >(ldrop_fwd_length … HLK1) -HLK1 normalize
+ #H lapply (yle_trans … H … Hdi) -d
+ /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/
+| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #_ #_ #_
+ >(ldrop_fwd_length … HLK1) -HLK1 normalize
+ #H lapply (yle_trans … H … Hdi) -d
+ /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/
+| #a #I #L1 <yplus_inj /4 width=1 by yle_succ/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/llneq_alt.ma".
+include "basic_2/computation/lpxs_ldrop.ma".
+include "basic_2/computation/lpxs_cpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
+
+(* Forward lemmas on negated lazy equivalence for local environments ********)
+
+lemma lpxs_llneq_fwd_cpxs: ∀h,g,G,L1,L2,T,d. L1 ⧣⧣[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃T1,T2. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] T1 & ⦃G, L2⦄ ⊢ T ▶*[d, ∞] T2 &
+ ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & (T1 = T2 → ⊥).
+#h #g #G #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+[ #I1 #I2 #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #_ #HnV12 #HL12
+ elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HY
+ elim (lpxs_inv_pair1 … H) -H #Z #X #_ #HV12 #H destruct
+ lapply (ldrop_mono … HY … HLK2) -HY #H destruct
+ lapply (ldrop_fwd_drop2 … HLK1) #H2LK1
+ elim (lift_total V1 0 (i+1)) #T1 #HVT1
+ elim (lift_total V2 0 (i+1)) #T2 #HVT2
+ @(ex4_2_intro … T1 T2) /3 width=10 by cpxs_lift, cpys_subst, lift_inj/ (**) (* explicit constructor *)
+| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #_ #IHK12 #HL12
+ elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HY
+ elim (lpxs_inv_pair1 … H) -H #Z #X #HK12 #_ #H destruct
+ lapply (ldrop_mono … HY … HLK2) -HY #H destruct
+ elim (IHK12 HK12) -IHK12 -HK12 #V1 #V2 #HV1 #HV2 #HV12 #HnV12
+ lapply (ldrop_fwd_drop2 … HLK1) #H2LK1
+ elim (lift_total V1 0 (i+1)) #T1 #HVT1
+ elim (lift_total V2 0 (i+1)) #T2 #HVT2
+ @(ex4_2_intro … T1 T2) /3 width=10 by cpxs_lift, cpys_subst_Y2, lift_inj/ (**) (* explicit constructor *)
+| #a #I #L1 #L2 #V #T #d #_ #IHV #HL12 elim (IHV HL12) -IHV -HL12
+ #V1 #V2 #HV1 #HV2 #HV12 #HnV12
+ @(ex4_2_intro … (ⓑ{a,I}V1.T) (ⓑ{a,I}V2.T)) /2 width=1 by cpys_bind, cpxs_pair_sn/
+ #H destruct /2 width=1 by/
+| #a #I #L1 #L2 #V #T #d #_ #IHT #HL12 elim (IHT ?) /2 width=1 by lpxs_pair_refl/ -IHT -HL12
+ #T1 #T2 #HT1 #HT2 #HT12 #HnT12
+ @(ex4_2_intro … (ⓑ{a,I}V.T1) (ⓑ{a,I}V.T2)) /2 width=1 by cpys_bind, cpxs_bind_dx/
+ #H destruct /2 width=1 by/
+| #I #L1 #L2 #V #T #d #_ #IHV #HL12 elim (IHV HL12) -IHV -HL12
+ #V1 #V2 #HV1 #HV2 #HV12 #HnV12
+ @(ex4_2_intro … (ⓕ{I}V1.T) (ⓕ{I}V2.T)) /2 width=1 by cpys_flat, cpxs_pair_sn/
+ #H destruct /2 width=1 by/
+| #I #L1 #L2 #V #T #d #_ #IHT #HL12 elim (IHT HL12) -IHT -HL12
+ #T1 #T2 #HT1 #HT2 #HT12 #HnT12
+ @(ex4_2_intro … (ⓕ{I}V.T1) (ⓕ{I}V.T2)) /2 width=1 by cpys_flat, cpxs_flat_dx/
+ #H destruct /2 width=1 by/
+]
+qed-.
+
+lemma lpxs_nlleq_fwd_cpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ (L1 ⋕[T, d] L2 → ⊥) →
+ ∃∃T1,T2. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] T1 & ⦃G, L2⦄ ⊢ T ▶*[d, ∞] T2 &
+ ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & (T1 = T2 → ⊥).
+/5 width=4 by lpxs_llneq_fwd_cpxs, lpxs_fwd_length, llneq_llneqa, conj/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/llprs_cprs.ma".
+include "basic_2/conversion/cpc_cpc.ma".
+include "basic_2/equivalence/cpcs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
+#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
+[ /3 width=3 by ex2_intro/
+| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
+ [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
+ | /3 width=5 by cprs_strap2, ex2_intro/
+ ]
+]
+qed-.
+
+(* Basic_1: was: pc3_gen_sort *)
+lemma cpcs_inv_sort: ∀G,L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
+#G #L #k1 #k2 #H elim (cpcs_inv_cprs … H) -H
+#T #H1 >(cprs_inv_sort1 … H1) -T #H2
+lapply (cprs_inv_sort1 … H2) -L #H destruct //
+qed-.
+
+lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+#a #G #L #W1 #T1 #T #H
+elim (cpcs_inv_cprs … H) -H #X #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
+/3 width=6 by cprs_bind, ex2_2_intro/
+qed-.
+
+lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
+
+(* Basic_1: was: pc3_gen_sort_abst *)
+lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
+#a #G #L #W #T #k #H
+elim (cpcs_inv_cprs … H) -H #X #H1
+>(cprs_inv_sort1 … H1) -X #H2
+elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: pc3_gen_lift *)
+lemma cpcs_inv_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
+#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
+elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
+elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
+elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
+>(lift_inj … HXU … HTU) -X -U -d -e /2 width=3 by cprs_div/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpr_cpcs_trans: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
+ ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, cprs_llpr_trans/
+qed-.
+
+lemma llprs_cpcs_trans: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡*[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡*[T2, 0] L2 →
+ ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, llprs_cprs_trans/
+qed-.
+
+lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cpr_cprs_div/
+qed-.
+
+lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_cpr_div/
+qed-.
+
+lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_div/
+qed-.
+
+(* Basic_1: was: pc3_wcpr0_t *)
+(* Basic_1: note: pc3_wcpr0_t should be renamed *)
+lemma llpr_cprs_conf: ∀G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #HL12 #T2 #HT12 elim (cprs_llpr_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
+qed-.
+
+(* Basic_1: was only: pc3_pr0_pr2_t *)
+(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
+lemma llpr_cpr_conf: ∀G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+/3 width=5 by llpr_cprs_conf, cpr_cprs/ qed-.
+
+(* Basic_1: was only: pc3_thin_dx *)
+lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+#G #L #V1 #V2 #HV12 #T1 #T2 #HT12
+elim (cpcs_inv_cprs … HV12) -HV12
+elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_flat, cprs_div/
+qed.
+
+lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
+
+lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
+#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, cprs_bind/
+qed.
+
+lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
+#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
+/3 width=5 by cprs_div, cprs_bind/
+qed.
+
+lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
+ ∀L2. L2 ⊑ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, lsubr_cprs_trans/
+qed-.
+
+(* Basic_1: was: pc3_lift *)
+lemma cpcs_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
+#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
+elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/
+qed.
+
+lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
+ ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
+#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
+
+(* More inversion lemmas ****************************************************)
+
+axiom cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+ ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+(*
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
+elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
+lapply (llprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by llprs_pair/ -HT2 #HT2
+lapply (llprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by llprs_pair/ -HT2 #HT2
+/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
+qed-.
+*)
+lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+ ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
+#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was pc3_t *)
+theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
+
+theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
+
+theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
+
+lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+ ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+
+lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+ ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+
+(* Basic_1: was: pc3_wcpr0 *)
+lemma llpr_cpcs_conf: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
+ ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
+/3 width=5 by cpcs_canc_dx, llpr_cprs_conf/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/llpr_llpr.ma".
+include "basic_2/computation/cprs_lift.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: pr3_t *)
+(* Basic_1: includes: pr1_t *)
+theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
+normalize /2 width=3 by trans_TC/ qed-.
+
+(* Basic_1: was: pr3_confluence *)
+(* Basic_1: includes: pr1_confluence *)
+theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
+normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
+
+theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=5 by cprs_trans, cprs_bind_dx/
+qed.
+
+(* Basic_1: was: pr3_flat *)
+theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
+#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
+qed.
+
+theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
+#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
+@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
+qed.
+
+theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
+#V #V2 #_ #HV2 #IHV1
+@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
+qed.
+
+theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+ ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2
+/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
+qed.
+
+theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+ ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr3_gen_appl *)
+lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓐV2. T2
+ | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
+ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
+ | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
+ ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
+ ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
+#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_appl1 … HU2) -HU2 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
+ | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
+ lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
+ lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
+ /5 width=5 by cprs_flat_dx, cpr_cprs, cprs_bind, lsubr_abst, ex2_3_intro, or3_intro1/
+ | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
+ @or3_intro2 @(ex4_5_intro … HV2 HT10) /3 width=3 by cprs_flat_sn, cprs_strap1, cpr_cprs, cprs_bind/ (**) (* full auto is too slow 11s *)
+ ]
+| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
+| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
+]
+qed-.
+
+(* Properties concerning sn parallel reduction on local environments ********)
+
+(* Basic_1: was just: pr3_pr2_pr2_t *)
+(* Basic_1: includes: pr3_pr0_pr2_t *)
+lemma llpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (llpr G 0).
+#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+ elim (llpr_inv_lref_ge_dx … HL12 … HLK2) -L2
+ /5 width=7 by cprs_delta, cprs_strap2, llpr_cpr_conf/
+| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (llpr_inv_bind_O … HL12) -HL12 /4 width=1 by cprs_bind/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) -HL12 /3 width=1 by cprs_flat/
+| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
+ elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_zeta/
+| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) /3 width=1 by cprs_tau/
+| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
+ elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_beta/
+| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
+ elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_theta/
+]
+qed-.
+
+lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=9 by llpr_cpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
+lemma cprs_llpr_trans: ∀G. s_rs_transitive … (cpr G) (llpr G 0).
+/3 width=6 by llpr_cpr_trans, llpr_cpr_conf, s_r_trans_LTC1/ qed-.
+
+(* Basic_1: was: pr3_strip *)
+(* Basic_1: includes: pr1_strip *)
+lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
+normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
+
+lemma cprs_llpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #H @(cprs_ind_dx … T0 H) -T0 /2 width=3 by ex2_intro/
+#T0 #T #HT0 #_ #IHT1 #L1 #HL01
+elim (IHT1 … L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2
+elim (llpr_cpr_conf_dx … HT0 … HL01) -L0 #T3 #HT03 #HT3
+elim (cprs_strip … HT2 … HT3) -T
+/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
+qed-.
+
+lemma cprs_llpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cprs_llpr_conf_dx … HT01 … HL01)
+/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/
+qed-.
+
+lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/llpr_ldrop.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Advanced properties ******************************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma cprs_delta: ∀G,L,K,V,V2,i.
+ ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡* V2 →
+ ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡* W2.
+#G #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=6 by cpr_cprs, cpr_delta/ ]
+#V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
+lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
+elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/
+qed.
+
+lemma cprs_llpr_conf: ∀G. s_r_confluent1 … (cprs G) (llpr G 0).
+/3 width=5 by llpr_cpr_conf, s_r_conf1_LTC1/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr3_gen_lref *)
+lemma cprs_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡* T2 →
+ T2 = #i ∨
+ ∃∃K,V1,T1. ⇩[i] L ≡ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡* T1 &
+ ⇧[0, i + 1] T1 ≡ T2.
+#G #L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1 by or_introl/
+#T #T2 #_ #HT2 *
+[ #H destruct
+ elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+ * /4 width=6 by cpr_cprs, ex3_3_intro, or_intror/
+| * #K #V1 #T1 #HLK #HVT1 #HT1
+ lapply (ldrop_fwd_drop2 … HLK) #H0LK
+ elim (cpr_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+ /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr3_lift *)
+lemma cprs_lift: ∀G. l_liftable (cprs G).
+/3 width=10 by l_liftable_LTC, cpr_lift/ qed.
+
+(* Basic_1: was: pr3_gen_lift *)
+lemma cprs_inv_lift1: ∀G. l_deliftable_sn (cprs G).
+/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡ break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyPRedSn $G $L1 $L2 $T $d }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡* break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyPRedSnStar $G $L1 $L2 $T $d }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazypredsn_5.ma".
+include "basic_2/relocation/llpx_sn.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
+
+definition llpr: genv → relation4 ynat term lenv lenv ≝ λG. llpx_sn (cpr G).
+
+interpretation "lazy parallel reduction (local environment, sn variant)"
+ 'LazyPRedSn G L1 L2 T d = (llpr G d T L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpr_inv_flat: ∀I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[T, d] L2.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpr_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpr_lref: ∀I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 → ⦃G, L1⦄ ⊢ ➡[#i, d] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+(* Note: lemma 250 *)
+lemma llpr_refl: ∀G,T,d. reflexive … (llpr G d T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+(* Basic_1: removed theorems 5: wcpr0_gen_sort wcpr0_gen_head
+ wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/fquq_alt.ma".
+include "basic_2/static/ssta_llpx_sn.ma".
+include "basic_2/reduction/cpr_lift.ma".
+include "basic_2/reduction/cpr_llpx_sn.ma".
+include "basic_2/reduction/llpr.ma".
+
+(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpr_inv_lref_ge_dx: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
+/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
+
+lemma llpr_inv_lref_ge_sn: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
+/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
+
+lemma llpr_inv_lref_ge_bi: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma llpr_inv_bind_O: ∀a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [ⓑ{a,I}V.T, 0] L2 →
+ ⦃G, L1⦄ ⊢ ➡[V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+lemma llpr_bind_repl_O: ∀I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V2 →
+ ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡ W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[T, 0] L2.ⓑ{J}W2.
+/2 width=4 by llpx_sn_bind_repl_O/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpr_ssta_conf: ∀h,g,G. s_r_confluent1 … (ssta h g G) (llpr G 0).
+/3 width=10 by ssta_llpx_sn_conf, cpr_lift/ qed-.
+
+lemma llpr_cpr_conf: ∀G. s_r_confluent1 … (cpr G) (llpr G 0).
+/3 width=10 by cpr_llpx_sn_conf, cpr_inv_lift1, cpr_lift/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=10 by llpr_lref, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
+#G #L #K #U #T #e #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (e+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=10 by llpr_lref, cpr_pair_sn, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
+#G #L #K #U #T #e #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (e+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/llpx_sn_llpx_sn.ma".
+include "basic_2/substitution/fqup.ma".
+include "basic_2/reduction/llpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties on context-sensitive parallel reduction for terms ********)
+
+fact cpr_conf_llpr_atom_atom:
+ ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
+/2 width=3 by cpr_atom, ex2_intro/ qed-.
+
+fact cpr_conf_llpr_atom_delta:
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+ ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01
+elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
+lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1))
+/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
+qed-.
+
+(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
+fact cpr_conf_llpr_delta_delta:
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+ ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
+ ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_
+lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1
+elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
+lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_bind_bind:
+ ∀a,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_bind_zeta:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_zeta_zeta:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_flat_flat:
+ ∀I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) //
+elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_flat_tau:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
+elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_tau_tau:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
+elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_flat_beta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
+/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
+qed-.
+
+(* Basic-1: includes:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact cpr_conf_llpr_flat_theta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2
+elim (cpr_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
+ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
+ /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
+| #T1 #HT01 #HXT1 #H destruct
+ elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+ elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
+ /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_llpr_beta_beta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
+/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+(* Basic_1: includes: pr0_upsilon_upsilon *)
+fact cpr_conf_llpr_theta_theta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+ ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/
+/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+theorem cpr_conf_llpr: ∀G. llpx_sn_confluent2 (cpr G) (cpr G).
+#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1 … H1) -H1
+ elim (cpr_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_llpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_llpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_llpr_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_conf_llpr_delta_delta/
+ ]
+| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpr_conf_llpr_bind_bind/
+ | /4 width=11 by ex2_commute, cpr_conf_llpr_bind_zeta/
+ | /3 width=11 by cpr_conf_llpr_bind_zeta/
+ | /3 width=12 by cpr_conf_llpr_zeta_zeta/
+ ]
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+ | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ]
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #HX2 #H2
+ |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+ |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+ ] destruct
+ [ /3 width=10 by cpr_conf_llpr_flat_flat/
+ | /4 width=8 by ex2_commute, cpr_conf_llpr_flat_tau/
+ | /4 width=12 by ex2_commute, cpr_conf_llpr_flat_beta/
+ | /4 width=14 by ex2_commute, cpr_conf_llpr_flat_theta/
+ | /3 width=8 by cpr_conf_llpr_flat_tau/
+ | /3 width=8 by cpr_conf_llpr_tau_tau/
+ | /3 width=12 by cpr_conf_llpr_flat_beta/
+ | /3 width=13 by cpr_conf_llpr_beta_beta/
+ | /3 width=14 by cpr_conf_llpr_flat_theta/
+ | /3 width=17 by cpr_conf_llpr_theta_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_confluence pr2_confluence *)
+theorem cpr_conf: ∀G,L. confluent … (cpr G L).
+/2 width=6 by cpr_conf_llpr/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma llpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_llpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+lemma llpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_llpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazypredsnstar_5.ma".
+include "basic_2/reduction/llpr.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+definition llprs: genv → relation4 ynat term lenv lenv ≝
+ λG,d. LTC … (llpr G d).
+
+interpretation "lazy parallel computation (local environment, sn variant)"
+ 'LazyPRedSnStar G L1 L2 T d = (llprs G d T L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma llprs_ind: ∀G,L1,T,d. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L2.
+#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma llprs_ind_dx: ∀G,L2,T,d. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L1.
+#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpr_llprs: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+/2 width=1 by inj/ qed.
+
+lemma llprs_refl: ∀G,L,T,d. ⦃G, L⦄ ⊢ ➡*[T, d] L.
+/2 width=1 by lpr_llprs/ qed.
+
+lemma llprs_strap1: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+normalize /2 width=3 by step/ qed-.
+
+lemma llprs_strap2: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llprs_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → |L1| = |L2|.
+#G #L1 #L2 #T #d #H @(llprs_ind … H) -L2
+/3 width=6 by llpr_fwd_length, trans_eq/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/llpx_sn_tc.ma".
+include "basic_2/computation/cprs_cprs.ma".
+include "basic_2/computation/llprs.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Advanced properties ******************************************************)
+
+lemma llprs_pair_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡*[T, 0] L.ⓑ{I}V2.
+/2 width=1 by llpx_sn_TC_pair_dx/ qed.
+
+(* Properties on context-sensitive parallel computation for terms ***********)
+
+lemma llprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (llprs G 0).
+/3 width=5 by cprs_llpr_trans, s_r_trans_LTC2/ qed-.
+
+(* Basic_1: was just: pr3_pr3_pr3_t *)
+lemma llprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (llprs G 0).
+#G @s_r_to_s_rs_trans @s_r_trans_LTC2
+/3 width=5 by cprs_llpr_trans, s_rs_trans_TC1/ (**) (* full auto too slow *)
+qed-.
+
+(* Note: this is an instance of a general theorem *)
+lemma llprs_cprs_conf_dx: ∀G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡* U2 →
+ ∀L0. ⦃G2, L0⦄ ⊢ ➡*[T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[U2, O] L2.
+#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx … H) -L0 //
+#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 … IHL2) -IHL2
+lapply (llprs_cprs_trans … HTU2 … HL2) -L2 #HTU2
+/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/
+qed-.
+
+(* Note: this is an instance of a general theorem *)
+lemma llprs_cpr_conf_dx: ∀G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∀L0. ⦃G2, L0⦄ ⊢ ➡*[T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[U2, O] L2.
+#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx … H) -L0 //
+#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 … IHL2) -IHL2
+lapply (llprs_cpr_trans … HTU2 … HL2) -L2 #HTU2
+/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/
+qed-.
+
+lemma llprs_cprs_conf_sn: ∀G,L0,L1,T0. ⦃G, L0⦄ ⊢ ➡*[T0, 0] L1 →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #L1 #T0 #H @(llprs_ind_dx … H) -L0 /2 width=3 by ex2_intro/
+#L0 #L #HL0 #HL1 #IHL1 #T1 #HT01 elim (cprs_llpr_conf_sn … HT01 … HL0)
+#T2 #HT12 #HT02 elim (IHL1 … HT02) -IHL1 -HT02
+lapply (cprs_trans … HT01 … HT12) #HT02
+lapply (cprs_llpr_conf … HT02 … HL0) -HT02 -HL0
+/4 width=5 by cprs_llpr_trans, cprs_trans, ex2_intro/
+qed-.
+
+lemma llprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡*[T0, 0] L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+/3 width=3 by llprs_cprs_conf_sn, cpr_cprs/ qed-.
+
+lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=3 by llprs_cprs_trans, llprs_pair_dx, cprs_bind/ qed-.
+
+(* Inversion lemmas on context-sensitive parallel computation for terms *****)
+
+(* Basic_1: was: pr3_gen_abst *)
+lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓛ{a}W2.T2.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (llprs_cpr_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro/
+qed-.
+
+lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
+ ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
+#a #G #L #W1 #W2 #T1 #T2 #H
+elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
+qed-.
+
+(* Basic_1: was pr3_gen_abbr *)
+lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓓ{a}V2.T2
+ ) ∨
+ ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_abbr1 … HU02) -HU02 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?)
+ /4 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
+ | #T2 #HT02 #HUT2
+ lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
+ /4 width=3 by llprs_pair_dx, cprs_trans, ex3_intro, or_intror/
+ ]
+| #U1 #HTU1 #HU01
+ elim (lift_total U2 0 1) #U #HU2
+ lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
+ /4 width=3 by cprs_strap1, ldrop_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* Note: we loose lprs_cprs_conf_dx and derivatives:
+ lprs_cpr_conf_dx lprs_cprs_conf
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/llprs.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Main properties **********************************************************)
+
+theorem llprs_trans: ∀G,T,d. Transitive … (llprs G d T).
+normalize /2 width=3 by trans_TC/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+definition llpx_sn_confluent2: relation (lenv→relation term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. llpx_sn R1 0 T0 L0 L1 → ∀L2. llpx_sn R2 0 T0 L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+(* Note: we miss llpx_sn_conf and derivatives: lpr_conf lprs_conf *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/llpx_ldrop.ma".
+include "basic_2/computation/cpxs_cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on lazy sn reduction on local environments ********************)
+
+lemma cpxs_llpx_conf: ∀h,g,G. s_r_confluent1 … (cpxs h g G) (llpx h g G 0).
+/3 width=5 by llpx_cpx_conf, s_r_conf1_LTC1/ qed-.
+
+lemma llpx_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (llpx h g G 0).
+#h #g #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| /3 width=2 by cpx_cpxs, cpx_sort/
+| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+ elim (llpx_inv_lref_ge_dx … HL12 … HLK2) -L2
+ /5 width=8 by cpxs_delta, cpxs_strap2, llpx_cpx_conf/
+| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (llpx_inv_bind_O … HL12) -HL12 /4 width=1 by cpxs_bind/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (llpx_inv_flat … HL12) -HL12 /3 width=1 by cpxs_flat/
+| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
+ elim (llpx_inv_bind_O … HL12) /3 width=3 by cpxs_zeta/
+| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
+ elim (llpx_inv_flat … HL12) /3 width=1 by cpxs_tau/
+| #G #L2 #V1 #V2 #T2 #HV12 #IHV12 #L1 #HL12
+ elim (llpx_inv_flat … HL12) /3 width=1 by cpxs_ti/
+| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
+ elim (llpx_inv_flat … HL12) -HL12 #HV1 #HL12
+ elim (llpx_inv_bind_O … HL12) /3 width=3 by cpxs_beta/
+| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
+ elim (llpx_inv_flat … HL12) -HL12 #HV1 #HL12
+ elim (llpx_inv_bind_O … HL12) /3 width=3 by cpxs_theta/
+]
+qed-.
+
+lemma llpx_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (llpx h g G 0).
+#h #g #G @s_r_trans_LTC1 /2 width=3 by llpx_cpx_trans, llpx_cpx_conf/ (**) (* full auto fails here but works in cprs_cprs *)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/cpxs_llpx.ma".
+include "basic_2/computation/csx_alt.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Properties on lazy sn extended reduction for local environments **********)
+
+lemma csx_llpx_conf: ∀h,g,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L1 #T #H @(csx_ind_alt … H) -T
+/5 width=3 by csx_intro_cpxs, llpx_cpxs_trans, cpxs_llpx_conf/ (* 2 cpxs_llpx_trans *)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/csx_llpx.ma".
+include "basic_2/computation/llpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Properties on lazy sn extended computation for local environments ********)
+
+lemma csx_llpxs_conf: ∀h,g,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L1 #T #HT #L2 #H @(llpxs_ind … H) -L2 /3 by llpxs_strap1, csx_llpx_conf/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cpx_lift.ma".
+include "basic_2/reduction/llpx_lpr.ma".
+include "basic_2/computation/fpbs.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+(* Properties on sn parallel reduction for local environments ***************)
+
+(* Note: this is used in the closure proof *)
+(* Note: original proof: /4 width=5 by fpbs_strap1, lpr_fpb, cpr_fpb/ *)
+(* Note: this should be moved *)
+lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
+/5 width=5 by fpbs_strap1, cpr_fpb, fpb_llpx, lpr_llpx/ qed.
+
+(* Note: this is used in the closure proof *)
+(* Note: this should be moved *)
+lemma cpr_lpr_ssta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,l2.
+ ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ T2 ▪[h, g] l2+1 → ⦃G, L2⦄ ⊢ T2 •[h, g] U2 →
+ ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
+/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, ssta_cpx, fpb_cpx/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡ break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyPRedSn $G $L1 $L2 $h $g $T $d }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡* break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyPRedSnStar $G $L1 $L2 $h $g $T $d }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ ⋕ ⬊ * break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L )"
+ non associative with precedence 45
+ for @{ 'LazySN $h $g $T $d $G $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ ⋕ ⬊ ⬊ * break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L )"
+ non associative with precedence 45
+ for @{ 'LazySNAlt $h $g $T $d $G $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazypredsn_7.ma".
+include "basic_2/relocation/llpx_sn.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
+
+definition llpx: ∀h. sd h → genv → relation4 ynat term lenv lenv ≝
+ λh,g,G. llpx_sn (cpx h g G).
+
+interpretation "lazy extended parallel reduction (local environment, sn variant)"
+ 'LazyPRedSn G L1 L2 h g T d = (llpx h g G d T L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_inv_flat: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_fwd_length: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+lemma llpx_fwd_flat_dx: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
+/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
+
+lemma llpx_fwd_pair_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ②{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
+/2 width=3 by llpx_sn_fwd_pair_sn/ qed-.
+
+(* Note: this might be removed *)
+lemma llpx_fwd_bind_sn: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
+/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
+
+(* Note: this might be removed *)
+lemma llpx_fwd_bind_dx: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, d] L2 →
+ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, ⫯d] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
+
+(* Note: this might be removed *)
+lemma llpx_fwd_flat_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
+/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_lref: ∀h,g,I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+lemma llpx_refl: ∀h,g,G,T,d. reflexive … (llpx h g G d T).
+/2 width=1 by llpx_sn_refl/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/aaa_lift.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/reduction/llpx_ldrop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+(* Note: lemma 500 *)
+lemma aaa_cpx_llpx_conf: ∀h,g,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T1, 0] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #g #L1 #k #X #H elim (cpx_inv_sort1 … H) -H // * //
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+ elim (cpx_inv_lref1 … H) -H
+ [ #H destruct
+ elim (llpx_inv_lref_ge_sn … HL12 … HLK1) -L1 /3 width=6 by aaa_lref/
+ | * #J #Y #Z #V2 #H #HV12 #HV2
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (llpx_inv_lref_ge_sn … HL12 … HLK1) -L1 /3 width=8 by aaa_lift, ldrop_fwd_drop2/
+ ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (llpx_inv_bind_O … HL12) -HL12 #HV1 #HT1
+ elim (cpx_inv_abbr1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=4 by llpx_bind_repl_O, aaa_abbr/
+ | #T2 #HT12 #HT2 #H destruct -IHV1 /3 width=8 by aaa_inv_lift, ldrop_drop/
+ ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (llpx_inv_bind_O … HL12) -HL12 #HV1 #HT1
+ elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=4 by llpx_bind_repl_O, aaa_abst/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (llpx_inv_flat … HL12) -HL12 #HV1 #HT1
+ elim (cpx_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
+ | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
+ lapply (IHV1 … HV12 … HV1) -IHV1 -HV12 #HV2
+ lapply (IHT1 (ⓛ{b}W2.U2) … HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+ elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
+ /5 width=6 by lsuba_aaa_trans, lsuba_abbr, aaa_abbr, aaa_cast/
+ | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+ lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=2 by ldrop_drop/ #HV2
+ lapply (IHT1 (ⓓ{b}W2.U2) … HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+ elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
+ ]
+| #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (llpx_inv_flat … HL12) -HL12 #HV1 #HT1
+ elim (cpx_inv_cast1 … H) -H
+ [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by aaa_cast/
+ | -IHV1 /2 width=1 by/
+ | -IHT1 /2 width=1 by/
+ ]
+]
+qed-.
+
+lemma aaa_cpx_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/2 width=7 by aaa_cpx_llpx_conf/ qed-.
+
+lemma aaa_llpx_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/2 width=7 by aaa_cpx_llpx_conf/ qed-.
+
+lemma aaa_cpr_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-.
+(*
+lamma aaa_llpr_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/3 width=5 by aaa_llpx_conf, llpr_llpx/ qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cpx_llpx_sn.ma".
+include "basic_2/reduction/cpx_lift.ma".
+include "basic_2/reduction/llpx.ma".
+
+(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpx_inv_lref_ge_dx: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
+/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
+
+lemma llpx_inv_lref_ge_sn: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
+/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
+
+lemma llpx_inv_lref_ge_bi: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma llpx_inv_bind_O: ∀h,g,a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [h, g, ⓑ{a,I}V.T, 0] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, g, V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+lemma llpx_bind_repl_O: ∀h,g,I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V2 →
+ ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[h, g, W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{J}W2.
+/2 width=4 by llpx_sn_bind_repl_O/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma llpx_fwd_bind_O_dx: ∀h,g,a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, 0] L2 →
+ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V.
+/2 width=3 by llpx_sn_fwd_bind_O_dx/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpx_cpx_conf: ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx h g G 0).
+/3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/ qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma llpx_ldrop_trans_O: ∀h,g,G,L1,L2,U. ⦃G, L1⦄ ⊢ ➡[h, g, U, 0] L2 →
+ ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
+ ∃∃K1. ⇩[e] L1 ≡ K1 & ⦃G, K1⦄ ⊢ ➡[h, g, T, 0] K2.
+/2 width=5 by llpx_sn_ldrop_trans_O/ qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma llpx_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=7 by llpx_fwd_bind_O_dx, llpx_fwd_pair_sn,llpx_fwd_flat_dx, ex3_2_intro/
+[ #I #G1 #L1 #V1 #K1 #H elim (llpx_inv_lref_ge_dx … H) -H [5,6: // |2,3,4: skip ]
+ #Y1 #W1 #HKL1 #HW1 #HWV1 elim (lift_total V1 0 1)
+ /4 width=7 by llpx_cpx_conf, cpx_delta, fqu_drop, ldrop_fwd_drop2, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HU1
+ elim (llpx_ldrop_trans_O … HU1 … HLK1) -L1
+ /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma llpx_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (llpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cpx_lleq.ma".
+include "basic_2/reduction/llpx.ma".
+
+(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma llpx_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
+/2 width=1 by llpx_sn_lrefl/ qed-.
+
+lemma lleq_llpx_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
+ ∀L.⦃G, L2⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L.
+/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_trans/ qed-.
+
+lemma lleq_llpx_conf: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
+ ∀L.⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L2⦄ ⊢ ➡[h, g, T, d] L.
+/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_conf/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/llpx_sn_lpx_sn.ma".
+include "basic_2/reduction/lpr.ma".
+include "basic_2/reduction/llpx.ma".
+
+(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
+
+(* Properties on sn parallel reduction **************************************)
+
+(* Note: this should be moved *)
+lemma lpr_llpx: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
+/3 width=4 by cpr_cpx, lpx_sn_llpx_sn, llpx_sn_co/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazypredsnstar_7.ma".
+include "basic_2/reduction/llpx.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
+
+definition llpxs: ∀h. sd h → genv → relation4 ynat term lenv lenv ≝
+ λh,g,G,d. LTC … (llpx h g G d).
+
+interpretation "lazy extended parallel computation (local environment, sn variant)"
+ 'LazyPRedSnStar G L1 L2 h g T d = (llpxs h g G d T L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma llpxs_ind: ∀h,g,G,L1,T,d. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡[h, g, T, d] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → R L2.
+#h #g #G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma llpxs_ind_dx: ∀h,g,G,L2,T,d. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → R L1.
+#h #g #G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+normalize /2 width=1 by inj/ qed.
+
+lemma llpxs_refl: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L.
+/2 width=1 by llpx_llpxs/ qed.
+
+lemma llpxs_strap1: ∀h,g,G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+normalize /2 width=3 by step/ qed.
+
+lemma llpxs_strap2: ∀h,g,G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+normalize /2 width=3 by TC_strap/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpxs_fwd_length: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → |L1| = |L2|.
+#h #g #G #L1 #L2 #T #d #H @(llpxs_ind … H) -L2
+/3 width=8 by llpx_fwd_length, trans_eq/
+qed-.
+
+(* Note: this might be moved *)
+lemma llpxs_fwd_bind_sn: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓑ{a,I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡*[h, g, V, d] L2.
+#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
+/3 width=6 by llpx_fwd_bind_sn, llpxs_strap1/
+qed-.
+
+(* Note: this might be moved *)
+lemma llpxs_fwd_bind_dx: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓑ{a,I}V.T, d] L2 →
+ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, g, T, ⫯d] L2.ⓑ{I}V.
+#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
+/3 width=6 by llpx_fwd_bind_dx, llpxs_strap1/
+qed-.
+
+(* Note: this might be moved *)
+lemma llpxs_fwd_flat_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡*[h, g, V, d] L2.
+#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
+/3 width=6 by llpx_fwd_flat_sn, llpxs_strap1/
+qed-.
+
+(* Note: this might be moved *)
+lemma llpxs_fwd_flat_dx: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind … H) -L2
+/3 width=6 by llpx_fwd_flat_dx, llpxs_strap1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/llpx_aaa.ma".
+include "basic_2/computation/llpxs.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_llpxs_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+#h #g #G #L1 #T #A #HT #L2 #HL12
+@(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_llpx_conf/
+qed-.
+(*
+lamma aaa_llprs_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/3 width=5 by aaa_llpxs_conf, llprs_llpxs/ qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/llpx_sn_tc.ma".
+include "basic_2/computation/cpxs_llpx.ma".
+include "basic_2/computation/llpxs.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
+
+(* Advanced properties ******************************************************)
+
+lemma llpxs_pair_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡*[h, g, T, 0] L.ⓑ{I}V2.
+/2 width=1 by llpx_sn_TC_pair_dx/ qed.
+
+(* Properties on context-sensitive extended parallel computation for terms **)
+
+lemma llpxs_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (llpxs h g G 0).
+/3 width=5 by s_r_trans_LTC2, llpx_cpxs_trans/ qed-.
+
+lemma llpxs_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (llpxs h g G 0).
+#h #g #G @s_r_to_s_rs_trans @s_r_trans_LTC2
+/3 width=5 by llpx_cpxs_trans, s_rs_trans_TC1/ (**) (* full auto too slow *)
+qed-.
+
+(* Note: this is an instance of a general theorem *)
+lemma llpxs_cpxs_conf_dx: ∀h,g,G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∀L0. ⦃G2, L0⦄ ⊢ ➡*[h, g, T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[h, g, U2, O] L2.
+#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx … H) -L0 //
+#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 … IHL2) -IHL2
+lapply (llpxs_cpxs_trans … HTU2 … HL2) -L2 #HTU2
+/3 width=3 by llpx_cpxs_trans, cpxs_llpx_conf/
+qed-.
+
+(* Note: this is an instance of a general theorem *)
+lemma llpxs_cpx_conf_dx: ∀h,g,G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+ ∀L0. ⦃G2, L0⦄ ⊢ ➡*[h, g, T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[h, g, U2, O] L2.
+#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx … H) -L0 //
+#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 … IHL2) -IHL2
+lapply (llpxs_cpx_trans … HTU2 … HL2) -L2 #HTU2
+/3 width=3 by llpx_cpxs_trans, cpxs_llpx_conf/
+qed-.
+
+lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+/4 width=5 by llpxs_cpxs_trans, llpxs_pair_dx, cpxs_bind/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+(* Note: this might be moved *)
+lemma llpxs_fwd_lref_ge_sn: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡*[h, g, #i, d] L2 → d ≤ i →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ ⦃G, K1⦄ ⊢ ➡*[h, g, V2, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2.
+#h #g #G #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 @(llpxs_ind … H) -L2 /2 width=5 by ex3_2_intro/ -HLK1
+#L #L2 #_ #HL2 * #K #V #HLK #HK1 #HV1 elim (llpx_inv_lref_ge_sn … HL2 … HLK) // -HL2 -HLK -Hdi
+#K2 #V2 #HLK2 #HK2 #HV2
+@(ex3_2_intro … HLK2) -HLK2
+[ /3 width=5 by llpxs_cpx_conf_dx, llpxs_strap1, llpx_cpx_conf/
+| /3 width=5 by llpxs_cpx_trans, cpxs_trans/
+]
+qed-.
+
+(* Inversion lemmas on context-sensitive ext parallel computation for terms *)
+
+lemma cpxs_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
+ U2 = ⓛ{a}V2.T2.
+#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (llpxs_cpx_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro/
+qed-.
+
+lemma cpxs_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
+ U2 = ⓓ{a}V2.T2
+ ) ∨
+ ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpx_inv_abbr1 … HU02) -HU02 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (llpxs_cpx_trans … HT02 (L.ⓓV1) ?)
+ /4 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
+ | #T2 #HT02 #HUT2
+ lapply (llpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
+ /4 width=3 by llpxs_pair_dx, cpxs_trans, ex3_intro, or_intror/
+ ]
+| #U1 #HTU1 #HU01
+ elim (lift_total U2 0 1) #U #HU2
+ /6 width=12 by cpxs_strap1, cpx_lift, ldrop_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma llpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (llpx_fqu_trans … H12 … HKL1) -L1
+ /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+ #L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fqu_trans … H2 … HL0) -L
+ #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
+ /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma llpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
+/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
+qed-.
+
+lemma llpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+ lapply (llpx_cpxs_trans … HT1 … HK1) -HT1 #HT1
+ lapply (cpxs_llpx_conf … HT1 … HK1) -HK1 #HK1
+ elim (llpx_fquq_trans … HT2 … HK1) -K
+ /3 width=7 by llpxs_strap2, cpxs_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma llpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+ lapply (llpx_cpxs_trans … HT1 … HK1) -HT1 #HT1
+ lapply (cpxs_llpx_conf … HT1 … HK1) -HK1 #HK1
+ elim (llpx_fqup_trans … HT2 … HK1) -K
+ /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/
+]
+qed-.
+
+lemma llpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g, T1, 0] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g, T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+ lapply (llpx_cpxs_trans … HT1 … HK1) -HT1 #HT1
+ lapply (cpxs_llpx_conf … HT1 … HK1) -HK1 #HK1
+ elim (llpx_fqus_trans … HT2 … HK1) -K
+ /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lleq_leq.ma".
+include "basic_2/reduction/llpx_lleq.ma".
+include "basic_2/computation/cpxs_lleq.ma".
+include "basic_2/computation/llpxs_cpxs.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS *************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma llpxs_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+/3 width=1 by llpx_lrefl, llpx_llpxs/ qed-.
+
+lemma lleq_llpxs_trans: ∀h,g,G,L2,L,T,d. ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L →
+ ∀L1. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L.
+#h #g #G #L2 #L #T #d #H @(llpxs_ind … H) -L
+/3 width=3 by llpxs_strap1, llpxs_lrefl/
+qed-.
+
+lemma lleq_llpxs_conf: ∀h,g,G,L1,L,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L →
+ ∀L2. L1 ⋕[T, d] L2 → ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L.
+/3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-.
+(*
+foct leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+ (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
+#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
+[ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
+ /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
+| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ lapply (ysucc_inv_Y_dx … He) -He #He
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
+]
+qed-.
+
+lamma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+ (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
+/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/llpxs.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
+
+(* Main properties **********************************************************)
+
+theorem llpxs_trans: ∀h,g,G,T,d. Transitive … (llpxs h g G d T).
+normalize /2 width=3 by trans_TC/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/llpx_lpr.ma".
+include "basic_2/computation/lprs.ma".
+include "basic_2/computation/llpxs.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
+
+(* Properties on sn parallel computation ************************************)
+
+(* Note: this should be moved *)
+lemma lprs_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+normalize /3 width=3 by lpr_llpx, monotonic_TC/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazysn_6.ma".
+include "basic_2/substitution/lleq.ma".
+include "basic_2/reduction/llpx.ma".
+
+(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
+
+definition llsx: ∀h. sd h → relation4 ynat term genv lenv ≝
+ λh,g,d,T,G. SN … (llpx h g G d T) (lleq d T).
+
+interpretation
+ "lazy extended strong normalization (local environment)"
+ 'LazySN h g d T G L = (llsx h g T d G L).
+
+(* Basic eliminators ********************************************************)
+
+lemma llsx_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
+ (∀L1. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+ (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
+ R L1
+ ) →
+ ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → R L.
+#h #g #G #T #d #R #H0 #L1 #H elim H -L1
+/5 width=1 by lleq_sym, SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llsx_intro: ∀h,g,G,L1,T,d.
+ (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T, d] L2) →
+ G ⊢ ⋕⬊*[h, g, T, d] L1.
+/5 width=1 by lleq_sym, SN_intro/ qed.
+
+lemma llsx_sort: ∀h,g,G,L,d,k. G ⊢ ⋕⬊*[h, g, ⋆k, d] L.
+#h #g #G #L1 #d #k @llsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=6 by llpx_fwd_length, lleq_sort/
+qed.
+
+lemma llsx_gref: ∀h,g,G,L,d,p. G ⊢ ⋕⬊*[h, g, §p, d] L.
+#h #g #G #L1 #d #p @llsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=6 by llpx_fwd_length, lleq_gref/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazysnalt_6.ma".
+include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/computation/llpxs_lleq.ma".
+include "basic_2/computation/llsx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* alternative definition of llsx *)
+definition llsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
+ λh,g,d,T,G. SN … (llpxs h g G d T) (lleq d T).
+
+interpretation
+ "lazy extended strong normalization (local environment) alternative"
+ 'LazySNAlt h g d T G L = (llsxa h g T d G L).
+
+(* Basic eliminators ********************************************************)
+
+lemma llsxa_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
+ (∀L1. G ⊢ ⋕⬊⬊*[h, g, T, d] L1 →
+ (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
+ R L1
+ ) →
+ ∀L. G ⊢ ⋕⬊⬊*[h, g, T, d] L → R L.
+#h #g #G #T #d #R #H0 #L1 #H elim H -L1
+/5 width=1 by lleq_sym, SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llsxa_intro: ∀h,g,G,L1,T,d.
+ (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
+ G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
+/5 width=1 by lleq_sym, SN_intro/ qed.
+
+fact llsxa_intro_aux: ∀h,g,G,L1,T,d.
+ (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → L1 ⋕[T, d] L → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
+ G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
+/4 width=3 by llsxa_intro/ qed-.
+
+lemma llsxa_llpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊⬊*[h, g, T, d] L1 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → G ⊢ ⋕⬊⬊*[h, g, T, d] L2.
+#h #g #G #L1 #T #d #H @(llsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 @llsxa_intro
+elim (lleq_dec T L1 L2 d) /4 width=4 by lleq_llpxs_trans, lleq_canc_sn/
+qed-.
+
+lemma llsxa_intro_llpx: ∀h,g,G,L1,T,d.
+ (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
+ G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
+#h #g #G #L1 #T #d #IH @llsxa_intro_aux
+#L #L2 #H @(llpxs_ind_dx … H) -L
+[ #H destruct #H elim H //
+| #L0 #L elim (lleq_dec T L1 L d)
+ /4 width=3 by llsxa_llpxs_trans, lleq_llpx_trans/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem llsx_llsxa: ∀h,g,G,L,T,d. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊⬊*[h, g, T, d] L.
+#h #g #G #L #T #d #H @(llsx_ind … H) -L
+/4 width=1 by llsxa_intro_llpx/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem llsxa_inv_llsx: ∀h,g,G,L,T,d. G ⊢ ⋕⬊⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, T, d] L.
+#h #g #G #L #T #d #H @(llsxa_ind … H) -L
+/4 width=1 by llsx_intro, llpx_llpxs/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llsx_intro_alt: ∀h,g,G,L1,T,d.
+ (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T, d] L2) →
+ G ⊢ ⋕⬊*[h, g, T, d] L1.
+/6 width=1 by llsxa_inv_llsx, llsx_llsxa, llsxa_intro/ qed.
+
+lemma llsx_llpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
+/4 width=3 by llsxa_inv_llsx, llsx_llsxa, llsxa_llpxs_trans/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma llsx_ind_alt: ∀h,g,G,T,d. ∀R:predicate lenv.
+ (∀L1. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+ (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
+ R L1
+ ) →
+ ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → R L.
+#h #g #G #T #d #R #IH #L #H @(llsxa_ind h g G T d … L)
+/4 width=1 by llsxa_inv_llsx, llsx_llsxa/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/csx_lift.ma".
+include "basic_2/computation/csx_llpxs.ma".
+include "basic_2/computation/llsx_ldrop.ma".
+include "basic_2/computation/llsx_llpx.ma".
+include "basic_2/computation/llsx_llpxs.ma".
+(*
+axiom cpx_llpx_trans: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. ⦃G, L1⦄⊢ ➡[h, g, T2, O] L2 →
+ ∃∃L. ⦃G, L1⦄⊢ ➡[h, g, T1, O] L & L ⋕[T2, 0] L2.
+(*
+fact llsx_cpx_trans_aux: ∀h,g,G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L1,d. G ⊢ ⋕⬊*[h, g, T1, d] L1 → d = 0 →
+ L0 ⋕[T1, d] L1 → ∀L2. L1 ⋕[T2, d] L2 →
+ G ⊢ ⋕⬊*[h, g, T2, d] L2.
+#h #g #G #L0 #T1 #T2 #HT12 #L1 #d #H @(llsx_ind … H) -L1
+#L1 #_ #IHL1 #Hd #He011 #L2 #He122 @llsx_intro
+#L3 #Hx223 #Hn223 destruct
+lapply (lleq_cpx_conf_sn … HT12 … He011) #He021
+lapply (lleq_cpx_conf … HT12 … He011) -HT12 #HT12
+lapply (lleq_llpx_trans … He122 … Hx223) -Hx223 #Hx123
+elim (cpx_llpx_trans … HT12 … Hx123) -Hx123 #L4 #Hx114 #He423
+(* lapply (lleq_cpx_conf … Hx114 … He011) #He120 *)
+@(IHL1 … Hx114) // -IHL1
+[ #HL13 @HnL2 -HnL2
+*)
+
+fact llsx_cpx_trans_aux: ∀h,g,G,L1,T1,d. G ⊢ ⋕⬊*[h, g, T1, d] L1 → d = 0 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. L1 ⋕[T1, d] L2 → G ⊢ ⋕⬊*[h, g, T2, 0] L2.
+#h #g #G #L1 #T1 #d #H @(llsx_ind … H) -L1
+#L1 #_ #IHL1 #Hd #T2 #HT12 #L2 #He112 @llsx_intro
+#L3 #Hx223 #Hn223 destruct
+lapply (lleq_cpx_conf_sn … HT12 … He112) #He122
+lapply (lleq_cpx_conf … HT12 … He112) -HT12 #HT12
+elim (cpx_llpx_trans … HT12 … Hx223) #L4 #Hx214 #He423
+@(IHL1 … L4) //
+*)
+axiom llsx_cpx_trans_O: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+ G ⊢ ⋕⬊*[h, g, T1, 0] L → G ⊢ ⋕⬊*[h, g, T2, 0] L.
+
+(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
+
+(* Advanced properties ******************************************************)
+
+lemma llsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,d. d ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
+ ∀K2. G ⊢ ⋕⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g, V, 0] K2 →
+ ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L2.
+#h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt … H) -V
+#V0 #_ #IHV0 #K2 #H @(llsx_ind … H) -K2
+#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @llsx_intro
+#L2 #HL02 #HnL02 elim (llpx_inv_lref_ge_sn … HL02 … HLK0) // -HL02
+#K2 #V2 #HLK2 #HK02 #HV02 elim (eq_term_dec V0 V2)
+#HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
+[ /4 width=7 by llpxs_strap1, lleq_lref/
+| lapply (llpx_cpx_conf … HV02 … HK02) -HK02 #HK02
+ @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
+ /3 width=3 by llsx_cpx_trans_O, llpxs_cpx_conf_dx, llsx_llpx_trans, llpxs_cpx_trans, llpxs_strap1/
+]
+qed.
+
+lemma llsx_lref_be: ∀h,g,I,G,K,V,i,d. d ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
+ G ⊢ ⋕⬊*[h, g, V, 0] K →
+ ∀L. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L.
+/2 width=8 by llsx_lref_be_lpxs/ qed.
+
+(* Main properties **********************************************************)
+
+theorem csx_llsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀d. G ⊢ ⋕⬊*[h, g, T, d] L.
+#h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
+#Z #Y #X #IH #G #L * * //
+[ #i #HG #HL #HT #H #d destruct
+ elim (lt_or_ge i (|L|)) /2 width=1 by llsx_lref_free/
+ elim (ylt_split i d) /2 width=1 by llsx_lref_skip/
+ #Hdi #Hi elim (ldrop_O1_lt … Hi) -Hi
+ #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
+ /4 width=6 by llsx_lref_be, fqup_lref/
+| #a #I #V #T #HG #HL #HT #H #d destruct
+ elim (csx_fwd_bind … H) -H /3 width=1 by llsx_bind/
+| #I #V #T #HG #HL #HT #H #d destruct
+ elim (csx_fwd_flat … H) -H /3 width=1 by llsx_flat/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/computation/llsx.ma".
+
+(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
+
+(* Advanced properties ******************************************************)
+
+lemma llsx_lref_free: ∀h,g,G,L,d,i. |L| ≤ i → G ⊢ ⋕⬊*[h, g, #i, d] L.
+#h #g #G #L1 #d #i #HL1 @llsx_intro
+#L2 #HL12 #H elim H -H
+/4 width=8 by llpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
+qed.
+
+lemma llsx_lref_skip: ∀h,g,G,L,d,i. yinj i < d → G ⊢ ⋕⬊*[h, g, #i, d] L.
+#h #g #G #L1 #d #i #HL1 @llsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=6 by llpx_fwd_length, lleq_skip/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/reduction/llpx_lleq.ma".
+include "basic_2/computation/llsx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma llsx_llpx_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
+#h #g #G #L1 #T #d #H @(llsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 @llsx_intro
+elim (lleq_dec T L1 L2 d) /4 width=4 by lleq_llpx_trans, lleq_canc_sn/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/llpxs_llpxs.ma".
+include "basic_2/computation/llsx_alt.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+fact llsx_bind_llpxs_aux: ∀h,g,a,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
+ ∀Y,T. G ⊢ ⋕⬊*[h, g, T, ⫯d] Y →
+ ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g, ⓑ{a,I}V.T, d] L2 →
+ G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L2.
+#h #g #a #I #G #L1 #V #d #H @(llsx_ind_alt … H) -L1
+#L1 #HL1 #IHL1 #Y #T #H @(llsx_ind_alt … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct @llsx_intro_alt
+#L0 #HL20 lapply (llpxs_fwd_bind_dx … HL20)
+lapply (llpxs_trans … HL12 … HL20)
+#HL10 #HT #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 d)
+ [ -HL20 #HV @(IHL1 … L0)
+ /3 width=4 by llsx_llpxs_trans, llpxs_fwd_bind_sn, lleq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10
+ /3 width=4 by llsx_llpxs_trans, llpxs_fwd_bind_sn/
+ ]
+| /3 width=4 by/
+]
+qed-.
+
+lemma llsx_bind: ∀h,g,a,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
+ ∀T. G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V →
+ G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L.
+/2 width=3 by llsx_bind_llpxs_aux/ qed.
+
+lemma llsx_flat_llpxs: ∀h,g,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
+ ∀L2,T. G ⊢ ⋕⬊*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, ⓕ{I}V.T, d] L2 →
+ G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L2.
+#h #g #I #G #L1 #V #d #H @(llsx_ind_alt … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(llsx_ind_alt … H) -L2
+#L2 #HL2 #IHL2 #HL12 @llsx_intro_alt
+#L0 #HL20 lapply (llpxs_fwd_flat_dx … HL20)
+lapply (llpxs_trans … HL12 … HL20)
+#HL10 #HT #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 d)
+ [ #HV @(IHL1 … L0) /3 width=3 by llsx_llpxs_trans, llpxs_fwd_flat_sn, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
+ | -HnV -HL10 /3 width=4 by llsx_llpxs_trans, llpxs_fwd_flat_sn/
+ ]
+| /3 width=1 by/
+]
+qed-.
+
+lemma llsx_flat: ∀h,g,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
+ ∀T. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L.
+/2 width=3 by llsx_flat_llpxs/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_leq.ma".
+include "basic_2/relocation/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+definition TC_llpx_sn_confluent1: relation (relation3 lenv term term) ≝ λS,R.
+ ∀Ls,T1,T2. S Ls T1 T2 →
+ ∀Ld. TC … (llpx_sn R 0 T1) Ls Ld → TC … (llpx_sn R 0 T2) Ls Ld.
+
+lemma TC_llpx_sn_s_confluent: ∀S,R. (llpx_sn_confluent1 S R) → TC_llpx_sn_confluent1 S R.
+#S #R #HSR #Ls #T1 #T2 #HT12 #Ld #H
+generalize in match HT12; -HT12
+@(TC_ind_dx … Ls H) -Ls
+[ /3 width=3 by inj/
+| #Ls #L #HLs #_ #IHLd #HT12
+ @(TC_strap … L) /2 width=3 by/ @IHLd -IHLd
+
+lemma TC_llpx_sn_lref_refl: ∀R. (∀L.reflexive … (R L)) →
+ ∀I,L1,K1,K2,V,d,i. d ≤ yinj i → ⇩[i] L1 ≡ K1.ⓑ{I}V →
+ TC lenv (llpx_sn R 0 V) K1 K2 →
+ ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → TC … (llpx_sn R d (#i)) L1 L2.
+#R #HR #I #L1 #K1 #K2 #V #d #i #Hdi #HLK1 #H @(TC_star_ind … K2 H) -K2
+[ /2 width=1 by llpx_sn_refl/
+| /4 width=9 by llpx_sn_refl, llpx_sn_lref, inj/
+| #K #K2 #_ #HV #IHK1 #L2 #HLK2 lapply (ldrop_fwd_length … HLK2)
+ #H elim (ldrop_O1_ex (K.ⓑ{I}V) i L2) [2: normalize in H ⊢ %; >(llpx_sn_fwd_length … HV) ]
+ /4 width=11 by llpx_sn_lref, step/
+]
+qed-.
+
+lemma TC_llpx_sn_lref: ∀R. (∀L.reflexive … (R L)) → (llpx_sn_confluent1 R R) →
+ ∀I,K1,V1,V2,d,i. d ≤ yinj i → LTC … R K1 V1 V2 →
+ ∀K2. TC lenv (llpx_sn R 0 V1) K1 K2 → ∀L1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 → TC … (llpx_sn R d (#i)) L1 L2.
+#R #H1R #H2R #I #K1 #V1 #V2 #d #i #Hdi #H @(TC_star_ind_dx … V1 H) -V1
+[ /2 width=1 by llpx_sn_refl/
+| /2 width=7 by TC_llpx_sn_lref_refl/
+| #V1 #V #HV1 #_ #IHV2 #K2 #HK12 #L1 #HLK1 #L2 #HLK2
+ lapply (ldrop_fwd_length … HLK1)
+ #H elim (ldrop_O1_ex (K1.ⓑ{I}V) i L1) [2: normalize in H ⊢ %; // ] -H
+ #L #_ #HLK @(TC_strap … L)
+ [ @(llpx_sn_lref … HLK1 … HLK) /2 width=1 by llpx_sn_refl/
+ | @(IHV2 … HLK … HLK2)
+ -HLK1 -HLK2 -HLK -IHV2 -Hdi @(TC_llpx_sn_s_confluent R R … HK12) //
+ ]
+]
+
+
+lemma llpx_sn_LTC_TC_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2,T,d. llpx_sn (LTC … R) d T L1 L2 →
+ TC … (llpx_sn R d T) L1 L2.
+#R #HR #L1 #L2 #T #d #H elim H -L1 -L2
+/3 width=3 by llpx_sn_gref, llpx_sn_free, llpx_sn_skip, llpx_sn_sort, inj/
+[ #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #_ #HV12 #IHV1
+
+(* Properties on transitive_closure *****************************************)
+
+lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀V1,V2. LTC … R L1 V1 V2 →
+ TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
+qed-.
+
+lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
+ TC … (lpx_sn R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+qed-.
+
+(* Inversion lemmas on transitive closure ***********************************)
+
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_trans … R (lpx_sn R) →
+ ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+ elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+ lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_ind: ∀R. s_rs_trans … R (lpx_sn R) →
+ ∀S:relation lenv.
+ S (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
+ S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
+#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
+[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
+| #L2 #I #V2 #IHL2 #X #H
+ elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
+ #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_trans … R (lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_trans … R (lpx_sn R) →
+ ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_trans … R (lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ lpx_sn (LTC … R) L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+qed-.
--- /dev/null
+include "basic_2/relocation/lleq_alt.ma".
+include "basic_2/reduction/cpx_lleq.ma".
+include "basic_2/reduction/lpx_lleq.ma".
+
+lemma not_ex_to_all_not: ∀A:Type[0]. ∀R:predicate A.
+ ((∃a. R a)→⊥) → (∀a. R a → ⊥).
+/3 width=2 by ex_intro/ qed-.
+
+lemma lt_repl_sn_trans_tw: ∀L1,L2,T1,T2. ♯{L1, T1} < ♯{L2, T2} →
+ ∀U1. ♯{U1} = ♯{T1} → ♯{L1, U1} < ♯{L2, T2}.
+normalize in ⊢ (?→?→?→?→?%%→?→?→?%%); //
+qed-.
+
+axiom cpx_fwd_lift1: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
+ ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → ∃T2. ⇧[d, e] T2 ≡ U2.
+(*
+#h #g #G #L #U1 #U2 #H elim H -G -L -U1 -U2
+[ * #i #G #L #T1 #d #e #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=2 by ex_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=2 by lift_lref_ge_minus, lift_lref_lt, ex_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=2 by ex_intro/
+ ]
+| #G #L #k #l #Hkl #T1 #d #e #H
+ lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by ex_intro/
+| #I #G #L #K #V1 #V2 #W2 #i #HLK #HV12 #HVW2 #IHV2 #T1 #d #e #H
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+ elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+ elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpx_delta, ex2_intro/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
+ elim (lift_split … HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpx_delta, ex2_intro/
+ ]
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpx_bind, ldrop_skip, lift_bind, ex2_intro/
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -V1
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpx_flat, lift_flat, ex2_intro/
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHU1 (K.ⓓW1) s … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
+ elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
+| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_tau, ex2_intro/
+| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ti, ex2_intro/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HLK … HV01) -V1 #V3 #HV32 #HV03
+ elim (IHT12 (K.ⓛW0) s … HT01) -T1 /2 width=1 by ldrop_skip/ #T3 #HT32 #HT03
+ elim (IHW12 … HLK … HW01) -W1
+ /4 width=7 by cpx_beta, lift_bind, lift_flat, ex2_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
+ elim (IHT12 (K.ⓓW0) s … HT01) -T1 /2 width=1 by ldrop_skip/ #T3 #HT32 #HT03
+ elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
+ elim (lift_trans_le … HV3 … HV2) -V
+ /4 width=9 by cpx_theta, lift_bind, lift_flat, ex2_intro/
+]
+qed-.
+*)
+lemma cpx_fwd_nlift2: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 → ∀d,e.
+ (∀T2. ⇧[d, e] T2 ≡ U2 → ⊥) → (∀T1. ⇧[d, e] T1 ≡ U1 → ⊥).
+#h #g #G #L #U1 #U2 #HU12 #d #e #HnU2 #T1 #HTU1
+elim (cpx_fwd_lift1 … HU12 … HTU1) -L -U1 /2 width=2 by/
+qed-.
+
+fact lleq_lpx_cpx_trans_aux: ∀h,g,G,T1,L1s,L1d,d. L1s ⋕[T1, d] L1d →
+ ∀Y,L2d. ⦃G, Y⦄ ⊢ ➡[h, g] L2d → Y = L1d → d = 0 →
+ ∀T2. ⦃G, L1d⦄ ⊢ T1 ➡[h, g] T2 →
+ ∃∃L2s. ⦃G, L1s⦄ ⊢ ➡[h, g] L2s & L2s ⋕[T2, d] L2d.
+#h #g #G #T1 #L1s @(f2_ind … rfw … L1s T1) -L1s -T1 #n #IH
+#Ys #U1 #Hn #Yd #d #HU1 elim (lleq_fwd_alt … HU1) #H #IHU1
+#Y #L2d * -Y -L2d
+[ -IH -IHU1 -HU1 #HY #Hd #U2 #HU12 destruct
+ >(length_inv_zero_dx … H) -Ys /2 width=3 by ex2_intro/
+| #Id #L1d #L2d #W1d #W2d #HL12d #HW12d #HY #Hd #U2 #HU12 destruct
+ elim (length_inv_pos_dx … H) -H #Is #L1s #W1s #_ #H destruct
+ elim (is_lift_dec U1 0 1) [ -IHU1 -HW12d | -HU1 ]
+ [ * #T1 #HTU1 lapply (lift_fwd_tw … HTU1) #H
+ lapply (lleq_inv_lift_le … HU1 L1s L1d … HTU1 ?) -HU1 /2 width=1 by ldrop_drop/
+ #HT1 elim (cpx_inv_lift1 … HU12 L1d … HTU1) -HU12 -HTU1 /2 width=4 by ldrop_drop/
+ #T2 #HTU2 #HT12 elim (IH … HT1 … HL12d … HT12) /2 width=3 by lt_repl_sn_trans_tw/ -IH -HT1 -HT12 -H
+ #L2s #HL12s #HT2 @(ex2_intro … (L2s.ⓑ{Is}W1s))
+ /3 width=10 by lleq_lift_le, lpx_pair, ldrop_drop/ (**) (* full auto too slow *)
+ | #HnU1 lapply (not_ex_to_all_not … HnU1) -HnU1 #HnU1
+ elim (IHU1 … HnU1) [2,3,4: // |5,6,7,8,9,10: skip ] -HnU1 #H1 #H2 #HW1s destruct
+ elim (IH … HW1s … HL12d … HW12d) // #L2s #HL12s #HW2d
+ @(ex2_intro … (L2s.ⓑ{Id}W2d)) /3 width=3 by lleq_cpx_trans, lpx_pair/
+ lapply (lleq_fwd_length … HW2d) #HL2sd -HW12d -HW1s
+ @lleq_intro_alt [ normalize // ] -HL2sd
+ #I2s #I2d #K2s #K2d #V2s #V2d #i @(nat_ind_plus … i) -i
+ [ #_ #_ #HLK2s #HLK2d -IH -IHU1 -HU12 -HL12s -HL12d
+ lapply (ldrop_inv_O2 … HLK2s) -HLK2s #H destruct
+ lapply (ldrop_inv_O2 … HLK2d) -HLK2d #H destruct /2 width=1 by and3_intro/
+ | #i #_ #_ #HnU2 #HLK2s #HLK2d
+ lapply (cpx_fwd_nlift2 … HU12 … HnU2) -HU12 -HnU2 #HnU1
+ lapply (ldrop_inv_drop1 … HLK2s) -HLK2s #HLK2s
+ lapply (ldrop_inv_drop1 … HLK2d) -HLK2d #HLK2d
+ elim (lpx_ldrop_trans_O1 … HL12s … HLK2s) -L2s #Y #HLK1s #H
+ elim (lpx_inv_pair2 … H) -H #K1s #V1s #HK12s #HV12s #H destruct
+ elim (lpx_ldrop_trans_O1 … HL12d … HLK2d) -L2d #Y #HLK1d #H
+ elim (lpx_inv_pair2 … H) -H #K1d #V1d #HK12d #HV12d #H destruct
+ elim (IHU1 … HnU1) [2,3,4: /2 width=2 by ldrop_drop/ | 5,6,7,8,9,10: skip ] -IHU1 -HnU1 -HLK1d
+ #H1 #H2 #HV1d destruct
+ lapply (ldrop_fwd_rfw … HLK1s) -HLK1s #H
+ elim (IH … HV1d … HK12d … HV12d) // -IH -HV1d -HK12d -HV12d
+ [ #Y #H1Y #H2Y
+
+
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvAlt $L1 $L2 }.
+
+include "basic_2/reducibility/lfpr.ma".
+
+(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************)
+
+definition lfpc: relation lenv ≝
+ λL1,L2. ⦃L1⦄ ➡ ⦃L2⦄ ∨ ⦃L2⦄ ➡ ⦃L1⦄.
+
+interpretation
+ "focalized parallel conversion (local environment)"
+ 'FocalizedPConv L1 L2 = (lfpc L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lfpc_refl: ∀L. ⦃L⦄ ⬌ ⦃L⦄.
+/2 width=1/ qed.
+
+lemma lfpc_sym: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L2⦄ ⬌ ⦃L1⦄.
+#L1 #L2 * /2 width=1/
+qed.
+
+lemma lfpc_lfpr: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄.
+#L1 #L2 * /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/conversion/lfpc.ma".
+
+(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************)
+
+(* Main properties **********************************************************)
+
+theorem lfpc_conf: ∀L0,L1,L2. ⦃L0⦄ ⬌ ⦃L1⦄ → ⦃L0⦄ ⬌ ⦃L2⦄ →
+ ∃∃L. ⦃L1⦄ ⬌ ⦃L⦄ & ⦃L2⦄ ⬌ ⦃L⦄.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvStarAlt $L1 $L2 }.
+
+include "basic_2/conversion/lfpc.ma".
+
+(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
+
+definition lfpcs: relation lenv ≝ TC … lfpc.
+
+interpretation "focalized parallel equivalence (local environment)"
+ 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma lfpcs_ind: ∀L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → R L → R L2) →
+ ∀L2. ⦃L1⦄ ⬌* ⦃L2⦄ → R L2.
+#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma lfpcs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → R L → R L1) →
+ ∀L1. ⦃L1⦄ ⬌* ⦃L2⦄ → R L1.
+#L2 #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lfpcs_refl: reflexive … lfpcs.
+/2 width=1/ qed.
+
+lemma lfpcs_sym: symmetric … lfpcs.
+/3 width=1/ qed.
+
+lemma lfpc_lfpcs: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/2 width=1/ qed.
+
+lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/2 width=3/ qed.
+
+lemma lfpcs_strap2: ∀L1,L,L2. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/2 width=3/ qed.
+
+lemma lfpcs_lfpr_dx: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=1/ qed.
+
+lemma lfpcs_lfpr_sn: ∀L1,L2. ⦃L2⦄ ➡ ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=1/ qed.
+
+lemma lfpcs_lfpr_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=3/ qed.
+
+lemma lfpcs_lfpr_strap2: ∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=3/ qed.
+
+lemma lfpcs_lfpr_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡ ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=3/ qed.
+
+lemma lfpcs_lfpr_conf: ∀L1,L. ⦃L⦄ ➡ ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/lfprs_aaa.ma".
+include "basic_2/equivalence/lfpcs_lfpcs.ma".
+
+(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
+
+(* Main properties about atomic arity assignment on terms *******************)
+
+theorem aaa_lfpcs_mono: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
+ ∀T,A1. L1 ⊢ T ⁝ A1 → ∀A2. L2 ⊢ T ⁝ A2 →
+ A1 = A2.
+#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2
+elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2
+lapply (aaa_lfprs_conf … HT1 … HL1) -L1 #HT1
+lapply (aaa_lfprs_conf … HT2 … HL2) -L2 #HT2
+lapply (aaa_mono … HT1 … HT2) -L -T //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fprs_cprs.ma".
+include "basic_2/computation/lfprs_fprs.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
+include "basic_2/equivalence/lfpcs_lfpcs.ma".
+
+(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
+
+(* Inversion lemmas on context-free parallel equivalence for closures *******)
+
+lemma lfpcs_inv_fpcs: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ → ∀T. ⦃L1, T⦄ ⬌* ⦃L2, T⦄.
+#L1 #L2 #HL12 #T
+elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2
+lapply (lfprs_inv_fprs … HL1 T) -HL1
+lapply (lfprs_inv_fprs … HL2 T) -HL2 /2 width=4/
+qed-.
+
+(* Properties on context-free parallel equivalence for closures *************)
+
+lemma fpcs_lfpcs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L2 #T1 #T2 #HT12
+elim (fpcs_inv_fprs … HT12) -HT12 /3 width=5 by fprs_lfprs, lfprs_div/ (**) (* auto too slow without trace *)
+qed.
+
+lemma fpcs_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ⬌* ⦃K2, T2⦄ →
+ ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
+ ∀d,e. ⇩[d, e] L1 ≡ K1 → ⇩[d, e] L2 ≡ K2 →
+ ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+#K1 #K2 #T1 #T2 #HT12 #L1 #L2 #HL12 #d #e #HLK1 #HLK2 #U1 #U2 #HTU1 #HTU2
+elim (fpcs_inv_fprs … HT12) -HT12 #K #T #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+elim (fprs_lift … HT1 … HLK1 … HTU1 HTU) -K1 -T1 #K1 #HU1 #_
+elim (fprs_lift … HT2 … HLK2 … HTU2 HTU) -K2 -T2 -T #K2 #HU2 #_ -K -d -e
+lapply (lfpcs_lfprs_conf K1 … HL12) -HL12 /2 width=3/ #H
+lapply (lfpcs_lfprs_strap1 … H K2 ?) -H /2 width=3/ #HK12
+lapply (lfpcs_inv_fpcs … HK12 U) -HK12 #HU
+/3 width=4 by fpcs_fprs_strap2, fpcs_fprs_div/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/lfprs_lfprs.ma".
+include "basic_2/conversion/lfpc_lfpc.ma".
+include "basic_2/equivalence/lfpcs_lfprs.ma".
+
+(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfpcs_inv_lfprs: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
+ ∃∃L. ⦃L1⦄ ➡* ⦃L⦄ & ⦃L2⦄ ➡* ⦃L⦄.
+#L1 #L2 #H @(lfpcs_ind … H) -L2
+[ /3 width=3/
+| #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0
+ [ elim (lfprs_strip … HL0 … HL2) -L #L #HL0 #HL2
+ lapply (lfprs_strap1 … HL10 … HL0) -L0 /2 width=3/
+ | lapply (lfprs_strap2 … HL2 … HL0) -L /2 width=3/
+ ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lfpcs_strip: ∀L,L1. ⦃L⦄ ⬌* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌ ⦃L2⦄ →
+ ∃∃L0. ⦃L1⦄ ⬌ ⦃L0⦄ & ⦃L2⦄ ⬌* ⦃L0⦄.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/2 width=3/ qed.
+
+theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
+
+theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/lfprs.ma".
+include "basic_2/equivalence/lfpcs.ma".
+
+(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
+
+(* Properties on focalized computation for local environments ***************)
+
+lemma lfpcs_lfprs_dx: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /3 width=3/
+qed.
+
+lemma lfpcs_lfprs_sn: ∀L1,L2. ⦃L2⦄ ➡* ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /3 width=3/
+qed.
+
+lemma lfpcs_lfprs_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L #HL1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /2 width=3/
+qed.
+
+lemma lfpcs_lfprs_strap2: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L #H #L2 #HL2 @(lfprs_ind_dx … H) -L1 /width=1/ /2 width=3/
+qed.
+
+lemma lfpcs_lfprs_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /2 width=3/
+qed.
+
+lemma lfpcs_lfprs_conf: ∀L1,L. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L #H #L2 #HL2 @(lfprs_ind … H) -L1 /width=1/ /2 width=3/
+qed.
+
+lemma lfprs_div: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /2 width=1/ /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/llpx_sn_ldrop.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+lemma cpr_llpx_sn_conf: ∀R. (∀I,L.reflexive … (R I L)) →
+ (∀I.l_liftable (R I)) →
+ (∀I.l_deliftable_sn (R I)) →
+ ∀G. s_r_confluent1 … (cpr G) (llpx_sn R 0).
+#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ //
+| #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /3 width=10 by llpx_sn_inv_lift_le, ldrop_drop/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ #HV1 #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/
+| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ #HV1 #H elim (llpx_sn_inv_bind_O … H) -H //
+ #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *)
+ [ /3 width=10 by llpx_sn_lift_le, ldrop_drop/
+ | /3 width=4 by llpx_sn_bind_repl_O/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/llpx_sn_ldrop.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+(* Note: lemma 1000 *)
+lemma cpx_llpx_sn_conf: ∀R. (∀I,L.reflexive … (R I L)) →
+ (∀I.l_liftable (R I)) →
+ (∀I.l_deliftable_sn (R I)) →
+ ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx_sn R 0).
+#R #H1R #H2R #H3R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ //
+| /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
+| #I #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /3 width=10 by llpx_sn_inv_lift_le, ldrop_drop/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+| #G #Ls #V1 #V2 #T #_ #IHV12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ #HV1 #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/
+| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ #HV1 #H elim (llpx_sn_inv_bind_O … H) -H //
+ #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *)
+ [ /3 width=10 by llpx_sn_lift_le, ldrop_drop/
+ | /3 width=4 by llpx_sn_bind_repl_O/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazyeq_4.ma".
+include "basic_2/substitution/llpx_sn.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+definition ceq: relation4 bind2 lenv term term ≝ λI,L,T1,T2. T1 = T2.
+
+definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
+
+interpretation
+ "lazy equivalence (local environment)"
+ 'LazyEq T d L1 L2 = (lleq d T L1 L2).
+
+definition lleq_transitive: predicate (relation4 bind2 lenv term term) ≝
+ λR. ∀I,L2,T1,T2. R I L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R I L1 T1 T2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
+ ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
+ ) → (
+ ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
+ ) → (
+ ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
+ ) → (
+ ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
+ ) → (
+ ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
+ ) → (
+ ∀a,I,L1,L2,V,T,d.
+ L1 ≡[V, d]L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
+ R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
+ ) → (
+ ∀I,L1,L2,V,T,d.
+ L1 ≡[V, d]L2 → L1 ≡[T, d] L2 →
+ R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
+ ) →
+ ∀d,T,L1,L2. L1 ≡[T, d] L2 → R d T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim H -L1 -L2 -T -d /2 width=8 by/
+qed-.
+
+lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,d. L1 ≡[ⓑ{a,I}V.T, d] L2 →
+ L1 ≡[V, d] L2 ∧ L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind/ qed-.
+
+lemma lleq_inv_flat: ∀I,L1,L2,V,T,d. L1 ≡[ⓕ{I}V.T, d] L2 →
+ L1 ≡[V, d] L2 ∧ L1 ≡[T, d] L2.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ≡[T, d] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+lemma lleq_fwd_lref: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V &
+ K1 ≡[V, yinj 0] K2 & d ≤ yinj i.
+#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref … H) /2 width=1/
+* /3 width=7 by or3_intro2, ex4_4_intro/
+qed-.
+
+lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K1,i. ⇩[i] L1 ≡ K1 →
+ ∃K2. ⇩[i] L2 ≡ K2.
+/2 width=7 by llpx_sn_fwd_ldrop_sn/ qed-.
+
+lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K2,i. ⇩[i] L2 ≡ K2 →
+ ∃K1. ⇩[i] L1 ≡ K1.
+/2 width=7 by llpx_sn_fwd_ldrop_dx/ qed-.
+
+lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
+ L1 ≡[ⓑ{a,I}V.T, d] L2 → L1 ≡[V, d] L2.
+/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
+
+lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,d.
+ L1 ≡[ⓑ{a,I}V.T, d] L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
+
+lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
+ L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[V, d] L2.
+/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
+
+lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
+ L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[T, d] L2.
+/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ≡[⋆k, d] L2.
+/2 width=1 by llpx_sn_sort/ qed.
+
+lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ≡[#i, d] L2.
+/2 width=1 by llpx_sn_skip/ qed.
+
+lemma lleq_lref: ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, 0] K2 → L1 ≡[#i, d] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, d] L2.
+/2 width=1 by llpx_sn_free/ qed.
+
+lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ≡[§p, d] L2.
+/2 width=1 by llpx_sn_gref/ qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
+ L1 ≡[V, d] L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
+ L1 ≡[ⓑ{a,I}V.T, d] L2.
+/2 width=1 by llpx_sn_bind/ qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,d.
+ L1 ≡[V, d] L2 → L1 ≡[T, d] L2 → L1 ≡[ⓕ{I}V.T, d] L2.
+/2 width=1 by llpx_sn_flat/ qed.
+
+lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
+/2 width=1 by llpx_sn_Y/ qed.
+
+lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
+#d #T #L1 #L2 #H @(lleq_ind … H) -d -T -L1 -L2
+/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
+qed-.
+
+lemma lleq_ge_up: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
+ ∀T,d,e. ⇧[d, e] T ≡ U →
+ dt ≤ d + e → L1 ≡[U, d] L2.
+/2 width=6 by llpx_sn_ge_up/ qed-.
+
+lemma lleq_ge: ∀L1,L2,T,d1. L1 ≡[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ≡[T, d2] L2.
+/2 width=3 by llpx_sn_ge/ qed-.
+
+lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
+ L1 ≡[ⓑ{a,I}V.T, 0] L2.
+/2 width=1 by llpx_sn_bind_O/ qed-.
+
+(* Advancded properties on lazy pointwise exyensions ************************)
+
+lemma llpx_sn_lrefl: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀L1,L2,T,d. L1 ≡[T, d] L2 → llpx_sn R d T L1 L2.
+/2 width=3 by llpx_sn_co/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_plus.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+inductive llpx_sn (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_sort: ∀L1,L2,d,k. |L1| = |L2| → llpx_sn R d (⋆k) L1 L2
+| llpx_sn_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn R d (#i) L1 L2
+| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn R (yinj 0) V1 K1 K2 → R I K1 V1 V2 → llpx_sn R d (#i) L1 L2
+| llpx_sn_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R d (#i) L1 L2
+| llpx_sn_gref: ∀L1,L2,d,p. |L1| = |L2| → llpx_sn R d (§p) L1 L2
+| llpx_sn_bind: ∀a,I,L1,L2,V,T,d.
+ llpx_sn R d V L1 L2 → llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn R d (ⓑ{a,I}V.T) L1 L2
+| llpx_sn_flat: ∀I,L1,L2,V,T,d.
+ llpx_sn R d V L1 L2 → llpx_sn R d T L1 L2 → llpx_sn R d (ⓕ{I}V.T) L1 L2
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
+ ∀a,I,V,T. X = ⓑ{a,I}V.T →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #L1 #L2 #X #d * -L1 -L2 -X -d
+[ #L1 #L2 #d #k #_ #b #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #b #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #d #p #_ #b #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #d #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
+| #I #L1 #L2 #V #T #d #_ #_ #b #J #W #U #H destruct
+]
+qed-.
+
+lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
+
+fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
+ ∀I,V,T. X = ⓕ{I}V.T →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
+#R #L1 #L2 #X #d * -L1 -L2 -X -d
+[ #L1 #L2 #d #k #_ #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #d #p #_ #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #d #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #V #T #d #HV #HT #J #W #U #H destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
+/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_fwd_length: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d //
+#I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #HLK1 #HLK2 #_ #_ #HK12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2
+normalize //
+qed-.
+
+lemma llpx_sn_fwd_ldrop_sn: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+ ∀K1,i. ⇩[i] L1 ≡ K1 → ∃K2. ⇩[i] L2 ≡ K2.
+#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
+#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
+qed-.
+
+lemma llpx_sn_fwd_ldrop_dx: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+ ∀K2,i. ⇩[i] L2 ≡ K2 → ∃K1. ⇩[i] L1 ≡ K1.
+#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H
+#HL12 lapply (ldrop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by ldrop_O1_le/
+qed-.
+
+fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 → ∀i. X = #i →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn R (yinj 0) V1 K1 K2 &
+ R I K1 V1 V2 & d ≤ yinj i.
+#R #L1 #L2 #X #d * -L1 -L2 -X -d
+[ #L1 #L2 #d #k #_ #j #H destruct
+| #L1 #L2 #d #i #_ #Hid #j #H destruct /2 width=1 by or3_intro1/
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
+ /3 width=9 by or3_intro2, ex5_5_intro/
+| #L1 #L2 #d #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
+| #L1 #L2 #d #p #_ #j #H destruct
+| #a #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
+| #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
+]
+qed-.
+
+lemma llpx_sn_fwd_lref: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn R (yinj 0) V1 K1 K2 &
+ R I K1 V1 V2 & d ≤ yinj i.
+/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
+
+lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2.
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (②{I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2.
+#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
+qed-.
+
+(* Basic_properties *********************************************************)
+
+lemma llpx_sn_refl: ∀R. (∀I,L. reflexive … (R I L)) → ∀T,L,d. llpx_sn R d T L L.
+#R #HR #T #L @(f2_ind … rfw … L T) -L -T
+#n #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
+#i #Hn elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
+#HiL #d elim (ylt_split i d) /2 width=1 by llpx_sn_skip/
+elim (ldrop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+qed-.
+
+lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/
+#a #I #V1 #T1 #Hn #L2 #HL12
+@llpx_sn_bind /2 width=1/ (**) (* explicit constructor *)
+@IH -IH // normalize /2 width=1 by eq_f2/
+qed-.
+
+lemma llpx_sn_ge_up: ∀R,L1,L2,U,dt. llpx_sn R dt U L1 L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
+ dt ≤ d + e → llpx_sn R d U L1 L2.
+#R #L1 #L2 #U #dt #H elim H -L1 -L2 -U -dt
+[ #L1 #L2 #dt #k #HL12 #X #d #e #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
+ elim (ylt_yle_false … Hidt) -Hidt
+ @(yle_trans … Hdtde) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
+| #I #L1 #L2 #K1 #K2 #W1 #W2 #dt #i #Hdti #HLK1 #HLK2 #HW1 #HW12 #_ #X #d #e #H #_
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12
+ lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
+ normalize in ⊢ (%→%→?); -I -W1 -W2 -dt /3 width=1 by llpx_sn_skip, ylt_inj/
+ | /4 width=9 by llpx_sn_lref, yle_inj, le_plus_b/
+ ]
+| /2 width=1 by llpx_sn_free/
+| #L1 #L2 #dt #p #HL12 #X #d #e #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
+ elim (lift_inv_bind2 … H) -H #V #T #HVW >commutative_plus #HTU #H destruct
+ @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
+ @(IHT … HTU) /2 width=1 by yle_succ/
+| #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
+ elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct
+ /3 width=4 by llpx_sn_flat/
+]
+qed-.
+
+(**) (* the minor premise comes first *)
+lemma llpx_sn_ge: ∀R,L1,L2,T,d1,d2. d1 ≤ d2 →
+ llpx_sn R d1 T L1 L2 → llpx_sn R d2 T L1 L2.
+#R #L1 #L2 #T #d1 #d2 * -d1 -d2 (**) (* destructed yle *)
+/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
+qed-.
+
+lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 →
+ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2.
+/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
+
+lemma llpx_sn_co: ∀R1,R2. (∀I,L,T1,T2. R1 I L T1 T2 → R2 I L T1 T2) →
+ ∀L1,L2,T,d. llpx_sn R1 d T L1 L2 → llpx_sn R2 d T L1 L2.
+#R1 #R2 #HR12 #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cofrees_alt.ma".
+include "basic_2/substitution/llpx_sn_alt_rec.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (not recursive) *)
+definition llpx_sn_alt: relation4 bind2 lenv term term → relation4 ynat term lenv lenv ≝
+ λR,d,T,L1,L2. |L1| = |L2| ∧
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R I1 K1 V1 V2
+ ).
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt R d T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r … H) -H
+#HL12 #IHU @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv_ge … H) -H //
+[ -n #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
+| * #J1 #K10 #W10 #j #Hdj #Hji #HLK10 #HnW10 #HnU destruct
+ lapply (ldrop_fwd_drop2 … HLK10) #H
+ lapply (ldrop_conf_ge … H … HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
+ elim (ldrop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by ldrop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
+ lapply (ldrop_fwd_drop2 … HLK20) #H
+ lapply (ldrop_conf_ge … H … HLK2 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK20
+ elim (IHn K10 W10 … K20 0) -IHn -HL12 /3 width=6 by ldrop_fwd_rfw/
+ elim (IHU … HnU HLK10 HLK20) -IHU -HnU -HLK10 -HLK20 //
+]
+qed.
+
+theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx_sn R d T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
+elim (IHU … HLK1 HLK2) /3 width=6 by nlift_frees/
+#H #HV12 @and3_intro // @IHn -IHn /3 width=6 by ldrop_fwd_rfw/
+lapply (ldrop_fwd_drop2 … HLK1) #H1
+lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #H2
+@conj [ @(ldrop_fwd_length_eq1 … H1 H2) // ] -HL12
+#Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #_
+>(minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus <minus_plus
+#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *)
+lapply (ldrop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1
+lapply (ldrop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2
+/4 width=14 by frees_be, yle_plus_dx2_trans, yle_succ_dx/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lift_neg.ma".
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (recursive) *)
+inductive llpx_sn_alt_r (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_alt_r_intro: ∀L1,L2,T,d.
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2
+ ) →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
+ ) → |L1| = |L2| → llpx_sn_alt_r R d T L1 L2
+.
+
+(* Compact definition of llpx_sn_alt_r **************************************)
+
+lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
+ ) → llpx_sn_alt_r R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → S d T L1 L2.
+#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+ /3 width=9 by nlift_bind_sn, and3_intro/
+| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
+ lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+ lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
+ @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H //
+qed-.
+
+lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
+ R I K1 V1 V2 & d ≤ yinj i.
+#R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i d) /3 width=1 by or3_intro1/
+#Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
+#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_sn_alt_r_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt_r R d (⋆k) L1 L2.
+#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
+qed.
+
+lemma llpx_sn_alt_r_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt_r R d (§p) L1 L2.
+#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
+qed.
+
+lemma llpx_sn_alt_r_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt_r R d (#i) L1 L2.
+#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
+/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
+qed.
+
+lemma llpx_sn_alt_r_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
+ llpx_sn_alt_r R d (#i) L1 L2.
+#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
+lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
+/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
+qed.
+
+lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn_alt_r R 0 V1 K1 K2 → R I K1 V1 V2 →
+ llpx_sn_alt_r R d (#i) L1 L2.
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
+[ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
+ @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+ elim (lt_or_eq_or_gt i j) #Hij destruct
+ [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+ | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+ lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+ | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+ ]
+]
+qed.
+
+lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,d.
+ llpx_sn_alt_r R d V L1 L2 → llpx_sn_alt_r R d T L1 L2 →
+ llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2.
+#R #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
+
+lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,d.
+ llpx_sn_alt_r R d V L1 L2 →
+ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2.
+#R #a #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt_r R d T L1 L2.
+#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 → llpx_sn R d T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
+[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
+| #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length … H)
+ #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
+ [ * /2 width=1 by llpx_sn_free/
+ | /2 width=1 by llpx_sn_skip/
+ | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+ ]
+| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind … H) -H
+ /3 width=1 by llpx_sn_bind/
+| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+(* Alternative definition of llpx_sn (recursive) ****************************)
+
+lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
+ ) → llpx_sn R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
+qed.
+
+lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
+#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
+qed-.
+
+lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/llpx_sn_leq.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ i < d ∨
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
+ R I K1 V1 V2 & d ≤ i.
+#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
+[ #_ #H elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
+ lapply (ldrop_mono … HLK22 … HLK2) -L2 #H destruct
+ /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ i < d ∨
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
+ R I K1 V1 V2 & d ≤ i.
+#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
+[ #H #_ elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
+ lapply (ldrop_mono … HLK11 … HLK1) -L1 #H destruct
+ /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ llpx_sn R 0 V1 K1 K2 & R I K1 V1 V2.
+#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
+[ #H elim (ylt_yle_false … H Hdi)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn R 0 V1 K1 K2 & R I K1 V1 V2.
+#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
+[ #H elim (ylt_yle_false … H Hdi)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R I1 K1 V1 V2.
+#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
+#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
+qed-.
+
+fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 →
+ ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn R 0 V1 K1 K2 → R I K1 V1 V2 → llpx_sn R d T L1 L2.
+#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
+/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
+[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
+ elim (yle_split_eq i d) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hid
+ #H destruct /2 width=9 by llpx_sn_lref/
+| #I #L1 #L2 #K11 #K22 #V1 #V2 #d0 #i #Hd0i #HLK11 #HLK22 #HK12 #HV12 #_ #d #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
+ /3 width=9 by llpx_sn_lref, yle_pred_sn/
+| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+ /4 width=9 by llpx_sn_bind, ldrop_drop/
+| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+ /3 width=9 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 →
+ ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn R 0 V1 K1 K2 → R I K1 V1 V2 → llpx_sn R d T L1 L2.
+/2 width=9 by llpx_sn_inv_S_aux/ qed-.
+
+lemma llpx_sn_inv_bind_O: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
+/3 width=9 by ldrop_pair, conj, llpx_sn_inv_S/
+qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
+ ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R J L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
+/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
+
+lemma llpx_sn_dec: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+ ∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
+| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
+ [ #HL12 #d elim (ylt_split i d) /3 width=1 by llpx_sn_skip, or_introl/
+ #Hdi elim (lt_or_ge i (|L1|)) #HiL1
+ elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
+ elim (ldrop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
+ elim (ldrop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
+ elim (eq_bind2_dec I2 I1)
+ [ #H2 elim (HR I1 K1 V1 V2) -HR
+ [ #H3 elim (IH K1 V1 … K2 0) destruct
+ /3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
+ ]
+ ]
+ -IH #H3 @or_intror
+ #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ]
+ [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
+ |7,8,9: /2 width=4 by ylt_yle_false/
+ ]
+ #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
+ lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
+ lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
+ #H #H0 destruct /2 width=1 by/
+ ]
+| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
+| #a #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V … L2 d) /2 width=1 by/
+ elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯d)) -IH /3 width=1 by or_introl, llpx_sn_bind/
+ #H1 #H2 @or_intror
+ #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
+| #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V … L2 d) /2 width=1 by/
+ elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, llpx_sn_flat/
+ #H1 #H2 @or_intror
+ #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+]
+-n /4 width=4 by llpx_sn_fwd_length, or_intror/
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma llpx_sn_lift_le: ∀R. (∀I. l_liftable (R I)) →
+ ∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2.
+#R #HR #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
+[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hdi #H destruct
+ [ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_skip/
+ | elim (ylt_yle_false … Hid0) -L1 -L2 -K1 -K2 -e -Hid0
+ /3 width=3 by yle_trans, yle_inj/
+ ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hdi #H destruct [ -HK12 | -IHK12 ]
+ [ elim (ldrop_trans_lt … HLK1 … HK11) // -K1
+ elim (ldrop_trans_lt … HLK2 … HK22) // -Hdi -K2
+ /3 width=18 by llpx_sn_lref/
+ | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
+ lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hdi -Hd0 -K2
+ /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
+ ]
+| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hid #H destruct
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
+ [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ ]
+| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
+ #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
+ #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_lift_ge: ∀R,K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → d ≤ d0 → llpx_sn R (d0+e) U L1 L2.
+#R #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
+[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
+ [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
+ | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
+ ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -e -Hid0
+ /3 width=3 by ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
+ lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hid -Hd0 -K2
+ /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
+ ]
+| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hid #H destruct
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
+ [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ ]
+| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
+ #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
+ #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma llpx_sn_inv_lift_le: ∀R. (∀I. l_deliftable_sn (R I)) →
+ ∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2.
+#R #HR #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+ [ /2 width=1 by llpx_sn_skip/
+ | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct [ -HK12 | -IHK12 ]
+ [ elim (ldrop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
+ elim (ldrop_conf_lt … HLK2 … HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
+ elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
+ lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
+ /3 width=10 by llpx_sn_lref/
+ | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
+ lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0
+ elim (le_inv_plus_l … Hid) -Hid /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind2 … H) -H
+ #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat2 … H) -H
+ #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d ≤ d0 → d0 ≤ yinj d + e → llpx_sn R d T K1 K2.
+#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct
+ [ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+ -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/
+ | elim (ylt_yle_false … Hid0) -L1 -L2 -Hd0 -Hid0
+ /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0
+ /3 width=3 by ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
+ lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e
+ elim (le_inv_plus_l … Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_bind2 … H) -H
+ >commutative_plus #V #T #HVW #HTU #H destruct
+ @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+ @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_flat2 … H) -H
+ #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ d0 → llpx_sn R (d0-e) T K1 K2.
+#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct [ -Hid0 | -Hded0 ]
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+ [ /4 width=3 by llpx_sn_skip, yle_plus_to_minus_inj2, ylt_yle_trans, ylt_inj/
+ | elim (le_inv_plus_l … Hid) -Hid #_
+ /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hid0
+ /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
+ lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hded0 -Hid
+ /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_bind2 … H) -H
+ #V #T #HVW #HTU #H destruct
+ @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+ <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
+ @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_flat2 … H) -H
+ #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+ ∀K1,K2,e. ⇩[e] L1 ≡ K1 → ⇩[e] L2 ≡ K2 →
+ ∀T. ⇧[0, e] T ≡ U → llpx_sn R 0 T K1 K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
+
+lemma llpx_sn_ldrop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+ ∀K1,e. ⇩[e] L1 ≡ K1 → ∀T. ⇧[0, e] T ≡ U →
+ ∃∃K2. ⇩[e] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K1 #e #HLK1 #T #HTU elim (llpx_sn_fwd_ldrop_sn … HU … HLK1)
+/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
+qed-.
+
+lemma llpx_sn_ldrop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+ ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
+ ∃∃K1. ⇩[e] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K2 #e #HLK2 #T #HTU elim (llpx_sn_fwd_ldrop_dx … HU … HLK2)
+/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
+qed-.
+
+(* Inversion lemmas on negated lazy pointwise extension *********************)
+
+lemma nllpx_sn_inv_bind: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+ ∀a,I,L1,L2,V,T,d. (llpx_sn R d (ⓑ{a,I}V.T) L1 L2 → ⊥) →
+ (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
+/4 width=1 by llpx_sn_bind, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_flat: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+ ∀I,L1,L2,V,T,d. (llpx_sn R d (ⓕ{I}V.T) L1 L2 → ⊥) →
+ (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R d T L1 L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
+/4 width=1 by llpx_sn_flat, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_bind_O: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+ ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
+ (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
+/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lpx_sn_ldrop.ma".
+include "basic_2/substitution/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on pointwise extensions ***************************************)
+
+lemma lpx_sn_llpx_sn: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀T,L1,L2,d. lpx_sn R L1 L2 → llpx_sn R d T L1 L2.
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
+| -HR #i elim (lt_or_ge i (|L1|))
+ [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
+ #Hi #Hn #L2 #d elim (ylt_split i d)
+ [ -n /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
+ #Hdi #HL12 elim (ldrop_O1_lt (Ⓕ) L1 i) //
+ #I #K1 #V1 #HLK1 elim (lpx_sn_ldrop_conf … HL12 … HLK1) -HL12
+ /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
+| /4 width=1 by llpx_sn_bind, lpx_sn_pair/
+| -HR /3 width=1 by llpx_sn_flat/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/llpx_sn_ldrop.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties about transitive closure **************************************)
+
+lemma llpx_sn_TC_pair_dx: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀I,L,V1,V2,T. LTC … (R I) L V1 V2 →
+ LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
+#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2
+/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predsn_3.ma".
+include "basic_2/relocation/lpx_sn.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (λ_.cpr G).
+
+interpretation "parallel reduction (local environment, sn variant)"
+ 'PRedSn G L1 L2 = (lpr G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: includes: wcpr0_gen_sort *)
+lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: wcpr0_gen_head *)
+lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Note: lemma 250 *)
+lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
+ ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predsn_5.ma".
+include "basic_2/reduction/lpr.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
+ λh,g,G. lpx_sn (λ_.cpx h g G).
+
+interpretation "extended parallel reduction (local environment, sn variant)"
+ 'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+lemma lpx_inv_pair1: ∀h,g,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpx_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpx_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
+/2 width=1 by lpx_sn_inv_pair/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡[h, g] L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
+ ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_length.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+inductive lpx_sn (R:relation4 bind2 lenv term term): relation lenv ≝
+| lpx_sn_atom: lpx_sn R (⋆) (⋆)
+| lpx_sn_pair: ∀I,K1,K2,V1,V2.
+ lpx_sn R K1 K2 → R I K1 V1 V2 →
+ lpx_sn R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_refl: ∀R. (∀I,L. reflexive ? (R I L)) → reflexive … (lpx_sn R).
+#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. lpx_sn R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #L1 #L2 * -L1 -L2
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. lpx_sn R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
+ ∃∃K1,V1. lpx_sn R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #L1 #L2 * -L1 -L2
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. lpx_sn R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
+ lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
+ ∧∧ lpx_sn R L1 L2 & R I1 L1 V1 V2 & I1 = I2.
+#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
+#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop.ma".
+include "basic_2/relocation/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* alternative definition of lpx_sn *)
+definition lpx_sn_alt: relation4 bind2 lenv term term → relation lenv ≝
+ λR,L1,L2. |L1| = |L2| ∧
+ (∀I1,I2,K1,K2,V1,V2,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R I1 K1 V1 V2
+ ).
+
+(* Basic forward lemmas ******************************************************)
+
+lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
+#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_sn/
+qed-.
+
+lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. lpx_sn_alt R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I1 #L2 #K1 #V1 #H elim H -H
+#H #IH elim (length_inv_pos_sn … H) -H
+#I2 #K2 #V2 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
+/2 width=1 by ldrop_drop, conj/
+qed-.
+
+lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
+#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_dx/
+qed-.
+
+lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. lpx_sn_alt R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I2 #L1 #K2 #V2 #H elim H -H
+#H #IH elim (length_inv_pos_dx … H) -H
+#I1 #K1 #V1 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
+/2 width=1 by ldrop_drop, conj/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
+#R @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
+#H destruct
+qed.
+
+lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
+ lpx_sn_alt R L1 L2 → R I L1 V1 V2 →
+ lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
+#HL12 #IH @conj normalize //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
+[ #HLK1 #HLK2
+ lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
+ lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
+ /2 width=1 by conj/
+| -HL12 -HV12 /3 width=6 by ldrop_inv_drop1/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
+#R #L1 elim L1 -L1
+[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
+| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
+]
+qed-.
+
+(* alternative definition of lpx_sn *****************************************)
+
+lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R I1 K1 V1 V2
+ ) → lpx_sn R L1 L2.
+/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
+
+lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R I1 K1 V1 V2.
+#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
+#H elim H -H /3 width=4 by conj/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_leq.ma".
+include "basic_2/relocation/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properies on dropping ****************************************************)
+
+lemma lpx_sn_ldrop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K1,V1,i. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R I K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by ldrop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_ldrop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K2,V2,i. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R I K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by ldrop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_deliftable_dropable: ∀R. (∀I. l_deliftable_sn (R I)) → dropable_sn (lpx_sn R).
+#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (HR … HV12 … HLK1 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/
+]
+qed-.
+
+lemma lpx_sn_liftable_dedropable: ∀R. (∀I,L. reflexive ? (R I L)) →
+ (∀I. l_liftable (R I)) → dedropable_sn (lpx_sn R).
+#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+ /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (lpx_sn_fwd_length … HK12)
+ #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
+ /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
+ @leq_O2 normalize //
+| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
+ /3 width=5 by ldrop_drop, leq_pair, lpx_sn_pair, ex3_intro/
+| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1
+ /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/
+]
+qed-.
+
+fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
+ #K1 #V1 #HK12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+ #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
+/2 width=5 by lpx_sn_dropable_aux/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. lpx_sn (λ_.R1) L0 L1 → ∀L2. lpx_sn (λ_.R2) L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn (λ_.R1) L1 L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn (λ_.R)).
+#R #HR #L1 #L #H elim H -L1 -L //
+#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
+elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
+qed-.
+
+theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
+ confluent2 … (lpx_sn (λ_.R1)) (lpx_sn (λ_.R2)).
+#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
+[ #_ #X1 #H1 #X2 #H2 -n
+ >(lpx_sn_inv_atom1 … H1) -X1
+ >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
+| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
+ elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+ elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
+ elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properties on transitive_closure *****************************************)
+
+lemma TC_lpx_sn_pair_refl: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,V. TC … (lpx_sn R) (L1.ⓑ{I}V) (L2. ⓑ{I}V).
+#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
+[ /2 width=1 by lpx_sn_refl/
+| /3 width=1 by TC_reflexive, lpx_sn_refl/
+| /3 width=5 by lpx_sn_pair, step/
+]
+qed-.
+
+lemma TC_lpx_sn_pair: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀V1,V2. LTC … (R I) L1 V1 V2 →
+ TC … (lpx_sn R) (L1.ⓑ{I}V1) (L2. ⓑ{I}V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
+qed-.
+
+lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀I,L. reflexive … (R I L)) →
+ ∀L1,L2. lpx_sn (λI.LTC … (R I)) L1 L2 →
+ TC … (lpx_sn R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+qed-.
+
+(* Inversion lemmas on transitive closure ***********************************)
+
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair2: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
+ ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … (R I) K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+ elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+ lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_ind: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
+ ∀S:relation lenv.
+ S (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ TC … (lpx_sn R) K1 K2 → LTC … (R I) K1 V1 V2 →
+ S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
+#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
+[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
+| #L2 #I #V2 #IHL2 #X #H
+ elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
+ #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … (R I) K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
+ ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … (R I) K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ lpx_sn (λI.LTC … (R I)) L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/llpx_sn_ldrop.ma".
+include "basic_2/static/ssta.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ******************************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+lemma ssta_llpx_sn_conf: ∀R. (∀I,L.reflexive … (R I L)) →
+ (∀I.l_liftable (R I)) →
+ ∀h,g,G. s_r_confluent1 … (ssta h g G) (llpx_sn R 0).
+#R #H1R #H2R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
+| #G #Ls #Ks #V1s #W2s #V2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #G #Ls #Ks #V1s #W1s #l #i #HLKs #Hl #HVW1s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW1s) -HLKs -HLKd -HVW1s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( h ⊢ break term 46 L1 : ⊑ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'CrSubEqN $h $L1 $L2 }.
+
+notation "hvbox( h ⊢ break term 46 L1 : : ⊑ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'CrSubEqNAlt $h $L1 $L2 }.
+
+include "basic_2/dynamic/nta.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
+
+(* Note: may not be transitive *)
+inductive lsubn (h:sh): relation lenv ≝
+| lsubn_atom: lsubn h (⋆) (⋆)
+| lsubn_pair: ∀I,L1,L2,W. lsubn h L1 L2 → lsubn h (L1. ⓑ{I} W) (L2. ⓑ{I} W)
+| lsubn_abbr: ∀L1,L2,V,W. ⦃h, L1⦄ ⊢ V : W → ⦃h, L2⦄ ⊢ V : W →
+ lsubn h L1 L2 → lsubn h (L1. ⓓV) (L2. ⓛW)
+.
+
+interpretation
+ "local environment refinement (native type assigment)"
+ 'CrSubEqN h L1 L2 = (lsubn h L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#h #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsubn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑ L2 → L2 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+ (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
+ h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr.
+#h #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
+]
+qed.
+
+lemma lsubn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑ L2 →
+ (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
+ h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsubn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#h #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsubc_inv_atom2: ∀h,L1. h ⊢ L1 :⊑ ⋆ → L1 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+ (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
+ h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst.
+#h #L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
+| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
+]
+qed.
+
+(* Basic_1: was: csubt_gen_bind *)
+lemma lsubn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑ K2. ⓑ{I} W →
+ (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
+ h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst.
+/2 width=3/ qed-.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma lsubn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L1|] L2.
+#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma lsubn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L2|] L2.
+#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: csubt_refl *)
+lemma lsubn_refl: ∀h,L. h ⊢ L :⊑ L.
+#h #L elim L -L // /2 width=1/
+qed.
+
+(* Basic_1: removed theorems 6:
+ csubt_gen_flat csubt_drop_flat csubt_clear_conf
+ csubt_getl_abbr csubt_getl_abst csubt_ty3_ld
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/dynamic/lsubn.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+(* Basic_1: was: csubt_pr2 *)
+lemma cpr_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
+/3 width=4 by lsubn_fwd_lsubs2, cpr_lsubs_trans/ qed.
+
+lemma cprs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+/3 width=4 by lsubn_fwd_lsubs2, cprs_lsubs_trans/ qed.
+
+(* Basic_1: was: csubt_pc3 *)
+lemma cpcs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 →
+ ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
+/3 width=4 by lsubn_fwd_lsubs2, cpcs_lsubs_trans/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/lsubn.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubn_ldrop_O1_conf: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
+ ∃∃K2. h ⊢ K1 :⊑ K2 & ⇩[0, e] L2 ≡ K2.
+#h #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+]
+qed.
+
+(* Note: the constant 0 cannot be generalized *)
+(* Basic_1: was only: csubt_drop_abbr csubt_drop_abst *)
+lemma lsubn_ldrop_O1_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. h ⊢ K1 :⊑ K2 & ⇩[0, e] L1 ≡ K1.
+#h #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/lsubn_nta.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
+
+(* Main properties **********************************************************)
+
+(* Note: new property *)
+theorem lsubn_trans: ∀h,L1,L. h ⊢ L1 :⊑ L → ∀L2. h ⊢ L :⊑ L2 → h ⊢ L1 :⊑ L2.
+#h #L1 #L #H elim H -L1 -L
+[ #X #H >(lsubn_inv_atom1 … H) -H //
+| #I #L1 #L #V #HL1 #H1W #IHL1 #X #H
+ elim (lsubn_inv_pair1 … H) -H * #L2
+ [ #HL2 #H #H2W destruct /4 width=1/
+ | #W #H1VW #H2VW #HL2 #H1 #H2 destruct /3 width=3/
+ ]
+| #L1 #L #V1 #W1 #H1VW1 #H2VW1 #HL1 #IHL1 #X #H
+ elim (lsubn_inv_pair1 … H) -H * #L2
+ [ #HL2 #H #HW destruct /3 width=1/
+ | #V #_ #_ #_ #_ #H destruct
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/nta_nta.ma".
+include "basic_2/dynamic/lsubn_ldrop.ma".
+include "basic_2/dynamic/lsubn_cpcs.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
+
+(* Properties concerning atomic arity assignment ****************************)
+
+(* Note: the corresponding confluence property does not hold *)
+(* Basic_1: was: csubt_ty3 *)
+lemma lsubn_nta_trans: ∀h,L2,T,U. ⦃h, L2⦄ ⊢ T : U → ∀L1. h ⊢ L1 :⊑ L2 →
+ ⦃h, L1⦄ ⊢ T : U.
+#h #L2 #T #U #H elim H -L2 -T -U
+[ //
+| #L2 #K2 #V2 #W2 #U2 #i #HLK2 #_ #WU2 #IHVW2 #L1 #HL12
+ elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubn_inv_pair2 … H) -H * #K1
+ [ #HK12 #H destruct /3 width=6/
+ | #V1 #_ #_ #_ #_ #H destruct
+ ]
+| #L2 #K2 #W2 #V2 #U2 #i #HLK2 #_ #HWU2 #IHWV2 #L1 #HL12
+ elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubn_inv_pair2 … H) -H * #K1 [ | -IHWV2 ]
+ [ #HK12 #H destruct /3 width=6/
+ | #V1 #H1V1W2 #_ #_ #H #_ destruct /2 width=6/
+ ]
+| /4 width=2/
+| /3 width=1/
+| /3 width=2/
+| /3 width=1/
+| /4 width=6/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lpss_ldrop.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel substitution ********************************)
+
+(* Note: lemma 500 *)
+lemma aaa_cpss_lpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. L1 ⊢ T1 ▶* T2 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #X #H
+ >(cpss_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
+ | * #Y #Z #V2 #H #HV12 #HV2
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
+]
+qed-.
+
+lemma aaa_cpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ▶* T2 → L ⊢ T2 ⁝ A.
+/2 width=5 by aaa_cpss_lpss_conf/ qed-.
+
+lemma aaa_lpss_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T ⁝ A.
+/2 width=5 by aaa_cpss_lpss_conf/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/cprs_lpss.ma".
+include "basic_2/equivalence/cpcs_cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties on sn parallel substitution for local environments ************)
+
+lemma cpcs_lpss_conf: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T1 ⬌* T2.
+#L1 #T1 #T2 #H #L2 #HL12
+elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
+elim (cprs_lpss_conf_dx … HT1 … HL12) -HT1 #U1 #H1 #HTU1
+elim (cprs_lpss_conf_dx … HT2 … HL12) -L1 #U2 #H2 #HTU2
+elim (cpss_conf … H1 … H2) -T #U #HU1 #HU2
+lapply (cprs_cpss_trans … HTU1 … HU1) -U1
+lapply (cprs_cpss_trans … HTU2 … HU2) -U2 /2 width=3/
+qed-.
+
+lemma cpcs_cpss_lpss_conf: ∀L1,T,T2. L1 ⊢ T ⬌* T2 → ∀T1. L1 ⊢ T ▶* T1 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T1 ⬌* T2.
+#L1 #T #T2 #HT2 #T1 #HT1 #L2 #HL12
+lapply (cpcs_lpss_conf … HT2 … HL12) -HT2 #HT2
+elim (lpss_cpss_conf_dx … HT1 … HL12) -L1 #T0 #HT0 #HT10
+lapply (cpcs_cpss_conf … HT0 … HT2) -T #HT02
+lapply (cpcs_cpss_strap2 … HT10 … HT02) -T0 //
+qed-.
+
+lemma cpcs_cpss2_lpss_conf: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
+ ∀T3. L1 ⊢ T1 ▶* T3 → ∀T4. L1 ⊢ T2 ▶* T4 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T3 ⬌* T4.
+#L1 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 #L2 #HL12
+lapply (cpcs_cpss_lpss_conf … HT12 … HT13 … HL12) -T1 #HT32
+elim (lpss_cpss_conf_dx … HT24 … HL12) -L1 #T #HT2 #HT4
+lapply (cpcs_cpss_strap1 … HT32 … HT2) -T2 #HT3
+lapply (cpcs_cpss_div … HT3 … HT4) -T //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 ➤ * break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRestStar $L $T1 $T2 }.
+
+include "basic_2/substitution/cpss.ma".
+
+(* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
+
+inductive cpqs: lenv → relation term ≝
+| cpqs_atom : ∀I,L. cpqs L (⓪{I}) (⓪{I})
+| cpqs_delta: ∀L,K,V,V2,W2,i.
+ ⇩[0, i] L ≡ K. ⓓV → cpqs K V V2 →
+ ⇧[0, i + 1] V2 ≡ W2 → cpqs L (#i) W2
+| cpqs_bind : ∀a,I,L,V1,V2,T1,T2.
+ cpqs L V1 V2 → cpqs (L. ⓑ{I} V1) T1 T2 →
+ cpqs L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| cpqs_flat : ∀I,L,V1,V2,T1,T2.
+ cpqs L V1 V2 → cpqs L T1 T2 →
+ cpqs L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+| cpqs_zeta : ∀L,V,T1,T,T2. cpqs (L.ⓓV) T1 T →
+ ⇧[0, 1] T2 ≡ T → cpqs L (+ⓓV. T1) T2
+| cpqs_tau : ∀L,V,T1,T2. cpqs L T1 T2 → cpqs L (ⓝV. T1) T2
+.
+
+interpretation "context-sensitive restricted parallel computation (term)"
+ 'PRestStar L T1 T2 = (cpqs L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpqs_lsubr_trans: lsub_trans … cpqs lsubr.
+#L1 #T1 #T2 #H elim H -L1 -T1 -T2
+[ //
+| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
+| /4 width=1/
+|4,6: /3 width=1/
+| /4 width=3/
+]
+qed-.
+
+lemma cpss_cpqs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➤* T2.
+#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
+qed.
+
+lemma cpqs_refl: ∀T,L. L ⊢ T ➤* T.
+/2 width=1/ qed.
+
+lemma cpqs_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
+ ∃∃T2,T. L ⊢ T1 ➤* T2 & ⇧[d, 1] T ≡ T2.
+#L #K #V #T1 #d #HLK
+elim (cpss_delift … T1 … HLK) -HLK /3 width=4/
+qed-.
+
+lemma cpqs_append: l_appendable_sn … cpqs.
+#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
+#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpqs_delta … (L@@K0) V1 … HVW2) //
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpqs_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➤* T2 → ∀I. T1 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ➤* V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+#L #T1 #T2 * -L -T1 -T2
+[ #I #L #J #H destruct /2 width=1/
+| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #L #V #T1 #T2 #_ #J #H destruct
+]
+qed-.
+
+lemma cpqs_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➤* T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ➤* V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+/2 width=3 by cpqs_inv_atom1_aux/ qed-.
+
+lemma cpqs_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➤* T2 → T2 = ⋆k.
+#L #T2 #k #H
+elim (cpqs_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+lemma cpqs_inv_lref1: ∀L,T2,i. L ⊢ #i ➤* T2 →
+ T2 = #i ∨
+ ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ➤* V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#L #T2 #i #H
+elim (cpqs_inv_atom1 … H) -H /2 width=1/
+* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
+qed-.
+
+lemma cpqs_inv_gref1: ∀L,T2,p. L ⊢ §p ➤* T2 → T2 = §p.
+#L #T2 #p #H
+elim (cpqs_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+fact cpqs_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 &
+ L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+#L #U1 #U2 * -L -U1 -U2
+[ #I #L #b #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
+| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
+| #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpqs_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➤* U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 &
+ L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+/2 width=3 by cpqs_inv_bind1_aux/ qed-.
+
+lemma cpqs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➤* U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 &
+ L. ⓓ V1 ⊢ T1 ➤* T2 &
+ U2 = ⓓ{a} V2. T2
+ ) ∨
+ ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true.
+#a #L #V1 #T1 #U2 #H
+elim (cpqs_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
+qed-.
+
+lemma cpqs_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➤* U2 →
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 &
+ L. ⓛ V1 ⊢ T1 ➤* T2 &
+ U2 = ⓛ{a} V2. T2.
+#a #L #V1 #T1 #U2 #H
+elim (cpqs_inv_bind1 … H) -H *
+[ /3 width=5/
+| #T #_ #_ #_ #H destruct
+]
+qed-.
+
+fact cpqs_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
+ ∀I,V1,T1. U1 = ⓕ{I} V1. T1 → (
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
+ U2 = ⓕ{I} V2. T2
+ ) ∨
+ (L ⊢ T1 ➤* U2 ∧ I = Cast).
+#L #U1 #U2 * -L -U1 -U2
+[ #I #L #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
+| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
+| #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
+| #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
+]
+qed-.
+
+lemma cpqs_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ➤* U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
+ U2 = ⓕ{I} V2. T2
+ ) ∨
+ (L ⊢ T1 ➤* U2 ∧ I = Cast).
+/2 width=3 by cpqs_inv_flat1_aux/ qed-.
+
+lemma cpqs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐ V1. T1 ➤* U2 →
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
+ U2 = ⓐ V2. T2.
+#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H *
+[ /3 width=5/
+| #_ #H destruct
+]
+qed-.
+
+lemma cpqs_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝ V1. T1 ➤* U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
+ U2 = ⓝ V2. T2
+ ) ∨
+ L ⊢ T1 ➤* U2.
+#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H * /2 width=1/ /3 width=5/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X
+ >shift_append_assoc normalize #H
+ elim (cpqs_inv_bind1 … H) -H *
+ [ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/unfold/cpqs.ma".
+
+(* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
+
+(* Relocation properties ****************************************************)
+
+lemma cpqs_lift: l_liftable cpqs.
+#K #T1 #T2 #H elim H -K -T1 -T2
+[ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
+ elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=8/
+ | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=6/
+ ]
+| #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
+| #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+| #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #H #U2 #HTU2
+ elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+ elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5/
+| #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
+]
+qed.
+
+lemma cpqs_inv_lift1: l_deliftable_sn cpqs.
+#L #U1 #U2 #H elim H -L -U1 -U2
+[ * #L #i #K #d #e #_ #T1 #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+ elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+ elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
+ elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=8/
+ ]
+| #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
+| #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -V1
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
+| #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHU1 (K.ⓓW1) … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
+ elim (lift_div_le … HU2 … HTU) -U // /3 width=5/
+| #L #V #U1 #U2 #_ #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lpss_lpss.ma".
+include "basic_2/reduction/lpr_lpss.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties on parallel substitution for terms ****************************)
+
+(* Basic_1: was: pr3_subst1 *)
+lemma cprs_cpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 → ∀T2. L ⊢ T0 ▶* T2 →
+ ∃∃T. L ⊢ T1 ▶* T & L ⊢ T2 ➡* T.
+#L @TC_strip1 /2 width=3 by cpr_cpss_conf/ qed-. (**) (* auto /3 width=3/ fails because a δ-expansion gets in the way *)
+
+(* Properties on sn parallel substitution for local environments ************)
+
+lemma cprs_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡* T.
+#L0 #T0 #T1 #H elim H -T1
+[ #T1 #HT01 #L1 #HL01
+ elim (cpr_lpss_conf_dx … HT01 … HL01) -L0 /3 width=3/
+| #T #T1 #_ #HT1 #IHT0 #L1 #HL01
+ elim (IHT0 … HL01) #T2 #HT2 #HT02
+ elim (cpr_lpss_conf_dx … HT1 … HL01) -L0 #T3 #HT13 #HT3
+ elim (cpr_cpss_conf … HT3 … HT2) -T #T #HT3 #HT2
+ lapply (cpss_trans … HT13 … HT3) -T3
+ lapply (cprs_strap1 … HT02 … HT2) -T2 /2 width=3/
+]
+qed-.
+
+lemma cprs_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cprs_lpss_conf_dx … HT01 … HL01) -HT01 #T #HT1
+lapply (lpss_cpss_trans … HL01 … HT1) -HT1 /2 width=3/
+qed-.
+
+lemma cprs_cpss_lpss_conf_sn: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. L2 ⊢ T2 ➡* U2 & L1 ⊢ U1 ▶* U2.
+#L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (cprs_cpss_conf … HTU1 … HT12) -T1 #U #HU1 #HT2U
+elim (cprs_lpss_conf_sn … HT2U … HL12) -HT2U -HL12 #U2 #HU2 #HTU2
+lapply (cpss_trans … HU1 … HU2) -U /2 width=3/
+qed-.
+
+lemma cprs_cpss_lpss_conf_dx: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. L2 ⊢ T2 ➡* U2 & L2 ⊢ U1 ▶* U2.
+#L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (cprs_lpss_conf_dx … HTU1 … HL12) -HTU1 #U2 #HU12 #HT1U2
+elim (lpss_cpss_conf_dx … HT12 … HL12) -L1 #T #HT1 #HT2
+elim (cprs_cpss_conf … HT1U2 … HT1) -T1 #U #HU2 #HTU
+lapply (cpss_trans … HU12 … HU2) -U2
+lapply (cpss_cprs_trans … HT2 … HTU) -T /2 width=3/
+qed-.
+
+
+lemma cprs_cpss2_lpss_conf_sn: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 → ∀T2. L1 ⊢ T1 ▶* T2 →
+ ∀U2. L1 ⊢ U1 ▶* U2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U. L2 ⊢ T2 ➡* U & L1 ⊢ U2 ▶* U.
+#L1 #T1 #U1 #HTU1 #T2 #HT12 #U2 #HU12 #L2 #HL12
+elim (cprs_cpss_lpss_conf_sn … HTU1 … HT12 … HL12) -T1 #T1 #HT21 #HUT1
+elim (cpss_conf … HU12 … HUT1) -U1 #U1 #HU21 #HTU1
+elim (lpss_cpss_conf_sn … HTU1 … HL12) -HTU1 -HL12 #U2 #HT1U2 #HU12
+lapply (cpss_trans … HU21 … HU12) -U1
+lapply (cprs_cpss_trans … HT21 … HT1U2) -T1 /2 width=3/
+qed-.
+
+lemma cprs_cpss2_lpss_conf_dx: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 → ∀T2. L1 ⊢ T1 ▶* T2 →
+ ∀U2. L1 ⊢ U1 ▶* U2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U. L2 ⊢ T2 ➡* U & L2 ⊢ U2 ▶* U.
+#L1 #T1 #U1 #HTU1 #T2 #HT12 #U2 #HU12 #L2 #HL12
+elim (cprs_cpss_lpss_conf_dx … HTU1 … HT12 … HL12) -T1 #T1 #HT21 #HUT1
+elim (lpss_cpss_conf_dx … HU12 … HL12) -L1 #U #HU1 #HU2
+elim (cpss_conf … HU1 … HUT1) -U1 #U1 #HU1 #HTU1
+lapply (cpss_trans … HU2 … HU1) -U
+lapply (cprs_cpss_trans … HT21 … HTU1) -T1 /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 ▶* break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $L $T1 $T2 }.
+
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/substitution/lsubr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
+
+inductive cpss: lenv → relation term ≝
+| cpss_atom : ∀I,L. cpss L (⓪{I}) (⓪{I})
+| cpss_delta: ∀L,K,V,V2,W2,i.
+ ⇩[0, i] L ≡ K. ⓓV → cpss K V V2 →
+ ⇧[0, i + 1] V2 ≡ W2 → cpss L (#i) W2
+| cpss_bind : ∀a,I,L,V1,V2,T1,T2.
+ cpss L V1 V2 → cpss (L. ⓑ{I} V1) T1 T2 →
+ cpss L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| cpss_flat : ∀I,L,V1,V2,T1,T2.
+ cpss L V1 V2 → cpss L T1 T2 →
+ cpss L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "context-sensitive parallel substitution (term)"
+ 'PSubstStar L T1 T2 = (cpss L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpss_lsubr_trans: lsub_trans … cpss lsubr.
+#L1 #T1 #T2 #H elim H -L1 -T1 -T2
+[ //
+| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
+| /4 width=1/
+| /3 width=1/
+]
+qed-.
+
+(* Basic_1: was by definition: subst1_refl *)
+lemma cpss_refl: ∀T,L. L ⊢ T ▶* T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic_1: was only: subst1_ex *)
+lemma cpss_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
+ ∃∃T2,T. L ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
+#K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK /2 width=4/
+ elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i) // /3 width=6/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+ ]
+]
+qed-.
+
+lemma cpss_append: l_appendable_sn … cpss.
+#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/
+#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpss_delta … (L@@K0) V1 … HVW2) //
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpss_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ∀I. T1 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+#L #T1 #T2 * -L -T1 -T2
+[ #I #L #J #H destruct /2 width=1/
+| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #I #H destruct /3 width=8/
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpss_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ▶* T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+/2 width=3 by cpss_inv_atom1_aux/ qed-.
+
+(* Basic_1: was only: subst1_gen_sort *)
+lemma cpss_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ▶* T2 → T2 = ⋆k.
+#L #T2 #k #H
+elim (cpss_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was only: subst1_gen_lref *)
+lemma cpss_inv_lref1: ∀L,T2,i. L ⊢ #i ▶* T2 →
+ T2 = #i ∨
+ ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
+ K ⊢ V ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#L #T2 #i #H
+elim (cpss_inv_atom1 … H) -H /2 width=1/
+* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
+qed-.
+
+lemma cpss_inv_gref1: ∀L,T2,p. L ⊢ §p ▶* T2 → T2 = §p.
+#L #T2 #p #H
+elim (cpss_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+fact cpss_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 ▶* V2 &
+ L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#L #U1 #U2 * -L -U1 -U2
+[ #I #L #b #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5/
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpss_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶* V2 &
+ L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
+ U2 = ⓑ{a,I} V2. T2.
+/2 width=3 by cpss_inv_bind1_aux/ qed-.
+
+fact cpss_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
+ ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
+ U2 = ⓕ{I} V2. T2.
+#L #U1 #U2 * -L -U1 -U2
+[ #I #L #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
+| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5/
+]
+qed-.
+
+lemma cpss_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
+ U2 = ⓕ{I} V2. T2.
+/2 width=3 by cpss_inv_flat1_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpss_fwd_tw: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #H elim H -L -T1 -T2 normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/ (**) (* auto is too slow without trace *)
+qed-.
+
+lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X
+ >shift_append_assoc normalize #H
+ elim (cpss_inv_bind1 … H) -H
+ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic_1: removed theorems 27:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+ subst1_gen_lift_eq subst1_confluence_neq
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/cpss.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was only: subst1_lift_lt subst1_lift_ge *)
+lemma cpss_lift: l_liftable cpss.
+#K #T1 #T2 #H elim H -K -T1 -T2
+[ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
+ elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=8/
+ | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=6/
+ ]
+| #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
+| #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+]
+qed.
+
+(* Basic_1: was only: subst1_gen_lift_lt subst1_gen_lift_ge *)
+lemma cpss_inv_lift1: l_deliftable_sn cpss.
+#L #U1 #U2 #H elim H -L -U1 -U2
+[ * #L #i #K #d #e #_ #T1 #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+ elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+ elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
+ elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=8/
+ ]
+| #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
+| #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1) -V1
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/sstas_lpss.ma".
+include "basic_2/computation/cprs_lpss.ma".
+include "basic_2/computation/dxprs.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
+
+(* Properties about sn parallel substitution for local environments *********)
+
+lemma dxprs_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*➡*[g] U1 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*➡*[g] U2 & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #HL12
+elim (sstas_lpss_conf … HTU … HL12) -HTU #U0 #HTU0 #HU0
+elim (cprs_cpss_conf … HU1 … HU0) -U #U #HU1 #HU0
+elim (cprs_lpss_conf_sn … HU0 … HL12) -HU0 -HL12 #U2 #HU2 #HU02
+lapply (cpss_trans … HU1 … HU2) -U /3 width=3/
+qed-.
+
+lemma dxprs_cpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*➡*[g] U1 → ∀T2. L ⊢ T1 ▶* T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*➡*[g] U2 & L ⊢ U1 ▶* U2.
+#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12
+elim (sstas_cpss_conf … HTW1 … HT12) -T1 #W2 #HTW2 #HW12
+elim (cprs_cpss_conf … HWU1 … HW12) -W1 /3 width=3/
+qed-.
+
+lemma dxprs_cpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (dxprs_cpss_conf … HTU1 … HT12) -T1 #U2 #HTU2 #HU12
+elim (dxprs_lpss_conf … HTU2 … HL12) -HTU2 -HL12 #U #HT2U #HU2
+lapply (cpss_trans … HU12 … HU2) -U2 /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 ⊢ ➤ * break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRestStarSn $T1 $T2 }.
+
+include "basic_2/substitution/lpss.ma".
+include "basic_2/unfold/cpqs.ma".
+
+(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
+
+definition lpqs: relation lenv ≝ lpx_sn cpqs.
+
+interpretation "restricted parallel computation (local environment, sn variant)"
+ 'PRestStarSn L1 L2 = (lpqs L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpqs_inv_atom1: ∀L2. ⋆ ⊢ ➤* L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+lemma lpqs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ➤* L2 →
+ ∃∃K2,V2. K1 ⊢ ➤* K2 & K1 ⊢ V1 ➤* V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpqs_inv_atom2: ∀L1. L1 ⊢ ➤* ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpqs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➤* K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ➤* K2 & K1 ⊢ V1 ➤* V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpqs_refl: ∀L. L ⊢ ➤* L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpqs_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ➤* K2 → K1 ⊢ V1 ➤* V2 →
+ K1.ⓑ{I}V1 ⊢ ➤* K2.ⓑ{I}V2.
+/2 width=1/ qed.
+
+lemma lpqs_append: ∀K1,K2. K1 ⊢ ➤* K2 → ∀L1,L2. L1 ⊢ ➤* L2 →
+ L1 @@ K1 ⊢ ➤* L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpqs_append/ qed.
+
+lemma lpss_lpqs: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➤* L2.
+#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpqs_fwd_length: ∀L1,L2. L1 ⊢ ➤* L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpqs_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➤* L →
+ ∃∃K2,L2. K1 ⊢ ➤* K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpqs_fwd_append2: ∀L,K2,L2. L ⊢ ➤* K2 @@ L2 →
+ ∃∃K1,L1. K1 ⊢ ➤* K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lpx_sn_lpx_sn.ma".
+include "basic_2/substitution/fsupp.ma".
+include "basic_2/unfold/lpqs_ldrop.ma".
+
+(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
+
+(* Main properties on context-sensitive rest parallel computation for terms *)
+
+theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L0 #T0 #IH #L1 * [|*]
+[ #I #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
+ elim (cpqs_inv_atom1 … H1) -H1
+ [ #H destruct
+ elim (cpqs_inv_atom1 … HT2) -HT2
+ [ #H destruct //
+ | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
+ elim (lpqs_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpqs_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
+ lapply (fsupp_lref … HLK1) /3 width=9/
+ ]
+ | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
+ elim (lpqs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpqs_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+ elim (cpqs_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
+ lapply (fsupp_lref … HLK1) /3 width=9/
+ ]
+| #a #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpqs_inv_bind1 … H1) -H1 *
+ [ #V #T #HV1 #HT1 #H destruct
+ elim (cpqs_inv_bind1 … H2) -H2 *
+ [ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
+ | #T2 #HT2 #HXT2 #H1 #H2 destruct /4 width=5/
+ ]
+ | #Y1 #HTY1 #HXY1 #H11 #H12 destruct
+ elim (lift_total X2 0 1) #Y2 #HXY2
+ lapply (cpqs_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
+ ]
+| #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpqs_inv_flat1 … H1) -H1 *
+ [ #V #T #HV1 #HT1 #H destruct
+ elim (cpqs_inv_flat1 … H2) -H2 *
+ [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
+ | #HX2 #H destruct /3 width=5/
+ ]
+ | #HX1 #H destruct /3 width=5/
+]
+qed-.
+
+theorem cpqs_trans: ∀L. Transitive … (cpqs L).
+/2 width=5 by cpqs_trans_lpqs/ qed-.
+
+(* Properties on context-sensitive rest. parallel computation for terms *****)
+
+lemma lpqs_cpqs_trans: ∀L1,L2. L1 ⊢ ➤* L2 →
+ ∀T1,T2. L2 ⊢ T1 ➤* T2 → L1 ⊢ T1 ➤* T2.
+/2 width=5 by cpqs_trans_lpqs/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/fsup.ma".
+include "basic_2/relocation/ldrop_lpx_sn.ma".
+include "basic_2/unfold/cpqs_lift.ma".
+include "basic_2/unfold/lpqs.ma".
+
+(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma lpqs_ldrop_conf: dropable_sn lpqs.
+/3 width=5 by lpx_sn_deliftable_dropable, cpqs_inv_lift1/ qed-.
+
+lemma ldrop_lpqs_trans: dedropable_sn lpqs.
+/3 width=9 by lpx_sn_liftable_dedropable, cpqs_lift/ qed-.
+
+lemma lpqs_ldrop_trans_O1: dropable_dx lpqs.
+/2 width=3 by lpx_sn_dropable/ qed-.
+
+(* Properties on context-sensitive rest. parallel computation for terms *****)
+
+lemma fsup_cpqs_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➤* U2 →
+ ∃∃L,U1. L1 ⊢ ➤* L & L ⊢ T1 ➤* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [2: * ] [1,2,3,4,5: /3 width=5/ ]
+[ #L #K #U #T #d #e #HLK #HUT #He #U2 #HU2
+ elim (lift_total U2 d e) #T2 #HUT2
+ lapply (cpqs_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9/
+| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+ elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+ elim (lift_total T d e) #U #HTU
+ elim (ldrop_lpqs_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (cpqs_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/lpqs_cpqs.ma".
+
+(* SN RESTRICTED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *****************)
+
+(* Main properties on context-sensitive rest parallel computation for terms *)
+
+fact cpqs_conf_lpqs_atom_atom:
+ ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T.
+/2 width=3/ qed-.
+
+fact cpqs_conf_lpqs_atom_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T.
+#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+qed-.
+
+fact cpqs_conf_lpqs_delta_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
+ ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
+lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+qed-.
+
+fact cpqs_conf_lpqs_bind_bind:
+ ∀a,I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 →
+ ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T.
+#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
+qed-.
+
+fact cpqs_conf_lpqs_bind_zeta:
+ ∀L0,V0,T0. (
+ ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 →
+ ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T.
+#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
+qed-.
+
+fact cpqs_conf_lpqs_zeta_zeta:
+ ∀L0,V0,T0. (
+ ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T.
+#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
+elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
+elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
+qed-.
+
+fact cpqs_conf_lpqs_flat_flat:
+ ∀I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 →
+ ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T.
+#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
+qed-.
+
+fact cpqs_conf_lpqs_flat_tau:
+ ∀L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T.
+#L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
+qed-.
+
+fact cpqs_conf_lpqs_tau_tau:
+ ∀L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
+ ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
+ ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
+ ) →
+ ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
+ ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
+ ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
+#L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
+qed-.
+
+theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
+[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpqs_inv_atom1 … H1) -H1
+ elim (cpqs_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpqs_conf_lpqs_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpqs_conf_lpqs_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpqs_conf_lpqs_delta_delta/
+ ]
+| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpqs_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpqs_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpqs_conf_lpqs_bind_bind/
+ | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/
+ | /3 width=11 by cpqs_conf_lpqs_bind_zeta/
+ | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/
+ ]
+| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpqs_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ ]
+ elim (cpqs_inv_flat1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #HX2 #H2
+ ] destruct
+ [ /3 width=10 by cpqs_conf_lpqs_flat_flat/
+ | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/
+ | /3 width=8 by cpqs_conf_lpqs_flat_tau/
+ | /3 width=7 by cpqs_conf_lpqs_tau_tau/
+ ]
+]
+qed-.
+
+theorem cpqs_conf: ∀L. confluent … (cpqs L).
+/2 width=6 by cpqs_conf_lpqs/ qed-.
+
+(* Properties on context-sensitive rest. parallel computation for terms *****)
+
+lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
+ ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
+qed-.
+
+lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
+ ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lpqs_conf: confluent … lpqs.
+/3 width=6 by lpx_sn_conf, cpqs_conf_lpqs/
+qed-.
+
+theorem lpqs_trans: Transitive … lpqs.
+/3 width=5 by lpx_sn_trans, cpqs_trans_lpqs/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
+ ∃∃L2,T2. L @@ L1 ⊢ ➤* L @@ L2 & L @@ L1 ⊢ T1 ➤* T2 &
+ T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #HT1
+ @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
+| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
+ elim (cpqs_inv_bind1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ lapply (lpqs_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lpx_sn_lpx_sn.ma".
+include "basic_2/substitution/fsupp.ma".
+include "basic_2/substitution/lpss_ldrop.ma".
+include "basic_2/reduction/lpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+fact cpr_cpss_conf_lpr_lpss_atom_atom:
+ ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T.
+/2 width=3/ qed-.
+
+fact cpr_cpss_conf_lpr_lpss_atom_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_delta_atom:
+ ∀L0,i. (
+ ∀L,T.⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_delta_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
+ ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
+lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_bind_bind:
+ ∀a,I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_bind_zeta:
+ ∀L0,V0,T0. (
+ ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T.
+#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_flat:
+ ∀I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_tau:
+ ∀L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T.
+#L0 #V0 #T0 #IH #T1 #HT01
+#V2 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_beta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T. ⦃L0,ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
+elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+fact cpr_cpss_conf_lpr_lpss_flat_theta:
+ ∀a,L0,V0,W0,T0. (
+ ∀L,T. ⦃L0,ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
+ ) →
+ ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+ ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 →
+ ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
+#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1
+elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
+/4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *)
+qed-.
+
+lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
+[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1 … H1) -H1
+ elim (cpss_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/
+ ]
+| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ] destruct
+ [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/
+ | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/
+ ]
+| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
+ | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ] destruct
+ [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/
+ | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/
+ | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/
+ | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_subst1 *)
+(* Basic_1: was: pr2_subst1 *)
+lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L).
+/2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-.
+
+lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/
+qed-.
+
+lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+qed-.
+
+(* Basic_1: includes: pr0_subst1_fwd *)
+lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 →
+ ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/
+qed-.
+
+(* Properties on sn parallel substitution on local environments *************)
+
+lemma lpr_lpss_conf: confluent2 … lpr lpss.
+/3 width=6 by lpx_sn_conf, cpr_cpss_conf_lpr_lpss/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/lpr_lpss.ma".
+include "basic_2/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Properties on sn parallel substitution on local environments *************)
+
+lemma lprs_lpss_conf: confluent2 … lprs lpss.
+/3 width=3 by TC_strip1, lpr_lpss_conf/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 ⊢ ▶ * break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarSn $T1 $T2 }.
+
+include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/substitution/cpss.ma".
+
+(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
+
+(* Basic_1: includes: csubst1_bind *)
+definition lpss: relation lenv ≝ lpx_sn cpss.
+
+interpretation "parallel substitution (local environment, sn variant)"
+ 'PSubstStarSn L1 L2 = (lpss L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpss_inv_atom1: ∀L2. ⋆ ⊢ ▶* L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+lemma lpss_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* L2 →
+ ∃∃K2,V2. K1 ⊢ ▶* K2 & K1 ⊢ V1 ▶* V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpss_inv_atom2: ∀L1. L1 ⊢ ▶* ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpss_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ▶* K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ▶* K2 & K1 ⊢ V1 ▶* V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was by definition: csubst1_refl *)
+lemma lpss_refl: ∀L. L ⊢ ▶* L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpss_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ▶* K2 → K1 ⊢ V1 ▶* V2 →
+ K1.ⓑ{I}V1 ⊢ ▶* K2.ⓑ{I}V2.
+/2 width=1/ qed.
+
+lemma lpss_append: ∀K1,K2. K1 ⊢ ▶* K2 → ∀L1,L2. L1 ⊢ ▶* L2 →
+ L1 @@ K1 ⊢ ▶* L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpss_append/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpss_fwd_length: ∀L1,L2. L1 ⊢ ▶* L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpss_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ▶* L →
+ ∃∃K2,L2. K1 ⊢ ▶* K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpss_fwd_append2: ∀L,K2,L2. L ⊢ ▶* K2 @@ L2 →
+ ∃∃K1,L1. K1 ⊢ ▶* K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
+
+(* Basic_1: removed theorems 28:
+ csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
+ csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
+ csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
+ csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
+ csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
+ csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
+ csubst1_head csubst1_flat csubst1_gen_head
+ csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
+ fsubst0_gen_base
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lpx_sn_lpx_sn.ma".
+include "basic_2/substitution/fsupp.ma".
+include "basic_2/substitution/lpss_ldrop.ma".
+
+(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
+
+(* Main properties on context-sensitive parallel substitution for terms *****)
+
+theorem cpss_trans_lpss: lpx_sn_transitive cpss cpss.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L0 #T0 #IH #L1 * [|*]
+[ #I #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
+ elim (cpss_inv_atom1 … H1) -H1
+ [ #H destruct
+ elim (cpss_inv_atom1 … HT2) -HT2
+ [ #H destruct //
+ | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
+ elim (lpss_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpss_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
+ lapply (fsupp_lref … HLK1) /3 width=9/
+ ]
+ | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+ elim (cpss_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
+ lapply (fsupp_lref … HLK1) /3 width=9/
+ ]
+| #a #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpss_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+ elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
+| #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpss_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+ elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
+]
+qed-.
+
+(* Basic_1: was only: subst1_trans *)
+theorem cpss_trans: ∀L. Transitive … (cpss L).
+/2 width=5 by cpss_trans_lpss/ qed-.
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+(* Basic_1: was only: subst1_subst1 *)
+lemma lpss_cpss_trans: ∀L1,L2. L1 ⊢ ▶* L2 →
+ ∀T1,T2. L2 ⊢ T1 ▶* T2 → L1 ⊢ T1 ▶* T2.
+/2 width=5 by cpss_trans_lpss/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/fsup.ma".
+include "basic_2/relocation/ldrop_lpx_sn.ma".
+include "basic_2/substitution/cpss_lift.ma".
+include "basic_2/substitution/lpss.ma".
+
+(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma lpss_ldrop_conf: dropable_sn lpss.
+/3 width=5 by lpx_sn_deliftable_dropable, cpss_inv_lift1/ qed-.
+
+lemma ldrop_lpss_trans: dedropable_sn lpss.
+/3 width=9 by lpx_sn_liftable_dedropable, cpss_lift/ qed-.
+
+lemma lpss_ldrop_trans_O1: dropable_dx lpss.
+/2 width=3 by lpx_sn_dropable/ qed-.
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+lemma fsup_cpss_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶* U2 →
+ ∃∃L,U1. L1 ⊢ ▶* L & L ⊢ T1 ▶* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [2: * ] [1,2,3,4,5: /3 width=5/ ]
+[ #L #K #U #T #d #e #HLK #HUT #He #U2 #HU2
+ elim (lift_total U2 d e) #T2 #HUT2
+ lapply (cpss_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9/
+| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+ elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+ elim (lift_total T d e) #U #HTU
+ elim (ldrop_lpss_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (cpss_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lpss_cpss.ma".
+
+(* SN PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ***************************)
+
+(* Main properties on context-sensitive parallel substitution for terms *****)
+
+fact cpss_conf_lpss_atom_atom:
+ ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ▶* T.
+/2 width=3/ qed-.
+
+fact cpss_conf_lpss_atom_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ▶* T.
+#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpss_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+qed-.
+
+fact cpss_conf_lpss_delta_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ▶* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
+ ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ▶* T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpss_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
+lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+qed-.
+
+fact cpss_conf_lpss_bind_bind:
+ ∀a,I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ▶* T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ▶* T.
+#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
+qed-.
+
+fact cpss_conf_lpss_flat_flat:
+ ∀I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0 ⊢ T0 ▶* T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ▶* T.
+#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
+qed-.
+
+theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
+[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_atom1 … H1) -H1
+ elim (cpss_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpss_conf_lpss_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpss_conf_lpss_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpss_conf_lpss_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpss_conf_lpss_delta_delta/
+ ]
+| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_bind1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct
+ elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
+ /3 width=10 by cpss_conf_lpss_bind_bind/
+| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_flat1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct
+ elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
+ /3 width=10 by cpss_conf_lpss_flat_flat/
+]
+qed-.
+
+(* Basic_1: was only: subst1_confluence_eq *)
+theorem cpss_conf: ∀L. confluent … (cpss L).
+/2 width=6 by cpss_conf_lpss/ qed-.
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+(* Basic_1: was only: subst1_subst1_back *)
+lemma lpss_cpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T0 ▶* T & L1 ⊢ T1 ▶* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpss_conf_lpss … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
+qed-.
+
+lemma lpss_cpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ▶* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpss_conf_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lpss_conf: confluent … lpss.
+/3 width=6 by lpx_sn_conf, cpss_conf_lpss/
+qed-.
+
+theorem lpss_trans: Transitive … lpss.
+/3 width=5 by lpx_sn_trans, cpss_trans_lpss/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
+ ∃∃L2,T2. L @@ L1 ⊢ ▶* L @@ L2 & L @@ L1 ⊢ T1 ▶* T2 &
+ T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #HT1
+ @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
+| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ lapply (lpss_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L1 ⓝ ⊑ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqT $L1 $L2 }.
+
+include "basic_2/relocation/ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
+
+inductive lsubr: relation lenv ≝
+| lsubr_sort: ∀L. lsubr L (⋆)
+| lsubr_abbr: ∀L1,L2,V. lsubr L1 L2 → lsubr (L1. ⓓV) (L2.ⓓV)
+| lsubr_abst: ∀I,L1,L2,V1,V2. lsubr L1 L2 → lsubr (L1. ⓑ{I}V1) (L2. ⓛV2)
+.
+
+interpretation
+ "local environment refinement (substitution)"
+ 'CrSubEq L1 L2 = (lsubr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_bind: ∀I,L1,L2,V. L1 ⊑ L2 → L1. ⓑ{I} V ⊑ L2.ⓑ{I} V.
+* /2 width=1/ qed.
+
+lemma lsubr_abbr: ∀I,L1,L2,V. L1 ⊑ L2 → L1. ⓓV ⊑ L2. ⓑ{I}V.
+* /2 width=1/ qed.
+
+lemma lsubr_refl: ∀L. L ⊑ L.
+#L elim L -L // /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #L1 #L2 #V #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_atom1: ∀L2. ⋆ ⊑ L2 → L2 = ⋆.
+/2 width=3 by lsubr_inv_atom1_aux/ qed-.
+
+fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
+ ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
+#L1 #L2 * -L1 -L2
+[ #L #K2 #W #H destruct
+| #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3/
+| #I #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⊑ K2.ⓓW →
+ ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
+/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
+
+fact lsubr_inv_abst2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W2. L2 = K2.ⓛW2 →
+ ∃∃I,K1,W1. K1 ⊑ K2 & L1 = K1.ⓑ{I}W1.
+#L1 #L2 * -L1 -L2
+[ #L #K2 #W2 #H destruct
+| #L1 #L2 #V #_ #K2 #W2 #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #K2 #W2 #H destruct /2 width=5/
+]
+qed-.
+
+lemma lsubr_inv_abst2: ∀L1,K2,W2. L1 ⊑ K2.ⓛW2 →
+ ∃∃I,K1,W1. K1 ⊑ K2 & L1 = K1.ⓑ{I}W1.
+/2 width=4 by lsubr_inv_abst2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubr_fwd_length: ∀L1,L2. L1 ⊑ L2 → |L2| ≤ |L1|.
+#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma lsubr_fwd_ldrop2_abbr: ∀L1,L2. L1 ⊑ L2 →
+ ∀K2,W,i. ⇩[0, i] L2 ≡ K2. ⓓW →
+ ∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1. ⓓW.
+#L1 #L2 #H elim H -L1 -L2
+[ #L #K2 #W #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #L2 #V #HL12 #IHL12 #K2 #W #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+ [ /2 width=3/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 /3 width=3/
+ ]
+| #I #L1 #L2 #V1 #V2 #_ #IHL12 #K2 #W #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct
+ elim (IHL12 … HLK2) -IHL12 -HLK2 /3 width=3/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+lemma lsubr_fwd_lsubr: ∀L1,L2. L1 ⓝ⊑ L2 → L1 ⊑ L2.
+#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma cpqs_cpr: ∀L,T1,T2. L ⊢ T1 ➤* T2 → L ⊢ T1 ➡ T2.
+#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
+qed.
+
+lemma cpss_cpr: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡ T2.
+/3 width=1/ qed.
+
+lemma lpqs_lpr: ∀L1,L2. L1 ⊢ ➤* L2 → L1 ⊢ ➡ L2.
+#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+qed.
+
+lemma lpss_lpr: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➡ L2.
+/3 width=1/ qed.
+
+lemma cpss_fwd_cir_eq: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ 𝐈⦃T1⦄ → T2 = T1.
+/3 width=3 by cpr_fwd_cir, cpss_cpr/ qed-.
+
+lemma cpss_cprs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡* T2.
+/3 width=1/ qed.
+
+lemma cprs_cpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ▶* T2 → L ⊢ T1 ➡* T2.
+/3 width=3/ qed-.
+
+lemma cpss_cprs_trans: ∀L,T1,T. L ⊢ T1 ▶* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
+/3 width=3/ qed-.
+
+lemma cpcs_cpss_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ▶* T2 → L ⊢ T1 ⬌* T2.
+#L #T1 #T #HT1 #T2 #HT2
+@(cpcs_cpr_strap1 … HT1) -T1 /2 width=3/
+qed-.
+
+lemma cpcs_cpss_strap2: ∀L,T1,T. L ⊢ T1 ▶* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+#L #T1 #T #HT1 #T2 #HT2
+@(cpcs_cpr_strap2 … HT2) -T2 /2 width=3/
+qed-.
+
+lemma cpcs_cpss_conf: ∀L,T,T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+#L #T #T1 #HT1 #T2 #HT2
+@(cpcs_cpr_conf … HT2) -T2 /2 width=3/
+qed-.
+
+lemma cpcs_cpss_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ▶* T → L ⊢ T1 ⬌* T2.
+#L #T1 #T #HT1 #T2 #HT2
+@(cpcs_cpr_div … HT1) -T1 /2 width=3/
+qed-.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lsubr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
+
+(* Auxiliary inversion lemmas ***********************************************)
+
+fact lsubr_inv_abbr1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓓW →
+ ∨∨ L2 = ⋆
+ | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
+ | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
+#L1 #L2 * -L1 -L2
+[ #L #K1 #W #H destruct /2 width=1/
+| #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
+| #I #L1 #L2 #V1 #V2 #HL12 #K1 #W #H destruct /3 width=4/
+]
+qed-.
+
+lemma lsubr_inv_abbr1: ∀K1,L2,W. K1.ⓓW ⊑ L2 →
+ ∨∨ L2 = ⋆
+ | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
+ | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
+/2 width=3 by lsubr_inv_abbr1_aux/ qed-.
+
+fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W1. L1 = K1.ⓛW1 →
+ L2 = ⋆ ∨
+ ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
+#L1 #L2 * -L1 -L2
+[ #L #K1 #W1 #H destruct /2 width=1/
+| #L1 #L2 #V #_ #K1 #W1 #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #K1 #W1 #H destruct /3 width=4/
+]
+qed-.
+
+lemma lsubr_inv_abst1: ∀K1,L2,W1. K1.ⓛW1 ⊑ L2 →
+ L2 = ⋆ ∨
+ ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
+/2 width=4 by lsubr_inv_abst1_aux/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lsubr_trans: Transitive … lsubr.
+#L1 #L #H elim H -L1 -L
+[ #L1 #X #H
+ lapply (lsubr_inv_atom1 … H) -H //
+| #L1 #L #V #_ #IHL1 #X #H
+ elim (lsubr_inv_abbr1 … H) -H // *
+ #L2 [2: #V2 ] #HL2 #H destruct /3 width=1/
+| #I #L1 #L #V1 #V #_ #IHL1 #X #H
+ elim (lsubr_inv_abst1 … H) -H // *
+ #L2 #V2 #HL2 #H destruct /3 width=1/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/dxprs_lpss.ma".
+include "basic_2/equivalence/cpcs_lpss.ma".
+include "basic_2/dynamic/snv_lift.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties about sn parallel substitution for local environments *********)
+
+lemma snv_cpss_lpss_conf: ∀h,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] → ∀T2. L1 ⊢ T1 ▶* T2 →
+ ∀L2. L1 ⊢ ▶* L2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+#h #g #L1 #T1 #H elim H -L1 -T1
+[ #L1 #k #X #H #L2 #_
+ >(cpss_inv_sort1 … H) -X //
+| #I #L1 #K1 #V1 #i #HLK1 #_ #IHV1 #W2 #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (IHV1 … HV12 … HK12) -IHV1 -HV12 -HK12 /2 width=5/
+ | * #K0 #V0 #V2 #HLK0 #HV12 #HVW2
+ lapply (ldrop_mono … HLK0 … HLK1) -HLK0 #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V #HLK2
+ lapply (IHV1 … HV12 … HK12) -IHV1 -HV12 -HK12 /2 width=7/
+ ]
+| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IHV1 … HV12 … HL12) -IHV1 #HV2
+ lapply (IHT1 … HT12 (L2.ⓑ{I}V2) ?) -IHT1 -HT12 /2 width=1/
+| #a #L1 #V1 #W1 #W0 #T1 #U1 #l #_ #_ #HVW1 #HW10 #HTU1 #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IHV1 … HV12 … HL12) -IHV1 #HV2
+ lapply (IHT1 … HT12 … HL12) -IHT1 #HT2
+ elim (ssta_cpss_lpss_conf … HVW1 … HV12 … HL12) -V1 #W2 #HVW2 #HW12
+ elim (dxprs_cpss_lpss_conf … HTU1 … HT12 … HL12) -T1 #X #HTU2 #H
+ elim (cpss_inv_bind1 … H) -H #W #U2 #HW0 #_ #H -U1 destruct
+ elim (cprs_cpss2_lpss_conf_dx … HW10 … HW12 … HW0 … HL12) -L1 -W1 -W0 #W0 #HW20 #HW0
+ lapply (dxprs_strap1 … (ⓛ{a}W0.U2) HTU2 ?) -HTU2 /3 width=3/ -HW0 /2 width=8/
+| #L1 #W1 #T1 #U1 #l #_ #_ #HTU1 #HUW1 #IHW1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (IHW1 … HW12 … HL12) -IHW1 #HW2
+ lapply (IHT1 … HT12 … HL12) -IHT1 #HT2
+ elim (ssta_cpss_lpss_conf … HTU1 … HT12 … HL12) -T1 #U2 #HTU2 #HU12
+ lapply (cpcs_cpss2_lpss_conf … HUW1 … HU12 … HW12 … HL12) -L1 -W1 -U1 /2 width=4/
+]
+qed-.
+
+lemma snv_lpss_conf: ∀h,g,L1,T. ⦃h, L1⦄ ⊢ T ¡[g] →
+ ∀L2. L1 ⊢ ▶* L2 → ⦃h, L2⦄ ⊢ T ¡[g].
+#h #g #L1 #T #HT #L2 #HL12
+@(snv_cpss_lpss_conf … HT … HL12) //
+qed-.
+
+lemma snv_cpss_conf: ∀h,g,L,T1. ⦃h, L⦄ ⊢ T1 ¡[g] →
+ ∀T2. L ⊢ T1 ▶* T2 → ⦃h, L⦄ ⊢ T2 ¡[g].
+#h #g #L #T1 #HT1 #T2 #HT12
+@(snv_cpss_lpss_conf … HT1 … HT12) //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lpss_ldrop.ma".
+include "basic_2/static/ssta_lift.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about sn parallel substitution ********************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma ssta_cpss_lpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
+[ #L1 #k1 #l1 #Hkl1 #X #H
+ >(cpss_inv_sort1 … H) -H /3 width=3/
+| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #_ #HWU1 #IHVW1 #X #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ elim (IHVW1 … HV12 … HK12) -IHVW1 -HV12 -HK12 #W2 #HVW2 #HW12
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
+ | * #Y #Z #V2 #H #HV12 #HV2
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #Z #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V0 #HLK2
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ elim (IHVW1 … HV12 … HK12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (ssta_lift … HVW2 … HLK2 … HV2 … HWU2) -HVW2 -HLK2 -HV2
+ lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 -HWU2 /3 width=3/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #_ #HWU1 #IHWV1 #X #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H [ | -IHWV1 -HWU1 -HL12 ]
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ elim (IHWV1 … HW12 … HK12) -IHWV1 -HK12 #V2 #HWV2 #_
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
+ | * #K2 #V2 #W2 #HLK2 #_ #_
+ lapply (ldrop_mono … HLK2 … HLK1) -HLK1 -HLK2 #H destruct
+ ]
+| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12 (L2.ⓑ{I}V2)) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
+| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=5/
+| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=3/
+]
+qed-.
+
+lemma ssta_cpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2. L ⊢ T1 ▶* T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* U2.
+/2 width=3 by ssta_cpss_lpss_conf/ qed-.
+
+lemma ssta_lpss_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
+ ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
+/2 width=3 by ssta_cpss_lpss_conf/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta_lpss.ma".
+include "basic_2/unfold/sstas.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
+
+(* Properties about sn parallel substitution for local environments *********)
+
+lemma sstas_cpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
+#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12
+elim (ssta_cpss_lpss_conf … HTU0 … HT0 … HL12) -HTU0 -HT0 #U #HTU #HU0
+elim (IHU01 … HU0 … HL12) -IHU01 -U0 -HL12 /3 width=4/
+qed-.
+
+lemma sstas_cpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2. L ⊢ T1 ▶* T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* U2.
+/2 width=3 by sstas_cpss_lpss_conf/ qed-.
+
+lemma sstas_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ▶* U2.
+/2 width=3 by sstas_cpss_lpss_conf/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( h ⊢ break term 46 L1 : ⊑ [ ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'StratifiedCrSubEqN $h $L1 $L2 }.
+
+include "basic_2/dynamic/snta.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
+
+(* Note: may not be transitive *)
+inductive lsubsn (h:sh): relation lenv ≝
+| lsubsn_atom: lsubsn h (⋆) (⋆)
+| lsubsn_pair: ∀I,L1,L2,W. lsubsn h L1 L2 →
+ lsubsn h (L1. ⓑ{I} W) (L2. ⓑ{I} W)
+| lsubsn_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V :[l+1] W → ⦃h, L2⦄ ⊢ V :[l+1] W →
+ lsubsn h L1 L2 → lsubsn h (L1. ⓓV) (L2. ⓛW)
+.
+
+interpretation
+ "local environment refinement (stratified native type assigment)"
+ 'StratifiedCrSubEqN h L1 L2 = (lsubsn h L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubsn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L1 = ⋆ → L2 = ⋆.
+#h #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsubsn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑[] L2 → L2 = ⋆.
+/2 width=5/ qed-.
+
+fact lsubsn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀I,K1,V. L1 = K1. ⓑ{I} V →
+ (∃∃K2. h ⊢ K1 :⊑[] K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
+ h ⊢ K1 :⊑[] K2 & L2 = K2. ⓛW & I = Abbr.
+#h #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
+]
+qed.
+
+lemma lsubsn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑[] L2 →
+ (∃∃K2. h ⊢ K1 :⊑[] K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
+ h ⊢ K1 :⊑[] K2 & L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsubsn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L2 = ⋆ → L1 = ⋆.
+#h #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsubsn_inv_atom2: ∀h,L1. h ⊢ L1 :⊑[] ⋆ → L1 = ⋆.
+/2 width=5/ qed-.
+
+fact lsubsn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀I,K2,W. L2 = K2. ⓑ{I} W →
+ (∃∃K1. h ⊢ K1 :⊑[] K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
+ h ⊢ K1 :⊑[] K2 & L1 = K1. ⓓV & I = Abst.
+#h #L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
+| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
+]
+qed.
+
+lemma lsubsn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑[] K2. ⓑ{I} W →
+ (∃∃K1. h ⊢ K1 :⊑[] K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V :[l+1] W & ⦃h, K2⦄ ⊢ V :[l+1] W &
+ h ⊢ K1 :⊑[] K2 & L1 = K1. ⓓV & I = Abst.
+/2 width=3/ qed-.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma lsubsn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L1 ≼[0, |L1|] L2.
+#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma lsubsn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 → L1 ≼[0, |L2|] L2.
+#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsubsn_refl: ∀h,L. h ⊢ L :⊑[] L.
+#h #L elim L -L // /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/dynamic/lsubsn.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+lemma cpr_lsubsn_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
+/3 width=5 by lsubsn_fwd_lsubs2, cpr_lsubs_trans/ qed-.
+
+lemma cprs_lsubsn_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+/3 width=5 by lsubsn_fwd_lsubs2, cprs_lsubs_trans/ qed-.
+
+lemma cpcs_lsubsn_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
+/3 width=5 by lsubsn_fwd_lsubs2, cpcs_lsubs_trans/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/lsubsn.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsn_ldrop_O1_conf: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀K1,e. ⇩[0, e] L1 ≡ K1 →
+ ∃∃K2. h ⊢ K1 :⊑[] K2 & ⇩[0, e] L2 ≡ K2.
+#h #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+]
+qed.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsn_ldrop_O1_trans: ∀h,L1,L2. h ⊢ L1 :⊑[] L2 →
+ ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. h ⊢ K1 :⊑[] K2 & ⇩[0, e] L1 ≡ K1.
+#h #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/snta_snta.ma".
+include "basic_2/dynamic/lsubsn_ldrop.ma".
+include "basic_2/dynamic/lsubsn_cpcs.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)
+
+(* Properties concerning stratified native type assignment ******************)
+
+(* Note: the corresponding confluence property does not hold *)
+lemma lsubsn_snta_trans: ∀h,L2,T,U,l. ⦃h, L2⦄ ⊢ T :[l] U →
+ ∀L1. h ⊢ L1 :⊑[] L2 → ⦃h, L1⦄ ⊢ T :[l] U.
+#h #L2 #T #U #l #H elim H -L2 -T -U -l
+[ //
+| #L2 #K2 #V2 #W2 #U2 #i #l #HLK2 #_ #WU2 #IHVW2 #L1 #HL12
+ elim (lsubsn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubsn_inv_pair2 … H) -H * #K1
+ [ #HK12 #H destruct /3 width=6/
+ | #V1 #l0 #_ #_ #_ #_ #H destruct
+ ]
+| #L2 #K2 #W2 #V2 #U2 #i #l #HLK2 #HWV2 #HWU2 #IHWV2 #L1 #HL12
+ elim (lsubsn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubsn_inv_pair2 … H) -H * #K1 [ -HWV2 | -IHWV2 ]
+ [ #HK12 #H destruct /3 width=6/
+ | #V1 #l0 #H1 #H2 #_ #H #_ destruct
+ elim (snta_fwd_correct … H2) -H2 #V #H
+ elim (snta_mono … HWV2 … H) -HWV2 -H /2 width=6/
+ ]
+| /4 width=3/
+| /3 width=2/
+| /3 width=2/
+| /3 width=1/
+| #L2 #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #IHUV2 #L1 #HL12
+ lapply (cpcs_lsubsn_trans … HL12 … HU12) -HU12 /3 width=3/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 : * break [ l ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'NativeTypeStar $h $l $L $T1 $T2 }.
+
+notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 : break [ l ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StratifiedNativeType $h $l $L $T1 $T2 }.
+
+include "basic_2/static/sh.ma".
+include "basic_2/equivalence/cpcs.ma".
+
+(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
+
+inductive snta (h:sh): nat → lenv → relation term ≝
+| snta_sort: ∀L,k. snta h 0 L (⋆k) (⋆(next h k))
+| snta_ldef: ∀L,K,V,W,U,i,l. ⇩[0, i] L ≡ K. ⓓV → snta h l K V W →
+ ⇧[0, i + 1] W ≡ U → snta h l L (#i) U
+| snta_ldec: ∀L,K,W,V,U,i,l. ⇩[0, i] L ≡ K. ⓛW → snta h l K W V →
+ ⇧[0, i + 1] W ≡ U → snta h (l+1) L (#i) U
+| snta_bind: ∀I,L,V,W,T,U,l1,l2. snta h l1 L V W → snta h l2 (L. ⓑ{I} V) T U →
+ snta h l2 L (ⓑ{I}V.T) (ⓑ{I}V.U)
+| snta_appl: ∀L,V,W1,W2,T,U,l1,l2. snta h (l1+1) L V W2 →
+ snta h l2 L (ⓛW1.T) (ⓛW2.U) →
+ snta h l2 L (ⓐV.ⓛW1.T) (ⓐV.ⓛW2.U)
+| snta_pure: ∀L,V,T,U,W,l. snta h (l+1) L T U → snta h l L (ⓐV.U) W →
+ snta h (l+1) L (ⓐV.T) (ⓐV.U)
+| snta_cast: ∀L,T,U,W,l1,l2. snta h l2 L T U → snta h l1 L U W →
+ snta h l2 L (ⓝU.T) U
+| snta_conv: ∀L,T,U1,U2,V2,l. snta h l L T U1 → L ⊢ U1 ⬌* U2 →
+ snta h (l-1) L U2 V2 → snta h l L T U2
+.
+
+interpretation "stratified native type assignment (term)"
+ 'StratifiedNativeType h l L T U = (snta h l L T U).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/dynamic/snta.ma".
+
+(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact snta_inv_sort1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀k0. T = ⋆k0 →
+ l = 0 ∧ L ⊢ ⋆(next h k0) ⬌* U.
+#h #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #k0 #H destruct /2 width=1/
+| #L #K #V #W #U #i #l #_ #_ #_ #_ #k0 #H destruct
+| #L #K #W #V #U #i #l #_ #_ #_ #_ #k0 #H destruct
+| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #k0 #H destruct
+| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #k0 #H destruct
+| #L #V #T #U #W #l #_ #_ #_ #_ #k0 #H destruct
+| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #k0 #H destruct
+| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #k0 #H destruct
+ elim (IHTU1 ??) -IHTU1 [3: // |2: skip ] #H #Hk0
+ lapply (cpcs_trans … Hk0 … HU12) -U1 /2 width=1/
+]
+qed.
+
+lemma snta_inv_sort1: ∀h,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k :[l] U →
+ l = 0 ∧ L ⊢ ⋆(next h k) ⬌* U.
+/2 width=4/ qed-.
+
+fact snta_inv_lref1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀j. T = #j →
+ (∃∃K,V,W,U0. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V :[l] W &
+ ⇧[0, j + 1] W ≡ U0 & L ⊢ U0 ⬌* U
+ ) ∨
+ (∃∃K,W,V,U0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W :[l-1] V &
+ ⇧[0, j + 1] W ≡ U0 & l > 0 & L ⊢ U0 ⬌* U
+ ).
+#h #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #j #H destruct
+| #L #K #V #W #U #i #l #HLK #HVW #HWU #_ #j #H destruct /3 width=8/
+| #L #K #W #V #U #i #l #HLK #HWV #HWU #_ #j #H destruct /3 width=8/
+| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct
+| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct
+| #L #V #T #U #W #l #_ #_ #_ #_ #j #H destruct
+| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #j #H destruct
+| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #j #H destruct
+ elim (IHTU1 ??) -IHTU1 [4: // |2: skip ] * #K #V #W #U0 #HLK #HVW #HWU0 [2: #H ] #HU01
+ lapply (cpcs_trans … HU01 … HU12) -U1 /3 width=8/
+]
+qed.
+
+lemma snta_inv_lref1: ∀h,L,U,i,l. ⦃h, L⦄ ⊢ #i :[l] U →
+ (∃∃K,V,W,U0. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V :[l] W &
+ ⇧[0, i + 1] W ≡ U0 & L ⊢ U0 ⬌* U
+ ) ∨
+ (∃∃K,W,V,U0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W :[l-1] V &
+ ⇧[0, i + 1] W ≡ U0 & l > 0 & L ⊢ U0 ⬌* U
+ ).
+/2 width=3/ qed-.
+
+fact snta_inv_bind1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀J,X,Y. T = ⓑ{J}Y.X →
+ ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 &
+ ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 &
+ L ⊢ ⓑ{J}Y.Z2 ⬌* U.
+#h #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #J #X #Y #H destruct
+| #L #K #V #W #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct
+| #L #K #W #V #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct
+| #I #L #V #W #T #U #l1 #l2 #HVW #HTU #_ #_ #J #X #Y #H destruct /2 width=3/
+| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct
+| #L #V #T #U #W #l #_ #_ #_ #_ #J #X #Y #H destruct
+| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct
+| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #J #X #Y #H destruct
+ elim (IHTU1 ????) -IHTU1 [5: // |2,3,4: skip ] #Z1 #Z2 #l0 #HZ1 #HZ2 #HU1
+ lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/
+]
+qed.
+
+lemma snta_inv_bind1: ∀h,J,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{J}Y.X :[l] U →
+ ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 &
+ L ⊢ ⓑ{J}Y.Z2 ⬌* U.
+/2 width=3/ qed-.
+
+fact snta_inv_cast1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀X,Y. T = ⓝY.X →
+ ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U.
+#h #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #X #Y #H destruct
+| #L #K #V #W #U #i #l #_ #_ #_ #_ #X #Y #H destruct
+| #L #K #W #V #U #i #l #_ #_ #_ #_ #X #Y #H destruct
+| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
+| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
+| #L #V #T #U #W #l #_ #_ #_ #_ #X #Y #H destruct
+| #L #T #U #W #l1 #l2 #HTU #_ #_ #_ #X #Y #H destruct /2 width=1/
+| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
+ elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #HXY #HU1
+ lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=1/
+]
+qed.
+
+lemma snta_inv_cast1: ∀h,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X :[l] U →
+ ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U.
+/2 width=3/ qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma snta_lift: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 →
+ ∀L2,d,e. ⇩[d, e] L2 ≡ L1 →
+ ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 →
+ ⦃h, L2⦄ ⊢ T2 :[l] U2.
+#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
+[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+ >(lift_inv_sort1 … H1) -X1
+ >(lift_inv_sort1 … H2) -X2 //
+| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2
+ elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=8/
+ | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
+ lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+ ]
+| #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // <minus_plus #W #HW1 #HWU2
+ elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
+ elim (lift_total V1 (d-i-1) e) /3 width=8/
+ | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
+ lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+ ]
+| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+ elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+ elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+ lapply (lift_mono … H1 … HV12) -H1 #H destruct
+ elim (lift_total W1 d e) /4 width=6/
+| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #V2 #X #HV12 #H1 #H destruct
+ elim (lift_inv_bind1 … H1) -H1 #W21 #T2 #HW121 #HT12 #H destruct
+ elim (lift_inv_flat1 … H2) -H2 #Y2 #X #HY #H2 #H destruct
+ elim (lift_inv_bind1 … H2) -H2 #W22 #U2 #HW122 #HU12 #H destruct
+ lapply (lift_mono … HY … HV12) -HY #H destruct /4 width=6/
+| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+ elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+ lapply (lift_mono … H1 … HV12) -H1 #H destruct
+ elim (lift_total W1 d e) #W2 #HW12 /4 width=6/
+| #L1 #T1 #U1 #W1 #l1 #l2 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL21 #X #H #U2 #HU12
+ elim (lift_inv_flat1 … H) -H #X2 #T2 #HUX2 #HT12 #H destruct
+ lapply (lift_mono … HUX2 … HU12) -HUX2 #H destruct
+ elim (lift_total W1 d e) /3 width=6/
+| #L1 #T1 #U11 #U12 #V12 #l #_ #HU112 #_ #IHTU11 #IHUV12 #L2 #d #e #HL21 #U1 #HTU1 #U2 #HU12
+ elim (lift_total U11 d e) #U #HU11
+ elim (lift_total V12 d e) #V22 #HV122
+ lapply (cpcs_lift … HL21 … HU11 … HU12 HU112) -HU112 /3 width=6/
+]
+qed.
+
+(* Advanced forvard lemmas **************************************************)
+
+fact snta_fwd_pure1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀X,Y. T = ⓐY.X →
+ ∃∃V,W,l0. ⦃h, L⦄ ⊢ Y :[l0+1] W & ⦃h, L⦄ ⊢ X :[l] V &
+ L ⊢ ⓐY.V ⬌* U.
+#h #L #T #U #l #H elim H -L -T -U -l
+[ #L #k #X #Y #H destruct
+| #L #K #V #W #U #i #l #_ #_ #_ #_ #X #Y #H destruct
+| #L #K #W #V #U #i #l #_ #_ #_ #_ #X #Y #H destruct
+| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
+| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #HTU #_ #_ #X #Y #H destruct /2 width=3/
+| #L #V #T #U #W #l #HTU #_ #_ #IHU #X #Y #H destruct
+ elim (IHU U Y ?) -IHU // /3 width=3/
+| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct
+| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
+ elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #V #W #l0 #HYW #HXV #HU1
+ lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/
+]
+qed.
+
+lemma snta_fwd_pure1: ∀h,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓐY.X :[l] U →
+ ∃∃V,W,l0. ⦃h, L⦄ ⊢ Y :[l0+1] W & ⦃h, L⦄ ⊢ X :[l] V &
+ L ⊢ ⓐY.V ⬌* U.
+/2 width=3/ qed-.
+
+lemma snta_fwd_correct: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U →
+ ∃T0. ⦃h, L⦄ ⊢ U :[l-1] T0.
+#h #L #T #U #l #H elim H -L -T -U -l
+[ /2 width=2/
+| #L #K #V #W #W0 #i #l #HLK #_ #HW0 * #V0 #HWV0
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+ elim (lift_total V0 0 (i+1)) /3 width=10/
+| #L #K #W #V #V0 #i #l #HLK #HWV #HWV0 #_
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+ elim (lift_total V 0 (i+1)) /3 width=10/
+| #I #L #V #W #T #U #l1 #l2 #HVW #_ #_ * /3 width=3/
+| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #_ #_ * #X #H
+ elim (snta_inv_bind1 … H) -H /4 width=5/
+| /3 width=2/
+| /2 width=2/
+| /2 width=2/
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma snta_cast_short: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ⦃h, L⦄ ⊢ ⓝU.T :[l] U.
+#h #L #T #U #l #HTU
+elim (snta_fwd_correct … HTU) /2 width=3/
+qed.
+
+lemma snta_typecheck: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U →
+ ∃T0. ⦃h, L⦄ ⊢ ⓝU.T :[l] T0.
+/3 width=2/ qed.
+
+lemma snta_cast_old: ∀h,L,W,T,U,l.
+ ⦃h, L⦄ ⊢ T :[l] U → ⦃h, L⦄ ⊢ U :[l-1] W → ⦃h, L⦄ ⊢ ⓝU.T :[l] ⓝW.U.
+#h #L #W #T #U #l #HTU #HUW
+@(snta_conv … U) /2 width=2/ /3 width=1/ (**) (* /4 width=3/ is a bit slow *)
+qed.
+
+lemma snta_appl_old: ∀h,L,V,W,T,U,l1,l2.
+ ⦃h, L⦄ ⊢ V :[l1+1] W → ⦃h, L⦄ ⊢ T :[l2+1] ⓛW.U →
+ ⦃h, L⦄ ⊢ ⓐV.T :[l2+1] ⓐV.ⓛW.U.
+#h #L #V #W #T #U #l1 #l2 #HVW #HTU
+elim (snta_fwd_correct … HTU) #X #H
+elim (snta_inv_bind1 … H) -H /4 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/snta_ltpss.ma".
+include "basic_2/dynamic/snta_thin.ma".
+include "basic_2/dynamic/lsubsn_snta.ma".
+
+(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
+(*
+lemma snta_fwd_abst: ∀h,L,W1,W2,T,U,l2. ⦃h, L⦄ ⊢ ⓛW1.T :[l2] ⓛW2.U →
+ ∃∃V1,V2,l1. ⦃h, L⦄ ⊢ W1 :[l1] V1 & ⦃h, L⦄ ⊢ W2 :[l1] V2 &
+ L ⊢ W1 ⬌* W2.
+#h #L #W1 #W2 #T #U #l2 #HTU
+elim (snta_fwd_correct … HTU) #X #H
+elim (snta_inv_bind1 … H) -H #W #T0 #l #HW2 #_ #_ -X
+elim (snta_inv_bind1 … HTU) -HTU #V1 #U0 #l0 #HWV1 #_ #H
+elim (cpcs_inv_abst … H Abst W1) -H
+#HW12 #_ -U0
+@(ex3_3_intro … HWV1 … HW12)
+[3: @(snta_conv … HTU0 HU0)
+
+ /3 width=3/
+
+*)
+(*
+#h #L #V #T #U #l2 #HTU
+elim (snta_fwd_correct … HTU) #X #H
+elim (snta_inv_bind1 … H) -H #W #T0 #l1 #HVW #HUT0 #_ -X
+elim (snta_inv_bind1 … HTU) -HTU #W0 #U0 #l0 #_ #HTU0 #H -l0
+elim (cpcs_inv_abst … H Abst V) -H /3 width=3/
+qed-.
+*)
+(*
+lemma snta_fwd_appl1_sound_aux: ∀h,l0. (∀L1,L2,T1,T2,U,l.
+ l < l0 → ⦃h, L1⦄ ⊢ T1 :[l] U →
+ L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
+ ) →
+ ∀L,T,U,l2. ⦃h, L⦄ ⊢ T :[l2] U →
+ ∀Z,Y,X1. T = ⓐZ.ⓛY.X1 → l0 = l2 →
+ ∃l1. ⦃h, L⦄ ⊢ Z :[l1+1] Y.
+#h #l0 #IH #L #T #U #l2 #H elim H -L -T -U -l2
+[
+|
+|
+|
+| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #HTU #_ #_ #Z #Y #X1 #H1 #H2 destruct -IH
+ elim (snta_fwd_abst … HTU) -X1 -U -l2 #Y0 #W0 #l0 #HY0 #H1 #HYW2
+ elim (snta_fwd_correct … HVW2) #W #H2
+ elim (snta_mono … H1 … H2) -H1 -H2 #H #_ destruct -W0 -W /4 width=6/
+| #L #V #T #U #W #l #HTU #HUW #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
+ elim (snta_inv_abst_sn … HTU) -HTU #Y0 #l0 #HY0 #HX12
+|
+| #L #T #U1 #U2 #V2 #l #HTU1 #HU12 #HUV2 #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+lemma snta_inv_appl_aux: ∀h,l0. (∀L1,L2,T1,T2,U,l.
+ l < l0 + 1 → ⦃h, L1⦄ ⊢ T1 :[l] U →
+ L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
+ ) →
+ ∀L,T,U,l2. ⦃h, L⦄ ⊢ T :[l2] U →
+ ∀Z,Y,X1,X2. T = ⓐZ.ⓛY.X1 → U = ⓐZ.ⓛY.X2 → l0 = l2 →
+ ∃∃l1. ⦃h, L⦄ ⊢ Z :[l1+1] Y & ⦃h, L.ⓛY⦄ ⊢ X1 :[l2] X2.
+#h #l0 #IH #L #T #U #l2 * -L -T -U -l2
+[
+|
+|
+|
+| #L #V #W1 #W2 #T #U #l1 #l2 #HVW2 #HTU #Z #Y #X1 #X2 #H1 #H2 #H3 destruct -IH
+ elim (snta_inv_abst … HTU) -HTU /2 width=2/
+| #L #V #T #U #W #l #HTU #HUW #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
+ elim (snta_inv_abst … HTU) -HTU #Y0 #l0 #HY0 #HX12
+|
+| #L #T #U1 #U2 #V2 #l #HTU1 #HU12 #HUV2 #Z #Y #X1 #X2 #H1 #H2 #H3 destruct
+
+ /2 width=2/
+
+
+axiom pippo: ∀h,l0. (∀L1,L2,T1,T2,U,l.
+ l < l0 + 1 → ⦃h, L1⦄ ⊢ T1 :[l] U →
+ L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
+ ) →
+ ∀L,T1,U1,l. ⦃h, L⦄ ⊢ T1 :[l] U1 →
+ ∀V2,W2,T2. L ⊢ T1 ➡* ⓐV2.ⓛW2.T2 → l0 = l →
+ ∃l0. ⦃h, L2⦄ ⊢ V2 :[l0+1] W2.
+(*
+#h #l #IH #L1 #T1 #U1 #l1 * -L1 -T1 -U1 -l1
+[
+|
+|
+|
+| #L1 #V1 #W1 #T1 #U1 #l1 #HVW1 #HTU1 #Y1 #X1 #H1 #L2 #Y2 #HL12 #HY12 #Z2 #X2 #HX12 #H2 destruct
+ elim (IH ??? Y2 … HVW1 HL12 ?) -HVW1 // [2: /3 width=1/ ] -HY12 #l21 #HY2W1 #H1l21 #H2l21
+ elim (IH … HTU1 HL12 HX12) -IH -HTU1 -HL12 -HX12 // #l22 #H #_ #H2l22
+ elim (snta_inv_bind1 … H) -H #Z #X #HZ2 #_ #H
+ elim (cpcs_inv_abst … H Abst W1) -H #H #_
+ lapply (transitive_le … (l21+l22) … H1l21 ?) -H1l21 // #Hl21
+ @(ex3_1_intro … Hl21) [2: /3 width=1/ ]
+ @(snta_conv … W1) /2 width=2/ (**) (* explicit constructors *)
+| #L1 #V1 #T1 #U1 #W1 #l1 #HTU1 #HUW1 #Y1 #X1 #H1 #L2 #Y2 #HL12 #HY12 #Z2 #X2 #HX12 #H2 destruct
+
+*)
+(* Properties on context-free parallel reduction for local environments *****)
+*)
+fact snta_ltpr_tpr_conf_aux: ∀h,l0. (∀L1,L2,T1,T2,U,l.
+ l < l0 → ⦃h, L1⦄ ⊢ T1 :[l] U →
+ L1 ➡ L2 → L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 :[l] U
+ ) →
+ ∀L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U → ∀L2. L1 ➡ L2 →
+ ∀T2. T1 ➡ T2 → l0 = l → ⦃h, L2⦄ ⊢ T2 :[l] U.
+#h #l0 #IH #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
+[ #L1 #k1 #L2 #_ #T2 #H #_ -l0
+ >(tpr_inv_atom1 … H) -H //
+| #L1 #K1 #V1 #W #U #i1 #l #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H #Hl -IH
+ >(tpr_inv_atom1 … H) -T2
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
+| #L1 #K1 #W1 #V1 #U1 #i1 #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #HL12 #T2 #H #Hl -IH
+(*
+ >(tpr_inv_atom1 … H) -T2
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total V1 0 (i+1)) #W #HW
+ lapply (snta_lift h … HLK … HWU1 … HW) /2 width=1/ -HLK -HW
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpr_lift … HW12 … HWU1 … HWU2) -HWU1 #HU12
+ @(snta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /3 width=6/ is too slow *)
+*)
+| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH
+(*
+ elim (tpr_inv_bind1 … H) -H *
+ [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
+ lapply (IHVW1 … HL12 … HV12) #HV2W1
+ lapply (IHVW1 L2 … V1 ?) // -IHVW1 #HWV1
+ lapply (IHTU1 (L2.ⓑ{I}V2) … HT1) -HT1 /2 width=1/ #HTU1
+ lapply (IHTU1 (L2.ⓑ{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H
+ lapply (tps_lsubs_trans … HT2 (L2.ⓑ{I}V2) ?) -HT2 /2 width=1/ #HT2
+ lapply (snta_tps_conf … HTU1 … HT2) -T #HT2U1
+ elim (snta_fwd_correct … H) -H #U2 #HU12
+ @(snta_conv … (ⓑ{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *)
+ | #T #HT1 #HTX #H destruct
+ lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HVW1
+ lapply (IHTU1 (L2.ⓓV1) … HT1) -T1 /2 width=1/ -L1 #H
+ elim (snta_fwd_correct … H) #T1 #HUT1
+ elim (snta_ldrop_conf … H L2 0 1 ? ?) -H // /2 width=1/ #T0 #U0 #HTU0 #H #HU10
+ lapply (delift_inv_lift1_eq … H L2 … HTX) -H -HTX /2 width=1/ #H destruct
+ @(snta_conv … HTU0) /2 width=2/
+ ]
+*)
+| #L1 #V1 #W11 #W2 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH
+(*
+ elim (tpr_inv_appl1 … H) -H *
+ [ #V2 #Y #HV12 #HY #H destruct
+ elim (tpr_inv_abst1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (IHTU1 L2 ? (ⓛW1.T1) ?) // #H
+ elim (snta_fwd_correct … H) -H #X #H
+ elim (snta_inv_bind1 … H) -H #W #U #HW #HU #_
+ @(snta_conv … (ⓐV2.ⓛW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *)
+ | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
+ lapply (IHVW1 … HL12 … HV12) #HVW2
+ lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HV1W2
+ lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1
+ elim (snta_fwd_correct … H1) #T #H2
+ elim (snta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H
+ elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21
+ elim (snta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0
+ lapply (lsubsn_snta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2
+ @(snta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *)
+ | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct
+ ]
+*)
+| #L1 #V1 #T1 #U1 #W1 #l #_ #HUW1 #IHTU1 #_ #L2 #HL12 #X #H #Hl
+ elim (tpr_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (cpr_tpr … HV12 L2) #HV
+ elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=6/ ] #U
+ @(snta_conv … (ⓐV2.U1)) /2 width=1/ -HV12 /4 width=8 by snta_pure, cprs_flat_dx/ (**) (* explicit constructor, /4 width=8/ is too slow without trace *)
+ | #V2 #W0 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
+ lapply (IHTU1 … HL12 (ⓛW0.T2) ? ?) -IHTU1 // /2 width=1/ -T0 #H1
+ lapply (IH … (ⓐV2.U1) … HUW1 HL12 ?) // /3 width=1/ #H2
+ lapply (snta_pure … H1 H2) -H2 #H
+ elim (snta_inv_bind1 … H1) -H1 #V0 #U2 #l1 #HWV0 #HTU2 #HU21
+ @(snta_conv … (ⓓV2.U2)) (**) (* explicit constructor *)
+ [2:
+(*
+ @snta_bind /3 width=2/ /3 width=6/ (**) (* /4 width=6/ is a bit slow *)
+*)
+ |3: @(cpcs_cpr_conf … (ⓐV1.ⓛW0.U2)) /2 width=1/
+ |4: /2 width=5/
+ | skip
+ ]
+(*
+ elim (snta_fwd_pure1 … H) -H #T1 #W2 #HVW2 #HUT1 #HTW1
+
+ elim (cpcs_inv_abst1 … HU21) #W3 #U3 #HU13 #H
+ elim (cprs_inv_abst … H Abst W0) -H #HW03 #_
+ elim (pippo … IH … HUW1 ? V2 W3 U3 HL12 ? ?) -IH -HUW1 -HL12 // /3 width=1/ -HU13 #l2 #HV2W3
+ lapply (snta_conv h L2 V2 W3 W0 V0 (l1+1) ? ? ?) /2 width=1/ -HV2W3 -HW03 -HWV0 #HV2W0
+*)
+(* SEGMENT 1.5
+ lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
+ lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
+ lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/
+
+axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y →
+ ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W.
+
+*)
+(* SEGMENT 2
+| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct
+ lapply (cpr_tpss … HU12) /4 width=4/
+| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12
+ @(snta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *)
+]
+qed.
+*)
+
+(* SEGMENT 3
+fact snta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U.
+
+
+ | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
+ elim (snta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0
+ lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
+ lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0
+ lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2
+ @(snta_abbr … HW2) -HW2
+ @(snta_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
+ ]
+| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct
+ elim (tpr_inv_cast1 … H) -H
+ [ * #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
+ lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
+ | -HV1 #HT1X
+ lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
+ ]
+]
+qed.
+
+lemma snta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 →
+ ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U.
+
+/2 width=9/ qed.
+
+axiom snta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A.
+/2 width=5/ qed.
+
+axiom snta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A.
+/2 width=5/ qed.
+*)
+*)*)
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/equivalence/cpcs_ltpss.ma".
+include "basic_2/dynamic/snta_snta.ma".
+
+(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about parallel unfold *****************************************)
+
+lemma snta_ltpss_tpss_conf: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U.
+#h #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
+[ #L1 #k #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H //
+| #L1 #K1 #V1 #W #U #i #l #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi
+ [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=7/
+ | elim (lt_or_ge i (d + e)) #Hide [ | -Hdi ]
+ [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=7/
+ | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=7/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=9/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
+ [ #H destruct
+ elim (lift_total V1 0 (i+1)) #W #HW
+ elim (lt_or_ge i d) #Hdi [ -HWV1 ]
+ [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12
+ lapply (cpr_tpss … HU12) -HU12 #HU12
+ @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *)
+ | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -HW -Hdi ]
+ [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12
+ lapply (cpr_tpss … HU12) -HU12 #HU12
+ @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *)
+ | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
+ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
+ ]
+| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (cpr_tpss … HV12) #HV
+ lapply (IHTU1 (L2.ⓑ{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H
+ elim (snta_fwd_correct … H) -H #U2 #HU12
+ @(snta_conv … (ⓑ{I}V2.U1)) /3 width=2/ /3 width=4/ /4 width=4/ (**) (* explicit constructor, /5 width=6/ is too slow *)
+| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
+ elim (tpss_inv_bind1 … HY) -HY #W21 #T2 #HW121 #HT12 #H destruct
+ lapply (cpr_tpss … HV12) #HVV12
+ lapply (IHTU1 L2 d e ? (ⓛW21.T2) ?) -IHTU1 // /2 width=1/ -HW121 -HT12 #H0
+ elim (snta_fwd_correct … H0) #X #H
+ elim (snta_inv_bind1 … H) -H #W #U #l0 #HW #HU #_
+ @(snta_conv … (ⓐV2.ⓛW12.U1)) /3 width=2/ /3 width=4/ /3 width=5/ (**) (* explicit constructor, /4 width=5/ is too slow *)
+| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (cpr_tpss … HV12) #HV
+ elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=4/ ] #U
+ @(snta_conv … (ⓐV2.U1)) /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *)
+| #L1 #T1 #U1 #W1 #l1 #l2 #HTU1 #HUW1 #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
+ elim (snta_fwd_correct … HTU1) -HTU1 #U #H
+ elim (snta_mono … HUW1 … H) -HUW1 -H #H #_ -U destruct
+ elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct
+ lapply (cpr_tpss … HU12) #HU /4 width=4/
+| #L1 #T1 #U11 #U12 #U #l #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12
+ @(snta_conv … U11) /2 width=5/ (**) (* explicit constructor, /3 width=7/ is too slow *)
+]
+qed.
+
+lemma snta_ltpss_tps_conf: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U.
+/3 width=7/ qed.
+
+lemma snta_ltpss_conf: ∀h,L1,T,U,l. ⦃h, L1⦄ ⊢ T :[l] U →
+ ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T :[l] U.
+/2 width=7/ qed.
+
+lemma snta_tpss_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U.
+/2 width=7/ qed.
+
+lemma snta_tps_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U.
+/2 width=7/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/snta_lift.ma".
+
+(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Main properties **********************************************************)
+
+theorem snta_mono: ∀h,L,T,U1,l1. ⦃h, L⦄ ⊢ T :[l1] U1 →
+ ∀U2,l2. ⦃h, L⦄ ⊢ T :[l2] U2 → l1 = l2 ∧ L ⊢ U1 ⬌* U2.
+#h #L #T #U1 #l1 #H elim H -L -T -U1 -l1
+[ #L #k #X #l2 #H
+ lapply (snta_inv_sort1 … H) -H * /2 width=1/
+| #L #K #V #W11 #W12 #i #l1 #HLK #_ #HW112 #IHVW11 #X #l2 #H
+ elim (snta_inv_lref1 … H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX
+ lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+ elim (IHVW11 … HVW21) -IHVW11 -HVW21 #Hl12 #HW121
+ lapply (cpcs_lift … HLK … HW112 … HW212 ?) // -K -W11 -W21 /3 width=3/
+| #L #K #W #V1 #V #i #l1 #HLK #_ #HWV #IHWV1 #X #l2 #H
+ elim (snta_inv_lref1 … H) -H * #K0 #W0 #V2 #V0 #HLK0 #HW0V2 #HWV0 [2: #HL2 ] #HX
+ lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H destruct
+ lapply (lift_mono … HWV0 … HWV) -HWV0 -HWV #H destruct
+ elim (IHWV1 … HW0V2) -IHWV1 -HW0V2 /3 width=1/
+| #I #L #V #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H
+ elim (snta_inv_bind1 … H) -H #W2 #U2 #l20 #_ #HTU2 #H
+ elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12
+ lapply (cpcs_trans … (ⓑ{I}V.U1) … H) -H /2 width=1/
+| #L #V #W #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H
+ elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H
+ elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12
+ lapply (cpcs_trans … (ⓐV.ⓛW1.U1) … H) -H /2 width=1/
+| #L #V #T #U1 #W1 #l1 #_ #_ #IHTU1 #_ #X #l2 #H
+ elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H
+ elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12
+ lapply (cpcs_trans … (ⓐV.U1) … H) -H /2 width=1/
+| #L #T #U1 #W1 #l10 #l1 #_ #_ #IHTU1 #_ #X #l2 #H
+ elim (snta_inv_cast1 … H) -H #HTU1
+ elim (IHTU1 … HTU1) -IHTU1 -HTU1 /2 width=1/
+| #L #T #U11 #U12 #V12 #l1 #_ #HU112 #_ #IHTU11 #_ #U2 #l2 #HTU2
+ elim (IHTU11 … HTU2) -IHTU11 -HTU2 #Hl12 #H
+ lapply (cpcs_canc_sn … HU112 … H) -U11 /2 width=1/
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma snta_cast_alt: ∀h,L,T,W,U,l. ⦃h, L⦄ ⊢ T :[l] W → ⦃h, L⦄ ⊢ T :[l] U →
+ ⦃h, L⦄ ⊢ ⓝW.T :[l] U.
+#h #L #T #W #U #l #HTW #HTU
+elim (snta_mono … HTW … HTU) #_ #HWU
+elim (snta_fwd_correct … HTU) -HTU /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/thin_ldrop.ma".
+include "basic_2/equivalence/cpcs_delift.ma".
+include "basic_2/dynamic/snta_lift.ma".
+
+(* STRATIFIED NATIVE TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties on basic local environment thinning ***************************)
+
+(* Note: this is known as the substitution lemma *)
+lemma snta_thin_conf: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 →
+ ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 →
+ ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 :[l] U2 &
+ L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2.
+#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
+[ /2 width=5/
+| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12
+ elim (lt_or_ge i d) #Hdi [ -HVW1 ]
+ [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H
+ lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1
+ elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2
+ elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12
+ lapply (delift_mono … H … HV12) -H -HV12 #H destruct
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1
+ lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 //
+ >minus_plus <plus_minus_m_m // /3 width=6/
+ | elim (lt_or_ge i (d+e)) #Hide [ -HVW1 | -Hdi -IHVW1 -HL1 ]
+ [ lapply (sfr_ldrop_trans_be_up … HLK1 … HL1 ? ?) -HL1 // /2 width=2/ <minus_n_O #H
+ elim (sfr_inv_bind … H ?) -H /2 width=1/ #HK1 #_
+ elim (thin_ldrop_conf_be … HL12 … HLK1 ? ?) -HL12 /2 width=2/ #K2 #H #HLK2
+ lapply (thin_inv_thin1 … H ?) -H /2 width=1/ #HK12
+ elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #V2 #W2 #HVW2 #HV12 #HW12
+ elim (lift_total V2 0 d) #T2 #HVT2
+ elim (lift_total W2 0 d) #U2 #HWU2
+ elim (lift_total W2 0 (i+1)) #U #HW2U
+ lapply (snta_lift … HVW2 … HLK2 … HVT2 … HWU2) -HVW2 -HLK2 #HTU2
+ lapply (ldrop_fwd_ldrop2 … HLK1) #HLK0
+ lapply (delift_lift_ge … HW12 … HLK0 HWU1 … HW2U) -HW12 -HLK0 -HWU1 // >minus_plus #HU1
+ lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2
+ lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/
+ | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei
+ lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1
+ elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W
+ <plus_minus in ⊢ (??%??→?); /2 width=2/ #HW1
+ <minus_minus // /2 width=2/ -Hdei >commutative_plus <minus_n_n /3 width=6/
+ ]
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL1 #HL12
+ elim (lt_or_ge i d) #Hdi [ -HWV1 | -IHWV1 ]
+ [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H
+ lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1
+ elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2
+ elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK1 HK12) -IHWV1 -HK1 -HK12 #X2 #V2 #HWV2 #H #_
+ lapply (delift_mono … H … HW12) -H #H destruct
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 //
+ >minus_plus <plus_minus_m_m // /3 width=6/
+ | elim (lt_or_ge i (d+e)) #Hide [ -HWV1 -HWU1 -HL12 | -Hdi -HL1 ]
+ [ lapply (sfr_inv_ldrop … HLK1 … HL1 ? ?) -HLK1 -HL1 // -Hdi -Hide #H destruct
+ | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei
+ lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1
+ elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W
+ <plus_minus in ⊢ (??%??→?); /2 width=2/ #HW1
+ <minus_minus // /2 width=2/ -Hdei >commutative_plus <minus_n_n /3 width=6/
+ ]
+ ]
+| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL1 #HL12
+ elim (IHVW1 … HL1 HL12) -IHVW1 #V2 #W2 #HVW2 #HV12 #_
+ elim (IHTU1 (L2.ⓑ{I}V2) (d+1) e ? ?) -IHTU1 /2 width=1/ -HL1 -HL12 #T2 #U2 #HTU2 #HT12 #HU12
+ lapply (delift_lsubs_trans … HT12 (L1.ⓑ{I}V2) ?) -HT12 /2 width=1/
+ lapply (delift_lsubs_trans … HU12 (L1.ⓑ{I}V2) ?) -HU12 /2 width=1/ /3 width=7/
+| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL1 #HL12
+ elim (IHVW1 … HL1 HL12) -IHVW1 #V2 #W22 #HVW2 #HV12 #HW122
+ elim (IHTU1 … HL1 HL12) -IHTU1 -HL1 -HL12 #X2 #Y2 #HXY2 #HX2 #HY2
+ elim (delift_inv_bind1 … HX2) -HX2 #W21 #T2 #W121 #HT12 #H destruct
+ elim (delift_inv_bind1 … HY2) -HY2 #X #U2 #HX #HU12 #H destruct
+ lapply (delift_mono … HX … HW122) -HX #H destruct
+ @(ex3_2_intro … (ⓐV2.ⓛW21.T2) (ⓐV2.ⓛW22.U2)) [ /2 width=2/ | 2,3: /3 width=1/ ] (**) (* explict constructor, /4 depth=?/ is too slow *)
+| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL1 #HL12
+ elim (IHTU1 … HL1 HL12) -IHTU1 #T2 #U2 #HTU2 #HT12 #HU12
+ elim (IHUW1 … HL1 HL12) -IHUW1 -HL1 -HL12 #X2 #W2 #HXW2 #H #HW12
+ elim (delift_inv_flat1 … H) -H #V2 #Y2 #HV12 #HY2 #H destruct
+ lapply (delift_mono … HY2 … HU12) -HY2 #H destruct /3 width=7/
+| #L1 #T1 #U1 #W1 #l1 #l2 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL1 #HL12
+ elim (IHTU1 … HL1 HL12) -IHTU1 #T2 #U2 #HTU2 #HT12 #HU12
+ elim (IHUW1 … HL1 HL12) -IHUW1 -HL1 -HL12 #Y2 #W2 #HUW2 #HY2 #HW12
+ lapply (delift_mono … HY2 … HU12) -HY2 #H destruct /3 width=5/
+| #L1 #T1 #U11 #U12 #V1 #l #_ #HU112 #_ #IHT1 #IHU12 #L2 #d #e #HL1 #HL12
+ elim (IHT1 … HL1 HL12) -IHT1 #T2 #U21 #HT2 #HT12 #HU121
+ elim (IHU12 … HL1 HL12) -IHU12 -HL1 #U22 #V2 #HU22 #HU122 #_
+ lapply (thin_cpcs_delift_mono … HU112 … HL12 … HU121 … HU122) -HU112 -HL12 -HU121 /3 width=5/
+]
+qed.
+
+lemma snta_ldrop_conf: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 →
+ ∀L2,d,e. ≽ [d, e] L1 → ⇩[d, e] L1 ≡ L2 →
+ ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 :[l] U2 &
+ L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2.
+/3 width=1/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/lstas_lstas.ma".
+include "basic_2/computation/fpbs_lift.ma".
+include "basic_2/computation/fpbg_fleq.ma".
+include "basic_2/equivalence/cpes_cpds.ma".
+include "basic_2/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Inductive premises for the preservation results **************************)
+
+definition IH_snv_cpx_lpx: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+
+definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
+
+definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+
+(* Properties for the preservation results **********************************)
+
+fact snv_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+/3 width=6 by lpr_lpx, cpr_cpx/ qed-.
+
+fact snv_sta_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ ∀G,L,T. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+ ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
+ ∀U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+/3 width=6 by sta_cpx/ qed-.
+
+fact snv_cpxs_lpx_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
+@(cpxs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cpxs_fpbs/
+qed-.
+
+fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+/3 width=10 by snv_cpxs_lpx_aux, cprs_cpxs, lpr_lpx/ qed-.
+
+fact snv_lstas_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 → ⦃G, L1⦄ ⊢ U1 ¡[h, g].
+/3 width=12 by snv_cpxs_lpx_aux, lstas_cpxs/ qed-.
+
+fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H
+@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
+qed-.
+
+fact da_cpcs_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] →
+ ∀l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ∀l2. ⦃G, L⦄ ⊢ T2 ▪[h, g] l2 →
+ ⦃G, L⦄ ⊢ T1 ⬌* T2 → l1 = l2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #l1 #Hl1 #l2 #Hl2 #H
+elim (cpcs_inv_cprs … H) -H /4 width=18 by da_cprs_lpr_aux, da_mono/
+qed-.
+
+fact sta_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l+1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •[h] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #H01 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (IH … H01 … 1 … Hl U1 … HT12 … HL12) -H01 -Hl -HT12 -HL12
+/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/
+qed-.
+
+fact lstas_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (IH1 … Hl21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
+[2: /3 width=12 by da_cprs_lpr_aux/
+|3: /3 width=10 by snv_cprs_lpr_aux/
+|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
+] -G0 -L0 -T0 -T1 -T -l1
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
+qed-.
+
+fact lstas_cpcs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l,l1. l ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l] U1 →
+ ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+ ∀l2. l ≤ l2 → ⦃G, L1⦄ ⊢ T2 ▪[h, g] l2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, l] U2 →
+ ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l #l1 #Hl1 #HTl1 #U1 #HTU1 #T2 #H02 #HT2 #l2 #Hl2 #HTl2 #U2 #HTU2 #H #L2 #HL12
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (lstas_cprs_lpr_aux … H01 HT1 … Hl1 HTl1 … HTU1 … H1 … HL12) -T1 /2 width=1 by/ #W1 #H1 #HUW1
+elim (lstas_cprs_lpr_aux … H02 HT2 … Hl2 HTl2 … HTU2 … H2 … HL12) -T2 /2 width=1 by/ #W2 #H2 #HUW2 -L0 -T0
+lapply (lstas_mono … H1 … H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/
+qed-.
+
+fact lstas_cpds_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
+ ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
+lapply (da_mono … H … Hl1) -H #H destruct
+lapply (lstas_da_conf … HTU1 … Hl1) #Hl12
+elim (le_or_ge l2 l) #Hl2
+[ lapply (lstas_conf_le … HTU1 … HT1T) -HT1T
+ /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
+| lapply (lstas_da_conf … HT1T … Hl1) #Hl1l
+ lapply (lstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
+ elim (lstas_cprs_lpr_aux … IH3 IH2 IH1 … Hl1l … HTU1 … HTT2 L) -IH2 -IH1 -Hl1l -HTU1 -HTT2
+ /3 width=12 by snv_lstas_aux, cpcs_cpes, fpbg_fpbs_trans, lstas_fpbs, monotonic_le_minus_l, ex3_2_intro/
+]
+qed-.
+
+fact cpds_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 * #W1 #l1 #l2 #Hl21 #Hl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
+lapply (IH2 … H01 … Hl1 … HT12 … HL12) -L0 -T0 // -T1
+lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
+elim (cpcs_inv_cprs … H) -H /3 width=7 by ex4_3_intro, ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lsubd_da.ma".
+include "basic_2/computation/cpds_cpds.ma".
+include "basic_2/dynamic/snv_aaa.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on degree assignment for terms ********************************)
+
+fact da_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #_ #_ #_ #_ #l #H2 #X3 #H3 #L2 #_ -IH2 -IH1
+ lapply (da_inv_sort … H2) -H2
+ lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1 by da_sort/
+| #i #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+ elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #H #HX0
+ elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #l1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (cpr_inv_lref1 … H3) -H3
+ [1,3: #H destruct
+ lapply (fqup_lref … G1 … HLK1)
+ elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ /4 width=10 by da_ldef, da_ldec, fqup_fpbg/
+ |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ lapply (fqup_lref … G1 … HLK1)
+ elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_drop2 … HLK2) -V2
+ /4 width=8 by da_lift, fqup_fpbg/
+ ]
+| #p #_ #_ #HT0 #H1 destruct -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+ elim (snv_inv_bind … H1) -H1 #_ #HT1
+ lapply (da_inv_bind … H2) -H2
+ elim (cpr_inv_bind1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=9 by da_bind, fqup_fpbg, lpr_pair/
+ | #T2 #HT12 #HT2 #H1 #H2 destruct
+ /4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, ldrop_drop/
+ ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #b0 #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10
+ lapply (da_inv_flat … H2) -H2 #Hl
+ elim (cpr_inv_appl1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -IH2 /4 width=7 by da_flat, fqup_fpbg/
+ | #b #V2 #W #W2 #U1 #U2 #HV12 #HW2 #HU12 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW #HU1
+ lapply (da_inv_bind … Hl) -Hl #Hl
+ elim (cpds_inv_abst1 … HT10) -HT10 #W3 #U3 #HW3 #_ #H destruct -U3
+ lapply (cprs_div … HW3 … HW10) -W3 #HWW1
+ lapply (da_sta_conf … HVW1 … Hl0) <minus_plus_m_m #H
+ elim (snv_fwd_da … HW) #l1 #Hl1
+ lapply (snv_lstas_aux … IH2 … HV1 … 1 … Hl0 W1 ?) /2 width=2 by fqup_fpbg, sta_lstas/ #HW1
+ lapply (da_cpcs_aux … IH2 IH1 … Hl1 … H … HWW1) -H
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, sta_fpbs/ #H destruct
+ lapply (IH1 … HV1 … Hl0 … HV12 … HL12) -HV1 -Hl0 -HV12 [ /2 by fqup_fpbg/ ] #Hl0
+ lapply (IH1 … Hl1 … HW2 … HL12) -Hl1 // /2 width=1 by fqup_fpbg/ -HW
+ lapply (IH1 … HU1 … Hl … HU12 (L2.ⓛW2) ?) -IH1 -HU1 -Hl -HU12 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
+ /4 width=6 by da_bind, lsubd_da_trans, lsubd_abbr/
+ | #b #V #V2 #W #W2 #U1 #U2 #HV1 #HV2 #HW2 #HU12 #H1 #H2 destruct -IH2 -V -W0 -T0 -l0 -HV1 -HVW1
+ elim (snv_inv_bind … HT1) -HT1 #_
+ lapply (da_inv_bind … Hl) -Hl
+ /5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
+ ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+ elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
+ lapply (da_inv_flat … H2) -H2 #Hl
+ elim (cpr_inv_cast1 … H3) -H3
+ [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
+ | /3 width=7 by fqup_fpbg/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/snv_lift.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+include "basic_2/dynamic/lsubsv_snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact snv_cpx_lpx_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpx_lpx h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH3 -IH2 -IH1 -H1
+ elim (cpx_inv_sort1 … H2) -H2 // * //
+| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+ elim (lpx_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (fqup_lref … G1 … HLK1) #HKL
+ elim (cpx_inv_lref1 … H2) -H2
+ [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/
+ | * #I0 #K0 #V0 #W0 #H #HVW0 #W0 -HV12
+ lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
+ lapply (ldrop_fwd_drop2 … HLK2) -HLK2 /4 width=8 by fqup_fpbg, snv_lift/
+ ]
+| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+ elim (cpx_inv_bind1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbg, snv_bind, lpx_pair/
+ | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
+ /4 width=10 by fqup_fpbg, snv_inv_lift, lpx_pair, ldrop_drop/
+ ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1
+ elim (cpx_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
+(* lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #H2l0
+ elim (sta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fqup_fpbg/ -HV1 #W2 #HVW2 #HW12
+ elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -HT12 -HTU1 #X #HTU2 #H
+ elim (cprs_inv_abst1 … H) -H #W20 #U2 #HW120 #_ #H destruct
+ lapply (lpr_cprs_conf … HL12 … HW10) -L1 #HW10
+ lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
+ lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
+ elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
+ lapply (cpds_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?)
+ /2 width=7 by snv_appl, cprs_bind/ *)
+ | #b #V2 #W20 #W2 #T20 #T2 #HV12 #HW202 #HT202 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
+ elim (cpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
+ lapply (cprs_div … HW10 … HW230) -W30 #HW120
+ lapply (snv_sta_aux … IH1 … Hl0 … HVW1) /2 width=1 by fqup_fpbg/ #HW10
+ lapply (da_sta_conf … HVW1 … Hl0) <minus_plus_m_m #HlW10
+ elim (snv_fwd_da … HW20) #l #Hl
+ lapply (da_cpcs_aux … IH1 IH2 … HlW10 … Hl … HW120) // -HlW10
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, sta_fpbs/ #H destruct
+(* lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HlV2
+ lapply (IH2 … Hl … HW202 … HL12) /2 width=1 by fqup_fpbg/ #HlW2
+ elim (sta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #W3 #HV2W3 #HW103
+ lapply (da_sta_conf … HV2W3 … HlV2) <minus_plus_m_m #HlW3
+ lapply (cpcs_cpr_strap1 … HW120 … HW202) -HW120 #HW102
+ lapply (lpr_cpcs_conf … HL12 … HW102) -HW102 #HW102
+ lapply (cpcs_canc_sn … HW103 … HW102) -W10 #HW32
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1 #HV2
+ lapply (IH1 … HW202 … HL12) /2 width=1 by fqup_fpbg/ -HW20 #HW2
+ lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT20 #HT2
+ lapply (snv_sta_aux … IH4 … HlV2 … HV2W3)
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/ #HW3
+ lapply (lsubsv_snv_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /3 width=3 by snv_bind, snv_cast/
+ @(lsubsv_abbr … l) /3 width=7 by fqup_fpbg/ #W #W0 #l0 #Hl0 #HV2W #HW20
+ lapply (lstas_sta_conf_pos … HV2W3 … HV2W) -HV2W #HW3W
+ @(lstas_cpcs_lpr_aux … IH1 IH2 IH3 … HlW3 … HW3W … HlW2 … HW20 … HW32) //
+ [ /3 width=9 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_sta_fpbs/
+ | /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/
+ ] *)
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
+ elim (cpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
+ elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
+(* lapply (lpr_cpr_conf … HL12 … HW10) -HW10 #HW10
+ elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbg, lpr_pair/ -HTU0 #X #HTU2 #H
+ elim (cprs_inv_abst1 … H) -H #W #U2 #HW1 #_ #H destruct -U3
+ elim (sta_cpr_lpr_aux … IH3 … HVW1 … HV10 … HL12) /2 width=2 by fqup_fpbg/ -IH3 -HVW1 #X #H1 #H2
+ lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
+ elim (lift_total X 0 1) #W20 #H3
+ lapply (sta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=2 by ldrop_drop/ -H1 #HVW20
+ lapply (cpcs_lift … (L2.ⓓW2) … H3 … HW13 H2) /2 width=2 by ldrop_drop/ -HW13 -H3 -H2 #HW320
+ lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
+ elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
+ lapply (cpds_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) /2 width=1 by cprs_bind/ -HW3 -HTU2 #HTU2
+ lapply (IH2 … Hl0 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -IH2 -Hl0 #Hl0
+ lapply (da_lift … Hl0 (L2.ⓓW2) … HV02) /2 width=2 by ldrop_drop/ -Hl0 #Hl0
+ lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ -HW0 #HW2
+ lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV10 #HV0
+ lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -L1 #HT2
+ lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /3 width=7 by snv_bind, snv_appl, ldrop_drop/ *)
+ ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
+ elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
+ elim (cpx_inv_cast1 … H2) -H2
+ [ * #W2 #T2 #HW12 #HT12 #H destruct
+(* lapply (cpcs_cprs_strap1 … HUW1 W2 ?) /2 width=1 by cpr_cprs/ -HUW1 #H1
+*) lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/ -HW1 -HW12 #HW2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -IH1 #HT2
+(* elim (sta_cpr_lpr_aux … IH3 … Hl0 … HTU1 … HT12 … HL12) /2 width=2 by fqup_fpbg/ -IH3 -HTU1 #U2 #HTU2 #HU12
+ lapply (IH2 … Hl0 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -IH2 -HT1 -HT12 -Hl0 #Hl0
+ /4 width=7 by snv_cast, lpr_cpcs_conf, cpcs_canc_sn/ *)
+ | #H -IH3 -IH2 -HW1 -HTU1 -HUW1
+ lapply (IH1 … H … HL12) -IH1 -H -HL12 /2 width=1 by fqup_fpbg/
+ | #H -IH3 -IH2 -HTU1 -HUW1
+ lapply (IH1 … H … HL12) -IH1 -H -HL12 /2 width=1 by fqup_fpbg/
+ ]
+]
+[4: @snv_cast //
+qed-.
+snv_lstas, snv_cpr_lpr
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/cpds_cpds.ma".
+include "basic_2/dynamic/snv_aaa.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+include "basic_2/dynamic/lsubsv_lstas.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on sn parallel reduction for local environments ***************)
+
+fact lstas_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpx_lpx h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_lstas_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #_ #_ #_ #_ #l1 #l2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
+ >(lstas_inv_sort1 … H2) -X2
+ >(cpr_inv_sort1 … H3) -X3 /2 width=3 by ex2_intro/
+| #i #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus … l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3
+ [ lapply (lstas_inv_O … H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ]
+ elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HK0 #HX0
+ elim (da_inv_lref … Hl1) -Hl1 * #K1 [ #V1 | #W1 #l0 ] #HLK1 [ #HVl1 | #HWl1 #H destruct ]
+ lapply (ldrop_mono … HK0 … HLK1) -HK0 #H destruct
+ elim (lstas_inv_lref1 … H2) -H2 * #K0 #V0 #W0 [2,4: #X0 ] #HK0 [1,2: #_ -X0 ] #HVW0 #HX2
+ lapply (ldrop_mono … HK0 … HLK1) -HK0 #H destruct
+ [ lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21 ]
+ lapply (fqup_lref … G1 … HLK1) #HKV1
+ elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #V2 ] #HK12 [ #HW12 | #HV12 ] #H destruct
+ lapply (ldrop_fwd_drop2 … HLK2) #H2
+ elim (cpr_inv_lref1 … H3) -H3
+ [1,3: #H destruct -HLK1
+ |2,4: * #K0 #V0 #X0 #H #HVX0 #HX0
+ lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
+ ]
+ [ lapply (IH2 … HWl1 … HW12 … HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
+ elim (da_inv_sta … H) -H
+ elim (IH1 … HWl1 … HVW0 … HW12 … HK12) -IH1 -HVW0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
+ elim (lift_total V2 0 (i+1))
+ /3 width=12 by cpcs_lift, lstas_ldec, ex2_intro/
+ | elim (IH1 … HVl1 … HVW0 … HV12 … HK12) -IH1 -HVl1 -HVW0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
+ elim (lift_total W2 0 (i+1))
+ /4 width=12 by cpcs_lift, lstas_ldef, ex2_intro/
+ | elim (IH1 … HVl1 … HVW0 … HVX0 … HK12) -IH1 -HVl1 -HVW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -l1 #W2 #HXW2 #HW02
+ elim (lift_total W2 0 (i+1))
+ /3 width=12 by cpcs_lift, lstas_lift, ex2_intro/
+ ]
+| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_bind … H1) -H1 #_ #HT1
+ lapply (da_inv_bind … Hl1) -Hl1 #Hl1
+ elim (lstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
+ elim (cpr_inv_bind1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
+ /4 width=5 by cpcs_bind2, lpr_cpr_conf, lstas_bind, ex2_intro/
+ | #T3 #HT13 #HXT3 #H1 #H2 destruct
+ elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
+ elim (lstas_inv_lift1 … HTU3 L2 … HXT3) -T3
+ /5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, ldrop_drop, ex2_intro/
+ ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU10
+ lapply (da_inv_flat … Hl1) -Hl1 #Hl1
+ elim (lstas_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
+ elim (cpr_inv_appl1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
+ elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1
+ /4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lstas_appl, ex2_intro/
+ | #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
+ lapply (da_inv_bind … Hl1) -Hl1 #Hl1
+ elim (lstas_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
+ elim (cpds_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW20 #_ #H destruct
+ lapply (cprs_div … HW10 … HW20) -W0 #HW12
+ lapply (da_sta_conf … HVW1 … Hl0) <minus_plus_m_m #H
+ elim (snv_fwd_da … HW2) #l #Hl
+ lapply (snv_lstas_aux … IH3 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fqup_fpbg, sta_lstas/ #HW1
+ lapply (da_cpcs_aux … IH3 IH2 … H … Hl … HW12) // -H
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, sta_fpbs/ #H destruct
+ lapply (snv_cpr_lpr_aux … IH3 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2l
+ elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fqup_fpbg, sta_lstas/ -HVW1 #W4 #H #HW14
+ lapply (lstas_inv_SO … H) #HV2W4
+ lapply (da_sta_conf … HV2W4 … HV2l) <minus_plus_m_m #HW4l
+ lapply (snv_lstas_aux … IH3 … HV2 … HV2l … H) -H /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/ #HW4
+ lapply (snv_cpr_lpr_aux … IH3 … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3
+ lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3l
+ elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
+ lapply (cpcs_cpr_strap1 … HW12 … HW23) #H
+ lapply (lpr_cpcs_conf … HL12 … H) -H #H
+ lapply (cpcs_canc_sn … HW14 H) -H #HW43
+ elim (lsubsv_lstas_trans … g … HTU3 … Hl21 … (L2.ⓓⓝW3.V2)) -HTU3
+ [ #U4 #HT3U4 #HU43 -HW12 -HW3 -HW3l -W4 -IH3 -IH2
+ @(ex2_intro … (ⓓ{b}ⓝW3.V2.U4)) /2 width=1 by lstas_bind/ -HT3U4
+ @(cpcs_canc_dx … (ⓓ{b}ⓝW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
+ @(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
+ /4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
+ | -U3
+ @(lsubsv_abbr … l) /3 width=7 by fqup_fpbg/
+ #W #W0 #l0 #Hl0 #HV2W #HW30
+ lapply (lstas_sta_conf_pos … HV2W4 … HV2W) -HV2W #HW4W
+ @(lstas_cpcs_lpr_aux … IH3 IH2 IH1 … Hl0 … HW4W … Hl0 … HW30 … HW43) //
+ [ /3 width=9 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_sta_fpbs/
+ | /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/
+ ]
+ | -IH3 -IH1 /3 width=9 by fqup_fpbg, lpr_pair/
+ ]
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
+ elim (snv_inv_bind … HT1) -HT1 #_ #HT0
+ lapply (da_inv_bind … Hl1) -Hl1 #Hl1
+ elim (lstas_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
+ elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
+ lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
+ lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
+ lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
+ lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
+ lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?)
+ /4 width=3 by lstas_appl, lstas_bind, cpr_theta, ex2_intro/
+ ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus … l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ [ lapply (lstas_inv_O … H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ]
+ elim (snv_inv_cast … H1) -H1 #U1 #l #_ #HT1 #_ #_ #_ -U1 -l
+ lapply (da_inv_flat … Hl1) -Hl1 #Hl1
+ lapply (lstas_inv_cast1 … H2) -H2 #HTU1
+ elim (cpr_inv_cast1 … H3) -H3
+ [ * #U2 #T2 #_ #HT12 #H destruct
+ elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1 -HL12
+ /3 width=3 by fqup_fpbg, lstas_cast, ex2_intro/
+ | #HT1X3
+ elim (IH1 … Hl1 … HTU1 … HT1X3 … HL12) -IH1 -Hl1 -HTU1 -HL12
+ /2 width=3 by fqup_fpbg, ex2_intro/
+ ]
+]
+qed-.
--- /dev/null
+lemma da_cpcs: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L⦄ ⊢ T2 ¡[h, g] →
+ ∀l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ∀l2. ⦃G, L⦄ ⊢ T2 ▪[h, g] l2 →
+ ⦃G, L⦄ ⊢ T1 ⬌* T2 → l1 = l2.
+#h #g #G #L #T1 #HT1 #T2 #HT2 #l1 #Hl1 #l2 #Hl2 #H
+elim (cpcs_inv_cprs … H) -H /3 width=12 by da_cprs_lpr, da_mono/
+qed-.
+
+lemma sta_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l+1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •[h] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G #L1 #T1 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (lstas_cpr_lpr … 1 … Hl U1 … HT12 … HL12) -Hl -HT12 -HL12
+/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/
+qed-.
+
+lemma snv_sta: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+ ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
+ ∀U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+/3 width=7 by lstas_inv_SO, sta_lstas, snv_lstas/ qed-.
+
+lemma lstas_cpds: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
+ ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
+#h #g #G #L #T1 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
+lapply (da_mono … H … Hl1) -H #H destruct
+lapply (lstas_da_conf … HTU1 … Hl1) #Hl12
+elim (le_or_ge l2 l) #Hl2
+[ lapply (lstas_conf_le … HTU1 … HT1T) -HT1T //
+ /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
+| lapply (lstas_da_conf … HT1T … Hl1) #Hl1l
+ lapply (lstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
+ elim (lstas_cprs_lpr … Hl1l … HTU1 … HTT2 L) -Hl1l -HTU1 -HTT2
+ /3 width=7 by snv_lstas, cpcs_cpes, monotonic_le_minus_l, ex3_2_intro/
+]
+qed-.
+
+lemma cpds_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G #L1 #T1 #HT1 #U1 * #W1 #l1 #l2 #Hl21 #Hl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (lstas_cpr_lpr … Hl1 … HTW1 … HT12 … HL12) // #W2 #HTW2 #HW12
+lapply (da_cpr_lpr … Hl1 … HT12 … HL12) // -T1
+lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
+elim (cpcs_inv_cprs … H) -H /3 width=7 by ex4_3_intro, ex2_intro/
+qed-.
+
+(* Note: missing da_scpds_lpr, da_scpes *)
+
+lemma scpds_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1,l. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, l] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g, l] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G #L1 #T1 #HT1 #U1 #l2 * #W1 #l1 #Hl21 #HTl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (lstas_cpr_lpr … HTl1 … HTW1 … HT12 … HL12) // #W2 #HTW2 #HW12
+lapply (da_cpr_lpr … HTl1 … HT12 … HL12) // -T1
+lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
+elim (cpcs_inv_cprs … H) -H /3 width=6 by ex4_2_intro, ex2_intro/
+qed-.
+
+lemma scpes_cpr_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+ ∀l1,l2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 ➡ U1 → ∀U2. ⦃G, L1⦄ ⊢ T2 ➡ U2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ U1 •*⬌*[h, g, l1, l2] U2.
+#h #g #G #L1 #T1 #HT1 #T2 #HT2 #l1 #l2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
+elim (scpds_cpr_lpr … HT10 … HTU1 … HL12) -HT10 -HTU1 // #X1 #HUX1 #H1
+elim (scpds_cpr_lpr … HT20 … HTU2 … HL12) -HT20 -HTU2 // #X2 #HUX2 #H2
+elim (cprs_conf … H1 … H2) -T0 /3 width=5 by scpds_div, scpds_cprs_trans/
+qed-.
+
+(* Note: missing lstas_scpds, scpes_le *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/da_sta.ma".
+include "basic_2/static/sta_aaa.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_da: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
+#h #g #G #L #T #A #H elim (aaa_sta h … H) -A /2 width=2 by sta_da/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/sta.ma".
+include "basic_2/static/da_da.ma".
+
+(* Properties on static type assignment for terms ***************************)
+
+lemma da_sta_conf: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
+ ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ U ▪[h, g] l-1.
+#h #g #G #L #T #U #H elim H -G -L -T -U
+[ #G #L #k #l #H
+ lapply (da_inv_sort … H) -H /3 width=1 by da_sort, deg_next/
+| #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #l #H
+ elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0
+ lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+ lapply (drop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
+| #G #L #K #W #V #U #i #HLK #_ #HWU #IHWV #l #H
+ elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0 [| #H0 ]
+ lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+ lapply (drop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
+| #a #I #G #L #V #T #U #_ #IHTU #l #H
+ lapply (da_inv_bind … H) -H /3 width=1 by da_bind/
+| #G #L #V #T #U #_ #IHTU #l #H
+ lapply (da_inv_flat … H) -H /3 width=1 by da_flat/
+| #G #L #W #T #U #_ #IHTU #l #H
+ lapply (da_inv_flat … H) -H /2 width=1 by/
+]
+qed-.
+
+lemma sta_da: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
+ ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
+#h #g #G #L #T #U #H elim H -G -L -T -U
+[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
+| #G #L #K #V #W #W0 #i #HLK #_ #_ * /3 width=5 by da_ldef, ex_intro/
+| #G #L #K #W #V #W0 #i #HLK #_ #_ * /3 width=5 by da_ldec, ex_intro/
+| #a #I #G #L #V #T #U #_ * /3 width=2 by da_bind, ex_intro/
+| #G #L #V #T #U #_ * /3 width=2 by da_flat, ex_intro/
+| #G #L #W #T #U #_ * /3 width=2 by da_flat, ex_intro/
+]
+qed-.
+
+lemma sta_da_ge: ∀h,G,L,T,U,l0. ⦃G, L⦄ ⊢ T •[h] U →
+ ∃∃g,l. ⦃G, L⦄ ⊢ T ▪[h, g] l & l0 ≤ l.
+#h #G #L #T #U #l0 #H elim H -G -L -T -U
+[ /3 width=4 by da_sort, ex2_2_intro/
+| #G #L #K #V #W #W0 #i #HLK #_ #_ * /3 width=5 by da_ldef, ex2_2_intro/
+| #G #L #K #W #V #W0 #i #HLK #_ #_ * /4 width=5 by da_ldec, lt_to_le, le_S_S, ex2_2_intro/
+| #a #I #G #L #V #T #U #_ * /3 width=4 by da_bind, ex2_2_intro/
+| #G #L #V #T #U #_ * /3 width=4 by da_flat, ex2_2_intro/
+| #G #L #W #T #U #_ * /3 width=4 by da_flat, ex2_2_intro/
+]
+qed-.
+
+(* Inversion lrmmas on static type assignment for terms *********************)
+
+lemma da_inv_sta: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l →
+ ∃U. ⦃G, L⦄ ⊢ T •[h] U.
+#h #g #G #L #T #l #H elim H -G -L -T -l
+[ /2 width=2/
+| #G #L #K #V #i #l #HLK #_ * #W #HVW
+ elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldef, ex_intro/
+| #G #L #K #W #i #l #HLK #_ * #V #HWV
+ elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldec, ex_intro/
+| #a #I #G #L #V #T #l #_ * /3 width=2 by sta_bind, ex_intro/
+| * #G #L #V #T #l #_ * /3 width=2 by sta_appl, sta_cast, ex_intro/
+]
+qed-.
+
+lemma sta_inv_refl_pos: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h] T → ⊥.
+#h #g #G #L #T #l #H1T #HTT
+lapply (da_sta_conf … HTT … H1T) -HTT <minus_plus_m_m #H2T
+lapply (da_mono … H2T … H1T) -h -G -L -T #H
+elim (plus_xySz_x_false 0 l 0) //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/statictypestar_6.ma".
+include "basic_2/static/sta.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+definition lstas: ∀h. genv → lenv → nat → relation term ≝
+ λh,G,L. lstar … (sta h G L).
+
+interpretation "nat-iterated static type assignment (term)"
+ 'StaticTypeStar h G L l T U = (lstas h G L l T U).
+
+(* Basic eliminators ********************************************************)
+
+lemma lstas_ind_sn: ∀h,G,L,U2. ∀R:relation2 nat term.
+ R 0 U2 → (
+ ∀l,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 → ⦃G, L⦄ ⊢ U1 •* [h, l] U2 →
+ R l U1 → R (l+1) T
+ ) →
+ ∀l,T. ⦃G, L⦄ ⊢ T •*[h, l] U2 → R l T.
+/3 width=5 by lstar_ind_l/ qed-.
+
+lemma lstas_ind_dx: ∀h,G,L,T. ∀R:relation2 nat term.
+ R 0 T → (
+ ∀l,U1,U2. ⦃G, L⦄ ⊢ T •* [h, l] U1 → ⦃G, L⦄ ⊢ U1 •[h] U2 →
+ R l U1 → R (l+1) U2
+ ) →
+ ∀l,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R l U.
+/3 width=5 by lstar_ind_r/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lstas_inv_O: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 0] U → T = U.
+/2 width=4 by lstar_inv_O/ qed-.
+
+lemma lstas_inv_SO: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 1] U → ⦃G, L⦄ ⊢ T •[h] U.
+/2 width=1 by lstar_inv_step/ qed-.
+
+lemma lstas_inv_step_sn: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •[h] T & ⦃G, L⦄ ⊢ T •*[h, l] T2.
+/2 width=3 by lstar_inv_S/ qed-.
+
+lemma lstas_inv_step_dx: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l] T & ⦃G, L⦄ ⊢ T •[h] T2.
+/2 width=3 by lstar_inv_S_dx/ qed-.
+
+lemma lstas_inv_sort1: ∀h,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, l] X → X = ⋆((next h)^l k).
+#h #G #L #X #k #l #H @(lstas_ind_dx … H) -X -l //
+#l #X #X0 #_ #H #IHX destruct
+lapply (sta_inv_sort1 … H) -H #H destruct
+>iter_SO //
+qed-.
+
+lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l+1] X → ⊥.
+#h #G #L #X #p #l #H elim (lstas_inv_step_sn … H) -H
+#U #H #HUX elim (sta_inv_gref1 … H)
+qed-.
+
+lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
+ ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
+#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
+#l #X #X0 #_ #HX0 * #U #HTU #H destruct
+elim (sta_inv_bind1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+qed-.
+
+lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
+ ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
+#h #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
+#l #X #X0 #_ #HX0 * #U #HTU #H destruct
+elim (sta_inv_appl1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+qed-.
+
+lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U → ⦃G, L⦄ ⊢ T •*[h, l+1] U.
+#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn … H) -H
+#U #H #HUX lapply (sta_inv_cast1 … H) -H /2 width=3 by lstar_S/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lstas_refl: ∀h,G,L. reflexive … (lstas h G L 0).
+// qed.
+
+lemma sta_lstas: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T •*[h, 1] U.
+/2 width=1 by lstar_step/ qed.
+
+lemma lstas_step_sn: ∀h,G,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h] U1 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2 →
+ ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
+/2 width=3 by lstar_S/ qed.
+
+lemma lstas_step_dx: ∀h,G,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T2 •[h] U2 →
+ ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
+/2 width=3 by lstar_dx/ qed.
+
+lemma lstas_split: ∀h,G,L. inv_ltransitive … (lstas h G L).
+/2 width=1 by lstar_inv_ltransitive/ qed-.
+
+lemma lstas_sort: ∀h,G,L,l,k. ⦃G, L⦄ ⊢ ⋆k •*[h, l] ⋆((next h)^l k).
+#h #G #L #l @(nat_ind_plus … l) -l //
+#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/
+qed.
+
+lemma lstas_bind: ∀h,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] ⓑ{a,I}V.U.
+#h #I #G #L #V #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/
+qed.
+
+lemma lstas_appl: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
+ ∀V.⦃G, L⦄ ⊢ ⓐV.T •*[h, l] ⓐV.U.
+#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/
+qed.
+
+lemma lstas_cast: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U →
+ ∀W. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U.
+#h #G #L #T #U #l #H elim (lstas_inv_step_sn … H) -H /3 width=3 by sta_cast, lstas_step_sn/
+qed.
+
+(* Basic_1: removed theorems 7:
+ sty1_abbr sty1_appl sty1_bind sty1_cast2
+ sty1_correct sty1_lift sty1_trans
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/sta_aaa.ma".
+include "basic_2/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma lstas_aaa_conf: ∀h,G,L,l. Conf3 … (aaa G L) (lstas h G L l).
+/3 width=6 by sta_aaa_conf, lstar_Conf3/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/statictypestaralt_6.ma".
+include "basic_2/unfold/lstas_lift.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* alternative definition of lstas *)
+inductive lstasa (h): genv → relation4 lenv nat term term ≝
+| lstasa_O : ∀G,L,T. lstasa h G L 0 T T
+| lstasa_sort: ∀G,L,l,k. lstasa h G L l (⋆k) (⋆((next h)^l k))
+| lstasa_ldef: ∀G,L,K,V,W,U,i,l. ⇩[i] L ≡ K.ⓓV → lstasa h G K (l+1) V W →
+ ⇧[0, i+1] W ≡ U → lstasa h G L (l+1) (#i) U
+| lstasa_ldec: ∀G,L,K,W,V,V0,U,i,l. ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W •[h] V0 →
+ lstasa h G K l W V → ⇧[0, i+1] V ≡ U → lstasa h G L (l+1) (#i) U
+| lstasa_bind: ∀a,I,G,L,V,T,U,l. lstasa h G (L.ⓑ{I}V) l T U →
+ lstasa h G L l (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| lstasa_appl: ∀G,L,V,T,U,l. lstasa h G L l T U → lstasa h G L l (ⓐV.T) (ⓐV.U)
+| lstasa_cast: ∀G,L,W,T,U,l. lstasa h G L (l+1) T U → lstasa h G L (l+1) (ⓝW.T) U
+.
+
+interpretation "nat-iterated static type assignment (term) alternative"
+ 'StaticTypeStarAlt h G L l T U = (lstasa h G L l T U).
+
+(* Base properties **********************************************************)
+
+lemma sta_lstasa: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T ••*[h, 1] U.
+#h #G #L #T #U #H elim H -G -L -T -U
+/2 width=8 by lstasa_O, lstasa_sort, lstasa_ldef, lstasa_ldec, lstasa_bind, lstasa_appl, lstasa_cast/
+qed.
+
+lemma lstasa_step_dx: ∀h,G,L,T1,T,l. ⦃G, L⦄ ⊢ T1 ••*[h, l] T →
+ ∀T2. ⦃G, L⦄ ⊢ T •[h] T2 → ⦃G, L⦄ ⊢ T1 ••*[h, l+1] T2.
+#h #G #L #T1 #T #l #H elim H -G -L -T1 -T -l
+[ /2 width=1 by sta_lstasa/
+| #G #L #l #k #X #H >(sta_inv_sort1 … H) -X >commutative_plus //
+| #G #L #K #V #W #U #i #l #HLK #_ #HWU #IHVW #U2 #HU2
+ lapply (drop_fwd_drop2 … HLK) #H
+ elim (sta_inv_lift1 … HU2 … H … HWU) -H -U /3 width=6 by lstasa_ldef/
+| #G #L #K #W #V #V0 #U #i #l #HLK #HWl0 #_ #HVU #IHWV #U2 #HU2
+ lapply (drop_fwd_drop2 … HLK) #H
+ elim (sta_inv_lift1 … HU2 … H … HVU) -H -U /3 width=8 by lstasa_ldec/
+| #a #I #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H
+ elim (sta_inv_bind1 … H) -H #U #HU1 #H destruct /3 width=1 by lstasa_bind/
+| #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H
+ elim (sta_inv_appl1 … H) -H #U #HU1 #H destruct /3 width=1 by lstasa_appl/
+| /3 width=1 by lstasa_cast/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lstas_lstasa: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ⦃G, L⦄ ⊢ T ••*[h, l] U.
+#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /2 width=3 by lstasa_step_dx, lstasa_O/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lstasa_inv_lstas: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T ••*[h, l] U → ⦃G, L⦄ ⊢ T •*[h, l] U.
+#h #G #L #T #U #l #H elim H -G -L -T -U -l
+/2 width=8 by lstas_inv_SO, lstas_ldec, lstas_ldef, lstas_cast, lstas_appl, lstas_bind/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lstas_ind_alt: ∀h. ∀R:genv→relation4 lenv nat term term.
+ (∀G,L,T. R G L O T T) →
+ (∀G,L,l,k. R G L l (⋆k) (⋆((next h)^l k))) → (
+ ∀G,L,K,V,W,U,i,l.
+ ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V •*[h, l+1] W → ⇧[O, i+1] W ≡ U →
+ R G K (l+1) V W → R G L (l+1) (#i) U
+ ) → (
+ ∀G,L,K,W,V,V0,U,i,l.
+ ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W •[h] V0 →
+ ⦃G, K⦄ ⊢ W •*[h, l]V → ⇧[O, i+1] V ≡ U →
+ R G K l W V → R G L (l+1) (#i) U
+ ) → (
+ ∀a,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
+ R G (L.ⓑ{I}V) l T U → R G L l (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+ ) → (
+ ∀G,L,V,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
+ R G L l T U → R G L l (ⓐV.T) (ⓐV.U)
+ ) → (
+ ∀G,L,W,T,U,l. ⦃G, L⦄⊢ T •*[h, l+1] U →
+ R G L (l+1) T U → R G L (l+1) (ⓝW.T) U
+ ) →
+ ∀G,L,l,T,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R G L l T U.
+#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #G #L #l #T #U #H
+elim (lstas_lstasa … H) /3 width=10 by lstasa_inv_lstas/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/lstas.ma".
+include "basic_2/static/da_sta.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on degree assignment for terms ********************************)
+
+lemma lstas_da_conf: ∀h,g,G,L,T,U,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U →
+ ∀l2. ⦃G, L⦄ ⊢ T ▪[h, g] l2 → ⦃G, L⦄ ⊢ U ▪[h, g] l2-l1.
+#h #g #G #L #T #U #l1 #H @(lstas_ind_dx … H) -U -l1 //
+#l1 #U #U0 #_ #HU0 #IHTU #l2 #HT
+<minus_plus /3 width=3 by da_sta_conf/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_inv_refl_pos: ∀h,G,L,T,l. ⦃G, L⦄ ⊢ T •*[h, l+1] T → ⊥.
+#h #G #L #T #l #H elim (lstas_inv_step_sn … H)
+#U #HTU #_ elim (sta_da_ge … (l+1) HTU) -U
+#g #l0 #HT #Hl0 lapply (lstas_da_conf … H … HT) -H
+#H0T lapply (da_mono … HT … H0T) -h -G -L -T
+#H elim (discr_x_minus_xy … H) -H
+[ #H destruct /2 width=3 by le_plus_xSy_O_false/
+| -Hl0 <plus_n_Sm #H destruct
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/sta_lift.ma".
+include "basic_2/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on relocation *************************************************)
+
+lemma lstas_lift: ∀h,G,l. l_liftable (llstar … (sta h G) l).
+/3 width=10 by l_liftable_llstar, sta_lift/ qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (llstar … (sta h G) l).
+/3 width=6 by l_deliftable_sn_llstar, sta_inv_lift1/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_inv_lref1: ∀h,G,L,U,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] U →
+ (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
+ ⇧[0, i+1] W ≡ U
+ ) ∨
+ (∃∃K,W,V,V0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V0 &
+ ⦃G, K⦄ ⊢ W •*[h, l] V & ⇧[0, i+1] V ≡ U
+ ).
+#h #G #L #U #i #l #H elim (lstas_inv_step_sn … H) -H
+#X #H #HXU elim (sta_inv_lref1 … H) -H
+* #K #V #W #HLK #HVW #HWX
+lapply (drop_fwd_drop2 … HLK) #H0LK
+elim (lstas_inv_lift1 … HXU … H0LK … HWX) -H0LK -X
+/4 width=8 by lstas_step_sn, ex4_4_intro, ex3_3_intro, or_introl, or_intror/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lstas_fwd_correct: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
+ ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
+ ∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2.
+#h #G #L #T1 #U1 #HTU1 #T2 #l #H @(lstas_ind_dx … H) -l -T2 /2 width=3 by ex_intro/ -HTU1
+#l #T #T2 #_ #HT2 #_ -T1 -U1 -l
+elim (sta_fwd_correct … HT2) -T /2 width=2 by ex_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_total: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
+ ∀l. ∃U0. ⦃G, L⦄ ⊢ T •*[h, l] U0.
+#h #G #L #T #U #HTU #l @(nat_ind_plus … l) -l /2 width=2 by ex_intro/
+#l * #U0 #HTU0 elim (lstas_fwd_correct … HTU … HTU0) -U
+/3 width=4 by lstas_step_dx, ex_intro/
+qed-.
+
+lemma lstas_ldef: ∀h,G,L,K,V,i. ⇩[i] L ≡ K.ⓓV →
+ ∀W,l. ⦃G, K⦄ ⊢ V •*[h, l+1] W →
+ ∀U. ⇧[0, i+1] W ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
+#h #G #L #K #V #i #HLK #W #l #HVW #U #HWU
+lapply (drop_fwd_drop2 … HLK)
+elim (lstas_inv_step_sn … HVW) -HVW #W0
+elim (lift_total W0 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldef, lstas_lift/
+qed.
+
+lemma lstas_ldec: ∀h,G,L,K,W,i. ⇩[i] L ≡ K.ⓛW → ∀V0. ⦃G, K⦄ ⊢ W •[h] V0 →
+ ∀V,l. ⦃G, K⦄ ⊢ W •*[h, l] V →
+ ∀U. ⇧[0, i+1] V ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
+#h #G #L #K #W #i #HLK #V0 #HWV0 #V #l #HWV #U #HVU
+lapply (drop_fwd_drop2 … HLK) #H
+elim (lift_total W 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldec, lstas_lift/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/sta_sta.ma".
+include "basic_2/unfold/lstas_lift.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Main properties **********************************************************)
+
+theorem lstas_trans: ∀h,G,L. ltransitive … (lstas h G L).
+/2 width=3 by lstar_ltransitive/ qed-.
+
+theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h G L l).
+/3 width=7 by sta_mono, lstar_singlevalued/ qed-.
+
+theorem lstas_conf_le: ∀h,G,L,T,U1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U1 →
+ ∀U2,l2. l1 ≤ l2 → ⦃G, L⦄ ⊢ T •*[h, l2] U2 →
+ ⦃G, L⦄ ⊢ U1 •*[h, l2-l1] U2.
+#h #G #L #T #U1 #l1 #HTU1 #U2 #l2 #Hl12
+>(plus_minus_m_m … Hl12) in ⊢ (%→?); -Hl12 >commutative_plus #H
+elim (lstas_split … H) -H #U #HTU
+>(lstas_mono … HTU … HTU1) -T //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_sta_conf_pos: ∀h,G,L,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 →
+ ∀U2,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U2 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2.
+#h #G #L #T #U1 #HTU1 #U2 #l #HTU2
+lapply (lstas_conf_le … T U1 1 … HTU2) -HTU2 /2 width=1 by sta_lstas/
+qed-.
+
+lemma lstas_strip_pos: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
+ ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+ ∃∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2 & ⦃G, L⦄ ⊢ U1 •*[h, l+1] U2.
+#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12
+elim (lstas_fwd_correct … HTU1 … HT12)
+lapply (lstas_sta_conf_pos … HTU1 … HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/statictype_5.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/substitution/drop.ma".
+include "basic_2/static/sh.ma".
+
+(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
+
+(* activate genv *)
+inductive sta (h:sh): relation4 genv lenv term term ≝
+| sta_sort: ∀G,L,k. sta h G L (⋆k) (⋆(next h k))
+| sta_ldef: ∀G,L,K,V,W,U,i. ⇩[i] L ≡ K.ⓓV → sta h G K V W →
+ ⇧[0, i + 1] W ≡ U → sta h G L (#i) U
+| sta_ldec: ∀G,L,K,W,V,U,i. ⇩[i] L ≡ K.ⓛW → sta h G K W V →
+ ⇧[0, i + 1] W ≡ U → sta h G L (#i) U
+| sta_bind: ∀a,I,G,L,V,T,U. sta h G (L.ⓑ{I}V) T U →
+ sta h G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| sta_appl: ∀G,L,V,T,U. sta h G L T U → sta h G L (ⓐV.T) (ⓐV.U)
+| sta_cast: ∀G,L,W,T,U. sta h G L T U → sta h G L (ⓝW.T) U
+.
+
+interpretation "static type assignment (term)"
+ 'StaticType h G L T U = (sta h G L T U).
+
+(* Basic inversion lemmas ************************************************)
+
+fact sta_inv_sort1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀k0. T = ⋆k0 →
+ U = ⋆(next h k0).
+#h #G #L #T #U * -G -L -T -U
+[ #G #L #k #k0 #H destruct //
+| #G #L #K #V #W #U #i #_ #_ #_ #k0 #H destruct
+| #G #L #K #W #V #U #i #_ #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #U #_ #k0 #H destruct
+| #G #L #V #T #U #_ #k0 #H destruct
+| #G #L #W #T #U #_ #k0 #H destruct
+qed-.
+
+(* Basic_1: was: sty0_gen_sort *)
+lemma sta_inv_sort1: ∀h,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k •[h] U → U = ⋆(next h k).
+/2 width=5 by sta_inv_sort1_aux/ qed-.
+
+fact sta_inv_lref1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀j. T = #j →
+ (∃∃K,V,W. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h] W &
+ ⇧[0, j+1] W ≡ U
+ ) ∨
+ (∃∃K,W,V. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V &
+ ⇧[0, j+1] W ≡ U
+ ).
+#h #G #L #T #U * -G -L -T -U
+[ #G #L #k #j #H destruct
+| #G #L #K #V #W #U #i #HLK #HVW #HWU #j #H destruct /3 width=6 by or_introl, ex3_3_intro/
+| #G #L #K #W #V #U #i #HLK #HWV #HWU #j #H destruct /3 width=6 by or_intror, ex3_3_intro/
+| #a #I #G #L #V #T #U #_ #j #H destruct
+| #G #L #V #T #U #_ #j #H destruct
+| #G #L #W #T #U #_ #j #H destruct
+]
+qed-.
+
+(* Basic_1: was sty0_gen_lref *)
+lemma sta_inv_lref1: ∀h,G,L,U,i. ⦃G, L⦄ ⊢ #i •[h] U →
+ (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h] W &
+ ⇧[0, i+1] W ≡ U
+ ) ∨
+ (∃∃K,W,V. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V &
+ ⇧[0, i+1] W ≡ U
+ ).
+/2 width=3 by sta_inv_lref1_aux/ qed-.
+
+fact sta_inv_gref1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀p0. T = §p0 → ⊥.
+#h #G #L #T #U * -G -L -T -U
+[ #G #L #k #p0 #H destruct
+| #G #L #K #V #W #U #i #_ #_ #_ #p0 #H destruct
+| #G #L #K #W #V #U #i #_ #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #U #_ #p0 #H destruct
+| #G #L #V #T #U #_ #p0 #H destruct
+| #G #L #W #T #U #_ #p0 #H destruct
+qed-.
+
+lemma sta_inv_gref1: ∀h,G,L,U,p. ⦃G, L⦄ ⊢ §p •[h] U → ⊥.
+/2 width=8 by sta_inv_gref1_aux/ qed-.
+
+fact sta_inv_bind1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
+ ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •[h] Z & U = ⓑ{b,J}Y.Z.
+#h #G #L #T #U * -G -L -T -U
+[ #G #L #k #b #J #X #Y #H destruct
+| #G #L #K #V #W #U #i #_ #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #U #i #_ #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #U #_ #b #J #X #Y #H destruct
+| #G #L #W #T #U #_ #b #J #X #Y #H destruct
+]
+qed-.
+
+(* Basic_1: was: sty0_gen_bind *)
+lemma sta_inv_bind1: ∀h,b,J,G,L,Y,X,U. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X •[h] U →
+ ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •[h] Z & U = ⓑ{b,J}Y.Z.
+/2 width=3 by sta_inv_bind1_aux/ qed-.
+
+fact sta_inv_appl1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀X,Y. T = ⓐY.X →
+ ∃∃Z. ⦃G, L⦄ ⊢ X •[h] Z & U = ⓐY.Z.
+#h #G #L #T #U * -G -L -T -U
+[ #G #L #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #_ #X #Y #H destruct
+| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #W #T #U #_ #X #Y #H destruct
+]
+qed-.
+
+(* Basic_1: was: sty0_gen_appl *)
+lemma sta_inv_appl1: ∀h,G,L,Y,X,U. ⦃G, L⦄ ⊢ ⓐY.X •[h] U →
+ ∃∃Z. ⦃G, L⦄ ⊢ X •[h] Z & U = ⓐY.Z.
+/2 width=3 by sta_inv_appl1_aux/ qed-.
+
+fact sta_inv_cast1_aux: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∀X,Y. T = ⓝY.X →
+ ⦃G, L⦄ ⊢ X •[h] U.
+#h #G #L #T #U * -G -L -T -U
+[ #G #L #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #_ #X #Y #H destruct
+| #G #L #V #T #U #_ #X #Y #H destruct
+| #G #L #W #T #U #HTU #X #Y #H destruct //
+]
+qed-.
+
+(* Basic_1: was: sty0_gen_cast *)
+lemma sta_inv_cast1: ∀h,G,L,X,Y,U. ⦃G, L⦄ ⊢ ⓝY.X •[h] U → ⦃G, L⦄ ⊢ X •[h] U.
+/2 width=4 by sta_inv_cast1_aux/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/sta.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* STATIC TYPE ASSIGNMENT FOR TERMS *****************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_sta: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G, L⦄ ⊢ T •[h] U.
+#h #G #L #T #A #H elim H -G -L -T -A
+[ /2 width=2 by sta_sort, ex_intro/
+| * #G #L #K [ #V | #W ] #B #i #HLK #_ * [ #W | #V ] #HVW
+ elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldef, sta_ldec, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_bind, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_bind, ex_intro/
+| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_appl, ex_intro/
+| #G #L #W #T #A #_ #_ #_ * /3 width=2 by sta_cast, ex_intro/
+]
+qed-.
+
+lemma sta_aaa_conf: ∀h,G,L. Conf3 … (aaa G L) (sta h G L).
+#h #G #L #T #A #H elim H -G -L -T -A
+[ #G #L #k #U #H
+ lapply (sta_inv_sort1 … H) -H #H destruct //
+| #I #G #L #K #V #B #i #HLK #HV #IHV #U #H
+ elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HU
+ lapply (drop_mono … HLK0 … HLK) -HLK0 #H0 destruct
+ lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+ @(aaa_lift … HLK … HU) -HU -L /2 width=2 by/
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
+ elim (sta_inv_bind1 … H) -H #U #HTU #H destruct /3 width=2 by aaa_abbr/
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
+ elim (sta_inv_bind1 … H) -H #U #HTU #H destruct /3 width=2 by aaa_abst/
+| #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
+ elim (sta_inv_appl1 … H) -H #U #HTU #H destruct /3 width=3 by aaa_appl/
+| #G #L #V #T #A #_ #_ #IHV #IHT #X #H
+ lapply (sta_inv_cast1 … H) -H /2 width=2 by/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/drop_drop.ma".
+include "basic_2/static/sta.ma".
+
+(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was: sty0_lift *)
+lemma sta_lift: ∀h,G. l_liftable (sta h G).
+#h #G #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1
+[ #G #L1 #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+ >(lift_inv_sort1 … H1) -X1
+ >(lift_inv_sort1 … H2) -X2 //
+| #G #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
+ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by sta_ldef/
+ | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+ lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by sta_ldef, drop_inv_gen/
+ ]
+| #G #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
+ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
+ elim (lift_total V1 (d-i-1) e) /3 width=9 by sta_ldec/
+ | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+ lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by sta_ldec, drop_inv_gen/
+ ]
+| #a #I #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+ elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+ elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+ lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by sta_bind, drop_skip/
+| #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+ elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+ elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+ lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by sta_appl/
+| #G #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
+ elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by sta_cast/
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+lemma sta_inv_lift1: ∀h,G. l_deliftable_sn (sta h G).
+#h #G #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2
+[ #G #L2 #k #L1 #s #d #e #_ #X #H
+ >(lift_inv_sort2 … H) -X /2 width=3 by sta_sort, lift_sort, ex2_intro/
+| #G #L2 #K2 #V2 #W2 #W #i #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
+ elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
+ elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by sta_ldef, ex2_intro/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+ elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
+ elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by sta_ldef, le_S, ex2_intro/
+ ]
+| #G #L2 #K2 #W2 #V2 #W #i #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+ elim (IHWV2 … HK21 … HW12) -K2 #V1 #_ #HWV1
+ elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by sta_ldec, ex2_intro/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+ elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
+ elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by sta_ldec, le_S, ex2_intro/
+ ]
+| #a #I #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by sta_bind, drop_skip, lift_bind, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by sta_appl, lift_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
+ elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by sta_cast, ex2_intro/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+(* Basic_1: was: sty0_correct *)
+lemma sta_fwd_correct: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∃T0. ⦃G, L⦄ ⊢ U •[h] T0.
+#h #G #L #T #U #H elim H -G -L -T -U
+[ /2 width=2/
+| #G #L #K #V #W #W0 #i #HLK #_ #HW0 * #V0 #HWV0
+ lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+ elim (lift_total V0 0 (i+1)) /3 width=11 by ex_intro, sta_lift/
+| #G #L #K #W #V #V0 #i #HLK #HWV #HWV0 #_
+ lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+ elim (lift_total V 0 (i+1)) /3 width=11 by ex_intro, sta_lift/
+| #a #I #G #L #V #T #U #_ * /3 width=2 by sta_bind, ex_intro/
+| #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2 by sta_appl, ex_intro/
+| #G #L #W #T #U #_ * /2 width=2 by ex_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/multiple/llpx_sn_drop.ma".
+include "basic_2/static/sta.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ******************************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+lemma sta_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R →
+ ∀h,G. s_r_confluent1 … (sta h G) (llpx_sn R 0).
+#R #H1R #H2R #h #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
+| #G #Ls #Ks #V1s #W2s #V2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #G #Ls #Ks #V1s #W1s #V2s #i #HLKs #_ #HV12s #IHVW1s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HV12s) -HLKs -HLKd -HV12s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/drop_drop.ma".
+include "basic_2/static/sta.ma".
+
+(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
+
+(* Main properties **********************************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+theorem sta_mono: ∀h,G,L. singlevalued … (sta h G L).
+#h #G #L #T #U1 #H elim H -G -L -T -U1
+[ #G #L #k #X #H >(sta_inv_sort1 … H) -X //
+| #G #L #K #V #W #U1 #i #HLK #_ #HWU1 #IHVW #U2 #H
+ elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HW0U2
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+ lapply (IHVW … HVW0) -IHVW -HVW0 #H destruct
+ >(lift_mono … HWU1 … HW0U2) -W0 -U1 //
+| #G #L #K #W #V #U1 #i #HLK #_ #HWU1 #IHWV #U2 #H
+ elim (sta_inv_lref1 … H) -H * #K0 #W0 #V0 #HLK0 #HWV0 #HV0U2
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+ lapply (IHWV … HWV0) -IHWV -HWV0 #H destruct
+ >(lift_mono … HWU1 … HV0U2) -W -U1 //
+| #a #I #G #L #V #T #U1 #_ #IHTU1 #X #H
+ elim (sta_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #V #T #U1 #_ #IHTU1 #X #H
+ elim (sta_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #W #T #U1 #_ #IHTU1 #U2 #H
+ lapply (sta_inv_cast1 … H) -H /2 width=1 by/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • break [ term 46 h ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StaticType $h $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • • * break [ term 46 h , break term 46 l ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StaticTypeStarAlt $h $G $L $l $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break 𝐇𝐍 ⦃ term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'HdNormal $L $T }.
+
+include "basic_2/reduction/cpr_tshf.ma".
+
+(* CONTEXT-SENSITIVE WEAK HEAD NORMAL TERMS *********************************)
+
+definition chnf: lenv → predicate term ≝ λL. NF … (cpr L) tshf.
+
+interpretation
+ "context-sensitive head normality (term)"
+ 'HdNormal L T = (chnf L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma chnf_inv_tshf: ∀L,T. L ⊢ 𝐇𝐍⦃T⦄ → T ≈ T.
+normalize /2 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tshf_thnf: ∀T. T ≈ T → ⋆ ⊢ 𝐇𝐍⦃T⦄.
+#T #HT #T2 #H elim (cpr_fwd_tshf1 … H) -H //
+#H elim H //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/tshf.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Forward lemmas on same head forms for terms ******************************)
+
+lemma cpr_fwd_tshf1: ∀L,T1,T2. L ⊢ T1 ➡ T2 → T1 ≈ T1 →
+ T2 ≈ T1 ∨ (L = ⋆ → ⊥).
+#L #T1 #T2 #H elim H -L -T1 -T2
+[ /2 width=1/
+| #L #K #V1 #V2 #W2 #i #HLK #_ #_ #_ #_
+ @or_intror #H destruct
+ lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H
+ elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct /2 width=1/
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
+ lapply (IHT12 HT1U2) -IHT12 -HT1U2 * #HUT2 /3 width=1/
+ lapply (simple_tshf_repl_sn … HUT2 HT1) /3 width=1/
+| #L #V #T #T1 #T2 #_ #_ #_ #H
+ elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct
+| #L #V #T1 #T2 #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
+| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #a #L #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H
+ elim (simple_inv_bind … H)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 ≈ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'Hom $L $T1 $T2 }.
+
+notation "hvbox( L ⊢ break 𝐇𝐑 ⦃ term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'HdReducible $L $T }.
+
+notation "hvbox( L ⊢ break 𝐇𝐈 ⦃ term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'NotHdReducible $L $T }.
+
+include "basic_2/grammar/term_simple.ma".
+
+(* SAME HEAD TERM FORMS *****************************************************)
+
+inductive tshf: relation term ≝
+ | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I})
+ | tshf_abbr: ∀V1,V2,T1,T2. tshf (-ⓓV1. T1) (-ⓓV2. T2)
+ | tshf_abst: ∀a,V1,V2,T1,T2. tshf (ⓛ{a}V1. T1) (ⓛ{a}V2. T2)
+ | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄ →
+ tshf (ⓐV1. T1) (ⓐV2. T2)
+.
+
+interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1.
+#T1 #T2 #H elim H -T1 -T2 /2 width=1/
+qed.
+
+lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2.
+#T1 #T2 #H elim H -T1 -T2 // /2 width=1/
+qed.
+
+lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1.
+/3 width=2/ qed.
+
+lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 #H elim H -T1 -T2 //
+[ #V1 #V2 #T1 #T2 #H
+ elim (simple_inv_bind … H)
+| #a #V1 #V2 #T1 #T2 #H
+ elim (simple_inv_bind … H)
+]
+qed. (**) (* remove from index *)
+
+lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+/3 width=3/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀a,I,W1,U1. T1 = ⓑ{a,I}W1.U1 →
+ ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
+ (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
+#T1 #T2 * -T1 -T2
+[ #J #a #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
+| #b #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
+| #V1 #V2 #T1 #T2 #_ #_ #_ #a #I #W1 #U1 #H destruct
+]
+qed.
+
+lemma tshf_inv_bind1: ∀a,I,W1,U1,T2. ⓑ{a,I}W1.U1 ≈ T2 →
+ ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
+ (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
+/2 width=5/ qed-.
+
+fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 →
+ ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
+ I = Appl & T2 = ⓐW2. U2.
+#T1 #T2 * -T1 -T2
+[ #J #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
+| #a #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/
+]
+qed.
+
+lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 →
+ ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
+ I = Appl & T2 = ⓐW2. U2.
+/2 width=4/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta_aaa.ma".
+include "basic_2/reducibility/cpr_aaa.ma".
+include "basic_2/reducibility/xpr.ma".
+
+(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma xpr_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •➡[g] U → L ⊢ U ⁝ A.
+#h #g #L #T #A #HT #U * /2 width=3/ /2 width=6/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta_lift.ma".
+include "basic_2/reducibility/cpr_lift.ma".
+include "basic_2/reducibility/xpr.ma".
+
+(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma xpr_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 •➡[g] U2 →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & ⦃h, L. ⓛV1⦄ ⊢ T1 •➡[g] T2 &
+ U2 = ⓛ{a}V2. T2.
+#h #g #a #L #V1 #T1 #U2 *
+[ #H elim (cpr_inv_abst1 … H Abst V1) /3 width=5/
+| #l #H elim (ssta_inv_bind1 … H) /3 width=5/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma xpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ ∀h,g. ⦃h, K⦄ ⊢ T1 •➡[g] T2 → ⦃h, L⦄ ⊢ U1 •➡[g] U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #h #g *
+/3 width=9/ /3 width=10/
+qed.
+
+lemma xpr_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •➡[g] U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •➡[g] T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 * [ #HU12 | #l #HU12 ]
+[ elim (cpr_inv_lift1 … HLK … HTU1 … HU12) -L -U1 /3 width=3/
+| elim (ssta_inv_lift1 … HU12 … HLK … HTU1) -L -U1 /3 width=4/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lsubss_ssta.ma".
+include "basic_2/reducibility/xpr.ma".
+
+(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
+
+(* Properties on lenv ref for stratified type assignment ********************)
+
+lemma lsubss_xpr_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
+ ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡[g] T2.
+#h #g #L1 #L2 #HL12 #T1 #T2 * [ | #l ] #HT12
+[ lapply (lsubss_fwd_lsubs2 … HL12) -HL12 /3 width=3/
+| /3 width=4/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • ➡ * break [ term 46 g ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'XPRedStar $h $g $L $T1 $T2 }.
+
+notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • ⬊ * break [ term 46 g ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'XSN $h $g $L $T }.
+
+include "basic_2/static/lsubss.ma".
+include "basic_2/reducibility/xpr.ma".
+(*
+include "basic_2/reducibility/cnf.ma".
+*)
+(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
+
+definition xprs: ∀h. sd h → lenv → relation term ≝
+ λh,g,L. TC … (xpr h g L).
+
+interpretation "extended parallel computation (term)"
+ 'XPRedStar h g L T1 T2 = (xprs h g L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma xprs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → R T → R T2) →
+ ∀T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T2.
+#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma xprs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → R T → R T1) →
+ ∀T1. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T1.
+#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma xprs_refl: ∀h,g,L. reflexive … (xprs h g L).
+/2 width=1/ qed.
+
+lemma xprs_strap1: ∀h,g,L,T1,T,T2.
+ ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2.
+/2 width=3/ qed.
+
+lemma xprs_strap2: ∀h,g,L,T1,T,T2.
+ ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2.
+/2 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+(*
+axiom xprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U.
+#L #T #U #H @(xprs_ind_dx … H) -T //
+#T0 #T #H1T0 #_ #IHT #H2T0
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
+qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/xpr_aaa.ma".
+include "basic_2/computation/xprs.ma".
+
+(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma xprs_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •➡*[g] U → L ⊢ U ⁝ A.
+#h #g #L #T #A #HT #U #H @(xprs_ind … H) -U // /2 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/cprs.ma".
+include "basic_2/computation/xprs.ma".
+
+(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
+
+(* properties on context sensitive parallel computation for terms ***********)
+
+lemma cprs_xprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2.
+#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/xpr_lift.ma".
+include "basic_2/computation/cprs.ma".
+include "basic_2/computation/xprs.ma".
+
+(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma xprs_fwd_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •➡*[g] U2 →
+ ∃∃V2,T2. L ⊢ V1 ➡* V2 & U2 = ⓛ{a}V2. T2.
+#h #g #a #L #V1 #T1 #U2 #H @(xprs_ind … H) -U2 /2 width=4/
+#U #U2 #_ #HU2 * #V #T #HV1 #H destruct
+elim (xpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #_ #H destruct /3 width=4/
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma xprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 →
+ ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •➡*[g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃h, L⦄ ⊢ U1 •➡*[g] U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 #HT12 @(xprs_ind … HT12) -T2
+[ -HLK #T2 #HT12
+ <(lift_mono … HTU1 … HT12) -T1 //
+| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (xpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/
+]
+qed.
+
+lemma xprs_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •➡*[g] U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •➡*[g] T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 #HU12 @(xprs_ind … HU12) -U2 /2 width=3/
+-HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1
+elim (xpr_inv_lift1 … HLK … HTU … HU2) -U -HLK /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reducibility/xpr_lsubss.ma".
+include "basic_2/computation/xprs.ma".
+
+(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
+
+(* Properties on lenv ref for stratified type assignment ********************)
+
+lemma lsubss_xprs_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
+ ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2.
+#h #g #L1 #L2 #HL12 #T1 #T2 #H @(xprs_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (lsubss_xpr_trans … HL12 … HT2) -L2 /2 width=3/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/xprs.ma".
+
+(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************)
+
+theorem xprs_trans: ∀h,g,L. transitive … (xprs h g L).
+/2 width=3/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/xoa/ex_5_4.ma".
+include "ground_2/xoa/or_3.ma".
+include "ground_2/ynat/ynat_minus_sn.ma".
+include "ground_2/ynat/ynat_minus_dx.ma".
+include "basic_2A/notation/relations/psubst_6.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* activate genv *)
+inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
+| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
+| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m →
+ ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W
+| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
+ cpy l m G L V1 V2 → cpy (↑l) m G (L.ⓑ{I}V1) T1 T2 →
+ cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
+ cpy l m G L V1 V2 → cpy l m G L T1 T2 →
+ cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation "context-sensitive extended ordinary substritution (term)"
+ 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
+[ //
+| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
+ elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_succ, cpy_bind/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
+#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
+#I #G #K #V #T1 elim T1 -T1
+[ * #i #L #l #HLK
+ /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
+ elim (lt_or_eq_or_gt i l) #Hil
+ /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i)
+ /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1
+ /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK
+ /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
+ ]
+]
+qed-.
+
+lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
+ ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
+ ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
+[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
+| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
+[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
+ lapply (drop_fwd_length_lt2 … HLK) #Hil
+ lapply (ylt_inj … Hil) -Hil #Hil
+ lapply (yle_ylt_trans … Hil Hli) #Hl
+ @(cpy_subst … Hli … HLK HVW)
+ <yplus_minus_dx /2 width=1 by ylt_fwd_le/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #HT12 #IHV normalize in match (|?|);
+ /2 width=1 by cpy_bind/
+| /2 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
+#G #L #T1 #T2 #l #m #HT12
+lapply (cpy_weak … HT12 0 (l + m) ? ?) -HT12
+/2 width=2 by cpy_weak_top/
+qed-.
+(*
+lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, i-l] T & ⦃G, L⦄ ⊢ T ▶[i, l+m-i] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+ elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+ elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #T1 #l #m #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
+ [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
+ | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+ elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim
+ #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hmi
+ @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 //
+ elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
+ <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
+ /3 width=2 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
+ /3 width=2 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
+ T2 = ⓪{J} ∨
+ ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{I}V &
+ ⬆[O, i+1] V ≡ T2 &
+ J = LRef i.
+#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
+[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{J}V &
+ ⬆[O, i+1] V ≡ T2 &
+ I = LRef i.
+/2 width=4 by cpy_inv_atom1_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
+#G #L #T2 #k #l #m #H
+elim (cpy_inv_atom1 … H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V. l ≤ i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{I}V &
+ ⬆[O, i+1] V ≡ T2.
+#G #L #T2 #i #l #m #H
+elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
+qed-.
+
+lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
+#G #L #T2 #p #l #m #H
+elim (cpy_inv_atom1 … H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[↑l, m] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[↑l, m] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+/2 width=3 by cpy_inv_bind1_aux/ qed-.
+
+fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+ ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
+ U2 = ⓕ{I}V2.T2.
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
+ U2 = ⓕ{I}V2.T2.
+/2 width=3 by cpy_inv_flat1_aux/ qed-.
+
+
+fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ //
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
+ elim (ylt_yle_false … Hli) -Hli //
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
+]
+qed-.
+
+lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
+/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
+ ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
+#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
+/2 width=4 by cpy_inv_refl_O2/
+qed-.
+
+(* Basic_1: removed theorems 25:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/substitution/cpy_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem cpy_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶[l1, m1] T1 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶[l1, m1] T.
+#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
+ elim (cpy_inv_lref1 … H) -H
+ [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
+ ]
+| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
+ /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem cpy_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶[l1, m1] T1 →
+ ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+ (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶[l1, m1] T.
+#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
+ elim (cpy_inv_lref1 … H1) -H1
+ [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
+ [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/
+ ]
+ ]
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0
+ [ -H #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+ | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
+ ]
+| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Note: the constant 1 comes from cpy_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 →
+ ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2.
+#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
+[ #I #G #L #l #m #T2 #H #Hm
+ elim (cpy_inv_atom1 … H) -H
+ [ #H destruct //
+ | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
+ lapply (ylt_yle_trans … (l+m) … Hilm2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
+ ]
+| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
+ lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+ >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
+| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
+ elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
+ elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
+]
+qed-.
+
+theorem cpy_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
+ lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1
+ lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
+ >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ elim (IHV10 … HV02) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02) -V0 //
+ elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+ ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+ lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
+ lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+ lapply (ylt_inv_inj … Hil) -Hil #Hil
+ lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
+ elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, drop_skip, yle_succ/
+| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+ ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+ lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
+ elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+ [ -Hltl
+ lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm
+ elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
+ elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
+ | -Hlti
+ elim (yle_inv_inj2 … Hltl) -Hltl #ltt #Hltl #H destruct
+ lapply (transitive_le … Hltl Hil) -Hltl #Hlti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
+ /4 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
+ ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, drop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+ ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+ l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
+ lapply (yle_trans … Hllt … Hlti) -Hllt #Hil
+ elim (yle_inv_inj2 … Hil) -Hil #ll #Hlli #H0 destruct
+ lapply (lift_inv_lref1_ge … H … Hlli) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hlli
+ /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=6 by cpy_bind, drop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=6 by cpy_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma cpy_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt + mt ≤ l →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
+ lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+ lapply (ylt_inv_inj … Hil) -Hil #Hil
+ lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct
+ elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -IHU12 -HTU1
+ /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt ≤ l → yinj l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
+ lapply (yle_fwd_plus_ge_inj … Hltl Hlmlmt) #Hmmt
+ elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
+ [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) /2 width=1 by ylt_inj/
+ [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
+ elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil
+ /3 width=5 by cpy_subst, ex2_intro/
+ | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+ lapply (yle_trans … Hltl (i-m) ?) /2 width=1 by yle_inj/ -Hltl #Hltim
+ lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
+ #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
+ >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -U1
+ /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK //
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ yinj l + m ≤ lt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
+ lapply (yle_trans … Hlmlt … Hlti) #Hlmi
+ elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt
+ elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi
+ lapply (lift_inv_lref2_ge … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+ lapply (drop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
+ elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hlmi -Hlim
+ #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
+ [ /2 width=1 by monotonic_yle_minus_dx/
+ | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (yle_inv_plus_inj2 … Hlmtl) #_ #Hmlt
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
+ >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation ***********************************)
+
+lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[l, lt + mt - (yinj l + m)] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
+lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt ≤ l → lt + mt ≤ yinj l + m →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
+lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
+elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
+elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2
+elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
+lapply (cpy_weak … HU2 l m ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_neg.ma".
+include "basic_2A/substitution/lift_lift.ma".
+include "basic_2A/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Inversion lemmas on negated relocation ***********************************)
+
+lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+ ∀i. l ≤ yinj i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) →
+ (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨
+ ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W &
+ (∀V. ⬆[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥).
+#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
+[ /3 width=2 by or_introl/
+| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
+ elim (lt_or_ge j i) #Hij
+ [ @or_intror @(ex5_4_intro … HLK) // -HLK
+ [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
+ #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
+ | -HnW /2 width=7 by lift_inv_lref2_be/
+ ]
+ | elim (lift_split … HVW i j) -HVW //
+ #X #_ #H elim HnW -HnW //
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ [ /4 width=9 by nlift_bind_sn, or_introl/
+ | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
+ ]
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
+ [ /4 width=9 by nlift_bind_dx, or_introl/
+ | * #J #K #W #j #Hlj #Hji #HLK #HnW
+ elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj
+ lapply (ylt_O … Hj) -Hj #Hj
+ lapply (drop_inv_drop1_lt … HLK ?) // -HLK #HLK
+ >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
+ #HnU1 <minus_le_minus_minus_comm in HnW;
+ /5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
+ ]
+ ]
+| #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ [ /4 width=9 by nlift_flat_sn, or_introl/
+ | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
+ ]
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
+ [ /4 width=9 by nlift_flat_dx, or_introl/
+ | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubst $G $L $T1 $l $m $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/psubststar_6.ma".
+include "basic_2A/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+definition cpys: ynat → ynat → relation4 genv lenv term term ≝
+ λl,m,G. LTC … (cpy l m G).
+
+interpretation "context-sensitive extended multiple substritution (term)"
+ 'PSubstStar G L T1 l m T2 = (cpys l m G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpys_ind: ∀G,L,T1,l,m. ∀R:predicate term. R T1 →
+ (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → R T → R T2) →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T2.
+#G #L #T1 #l #m #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma cpys_ind_dx: ∀G,L,T2,l,m. ∀R:predicate term. R T2 →
+ (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → R T → R T1) →
+ ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T1.
+#G #L #T2 #l #m #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cpy_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+/2 width=1 by inj/ qed.
+
+lemma cpys_strap1: ∀G,L,T1,T,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by step/ qed-.
+
+lemma cpys_strap2: ∀G,L,T1,T,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+lemma lsuby_cpys_trans: ∀G,l,m. lsub_trans … (cpys l m G) (lsuby l m).
+/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
+qed-.
+
+lemma cpys_refl: ∀G,L,l,m. reflexive … (cpys l m G L).
+/2 width=1 by cpy_cpys/ qed.
+
+lemma cpys_bind: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[↑l, m] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] ⓑ{a,I}V2.T2.
+#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2
+[ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
+| /3 width=5 by cpys_strap1, cpy_bind/
+]
+qed.
+
+lemma cpys_flat: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+ ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] ⓕ{I}V2.T2.
+#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2
+[ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
+| /3 width=5 by cpys_strap1, cpy_flat/
+qed.
+
+lemma cpys_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T2 →
+ ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
+ ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H #l1 #l2 #Hl21 #Hlm12 @(cpys_ind … H) -T2
+/3 width=7 by cpys_strap1, cpy_weak/
+qed-.
+
+lemma cpys_weak_top: ∀G,L,T1,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2
+/3 width=4 by cpys_strap1, cpy_weak_top/
+qed-.
+
+lemma cpys_weak_full: ∀G,L,T1,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
+#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2
+/3 width=5 by cpys_strap1, cpy_weak_full/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpys_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #T1 #l #m #HTU1 #Hllt #Hlmlmt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
+ elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
+/2 width=3 by transitive_le/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶*[l, m] T2 → T2 = ⋆k.
+#G #L #T2 #k #l #m #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_sort1 … HT2) -HT2 //
+qed-.
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶*[l, m] T2 → T2 = §p.
+#G #L #T2 #p #l #m #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_gref1 … HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 &
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[↑l, m] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2
+ /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 &
+ U2 = ⓕ{I}V2.T2.
+#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (cpy_inv_flat1 … HU2) -HU2
+ /3 width=5 by cpys_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 0] T2 → T1 = T2.
+#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀l,m:nat.
+ ⦃G, L⦄ ⊢ U1 ▶*[l, m] U2 → ∀T1. ⬆[l, m] T1 ≡ U1 → U1 = U2.
+#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind … H) -U2
+/2 width=7 by cpy_inv_lift1_eq/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/psubststaralt_6.ma".
+include "basic_2A/multiple/cpys_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* alternative definition of cpys *)
+inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
+| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I})
+| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m →
+ ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (↓(l+m-i)) G K V1 V2 →
+ ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2
+| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
+ cpysa l m G L V1 V2 → cpysa (↑l) m G (L.ⓑ{I}V1) T1 T2 →
+ cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
+ cpysa l m G L V1 V2 → cpysa l m G L T1 T2 →
+ cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+ "context-sensitive extended multiple substritution (term) alternative"
+ 'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
+[ //
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
+| /4 width=1 by lsuby_succ, cpysa_bind/
+| /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T.
+#G #T elim T -T //
+#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
+qed.
+
+lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T →
+ ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2.
+#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m
+[ #I #G #L #l #m #X #H
+ elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
+| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H
+ lapply (drop_fwd_drop2 … HLK) #H0LK
+ lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H
+ elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
+ /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
+| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H
+ elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
+| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H
+ elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2.
+/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
+ (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) →
+ (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m →
+ ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ↓(l+m-i)] V2 →
+ ⬆[O, i+1] V2 ≡ W2 → R O (↓(l+m-i)) G K V1 V2 → R l m G L (#i) W2
+ ) →
+ (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[↑l, m] T2 → R l m G L V1 V2 →
+ R (↑l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 →
+ R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2.
+#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m
+/3 width=8 by cpysa_inv_cpys/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/substitution/cpy_cpy.ma".
+include "basic_2A/multiple/cpys_alt.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2.
+#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
+
+lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+ (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
+
+lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
+
+lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
+
+lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
+ ∀i. l ≤ i → i ≤ l + m →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2.
+#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2
+[ /2 width=3 by ex2_intro/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02
+ elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+ >ymax_pre_sn_comm //
+]
+qed-.
+
+lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj l + m)] T2 &
+ ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
+lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
+
+theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 →
+ (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
+
+theorem cpys_trans_eq: ∀G,L,T1,T,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 →
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by trans_TC/ qed-.
+
+theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
+
+theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 →
+ ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2.
+#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 //
+[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2
+ elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ]
+ [ lapply (cpys_weak_full … HW2) -HW2 #HW2
+ lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 //
+ [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
+ #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 //
+ | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2
+ elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2
+ /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hli -Hilm
+ #X #_ #H elim (lift_inv_lref2_be … H) -H //
+ ]
+| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H
+ #V #T #HV2 #HT2 #H destruct
+ lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
+| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H
+ #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/xoa/ex_4_4.ma".
+include "ground_2/xoa/ex_6_5.ma".
+include "basic_2A/substitution/cpy_lift.ma".
+include "basic_2A/multiple/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m.
+ l ≤ yinj i → i < l + m →
+ ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ↓(l+m-i)] U1 →
+ ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2.
+#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1
+[ /3 width=5 by cpy_cpys, cpy_subst/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+ elim (lift_total U 0 (i+1)) #U0 #HU0
+ lapply (IHU … HU0) -IHU #H
+ lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+ lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+ lapply (cpy_weak … HU02 l m ? ?) -HU02
+ [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+ >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
+]
+qed.
+
+lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l.
+ l ≤ yinj i →
+ ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 →
+ ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2.
+#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12
+@(cpys_subst … HLK … HU12) >yminus_Y_inj //
+qed.
+
+(* Advanced inversion lemmas *************************************************)
+
+lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{J}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[0, ↓(l+m-i)] V2 &
+ ⬆[O, i+1] V2 ≡ T2 &
+ I = LRef i.
+#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2
+[ /2 width=1 by or_introl/
+| #T #T2 #_ #HT2 *
+ [ #H destruct
+ elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
+ | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI
+ lapply (drop_fwd_drop2 … HLK) #H
+ elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
+ [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+ /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+ ]
+]
+qed-.
+
+lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V1,V2. l ≤ i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{I}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[0, ↓(l+m-i)] V2 &
+ ⬆[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
+qed-.
+
+lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/
+* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/
+qed-.
+
+lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 →
+ ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⬆[O, i+1] V2 ≡ T2 →
+ ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ↓(l+m-i)] V2
+ & l ≤ i
+ & i < l + m.
+#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H
+[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK //
+| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2
+ lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct
+ lapply (drop_mono … HLY … HLK) -L #H destruct
+ /2 width=1 by and3_intro/
+]
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
+ ∀L,U1,s,l,m. lt + mt ≤ yinj l → ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T l m) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
+ ∀L,U1,s,l,m. lt ≤ yinj l → l ≤ lt + mt →
+ ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 →
+ ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T l m) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
+ ∀L,U1,s,l,m. yinj l ≤ lt → ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T l m) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+(* Inversion lemmas for relocation ******************************************)
+
+lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt + mt ≤ l →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt ≤ l → yinj l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ yinj l + m ≤ lt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj l + m)] T2 &
+ ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt ≤ l → lt + mt ≤ yinj l + m →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 →
+ ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 →
+ l ≤ yinj i → i < l + m →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ↓(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm
+elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 /2 width=3 by ex2_intro/
+| /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $G $L $T1 $l $m $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarAlt $G $L $T1 $l $m $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 break ⊆ [ term 46 l , break term 46 m ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEq $L1 $l $m $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_minus_sn.ma".
+include "basic_2A/notation/relations/lrsubeq_4.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+inductive lsuby: relation4 ynat ynat lenv lenv ≝
+| lsuby_atom: ∀L,l,m. lsuby l m L (⋆)
+| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
+ lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 →
+ lsuby 0 (↑m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
+ lsuby l m L1 L2 → lsuby (↑l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+.
+
+interpretation
+ "local environment refinement (extended substitution)"
+ 'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ↓m] L2 → 0 < m →
+ L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V.
+#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/
+qed.
+
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[↓l, m] L2 → 0 < l →
+ L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/
+qed.
+
+lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
+ ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
+#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
+qed.
+
+lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L.
+#L elim L -L //
+#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
+#Hl destruct /2 width=1 by lsuby_succ/
+#m elim (ynat_cases … m) [| * #x ]
+#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
+qed.
+
+lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, yinj 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
+[ #l #H elim (le_plus_xSy_O_false … H)
+| #L1 #I1 #V1 #l #H lapply (le_plus_to_le_r … H) -H #HL12
+ elim (ynat_cases l) /3 width=1 by lsuby_zero/
+ * /3 width=1 by lsuby_succ/
+]
+qed.
+
+lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1.
+#l #m #L1 #L2 #H elim H -l -m -L1 -L2
+[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 //
+| /2 width=1 by lsuby_O2/
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l … H)
+ /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l … H)
+ /3 width=1 by lsuby_succ/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
+]
+qed-.
+
+lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆.
+/2 width=5 by lsuby_inv_atom1_aux/ qed-.
+
+fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
+ L2 = ⋆ ∨
+ ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
+ L2 = ⋆ ∨
+ ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=9 by lsuby_inv_zero1_aux/ qed-.
+
+fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m →
+ L2 = ⋆ ∨
+ ∃∃J2,K2. K1 ⊆[0, ↓m] K2 & L2 = K2.ⓑ{J2}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct
+ /3 width=4 by ex2_2_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m →
+ L2 = ⋆ ∨
+ ∃∃I2,K2. K1 ⊆[0, ↓m] K2 & L2 = K2.ⓑ{I2}V.
+/2 width=6 by lsuby_inv_pair1_aux/ qed-.
+
+fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
+ L2 = ⋆ ∨
+ ∃∃J2,K2,W2. K1 ⊆[↓l, m] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+]
+qed-.
+
+lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l →
+ L2 = ⋆ ∨
+ ∃∃I2,K2,V2. K1 ⊆[↓l, m] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5 by lsuby_inv_succ1_aux/ qed-.
+
+fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 →
+ ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
+ /2 width=5 by ex2_3_intro/
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
+ ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=9 by lsuby_inv_zero2_aux/ qed-.
+
+fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m →
+ ∃∃J1,K1. K1 ⊆[0, ↓m] K2 & L1 = K1.ⓑ{J1}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
+ /2 width=4 by ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m →
+ ∃∃I1,K1. K1 ⊆[0, ↓m] K2 & L1 = K1.ⓑ{I1}V.
+/2 width=6 by lsuby_inv_pair2_aux/ qed-.
+
+fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l →
+ ∃∃J1,K1,W1. K1 ⊆[↓l, m] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W2 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct
+ /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l →
+ ∃∃I1,K1,V1. K1 ⊆[↓l, m] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=5 by lsuby_inv_succ2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize /2 width=1 by le_S_S/
+qed-.
+
+(* Properties on basic slicing **********************************************)
+
+lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+ l ≤ i → i < l + m →
+ ∃∃I1,K1. K1 ⊆[0, ↓(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #s #i #H
+ elim (drop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
+ elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct -I2 >ypred_succ
+ /2 width=4 by drop_pair, ex2_2_intro/
+ | <(S_pred … Hi) <ysucc_inj #Him
+ lapply (ylt_inv_succ … Him) -Him #Him
+ elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
+ >yminus_succ /3 width=4 by drop_drop_lt, ex2_2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli
+ elim (yle_inv_succ_sn_lt … Hli) -Hli #Hli #Hi
+ lapply (ylt_inv_inj … Hi) -Hi #Hi
+ <(S_pred … Hi) >yplus_succ1 <ysucc_inj #H
+ lapply (ylt_inv_succ … H) -H #Hilm
+ lapply (drop_inv_drop1_lt … HLK2 Hi) -HLK2 #HLK1
+ elim (IHL12 … HLK1) -IHL12 -HLK1 >minus_SO_dx
+ /3 width=4 by drop_drop_lt, ex2_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: ∀l,m. Transitive … (lsuby l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 … H) -H
+ #H destruct //
+| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
+ * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
+]
+qed-.
* Initial invocation: - Patience on me to gain peace and perfection! -
*)
-include "ground_2A/lib/star.ma".
+include "ground_2/lib/star.ma".
include "basic_2A/notation/constructors/item0_0.ma".
include "basic_2A/notation/constructors/snitem2_2.ma".
(* *)
(**************************************************************************)
-include "ground_2A/lib/list.ma".
include "basic_2A/notation/constructors/star_0.ma".
include "basic_2A/notation/constructors/dxbind2_3.ma".
include "basic_2A/notation/constructors/dxabbr_2.ma".
(* GLOBAL ENVIRONMENTS ******************************************************)
(* global environments *)
-definition genv ≝ list2 bind2 term.
+inductive genv: Type[0] ≝
+| GAtom: genv (* empty *)
+| GPair: genv → bind2 → term → genv (* binary binding construction *)
+.
interpretation "sort (global environment)"
- 'Star = (nil2 bind2 term).
+ 'Star = (GAtom).
interpretation "global environment binding construction (binary)"
- 'DxBind2 L I T = (cons2 bind2 term I T L).
+ 'DxBind2 G I T = (GPair G I T).
interpretation "abbreviation (global environment)"
- 'DxAbbr L T = (cons2 bind2 term Abbr T L).
+ 'DxAbbr G T = (GPair G Abbr T).
interpretation "abstraction (global environment)"
- 'DxAbst L T = (cons2 bind2 term Abst T L).
+ 'DxAbst G T = (GPair G Abst T).
(* Basic properties *********************************************************)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/grammar/genv.ma".
+
+(* LENGTH OF A GLOBAL ENVIRONMENT *******************************************)
+
+let rec glength L ≝ match L with
+[ GAtom ⇒ 0
+| GPair G _ _ ⇒ glength G + 1
+].
+
+interpretation "length (global environment)" 'card G = (glength G).
(* *)
(**************************************************************************)
-include "ground_2A/lib/bool.ma".
-include "ground_2A/lib/arith.ma".
+include "ground_2/lib/bool.ma".
+include "ground_2/lib/arith.ma".
(* ITEMS ********************************************************************)
(* *)
(**************************************************************************)
-include "ground_2A/notation/functions/append_2.ma".
+include "ground_2/notation/functions/append_2.ma".
include "basic_2A/notation/functions/snbind2_3.ma".
include "basic_2A/notation/functions/snabbr_2.ma".
include "basic_2A/notation/functions/snabst_2.ma".
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_2_3.ma".
include "basic_2A/grammar/lenv.ma".
(* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
(* *)
(**************************************************************************)
-include "ground_2A/ynat/ynat_lt.ma".
+include "ground_2/ynat/ynat_lt.ma".
include "basic_2A/notation/relations/midiso_4.ma".
include "basic_2A/grammar/lenv_length.ma".
| lreq_zero: ∀I1,I2,L1,L2,V1,V2.
lreq 0 0 L1 L2 → lreq 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
| lreq_pair: ∀I,L1,L2,V,m. lreq 0 m L1 L2 →
- lreq 0 (⫯m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+ lreq 0 (â\86\91m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
| lreq_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
- lreq l m L1 L2 â\86\92 lreq (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+ lreq l m L1 L2 â\86\92 lreq (â\86\91l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
.
interpretation
(* Basic properties *********************************************************)
-lemma lreq_pair_lt: â\88\80I,L1,L2,V,m. L1 ⩬[0, â«°m] L2 → 0 < m →
+lemma lreq_pair_lt: â\88\80I,L1,L2,V,m. L1 ⩬[0, â\86\93m] L2 → 0 < m →
L1.ⓑ{I}V ⩬[0, m] L2.ⓑ{I}V.
#I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lreq_pair/
qed.
-lemma lreq_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[â«°l, m] L2 → 0 < l →
+lemma lreq_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[â\86\93l, m] L2 → 0 < l →
L1.ⓑ{I1}V1 ⩬[l, m] L2. ⓑ{I2}V2.
#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lreq_succ/
qed.
fact lreq_inv_pair1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
∀J,K1,W. L1 = K1.ⓑ{J}W → l = 0 → 0 < m →
- â\88\83â\88\83K2. K1 ⩬[0, â«°m] K2 & L2 = K2.ⓑ{J}W.
+ â\88\83â\88\83K2. K1 ⩬[0, â\86\93m] K2 & L2 = K2.ⓑ{J}W.
#L1 #L2 #l #m * -L1 -L2 -l -m
[ #l #m #J #K1 #W #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
qed-.
lemma lreq_inv_pair1: ∀I,K1,L2,V,m. K1.ⓑ{I}V ⩬[0, m] L2 → 0 < m →
- â\88\83â\88\83K2. K1 ⩬[0, â«°m] K2 & L2 = K2.ⓑ{I}V.
+ â\88\83â\88\83K2. K1 ⩬[0, â\86\93m] K2 & L2 = K2.ⓑ{I}V.
/2 width=6 by lreq_inv_pair1_aux/ qed-.
lemma lreq_inv_pair: ∀I1,I2,L1,L2,V1,V2,m. L1.ⓑ{I1}V1 ⩬[0, m] L2.ⓑ{I2}V2 → 0 < m →
- â\88§â\88§ L1 ⩬[0, â«°m] L2 & I1 = I2 & V1 = V2.
+ â\88§â\88§ L1 ⩬[0, â\86\93m] L2 & I1 = I2 & V1 = V2.
#I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 … H) -H //
#Y #HL12 #H destruct /2 width=1 by and3_intro/
qed-.
fact lreq_inv_succ1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
- â\88\83â\88\83J2,K2,W2. K1 ⩬[â«°l, m] K2 & L2 = K2.ⓑ{J2}W2.
+ â\88\83â\88\83J2,K2,W2. K1 ⩬[â\86\93l, m] K2 & L2 = K2.ⓑ{J2}W2.
#L1 #L2 #l #m * -L1 -L2 -l -m
[ #l #m #J1 #K1 #W1 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
qed-.
lemma lreq_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⩬[l, m] L2 → 0 < l →
- â\88\83â\88\83I2,K2,V2. K1 ⩬[â«°l, m] K2 & L2 = K2.ⓑ{I2}V2.
+ â\88\83â\88\83I2,K2,V2. K1 ⩬[â\86\93l, m] K2 & L2 = K2.ⓑ{I2}V2.
/2 width=5 by lreq_inv_succ1_aux/ qed-.
lemma lreq_inv_atom2: ∀L1,l,m. L1 ⩬[l, m] ⋆ → L1 = ⋆.
qed-.
lemma lreq_inv_succ: ∀I1,I2,L1,L2,V1,V2,l,m. L1.ⓑ{I1}V1 ⩬[l, m] L2.ⓑ{I2}V2 → 0 < l →
- L1 ⩬[⫰l, m] L2.
+ L1 ⩬[â\86\93l, m] L2.
#I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … H) -H //
#Z #Y #X #HL12 #H destruct //
qed-.
qed-.
lemma lreq_inv_pair2: ∀I,K2,L1,V,m. L1 ⩬[0, m] K2.ⓑ{I}V → 0 < m →
- â\88\83â\88\83K1. K1 ⩬[0, â«°m] K2 & L1 = K1.ⓑ{I}V.
+ â\88\83â\88\83K1. K1 ⩬[0, â\86\93m] K2 & L1 = K1.ⓑ{I}V.
#I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 … (lreq_sym … H)) -H
/3 width=3 by lreq_sym, ex2_intro/
qed-.
lemma lreq_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⩬[l, m] K2.ⓑ{I2}V2 → 0 < l →
- â\88\83â\88\83I1,K1,V1. K1 ⩬[â«°l, m] K2 & L1 = K1.ⓑ{I1}V1.
+ â\88\83â\88\83I1,K1,V1. K1 ⩬[â\86\93l, m] K2 & L1 = K1.ⓑ{I1}V1.
#I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … (lreq_sym … H)) -H
/3 width=5 by lreq_sym, ex2_3_intro/
qed-.
(* *)
(**************************************************************************)
-include "ground_2A/ynat/ynat_plus.ma".
+include "ground_2/ynat/ynat_plus.ma".
include "basic_2A/grammar/lreq.ma".
(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
(* *)
(**************************************************************************)
-include "ground_2A/lib/list.ma".
+include "ground_2/lib/list.ma".
include "basic_2A/notation/functions/snapplvector_2.ma".
include "basic_2A/grammar/term_simple.ma".
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_1_2.ma".
include "basic_2A/notation/relations/topiso_2.ma".
include "basic_2A/grammar/term_simple.ma".
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/notation/relations/psubststar_6.ma".
-include "basic_2A/substitution/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-definition cpys: ynat → ynat → relation4 genv lenv term term ≝
- λl,m,G. LTC … (cpy l m G).
-
-interpretation "context-sensitive extended multiple substritution (term)"
- 'PSubstStar G L T1 l m T2 = (cpys l m G L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma cpys_ind: ∀G,L,T1,l,m. ∀R:predicate term. R T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → R T → R T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T2.
-#G #L #T1 #l #m #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-lemma cpys_ind_dx: ∀G,L,T2,l,m. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → R T → R T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T1.
-#G #L #T2 #l #m #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma cpy_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
-/2 width=1 by inj/ qed.
-
-lemma cpys_strap1: ∀G,L,T1,T,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
-normalize /2 width=3 by step/ qed-.
-
-lemma cpys_strap2: ∀G,L,T1,T,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
-normalize /2 width=3 by TC_strap/ qed-.
-
-lemma lsuby_cpys_trans: ∀G,l,m. lsub_trans … (cpys l m G) (lsuby l m).
-/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
-qed-.
-
-lemma cpys_refl: ∀G,L,l,m. reflexive … (cpys l m G L).
-/2 width=1 by cpy_cpys/ qed.
-
-lemma cpys_bind: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] ⓑ{a,I}V2.T2.
-#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2
-[ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
-| /3 width=5 by cpys_strap1, cpy_bind/
-]
-qed.
-
-lemma cpys_flat: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] ⓕ{I}V2.T2.
-#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2
-[ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
-| /3 width=5 by cpys_strap1, cpy_flat/
-qed.
-
-lemma cpys_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T2 →
- ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
- ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T2.
-#G #L #T1 #T2 #l1 #m1 #H #l1 #l2 #Hl21 #Hlm12 @(cpys_ind … H) -T2
-/3 width=7 by cpys_strap1, cpy_weak/
-qed-.
-
-lemma cpys_weak_top: ∀G,L,T1,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, |L| - l] T2.
-#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2
-/3 width=4 by cpys_strap1, cpy_weak_top/
-qed-.
-
-lemma cpys_weak_full: ∀G,L,T1,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
-#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2
-/3 width=5 by cpys_strap1, cpy_weak_full/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpys_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
- l ≤ lt → l + m ≤ lt + mt →
- ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #T1 #l #m #HTU1 #Hllt #Hlmlmt @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
- elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ♯{T1} ≤ ♯{T2}.
-#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
-/2 width=3 by transitive_le/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶*[l, m] T2 → T2 = ⋆k.
-#G #L #T2 #k #l #m #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct
->(cpy_inv_sort1 … HT2) -HT2 //
-qed-.
-
-(* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶*[l, m] T2 → T2 = §p.
-#G #L #T2 #p #l #m #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct
->(cpy_inv_gref1 … HT2) -HT2 //
-qed-.
-
-lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 &
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 &
- U2 = ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2
-[ /2 width=5 by ex3_2_intro/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2
- /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
-]
-qed-.
-
-lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 &
- U2 = ⓕ{I}V2.T2.
-#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2
-[ /2 width=5 by ex3_2_intro/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (cpy_inv_flat1 … HU2) -HU2
- /3 width=5 by cpys_strap1, ex3_2_intro/
-]
-qed-.
-
-lemma cpys_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 0] T2 → T1 = T2.
-#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀l,m:nat.
- ⦃G, L⦄ ⊢ U1 ▶*[l, m] U2 → ∀T1. ⬆[l, m] T1 ≡ U1 → U1 = U2.
-#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind … H) -U2
-/2 width=7 by cpy_inv_lift1_eq/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/notation/relations/psubststaralt_6.ma".
-include "basic_2A/multiple/cpys_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* alternative definition of cpys *)
-inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
-| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I})
-| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m →
- ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(l+m-i)) G K V1 V2 →
- ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2
-| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
- cpysa l m G L V1 V2 → cpysa (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
- cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
- cpysa l m G L V1 V2 → cpysa l m G L T1 T2 →
- cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation
- "context-sensitive extended multiple substritution (term) alternative"
- 'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m).
-#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
-[ //
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
-| /4 width=1 by lsuby_succ, cpysa_bind/
-| /3 width=1 by cpysa_flat/
-]
-qed-.
-
-lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T.
-#G #T elim T -T //
-#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
-qed.
-
-lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T →
- ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2.
-#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m
-[ #I #G #L #l #m #X #H
- elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
-| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H
- lapply (drop_fwd_drop2 … HLK) #H0LK
- lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H
- elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
- /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
-| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H
- elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
-| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H
- elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
-]
-qed-.
-
-lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2.
-/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
-/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
- (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) →
- (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m →
- ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 →
- ⬆[O, i+1] V2 ≡ W2 → R O (⫰(l+m-i)) G K V1 V2 → R l m G L (#i) W2
- ) →
- (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → R l m G L V1 V2 →
- R (⫯l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
- ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 →
- R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2.
-#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m
-/3 width=8 by cpysa_inv_cpys/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/substitution/cpy_cpy.ma".
-include "basic_2A/multiple/cpys_alt.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2.
-#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
-normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
-
-lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
- ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
- (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
-normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
-
-lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
-
-lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
-
-lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
- ∀i. l ≤ i → i ≤ l + m →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2.
-#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2
-[ /2 width=3 by ex2_intro/
-| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
- elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02
- elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
- >ymax_pre_sn_comm //
-]
-qed-.
-
-lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj l + m)] T2 &
- ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
-elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
-lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
-lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
->yplus_minus_inj /2 width=3 by ex2_intro/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
-normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
-
-theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
- ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 →
- (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
-normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
-
-theorem cpys_trans_eq: ∀G,L,T1,T,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 →
- ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
-normalize /2 width=3 by trans_TC/ qed-.
-
-theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
-
-theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 →
- ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2.
-#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 //
-[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2
- elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ]
- [ lapply (cpys_weak_full … HW2) -HW2 #HW2
- lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 //
- [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
- #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 //
- | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2
- elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2
- /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hli -Hilm
- #X #_ #H elim (lift_inv_lref2_be … H) -H //
- ]
-| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H
- #V #T #HV2 #HT2 #H destruct
- lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
-| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H
- #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/substitution/cpy_lift.ma".
-include "basic_2A/multiple/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m.
- l ≤ yinj i → i < l + m →
- ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(l+m-i)] U1 →
- ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2.
-#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1
-[ /3 width=5 by cpy_cpys, cpy_subst/
-| #U #U1 #_ #HU1 #IHU #U2 #HU12
- elim (lift_total U 0 (i+1)) #U0 #HU0
- lapply (IHU … HU0) -IHU #H
- lapply (drop_fwd_drop2 … HLK) -HLK #HLK
- lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
- lapply (cpy_weak … HU02 l m ? ?) -HU02
- [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
- >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
-]
-qed.
-
-lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l.
- l ≤ yinj i →
- ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 →
- ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2.
-#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12
-@(cpys_subst … HLK … HU12) >yminus_Y_inj //
-qed.
-
-(* Advanced inversion lemmas *************************************************)
-
-lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 →
- T2 = ⓪{I} ∨
- ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{J}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 &
- ⬆[O, i+1] V2 ≡ T2 &
- I = LRef i.
-#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2
-[ /2 width=1 by or_introl/
-| #T #T2 #_ #HT2 *
- [ #H destruct
- elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
- | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI
- lapply (drop_fwd_drop2 … HLK) #H
- elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
- [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
- /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
- ]
-]
-qed-.
-
-lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 →
- T2 = #i ∨
- ∃∃I,K,V1,V2. l ≤ i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{I}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 &
- ⬆[O, i+1] V2 ≡ T2.
-#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
-* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
-qed-.
-
-lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 →
- T2 = #i ∨
- ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2.
-#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/
-* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/
-qed-.
-
-lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 →
- ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⬆[O, i+1] V2 ≡ T2 →
- ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2
- & l ≤ i
- & i < l + m.
-#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H
-[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK //
-| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2
- lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct
- lapply (drop_mono … HLY … HLK) -L #H destruct
- /2 width=1 by and3_intro/
-]
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
- ∀L,U1,s,l,m. lt + mt ≤ yinj l → ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 →
- ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2.
-#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T l m) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
- ∀L,U1,s,l,m. lt ≤ yinj l → l ≤ lt + mt →
- ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 →
- ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2.
-#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T l m) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
- ∀L,U1,s,l,m. yinj l ≤ lt → ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 →
- ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2.
-#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T l m) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-(* Inversion lemmas for relocation ******************************************)
-
-lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt + mt ≤ l →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt ≤ l → yinj l + m ≤ lt + mt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- yinj l + m ≤ lt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj l + m)] T2 &
- ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt ≤ l → lt + mt ≤ yinj l + m →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 →
- ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 →
- l ≤ yinj i → i < l + m →
- ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2.
-#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm
-elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
->yplus_O1 <yplus_inj >yplus_SO2
-[ >yminus_succ2 /2 width=3 by ex2_intro/
-| /2 width=1 by ylt_fwd_le_succ1/
-| /2 width=3 by yle_trans/
-]
-qed-.
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/relocation/mr2_minus.ma".
include "basic_2A/notation/relations/rdropstar_3.ma".
include "basic_2A/notation/relations/rdropstar_4.ma".
include "basic_2A/substitution/drop.ma".
-include "basic_2A/multiple/mr2_minus.ma".
include "basic_2A/multiple/lifts_vector.ma".
(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
-inductive drops (s:bool): list2 nat nat → relation lenv ≝
+inductive drops (s:bool): mr2 → relation lenv ≝
| drops_nil : ∀L. drops s (◊) L L
| drops_cons: ∀L1,L,L2,cs,l,m.
- drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s ({l, m} @ cs) L1 L2
+ drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s (❨l, m❩; cs) L1 L2
.
interpretation "iterated slicing (local environment) abstract"
/2 width=4 by drops_inv_nil_aux/ qed-.
fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 →
- ∀l,m,tl. cs = {l, m} @ tl →
+ ∀l,m,tl. cs = ❨l, m❩; tl →
∃∃L. ⬇*[s, tl] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
#L1 #L2 #s #cs * -L1 -L2 -cs
[ #L #l #m #tl #H destruct
qed-.
(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 →
+lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, ❨l, m❩; cs] L1 ≡ L2 →
∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
/2 width=3 by drops_inv_cons_aux/ qed-.
-lemma drops_inv_skip2: â\88\80I,s,cs,cs2,i. cs â\96 i â\89¡ cs2 →
+lemma drops_inv_skip2: â\88\80I,s,cs,cs2,i. cs â\96 i â\89\98 cs2 →
∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
- â\88\83â\88\83K1,V1,cs1. cs + 1 â\96 i + 1 â\89¡ cs1 + 1 &
+ â\88\83â\88\83K1,V1,cs1. cs + 1 â\96 i + 1 â\89\98 cs1 + 1 &
⬇*[s, cs1] K1 ≡ K2 &
⬆*[cs1] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
lemma drops_drop_trans: ∀L1,L,cs. ⬇*[Ⓕ, cs] L1 ≡ L → ∀L2,i. ⬇[i] L ≡ L2 →
∃∃L0,cs0,i0. ⬇[i0] L1 ≡ L0 & ⬇*[Ⓕ, cs0] L0 ≡ L2 &
- @â¦\83i, csâ¦\84 â\89¡ i0 & cs â\96 i â\89¡ cs0.
+ @â\9dªi, csâ\9d« â\89\98 i0 & cs â\96 i â\89\98 cs0.
#L1 #L #cs #H elim H -L1 -L -cs
[ /2 width=7 by drops_nil, minuss_nil, at_nil, ex4_3_intro/
| #L1 #L0 #L #cs #l #m #_ #HL0 #IHL0 #L2 #i #HL2
(* *)
(**************************************************************************)
+include "ground_2/relocation/mr2_append.ma".
include "basic_2A/multiple/drops_drop.ma".
(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
(* *)
(**************************************************************************)
-include "ground_2A/ynat/ynat_plus.ma".
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/xoa/ex_5_4.ma".
+include "ground_2/ynat/ynat_plus.ma".
include "basic_2A/notation/relations/freestar_4.ma".
include "basic_2A/substitution/lift_neg.ma".
include "basic_2A/substitution/drop.ma".
qed-.
lemma frees_inv_bind: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄ →
- L â\8a¢ i ϵ ð\9d\90\85*[l]â¦\83Wâ¦\84 â\88¨ L.â\93\91{I}W â\8a¢ i+1 ϵ ð\9d\90\85*[⫯l]⦃U⦄ .
+ L â\8a¢ i ϵ ð\9d\90\85*[l]â¦\83Wâ¦\84 â\88¨ L.â\93\91{I}W â\8a¢ i+1 ϵ ð\9d\90\85*[â\86\91l]⦃U⦄ .
#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
[ #HnX elim (nlift_inv_bind … HnX) -HnX
/4 width=2 by frees_eq, or_intror, or_introl/
/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
qed.
-lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ →
+lemma pippo (j) (i): O < j → j < i+1 → ↓j< i.
+/2 width=1 by lt_plus_to_minus/ qed-.
+
+lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[↑l]⦃U⦄ →
L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
[ /4 width=9 by frees_eq, nlift_bind_dx/
| * #I #K #W #j #Hlj #Hji #HnU #HLK #HW
- elim (yle_inv_succ1 … Hlj) -Hlj <yminus_SO2 #Hyj #H
- lapply (ylt_O … H) -H #Hj
+ elim (yle_inv_succ_sn_lt … Hlj) -Hlj #Hyj #H
+ lapply (ylt_inv_inj … H) -H #Hi
>(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
- /4 width=9 by frees_be, nlift_bind_dx, drop_inv_drop1_lt, lt_plus_to_minus/
+ /4 width=9 by frees_be, nlift_bind_dx, drop_inv_drop1_lt, yle_plus_dx1_trans, monotonic_lt_minus_l/
]
qed.
(* *)
(**************************************************************************)
+include "ground_2/lib/arith_2a.ma".
+include "ground_2/ynat/ynat_minus_sn.ma".
include "basic_2A/substitution/drop_drop.ma".
include "basic_2A/multiple/frees.ma".
| -IH /3 width=5 by frees_inv_gref, or_intror/
| #a #I #W #U #Hx #l #i destruct
elim (IH L W … l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
- elim (IH (L.â\93\91{I}W) U â\80¦ (⫯l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
+ elim (IH (L.â\93\91{I}W) U â\80¦ (â\86\91l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
@or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
| #I #W #U #Hx #l #i destruct
elim (IH L W … l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
]
qed-.
+alias symbol "UpArrow" (instance 2) = "ynat successor".
+alias symbol "minus" (instance 6) = "natural minus".
+alias symbol "minus" (instance 5) = "natural minus".
+alias symbol "RDrop" (instance 9) = "basic slicing (local environment) lget".
lemma frees_S: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[yinj l]⦃U⦄ → ∀I,K,W. ⬇[l] L ≡ K.ⓑ{I}W →
- (K â\8a¢ i-l-1 ϵ ð\9d\90\85*[0]â¦\83Wâ¦\84 â\86\92 â\8a¥) â\86\92 L â\8a¢ i ϵ ð\9d\90\85*[⫯l]⦃U⦄.
+ (K â\8a¢ i-l-1 ϵ ð\9d\90\85*[0]â¦\83Wâ¦\84 â\86\92 â\8a¥) â\86\92 L â\8a¢ i ϵ ð\9d\90\85*[â\86\91l]⦃U⦄.
#L #U #l #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
* #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
lapply (yle_inv_inj … Hlj) -Hlj #Hlj
lemma frees_inv_lift_ge: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
∀K,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
∀T. ⬆[l0, m0] T ≡ U → l0 + m0 ≤ i →
- K ⊢ i-m0 ϵ𝐅*[l-yinj m0]⦃T⦄.
+ K ⊢ i-m0 ϵ𝐅*[l-m0]⦃T⦄.
#L #U #l #i #H elim H -L -U -l -i
[ #L #U #l #i #HnU #K #s #l0 #m0 #HLK #T #HTU #Hlm0i -L -s
elim (le_inv_plus_l … Hlm0i) -Hlm0i #Hl0im0 #Hm0i @frees_eq #X #HXT -K
(* *)
(**************************************************************************)
+include "ground_2/relocation/mr2_at.ma".
+include "ground_2/relocation/mr2_plus.ma".
include "basic_2A/notation/relations/rliftstar_3.ma".
include "basic_2A/substitution/lift.ma".
-include "basic_2A/multiple/mr2_plus.ma".
(* GENERIC TERM RELOCATION **************************************************)
-inductive lifts: list2 nat nat → relation term ≝
+inductive lifts: mr2 → relation term ≝
| lifts_nil : ∀T. lifts (◊) T T
| lifts_cons: ∀T1,T,T2,cs,l,m.
- ⬆[l,m] T1 ≡ T → lifts cs T T2 → lifts ({l, m} @ cs) T1 T2
+ ⬆[l,m] T1 ≡ T → lifts cs T T2 → lifts (❨l, m❩; cs) T1 T2
.
interpretation "generic relocation (term)"
/2 width=3 by lifts_inv_nil_aux/ qed-.
fact lifts_inv_cons_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 →
- ∀l,m,tl. cs = {l, m} @ tl →
+ ∀l,m,tl. cs = ❨l, m❩; tl →
∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[tl] T ≡ T2.
#T1 #T2 #cs * -T1 -T2 -cs
[ #T #l #m #tl #H destruct
/2 width=3 by ex2_intro/
qed-.
-lemma lifts_inv_cons: ∀T1,T2,l,m,cs. ⬆*[{l, m} @ cs] T1 ≡ T2 →
+lemma lifts_inv_cons: ∀T1,T2,l,m,cs. ⬆*[❨l, m❩; cs] T1 ≡ T2 →
∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[cs] T ≡ T2.
/2 width=3 by lifts_inv_cons_aux/ qed-.
(* Basic_1: was: lift1_lref *)
lemma lifts_inv_lref1: ∀T2,cs,i1. ⬆*[cs] #i1 ≡ T2 →
- â\88\83â\88\83i2. @â¦\83i1, csâ¦\84 â\89¡ i2 & T2 = #i2.
+ â\88\83â\88\83i2. @â\9dªi1, csâ\9d« â\89\98 i2 & T2 = #i2.
#T2 #cs elim cs -cs
[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/
| #l #m #cs #IH #i1 #H
(* *)
(**************************************************************************)
+include "ground_2/relocation/mr2_minus.ma".
include "basic_2A/substitution/lift_lift.ma".
-include "basic_2A/multiple/mr2_minus.ma".
include "basic_2A/multiple/lifts.ma".
(* GENERIC TERM RELOCATION **************************************************)
qed-.
(* Basic_1: was: lift1_free (right to left) *)
-lemma lifts_lift_trans: â\88\80cs,i,i0. @â¦\83i, csâ¦\84 â\89¡ i0 →
- â\88\80cs0. cs + 1 â\96 i + 1 â\89¡ cs0 + 1 →
+lemma lifts_lift_trans: â\88\80cs,i,i0. @â\9dªi, csâ\9d« â\89\98 i0 →
+ â\88\80cs0. cs + 1 â\96 i + 1 â\89\98 cs0 + 1 →
∀T1,T0. ⬆*[cs0] T1 ≡ T0 →
∀T2. ⬆[O, i0 + 1] T0 ≡ T2 →
∃∃T. ⬆[0, i + 1] T1 ≡ T & ⬆*[cs] T ≡ T2.
(* *)
(**************************************************************************)
+include "ground_2/relocation/mr2_append.ma".
include "basic_2A/multiple/lifts_lift.ma".
(* GENERIC RELOCATION *******************************************************)
(* GENERIC TERM VECTOR RELOCATION *******************************************)
-inductive liftsv (cs:list2 nat nat) : relation (list term) ≝
-| liftsv_nil : liftsv cs (â\97\8a) (â\97\8a)
+inductive liftsv (cs:mr2) : relation (list term) ≝
+| liftsv_nil : liftsv cs (â\92º) (â\92º)
| liftsv_cons: ∀T1s,T2s,T1,T2.
⬆*[cs] T1 ≡ T2 → liftsv cs T1s T2s →
- liftsv cs (T1 @ T1s) (T2 @ T2s)
+ liftsv cs (T1 ⨮ T1s) (T2 ⨮ T2s)
.
interpretation "generic relocation (vector)"
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_4.ma".
include "basic_2A/notation/relations/lazyeq_4.ma".
include "basic_2A/multiple/llpx_sn.ma".
∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2
) → (
∀a,I,L1,L2,V,T,l.
- L1 â\89¡[V, l]L2 â\86\92 L1.â\93\91{I}V â\89¡[T, ⫯l] L2.ⓑ{I}V →
- R l V L1 L2 â\86\92 R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2
+ L1 â\89¡[V, l]L2 â\86\92 L1.â\93\91{I}V â\89¡[T, â\86\91l] L2.ⓑ{I}V →
+ R l V L1 L2 â\86\92 R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2
) → (
∀I,L1,L2,V,T,l.
L1 ≡[V, l]L2 → L1 ≡[T, l] L2 →
qed-.
lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,l. L1 ≡[ⓑ{a,I}V.T, l] L2 →
- L1 â\89¡[V, l] L2 â\88§ L1.â\93\91{I}V â\89¡[T, ⫯l] L2.ⓑ{I}V.
+ L1 â\89¡[V, l] L2 â\88§ L1.â\93\91{I}V â\89¡[T, â\86\91l] L2.ⓑ{I}V.
/2 width=2 by llpx_sn_inv_bind/ qed-.
lemma lleq_inv_flat: ∀I,L1,L2,V,T,l. L1 ≡[ⓕ{I}V.T, l] L2 →
/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,l.
- L1 â\89¡[â\93\91{a,I}V.T, l] L2 â\86\92 L1.â\93\91{I}V â\89¡[T, ⫯l] L2.ⓑ{I}V.
+ L1 â\89¡[â\93\91{a,I}V.T, l] L2 â\86\92 L1.â\93\91{I}V â\89¡[T, â\86\91l] L2.ⓑ{I}V.
/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,l.
/2 width=1 by llpx_sn_gref/ qed.
lemma lleq_bind: ∀a,I,L1,L2,V,T,l.
- L1 â\89¡[V, l] L2 â\86\92 L1.â\93\91{I}V â\89¡[T, ⫯l] L2.ⓑ{I}V →
+ L1 â\89¡[V, l] L2 â\86\92 L1.â\93\91{I}V â\89¡[T, â\86\91l] L2.ⓑ{I}V →
L1 ≡[ⓑ{a,I}V.T, l] L2.
/2 width=1 by llpx_sn_bind/ qed.
(* Inversion lemmas on negated lazy quivalence for local environments *******)
lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) →
- (L1 â\89¡[V, l] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V â\89¡[T, ⫯l] L2.ⓑ{I}V → ⊥).
+ (L1 â\89¡[V, l] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V â\89¡[T, â\86\91l] L2.ⓑ{I}V → ⊥).
/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) →
(* *)
(**************************************************************************)
+include "ground_2/xoa/and_4.ma".
include "basic_2A/notation/relations/lazyor_5.ma".
include "basic_2A/multiple/frees.ma".
(* *)
(**************************************************************************)
-include "ground_2A/ynat/ynat_plus.ma".
+include "ground_2/xoa/ex_5_5.ma".
+include "ground_2/ynat/ynat_plus.ma".
include "basic_2A/substitution/drop.ma".
(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
| llpx_sn_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R l (#i) L1 L2
| llpx_sn_gref: ∀L1,L2,l,p. |L1| = |L2| → llpx_sn R l (§p) L1 L2
| llpx_sn_bind: ∀a,I,L1,L2,V,T,l.
- llpx_sn R l V L1 L2 â\86\92 llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn R l V L1 L2 â\86\92 llpx_sn R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
llpx_sn R l (ⓑ{a,I}V.T) L1 L2
| llpx_sn_flat: ∀I,L1,L2,V,T,l.
llpx_sn R l V L1 L2 → llpx_sn R l T L1 L2 → llpx_sn R l (ⓕ{I}V.T) L1 L2
fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 →
∀a,I,V,T. X = ⓑ{a,I}V.T →
- llpx_sn R l V L1 L2 â\88§ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+ llpx_sn R l V L1 L2 â\88§ llpx_sn R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #L1 #L2 #X #l * -L1 -L2 -X -l
[ #L1 #L2 #l #k #_ #b #J #W #U #H destruct
| #L1 #L2 #l #i #_ #_ #b #J #W #U #H destruct
qed-.
lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R l V L1 L2 â\88§ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+ llpx_sn R l V L1 L2 â\88§ llpx_sn R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 →
qed-.
lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+ llpx_sn R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H //
qed-.
(* *)
(**************************************************************************)
+include "ground_2/xoa/and_4.ma".
include "basic_2A/substitution/lift_neg.ma".
include "basic_2A/substitution/drop_drop.ma".
include "basic_2A/multiple/llpx_sn.ma".
qed-.
lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn_alt_r R l V L1 L2 â\88§ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+ llpx_sn_alt_r R l V L1 L2 â\88§ llpx_sn_alt_r R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H
#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2
[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
/3 width=9 by nlift_bind_sn, and3_intro/
-| lapply (yle_inv_succ1 … Hli) -Hli * #Hli #Hi
- lapply (drop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
- lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
- elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
- @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+| lapply (yle_inv_succ_sn_lt … Hli) -Hli * #Hli #Hi
+ lapply (drop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_inv_inj/ #HLK1
+ lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_inv_inj/ #HLK2
+ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by yle_plus_dx1_trans, and3_intro/
+ @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_inv_inj/
]
qed-.
lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,l.
llpx_sn_alt_r R l V L1 L2 →
- llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn_alt_r R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2.
#R #a #I #L1 #L2 #V #T #l #HV #HT
elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_2.ma".
include "basic_2A/substitution/drop_drop.ma".
include "basic_2A/multiple/llpx_sn_lreq.ma".
| #p #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
| #a #I #V #T #Hx #L2 #l destruct
elim (IH L1 V … L2 l) /2 width=1 by/
- elim (IH (L1.â\93\91{I}V) T â\80¦ (L2.â\93\91{I}V) (⫯l)) -IH /3 width=1 by or_introl, llpx_sn_bind/
+ elim (IH (L1.â\93\91{I}V) T â\80¦ (L2.â\93\91{I}V) (â\86\91l)) -IH /3 width=1 by or_introl, llpx_sn_bind/
#H1 #H2 @or_intror
#H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
| #I #V #T #Hx #L2 #l destruct
| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
* #_ #H destruct
lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
- [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
+ [ /3 width=3 by llpx_sn_skip, ylt_plus_dx1_trans/
| /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
]
| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀a,I,L1,L2,V,T,l. (llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → ⊥) →
- (llpx_sn R l V L1 L2 â\86\92 â\8a¥) â\88¨ (llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+ (llpx_sn R l V L1 L2 â\86\92 â\8a¥) â\88¨ (llpx_sn R (â\86\91l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
#R #HR #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
/4 width=1 by llpx_sn_bind, or_intror, or_introl/
qed-.
(* *)
(**************************************************************************)
+include "ground_2/lib/arith_2a.ma".
include "basic_2A/substitution/lpx_sn_drop.ma".
include "basic_2A/multiple/llpx_sn.ma".
(* *)
(**************************************************************************)
+include "ground_2/lib/arith_2a.ma".
include "basic_2A/substitution/drop_lreq.ma".
include "basic_2A/multiple/llpx_sn.ma".
(* Properties about transitive closure **************************************)
lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L,V1,V2,T. LTC … R L V1 V2 →
- LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
+ ∀I,L,V1,V2,T. CTC … R L V1 V2 →
+ CTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2
/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/
qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/notation/relations/rat_3.ma".
-include "basic_2A/grammar/term_vector.ma".
-
-(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
-
-inductive at: list2 nat nat → relation nat ≝
-| at_nil: ∀i. at (◊) i i
-| at_lt : ∀cs,l,m,i1,i2. i1 < l →
- at cs i1 i2 → at ({l, m} @ cs) i1 i2
-| at_ge : ∀cs,l,m,i1,i2. l ≤ i1 →
- at cs (i1 + m) i2 → at ({l, m} @ cs) i1 i2
-.
-
-interpretation "application (multiple relocation with pairs)"
- 'RAt i1 cs i2 = (at cs i1 i2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact at_inv_nil_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → cs = ◊ → i1 = i2.
-#cs #i1 #i2 * -cs -i1 -i2
-[ //
-| #cs #l #m #i1 #i2 #_ #_ #H destruct
-| #cs #l #m #i1 #i2 #_ #_ #H destruct
-]
-qed-.
-
-lemma at_inv_nil: ∀i1,i2. @⦃i1, ◊⦄ ≡ i2 → i1 = i2.
-/2 width=3 by at_inv_nil_aux/ qed-.
-
-fact at_inv_cons_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 →
- ∀l,m,cs0. cs = {l, m} @ cs0 →
- i1 < l ∧ @⦃i1, cs0⦄ ≡ i2 ∨
- l ≤ i1 ∧ @⦃i1 + m, cs0⦄ ≡ i2.
-#cs #i1 #i2 * -cs -i1 -i2
-[ #i #l #m #cs #H destruct
-| #cs1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
-| #cs1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_intror, conj/
-]
-qed-.
-
-lemma at_inv_cons: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
- i1 < l ∧ @⦃i1, cs⦄ ≡ i2 ∨
- l ≤ i1 ∧ @⦃i1 + m, cs⦄ ≡ i2.
-/2 width=3 by at_inv_cons_aux/ qed-.
-
-lemma at_inv_cons_lt: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
- i1 < l → @⦃i1, cs⦄ ≡ i2.
-#cs #l #m #i1 #m2 #H
-elim (at_inv_cons … H) -H * // #Hli1 #_ #Hi1l
-lapply (le_to_lt_to_lt … Hli1 Hi1l) -Hli1 -Hi1l #Hl
-elim (lt_refl_false … Hl)
-qed-.
-
-lemma at_inv_cons_ge: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
- l ≤ i1 → @⦃i1 + m, cs⦄ ≡ i2.
-#cs #l #m #i1 #m2 #H
-elim (at_inv_cons … H) -H * // #Hi1l #_ #Hli1
-lapply (le_to_lt_to_lt … Hli1 Hi1l) -Hli1 -Hi1l #Hl
-elim (lt_refl_false … Hl)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/notation/relations/rminus_3.ma".
-include "basic_2A/multiple/mr2.ma".
-
-(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
-
-inductive minuss: nat → relation (list2 nat nat) ≝
-| minuss_nil: ∀i. minuss i (◊) (◊)
-| minuss_lt : ∀cs1,cs2,l,m,i. i < l → minuss i cs1 cs2 →
- minuss i ({l, m} @ cs1) ({l - i, m} @ cs2)
-| minuss_ge : ∀cs1,cs2,l,m,i. l ≤ i → minuss (m + i) cs1 cs2 →
- minuss i ({l, m} @ cs1) cs2
-.
-
-interpretation "minus (multiple relocation with pairs)"
- 'RMinus cs1 i cs2 = (minuss i cs1 cs2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact minuss_inv_nil1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 → cs1 = ◊ → cs2 = ◊.
-#cs1 #cs2 #i * -cs1 -cs2 -i
-[ //
-| #cs1 #cs2 #l #m #i #_ #_ #H destruct
-| #cs1 #cs2 #l #m #i #_ #_ #H destruct
-]
-qed-.
-
-lemma minuss_inv_nil1: ∀cs2,i. ◊ ▭ i ≡ cs2 → cs2 = ◊.
-/2 width=4 by minuss_inv_nil1_aux/ qed-.
-
-fact minuss_inv_cons1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 →
- ∀l,m,cs. cs1 = {l, m} @ cs →
- l ≤ i ∧ cs ▭ m + i ≡ cs2 ∨
- ∃∃cs0. i < l & cs ▭ i ≡ cs0 &
- cs2 = {l - i, m} @ cs0.
-#cs1 #cs2 #i * -cs1 -cs2 -i
-[ #i #l #m #cs #H destruct
-| #cs1 #cs #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=3 by ex3_intro, or_intror/
-| #cs1 #cs #l1 #m1 #i1 #Hli1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
-]
-qed-.
-
-lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
- l ≤ i ∧ cs1 ▭ m + i ≡ cs2 ∨
- ∃∃cs. i < l & cs1 ▭ i ≡ cs &
- cs2 = {l - i, m} @ cs.
-/2 width=3 by minuss_inv_cons1_aux/ qed-.
-
-lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
- l ≤ i → cs1 ▭ m + i ≡ cs2.
-#cs1 #cs2 #l #m #i #H
-elim (minuss_inv_cons1 … H) -H * // #cs #Hil #_ #_ #Hli
-lapply (lt_to_le_to_lt … Hil Hli) -Hil -Hli #Hi
-elim (lt_refl_false … Hi)
-qed-.
-
-lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
- i < l →
- ∃∃cs. cs1 ▭ i ≡ cs & cs2 = {l - i, m} @ cs.
-#cs1 #cs2 #l #m #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
-#Hli #_ #Hil lapply (lt_to_le_to_lt … Hil Hli) -Hil -Hli
-#Hi elim (lt_refl_false … Hi)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/multiple/mr2.ma".
-
-(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
-
-(* Main properties **********************************************************)
-
-theorem at_mono: ∀cs,i,i1. @⦃i, cs⦄ ≡ i1 → ∀i2. @⦃i, cs⦄ ≡ i2 → i1 = i2.
-#cs #i #i1 #H elim H -cs -i -i1
-[ #i #x #H <(at_inv_nil … H) -x //
-| #cs #l #m #i #i1 #Hil #_ #IHi1 #x #H
- lapply (at_inv_cons_lt … H Hil) -H -Hil /2 width=1 by/
-| #cs #l #m #i #i1 #Hli #_ #IHi1 #x #H
- lapply (at_inv_cons_ge … H Hli) -H -Hli /2 width=1 by/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/multiple/mr2.ma".
-
-(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
-
-let rec pluss (cs:list2 nat nat) (i:nat) on cs ≝ match cs with
-[ nil2 ⇒ ◊
-| cons2 l m cs ⇒ {l + i, m} @ pluss cs i
-].
-
-interpretation "plus (multiple relocation with pairs)"
- 'plus x y = (pluss x y).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma pluss_inv_nil2: ∀i,cs. cs + i = ◊ → cs = ◊.
-#i * // normalize
-#l #m #cs #H destruct
-qed.
-
-lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l, m} @ cs2 →
- ∃∃cs1. cs1 + i = cs2 & cs = {l - i, m} @ cs1.
-#i #l #m #cs2 * normalize
-[ #H destruct
-| #l1 #m1 #cs1 #H destruct /2 width=3 by ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 break ⊆ [ term 46 l , break term 46 m ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEq $L1 $l $m $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubst $G $L $T1 $l $m $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $G $L $T1 $l $m $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarAlt $G $L $T1 $l $m $T2 }.
(* *)
(**************************************************************************)
+include "ground_2/xoa/and_4.ma".
include "basic_2A/notation/relations/prednotreducible_3.ma".
include "basic_2A/reduction/crr.ma".
lemma cnr_abst: ∀a,G,L,W,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}W.T⦄.
#a #G #L #W #T #HW #HT #X #H
elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
->(HW … HW0) -W0 >(HT … HT0) -T0 //
+<(HW … HW0) -W0 <(HT … HT0) -T0 //
qed.
(* Basic_1: was only: nf2_appl_lref *)
lemma cnr_appl_simple: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄.
#G #L #V #T #HV #HT #HS #X #H
elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
->(HV … HV0) -V0 >(HT … HT0) -T0 //
+<(HV … HV0) -V0 <(HT … HT0) -T0 //
qed.
(* Basic_1: was: nf2_dec *)
(* Main properties on irreducibility ****************************************)
theorem cir_cnr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
-/2 width=4 by cpr_fwd_cir/ qed.
+/3 width=4 by cpr_fwd_cir, sym_eq/ qed.
(* Main inversion lemmas on irreducibility **********************************)
⬇[s, l, m] L0 ≡ L → ⬆[l, m] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄.
#G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
-<(HLT … HT1) in HT0; -L #HT0
+>(HLT … HT1) in HT0; -L #HT0
>(lift_mono … HT10 … HT0) -T1 -X //
qed.
#G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
elim (lift_total X l m) #X0 #HX0
lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0
->(HLT0 … HTX0) in HX0; -L0 -X0 #H
+<(HLT0 … HTX0) in HX0; -L0 -X0 #H
>(lift_inj … H … HT0) -T0 -X -s -l -m //
qed-.
lapply (H (⋆(next h k)) ?) -H /2 width=2 by cpx_st/ -L -d #H
lapply (destruct_tatom_tatom_aux … H) -H #H (**) (* destruct lemma needed *)
lapply (destruct_sort_sort_aux … H) -H #H (**) (* destruct lemma needed *)
-lapply (next_lt h k) >H -H #H elim (lt_refl_false … H)
+lapply (next_lt h k) <H -H #H elim (lt_refl_false … H)
qed-.
lemma cnx_inv_delta: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄ → ⊥.
⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}W.T⦄.
#h #g #a #G #L #W #T #HW #HT #X #H
elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
->(HW … HW0) -W0 >(HT … HT0) -T0 //
+<(HW … HW0) -W0 <(HT … HT0) -T0 //
qed.
lemma cnx_appl_simple: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄.
#h #g #G #L #V #T #HV #HT #HS #X #H
elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
->(HV … HV0) -V0 >(HT … HT0) -T0 //
+<(HV … HV0) -V0 <(HT … HT0) -T0 //
qed.
axiom cnx_dec: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T1⦄ ∨
(* Main properties on irreducibility ****************************************)
theorem cix_cnx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
-/2 width=6 by cpx_fwd_cix/ qed.
+/3 width=6 by cpx_fwd_cix, sym_eq/ qed.
(* Main inversion lemmas on irreducibility **********************************)
⬆[l, m] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄.
#h #g #G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
elim (cpx_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
-<(HLT … HT1) in HT0; -L #HT0
+>(HLT … HT1) in HT0; -L #HT0
>(lift_mono … HT10 … HT0) -T1 -X //
qed.
#h #g #G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
elim (lift_total X l m) #X0 #HX0
lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0
->(HLT0 … HTX0) in HX0; -L0 -X0 #H
+<(HLT0 … HTX0) in HX0; -L0 -X0 #H
>(lift_inj … H … HT0) -T0 -X -l -m //
qed-.
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_1.ma".
+include "ground_2/xoa/ex_5_6.ma".
+include "ground_2/xoa/ex_6_6.ma".
+include "ground_2/xoa/ex_6_7.ma".
+include "ground_2/xoa/ex_7_7.ma".
+include "ground_2/xoa/or_4.ma".
include "basic_2A/notation/relations/pred_4.ma".
include "basic_2A/static/lsubr.ma".
include "basic_2A/unfold/lstas.ma".
d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
/3 width=1 by cpr_eps, cpr_flat, cpr_bind/
-[ #G #L #d #k #H0 destruct normalize //
-| #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
+[ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
/3 width=6 by cpr_delta/
| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
]
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_3_4.ma".
+include "ground_2/xoa/ex_4_5.ma".
+include "ground_2/xoa/or_5.ma".
include "basic_2A/notation/relations/pred_6.ma".
include "basic_2A/static/sd.ma".
include "basic_2A/reduction/cpr.ma".
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_5_3.ma".
+include "ground_2/xoa/ex_6_4.ma".
include "basic_2A/notation/relations/lrsubeqa_3.ma".
include "basic_2A/static/lsubr.ma".
include "basic_2A/static/aaa.ma".
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_5_3.ma".
+include "ground_2/xoa/ex_6_4.ma".
include "basic_2A/notation/relations/lrsubeqd_5.ma".
include "basic_2A/static/lsubr.ma".
include "basic_2A/static/da.ma".
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_3.ma".
include "basic_2A/static/lsubr.ma".
(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
(* *)
(**************************************************************************)
-include "ground_2A/lib/arith.ma".
+include "ground_2/lib/arith_2a.ma".
(* SORT HIERARCHY ***********************************************************)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_max.ma".
-include "basic_2A/notation/relations/psubst_6.ma".
-include "basic_2A/grammar/genv.ma".
-include "basic_2A/substitution/lsuby.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* activate genv *)
-inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
-| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
-| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m →
- ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W
-| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
- cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
- cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
- cpy l m G L V1 V2 → cpy l m G L T1 T2 →
- cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation "context-sensitive extended ordinary substritution (term)"
- 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
-#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
-[ //
-| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
- elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
-| /4 width=1 by lsuby_succ, cpy_bind/
-| /3 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
-#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
-qed.
-
-(* Basic_1: was: subst1_ex *)
-lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
-#I #G #K #V #T1 elim T1 -T1
-[ * #i #L #l #HLK
- /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i l) #Hil
- /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i)
- /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
-| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1
- /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK
- /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
- ]
-]
-qed-.
-
-lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
- ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
- ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
-#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
-[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
-| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
-| /3 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
-[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
- lapply (drop_fwd_length_lt2 … HLK)
- /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
-| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
- /2 width=1 by cpy_bind/
-| /2 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
-#G #L #T1 #T2 #l #m #HT12
-lapply (cpy_weak … HT12 0 (l + m) ? ?) -HT12
-/2 width=2 by cpy_weak_top/
-qed-.
-
-lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, i-l] T & ⦃G, L⦄ ⊢ T ▶[i, l+m-i] T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
- elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
- elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
- l ≤ lt → l + m ≤ lt + mt →
- ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #T1 #l #m #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
- [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
- | elim (le_inv_plus_l … Hil) #Hlim #Hmi
- elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim
- #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hmi
- @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 //
- elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
- <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
- /3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
- /3 width=2 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
-/3 width=1 by monotonic_le_plus_l, le_plus/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
- T2 = ⓪{J} ∨
- ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{I}V &
- ⬆[O, i+1] V ≡ T2 &
- J = LRef i.
-#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
-[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
-| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
-]
-qed-.
-
-lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
- T2 = ⓪{I} ∨
- ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{J}V &
- ⬆[O, i+1] V ≡ T2 &
- I = LRef i.
-/2 width=4 by cpy_inv_atom1_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
-#G #L #T2 #k #l #m #H
-elim (cpy_inv_atom1 … H) -H //
-* #I #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
- T2 = #i ∨
- ∃∃I,K,V. l ≤ i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{I}V &
- ⬆[O, i+1] V ≡ T2.
-#G #L #T2 #i #l #m #H
-elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
-* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
-qed-.
-
-lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
-#G #L #T2 #p #l #m #H
-elim (cpy_inv_atom1 … H) -H //
-* #I #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
- U2 = ⓑ{a,I}V2.T2.
-#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
-[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
-| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
-]
-qed-.
-
-lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
- U2 = ⓑ{a,I}V2.T2.
-/2 width=3 by cpy_inv_bind1_aux/ qed-.
-
-fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
- ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
- U2 = ⓕ{I}V2.T2.
-#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
-[ #I #G #L #l #m #J #W1 #U1 #H destruct
-| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
- U2 = ⓕ{I}V2.T2.
-/2 width=3 by cpy_inv_flat1_aux/ qed-.
-
-
-fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
-[ //
-| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
- elim (ylt_yle_false … Hli) -Hli //
-| /3 width=1 by eq_f2/
-| /3 width=1 by eq_f2/
-]
-qed-.
-
-lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
-/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
- ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
-#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
-/2 width=4 by cpy_inv_refl_O2/
-qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/substitution/cpy_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem cpy_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶[l1, m1] T1 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶[l1, m1] T.
-#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
- elim (cpy_inv_lref1 … H) -H
- [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
- | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
- ]
-| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
- /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -V0
- elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_confluence_neq *)
-theorem cpy_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶[l1, m1] T1 →
- ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
- (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶[l1, m1] T.
-#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
- elim (cpy_inv_lref1 … H1) -H1
- [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
- [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/
- | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/
- ]
- ]
-| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0
- [ -H #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
- | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
- ]
-| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0
- elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Note: the constant 1 comes from cpy_subst *)
-(* Basic_1: was: subst1_trans *)
-theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 →
- ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2.
-#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
-[ #I #G #L #l #m #T2 #H #Hm
- elim (cpy_inv_atom1 … H) -H
- [ #H destruct //
- | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
- lapply (ylt_yle_trans … (l+m) … Hilm2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
- ]
-| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
- lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
- >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
-| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
- elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/
-| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
- elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
-]
-qed-.
-
-theorem cpy_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
- ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
-#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
- lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1
- lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
- >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
-| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV10 … HV02) -IHV10 -HV02 // #V
- elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV10 … HV02) -V0 //
- elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/substitution/drop_drop.ma".
-include "basic_2A/substitution/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Properties on relocation *************************************************)
-
-(* Basic_1: was: subst1_lift_lt *)
-lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
- ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
- lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2.
-#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
-[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
- lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
- lapply (ylt_inv_inj … Hil) -Hil #Hil
- lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct
- elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
- elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
-| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
- ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
- lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2.
-#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
-[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
- elim (lift_inv_lref1 … H) -H * #Hil #H destruct
- [ -Hltl
- lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm
- elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
- elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
- | -Hlti
- elim (yle_inv_inj2 … Hltl) -Hltl #ltt #Hltl #H destruct
- lapply (transitive_le … Hltl Hil) -Hltl #Hlti
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
- /4 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
- ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-(* Basic_1: was: subst1_lift_ge *)
-lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
- ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
- l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2.
-#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
-[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
- lapply (yle_trans … Hllt … Hlti) -Hllt #Hil
- elim (yle_inv_inj2 … Hil) -Hil #ll #Hlli #H0 destruct
- lapply (lift_inv_lref1_ge … H … Hlli) -H #H destruct
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hlli
- /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=6 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=6 by cpy_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma cpy_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt + mt ≤ l →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
- lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
- lapply (ylt_inv_inj … Hil) -Hil #Hil
- lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct
- elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil /3 width=5 by cpy_subst, ex2_intro/
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -IHU12 -HTU1
- /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt ≤ l → yinj l + m ≤ lt + mt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
- lapply (yle_fwd_plus_ge_inj … Hltl Hlmlmt) #Hmmt
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
- [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) /2 width=1 by ylt_inj/
- [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
- elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil
- /3 width=5 by cpy_subst, ex2_intro/
- | elim (le_inv_plus_l … Hil) #Hlim #Hmi
- lapply (yle_trans … Hltl (i-m) ?) /2 width=1 by yle_inj/ -Hltl #Hltim
- lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
- #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
- >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -U1
- /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK //
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- yinj l + m ≤ lt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
- lapply (yle_trans … Hlmlt … Hlti) #Hlmi
- elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt
- elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi
- lapply (lift_inv_lref2_ge … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
- lapply (drop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
- elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hlmi -Hlim
- #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
- [ /2 width=1 by monotonic_yle_minus_dx/
- | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (yle_inv_plus_inj2 … Hlmtl) #_ #Hmlt
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
- >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation ***********************************)
-
-lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[l, lt + mt - (yinj l + m)] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
-elim (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
-lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
-lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
-elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt ≤ l → lt + mt ≤ yinj l + m →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
-lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 //
-[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
-elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
- lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
-elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2
-elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
-[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
-lapply (cpy_weak … HU2 l m ? ?) -HU2 //
-[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
-lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/substitution/lift_neg.ma".
-include "basic_2A/substitution/lift_lift.ma".
-include "basic_2A/substitution/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Inversion lemmas on negated relocation ***********************************)
-
-lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
- ∀i. l ≤ yinj i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) →
- (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨
- ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W &
- (∀V. ⬆[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥).
-#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
-[ /3 width=2 by or_introl/
-| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
- elim (lt_or_ge j i) #Hij
- [ @or_intror @(ex5_4_intro … HLK) // -HLK
- [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
- #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
- | -HnW /2 width=7 by lift_inv_lref2_be/
- ]
- | elim (lift_split … HVW i j) -HVW //
- #X #_ #H elim HnW -HnW //
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
- [ /4 width=9 by nlift_bind_sn, or_introl/
- | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
- [ /4 width=9 by nlift_bind_dx, or_introl/
- | * #J #K #W #j #Hlj #Hji #HLK #HnW
- elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj
- lapply (ylt_O … Hj) -Hj #Hj
- lapply (drop_inv_drop1_lt … HLK ?) // -HLK #HLK
- >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
- #HnU1 <minus_le_minus_minus_comm in HnW;
- /5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
- ]
- ]
-| #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
- [ /4 width=9 by nlift_flat_sn, or_introl/
- | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
- [ /4 width=9 by nlift_flat_dx, or_introl/
- | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
- ]
-]
-qed-.
(* *)
(**************************************************************************)
-include "ground_2A/lib/lstar.ma".
+include "ground_2/xoa/ex_1_2.ma".
+include "ground_2/xoa/ex_1_3.ma".
+include "ground_2/xoa/ex_3_3.ma".
+include "ground_2/lib/star.ma".
+include "ground_2/lib/lstar_2a.ma".
include "basic_2A/notation/relations/rdrop_5.ma".
include "basic_2A/notation/relations/rdrop_4.ma".
include "basic_2A/notation/relations/rdrop_3.ma".
#L1 #L2 * /2 width=1 by drop_FT/
qed-.
-lemma d_liftable_LTC: ∀R. d_liftable R → d_liftable (LTC … R).
+lemma d_liftable_LTC: ∀R. d_liftable R → d_liftable (CTC … R).
#R #HR #K #T1 #T2 #H elim H -T2
[ /3 width=10 by inj/
| #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
]
qed-.
-lemma d_deliftable_sn_LTC: ∀R. d_deliftable_sn R → d_deliftable_sn (LTC … R).
+lemma d_deliftable_sn_LTC: ∀R. d_deliftable_sn R → d_deliftable_sn (CTC … R).
#R #HR #L #U1 #U2 #H elim H -U2
[ #U2 #HU12 #K #s #l #m #HLK #T1 #HTU1
elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
qed-.
lemma drop_fwd_length: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| + m.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m // normalize /2 width=1 by/
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m // normalize
+/2 width=1 by le_to_le_to_eq, le_n/
qed-.
lemma drop_fwd_length_minus2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| = |L1| - m.
(* *)
(**************************************************************************)
+include "ground_2/lib/arith_2a.ma".
include "basic_2A/grammar/lenv_append.ma".
include "basic_2A/substitution/drop.ma".
(* *)
(**************************************************************************)
+include "ground_2/ynat/ynat_minus_sn.ma".
include "basic_2A/grammar/lreq_lreq.ma".
include "basic_2A/substitution/drop.ma".
lemma lreq_drop_trans_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
∀I,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W →
l ≤ i → i < l + m →
- â\88\83â\88\83K1. K1 ⩬[0, â«°(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W.
+ â\88\83â\88\83K1. K1 ⩬[0, â\86\93(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W.
#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
[ #l #m #J #K2 #W #s #i #H
elim (drop_inv_atom1 … H) -H #H destruct
elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
[ #_ destruct >ypred_succ
/2 width=3 by drop_pair, ex2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ … H) -H
- #Him elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
- >yminus_succ <yminus_inj /3 width=3 by drop_drop_lt, ex2_intro/
+ | <(S_pred … Hi) <ysucc_inj #Him
+ lapply (ylt_inv_succ … Him) -Him #Him
+ elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
+ >yminus_succ /3 width=3 by drop_drop_lt, ex2_intro/
]
| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J #K2 #W #s #i #HLK2 #Hli
- elim (yle_inv_succ1 … Hli) -Hli
- #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
- #Hilm lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/
+ elim (yle_inv_succ_sn_lt … Hli) -Hli #Hli #Hi
+ lapply (ylt_inv_inj … Hi) -Hi #Hi
+ <(S_pred … Hi) >yplus_succ1 <ysucc_inj #H
+ lapply (ylt_inv_succ … H) -H #Hilm
+ lapply (drop_inv_drop1_lt … HLK2 Hi) -HLK2 #HLK1
+ elim (IHL12 … HLK1) -IHL12 -HLK1 >minus_SO_dx
+ /3 width=3 by drop_drop_lt, ex2_intro/
]
qed-.
lemma lreq_drop_conf_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
∀I,K1,W,s,i. ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W →
l ≤ i → i < l + m →
- â\88\83â\88\83K2. K1 ⩬[0, â«°(l+m-i)] K2 & ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W.
+ â\88\83â\88\83K2. K1 ⩬[0, â\86\93(l+m-i)] K2 & ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W.
#L1 #L2 #l #m #HL12 #I #K1 #W #s #i #HLK1 #Hli #Hilm
elim (lreq_drop_trans_be … (lreq_sym … HL12) … HLK1) // -L1 -Hli -Hilm
/3 width=3 by lreq_sym, ex2_intro/
(**************************************************************************)
include "basic_2A/notation/relations/rdrop_3.ma".
-include "basic_2A/grammar/genv.ma".
+include "basic_2A/grammar/genv_length.ma".
(* GLOBAL ENVIRONMENT READING ***********************************************)
qed-.
lemma gget_inv_lt: ∀I,G1,G2,V,m.
- ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2.
+ ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2.
/2 width=5 by gget_inv_lt_aux/ qed-.
(* Basic properties *********************************************************)
lemma gget_total: ∀m,G1. ∃G2. ⬇[m] G1 ≡ G2.
#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/
-#I #V #G1 * #G2 #HG12
+#G1 #I #V * #G2 #HG12
elim (lt_or_eq_or_gt m (|G1|)) #Hm
[ /3 width=2 by gget_lt, ex_intro/
| destruct /3 width=2 by gget_eq, ex_intro/
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_3_2.ma".
include "basic_2A/notation/relations/rlift_4.ma".
include "basic_2A/grammar/term_weight.ma".
include "basic_2A/grammar/term_simple.ma".
| /3 width=3 by lift_gref, ex2_intro/
| #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
- elim (IHT (l2+1) … ? ? Hm12) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+ elim (IHT (l2+1) … ? ? Hm12) /3 width=5 by lift_bind, monotonic_le_plus_l, ex2_intro/
| #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, monotonic_le_plus_l, le_S_S/ (* full auto a bit slow *)
| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T
+ elim (IHT12 … HT20) -IHT12 -HT20 [2: <assoc_plus1 /2 width=1 by le_S_S/ ]
<plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
(* BASIC TERM VECTOR RELOCATION *********************************************)
inductive liftv (l,m:nat) : relation (list term) ≝
-| liftv_nil : liftv l m (â\97\8a) (â\97\8a)
+| liftv_nil : liftv l m (â\92º) (â\92º)
| liftv_cons: ∀T1s,T2s,T1,T2.
⬆[l, m] T1 ≡ T2 → liftv l m T1s T2s →
- liftv l m (T1 @ T1s) (T2 @ T2s)
+ liftv l m (T1 ⨮ T1s) (T2 ⨮ T2s)
.
interpretation "relocation (vector)" 'RLift l m T1s T2s = (liftv l m T1s T2s).
(* Basic inversion lemmas ***************************************************)
-fact liftv_inv_nil1_aux: â\88\80T1s,T2s,l,m. â¬\86[l, m] T1s â\89¡ T2s â\86\92 T1s = â\97\8a â\86\92 T2s = â\97\8a.
+fact liftv_inv_nil1_aux: â\88\80T1s,T2s,l,m. â¬\86[l, m] T1s â\89¡ T2s â\86\92 T1s = â\92º â\86\92 T2s = â\92º.
#T1s #T2s #l #m * -T1s -T2s //
#T1s #T2s #T1 #T2 #_ #_ #H destruct
qed-.
-lemma liftv_inv_nil1: â\88\80T2s,l,m. â¬\86[l, m] â\97\8a â\89¡ T2s â\86\92 T2s = â\97\8a.
+lemma liftv_inv_nil1: â\88\80T2s,l,m. â¬\86[l, m] â\92º â\89¡ T2s â\86\92 T2s = â\92º.
/2 width=5 by liftv_inv_nil1_aux/ qed-.
fact liftv_inv_cons1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s →
- ∀U1,U1s. T1s = U1 @ U1s →
+ ∀U1,U1s. T1s = U1 ⨮ U1s →
∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
- T2s = U2 @ U2s.
+ T2s = U2 ⨮ U2s.
#T1s #T2s #l #m * -T1s -T2s
[ #U1 #U1s #H destruct
| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma liftv_inv_cons1: ∀U1,U1s,T2s,l,m. ⬆[l, m] U1 @ U1s ≡ T2s →
+lemma liftv_inv_cons1: ∀U1,U1s,T2s,l,m. ⬆[l, m] U1 ⨮ U1s ≡ T2s →
∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
- T2s = U2 @ U2s.
+ T2s = U2 ⨮ U2s.
/2 width=3 by liftv_inv_cons1_aux/ qed-.
(* Basic properties *********************************************************)
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_3_2.ma".
include "basic_2A/grammar/lenv_length.ma".
(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
(* *)
(**************************************************************************)
+include "ground_2/lib/star.ma".
include "basic_2A/substitution/lpx_sn.ma".
(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … R L1 V1 V2 →
+ ∀V1,V2. CTC … R L1 V1 V2 →
TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
[ /2 width=1 by TC_lpx_sn_pair_refl/
qed-.
lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
+ ∀L1,L2. lpx_sn (CTC … R) L1 L2 →
TC … (lpx_sn R) L1 L2.
#R #HR #L1 #L2 #H elim H -L1 -L2
/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+ ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & CTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
∀S:relation lenv.
S (⋆) (⋆) → (
∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
+ TC … (lpx_sn R) K1 K2 → CTC … R K1 V1 V2 →
S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
) →
∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀L1,L2. TC … (lpx_sn R) L1 L2 →
∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & CTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
[ #J #K #W #H destruct
| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & CTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (LTC … R) L1 L2.
+ lpx_sn (CTC … R) L1 L2.
/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
(* Forward lemmas on transitive closure *************************************)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_plus.ma".
-include "basic_2A/notation/relations/lrsubeq_4.ma".
-include "basic_2A/substitution/drop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-inductive lsuby: relation4 ynat ynat lenv lenv ≝
-| lsuby_atom: ∀L,l,m. lsuby l m L (⋆)
-| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
- lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 →
- lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
-| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
- lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-.
-
-interpretation
- "local environment refinement (extended substitution)"
- 'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m →
- L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V.
-#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/
-qed.
-
-lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l →
- L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/
-qed.
-
-lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
- ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
-#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
-qed.
-
-lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L.
-#L elim L -L //
-#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
-#Hl destruct /2 width=1 by lsuby_succ/
-#m elim (ynat_cases … m) [| * #x ]
-#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
-qed.
-
-lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, yinj 0] L2.
-#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
-[ #l #H elim (le_plus_xSy_O_false … H)
-| #L1 #I1 #V1 #l #H lapply (le_plus_to_le_r … H) -H #HL12
- elim (ynat_cases l) /3 width=1 by lsuby_zero/
- * /3 width=1 by lsuby_succ/
-]
-qed.
-
-lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1.
-#l #m #L1 #L2 #H elim H -l -m -L1 -L2
-[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 //
-| /2 width=1 by lsuby_O2/
-| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l … H)
- /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l … H)
- /3 width=1 by lsuby_succ/
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #l #m * -L1 -L2 -l -m //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
-]
-qed-.
-
-lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆.
-/2 width=5 by lsuby_inv_atom1_aux/ qed-.
-
-fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
- L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
- L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=9 by lsuby_inv_zero1_aux/ qed-.
-
-fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m →
- L2 = ⋆ ∨
- ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W.
-#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct
- /3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m →
- L2 = ⋆ ∨
- ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V.
-/2 width=6 by lsuby_inv_pair1_aux/ qed-.
-
-fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
- L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-]
-qed-.
-
-lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l →
- L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=5 by lsuby_inv_succ1_aux/ qed-.
-
-fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 →
- ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
- ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=9 by lsuby_inv_zero2_aux/ qed-.
-
-fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m →
- ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
- /2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m →
- ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V.
-/2 width=6 by lsuby_inv_pair2_aux/ qed-.
-
-fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l →
- ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W2 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct
- /2 width=5 by ex2_3_intro/
-]
-qed-.
-
-lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l →
- ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=5 by lsuby_inv_succ2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize /2 width=1 by le_S_S/
-qed-.
-
-(* Properties on basic slicing **********************************************)
-
-lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
- l ≤ i → i < l + m →
- ∃∃I1,K1. K1 ⊆[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W #s #i #H
- elim (drop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct -I2 >ypred_succ
- /2 width=4 by drop_pair, ex2_2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ … H) -H
- #Him elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
- >yminus_succ <yminus_inj /3 width=4 by drop_drop_lt, ex2_2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli
- elim (yle_inv_succ1 … Hli) -Hli
- #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
- #Hilm lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=4 by ylt_O, drop_drop_lt, ex2_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2A/substitution/lsuby.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-(* Main properties **********************************************************)
-
-theorem lsuby_trans: ∀l,m. Transitive … (lsuby l m).
-#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
-[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 … H) -H
- #H destruct //
-| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
-| #I1 #I2 #L1 #L2 #V #m #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
- * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
-]
-qed-.
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/xoa/ex_4_4.ma".
include "basic_2A/notation/relations/statictypestar_6.ma".
include "basic_2A/grammar/genv.ma".
include "basic_2A/substitution/drop.ma".
--- /dev/null
+<?xml version="1.0" encoding="UTF-8"?>
+
+<page xmlns="http://lambdadelta.info/"
+ description = "\lambda\delta home page"
+ title = "\lambda\delta home page"
+ logo = "crux"
+ head = "cic:/matita/lambdadelta/basic_2A/ (core λδ version 2A)"
+>
+ <sitemap name="sitemap"/>
+
+ <section4 name="summary">Summary of the Specification</section4>
+ <body>Here is a numerical account of the specification's contents
+ and its timeline.
+ </body>
+ <table name="basic_2A_sum"/>
+
+ <news class="delta" date="2020-02-27.">
+ λδ-2A is repackaged on the basis of λδ-ground-2.
+ </news>
+ <news class="delta" date="2019-11-20.">
+ λδ-2A is repackaged (was λδ-2A1).
+ </news>
+ <news class="delta" date="2015-08-27.">
+ λδ-2A appears too complex and is dismissed.
+ </news>
+ <news class="gamma" date="2014-10-28.">
+ λδ-2A is released.
+ </news>
+ <news class="beta" date="2014-09-09.">
+ Iterated static type assignment defined (more elegantly)
+ as a primitive notion.
+ </news>
+ <news class="beta" date="2014-06-18.">
+ Preservation of stratified native validity
+ for context-sensitive computation on terms.
+ </news>
+ <news class="alpha" date="2014-06-09.">
+ Strong qrst-normalization
+ for simply typed terms.
+ </news>
+ <news class="alpha" date="2014-04-16.">
+ Lazy equivalence on local environments
+ added as q-step to rst-computation on closures
+ (anniversary milestone).
+ </news>
+ <news class="alpha" date="2014-01-20.">
+ Parametrized slicing on local environments
+ comprises both versions of this operation
+ (one from basic_1, the other used in basic_2 till now).
+ </news>
+ <news class="alpha" date="2013-08-07.">
+ Passive support for global environments.
+ </news>
+ <news class="alpha" date="2013-07-27.">
+ Reaxiomatized β-reductum as in rt-reduction.
+ </news>
+ <news class="alpha" date="2013-07-20.">
+ Context-sensitive strong rt-normalization
+ for simply typed terms.
+ </news>
+ <news class="alpha" date="2013-04-16.">
+ Reaxiomatized substitution and reduction
+ commute with respect to subclosure
+ (anniversary milestone).
+ </news>
+ <news class="alpha" date="2013-03-16.">
+ Mutual recursive preservation of stratified native validity
+ for rst-computation on closures.
+ </news>
+ <news class="alpha" date="2012-10-16.">
+ Confluence for context-free parallel reduction on closures.
+ </news>
+ <news class="alpha" date="2012-07-26.">
+ Term binders polarized to control ζ-reduction (not released).
+ </news>
+ <news class="alpha" date="2012-04-16.">
+ Context-sensitive subject equivalence
+ for atomic arity assignment
+ (anniversary milestone).
+ </news>
+ <news class="alpha" date="2012-03-15.">
+ Context-sensitive strong normalization
+ for simply typed terms.
+ </news>
+ <news class="alpha" date="2012-01-27.">
+ Generic candidates of reducibility.
+ </news>
+ <news class="alpha" date="2011-09-21.">
+ Confluence for context-sensitive parallel reduction on terms.
+ </news>
+ <news class="alpha" date="2011-09-06.">
+ Confluence for context-free parallel reduction on terms.
+ </news>
+ <news class="alpha" date="2011-04-17.">
+ λδ-2A is started.
+ </news>
+<!--
+ <section4 name="structure">Logical Structure of the Specification</section4>
+ <body>This table reports the specification's components and their planes.
+ </body>
+ <table name="basic_2A_src"/>
+-->
+ <footer/>
+</page>
include ../Makefile.common
test:
- @./inline.native -p ../lambdadelta/*/deps.txt > deps.txt
- @../matitadep/matitadep.native -c ../lambdadelta/.depend deps.txt > redundant.txt
- @./inline.native -i -b ../lambdadelta redundant.txt ../lambdadelta/*/deps.txt
+ @./inline.native -p ../../*/deps.txt > deps.txt
+ @../../../../../components/binaries/matitadep/matitadep.native -c ../../.depend deps.txt > redundant.txt
+ @./inline.native -i -b ../.. redundant.txt ../../*/deps.txt
.PHONY: test
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* right subtraction *)
+definition yminus_dx: nat → ynat → nat ≝ λx,y. match y with
+[ yinj n ⇒ x-n
+| Y ⇒ 0
+].
+
+interpretation "ynat right minus" 'minus x y = (yminus_dx x y).
+
+lemma yminus_dx_inj (m) (n): m - yinj n = m - n.
+// qed.
+
+lemma yminus_dx_Y2: ∀m. m - (∞) = 0.
+// qed.
+
+lemma yminus_dx_succ_bi (n:ynat) (m): ↑m - ↑n = m - n.
+* // qed.
+
+lemma yminus_dx_S2: ∀n:ynat. ∀m. m - ↑n = ↓(m - n).
+* // qed.
+
+lemma yplus_minus_dx (x) (y): x ≤ yinj y → yinj y = x + yinj (y-x).
+* [ #x #y #H >yplus_inj /4 width=1 by yle_inv_inj, minus_to_plus, eq_f/ ]
+#y #H lapply (yle_inv_Y1 … H) -H #H destruct
+qed-.
+
+(*
+
+
+(* Basic properties *********************************************************)
+
+lemma yminus_inj: ∀m,n. yinj m - yinj n = yinj (m - n).
+#m #n elim n -n //
+#n #IH >yminus_S2 >IH -IH >eq_minus_S_pred //
+qed.
+
+lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
+#n elim n -n //
+qed.
+
+lemma yminus_O1: ∀x:ynat. 0 - x = 0.
+* // qed.
+
+lemma yminus_refl: ∀x:ynat. x - x = 0.
+* // qed.
+
+lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
+* #y [ * #z [ * // ] ] >yminus_O1 //
+qed.
+
+(* Properties on predecessor ************************************************)
+
+lemma yminus_SO2: ∀m. m - 1 = ⫰m.
+* //
+qed.
+
+lemma yminus_pred1: ∀x,y. ⫰x - y = ⫰(x-y).
+#x * // #y elim y -y //
+qed.
+
+lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
+* // #n *
+[ #m #Hm #Hn >yminus_inj >yminus_inj
+ /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+(* Properties on successor **************************************************)
+
+lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
+#n *
+[ #m #Hmn >yminus_inj >yminus_inj
+ /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
+* //
+qed.
+
+(* Properties on order ******************************************************)
+
+lemma yle_minus_sn: ∀n,m. m - n ≤ m.
+* // #n * /2 width=1 by yle_inj/
+qed.
+
+lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
+#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
+qed-.
+
+lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
+* // #n *
+[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+ /2 width=1 by yle_inj/
+| >yminus_Y_inj #H destruct
+]
+qed.
+
+lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
+#x #y #Hxy * //
+#z elim z -z /3 width=1 by yle_pred/
+qed.
+
+(* Properties on strict order ***********************************************)
+
+lemma ylt_to_minus: ∀x,y:ynat. x < y → 0 < y - x.
+#x #y #H elim H -x -y /3 width=1 by ylt_inj, lt_plus_to_minus_r/
+qed.
+
+lemma yminus_to_lt: ∀x,y:ynat. 0 < y - x → x < y.
+* [2: #y #H elim (ylt_yle_false … H) // ]
+#m * /4 width=1 by ylt_inv_inj, ylt_inj, lt_minus_to_plus_r/
+qed-.
+
+lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
+#x #y * -x -y
+/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
+qed.
+
+(* Properties on minus ******************************************************)
+
+lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
+#m #n elim n -n //
+#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
+qed.
+
+lemma yplus_minus: ∀m,n. m + n - n ≤ m.
+#m * //
+qed.
+
+lemma yminus_plus2: ∀z,y,x:ynat. x - (y + z) = x - y - z.
+* // #z * [2: >yplus_Y1 >yminus_O1 // ] #y *
+[ #x >yplus_inj >yminus_inj >yminus_inj >yminus_inj /2 width=1 by eq_f/
+| >yplus_inj >yminus_Y_inj //
+]
+qed.
+
+(* Forward lemmas on minus **************************************************)
+
+lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
+#x #z #y #H lapply (monotonic_yle_minus_dx … H y) -H //
+qed-.
+
+lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
+/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
+
+lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
+/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
+
+lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
+#x *
+[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
+ /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yplus_minus_assoc_comm_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z - (y - x) = z + x - y.
+#x *
+[ #y *
+ [ #z >yminus_inj >yminus_inj >yplus_inj >yminus_inj
+ /4 width=1 by yle_inv_inj, minus_le_minus_minus_comm, eq_f/
+ | >yminus_inj >yminus_Y_inj //
+ ]
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
+#y * // #x * //
+#z #Hxy >yplus_inj >yminus_inj <plus_minus
+/2 width=1 by yle_inv_inj/
+qed-.
+
+lemma ylt_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y < z → x < z - y.
+#x #z #y #H lapply (monotonic_ylt_minus_dx … H y ?) -H //
+qed-.
+
+lemma ylt_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x < z → x < z - y.
+/2 width=1 by ylt_plus1_to_minus_inj2/ qed-.
+
+lemma ylt_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. z ≤ x → x < y + z → x - z < y.
+/2 width=1 by monotonic_ylt_minus_dx/ qed-.
+
+lemma ylt_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. z ≤ x → x < z + y → x - z < y.
+/2 width=1 by ylt_plus2_to_minus_inj2/ qed-.
+
+lemma yplus_inv_minus: ∀x1,y1. y1 ≤ yinj x1 →
+ ∀x2,y2. yinj x1 + x2 = yinj y2 + y1 →
+ yinj x1 - y1 = yinj y2 - x2 ∧ x2 ≤ yinj y2.
+#x1 #y1 #Hyx1 #x2 #y2 #H0
+lapply (yle_fwd_plus_ge_inj … x2 y2 Hyx1 ?) // #Hxy2
+elim (yle_inv_inj2 … Hyx1) -Hyx1 #m #Hyx1 #H destruct
+elim (yle_inv_inj2 … Hxy2) #n #H1 #H destruct
+>yplus_inj in H0; >yplus_inj >yminus_inj >yminus_inj #H0
+@conj // lapply (yinj_inj … H0) -H0 /3 width=1 by arith_b1, eq_f/
+qed-.
+
+(* Inversion lemmas on minus ************************************************)
+
+lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
+/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
+
+lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
+/2 width=1 by yle_inv_plus_inj2/ qed-.
+
+lemma yle_inv_plus_inj_dx: ∀x,y:ynat. ∀z:nat. x + y ≤ z →
+ ∃∃m,n. x = yinj m & y = yinj n & x ≤ z - y & y ≤ z.
+#x #y #z #Hz elim (yle_inv_inj2 … Hz)
+#z0 #_ #H elim (yplus_inv_inj … H) -H
+#m #n #H1 #H2 #H3 destruct
+elim (yle_inv_plus_inj2 … Hz) -Hz /2 width=2 by ex4_2_intro/
+qed-.
+*)
lemma plus_SO_dx (n): n + 1 = ↑n.
// qed.
+lemma minus_SO_dx (n): n-1 = ↓n.
+// qed.
+
lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
// qed-.
lemma discr_plus_x_xy: ∀x,y. x = x + y → y = 0.
/2 width=2 by le_plus_minus_comm/ qed-.
-lemma plus2_inv_le_sn: ∀m1,m2,n1,n2. m1 + n1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
+lemma plus2_le_sn_sn: ∀m1,m2,n1,n2. m1 + n1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
#m1 #m2 #n1 #n2 #H #Hm
lapply (monotonic_le_plus_l n1 … Hm) -Hm >H -H
/2 width=2 by le_plus_to_le/
qed-.
+lemma plus2_le_sn_dx: ∀m1,m2,n1,n2. m1 + n1 = n2 + m2 → m1 ≤ m2 → n2 ≤ n1.
+/2 width=4 by plus2_le_sn_sn/ qed-.
+
+lemma plus2_le_dx_sn: ∀m1,m2,n1,n2. n1 + m1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
+/2 width=4 by plus2_le_sn_sn/ qed-.
+
+lemma plus2_le_dx_dx: ∀m1,m2,n1,n2. n1 + m1 = n2 + m2 → m1 ≤ m2 → n2 ≤ n1.
+/2 width=4 by plus2_le_sn_sn/ qed-.
+
lemma lt_S_S_to_lt: ∀x,y. ↑x < ↑y → x < y.
/2 width=1 by le_S_S_to_le/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "arithmetics/lstar.ma".
+
+(* PROPERTIES OF NAT-LABELED REFLEXIVE AND TRANSITIVE CLOSURE ***************)
+
+definition llstar: ∀A:Type[0]. ∀B. (A→relation B) → nat → (A→relation B) ≝
+ λA,B,R,l,a. lstar … (R a) l.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( l1 @@ break l2 )"
+ right associative with precedence 47
+ for @{ 'Append $l1 $l2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (6, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex3 (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex3 (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) }.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (7, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex3 (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P6) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex3 (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P6) }.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (7, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex4 (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex4 (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) }.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (8, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
+ non associative with precedence 20
+ for @{ 'Ex4 (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P7) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
+ non associative with precedence 20
+ for @{ 'Ex4 (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P7) }.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/functions/append_2.ma".
+include "ground_2/relocation/mr2.ma".
+
+(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
+
+let rec mr2_append cs1 cs2 on cs1 ≝
+ match cs1 with
+ [ nil2 ⇒ cs2
+ | cons2 l m cs1 ⇒ ❨l, m❩; mr2_append cs1 cs2
+ ].
+
+interpretation "append (multiple relocation with pairs)"
+ 'Append cs1 cs2 = (mr2_append cs1 cs2).
+
+(* Basic properties *********************************************************)
+
+lemma mr2_append_nil (cs2): cs2 = ◊ @@ cs2.
+// qed.
+
+lemma mr2_append_cons (l) (m) (cs1) (cs2):
+ ❨l, m❩; (cs1 @@ cs2) = (❨l, m❩; cs1) @@ cs2.
+// qed.
</body>
<table name="ground_2_sum"/>
+ <news class="alpha" date="2020 February 27.">
+ Specification becomes an independent package.
+ </news>
<news class="alpha" date="2020 January 6.">
Centralized xoa infrastructure removed.
</news>
[ "trace ( ∥?∥ )" "trace_at ( @❪?,?❫ ≘ ? )" "trace_after ( ? ⊚ ? ≘ ? )" "trace_isid ( 𝐈❪?❫ )" "trace_isun ( 𝐔❪?❫ )"
"trace_sle ( ? ⊆ ? )" "trace_sor ( ? ⋓ ? ≘ ? )" "trace_snot ( ∁ ? )" * ]
*)
- [ "mr2 ( ◊ ) ( ❨?,?❩;? )" "mr2_at ( @❪?,?❫ ≘ ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? ▭ ? ≘ ? )" * ]
+ [ "mr2 ( ◊ ) ( ❨?,?❩;? )" "mr2_append ( ? @@? )" "mr2_at ( @❪?,?❫ ≘ ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? ▭ ? ≘ ? )" * ]
}
]
}
[ { "" * } {
[ "ynat ( ∞ )" "ynat_pred ( ↓? )" "ynat_succ ( ↑? )"
"ynat_le ( ? ≤ ? )" "ynat_lt ( ? < ? )"
- "ynat_plus ( ? + ? )" *
+ "ynat_plus ( ? + ? )" "ynat_minus_sn ( ? - ? )" *
]
}
]
[ "list ( Ⓔ{?} ) ( ? ⨮{?} ? )" "list_length ( |?| )" * ]
[ "bool ( Ⓕ ) ( Ⓣ )" "arith ( ?^? ) ( ↑? ) ( ↓? ) ( ? ∨ ? ) ( ? ∧ ? )" "arith_2a" "arith_2b" * ]
[ "ltc" "ltc_ctc" * ]
- [ "logic ( ⊥ ) ( ⊤ )" "relations ( ? ⊆ ? )" "functions" "exteq ( ? ≐{?,?} ? )" "star" * ]
+ [ "logic ( ⊥ ) ( ⊤ )" "relations ( ? ⊆ ? )" "functions" "exteq ( ? ≐{?,?} ? )" "star" "lstar_2a" * ]
}
]
}
<key name="ex">5 5</key>
<key name="ex">5 6</key>
<key name="ex">5 7</key>
+ <key name="ex">6 3</key>
<key name="ex">6 4</key>
<key name="ex">6 5</key>
<key name="ex">6 6</key>
<key name="ex">6 7</key>
<key name="ex">6 8</key>
<key name="ex">6 9</key>
+ <key name="ex">7 3</key>
+ <key name="ex">7 4</key>
<key name="ex">7 5</key>
<key name="ex">7 6</key>
<key name="ex">7 7</key>
<key name="ex">7 10</key>
+ <key name="ex">8 4</key>
<key name="ex">8 5</key>
<key name="ex">9 3</key>
<key name="or">3</key>
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2/notation/xoa/ex_6_3.ma".
+
+(* multiple existental quantifier (6, 3) *)
+
+inductive ex6_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→Prop) : Prop ≝
+ | ex6_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → ex6_3 ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 3)" 'Ex3 P0 P1 P2 P3 P4 P5 = (ex6_3 ? ? ? P0 P1 P2 P3 P4 P5).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2/notation/xoa/ex_7_3.ma".
+
+(* multiple existental quantifier (7, 3) *)
+
+inductive ex7_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→Prop) : Prop ≝
+ | ex7_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → P6 x0 x1 x2 → ex7_3 ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (7, 3)" 'Ex3 P0 P1 P2 P3 P4 P5 P6 = (ex7_3 ? ? ? P0 P1 P2 P3 P4 P5 P6).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2/notation/xoa/ex_7_4.ma".
+
+(* multiple existental quantifier (7, 4) *)
+
+inductive ex7_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→Prop) : Prop ≝
+ | ex7_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → ex7_4 ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (7, 4)" 'Ex4 P0 P1 P2 P3 P4 P5 P6 = (ex7_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2/notation/xoa/ex_8_4.ma".
+
+(* multiple existental quantifier (8, 4) *)
+
+inductive ex8_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→Prop) : Prop ≝
+ | ex8_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → P7 x0 x1 x2 x3 → ex8_4 ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (8, 4)" 'Ex4 P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
+
#H destruct
qed.
+lemma yle_inv_succ_sn_lt (x:ynat) (y:ynat):
+ ↑x ≤ y → ∧∧ x ≤ ↓y & 0 < y.
+#x #y #H elim (yle_inv_succ1 … H) -H /3 width=2 by ylt_O1, conj/
+qed-.
+
(* Properties on predecessor ************************************************)
lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ↓m < ↓n.
(* Properties on successor **************************************************)
-lemma ylt_O_succ: ∀n. 0 < ↑n.
+lemma ylt_O_succ: ∀x:ynat. 0 < ↑x.
* /2 width=1 by ylt_inj/
qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* left subtraction *)
+definition yminus_sn (x) (y): ynat ≝ ypred^y x.
+
+interpretation "ynat left minus" 'minus x y = (yminus_sn x y).
+
+lemma yminus_O2: ∀m:ynat. m - 0 = m.
+// qed.
+
+lemma yminus_S2: ∀m:ynat. ∀n:nat. m - S n = ↓(m - n).
+// qed.
+
+(* Basic properties *********************************************************)
+
+lemma yminus_inj: ∀m,n. yinj m - n = yinj (m - n).
+#m #n elim n -n //
+#n #IH >yminus_S2 >IH -IH >eq_minus_S_pred //
+qed.
+
+lemma yminus_Y_inj: ∀n. ∞ - n = ∞.
+#n elim n -n //
+qed.
+
+lemma yminus_O1: ∀x:nat. yinj 0 - x = 0.
+// qed.
+
+lemma yminus_refl: ∀x:nat. yinj x - x = 0.
+// qed.
+
+lemma yminus_minus_comm: ∀x:ynat. ∀y,z. x - y - z = x - z - y.
+* // qed.
+
+(* Properties on predecessor ************************************************)
+
+lemma yminus_SO2: ∀m:ynat. m - 1 = ↓m.
+// qed.
+
+lemma yminus_pred1: ∀x,y. ↓x - y = ↓(x-y).
+#x * // #y elim y -y //
+qed.
+
+lemma yminus_pred: ∀m:ynat. ∀n. 0 < m → 0 < n → ↓m - ↓n = m - n.
+* // #m #n >yminus_inj >yminus_inj
+/4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
+qed-.
+
+(* Properties on successor **************************************************)
+
+lemma yminus_succ: ∀m:ynat. ∀n. ↑m - ↑n = m - n.
+* // qed.
+
+lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ↑m - n = ↑(m - n).
+#n *
+[ #m #Hmn >yminus_inj >yminus_inj
+ /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yminus_succ2: ∀x:ynat. ∀y. x - ↑y = ↓(x-y).
+* //
+qed.
+
+(* Properties on order ******************************************************)
+
+lemma yle_minus_sn: ∀m:ynat. ∀n. m - n ≤ m.
+* // #n /2 width=1 by yle_inj/
+qed.
+
+lemma yle_to_minus: ∀m:ynat. ∀n:nat. m ≤ n → m - n = 0.
+*
+[ #m #n #H >yminus_inj /4 width=1 by yle_inv_inj, eq_minus_O, eq_f/
+| #n #H lapply (yle_inv_Y1 … H) -H #H destruct
+]
+qed-.
+
+lemma yminus_to_le: ∀m:ynat. ∀n. m - n = 0 → m ≤ n.
+* [2: #n >yminus_Y_inj #H destruct ]
+#m #n >yminus_inj #H
+lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+/2 width=1 by yle_inj/
+qed.
+
+lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
+#x #y * /3 width=1 by yle_inj, monotonic_le_minus_l2/
+qed.
+
+(* Properties on strict order ***********************************************)
+
+lemma ylt_to_minus: ∀y:ynat. ∀x. yinj x < y → 0 < y - x.
+* // #y #x #H >yminus_inj
+/4 width=1 by ylt_inj, ylt_inv_inj, lt_plus_to_minus_r/
+qed.
+
+lemma yminus_to_lt: ∀y:ynat. ∀x. 0 < y - x → x < y.
+* // #y #x >yminus_inj #H
+/4 width=1 by ylt_inv_inj, ylt_inj, lt_minus_to_plus_r/
+qed-.
+
+lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
+#x #y * -x -y
+/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
+qed.
+
+(* Properties on minus ******************************************************)
+
+lemma yplus_minus: ∀m:ynat. ∀n:nat. m + n - n = m.
+#m #n elim n -n //
+#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
+qed.
+
+lemma yminus_plus2: ∀x:ynat. ∀y,z. x - (y + z) = x - y - z.
+* // qed.
+
+(* Forward lemmas on minus **************************************************)
+
+lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
+#x #z #y #H lapply (monotonic_yle_minus_dx … H y) -H //
+qed-.
+
+lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
+/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
+
+lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
+/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
+
+lemma yminus_plus (x:ynat) (y:nat): y ≤ x → x = (x-y)+y.
+* // #x #y #H >yminus_inj >yplus_inj
+/4 width=1 by yle_inv_inj, plus_minus, eq_f/
+qed-.
+
+lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
+#x *
+[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
+ /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+alias symbol "plus" (instance 5) = "ynat plus".
+alias symbol "minus" (instance 4) = "ynat left minus".
+alias symbol "minus" (instance 3) = "natural minus".
+alias symbol "minus" (instance 2) = "ynat left minus".
+alias symbol "leq" (instance 6) = "natural 'less or equal to'".
+lemma yplus_minus_assoc_comm_inj: ∀z:ynat. ∀x,y:nat. x ≤ y → z - (y - x) = z + x - y.
+* // #z #x #y >yminus_inj >yplus_inj >yminus_inj
+/4 width=1 by yle_inv_inj, minus_le_minus_minus_comm, eq_f/
+qed-.
+
+lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
+#y * // #x * //
+#z #Hxy >yplus_inj >yminus_inj <plus_minus
+/2 width=1 by yle_inv_inj/
+qed-.
+
+lemma ylt_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y < z → x < z - y.
+#x #z #y #H lapply (monotonic_ylt_minus_dx … H y ?) -H //
+qed-.
+
+lemma ylt_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x < z → x < z - y.
+/2 width=1 by ylt_plus1_to_minus_inj2/ qed-.
+
+lemma ylt_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. z ≤ x → x < y + z → x - z < y.
+/2 width=1 by monotonic_ylt_minus_dx/ qed-.
+
+lemma ylt_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. z ≤ x → x < z + y → x - z < y.
+/2 width=1 by ylt_plus2_to_minus_inj2/ qed-.
+
+lemma yplus_inv_Y1: ∀x,y. ∞ = x + y → ∨∨ ∞ = x | ∞ = y.
+* /2 width=1 by or_introl/ #x * // #y >yplus_inj #H destruct
+qed-.
+
+lemma yplus_inv_minus:
+ ∀x1,y2:ynat.∀y1,x2:nat.
+ y1 ≤ x1 → x1 + x2 = y2 + y1 → ∧∧ x1 - y1 = y2 - x2 & x2 ≤ y2.
+*
+[ #x1 * [| #y1 #x2 #_ >yplus_inj >yplus_Y1 #H destruct ]
+ #y2 #y1 #x2 #H1 >yplus_inj >yplus_inj #H2 >yminus_inj >yminus_inj
+ lapply (yle_inv_inj … H1) -H1 #Hyx1
+ lapply (yinj_inj … H2) -H2 #Hxy (**) (* destruct lemma needed *)
+ /5 width=4 by yle_inj, plus2_le_sn_sn, plus_to_minus_2, conj, eq_f2/
+| #y2 #y1 #x2 #_ >yplus_Y1 #H
+ elim (yplus_inv_Y1 … H) -H #H destruct /2 width=1 by conj/
+]
+qed-.
+
+(* Inversion lemmas on minus ************************************************)
+
+lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
+/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
+
+lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
+/2 width=1 by yle_inv_plus_inj2/ qed-.
+
+lemma yle_inv_plus_inj_dx: ∀z,x:ynat. ∀y:nat. x + y ≤ z →
+ ∧∧ x ≤ z - y & y ≤ z.
+* [| /2 width=1 by conj/ ]
+#z * [| #y >yplus_Y1 #H >(yle_inv_Y1 … H) -z /2 width=1 by conj/ ]
+#x #y >yplus_inj #H >yminus_inj
+/5 width=2 by yle_inv_inj, yle_inj, le_plus_to_minus_r, le_plus_b, conj/
+qed-.
lemma yplus_succ_swap: ∀m,n. m + ↑n = ↑m + n.
// qed.
-lemma yplus_SO2: ∀m. m + 1 = ↑m.
+lemma yplus_SO2: ∀m:ynat. m + 1 = ↑m.
* //
qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "arithmetics/nat.ma".
-include "ground_2A/lib/star.ma".
-
-(* ARITHMETICAL PROPERTIES **************************************************)
-
-(* Equations ****************************************************************)
-
-lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
-// qed-.
-
-(* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
-lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
-#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
-qed-.
-
-fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
-/2 width=1 by plus_minus_minus_be/ qed-.
-
-lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
-// qed.
-
-lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
-/2 by plus_minus/ qed.
-
-lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
-/2 by plus_minus/ qed.
-
-lemma minus_minus_comm3: ∀n,x,y,z. n-x-y-z = n-y-z-x.
-// qed.
-
-lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
-#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
-qed.
-
-lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
-#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1 by arith_b1/
-qed.
-
-lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
-/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
-
-lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
- a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
-#a1 #a2 #b #c1 #H1 #H2 >plus_minus /2 width=1 by arith_b2/
-qed.
-
-lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
-/2 width=1 by plus_minus/ qed-.
-
-(* Properties ***************************************************************)
-
-lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
-#n1 elim n1 -n1 [| #n1 #IHn1 ] * [2,4: #n2 ]
-[1,4: @or_intror #H destruct
-| elim (IHn1 n2) -IHn1 /3 width=1 by or_intror, or_introl/
-| /2 width=1 by or_introl/
-]
-qed-.
-
-lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
-#m #n elim (lt_or_ge m n) /2 width=1 by or3_intro0/
-#H elim H -m /2 width=1 by or3_intro1/
-#m #Hm * /3 width=1 by not_le_to_lt, le_S_S, or3_intro2/
-qed-.
-
-fact le_repl_sn_conf_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
-// qed-.
-
-fact le_repl_sn_trans_aux: ∀x,y,z:nat. x ≤ z → y = x → y ≤ z.
-// qed-.
-
-lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
-/3 width=1 by monotonic_le_minus_l/ qed.
-
-(* Note: this might interfere with nat.ma *)
-lemma monotonic_lt_pred: ∀m,n. m < n → O < m → pred m < pred n.
-#m #n #Hmn #Hm whd >(S_pred … Hm)
-@le_S_S_to_le >S_pred /2 width=3 by transitive_lt/
-qed.
-
-lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
-/3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
-
-lemma arith_k_sn: ∀z,x,y,n. z < x → x+n ≤ y → x-z-1+n ≤ y-z-1.
-#z #x #y #n #Hzx #Hxny
->plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
->plus_minus [2: /2 width=1 by lt_to_le/ ]
-/2 width=1 by monotonic_le_minus_l2/
-qed.
-
-lemma arith_k_dx: ∀z,x,y,n. z < x → y ≤ x+n → y-z-1 ≤ x-z-1+n.
-#z #x #y #n #Hzx #Hyxn
->plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
->plus_minus [2: /2 width=1 by lt_to_le/ ]
-/2 width=1 by monotonic_le_minus_l2/
-qed.
-
-(* Inversion & forward lemmas ***********************************************)
-
-lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
-// qed-.
-
-lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
-/2 width=1 by monotonic_pred/ qed-.
-
-lemma lt_refl_false: ∀n. n < n → ⊥.
-#n #H elim (lt_to_not_eq … H) -H /2 width=1 by/
-qed-.
-
-lemma lt_zero_false: ∀n. n < 0 → ⊥.
-#n #H elim (lt_to_not_le … H) -H /2 width=1 by/
-qed-.
-
-lemma pred_inv_refl: ∀m. pred m = m → m = 0.
-* // normalize #m #H elim (lt_refl_false m) //
-qed-.
-
-lemma le_plus_xSy_O_false: ∀x,y. x + S y ≤ 0 → ⊥.
-#x #y #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
-qed-.
-
-lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥.
-#y #z #x elim x -x /3 width=1 by le_S_S_to_le/
-#H elim (le_plus_xSy_O_false … H)
-qed-.
-
-lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
-/2 width=4 by le_plus_xySz_x_false/ qed-.
-
-lemma plus_xSy_x_false: ∀y,x. x + S y = x → ⊥.
-/2 width=4 by plus_xySz_x_false/ qed-.
-
-(* Note this should go in nat.ma *)
-lemma discr_x_minus_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
-#x @(nat_ind_plus … x) -x /2 width=1 by or_introl/
-#x #_ #y @(nat_ind_plus … y) -y /2 width=1 by or_intror/
-#y #_ >minus_plus_plus_l
-#H lapply (discr_plus_xy_minus_xz … H) -H
-#H destruct
-qed-.
-
-lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
-* /2 width=1 by conj/ #x #y normalize #H destruct
-qed-.
-
-(* Iterators ****************************************************************)
-
-(* Note: see also: lib/arithemetics/bigops.ma *)
-let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
- match n with
- [ O ⇒ nil
- | S k ⇒ op (iter k B op nil)
- ].
-
-interpretation "iterated function" 'exp op n = (iter n ? op).
-
-lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
-#B #f #b #l >commutative_plus //
-qed.
-
-lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b).
-#B #f #b #l elim l -l normalize //
-qed.
-
-lemma iter_plus: ∀B:Type[0]. ∀f:B→B. ∀b,l1,l2. f^(l1+l2) b = f^l1 (f^l2 b).
-#B #f #b #l1 elim l1 -l1 normalize //
-qed.
-
-(* Trichotomy operator ******************************************************)
-
-(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
-let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
- match n1 with
- [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
- | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
- ].
-
-lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
-#A #a1 #a2 #a3 #n2 elim n2 -n2
-[ #n1 #H elim (lt_zero_false … H)
-| #n2 #IH #n1 elim n1 -n1 /3 width=1 by monotonic_lt_pred/
-]
-qed.
-
-lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
-#A #a1 #a2 #a3 #n elim n -n normalize //
-qed.
-
-lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
-#A #a1 #a2 #a3 #n1 elim n1 -n1
-[ #n2 #H elim (lt_zero_false … H)
-| #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basics/bool.ma".
-include "ground_2A/lib/star.ma".
-include "ground_2A/notation/constructors/no_0.ma".
-include "ground_2A/notation/constructors/yes_0.ma".
-
-(* BOOLEAN PROPERTIES *******************************************************)
-
-interpretation "boolean false" 'no = false.
-
-interpretation "boolean true" 'yes = true.
-
-(* Basic properties *********************************************************)
-
-lemma orb_false_r: ∀b1,b2:bool. (b1 ∨ b2) = false → b1 = false ∧ b2 = false.
-* normalize /2 width=1 by conj/ #b2 #H destruct
-qed-.
-
-lemma commutative_orb: commutative … orb.
-* * // qed.
-
-lemma eq_bool_dec: ∀b1,b2:bool. Decidable (b1 = b2).
-* * /2 width=1 by or_introl/
-@or_intror #H destruct
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/notation/constructors/nil_0.ma".
-include "ground_2A/notation/constructors/cons_2.ma".
-include "ground_2A/notation/constructors/cons_3.ma".
-include "ground_2A/notation/functions/append_2.ma".
-include "ground_2A/lib/arith.ma".
-
-(* LISTS ********************************************************************)
-
-inductive list (A:Type[0]) : Type[0] :=
- | nil : list A
- | cons: A → list A → list A.
-
-interpretation "nil (list)" 'Nil = (nil ?).
-
-interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl).
-
-let rec all A (R:predicate A) (l:list A) on l ≝
- match l with
- [ nil ⇒ ⊤
- | cons hd tl ⇒ R hd ∧ all A R tl
- ].
-
-inductive list2 (A1,A2:Type[0]) : Type[0] :=
- | nil2 : list2 A1 A2
- | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2.
-
-interpretation "nil (list of pairs)" 'Nil = (nil2 ? ?).
-
-interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl).
-
-let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with
-[ nil2 ⇒ l2
-| cons2 a1 a2 tl ⇒ {a1, a2} @ append2 A1 A2 tl l2
-].
-
-interpretation "append (list of pairs)"
- 'Append l1 l2 = (append2 ? ? l1 l2).
-
-let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with
-[ nil2 ⇒ 0
-| cons2 _ _ l ⇒ length2 A1 A2 l + 1
-].
-
-interpretation "length (list of pairs)"
- 'card l = (length2 ? ? l).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "arithmetics/lstar.ma".
-
-(* PROPERTIES OF NAT-LABELED REFLEXIVE AND TRANSITIVE CLOSURE ***************)
-
-definition llstar: ∀A:Type[0]. ∀B. (A→relation B) → nat → (A→relation B) ≝
- λA,B,R,l,a. lstar … (R a) l.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basics/star1.ma".
-include "ground_2A/xoa/xoa_props.ma".
-
-(* PROPERTIES OF RELATIONS **************************************************)
-
-definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
-
-definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
- ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
-
-definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
- ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & R1 a2 a.
-
-definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
- ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & R1 a a2.
-
-definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
- ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
- ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
-
-definition LTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝
- λA,B,R,a. TC … (R a).
-
-definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
- ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
-
-definition s_r_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
- ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2.
-
-definition s_rs_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
- ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2.
-
-definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
- ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
-
-lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
- ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & TC … R1 a2 a.
-#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
-[ #a1 #Ha01 #a2 #Ha02
- elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
-| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
- elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
- elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5 by step, ex2_intro/
-]
-qed.
-
-lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
- ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
- ∃∃a. TC … R2 a1 a & R1 a2 a.
-#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
-[ #a2 #Ha02 #a1 #Ha01
- elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
-| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
- elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
- elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3 by step, ex2_intro/
-]
-qed.
-
-lemma TC_confluent2: ∀A,R1,R2.
- confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
-#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
-[ #a1 #Ha01 #a2 #Ha02
- elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3 by inj, ex2_intro/
-| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
- elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
- elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5 by step, ex2_intro/
-]
-qed.
-
-lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
- ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & TC … R1 a a2.
-#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
-[ #a0 #Ha10 #a2 #Ha02
- elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
-| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
- elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
- elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/
-]
-qed.
-
-lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
- ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
- ∃∃a. TC … R2 a1 a & R1 a a2.
-#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
-[ #a2 #Ha02 #a1 #Ha10
- elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
-| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
- elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
- elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3 by step, ex2_intro/
-]
-qed.
-
-lemma TC_transitive2: ∀A,R1,R2.
- transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
-#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
-[ #a0 #Ha10 #a2 #Ha02
- elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3 by inj, ex2_intro/
-| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
- elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
- elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/
-]
-qed.
-
-definition NF: ∀A. relation A → relation A → predicate A ≝
- λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
-
-definition NF_dec: ∀A. relation A → relation A → Prop ≝
- λA,R,S. ∀a1. NF A R S a1 ∨
- ∃∃a2. R … a1 a2 & (S a2 a1 → ⊥).
-
-inductive SN (A) (R,S:relation A): predicate A ≝
-| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
-.
-
-lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
-#A #R #S #a1 #Ha1
-@SN_intro #a2 #HRa12 #HSa12
-elim HSa12 -HSa12 /2 width=1 by/
-qed.
-
-lemma SN_to_NF: ∀A,R,S. NF_dec A R S →
- ∀a1. SN A R S a1 →
- ∃∃a2. star … R a1 a2 & NF A R S a2.
-#A #R #S #HRS #a1 #H elim H -a1
-#a1 #_ #IHa1 elim (HRS a1) -HRS /2 width=3 by srefl, ex2_intro/
-* #a0 #Ha10 #Ha01 elim (IHa1 … Ha10 Ha01) -IHa1 -Ha01 /3 width=3 by star_compl, ex2_intro/
-qed-.
-
-definition NF_sn: ∀A. relation A → relation A → predicate A ≝
- λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
-
-inductive SN_sn (A) (R,S:relation A): predicate A ≝
-| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
-.
-
-lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
-#A #R #S #a2 #Ha2
-@SN_sn_intro #a1 #HRa12 #HSa12
-elim HSa12 -HSa12 /2 width=1 by/
-qed.
-
-lemma LTC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S.
-#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 /3 width=3 by inj/
-#T #T2 #_ #HT2 #IHT1 #L1 #HL12
-lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3 by step/
-qed-.
-
-lemma s_r_conf1_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (LTC … S) R.
-#A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3 by/
-qed-.
-
-lemma s_r_trans_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R →
- s_r_transitive A B S R → s_rs_transitive A B S R.
-#A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /2 width=3 by/
-#T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR … HT1 … HL12) -H2SR -HT1
-/4 width=5 by s_r_conf1_LTC1, trans_TC/
-qed-.
-
-lemma s_r_trans_LTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (LTC … R).
-#A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /3 width=3 by inj/
-qed-.
-
-lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (LTC … S) R →
- s_rs_transitive A B S R.
-#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
-elim (TC_idem … (S L1) … T1 T2)
-#_ #H @H @HSR //
-qed-.
-
-lemma s_rs_to_s_r_trans: ∀A,B,S,R. s_rs_transitive A B S R →
- s_r_transitive A B (LTC … S) R.
-#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
-elim (TC_idem … (S L1) … T1 T2)
-#H #_ @H @HSR //
-qed-.
-
-lemma s_rs_trans_TC1: ∀A,B,S,R. s_rs_transitive A B S R →
- s_rs_transitive A B (LTC … S) R.
-#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
-elim (TC_idem … (S L1) … T1 T2)
-elim (TC_idem … (S L2) … T1 T2)
-#_ #H1 #H2 #_ @H2 @HSR /3 width=3 by/
-qed-.
-
-(* relations on unboxed pairs ***********************************************)
-
-lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
- ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
- ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
-#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
-[ #a2 #b2 #H02
- elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
-| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2
- elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4 by ex2_2_intro, bi_step/
-]
-qed.
-
-lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
- bi_confluent A B (bi_TC … R).
-#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
-[ #a1 #b1 #H01 #a2 #b2 #H02
- elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
-| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02
- elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20
- elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7 by ex2_2_intro, bi_step/
-]
-qed.
-
-lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
- ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
- R a1 b1 a2 b2 ∨
- ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
-#A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4 by ex2_2_intro, or_intror/
-qed-.
-
-lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
- ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
- R a1 b1 a2 b2 ∨
- ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
-#A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
-[ /2 width=1 by or_introl/
-| #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4 by ex2_2_intro, or_intror/ (**) (* auto fails without #_ *)
-]
-qed-.
-
-(* relations on unboxed triples *********************************************)
-
-definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
- λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
- ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
-
-lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
-/3 width=1 by and3_intro, or_intror/ qed.
-
-definition tri_star: ∀A,B,C,R. tri_relation A B C ≝
- λA,B,C,R. tri_RC A B C (tri_TC … R).
-
-lemma tri_star_tri_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_star … R).
-/2 width=1 by/ qed.
-
-lemma tri_TC_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
- tri_TC A B C R a1 b1 c1 a2 b2 c2 →
- tri_star A B C R a1 b1 c1 a2 b2 c2.
-/2 width=1 by or_introl/ qed.
-
-lemma tri_R_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
- R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
-/3 width=1 by tri_TC_to_tri_star, tri_inj/ qed.
-
-lemma tri_star_strap1: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
- tri_star A B C R a1 b1 c1 a b c →
- R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
-#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
-[ /3 width=5 by tri_TC_to_tri_star, tri_step/
-| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
-]
-qed.
-
-lemma tri_star_strap2: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c →
- tri_star A B C R a b c a2 b2 c2 →
- tri_star A B C R a1 b1 c1 a2 b2 c2.
-#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
-[ /3 width=5 by tri_TC_to_tri_star, tri_TC_strap/
-| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
-]
-qed.
-
-lemma tri_star_to_tri_TC_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
- tri_star A B C R a1 b1 c1 a b c →
- tri_TC A B C R a b c a2 b2 c2 →
- tri_TC A B C R a1 b1 c1 a2 b2 c2.
-#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
-[ /2 width=5 by tri_TC_transitive/
-| * #H1 #H2 #H3 destruct /2 width=1 by/
-]
-qed.
-
-lemma tri_TC_to_tri_star_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
- tri_TC A B C R a1 b1 c1 a b c →
- tri_star A B C R a b c a2 b2 c2 →
- tri_TC A B C R a1 b1 c1 a2 b2 c2.
-#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
-[ /2 width=5 by tri_TC_transitive/
-| * #H1 #H2 #H3 destruct /2 width=1 by/
-]
-qed.
-
-lemma tri_tansitive_tri_star: ∀A,B,C,R. tri_transitive A B C (tri_star … R).
-#A #B #C #R #a1 #a #b1 #b #c1 #c #H #a2 #b2 #c2 *
-[ /3 width=5 by tri_star_to_tri_TC_to_tri_TC, tri_TC_to_tri_star/
-| * #H1 #H2 #H3 destruct /2 width=1 by/
-]
-qed.
-
-lemma tri_star_ind: ∀A,B,C,R,a1,b1,c1. ∀P:relation3 A B C. P a1 b1 c1 →
- (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
- ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
-#A #B #C #R #a1 #b1 #c1 #P #H #IH #a2 #b2 #c2 *
-[ #H12 elim H12 -a2 -b2 -c2 /3 width=6 by tri_TC_to_tri_star/
-| * #H1 #H2 #H3 destruct //
-]
-qed-.
-
-lemma tri_star_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
- (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
- ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
-#A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 *
-[ #H12 @(tri_TC_ind_dx … a1 b1 c1 H12) -a1 -b1 -c1 /3 width=6 by tri_TC_to_tri_star/
-| * #H1 #H2 #H3 destruct //
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( hd @ break tl )"
- right associative with precedence 47
- for @{ 'Cons $hd $tl }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( { term 46 hd1 , break term 46 hd2 } @ break term 46 tl )"
- non associative with precedence 47
- for @{ 'Cons $hd1 $hd2 $tl }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "∞"
- non associative with precedence 55
- for @{ 'Infinity }.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "◊"
- non associative with precedence 46
- for @{ 'Nil }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "Ⓕ"
- non associative with precedence 55
- for @{'no}.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "Ⓣ"
- non associative with precedence 55
- for @{'yes}.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( l1 @@ break l2 )"
- right associative with precedence 47
- for @{ 'Append $l1 $l2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( ⫰ term 70 T )"
- non associative with precedence 70
- for @{ 'Predecessor $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( ⫯ term 70 T )"
- non associative with precedence 70
- for @{ 'Successor $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "⊥"
- non associative with precedence 19
- for @{'false}.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "⊤"
- non associative with precedence 19
- for @{'true}.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was generated by xoa.native: do not edit *********************)
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was generated by xoa.native: do not edit *********************)
-
-(* multiple existental quantifier (1, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }.
-
-(* multiple existental quantifier (1, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) }.
-
-(* multiple existental quantifier (2, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }.
-
-(* multiple existental quantifier (2, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) }.
-
-(* multiple existental quantifier (3, 1) *)
-
-notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }.
-
-(* multiple existental quantifier (3, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }.
-
-(* multiple existental quantifier (3, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }.
-
-(* multiple existental quantifier (3, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) }.
-
-(* multiple existental quantifier (4, 1) *)
-
-notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) (λ${ident x0}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) (λ${ident x0}:$T0.$P3) }.
-
-(* multiple existental quantifier (4, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) }.
-
-(* multiple existental quantifier (4, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }.
-
-(* multiple existental quantifier (4, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }.
-
-(* multiple existental quantifier (4, 5) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) }.
-
-(* multiple existental quantifier (5, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) (λ${ident x0}.λ${ident x1}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }.
-
-(* multiple existental quantifier (5, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }.
-
-(* multiple existental quantifier (5, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }.
-
-(* multiple existental quantifier (5, 5) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }.
-
-(* multiple existental quantifier (5, 6) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) }.
-
-(* multiple existental quantifier (6, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P5) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) }.
-
-(* multiple existental quantifier (6, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }.
-
-(* multiple existental quantifier (6, 5) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) }.
-
-(* multiple existental quantifier (6, 6) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }.
-
-(* multiple existental quantifier (6, 7) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) }.
-
-(* multiple existental quantifier (7, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P6) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P6) }.
-
-(* multiple existental quantifier (7, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) }.
-
-(* multiple existental quantifier (7, 7) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P6) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P6) }.
-
-(* multiple existental quantifier (8, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P7) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P7) }.
-
-(* multiple existental quantifier (8, 5) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P6) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P7) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P6) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P7) }.
-
-(* multiple disjunction connective (3) *)
-
-notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)"
- non associative with precedence 30
- for @{ 'Or $P0 $P1 $P2 }.
-
-(* multiple disjunction connective (4) *)
-
-notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)"
- non associative with precedence 30
- for @{ 'Or $P0 $P1 $P2 $P3 }.
-
-(* multiple disjunction connective (5) *)
-
-notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3 break | term 29 P4)"
- non associative with precedence 30
- for @{ 'Or $P0 $P1 $P2 $P3 $P4 }.
-
-(* multiple conjunction connective (3) *)
-
-notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)"
- non associative with precedence 35
- for @{ 'And $P0 $P1 $P2 }.
-
-(* multiple conjunction connective (4) *)
-
-notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2 break & term 34 P3)"
- non associative with precedence 35
- for @{ 'And $P0 $P1 $P2 $P3 }.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was generated by xoa.native: do not edit *********************)
-
-include "basics/pts.ma".
-
-include "ground_2A/notation/xoa_notation.ma".
-
-(* multiple existental quantifier (1, 2) *)
-
-inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝
- | ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ?
-.
-
-interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0).
-
-(* multiple existental quantifier (1, 3) *)
-
-inductive ex1_3 (A0,A1,A2:Type[0]) (P0:A0→A1→A2→Prop) : Prop ≝
- | ex1_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → ex1_3 ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (1, 3)" 'Ex P0 = (ex1_3 ? ? ? P0).
-
-(* multiple existental quantifier (2, 2) *)
-
-inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝
- | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1).
-
-(* multiple existental quantifier (2, 3) *)
-
-inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝
- | ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? P0 P1).
-
-(* multiple existental quantifier (3, 1) *)
-
-inductive ex3 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝
- | ex3_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3 ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3 ? P0 P1 P2).
-
-(* multiple existental quantifier (3, 2) *)
-
-inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝
- | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2).
-
-(* multiple existental quantifier (3, 3) *)
-
-inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝
- | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2).
-
-(* multiple existental quantifier (3, 4) *)
-
-inductive ex3_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2:A0→A1→A2→A3→Prop) : Prop ≝
- | ex3_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → ex3_4 ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 4)" 'Ex P0 P1 P2 = (ex3_4 ? ? ? ? P0 P1 P2).
-
-(* multiple existental quantifier (4, 1) *)
-
-inductive ex4 (A0:Type[0]) (P0,P1,P2,P3:A0→Prop) : Prop ≝
- | ex4_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → P3 x0 → ex4 ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 1)" 'Ex P0 P1 P2 P3 = (ex4 ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (4, 2) *)
-
-inductive ex4_2 (A0,A1:Type[0]) (P0,P1,P2,P3:A0→A1→Prop) : Prop ≝
- | ex4_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → ex4_2 ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (4, 3) *)
-
-inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝
- | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (4, 4) *)
-
-inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝
- | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (4, 5) *)
-
-inductive ex4_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→A4→Prop) : Prop ≝
- | ex4_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → ex4_5 ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 5)" 'Ex P0 P1 P2 P3 = (ex4_5 ? ? ? ? ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (5, 2) *)
-
-inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝
- | ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (5, 3) *)
-
-inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝
- | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (5, 4) *)
-
-inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝
- | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (5, 5) *)
-
-inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝
- | ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (5, 6) *)
-
-inductive ex5_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
- | ex5_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → ex5_6 ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 6)" 'Ex P0 P1 P2 P3 P4 = (ex5_6 ? ? ? ? ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (6, 3) *)
-
-inductive ex6_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→Prop) : Prop ≝
- | ex6_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → ex6_3 ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (6, 3)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_3 ? ? ? P0 P1 P2 P3 P4 P5).
-
-(* multiple existental quantifier (6, 4) *)
-
-inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝
- | ex6_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → ex6_4 ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5).
-
-(* multiple existental quantifier (6, 5) *)
-
-inductive ex6_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→Prop) : Prop ≝
- | ex6_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → ex6_5 ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-
-(* multiple existental quantifier (6, 6) *)
-
-inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
- | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-
-(* multiple existental quantifier (6, 7) *)
-
-inductive ex6_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
- | ex6_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → ex6_7 ? ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-
-(* multiple existental quantifier (7, 3) *)
-
-inductive ex7_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→Prop) : Prop ≝
- | ex7_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → P6 x0 x1 x2 → ex7_3 ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (7, 3)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_3 ? ? ? P0 P1 P2 P3 P4 P5 P6).
-
-(* multiple existental quantifier (7, 4) *)
-
-inductive ex7_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→Prop) : Prop ≝
- | ex7_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → ex7_4 ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (7, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
-
-(* multiple existental quantifier (7, 7) *)
-
-inductive ex7_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
- | ex7_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → P6 x0 x1 x2 x3 x4 x5 x6 → ex7_7 ? ? ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (7, 7)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
-
-(* multiple existental quantifier (8, 4) *)
-
-inductive ex8_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→Prop) : Prop ≝
- | ex8_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → P7 x0 x1 x2 x3 → ex8_4 ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (8, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
-
-(* multiple existental quantifier (8, 5) *)
-
-inductive ex8_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→A4→Prop) : Prop ≝
- | ex8_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → P6 x0 x1 x2 x3 x4 → P7 x0 x1 x2 x3 x4 → ex8_5 ? ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (8, 5)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
-
-(* multiple disjunction connective (3) *)
-
-inductive or3 (P0,P1,P2:Prop) : Prop ≝
- | or3_intro0: P0 → or3 ? ? ?
- | or3_intro1: P1 → or3 ? ? ?
- | or3_intro2: P2 → or3 ? ? ?
-.
-
-interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2).
-
-(* multiple disjunction connective (4) *)
-
-inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝
- | or4_intro0: P0 → or4 ? ? ? ?
- | or4_intro1: P1 → or4 ? ? ? ?
- | or4_intro2: P2 → or4 ? ? ? ?
- | or4_intro3: P3 → or4 ? ? ? ?
-.
-
-interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3).
-
-(* multiple disjunction connective (5) *)
-
-inductive or5 (P0,P1,P2,P3,P4:Prop) : Prop ≝
- | or5_intro0: P0 → or5 ? ? ? ? ?
- | or5_intro1: P1 → or5 ? ? ? ? ?
- | or5_intro2: P2 → or5 ? ? ? ? ?
- | or5_intro3: P3 → or5 ? ? ? ? ?
- | or5_intro4: P4 → or5 ? ? ? ? ?
-.
-
-interpretation "multiple disjunction connective (5)" 'Or P0 P1 P2 P3 P4 = (or5 P0 P1 P2 P3 P4).
-
-(* multiple conjunction connective (3) *)
-
-inductive and3 (P0,P1,P2:Prop) : Prop ≝
- | and3_intro: P0 → P1 → P2 → and3 ? ? ?
-.
-
-interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2).
-
-(* multiple conjunction connective (4) *)
-
-inductive and4 (P0,P1,P2,P3:Prop) : Prop ≝
- | and4_intro: P0 → P1 → P2 → P3 → and4 ? ? ? ?
-.
-
-interpretation "multiple conjunction connective (4)" 'And P0 P1 P2 P3 = (and4 P0 P1 P2 P3).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was generated by xoa.native: do not edit *********************)
-
-include "basics/pts.ma".
-
-include "ground_2A/notation/xoa2_notation.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basics/logic.ma".
-include "ground_2A/notation/xoa/false_0.ma".
-include "ground_2A/notation/xoa/true_0.ma".
-include "ground_2A/xoa/xoa.ma".
-
-interpretation "logical false" 'false = False.
-
-interpretation "logical true" 'true = True.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "arithmetics/nat.ma".
-include "ground_2A/notation/constructors/infinity_0.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* the type of natural numbers with infinity *)
-inductive ynat: Type[0] ≝
-| yinj: nat → ynat
-| Y : ynat
-.
-
-coercion yinj.
-
-interpretation "ynat infinity" 'Infinity = Y.
-
-(* Inversion lemmas *********************************************************)
-
-lemma yinj_inj: ∀m,n. yinj m = yinj n → m = n.
-#m #n #H destruct //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_succ.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* order relation *)
-inductive yle: relation ynat ≝
-| yle_inj: ∀m,n. m ≤ n → yle m n
-| yle_Y : ∀m. yle m (∞)
-.
-
-interpretation "ynat 'less or equal to'" 'leq x y = (yle x y).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact yle_inv_inj2_aux: ∀x,y. x ≤ y → ∀n. y = yinj n →
- ∃∃m. m ≤ n & x = yinj m.
-#x #y * -x -y
-[ #x #y #Hxy #n #Hy destruct /2 width=3 by ex2_intro/
-| #x #n #Hy destruct
-]
-qed-.
-
-lemma yle_inv_inj2: ∀x,n. x ≤ yinj n → ∃∃m. m ≤ n & x = yinj m.
-/2 width=3 by yle_inv_inj2_aux/ qed-.
-
-lemma yle_inv_inj: ∀m,n. yinj m ≤ yinj n → m ≤ n.
-#m #n #H elim (yle_inv_inj2 … H) -H
-#x #Hxn #H destruct //
-qed-.
-
-fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
-#m #x * -m -x
-[ #m #n #Hmn #H destruct /3 width=1 by le_n_O_to_eq, eq_f/
-| #m #H destruct
-]
-qed-.
-
-lemma yle_inv_O2: ∀m:ynat. m ≤ 0 → m = 0.
-/2 width =3 by yle_inv_O2_aux/ qed-.
-
-fact yle_inv_Y1_aux: ∀x,n. x ≤ n → x = ∞ → n = ∞.
-#x #n * -x -n //
-#x #n #_ #H destruct
-qed-.
-
-lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
-/2 width=3 by yle_inv_Y1_aux/ qed-.
-
-(* Inversion lemmas on successor ********************************************)
-
-fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ ⫯⫰y = y.
-#x #y * -x -y
-[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
- #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
- #m #Hnm #H destruct /3 width=1 by yle_inj, conj/
-| #x #y #H destruct /2 width=1 by yle_Y, conj/
-]
-qed-.
-
-lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → m ≤ ⫰y ∧ ⫯⫰y = y.
-/2 width=3 by yle_inv_succ1_aux/ qed-.
-
-lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
-#m #n #H elim (yle_inv_succ1 … H) -H //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma le_O1: ∀n:ynat. 0 ≤ n.
-* /2 width=1 by yle_inj/
-qed.
-
-lemma yle_refl: reflexive … yle.
-* /2 width=1 by le_n, yle_inj/
-qed.
-
-lemma yle_split: ∀x,y:ynat. x ≤ y ∨ y ≤ x.
-* /2 width=1 by or_intror/
-#x * /2 width=1 by or_introl/
-#y elim (le_or_ge x y) /3 width=1 by yle_inj, or_introl, or_intror/
-qed-.
-
-(* Properties on predecessor ************************************************)
-
-lemma yle_pred_sn: ∀m,n. m ≤ n → ⫰m ≤ n.
-#m #n * -m -n /3 width=3 by transitive_le, yle_inj/
-qed.
-
-lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
-/2 width=1 by yle_refl, yle_pred_sn/ qed.
-
-lemma yle_pred: ∀m,n. m ≤ n → ⫰m ≤ ⫰n.
-#m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/
-qed.
-
-(* Properties on successor **************************************************)
-
-lemma yle_succ: ∀m,n. m ≤ n → ⫯m ≤ ⫯n.
-#m #n * -m -n /3 width=1 by yle_inj, le_S_S/
-qed.
-
-lemma yle_succ_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
-#m #n * -m -n /3 width=1 by le_S, yle_inj/
-qed.
-
-lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
-/2 width=1 by yle_succ_dx/ qed.
-
-lemma yle_refl_SP_dx: ∀x. x ≤ ⫯⫰x.
-* // * //
-qed.
-
-(* Main properties **********************************************************)
-
-theorem yle_trans: Transitive … yle.
-#x #y * -x -y
-[ #x #y #Hxy * //
- #z #H lapply (yle_inv_inj … H) -H
- /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
-| #x #z #H lapply (yle_inv_Y1 … H) //
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_le.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* strict order relation *)
-inductive ylt: relation ynat ≝
-| ylt_inj: ∀m,n. m < n → ylt m n
-| ylt_Y : ∀m:nat. ylt m (∞)
-.
-
-interpretation "ynat 'less than'" 'lt x y = (ylt x y).
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m.
-#x #y * -x -y /2 width=2 by ex_intro/
-qed-.
-
-lemma ylt_fwd_le_succ: ∀x,y. x < y → ⫯x ≤ y.
-#x #y * -x -y /2 width=1 by yle_inj/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
- ∃∃m. m < n & x = yinj m.
-#x #y * -x -y
-[ #x #y #Hxy #n #Hy elim (le_inv_S1 … Hxy) -Hxy
- #m #Hm #H destruct /3 width=3 by le_S_S, ex2_intro/
-| #x #n #Hy destruct
-]
-qed-.
-
-lemma ylt_inv_inj2: ∀x,n. x < yinj n →
- ∃∃m. m < n & x = yinj m.
-/2 width=3 by ylt_inv_inj2_aux/ qed-.
-
-lemma ylt_inv_inj: ∀m,n. yinj m < yinj n → m < n.
-#m #n #H elim (ylt_inv_inj2 … H) -H
-#x #Hx #H destruct //
-qed-.
-
-lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
-#n #H elim (ylt_fwd_gen … H) -H
-#y #H destruct
-qed-.
-
-lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
-* // #n #H lapply (ylt_inv_inj … H) -H normalize
-/3 width=1 by S_pred, eq_f/
-qed-.
-
-(* Inversion lemmas on successor ********************************************)
-
-fact ylt_inv_succ1_aux: ∀x,y. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰y = y.
-#x #y * -x -y
-[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
- #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
- #m #Hnm #H destruct /3 width=1 by ylt_inj, conj/
-| #x #y #H elim (ysucc_inv_inj_sn … H) -H
- #m #H #_ destruct /2 width=1 by ylt_Y, conj/
-]
-qed-.
-
-lemma ylt_inv_succ1: ∀m,y. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
-/2 width=3 by ylt_inv_succ1_aux/ qed-.
-
-lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
-#m #n #H elim (ylt_inv_succ1 … H) -H //
-qed-.
-
-(* Forward lemmas on successor **********************************************)
-
-fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ⫯n → x ≤ n.
-#x #y * -x -y
-[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
- #n #H1 #H2 destruct /3 width=1 by yle_inj, le_S_S_to_le/
-| #x #n #H lapply (ysucc_inv_Y_sn … H) -H //
-]
-qed-.
-
-lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
-/2 width=3 by ylt_fwd_succ2_aux/ qed-.
-
-(* inversion and forward lemmas on yle **************************************)
-
-lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
-#m #n * -m -n /2 width=1 by yle_inj/
-qed-.
-
-lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
-#m #n * -m -n /3 width=1 by lt_to_le, yle_inj/
-qed-.
-
-lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
-#m #n * -m -n
-[ #m #n #Hmn #H lapply (yle_inv_inj … H) -H
- #H elim (lt_refl_false n) /2 width=3 by le_to_lt_to_lt/
-| #m #H lapply (yle_inv_Y1 … H) -H
- #H destruct
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ylt_O: ∀x. ⫯⫰(yinj x) = yinj x → 0 < x.
-* /2 width=1 by/ normalize
-#H destruct
-qed.
-
-(* Properties on predecessor ************************************************)
-
-lemma ylt_pred: ∀m,n. m < n → 0 < m → ⫰m < ⫰n.
-#m #n * -m -n
-/4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/
-qed.
-
-(* Properties on successor **************************************************)
-
-lemma ylt_O_succ: ∀n. 0 < ⫯n.
-* /2 width=1 by ylt_inj/
-qed.
-
-lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
-#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
-qed.
-
-(* Properties on order ******************************************************)
-
-lemma yle_split_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.
-#m #n * -m -n
-[ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn
- /3 width=1 by or_introl, ylt_inj/
-| * /2 width=1 by or_introl, ylt_Y/
-]
-qed-.
-
-lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m..
-#m #n elim (yle_split m n) /2 width=1 by or_intror/
-#H elim (yle_split_eq … H) -H /2 width=1 by or_introl, or_intror/
-qed-.
-
-lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z.
-#x #y #z * -y -z
-[ #y #z #Hyz #H elim (ylt_inv_inj2 … H) -H
- #m #Hm #H destruct /3 width=3 by ylt_inj, lt_to_le_to_lt/
-| #y * //
-]
-qed-.
-
-lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x < z.
-#x #y #z * -y -z
-[ #y #z #Hyz #H elim (yle_inv_inj2 … H) -H
- #m #Hm #H destruct /3 width=3 by ylt_inj, le_to_lt_to_lt/
-| #y #H elim (yle_inv_inj2 … H) -H //
-]
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem ylt_trans: Transitive … ylt.
-#x #y * -x -y
-[ #x #y #Hxy * //
- #z #H lapply (ylt_inv_inj … H) -H
- /3 width=3 by transitive_lt, ylt_inj/ (**) (* full auto too slow *)
-| #x #z #H elim (ylt_yle_false … H) //
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_plus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-lemma ymax_pre_dx: ∀x,y. x ≤ y → x - y + y = y.
-#x #y * -x -y //
-#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
-qed-.
-
-lemma ymax_pre_sn: ∀x,y. y ≤ x → x - y + y = x.
-#x #y * -x -y
-[ #x #y #Hxy >yminus_inj /3 width=3 by plus_minus, eq_f/
-| * //
-]
-qed-.
-
-lemma ymax_pre_i_dx: ∀y,x. y ≤ x - y + y.
-// qed.
-
-lemma ymax_pre_i_sn: ∀y,x. x ≤ x - y + y.
-* // #y * /2 width=1 by yle_inj/
-qed.
-
-lemma ymax_pre_e: ∀x,z. x ≤ z → ∀y. y ≤ z → x - y + y ≤ z.
-#x #z #Hxz #y #Hyz elim (yle_split x y)
-[ #Hxy >(ymax_pre_dx … Hxy) -x //
-| #Hyx >(ymax_pre_sn … Hyx) -y //
-]
-qed.
-
-lemma ymax_pre_dx_comm: ∀x,y. x ≤ y → y + (x - y) = y.
-/2 width=1 by ymax_pre_dx/ qed-.
-
-lemma ymax_pre_sn_comm: ∀x,y. y ≤ x → y + (x - y) = x.
-/2 width=1 by ymax_pre_sn/ qed-.
-
-lemma ymax_pre_i_dx_comm: ∀y,x. y ≤ y + (x - y).
-// qed.
-
-lemma ymax_pre_i_sn_comm: ∀y,x. x ≤ y + (x - y).
-/2 width=1 by ymax_pre_i_sn/ qed.
-
-lemma ymax_pre_e_comm: ∀x,z. x ≤ z → ∀y. y ≤ z → y + (x - y) ≤ z.
-/2 width=1 by ymax_pre_e/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_plus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-fact ymin_pre_dx_aux: ∀x,y. y ≤ x → x - (x - y) ≤ y.
-#x #y * -x -y
-[ #x #y #Hxy >yminus_inj
- /3 width=4 by yle_inj, monotonic_le_minus_l/
-| * //
-]
-qed-.
-
-lemma ymin_pre_sn: ∀x,y. x ≤ y → x - (x - y) = x.
-#x #y * -x -y //
-#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
-qed-.
-
-lemma ymin_pre_i_dx: ∀x,y. x - (x - y) ≤ y.
-#x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/
-#Hxy >(ymin_pre_sn … Hxy) //
-qed.
-
-lemma ymin_pre_i_sn: ∀x,y. x - (x - y) ≤ x.
-// qed.
-
-lemma ymin_pre_dx: ∀x,y. y ≤ yinj x → yinj x - (yinj x - y) = y.
-#x #y #H elim (yle_inv_inj2 … H) -H
-#z #Hzx #H destruct >yminus_inj
-/3 width=4 by minus_le_minus_minus_comm, eq_f/
-qed-.
-
-lemma ymin_pre_e: ∀z,x. z ≤ yinj x → ∀y. z ≤ y →
- z ≤ yinj x - (yinj x - y).
-#z #x #Hzx #y #Hzy elim (yle_split x y)
-[ #H >(ymin_pre_sn … H) -y //
-| #H >(ymin_pre_dx … H) -x //
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_lt.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* subtraction *)
-definition yminus: ynat → ynat → ynat ≝ λx,y. match y with
-[ yinj n ⇒ ypred^n x
-| Y ⇒ yinj 0
-].
-
-interpretation "ynat minus" 'minus x y = (yminus x y).
-
-(* Basic properties *********************************************************)
-
-lemma yminus_inj: ∀n,m. yinj m - yinj n = yinj (m - n).
-#n elim n -n /2 width=3 by trans_eq/
-qed.
-
-lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
-#n elim n -n // normalize
-#n #IHn >IHn //
-qed.
-
-lemma yminus_O1: ∀x:ynat. 0 - x = 0.
-* // qed.
-
-lemma yminus_refl: ∀x:ynat. x - x = 0.
-* // qed.
-
-lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
-* #y [ * #z [ * // ] ] >yminus_O1 //
-qed.
-
-(* Properties on predecessor ************************************************)
-
-lemma yminus_SO2: ∀m. m - 1 = ⫰m.
-* //
-qed.
-
-lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
-* // #n *
-[ #m #Hm #Hn >yminus_inj >yminus_inj
- /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-(* Properties on successor **************************************************)
-
-lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
-* // #n * [2: >yminus_Y_inj // ]
-#m >yminus_inj //
-qed.
-
-lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
-#n *
-[ #m #Hmn >yminus_inj >yminus_inj
- /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
-* //
-qed.
-
-(* Properties on order ******************************************************)
-
-lemma yle_minus_sn: ∀n,m. m - n ≤ m.
-* // #n * /2 width=1 by yle_inj/
-qed.
-
-lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
-#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
-qed-.
-
-lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
-* // #n *
-[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
- /2 width=1 by yle_inj/
-| >yminus_Y_inj #H destruct
-]
-qed.
-
-lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
-#x #y #Hxy * //
-#z elim z -z /3 width=1 by yle_pred/
-qed.
-
-(* Properties on strict order ***********************************************)
-
-lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
-#x #y * -x -y
-/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/ynat/ynat_minus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* addition *)
-definition yplus: ynat → ynat → ynat ≝ λx,y. match y with
-[ yinj n ⇒ ysucc^n x
-| Y ⇒ Y
-].
-
-interpretation "ynat plus" 'plus x y = (yplus x y).
-
-(* Properties on successor **************************************************)
-
-lemma yplus_succ2: ∀m,n. m + ⫯n = ⫯(m + n).
-#m * //
-qed.
-
-lemma yplus_succ1: ∀m,n. ⫯m + n = ⫯(m + n).
-#m * normalize //
-qed.
-
-lemma yplus_succ_swap: ∀m,n. m + ⫯n = ⫯m + n.
-// qed.
-
-lemma yplus_SO2: ∀m. m + 1 = ⫯m.
-* //
-qed.
-
-(* Basic properties *********************************************************)
-
-lemma yplus_inj: ∀n,m. yinj m + yinj n = yinj (m + n).
-#n elim n -n [ normalize // ]
-#n #IHn #m >(yplus_succ2 ? n) >IHn -IHn
-<plus_n_Sm //
-qed.
-
-lemma yplus_Y1: ∀m. ∞ + m = ∞.
-* normalize //
-qed.
-
-lemma yplus_comm: commutative … yplus.
-* [ #m ] * [1,3: #n ] //
-normalize >ysucc_iter_Y //
-qed.
-
-lemma yplus_assoc: associative … yplus.
-#x #y * // #z cases y -y
-[ #y >yplus_inj whd in ⊢ (??%%); <iter_plus //
-| >yplus_Y1 //
-]
-qed.
-
-lemma yplus_O1: ∀n:ynat. 0 + n = n.
-#n >yplus_comm // qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma yplus_inv_inj: ∀z,y,x. x + y = yinj z →
- ∃∃m,n. m + n = z & x = yinj m & y = yinj n.
-#z * [2: normalize #x #H destruct ]
-#y * [2: >yplus_Y1 #H destruct ]
-/3 width=5 by yinj_inj, ex3_2_intro/
-qed-.
-
-(* Properties on order ******************************************************)
-
-lemma yle_plus_dx2: ∀n,m. n ≤ m + n.
-* //
-#n elim n -n //
-#n #IHn #m >(yplus_succ2 ? n) @(yle_succ n) // (**) (* full auto fails *)
-qed.
-
-lemma yle_plus_dx1: ∀n,m. m ≤ m + n.
-// qed.
-
-lemma yle_plus_dx1_trans: ∀x,z. z ≤ x → ∀y. z ≤ x + y.
-/2 width=3 by yle_trans/ qed.
-
-lemma yle_plus_dx2_trans: ∀y,z. z ≤ y → ∀x. z ≤ x + y.
-/2 width=3 by yle_trans/ qed.
-
-lemma monotonic_yle_plus_dx: ∀x,y. x ≤ y → ∀z. x + z ≤ y + z.
-#x #y #Hxy * //
-#z elim z -z /3 width=1 by yle_succ/
-qed.
-
-lemma monotonic_yle_plus_sn: ∀x,y. x ≤ y → ∀z. z + x ≤ z + y.
-/2 width=1 by monotonic_yle_plus_dx/ qed.
-
-lemma monotonic_yle_plus: ∀x1,y1. x1 ≤ y1 → ∀x2,y2. x2 ≤ y2 →
- x1 + x2 ≤ y1 + y2.
-/3 width=3 by monotonic_yle_plus_dx, yle_trans/ qed.
-
-(* Forward lemmas on order **************************************************)
-
-lemma yle_fwd_plus_sn2: ∀x,y,z. x + y ≤ z → y ≤ z.
-/2 width=3 by yle_trans/ qed-.
-
-lemma yle_fwd_plus_sn1: ∀x,y,z. x + y ≤ z → x ≤ z.
-/2 width=3 by yle_trans/ qed-.
-
-lemma yle_inv_monotonic_plus_dx: ∀x,y:ynat.∀z:nat. x + z ≤ y + z → x ≤ y.
-#x #y #z elim z -z /3 width=1 by yle_inv_succ/
-qed-.
-
-lemma yle_inv_monotonic_plus_sn: ∀x,y:ynat.∀z:nat. z + x ≤ z + y → x ≤ y.
-/2 width=2 by yle_inv_monotonic_plus_dx/ qed-.
-
-lemma yle_fwd_plus_ge: ∀m1,m2:nat. m2 ≤ m1 → ∀n1,n2:ynat. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
-#m1 #m2 #Hm12 #n1 #n2 #H
-lapply (monotonic_yle_plus … Hm12 … H) -Hm12 -H
-/2 width=2 by yle_inv_monotonic_plus_sn/
-qed-.
-
-lemma yle_fwd_plus_ge_inj: ∀m1:nat. ∀m2,n1,n2:ynat. m2 ≤ m1 → m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
-#m2 #m1 #n1 #n2 #H elim (yle_inv_inj2 … H) -H
-#x #H0 #H destruct /3 width=4 by yle_fwd_plus_ge, yle_inj/
-qed-.
-
-(* Forward lemmas on strict order *******************************************)
-
-lemma ylt_inv_monotonic_plus_dx: ∀x,y,z. x + z < y + z → x < y.
-* [2: #y #z >yplus_comm #H elim (ylt_inv_Y1 … H) ]
-#x * // #y * [2: #H elim (ylt_inv_Y1 … H) ]
-/4 width=3 by ylt_inv_inj, ylt_inj, lt_plus_to_lt_l/
-qed-.
-
-(* Properties on strict order ***********************************************)
-
-lemma ylt_plus_dx1_trans: ∀x,z. z < x → ∀y. z < x + yinj y.
-/2 width=3 by ylt_yle_trans/ qed.
-
-lemma ylt_plus_dx2_trans: ∀y,z. z < y → ∀x. z < yinj x + y.
-/2 width=3 by ylt_yle_trans/ qed.
-
-lemma monotonic_ylt_plus_dx: ∀x,y. x < y → ∀z:nat. x + yinj z < y + yinj z.
-#x #y #Hxy #z elim z -z /3 width=1 by ylt_succ/
-qed.
-
-lemma monotonic_ylt_plus_sn: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z + y.
-/2 width=1 by monotonic_ylt_plus_dx/ qed.
-
-(* Properties on minus ******************************************************)
-
-lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
-#m #n elim n -n //
-#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
-qed.
-
-lemma yplus_minus: ∀m,n. m + n - n ≤ m.
-#m * //
-qed.
-
-(* Forward lemmas on minus **************************************************)
-
-lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
-/2 width=1 by monotonic_yle_minus_dx/ qed-.
-
-lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
-/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
-
-lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
-/2 width=1 by monotonic_yle_minus_dx/ qed-.
-
-lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
-/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
-
-lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
-#x *
-[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
- /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
-#y * // #x * //
-#z #Hxy >yplus_inj >yminus_inj <plus_minus
-/2 width=1 by yle_inv_inj/
-qed-.
-
-(* Inversion lemmas on minus ************************************************)
-
-lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
-/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
-
-lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
-/2 width=1 by yle_inv_plus_inj2/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/notation/functions/predecessor_1.ma".
-include "ground_2A/lib/arith.ma".
-include "ground_2A/ynat/ynat.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* the predecessor function *)
-definition ypred: ynat → ynat ≝ λm. match m with
-[ yinj m ⇒ pred m
-| Y ⇒ Y
-].
-
-interpretation "ynat predecessor" 'Predecessor m = (ypred m).
-
-(* Properties ***************************************************************)
-
-lemma ypred_inj_rew: ∀m:nat. ⫰m = pred m.
-// qed.
-
-(* Inversion lemmas *********************************************************)
-
-lemma ypred_inv_refl: ∀m. ⫰m = m → m = 0 ∨ m = ∞.
-* // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
-/4 width=1 by pred_inv_refl, or_introl, eq_f/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2A/notation/functions/successor_1.ma".
-include "ground_2A/ynat/ynat_pred.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* the successor function *)
-definition ysucc: ynat → ynat ≝ λm. match m with
-[ yinj m ⇒ S m
-| Y ⇒ Y
-].
-
-interpretation "ynat successor" 'Successor m = (ysucc m).
-
-(* Properties ***************************************************************)
-
-lemma ypred_succ: ∀m. ⫰⫯m = m.
-* // qed.
-
-lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m. n = ⫯m.
-*
-[ * /2 width=1 by or_introl/
- #n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
-| @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
-]
-qed-.
-
-lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
-#m elim m -m //
-#m #IHm whd in ⊢ (??%?); >IHm //
-qed.
-
-(* Inversion lemmas *********************************************************)
-
-lemma ysucc_inj: ∀m,n. ⫯m = ⫯n → m = n.
-#m #n #H <(ypred_succ m) <(ypred_succ n) //
-qed-.
-
-lemma ysucc_inv_refl: ∀m. ⫯m = m → m = ∞.
-* // normalize
-#m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
-#H elim (lt_refl_false m) //
-qed-.
-
-lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
- ∃∃m1. n1 = yinj m1 & m2 = S m1.
-#m2 * normalize
-[ #n1 #H destruct /2 width=3 by ex2_intro/
-| #H destruct
-]
-qed-.
-
-lemma ysucc_inv_inj_dx: ∀m2,n1. ⫯n1 = yinj m2 →
- ∃∃m1. n1 = yinj m1 & m2 = S m1.
-/2 width=1 by ysucc_inv_inj_sn/ qed-.
-
-lemma ysucc_inv_Y_sn: ∀m. ∞ = ⫯m → m = ∞.
-* // normalize
-#m #H destruct
-qed-.
-
-lemma ysucc_inv_Y_dx: ∀m. ⫯m = ∞ → m = ∞.
-/2 width=1 by ysucc_inv_Y_sn/ qed-.
-
-lemma ysucc_inv_O_sn: ∀m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
-#m #H elim (ysucc_inv_inj_sn … H) -H
-#n #_ #H destruct
-qed-.
-
-lemma ysucc_inv_O_dx: ∀m. ⫯m = 0 → ⊥.
-/2 width=2 by ysucc_inv_O_sn/ qed-.
∃∃m2. L1 ≋ ⓧ*[n1,m2] L2 & 0 = n1 & ↑m2 = n2.
#L1 #L2 #n1 #n2 #H #HL12
lapply (lveq_fwd_length_plus … H) normalize >plus_n_Sm #H0
-lapply (plus2_inv_le_sn … H0 HL12) -H0 -HL12 #H0
+lapply (plus2_le_sn_sn … H0 HL12) -H0 -HL12 #H0
elim (le_inv_S1 … H0) -H0 #m2 #_ #H0 destruct
elim (lveq_inv_void_succ_dx … H) -H /2 width=3 by ex3_intro/
qed-.
projects / helm.git / commitdiff