From: Ferruccio Guidi Date: Thu, 27 Feb 2020 15:18:06 +0000 (+0100) Subject: milestone update in ground_2 and basic_2A X-Git-Tag: make_still_working~187 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=1fd63df4c77f5c24024769432ea8492748b4ac79;p=helm.git milestone update in ground_2 and basic_2A + basic_2A ported to ground_2 + cpy, cpys and lsuby parked in basic_2A + ground_2 updated for basic_2A becomes an independent package + ground_2A removed + parked files for basic_2A moved in basic_2A + web site update --- diff --git a/.gitignore b/.gitignore index c0ab80566..c9ebefc8e 100644 --- a/.gitignore +++ b/.gitignore @@ -57,6 +57,8 @@ helm/www/lambdadelta/xslt/basic_2_src.xsl 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 diff --git a/helm/www/lambdadelta/Makefile b/helm/www/lambdadelta/Makefile index 1acebccc5..c654d6abf 100644 --- a/helm/www/lambdadelta/Makefile +++ b/helm/www/lambdadelta/Makefile @@ -1,14 +1,14 @@ 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 @@ -185,6 +185,15 @@ $(LDWIXSTAMP): $(INDEX) ldw-ix: $(LDWIXSTAMP) +# INSTALL Î»Î´-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)/%) @@ -221,10 +230,6 @@ install-bib: $(BIB:%=$(BIBDIR)/%) $(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) diff --git a/helm/www/lambdadelta/download/lambdadelta_2A.tar.bz2 b/helm/www/lambdadelta/download/lambdadelta_2A.tar.bz2 index d82582bbc..456cecfc3 100644 Binary files a/helm/www/lambdadelta/download/lambdadelta_2A.tar.bz2 and b/helm/www/lambdadelta/download/lambdadelta_2A.tar.bz2 differ diff --git a/helm/www/lambdadelta/download/lambdadelta_2B.tar.bz2 b/helm/www/lambdadelta/download/lambdadelta_2B.tar.bz2 index 4a4e63330..237ba25be 100644 Binary files a/helm/www/lambdadelta/download/lambdadelta_2B.tar.bz2 and b/helm/www/lambdadelta/download/lambdadelta_2B.tar.bz2 differ diff --git a/helm/www/lambdadelta/download/lambdadelta_ground.tar.bz2 b/helm/www/lambdadelta/download/lambdadelta_ground.tar.bz2 new file mode 100644 index 000000000..8cc8f8814 Binary files /dev/null and b/helm/www/lambdadelta/download/lambdadelta_ground.tar.bz2 differ diff --git a/helm/www/lambdadelta/web/home/specification.ldw.xml b/helm/www/lambdadelta/web/home/specification.ldw.xml index 3b7e71aa5..682bbda3b 100644 --- a/helm/www/lambdadelta/web/home/specification.ldw.xml +++ b/helm/www/lambdadelta/web/home/specification.ldw.xml @@ -67,19 +67,6 @@ is available in the following formats. - - Informational pages on the parts of the specification: - Background, - Syntax, - Core, - Applications. - - - - - The scripts are grouped in directories, first by part, then by component. - - the scripts are checked by the latest version of Matita from @@ -89,25 +76,66 @@ lambdadelta_2B for Matita 0.99.4 - (revised ). - Source scripts [Git revision: 2019-11-19 20:45:15]. + (revised ). + Source scripts. Documentation (J2a). + + + repackaged without Î»Î´-ground-2. + + + released [Git revision: 2019-11-19 20:45:15]. + + + Informational pages on the parts of the specification: + Background, + Syntax, + Core, + Applications. + lambdadelta_2A for Matita 0.99.2 - (revised ). - Source scripts [Git revision: 2014-10-28 17:46:26]. + (revised ). + Source scripts. Documentation (R2c). - - repackaging (was lambdadelta_2A1). + + repackaged on the basis of Î»Î´-ground-2. + + + repackaged (was lambdadelta_2A1). + + + released [Git revision: 2014-10-28 17:46:26]. + + + + Informational pages on the parts of the specification: + Core. + + + + + + lambdadelta_ground for Matita 0.99.4 + (revised ). + Source scripts. + + + released [Git revision: 2014-10-28 17:46:26]. + + Informational pages on the parts of the specification: + Core. + + Î»Î´ version 1 (superseded) @@ -116,17 +144,6 @@ is available in the following formats. - - Informational pages on the parts of the specification: - Background, - Core. - - - - - The scripts are grouped in directories, one for each part. - - lambdadelta_1A for Coq 7.3.1 @@ -134,13 +151,18 @@ Source scripts. Documentation (J1a). - + repackaging (was lambdadelta_1). - + 17 new lemmas and former lemma "eq_nat_dec" renamed as "nat_dec_neg". + + Informational pages on the parts of the specification: + Background, + Core. + diff --git a/matita/matita/contribs/lambdadelta/Makefile b/matita/matita/contribs/lambdadelta/Makefile index d08583562..41c7acdd1 100644 --- a/matita/matita/contribs/lambdadelta/Makefile +++ b/matita/matita/contribs/lambdadelta/Makefile @@ -20,15 +20,13 @@ PRB_OPTS := $(XOA_OPTS) -g -i 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") @@ -55,14 +53,6 @@ endef $(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 @@ -254,11 +244,41 @@ TRIMS := $(MAS) $(TBLS) $(LDWS) 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 ###################################################################### diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cir_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cir_append.etc deleted file mode 100644 index efd097f84..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cir_append.etc +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,T,W. â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, â.âW @@ Lâ¦ â¢ â¡ ðâ¦Tâ¦. -/3 width=2 by crr_inv_labst_last/ qed. - -lemma cir_tif: âG,T,W. â¦G, ââ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, â.âWâ¦ â¢ â¡ ðâ¦Tâ¦. -/3 width=2 by crr_inv_trr/ qed. - -(* Advanced inversion lemmas ************************************************) - -lemma cir_inv_append_sn: âG,L,K,T. â¦G, K @@ Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦. -/3 width=1/ qed-. - -lemma cir_inv_tir: âG,T,W. â¦G, â.âWâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, ââ¦ â¢ â¡ ðâ¦Tâ¦. -/3 width=1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cix_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cix_append.etc deleted file mode 100644 index 7b5522ebd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cix_append.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L,K,T. â¦G, K @@ Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦. -/3 width=1 by crx_append_sn/ qed-. - -lemma cix_inv_tix: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â â¦G, ââ¦ â¢ â¡[h, g] ðâ¦Tâ¦. -/3 width=1 by trx_crx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cl_shift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cl_shift.etc deleted file mode 100644 index 5fdab7148..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cl_shift.etc +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â match L with -[ LAtom â T -| LPair L I V â shift L (-â{I} V. T) -]. - -interpretation "shift (closure)" 'Append L T = (shift L T). - -(* Basic properties *********************************************************) - -lemma shift_append_assoc: âL,K. âT:term. (L @@ K) @@ T = L @@ K @@ T. -#L #K elim K -K // normalize // -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma shift_inj: âL1,L2. âT1,T2:term. L1 @@ T1 = L2 @@ T2 â |L1| = |L2| â - L1 = L2 â§ T1 = T2. -#L1 elim L1 -L1 -[ * normalize /2 width=1/ - #L2 #I2 #V2 #T1 #T2 #_ shift_append_assoc normalize #H - elim (cpr_inv_bind1 â¦ H) -H * - [ #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *) - | #T #_ #_ #H destruct - ] -] -qed-. - diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpx.etc deleted file mode 100644 index 5f8bf520c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpx.etc +++ /dev/null @@ -1,25 +0,0 @@ -lemma cpx_append: âh,g,G. l_appendable_sn â¦ (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 â¦ HK0) #H -@(cpx_delta â¦ I â¦ (L@@K0) V1 â¦ HVW2) // -@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *) -qed. - -lemma cpx_fwd_shift1: âh,g,G,L1,L,T1,T. â¦G, Lâ¦ â¢ L1 @@ T1 â¡[h, g] T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#h #g #G #L1 @(lenv_ind_dx â¦ L1) -L1 normalize -[ #L #T1 #T #HT1 - @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) -| #I #L1 #V1 #IH #L #T1 #X - >shift_append_assoc normalize #H - elim (cpx_inv_bind1 â¦ H) -H * - [ #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *) - | #T #_ #_ #H destruct - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpy2.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpy2.etc deleted file mode 100644 index e941c9653..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpy2.etc +++ /dev/null @@ -1,27 +0,0 @@ -include "basic_2/grammar/cl_shift.ma". -include "basic_2/relocation/ldrop_append.ma". - -lemma cpy_append: âG,d,e. l_appendable_sn â¦ (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 â¦ HK0) #H -@(cpy_subst I â¦ (L@@K0) â¦ HVW) // (**) (* /4/ does not work *) -@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2 by lt_to_le/ -qed-. - -lemma cpy_fwd_shift1: âG,L1,L,T1,T,d,e. â¦G, Lâ¦ â¢ L1 @@ T1 â¶[d, e] T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#G #L1 @(lenv_ind_dx â¦ L1) -L1 normalize -[ #L #T1 #T #d #e #HT1 - @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) -| #I #L1 #V1 #IH #L #T1 #X #d #e - >shift_append_assoc normalize #H - elim (cpy_inv_bind1 â¦ H) -H - #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *) - /2 width=3 by trans_eq/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpys0.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpys0.etc deleted file mode 100644 index fa9bcf6de..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpys0.etc +++ /dev/null @@ -1,24 +0,0 @@ -include "basic_2/grammar/cl_shift.ma". -include "basic_2/relocation/ldrop_append.ma". - -lemma cpys_append: âG. l_appendable_sn â¦ (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 â¦ HK0) #H -@(cpys_delta â¦ I â¦ (L@@K0) V1 â¦ HVW2) // -@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *) -qed. - -lemma cpys_fwd_shift1: âG,L1,L,T1,T. â¦G, Lâ¦ â¢ L1 @@ T1 â¶*Ã T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#G #L1 @(lenv_ind_dx â¦ L1) -L1 normalize -[ #L #T1 #T #HT1 @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) -| #I #L1 #V1 #IH #L #T1 #X >shift_append_assoc normalize - #H elim (cpys_inv_bind1 â¦ H) -H - #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpys2.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpys2.etc deleted file mode 100644 index 22f74d0a8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/cpys2.etc +++ /dev/null @@ -1,14 +0,0 @@ -lemma cpys_append: âG,d,e. l_appendable_sn â¦ (cpys d e G). -#G #d #e #K #T1 #T2 #H @(cpys_ind â¦ H) -T2 -/3 width=3 by cpys_strap1, cpy_append/ -qed-. - -lemma cpys_fwd_shift1: âG,L,L1,T1,T,d,e. â¦G, Lâ¦ â¢ L1 @@ T1 â¶*[d, e] T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#G #L #L1 #T1 #T #d #e #H @(cpys_ind â¦ H) -T -[ /2 width=4 by ex2_2_intro/ -| #T #X #_ #HX * #L0 #T0 #HL10 #H destruct - elim (cpy_fwd_shift1 â¦ HX) -HX #L2 #T2 #HL02 #H destruct - /2 width=4 by ex2_2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/crr_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/crr_append.etc deleted file mode 100644 index dbf794d99..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/append/crr_append.etc +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,K,T. â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, K @@ Lâ¦ â¢ â¡ ðâ¦Tâ¦. -#G #L #K0 #T #H elim H -L -T /2 width=1/ -#L #K #V #i #HLK -lapply (ldrop_fwd_length_lt2 â¦ HLK) #Hi -lapply (ldrop_O1_append_sn_le â¦ HLK â¦ K0) -HLK /2 width=2/ -Hi /2 width=3/ -qed. - -lemma trr_crr: âG,L,T. â¦G, ââ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦. -#G #L #T #H lapply (crr_append_sn â¦ H) // -qed. - -(* Advanced inversion lemmas ************************************************) - -fact crr_inv_labst_last_aux: âG,L1,T,W. â¦G, L1â¦ â¢ â¡ ðâ¦Tâ¦ â - âL2. L1 = â.âW @@ L2 â â¦G, L2â¦ â¢ â¡ ðâ¦Tâ¦. -#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 â¦ HLK1) - >append_length >commutative_plus normalize in â¢ (??% â ?); #H - elim (le_to_or_lt_eq i (|L2|)) /2 width=1/ -H #Hi destruct - [ elim (ldrop_O1_lt â¦ Hi) #I2 #K2 #V2 #HLK2 - lapply (ldrop_O1_inv_append1_le â¦ HLK1 â¦ HLK2) -HLK1 /2 width=2/ -Hi - normalize #H destruct /2 width=3/ - | lapply (ldrop_O1_inv_append1_ge â¦ HLK1 ?) -HLK1 // shift_append_assoc #H (ldrop_fwd_length â¦ HL) -HL // -qed. - -lemma cny_lref_top: âG,L,d,e,i. d+e â¤ yinj i â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦#iâ¦. -#G #L #d #e #i #Hdei #X #H elim (cpy_inv_lref1 â¦ H) -H // * -#I #K #V #_ #H elim (ylt_yle_false â¦ H) // -qed. - -lemma cny_lref_skip: âG,L,d,e,i. yinj i < d â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦#iâ¦. -#G #L #d #e #i #Hid #X #H elim (cpy_inv_lref1 â¦ H) -H // * -#I #K #V #H elim (ylt_yle_false â¦ H) // -qed. - -lemma cny_gref: âG,L,d,e,p. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Â§pâ¦. -#G #L #d #e #p #X #H elim (cpy_inv_gref1 â¦ H) -H // -qed. - -lemma cny_bind: âG,L,V,d,e. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Vâ¦ â - âI,T. â¦G, L.â{I}Vâ¦ â¢ â¶[â«¯d, e] ðâ¦Tâ¦ â - âa. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦â{a,I}V.Tâ¦. -#G #L #V1 #d #e #HV1 #I #T1 #HT1 #a #X #H -elim (cpy_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct ->(HV1 â¦ HV12) -V2 >(HT1 â¦ HT12) -T2 // -qed. - -lemma cny_flat: âG,L,V,d,e. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Vâ¦ â - âT. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Tâ¦ â - âI. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦â{I}V.Tâ¦. -#G #L #V1 #d #e #HV1 #T1 #HT1 #I #X #H -elim (cpy_inv_flat1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct ->(HV1 â¦ HV12) -V2 >(HT1 â¦ HT12) -T2 // -qed. - -lemma cny_narrow: âG,L,T,d1,e1. â¦G, Lâ¦ â¢ â¶[d1, e1] ðâ¦Tâ¦ â - âd2,e2. d1 â¤ d2 â d2 + e2 â¤ d1 + e1 â â¦G, Lâ¦ â¢ â¶[d2, e2] ðâ¦Tâ¦. -#G #L #T1 #d1 #e1 #HT1 #d2 #e2 #Hd12 #Hde21 #T2 #HT12 -@HT1 /2 width=5 by cpy_weak/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cny_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cny_lift.etc deleted file mode 100644 index 213fa6abb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cny_lift.etc +++ /dev/null @@ -1,118 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â dt + et â¤ d â â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdetd #U2 #HU12 -elim (cpy_inv_lift1_le â¦ HU12 â¦ HLK â¦ HTU1) // -L -Hdetd #T2 #HT12 ->(HT1 â¦ HT12) -K /2 width=5 by lift_mono/ -qed-. - -lemma cny_lift_be: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â dt â¤ d â yinj d â¤ dt + et â â¦G, Lâ¦ â¢ â¶[dt, et+e] ðâ¦Uâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdtd #Hddet #U2 #HU12 -elim (cpy_inv_lift1_be â¦ HU12 â¦ HLK â¦ HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hdtd -Hddet #T2 ->yplus_minus_inj #HT12 >(HT1 â¦ HT12) -K /2 width=5 by lift_mono/ -qed-. - -lemma cny_lift_ge: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â d â¤ dt â â¦G, Lâ¦ â¢ â¶[dt+e, et] ðâ¦Uâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #U2 #HU12 -elim (cpy_inv_lift1_ge â¦ HU12 â¦ HLK â¦ HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hddt #T2 ->yplus_minus_inj #HT12 >(HT1 â¦ HT12) -K /2 width=5 by lift_mono/ -qed-. - -(* Inversion lemmas on relocation *******************************************) - -lemma cny_inv_lift_le: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â dt + et â¤ d â â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦. -#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 â¦ HT12 â¦ HLK â¦ HTU1 â¦ HTU2 ?) // -K -Hdetd #HU12 -lapply (HU1 â¦ HU12) -L /2 width=5 by lift_inj/ -qed-. - -lemma cny_inv_lift_be: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â dt â¤ d â yinj d + e â¤ dt + et â â¦G, Kâ¦ â¢ â¶[dt, et-e] ðâ¦Tâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdtd #Hdedet #T2 #HT12 -lapply (yle_fwd_plus_ge_inj â¦ Hdedet) // #Heet -elim (yle_inv_plus_inj2 â¦ Hdedet) -Hdedet #Hddete #Hedet -elim (lift_total T2 d e) #U2 #HTU2 -lapply (cpy_lift_be â¦ HT12 â¦ HLK â¦ HTU1 â¦ HTU2 ? ?) // [ >yplus_minus_assoc_inj // ] -K -Hdtd -Hddete ->ymax_pre_sn // -Heet #HU12 -lapply (HU1 â¦ HU12) -L /2 width=5 by lift_inj/ -qed-. - -lemma cny_inv_lift_ge: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â yinj d + e â¤ dt â â¦G, Kâ¦ â¢ â¶[dt-e, et] ðâ¦Tâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdedt #T2 #HT12 -elim (yle_inv_plus_inj2 â¦ Hdedt) -Hdedt #Hddte #Hedt -elim (lift_total T2 d e) #U2 #HTU2 -lapply (cpy_lift_ge â¦ HT12 â¦ HLK â¦ HTU1 â¦ HTU2 ?) // -K -Hddte ->ymax_pre_sn // -Hedt #HU12 -lapply (HU1 â¦ HU12) -L /2 width=5 by lift_inj/ -qed-. - -(* Advanced inversion lemmas on relocation **********************************) - -lemma cny_inv_lift_ge_up: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â - â§[d, e] T â¡ U â d â¤ dt â dt â¤ yinj d + e â yinj d + e â¤ dt + et â - â¦G, Kâ¦ â¢ â¶[d, dt + et - (yinj d + e)] ðâ¦Tâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hddt #Hdtde #Hdedet -lapply (cny_narrow â¦ HU1 (d+e) (dt+et-(d+e)) ? ?) -HU1 [ >ymax_pre_sn_comm ] // #HU1 -lapply (cny_inv_lift_ge â¦ HU1 â¦ HLK â¦ HTU1 ?) // -L -U1 ->yplus_minus_inj // -qed-. - -lemma cny_inv_lift_subst: âG,L,K,V,W,i,d,e. d â¤ yinj i â i < d + e â - â©[i+1] L â¡ K â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Wâ¦ â - â§[O, i+1] V â¡ W â â¦G, Kâ¦ â¢ â¶[O, â«°(d+e-i)] ðâ¦Vâ¦. -#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW -lapply (cny_inv_lift_ge_up â¦ HW â¦ HLK â¦ HVW ? ? ?) // ->yplus_O1 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: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[yinj d, dt + et - (yinj d + yinj e)] ðâ¦Tâ¦ â - â©[s, d, e] L â¡ K â â§[d, e] T â¡ U â - yinj d â¤ dt â dt â¤ yinj d + yinj e â yinj d + yinj e â¤ dt + et â - â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦. -#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #Hdtde #Hdedet -lapply (cny_lift_be â¦ HT1 â¦ HLK â¦ HTU1 ? ?) // -K -T1 -#HU1 @(cny_narrow â¦ HU1) -HU1 // (**) (* auto fails *) -qed-. - -lemma cny_lift_subst: âG,L,K,V,W,i,d,e. d â¤ yinj i â i < d + e â - â©[i+1] L â¡ K â â¦G, Kâ¦ â¢ â¶[O, â«°(d+e-i)] ðâ¦Vâ¦ â - â§[O, i+1] V â¡ W â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Wâ¦. -#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW -@(cny_lift_ge_up â¦ HLK â¦ HVW) // >yplus_O1 yplus_SO2 -[ >yminus_succ2 // -| /2 width=3 by yle_trans/ -| /2 width=1 by ylt_fwd_le_succ1/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpx_cpzs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpx_cpzs.etc deleted file mode 100644 index 6c208d51b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpx_cpzs.etc +++ /dev/null @@ -1,73 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âk1,k2. âk1 = âk2 â k1 = k2. -#k1 #k2 #H destruct // -qed-. - -axiom cpzs_inv_subst: âI,G,L,K,V1,V2,W2,i. - â©[i] L â¡ K.â{I}V1 â â§[O, i+1] V2 â¡ W2 â - â¦G, Lâ¦ â¢ #i â*[O, â] W2 â â¦G, Kâ¦â¢ V1 â*[O, â] V2. - -axiom cpzs_subst: âI,G,L,K,V1,V2,W2,i. - â©[i] L â¡ K.â{I}V1 â â§[O, i+1] V2 â¡ W2 â - â¦G, Kâ¦â¢ V1 â*[O, â] V2 â â¦G, Lâ¦ â¢ #i â*[O, â] W2. - -(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************) - -(* Forward lemmas on delta-equivalence for terms ****************************) - -lemma cpx_fwd_cpys_cpzs: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â - âd,e. â¦G, Lâ¦ â¢ T1 â*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¶*[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 â¦ H) - | lapply (cpys_inv_sort1 â¦ H) -H #H >(destruct_tsort_tsort â¦ H) - ] -H #H elim (lt_refl_false â¦ H) -| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #d #e @conj #H -(* - [ @(cpys_subst â¦ HLK â¦ HVW2) // >yminus_Y_inj /3 width=7 by cpzs_inv_subst/ - | elim (cpys_inv_lref1_ldrop â¦ H â¦ HLK â¦ 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 (â«¯d) e) -IHT12 - #IHTdx #IHTsn #IHVdx #IHVsn @conj #H - [ elim (cpzs_inv_bind â¦ H) -H /3 width=1 by cpys_bind/ - | elim (cpys_inv_bind1 â¦ 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 â¦ H) -H /3 width=1 by cpys_flat/ - | elim (cpys_inv_flat1 â¦ 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 â¦ H) - | elim (cpys_inv_flat1 â¦ 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 â¦ H) - | elim (cpys_inv_flat1 â¦ H) -H #X1 #X2 #H1 #H2 #H destruct - ] -] - - \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye.etc deleted file mode 100644 index 71e06ea72..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye.etc +++ /dev/null @@ -1,89 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â ynat â relation4 genv lenv term term â - Î»d,e,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â§ â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦T2â¦. - -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: âG,L2,T1,T2,d,e. â¦G, L2â¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â - âL1. L1 âÃ[d, e] L2 â L2 âÃ[d, e] L1 â â¦G, L1â¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦. -#G #L2 #T1 #T2 #d #e * -/4 width=8 by lsuby_cpys_trans, lsuby_cny_conf, conj/ -qed-. - -lemma cpye_sort: âG,L,d,e,k. â¦G, Lâ¦ â¢ âk â¶*[d, e] ðâ¦âkâ¦. -/3 width=5 by cny_sort, conj/ qed. - -lemma cpye_free: âG,L,d,e,i. |L| â¤ i â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦#iâ¦. -/3 width=6 by cny_lref_free, conj/ qed. - -lemma cpye_top: âG,L,d,e,i. d + e â¤ yinj i â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦#iâ¦. -/3 width=6 by cny_lref_top, conj/ qed. - -lemma cpye_skip: âG,L,d,e,i. yinj i < d â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦#iâ¦. -/3 width=6 by cny_lref_skip, conj/ qed. - -lemma cpye_gref: âG,L,d,e,p. â¦G, Lâ¦ â¢ Â§p â¶*[d, e] ðâ¦Â§pâ¦. -/3 width=5 by cny_gref, conj/ qed. - -lemma cpye_bind: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â - âI,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯d, e] ðâ¦T2â¦ â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[d, e] ðâ¦â{a,I}V2.T2â¦. -#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: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â - âT1,T2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â - âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[d, e] ðâ¦â{I}V2.T2â¦. -#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: âG,L,X,d,e,k. â¦G, Lâ¦ â¢ âk â¶*[d, e] ðâ¦Xâ¦ â X = âk. -#G #L #X #d #e #k * /2 width=5 by cpys_inv_sort1/ -qed-. - -lemma cpye_inv_gref1: âG,L,X,d,e,p. â¦G, Lâ¦ â¢ Â§p â¶*[d, e] ðâ¦Xâ¦ â X = Â§p. -#G #L #X #d #e #p * /2 width=5 by cpys_inv_gref1/ -qed-. - -lemma cpye_inv_bind1: âa,I,G,L,V1,T1,X,d,e. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[d, e] ðâ¦Xâ¦ â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ & â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯d, e] ðâ¦T2â¦ & - X = â{a,I}V2.T2. -#a #I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_bind1 â¦ H1) -H1 -#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_bind â¦ H2) -H2 -/5 width=8 by lsuby_cny_conf, lsuby_succ, ex3_2_intro, conj/ -qed-. - -lemma cpye_inv_flat1: âI,G,L,V1,T1,X,d,e. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[d, e] ðâ¦Xâ¦ â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ & â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ & - X = â{I}V2.T2. -#I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_flat1 â¦ H1) -H1 -#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_flat â¦ H2) -H2 -/3 width=5 by ex3_2_intro, conj/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_alt.etc deleted file mode 100644 index a9dc53cf5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_alt.etc +++ /dev/null @@ -1,89 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â ynat â relation4 genv lenv term term â -| cpyea_sort : âG,L,d,e,k. cpyea d e G L (âk) (âk) -| cpyea_free : âG,L,d,e,i. |L| â¤ i â cpyea d e G L (#i) (#i) -| cpyea_top : âG,L,d,e,i. d + e â¤ yinj i â cpyea d e G L (#i) (#i) -| cpyea_skip : âG,L,d,e,i. yinj i < d â cpyea d e G L (#i) (#i) -| cpyea_subst: âI,G,L,K,V1,V2,W2,i,d,e. d â¤ yinj i â yinj i < d+e â - â©[i] L â¡ K.â{I}V1 â cpyea (yinj 0) (â«°(d+e-yinj i)) G K V1 V2 â - â§[0, i+1] V2 â¡ W2 â cpyea d e G L (#i) W2 -| cpyea_gref : âG,L,d,e,p. cpyea d e G L (Â§p) (Â§p) -| cpyea_bind : âa,I,G,L,V1,V2,T1,T2,d,e. - cpyea d e G L V1 V2 â cpyea (â«¯d) e G (L.â{I}V1) T1 T2 â - cpyea d e G L (â{a,I}V1.T1) (â{a,I}V2.T2) -| cpyea_flat : âI,G,L,V1,V2,T1,T2,d,e. - cpyea d e G L V1 V2 â cpyea d e G L T1 T2 â - cpyea d e G L (â{I}V1.T1) (â{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: âG,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â â¦G, Lâ¦ â¢ T1 â¶â¶*[d, e] ðâ¦T2â¦. -#G #L #T1 @(fqup_wf_ind_eq â¦ G L T1) -G -L -T1 -#Z #Y #X #IH #G #L * * -[ #k #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_sort1 â¦ H) -H // -| #i #HG #HL #HT #T2 #d #e #H destruct - elim (cpye_inv_lref1 â¦ 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 â¦ H) -H // -| #a #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct - elim (cpye_inv_bind1 â¦ H) -H /3 width=1 by cpyea_bind/ -| #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct - elim (cpye_inv_flat1 â¦ H) -H /3 width=1 by cpyea_flat/ -] -qed. - -(* Main inversion properties ************************************************) - -theorem cpyea_inv_cpye: âG,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶â¶*[d, e] ðâ¦T2â¦ â â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦. -#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: âR:ynatâynatârelation4 genv lenv term term. - (âG,L,d,e,k. R d e G L (âk) (âk)) â - (âG,L,d,e,i. |L| â¤ i â R d e G L (#i) (#i)) â - (âG,L,d,e,i. d + e â¤ yinj i â R d e G L (#i) (#i)) â - (âG,L,d,e,i. yinj i < d â R d e G L (#i) (#i)) â - (âI,G,L,K,V1,V2,W2,i,d,e. d â¤ yinj i â yinj i < d + e â - â©[i] L â¡ K.â{I}V1 â â¦G, Kâ¦ â¢ V1 â¶*[yinj O, â«°(d+e-yinj i)] ðâ¦V2â¦ â - â§[O, i+1] V2 â¡ W2 â R (yinj O) (â«°(d+e-yinj i)) G K V1 V2 â R d e G L (#i) W2 - ) â - (âG,L,d,e,p. R d e G L (Â§p) (Â§p)) â - (âa,I,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â - â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯d, e] ðâ¦T2â¦ â R d e G L V1 V2 â - R (â«¯d) e G (L.â{I}V1) T1 T2 â R d e G L (â{a,I}V1.T1) (â{a,I}V2.T2) - ) â - (âI,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â - â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â R d e G L V1 V2 â - R d e G L T1 T2 â R d e G L (â{I}V1.T1) (â{I}V2.T2) - ) â - âd,e,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â 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 â¦ H) -G -L -T1 -T2 -d -e -/3 width=8 by cpyea_inv_cpye/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_cpye.etc deleted file mode 100644 index 42bd19b8e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_cpye.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,T,T2,d,e. â¦G, Lâ¦ â¢ T â¶*[d, e] ðâ¦T2â¦ â - âT1. â¦G, Lâ¦ â¢ T â¶*[d, e] T1 â â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2. -#G #L #T #T2 #d #e * #H2 #HT2 #T1 #H1 elim (cpys_conf_eq â¦ H1 â¦ H2) -T -#T0 #HT10 #HT20 >(cpys_inv_cny1 â¦ HT2 â¦ HT20) -T2 // -qed-. - \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_lift.etc deleted file mode 100644 index 3c0abe617..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpye_lift.etc +++ /dev/null @@ -1,169 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âI,G,L,K,V1,V2,W2,i,d,e. d â¤ yinj i â i < d + e â - â©[i] L â¡ K.â{I}V1 â â¦G, Kâ¦ â¢ V1 â¶*[O, â«°(d+e-i)] ðâ¦V2â¦ â - â§[O, i+1] V2 â¡ W2 â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦W2â¦. -#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: âG,L,T1,d,e. âT2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦. -#G #L #T1 @(fqup_wf_ind_eq â¦ 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 â¦ 0 (â«°(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 â¦ d e) // elim (IH G (L.â{I}V1) T1 â¦ (â«¯d) e) // - /3 width=2 by cpye_bind, ex_intro/ -| #I #V1 #T1 #HG #HL #HT #d #e destruct - elim (IH G L V1 â¦ d e) // elim (IH G L T1 â¦ d e) // - /3 width=2 by cpye_flat, ex_intro/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma cpye_inv_lref1: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â - â¨â¨ |L| â¤ i â§ T2 = #i - | d + e â¤ yinj i â§ T2 = #i - | yinj i < d â§ T2 = #i - | ââI,K,V1,V2. d â¤ yinj i & yinj i < d + e & - â©[i] L â¡ K.â{I}V1 & - â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦ & - â§[O, i+1] V2 â¡ T2. -#G #L #T2 #i #d #e * #H1 #H2 elim (cpys_inv_lref1 â¦ H1) -H1 -[ #H destruct elim (cny_inv_lref â¦ 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 â¦ HLK â¦ HVT2) (**) (* explicit constructor *) - /4 width=13 by cny_inv_lift_subst, ldrop_fwd_drop2, conj/ -] -qed-. - -lemma cpye_inv_lref1_free: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â - (â¨â¨ |L| â¤ i | d + e â¤ yinj i | yinj i < d) â T2 = #i. -#G #L #T2 #d #e #i #H * elim (cpye_inv_lref1 â¦ H) -H * // -#I #K #V1 #V2 #Hdi #Hide #HLK #_ #_ #H -[ elim (lt_refl_false i) -d - @(lt_to_le_to_lt â¦ H) -H /2 width=5 by ldrop_fwd_length_lt2/ (**) (* full auto slow: 19s *) -] -elim (ylt_yle_false â¦ H) // -qed-. - -lemma cpye_inv_lref1_lget: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â - âI,K,V1. â©[i] L â¡ K.â{I}V1 â - â¨â¨ d + e â¤ yinj i â§ T2 = #i - | yinj i < d â§ T2 = #i - | ââV2. d â¤ yinj i & yinj i < d + e & - â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦ & - â§[O, i+1] V2 â¡ T2. -#G #L #T2 #d #e #i #H #I #K #V1 #HLK elim (cpye_inv_lref1 â¦ H) -H * -[ #H elim (lt_refl_false i) -T2 -d - @(lt_to_le_to_lt â¦ 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 â¦ HLY â¦ HLK) -HLY -HLK #H destruct - /3 width=3 by or3_intro2, ex4_intro/ -] -qed-. - -lemma cpye_inv_lref1_subst_ex: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â - âI,K,V1. d â¤ yinj i â yinj i < d + e â - â©[i] L â¡ K.â{I}V1 â - ââV2. â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦ & - â§[O, i+1] V2 â¡ T2. -#G #L #T2 #d #e #i #H #I #K #V1 #Hdi #Hide #HLK -elim (cpye_inv_lref1_lget â¦ H â¦ HLK) -H * /2 width=3 by ex2_intro/ -#H elim (ylt_yle_false â¦ H) // -qed-. - -lemma cpye_inv_lref1_subst: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â - âI,K,V1,V2. d â¤ yinj i â yinj i < d + e â - â©[i] L â¡ K.â{I}V1 â â§[O, i+1] V2 â¡ T2 â - â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦. -#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2 -elim (cpye_inv_lref1_subst_ex â¦ H â¦ HLK) -H -HLK // -#X2 #H0 #HXT2 lapply (lift_inj â¦ HXT2 â¦ HVT2) -HXT2 -HVT2 #H destruct // -qed-. - -(* Inversion lemmas on relocation *******************************************) - -lemma cpye_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â - âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt + et â¤ d â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[dt, et] ðâ¦T2â¦ & â§[d, e] T2 â¡ U2. -#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdetd -elim (cpys_inv_lift1_le â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 -lapply (cny_inv_lift_le â¦ HU2 â¦ HLK â¦ HTU2 ?) -L -/3 width=3 by ex2_intro, conj/ -qed-. - -lemma cpye_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â - âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt â¤ d â yinj d + e â¤ dt + et â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[dt, et - e] ðâ¦T2â¦ & â§[d, e] T2 â¡ U2. -#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet -elim (cpys_inv_lift1_be â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 -lapply (cny_inv_lift_be â¦ HU2 â¦ HLK â¦ HTU2 ? ?) -L -/3 width=3 by ex2_intro, conj/ -qed-. - -lemma cpye_inv_lift1_ge: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â - âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - yinj d + e â¤ dt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[dt - e, et] ðâ¦T2â¦ & â§[d, e] T2 â¡ U2. -#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdedt -elim (cpys_inv_lift1_ge â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 -lapply (cny_inv_lift_ge â¦ HU2 â¦ HLK â¦ HTU2 ?) -L -/3 width=3 by ex2_intro, conj/ -qed-. - -lemma cpye_inv_lift1_ge_up: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â - âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - d â¤ dt â dt â¤ yinj d + e â yinj d + e â¤ dt + et â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[d, dt + et - (yinj d + e)] ðâ¦T2â¦ & - â§[d, e] T2 â¡ 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 â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 -lapply (cny_inv_lift_ge_up â¦ HU2 â¦ HLK â¦ HTU2 ? ? ?) -L -/3 width=3 by ex2_intro, conj/ -qed-. - -lemma cpye_inv_lift1_subst: âG,L,W1,W2,d,e. â¦G, Lâ¦ â¢ W1 â¶*[d, e] ðâ¦W2â¦ â - âK,V1,i. â©[i+1] L â¡ K â â§[O, i+1] V1 â¡ W1 â - d â¤ yinj i â i < d + e â - ââV2. â¦G, Kâ¦ â¢ V1 â¶*[O, â«°(d+e-i)] ðâ¦V2â¦ & â§[O, i+1] V2 â¡ W2. -#G #L #W1 #W2 #d #e * #HW12 #HW2 #K #V1 #i #HLK #HVW1 #Hdi #Hide -elim (cpys_inv_lift1_subst â¦ HW12 â¦ HLK â¦ HVW1) -W1 // #V2 #HV12 #HVW2 -lapply (cny_inv_lift_subst â¦ HLK HW2 HVW2) -L -/3 width=3 by ex2_intro, conj/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpys_cny.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpys_cny.etc deleted file mode 100644 index 1499ceea5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpys_cny.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,T1,d,e. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦T1â¦ â - âT2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â T1 = T2. -#G #L #T1 #d #e #HT1 #T2 #H @(cpys_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 destruct <(HT1 â¦ HT2) -T // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpzs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpzs.etc deleted file mode 100644 index 0fada97a0..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/cpzs.etc +++ /dev/null @@ -1,83 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â ynat â relation4 genv lenv term term â - Î»d,e,G,L,T1,T2. - ââT. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦Tâ¦ & â¦G, Lâ¦ â¢ T2 â¶*[d, e] ðâ¦Tâ¦. - -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: âG,L,T1,T,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦Tâ¦ â - âT2. â¦G, Lâ¦ â¢ T2 â¶*[d, e] ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ T1 â*[d, e] T2. -/2 width=3 by ex2_intro/ qed. - -lemma cpzs_refl: âG,L,d,e. reflexive â¦ (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: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â*[d, e] V2 â - âI,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â*[â«¯d, e] T2 â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â*[d, e] â{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: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â*[d, e] V2 â - âT1,T2. â¦G, Lâ¦ â¢ T1 â*[d, e] T2 â - âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â*[d, e] â{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: âG,L,d,e,k1,k2. â¦G, Lâ¦ â¢ âk1 â*[d, e] âk2 â k1 = k2. -#G #L #d #e #k1 #k2 * #X #H1 #H2 -lapply (cpye_inv_sort1 â¦ H1) -H1 #H1 -lapply (cpye_inv_sort1 â¦ H2) -H2 #H2 -destruct // -qed-. - -lemma cpzs_inv_bind: âa1,a2,I1,I2,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ â{a1,I1}V1.T1 â*[d, e] â{a2,I2}V2.T2 â - â§â§ a1 = a2 & I1 = I2 - & â¦G, Lâ¦ â¢ V1 â*[d, e] V2 & â¦G, L.â{I1}V1â¦ â¢ T1 â*[â«¯d, e] T2. -#a1 #a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2 -elim (cpye_inv_bind1 â¦ H1) -H1 #V2 #T2 #HV12 #HT12 #H1 -elim (cpye_inv_bind1 â¦ 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: âI1,I2,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ â{I1}V1.T1 â*[d, e] â{I2}V2.T2 â - â§â§ I1 = I2 - & â¦G, Lâ¦ â¢ V1 â*[d, e] V2 & â¦G, Lâ¦ â¢ T1 â*[d, e] T2. -#I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2 -elim (cpye_inv_flat1 â¦ H1) -H1 #V2 #T2 #HV12 #HT12 #H1 -elim (cpye_inv_flat1 â¦ H2) -H2 #W2 #U2 #HW12 #HU12 #H2 -destruct /3 width=3 by cpye_div, and3_intro/ -qed-. - -lemma cpzs_inv_flat_bind: âa2,I1,I2,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ â{I1}V1.T1 â*[d, e] â{a2,I2}V2.T2 â â¥. -#a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2 -elim (cpye_inv_flat1 â¦ H1) -H1 #V2 #T2 #HV12 #HT12 #H1 -elim (cpye_inv_bind1 â¦ H2) -H2 #W2 #U2 #HW12 #HU12 #H2 -destruct -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lleq_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lleq_cpye.etc deleted file mode 100644 index e6890e637..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lleq_cpye.etc +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T,d. L1 â[T, d] L2 â âG,T1. â¦G, L1â¦ â¢ T â¶*[d, â] ðâ¦T1â¦ â - âT2. â¦G, L2â¦ â¢ T â¶*[d, â] ðâ¦T2â¦ â T1 = T2. -#L1 #L2 #T #d #H @(lleq_ind_alt â¦ H) -L1 -L2 -T -d -[ #L1 #L2 #d #k #_ #G #T1 #H1 #T2 #H2 - >(cpye_inv_sort1 â¦ H1) -H1 >(cpye_inv_sort1 â¦ H2) -H2 // -| #L1 #L2 #d #i #_ #Hid #G #T1 #H1 #T2 #H2 - >(cpye_inv_lref1_free â¦ H1) -H1 [ >(cpye_inv_lref1_free â¦ 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 â¦ H1 â¦ HLK1) -H1 -HLK1 // - elim (cpye_inv_lref1_subst_ex â¦ H2 â¦ HLK2) -H2 -HLK2 // - >yminus_Y_inj #V2 #HV2 #HVT2 #V1 #HV1 #HVT1 - lapply (IHV â¦ HV1 â¦ 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 â¦ H1) -H1 [ >(cpye_inv_lref1_free â¦ H2) -H2 ] - /2 width=1 by or3_intro0/ -| #L1 #L2 #d #p #_ #G #T1 #H1 #T2 #H2 - >(cpye_inv_gref1 â¦ H1) -H1 >(cpye_inv_gref1 â¦ H2) -H2 // -| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2 - elim (cpye_inv_bind1 â¦ H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct - elim (cpye_inv_bind1 â¦ 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 â¦ H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct - elim (cpye_inv_flat1 â¦ H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct - /3 width=3 by eq_f2/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lpx_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lpx_cpye.etc deleted file mode 100644 index 48816c85d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lpx_cpye.etc +++ /dev/null @@ -1,44 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,d,e. - âL0,T0,T1. â¦G, L0â¦ â¢ T0 â¡[h, g] T1 â âL1. â¦G, L0â¦ â¢ â¡[h, g] L1 â - âT2. â¦G, L0â¦ â¢ T0 â¶*[d, e] T2 â - ââT. â¦G, L1â¦ â¢ T1 â¶*[d, e] T & â¦G, L0â¦ â¢ T2 â¡[h, g] T. - -(* SN EXTENDED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *********************) - -(* Forward lemmas on evaluation for extended substitution *******************) - -lemma cpx_cpys_cpye_fwd_lpx: âh,g,G,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â - âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â - âU1,d,e. â¦G, L1â¦ â¢ T1 â¶*[d, e] U1 â - âU2. â¦G, L2â¦ â¢ T2 â¶*[d, e] ðâ¦U2â¦ â - â¦G, L1â¦ â¢ U1 â¡[h, g] U2. -#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e #HTU1 -elim (cpx_cpys_conf_lpx â¦ HT12 â¦ HL12 â¦ HTU1) -T1 -/3 width=9 by cpx_cpys_trans_lpx, cpye_cpys_conf/ -qed-. - -lemma cpx_cpye_fwd_lpx: âh,g,G,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â - âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â - âU1,d,e. â¦G, L1â¦ â¢ T1 â¶*[d, e] ðâ¦U1â¦ â - âU2. â¦G, L2â¦ â¢ T2 â¶*[d, e] ðâ¦U2â¦ â - â¦G, L1â¦ â¢ U1 â¡[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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lpxs_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lpxs_cpye.etc deleted file mode 100644 index a6f0f86bd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/lpxs_cpye.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2. â¦G, L1â¦ â¢ â¡*[h, g] L2 â - âT1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â - âU1,d,e. â¦G, L1â¦ â¢ T1 â¶*[d, e] ðâ¦U1â¦ â - âU2. â¦G, L2â¦ â¢ T2 â¶*[d, e] ðâ¦U2â¦ â - â¦G, L1â¦ â¢ U1 â¡*[h, g] U2. -#h #g #G #L1 #L2 #H @(lpxs_ind_dx â¦ 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 â¦ HT12 â¦ HL1 â¦ HTU1 â¦ HTX2) -T1 - /4 width=9 by lpx_cpxs_trans, cpxs_strap2/ (**) (* full auto too long: 41s *) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/pdeltaconvstar_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/pdeltaconvstar_6.etc deleted file mode 100644 index b7de5f543..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/pdeltaconvstar_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 break â * [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PDeltaConvStar $G $L $T1 $d $e $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubsteval_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubsteval_6.etc deleted file mode 100644 index 360f43d29..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubsteval_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G, break term 46 L â¦ â¢ break term 46 T1 â¶ * break [ term 46 d , break term 46 e ] break ð â¦ term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'PSubstEval $G $L $T1 $T2 $d $e }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubstevalalt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubstevalalt_6.etc deleted file mode 100644 index 18cdf608f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubstevalalt_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G, break term 46 L â¦ â¢ break term 46 T1 â¶ â¶ * break [ term 46 d , break term 46 e ] break ð â¦ term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'PSubstEvalAlt $G $L $T1 $T2 $d $e }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubstnormal_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubstnormal_5.etc deleted file mode 100644 index 4a7a78afc..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cny/psubstnormal_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G, break term 46 L â¦ â¢ â¶ break [ term 46 d , break term 46 e ] ð break â¦ term 46 T â¦ )" - non associative with precedence 45 - for @{ 'PSubstNormal $G $L $T $d $e }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees.etc deleted file mode 100644 index e1b9b06b2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees.etc +++ /dev/null @@ -1,130 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â - Î»d,i,L,U1. âU2. â¦â, Lâ¦ â¢ U1 â¶*[d, â] U2 â âT2. â§[i, 1] T2 â¡ 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: âL,U,d,i. L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â âT. â§[i, 1] T â¡ U. -/2 width=1 by/ qed-. - -lemma cofrees_fwd_bind_sn: âa,I,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{a,I}W.Uâ¦ â - L â¢ i ~Ïµ ð*[d]â¦Wâ¦. -#a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (â{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1 -#X #H elim (lift_inv_bind2 â¦ H) -H /2 width=2 by ex_intro/ -qed-. - -lemma cofrees_fwd_bind_dx: âa,I,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{a,I}W.Uâ¦ â - L.â{I}W â¢ i+1 ~Ïµ ð*[â«¯d]â¦Uâ¦. -#a #I #L #W #U1 #i #d #H #U2 #HU12 elim (H (â{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1 -#X #H elim (lift_inv_bind2 â¦ H) -H /2 width=2 by ex_intro/ -qed-. - -lemma cofrees_fwd_flat_sn: âI,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{I}W.Uâ¦ â - L â¢ i ~Ïµ ð*[d]â¦Wâ¦. -#I #L #W1 #U #i #d #H #W2 #HW12 elim (H (â{I}W2.U)) /2 width=1 by cpys_flat/ -W1 -#X #H elim (lift_inv_flat2 â¦ H) -H /2 width=2 by ex_intro/ -qed-. - -lemma cofrees_fwd_flat_dx: âI,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{I}W.Uâ¦ â - L â¢ i ~Ïµ ð*[d]â¦Uâ¦. -#I #L #W #U1 #i #d #H #U2 #HU12 elim (H (â{I}W.U2)) /2 width=1 by cpys_flat/ -U1 -#X #H elim (lift_inv_flat2 â¦ H) -H /2 width=2 by ex_intro/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma cofrees_inv_gen: âL,U,U0,d,i. â¦â, Lâ¦ â¢ U â¶*[d, â] U0 â (âT. â§[i, 1] T â¡ U0 â â¥) â - L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥. -#L #U #U0 #d #i #HU0 #HnU0 #HU elim (HU â¦ HU0) -L -U -d /2 width=2 by/ -qed-. - -lemma cofrees_inv_lref_eq: âL,d,i. L â¢ i ~Ïµ ð*[d]â¦#iâ¦ â â¥. -#L #d #i #H elim (H (#i)) -H // -#X #H elim (lift_inv_lref2_be â¦ H) -H // -qed-. - -lemma cofrees_inv_bind: âa,I,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{a,I}W.Uâ¦ â - L â¢ i ~Ïµ ð*[d]â¦Wâ¦ â§ L.â{I}W â¢ i+1 ~Ïµ ð*[â«¯d]â¦Uâ¦. -/3 width=8 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-. - -lemma cofrees_inv_flat: âI,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{I}W.Uâ¦ â - L â¢ i ~Ïµ ð*[d]â¦Wâ¦ â§ L â¢ i ~Ïµ ð*[d]â¦Uâ¦. -/3 width=7 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-. - -(* Basic Properties *********************************************************) - -lemma cofrees_lsuby_conf: âL1,U,d,i. L1 â¢ i ~Ïµ ð*[d]â¦Uâ¦ â - âL2. L1 â[d, â] L2 â L2 â¢ i ~Ïµ ð*[d]â¦Uâ¦. -/3 width=3 by lsuby_cpys_trans/ qed-. - -lemma cofrees_sort: âL,d,i,k. L â¢ i ~Ïµ ð*[d]â¦âkâ¦. -#L #d #i #k #X #H >(cpys_inv_sort1 â¦ H) -X /2 width=2 by ex_intro/ -qed. - -lemma cofrees_gref: âL,d,i,p. L â¢ i ~Ïµ ð*[d]â¦Â§pâ¦. -#L #d #i #p #X #H >(cpys_inv_gref1 â¦ H) -X /2 width=2 by ex_intro/ -qed. - -lemma cofrees_bind: âL,V,d,i. L â¢ i ~Ïµ ð*[d] â¦Vâ¦ â - âI,T. L.â{I}V â¢ i+1 ~Ïµ ð*[â«¯d]â¦Tâ¦ â - âa. L â¢ i ~Ïµ ð*[d]â¦â{a,I}V.Tâ¦. -#L #W1 #d #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 â¦ H) -H -#W2 #U2 #HW12 #HU12 #H destruct -elim (HW1 â¦ HW12) elim (HU1 â¦ HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/ -qed. - -lemma cofrees_flat: âL,V,d,i. L â¢ i ~Ïµ ð*[d]â¦Vâ¦ â âT. L â¢ i ~Ïµ ð*[d]â¦Tâ¦ â - âI. L â¢ i ~Ïµ ð*[d]â¦â{I}V.Tâ¦. -#L #W1 #d #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 â¦ H) -H -#W2 #U2 #HW12 #HU12 #H destruct -elim (HW1 â¦ HW12) elim (HU1 â¦ HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/ -qed. - -lemma cofrees_cpy_trans: âL,U1,U2,d. â¦â, Lâ¦ â¢ U1 â¶[d, â] U2 â - âi. L â¢ i ~Ïµ ð*[d]â¦U1â¦ â L â¢ i ~Ïµ ð*[d]â¦U2â¦. -/3 width=3 by cpys_strap2/ qed-. - -axiom cofrees_dec: âL,T,d,i. Decidable (L â¢ i ~Ïµ ð*[d]â¦Tâ¦). - -(* Basic negated properties *************************************************) - -lemma frees_cpy_div: âL,U1,U2,d. â¦â, Lâ¦ â¢ U1 â¶[d, â] U2 â - âi. (L â¢ i ~Ïµ ð*[d]â¦U2â¦ â â¥) â (L â¢ i ~Ïµ ð*[d]â¦U1â¦ â â¥). -/3 width=7 by cofrees_cpy_trans/ qed-. - -(* Basic negated inversion lemmas *******************************************) - -lemma frees_inv_bind: âa,I,L,V,T,d,i. (L â¢ i ~Ïµ ð*[d]â¦â{a,I}V.Tâ¦ â â¥) â - (L â¢ i ~Ïµ ð*[d]â¦Vâ¦ â â¥) â¨ (L.â{I}V â¢ i+1 ~Ïµ ð*[â«¯d]â¦Tâ¦ â â¥). -#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: âI,L,V,T,d,i. (L â¢ i ~Ïµ ð*[d]â¦â{I}V.Tâ¦ â â¥) â - (L â¢ i ~Ïµ ð*[d]â¦Vâ¦ â â¥) â¨ (L â¢ i ~Ïµ ð*[d]â¦Tâ¦ â â¥). -#I #L #W #U #d #H elim (cofrees_dec L W d) -/4 width=8 by cofrees_flat, or_intror, or_introl/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_alt.etc deleted file mode 100644 index 7bb9c9ca7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_alt.etc +++ /dev/null @@ -1,54 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,U,d,i. (âT. â§[i, 1] T â¡ U â â¥) â (L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥). -#L #U #d #i #HnTU #H elim (cofrees_fwd_lift â¦ H) -H /2 width=2 by/ -qed-. - -lemma frees_inv_ge: âL,U,d,i. d â¤ yinj i â (L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥) â - (âT. â§[i, 1] T â¡ U â â¥) â¨ - ââI,K,W,j. d â¤ yinj j & j < i & â©[j]L â¡ K.â{I}W & - (K â¢ i-j-1 ~Ïµ ð*[yinj 0]â¦Wâ¦ â â¥) & (âT. â§[j, 1] T â¡ U â â¥). -#L #U #d #i #Hdi #H @(frees_ind â¦ H) -U /3 width=2 by or_introl/ -#U1 #U2 #HU12 #HU2 * -[ #HnU2 elim (cpy_fwd_nlift2_ge â¦ HU12 â¦ 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 â¦ HU12 â¦ 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 â¦ HLK1 â¦ HLK2 ?) -HLK2 /2 width=1 by lt_to_le/ - #HK12 lapply (ldrop_inv_drop1_lt â¦ HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12 - #HK12 - @or_intror @(ex5_4_intro â¦ HLK1 â¦ HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/ - @(frees_be â¦ HK12 â¦ HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1 - >minus_plus in â¢ (??(?(?%?)?)??â?); >minus_plus in â¢ (??(?(??%)?)??â?); >arith_b1 /2 width=1 by/ -] -qed-. - -lemma frees_ind_ge: âR:relation4 ynat nat lenv term. - (âd,i,L,U. d â¤ yinj i â (âT. â§[i, 1] T â¡ U â â¥) â R d i L U) â - (âd,i,j,I,L,K,W,U. d â¤ yinj j â j < i â â©[j]L â¡ K.â{I}W â (K â¢ i-j-1 ~Ïµ ð*[0]â¦Wâ¦ â â¥) â (âT. â§[j, 1] T â¡ U â â¥) â R 0 (i-j-1) K W â R d i L U) â - âd,i,L,U. d â¤ yinj i â (L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥) â R d i L U. -#R #IH1 #IH2 #d #i #L #U -generalize in match d; -d generalize in match i; -i -@(f2_ind â¦ rfw â¦ L U) -L -U -#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_lift.etc deleted file mode 100644 index f93f72216..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_lift.etc +++ /dev/null @@ -1,180 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,d,i,j. L â¢ i ~Ïµ ð*[d]â¦#jâ¦ â d â¤ yinj j â j < i â - âI,K,W. â©[j]L â¡ K.â{I}W â K â¢ i-j-1 ~Ïµ ð*[yinj 0]â¦Wâ¦. -#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 â¦ HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/ ->minus_plus minus_plus_plus_l // -| #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct - elim (cofrees_inv_flat â¦ H1) -H1 #HW #HU - elim (nlift_inv_flat â¦ H2) -H2 [ /3 width=9 by/ ] - #HnU @(IH â¦ HU â¦ HnU â¦ HLK) // (**) (* full auto fails *) -] -qed-. - -(* Advanced properties ******************************************************) - -lemma cofrees_lref_skip: âL,d,i,j. j < i â yinj j < d â L â¢ i ~Ïµ ð*[d]â¦#jâ¦. -#L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 â¦ H) -H -[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/ -| * #I #K #W1 #W2 #Hdj elim (ylt_yle_false â¦ Hdj) -i -I -L -K -W1 -W2 -X // -] -qed. - -lemma cofrees_lref_lt: âL,d,i,j. i < j â L â¢ i ~Ïµ ð*[d]â¦#jâ¦. -#L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 â¦ 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 â¦ H i j) /2 width=2 by lt_to_le, ex_intro/ -] -qed. - -lemma cofrees_lref_gt: âI,L,K,W,d,i,j. j < i â â©[j] L â¡ K.â{I}W â - K â¢ (i-j-1) ~Ïµ ð*[O]â¦Wâ¦ â L â¢ i ~Ïµ ð*[d]â¦#jâ¦. -#I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 â¦ H) -H -[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/ -| * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono â¦ HLK0 â¦ HLK) -L - #H destruct elim (HW1 â¦ HW12) -I -K -W1 -d - #V2 #HVW2 elim (lift_trans_le â¦ HVW2 â¦ HW2) -W2 // - >minus_plus minus_plus yplus_inj >yminus_Y_inj #T2 #HT12 - lapply (cpys_weak â¦ HT12 (d-yinj e0) (â) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12 - | elim (cpys_inv_lift1_ge â¦ HU12 â¦ HLK â¦ HTU1) // #T2 - ] -| elim (cpys_inv_lift1_be â¦ HU12 â¦ HLK â¦ HTU1) // >yminus_Y_inj #T2 #HT12 - lapply (cpys_weak â¦ HT12 (d-yinj e0) (â) ? ?) // -HT12 -] --s -L #HT12 #HTU2 -elim (HT1 â¦ HT12) -T1 #V2 #HVT2 -elim (lift_trans_le â¦ HVT2 â¦ HTU2 ?) // (tpss_inv_sort1 â¦ H) -H // -| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 â¦ H) -H - [ #H destruct - elim (lt_or_ge i d) #Hdi - [ elim (ltpss_dx_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss11 â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss21 â¦ 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 â¦ HL12 â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_dx_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct - lapply (ldrop_mono â¦ HL2K0 â¦ HLK2) -HL2K0 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) -HLK2 #HLK2 - lapply (tpss_trans_eq â¦ HV12 HVW2) -V2 /3 width=7/ - ] -| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_bind1 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ -] -qed. - -lemma aaa_ltpss_dx_tps_conf: âL1,T1,A. L1 â¢ T1 â A â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶ [d, e] T2 â L2 â¢ T2 â A. -/3 width=7/ qed. - -lemma aaa_ltpss_dx_conf: âL1,T,A. L1 â¢ T â A â - âL2,d,e. L1 â¶* [d, e] L2 â L2 â¢ T â A. -/2 width=7/ qed. - -lemma aaa_tpss_conf: âL,T1,A. L â¢ T1 â A â - âT2,d,e. L â¢ T1 â¶* [d, e] T2 â L â¢ T2 â A. -/2 width=7/ qed. - -lemma aaa_tps_conf: âL,T1,A. L â¢ T1 â A â - âT2,d,e. L â¢ T1 â¶ [d, e] T2 â L â¢ T2 â A. -/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/aaa_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/aaa_ltpss_sn.etc deleted file mode 100644 index 4f2a44827..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/aaa_ltpss_sn.etc +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T,A. L1 â¢ T â A â - âL2,d,e. L1 â¢ â¶* [d, e] L2 â L2 â¢ T â A. -#L1 #T #A #HT #L2 #d #e #HL12 -lapply (ltpss_sn_ltpssa â¦ HL12) -HL12 #HL12 -@(TC_Conf3 â¦ (Î»L,A. L â¢ T â A) â¦ HT ? HL12) /2 width=5/ -qed. - -lemma aaa_ltpss_sn_tpss_conf: âL1,T1,A. L1 â¢ T1 â A â - âL2,d,e. L1 â¢ â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â L2 â¢ T2 â A. -/3 width=5/ qed. - -lemma aaa_ltpss_sn_tps_conf: âL1,T1,A. L1 â¢ T1 â A â - âL2,d,e. L1 â¢ â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶ [d, e] T2 â L2 â¢ T2 â A. -/3 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cnf_cif.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cnf_cif.etc deleted file mode 100644 index 5a1ae5a5b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cnf_cif.etc +++ /dev/null @@ -1,106 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,T2,d,e. L â¢ T1 â¶[d, e] T2 â L â¢ ðâ¦T1â¦ â 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 â¦ HLK ?) // -| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H - elim (cif_inv_bind â¦ H) -H #HV1 #HT1 * #H destruct - lapply (IHV12 â¦ HV1) -IHV12 -HV1 #H destruct /3 width=1/ -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H - elim (cif_inv_flat â¦ H) -H #HV1 #HT1 #_ #_ /3 width=1/ -] -qed. - -lemma tpss_cif_eq: âL,T1,T2,d,e. L â¢ T1 â¶*[d, e] T2 â L â¢ ðâ¦T1â¦ â T1 = T2. -#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 #HT1 -lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/ -qed. - -lemma tpr_cif_eq: âT1,T2. T1 â¡ T2 â âL. L â¢ ðâ¦T1â¦ â T1 = T2. -#T1 #T2 #H elim H -T1 -T2 -[ // -| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H - [ elim (cif_inv_appl â¦ H) -H #HV1 #HT1 #_ - >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // - | elim (cif_inv_ri2 â¦ H) /2 width=1/ - ] -| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H - elim (cif_inv_appl â¦ H) -H #_ #_ #H - elim (simple_inv_bind â¦ H) -| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H - [ lapply (tps_lsubr_trans â¦ HT2 (L.âV2) ?) -HT2 /2 width=1/ #HT2 - elim (cif_inv_bind â¦ H) -H #HV1 #HT1 * #H destruct - lapply (IHV1 â¦ HV1) -IHV1 -HV1 #H destruct - lapply (IHT1 â¦ HT1) -IHT1 #H destruct - lapply (tps_cif_eq â¦ HT2 ?) -HT2 // - | <(tps_inv_refl_SO2 â¦ HT2 ?) -HT2 // - elim (cif_inv_ib2 â¦ H) -H /2 width=1/ /3 width=2/ - ] -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H - elim (cif_inv_appl â¦ H) -H #_ #_ #H - elim (simple_inv_bind â¦ H) -| #V1 #T1 #T #T2 #_ #_ #_ #L #H - elim (cif_inv_ri2 â¦ H) /2 width=1/ -| #V1 #T1 #T2 #_ #_ #L #H - elim (cif_inv_ri2 â¦ H) /2 width=1/ -] -qed. - -lemma cpr_cif_eq: âL,T1,T2. L â¢ T1 â¡ T2 â L â¢ ðâ¦T1â¦ â T1 = T2. -#L #T1 #T2 * #T0 #HT10 #HT02 #HT1 -lapply (tpr_cif_eq â¦ HT10 â¦ HT1) -HT10 #H destruct /2 width=5/ -qed. - -theorem cif_cnf: âL,T. L â¢ ðâ¦Tâ¦ â L â¢ ðâ¦Tâ¦. -/3 width=3/ qed. - -(* Note: this property is unusual *) -lemma cnf_crf_false: âL,T. L â¢ ðâ¦Tâ¦ â L â¢ ðâ¦Tâ¦ â â¥. -#L #T #H elim H -L -T -[ #L #K #V #i #HLK #H - elim (cnf_inv_delta â¦ HLK H) -| #L #V #T #_ #IHV #H - elim (cnf_inv_appl â¦ H) -H /2 width=1/ -| #L #V #T #_ #IHT #H - elim (cnf_inv_appl â¦ H) -H /2 width=1/ -| #I #L #V #T * #H1 #H2 destruct - [ elim (cnf_inv_zeta â¦ H2) - | elim (cnf_inv_tau â¦ H2) - ] -|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct - [1,3: elim (cnf_inv_abbr â¦ H2) -H2 /2 width=1/ - |*: elim (cnf_inv_abst â¦ H2) -H2 /2 width=1/ - ] -| #a #L #V #W #T #H - elim (cnf_inv_appl â¦ H) -H #_ #_ #H - elim (simple_inv_bind â¦ H) -| #a #L #V #W #T #H - elim (cnf_inv_appl â¦ H) -H #_ #_ #H - elim (simple_inv_bind â¦ H) -] -qed. - -theorem cnf_cif: âL,T. L â¢ ðâ¦Tâ¦ â L â¢ ðâ¦Tâ¦. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpcs_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpcs_ltpss_dx.etc deleted file mode 100644 index 80f418b08..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpcs_ltpss_dx.etc +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,d,e. L1 â¶* [d, e] L2 â - âT1,T2. L1 â¢ T1 â¬* T2 â L2 â¢ T1 â¬* T2. -#L1 #L2 #d #e #HL12 #T1 #T2 #H -elim (cpcs_inv_cprs â¦ H) -H #T #HT1 #HT2 -elim (cprs_ltpss_dx_conf â¦ HT1 â¦ HL12) -HT1 #U1 #HTU1 #H1 -elim (cprs_ltpss_dx_conf â¦ HT2 â¦ HL12) -L1 #U2 #HTU2 #H2 -elim (tpss_conf_eq â¦ H1 â¦ H2) -T #U #HU1 #HU2 -lapply (cprs_tpss_trans â¦ HTU1 â¦ HU1) -U1 -lapply (cprs_tpss_trans â¦ HTU2 â¦ HU2) -U2 /2 width=3/ -qed-. - -lemma cpcs_ltpss_dx_tpss_conf: âL1,L2,d,e. L1 â¶* [d, e] L2 â - âT,T2. L1 â¢ T â¬* T2 â - âT1. L2 â¢ T â¶* [d, e] T1 â - L2 â¢ T1 â¬* T2. -#L1 #L2 #d #e #HL12 #T #T2 #HT2 #T1 #HT1 -lapply (cpcs_ltpss_dx_conf â¦ HL12 â¦ HT2) -L1 #HT2 -lapply (cpcs_tpss_conf â¦ HT1 â¦ HT2) -T // -qed-. - -lemma cpcs_ltpss_dx_tpss2_conf: âL1,L2,d,e. L1 â¶* [d, e] L2 â - âT1,T2. L1 â¢ T1 â¬* T2 â - âT3. L2 â¢ T1 â¶* [d, e] T3 â - âT4. L2 â¢ T2 â¶* [d, e] T4 â - L2 â¢ T3 â¬* T4. -#L1 #L2 #d #e #HL12 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 -lapply (cpcs_ltpss_dx_tpss_conf â¦ HL12 â¦ HT12 â¦ HT13) -L1 -T1 #HT32 -lapply (cpcs_tpss_strap1 â¦ HT32 â¦ HT24) -T2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpcs_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpcs_ltpss_sn.etc deleted file mode 100644 index 5bf1ed57f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpcs_ltpss_sn.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â - âT1,T2. L1 â¢ T1 â¬* T2 â L2 â¢ T1 â¬* T2. -#L1 #L2 #d #e #H -lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 // -#L #L2 #_ #HL2 #IHL1 #T1 #U1 #HTU1 -lapply (IHL1 â¦ HTU1) -IHL1 -HTU1 #HTU1 -lapply (cpcs_ltpss_dx_conf â¦ HL2 â¦ HTU1) -HTU1 -HL2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr.etc deleted file mode 100644 index d82299c38..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr.etc +++ /dev/null @@ -1,111 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â relation term â - Î»L,T1,T2. ââT. T1 â¡ T & L â¢ T â¶* [0, |L|] T2. - -interpretation - "context-sensitive parallel reduction (term)" - 'PRed L T1 T2 = (cpr L T1 T2). - -(* Basic properties *********************************************************) - -lemma cpr_intro: âL,T1,T,T2,d,e. T1 â¡ T â L â¢ T â¶* [d, e] T2 â L â¢ T1 â¡ T2. -/3 width=5/ qed-. - -(* Basic_1: was by definition: pr2_free *) -lemma cpr_tpr: âT1,T2. T1 â¡ T2 â âL. L â¢ T1 â¡ T2. -/2 width=3/ qed. - -lemma cpr_tpss: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¡ T2. -/3 width=5/ qed. - -lemma cpr_refl: âL,T. L â¢ T â¡ T. -/2 width=1/ qed. - -(* Note: new property *) -(* Basic_1: was only: pr2_thin_dx *) -lemma cpr_flat: âI,L,V1,V2,T1,T2. - L â¢ V1 â¡ V2 â L â¢ T1 â¡ T2 â L â¢ â{I} V1. T1 â¡ â{I} V2. T2. -#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/ -qed. - -lemma cpr_cast: âL,V,T1,T2. - L â¢ T1 â¡ T2 â L â¢ âV. T1 â¡ 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: âL1,T1,T2. L1 â¢ T1 â¡ T2 â - âL2. L2 â [0, |L1|] L1 â L2 â¢ T1 â¡ T2. -#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12 -lapply (tpss_lsubr_trans â¦ HT2 â¦ HL12) -HT2 -HL12 /3 width=4/ -qed. - -(* Basic inversion lemmas ***************************************************) - -(* Basic_1: was: pr2_gen_csort *) -lemma cpr_inv_atom: âT1,T2. â â¢ T1 â¡ T2 â T1 â¡ T2. -#T1 #T2 * #T #HT normalize #HT2 -<(tpss_inv_refl_O2 â¦ HT2) -HT2 // -qed-. - -(* Basic_1: was: pr2_gen_sort *) -lemma cpr_inv_sort1: âL,T2,k. L â¢ âk â¡ T2 â T2 = âk. -#L #T2 #k * #X #H ->(tpr_inv_atom1 â¦ H) -H #H ->(tpss_inv_sort1 â¦ H) -H // -qed-. - -(* Basic_1: was: pr2_gen_cast *) -lemma cpr_inv_cast1: âL,V1,T1,U2. L â¢ âV1. T1 â¡ U2 â ( - ââV2,T2. L â¢ V1 â¡ V2 & L â¢ T1 â¡ T2 & - U2 = âV2. T2 - ) â¨ L â¢ T1 â¡ U2. -#L #V1 #T1 #U2 * #X #H #HU2 -elim (tpr_inv_cast1 â¦ H) -H /3 width=3/ -* #V #T #HV1 #HT1 #H destruct -elim (tpss_inv_flat1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cpr_fwd_bind1_minus: âI,L,V1,T1,T. L â¢ -â{I}V1.T1 â¡ T â âb. - ââV2,T2. L â¢ â{b,I}V1.T1 â¡ â{b,I}V2.T2 & - T = -â{I}V2.T2. -#I #L #V1 #T1 #T * #X #H1 #H2 #b -elim (tpr_fwd_bind1_minus â¦ H1 b) -H1 #V0 #T0 #HT10 #H destruct -elim (tpss_inv_bind1 â¦ H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ -qed-. - -lemma cpr_fwd_shift1: âL,L1,T1,T. L â¢ L1 @@ T1 â¡ T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#L #L1 #T1 #T * #X #H1 #H2 -elim (tpr_fwd_shift1 â¦ H1) -H1 #L0 #T0 #HL10 #H destruct -elim (tpss_fwd_shift1 â¦ 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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_aaa.etc deleted file mode 100644 index 7c1273334..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_aaa.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,A. L â¢ T1 â A â âT2. L â¢ T1 â¡ T2 â L â¢ T2 â A. -#L #T1 #A #HT1 #T2 * /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_cpr.etc deleted file mode 100644 index d231c849f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_cpr.etc +++ /dev/null @@ -1,63 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âa,I,L,V1,V2,T1,T2. L â¢ V1 â¡ V2 â T1 â¡ T2 â - L â¢ â{a,I} V1. T1 â¡ â{a,I} V2. T2. -#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12 -@ex2_intro [2: @(tpr_delta â¦ HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *) -qed. - -(* Basic_1: was only: pr2_gen_cbind *) -lemma cpr_bind_dx: âa,I,L,V1,V2,T1,T2. V1 â¡ V2 â L. â{I} V2 â¢ T1 â¡ T2 â - L â¢ â{a,I} V1. T1 â¡ â{a,I} V2. T2. -#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2 -elim (tpss_split_up â¦ HT2 1 ? ?) -HT2 // #T0 (tpr_inv_atom1 â¦ H) -H #H -elim (tpss_inv_lref1 â¦ H) -H /2 width=1/ -* /3 width=6/ -qed-. - -(* Basic_1: was pr2_gen_abbr *) -lemma cpr_inv_abbr1: âa,L,V1,T1,U2. L â¢ â{a}V1. T1 â¡ U2 â - (ââV,V2,T2. V1 â¡ V & L â¢ V â¶* [O, |L|] V2 & - L. âV â¢ T1 â¡ T2 & - U2 = â{a}V2. T2 - ) â¨ - ââT2. L.âV1 â¢ T1 â¡ T2 & â§[0,1] U2 â¡ T2 & a = true. -#a #L #V1 #T1 #Y * #X #H1 #H2 -elim (tpr_inv_abbr1 â¦ H1) -H1 * -[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct - elim (tpss_inv_bind1 â¦ H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct - lapply (tps_lsubr_trans â¦ HT0 (L. âV) ?) -HT0 /2 width=1/ #HT0 - lapply (tps_weak_full â¦ HT0) -HT0 #HT0 - lapply (tpss_lsubr_trans â¦ HT02 (L. âV) ?) -HT02 /2 width=1/ #HT02 - lapply (tpss_weak_full â¦ HT02) -HT02 #HT02 - lapply (tpss_strap2 â¦ HT0 HT02) -T0 /4 width=7/ -| #T2 #HT12 #HXT2 #H destruct - elim (lift_total Y 0 1) #Z #HYZ - lapply (tpss_lift_ge â¦ H2 (L.âV1) â¦ HXT2 â¦ HYZ) -X // /2 width=1/ #H - lapply (cpr_intro â¦ HT12 â¦ H) -T2 /3 width=3/ -] -qed-. - -(* Basic_1: was: pr2_gen_abst *) -lemma cpr_inv_abst1: âa,L,V1,T1,U2. L â¢ â{a}V1. T1 â¡ U2 â âI,W. - ââV2,T2. L â¢ V1 â¡ V2 & L. â{I} W â¢ T1 â¡ T2 & U2 = â{a}V2. T2. -#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W -elim (tpr_inv_abst1 â¦ H1) -H1 #V #T #HV1 #HT1 #H destruct -elim (tpss_inv_bind1 â¦ H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct -lapply (tpss_lsubr_trans â¦ HT2 (L. â{I} W) ?) -HT2 /2 width=1/ /4 width=5/ -qed-. - -(* Basic_1: was pr2_gen_appl *) -lemma cpr_inv_appl1: âL,V1,U0,U2. L â¢ âV1. U0 â¡ U2 â - â¨â¨ ââV2,T2. L â¢ V1 â¡ V2 & L â¢ U0 â¡ T2 & - U2 = âV2. T2 - | ââa,V2,W,T1,T2. L â¢ V1 â¡ V2 & L. âV2 â¢ T1 â¡ T2 & - U0 = â{a}W. T1 & - U2 = â{a}V2. T2 - | ââa,V2,V,W1,W2,T1,T2. L â¢ V1 â¡ V2 & L â¢ W1 â¡ W2 & L. âW2 â¢ T1 â¡ T2 & - â§[0,1] V2 â¡ V & - U0 = â{a}W1. T1 & - U2 = â{a}W2. âV. T2. -#L #V1 #U0 #Y * #X #H1 #H2 -elim (tpr_inv_appl1 â¦ H1) -H1 * -[ #V #U #HV1 #HU0 #H destruct - elim (tpss_inv_flat1 â¦ 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 â¦ H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct - lapply (tpss_weak â¦ 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 â¦ H2) -H2 #W2 #X #HW02 #HX #HY destruct - elim (tpss_inv_flat1 â¦ HX) -HX #V2 #T2 #HV2 #HT2 #H destruct - elim (tpss_inv_lift1_ge â¦ HV2 â¦ HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V (ltpr_fwd_length â¦ HL12) in HT2; #HT2 -elim (ltpr_tpr_tpss_conf â¦ HL12 â¦ HT2) -L1 /3 width=3/ -qed. - -lemma cpr_ltpr_conf_tpss: âL1,L2. L1 â¡ L2 â âT1,T2. L1 â¢ T1 â¡ T2 â - âd,e,U1. L1 â¢ T1 â¶* [d, e] U1 â - ââU2. L2 â¢ U1 â¡ U2 & L2 â¢ T2 â¡ U2. -#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 -elim (cpr_ltpr_conf_eq â¦ HT12 â¦ HL12) -HT12 #T #HT1 #HT2 -elim (cpr_tpss_ltpr â¦ HL12 â¦ HT1 â¦ HTU1) -L1 -HT1 #U2 #HU12 #HTU2 -lapply (tpss_weak_full â¦ HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *) -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_ltpss_dx.etc deleted file mode 100644 index 5d90e6865..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_ltpss_dx.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T,U1. L1 â¢ T â¡ U1 â - âL2,d,e. L1 â¶* [d, e] L2 â - ââU2. L2 â¢ T â¡ U2 & L2 â¢ U1 â¶* [d, e] U2. -#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12 -lapply (ltpss_dx_fwd_length â¦ HL12) #H >H in HU1; -H #HU1 -elim (ltpss_dx_tpss_conf â¦ HU1 â¦ HL12) -L1 /3 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_ltpss_sn.etc deleted file mode 100644 index f946e4cfb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_ltpss_sn.etc +++ /dev/null @@ -1,35 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â - âT1,T2. L2 â¢ T1 â¡ T2 â L1 â¢ T1 â¡ T2. -#L1 #L2 #d #e #HL12 #T1 #T2 * -lapply (ltpss_sn_weak_full â¦ HL12) -<(ltpss_sn_fwd_length â¦ HL12) -HL12 /3 width=5/ -qed. - -lemma ltpss_sn_cpr_conf: âL1,T,U1. L1 â¢ T â¡ U1 â - âL2,d,e. L1 â¢ â¶* [d, e] L2 â - ââU2. L2 â¢ T â¡ U2 & L1 â¢ U1 â¶* [d, e] U2. -#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12 -lapply (ltpss_sn_fwd_length â¦ HL12) #H >H in HU1; -H #HU1 -elim (ltpss_sn_tpss_conf â¦ HU1 â¦ HL12) -HU1 -HL12 /3 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_tpss.etc deleted file mode 100644 index d8c7225d9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cpr_tpss.etc +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,T. L â¢ T1 â¡ T â - âT2,d,e. L â¢ T â¶* [d, e] T2 â L â¢ T1 â¡ T2. -#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2 -lapply (tpss_weak_full â¦ HT2) -HT2 #HT2 -lapply (tpss_trans_eq â¦ HT0 HT2) -T /2 width=3/ -qed. - -lemma cpr_tps_trans: âL,T1,T. L â¢ T1 â¡ T â - âT2,d,e. L â¢ T â¶ [d, e] T2 â L â¢ T1 â¡ T2. -/3 width=5/ qed. - -lemma cpr_tpss_conf: âL,T0,T1. L â¢ T0 â¡ T1 â - âT2,d,e. L â¢ T0 â¶* [d, e] T2 â - ââT. L â¢ T1 â¶* [d, e] T & L â¢ T2 â¡ T. -#L #T0 #T1 * #U0 #HTU0 #HU0T1 #T2 #d #e #HT02 -elim (tpr_tpss_conf â¦ HTU0 â¦ HT02) -T0 #T0 #HT20 #HUT0 -elim (tpss_conf_eq â¦ HU0T1 â¦ HUT0) -U0 /3 width=5/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpr.etc deleted file mode 100644 index 2682a7609..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpr.etc +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T1,U1,d,e. L1 â¢ T1 â¶* [d, e] U1 â - âL2. L1 â¡ L2 â âT2. L2 â¢ T1 â¡* T2 â - ââU2. L2 â¢ U1 â¡* U2 & L2 â¢ T2 â¶* [d, e] U2. -#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2 -[ #T2 #HT12 - elim (cpr_tpss_ltpr â¦ HL12 â¦ HT12 â¦ HTU1) -L1 -T1 /3 width=3/ -| #T #T2 #_ #HT2 * #U #HU1 #HTU - elim (cpr_tpss_ltpr â¦ HT2 â¦ HTU) -L1 -T // /3 width=3/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpss_dx.etc deleted file mode 100644 index 7f8618007..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpss_dx.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T,U1. L1 â¢ T â¡* U1 â - âL2,d,e. L1 â¶* [d, e] L2 â - ââU2. L2 â¢ T â¡* U2 & L2 â¢ U1 â¶* [d, e] U2. -#L1 #T #U1 #H @(cprs_ind â¦ H) -U1 /2 width=3/ -#T1 #U1 #_ #HTU1 #IHT1 #L2 #d #e #HL12 -elim (IHT1 â¦ HL12) -IHT1 #U #HTU #HT1U -elim (ltpss_dx_cpr_conf â¦ HTU1 â¦ HL12) -L1 #U0 #HT1U0 #HU10 -elim (cpr_tpss_conf â¦ HT1U0 â¦ HT1U) -T1 #U2 #HU02 #HU2 -lapply (tpss_trans_eq â¦ HU10 HU02) -U0 /3 width=3/ -qed-. - -lemma cprs_ltpss_dx_tpss_conf: âL1,T1,U1. L1 â¢ T1 â¡* U1 â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â - ââU2. L2 â¢ T2 â¡* U2 & L2 â¢ U1 â¶* [d, e] U2. -#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 -elim (cprs_ltpss_dx_conf â¦ HTU1 â¦ HL12) -L1 #U #HT1U #HU1 -elim (cprs_tpss_conf â¦ HT1U â¦ HT12) -T1 #T #HUT #HT2 -lapply (tpss_trans_eq â¦ HU1 HUT) -U /2 width=3/ -qed-. - -lemma cprs_ltpss_dx_tpss2_conf: âL1,T1,U1. L1 â¢ T1 â¡* U1 â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â - âU2. L2 â¢ U1 â¶* [d, e] U2 â - ââU. L2 â¢ T2 â¡* U & L2 â¢ U2 â¶* [d, e] U. -#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 #U2 #HU12 -elim (cprs_ltpss_dx_tpss_conf â¦ HTU1 â¦ HL12 â¦ HT12) -L1 -T1 #U #HT2U #HU1 -elim (tpss_conf_eq â¦ HU12 â¦ HU1) -U1 #U0 #HU20 #HU0 -lapply (cprs_tpss_trans â¦ HT2U â¦ HU0) -U /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpss_sn.etc deleted file mode 100644 index 32009d11d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_ltpss_sn.etc +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â - âT,U1. L1 â¢ T â¡* U1 â - ââU2. L2 â¢ T â¡* U2 & L1 â¢ U1 â¶* [d, e] U2. -#L1 #L2 #d #e #H -lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 /2 width=3/ -#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1 -lapply (ltpssa_ltpss_sn â¦ HL1) -HL1 #HL1 -lapply (ltpss_sn_dx_trans_eq â¦ HL1 â¦ HL2) -HL1 #HL12 -elim (IHL1 â¦ HTU1) -IHL1 -HTU1 #U #HTU #HU1 -elim (cprs_ltpss_dx_conf â¦ HTU â¦ HL2) -HTU -HL2 #U2 #HTU2 #HU2 -lapply (ltpss_sn_tpss_trans_eq â¦ HU2 â¦ HL12) -HU2 -HL12 #HU2 -lapply (tpss_trans_eq â¦ HU1 HU2) -U /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_tpss.etc deleted file mode 100644 index 0c575ae02..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cprs_tpss.etc +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,T. L â¢ T1 â¡* T â - âT2,d,e. L â¢ T â¶* [d, e] T2 â L â¢ T1 â¡* T2. -#L #T1 #T #H @(cprs_ind â¦ H) -T /2 width=3/ /3 width=5/ -qed. - -lemma cprs_tps_trans: âL,T1,T. L â¢ T1 â¡* T â - âT2,d,e. L â¢ T â¶ [d, e] T2 â L â¢ T1 â¡* T2. -/3 width=5 by inj, cprs_tpss_trans/ qed. (**) (* auto too slow without trace *) - -lemma cprs_tpss_conf: âL,T0,T1. L â¢ T0 â¡* T1 â - âT2,d,e. L â¢ T0 â¶* [d, e] T2 â - ââT. L â¢ T1 â¶* [d, e] T & L â¢ T2 â¡* T. -#L #T0 #T1 #H @(cprs_ind â¦ H) -T1 /2 width=3/ -#T #T1 #_ #HT1 #IHT0 #T2 #d #e #HT02 -elim (IHT0 â¦ HT02) -T0 #T0 #HT0 #HT20 -elim (cpr_tpss_conf â¦ HT1 â¦ HT0) -T /3 width=5/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/dxprs_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/dxprs_ltpss_dx.etc deleted file mode 100644 index 8e3260dd4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/dxprs_ltpss_dx.etc +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,T,U1. â¦h, L1â¦ â¢ T â¢*â¡*[g] U1 â - âL2,d,e. L1 â¶* [d, e] L2 â - ââU2. â¦h, L2â¦ â¢ T â¢*â¡*[g] U2 & L2 â¢ U1 â¶* [d, e] U2. -#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12 -elim (sstas_ltpss_dx_conf â¦ HTU â¦ HL12) -HTU #U0 #HTU0 #HU0 -elim (cprs_ltpss_dx_conf â¦ HU1 â¦ HL12) -L1 #U2 #HU2 #HU12 -elim (cprs_tpss_conf â¦ HU2 â¦ HU0) -U #U #HU2 #HU0 -lapply (tpss_trans_eq â¦ HU12 HU2) -U2 /3 width=3/ -qed-. - -lemma dxprs_tpss_conf: âh,g,L,T1,U1. â¦h, Lâ¦ â¢ T1 â¢*â¡*[g] U1 â - âT2,d,e. L â¢ T1 â¶* [d, e] T2 â - ââU2. â¦h, Lâ¦ â¢ T2 â¢*â¡*[g] U2 & L â¢ U1 â¶* [d, e] U2. -#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #d #e #HT12 -elim (sstas_tpss_conf â¦ HTW1 â¦ HT12) -T1 #W2 #HTW2 #HW12 -elim (cprs_tpss_conf â¦ HWU1 â¦ HW12) -W1 /3 width=3/ -qed-. - -lemma dxprs_ltpss_dx_tpss_conf: âh,g,L1,T1,U1. â¦h, L1â¦ â¢ T1 â¢*â¡*[g] U1 â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â - ââU2. â¦h, L2â¦ â¢ T2 â¢*â¡*[g] U2 & - L2 â¢ U1 â¶* [d, e] U2. -#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 -elim (dxprs_ltpss_dx_conf â¦ HTU1 â¦ HL12) -L1 #U #HT1U #HU1 -elim (dxprs_tpss_conf â¦ HT1U â¦ HT12) -T1 #U2 #HTU2 #HU2 -lapply (tpss_trans_eq â¦ HU1 HU2) -U /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/dxprs_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/dxprs_ltpss_sn.etc deleted file mode 100644 index 553e4658b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/dxprs_ltpss_sn.etc +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,L2,d,e. L1 â¢ â¶* [d, e] L2 â - âT,U1. â¦h, L1â¦ â¢ T â¢*â¡*[g] U1 â - ââU2. â¦h, L2â¦ â¢ T â¢*â¡*[g] U2 & L1 â¢ U1 â¶* [d, e] U2. -#h #g #L1 #L2 #d #e #H -lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 [ /2 width=3/ ] -#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1 -lapply (ltpssa_ltpss_sn â¦ HL1) -HL1 #HL1 -lapply (ltpss_sn_dx_trans_eq â¦ HL1 â¦ HL2) -HL1 #HL12 -elim (IHL1 â¦ HTU1) -IHL1 -HTU1 #U #HTU #HU1 -elim (dxprs_ltpss_dx_conf â¦ HTU â¦ HL2) -HTU -HL2 #U2 #HTU2 #HU2 -lapply (ltpss_sn_tpss_trans_eq â¦ HU2 â¦ HL12) -HU2 -HL12 #HU2 -lapply (tpss_trans_eq â¦ HU1 HU2) -U /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ldrop_lbotr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ldrop_lbotr.etc deleted file mode 100644 index df6b2c11e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ldrop_lbotr.etc +++ /dev/null @@ -1,94 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âI,L,K,V,i. â©[0, i] L â¡ K. â{I}V â âd,e. â[d, e] L â - d â¤ i â i < d + e â I = Abbr. -#I #L elim L -L -[ #K #V #i #H - lapply (ldrop_inv_atom1 â¦ H) -H #H destruct -| #L #J #W #IHL #K #V #i #H - elim (ldrop_inv_O1 â¦ H) -H * - [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct - lapply (le_n_O_to_eq â¦ Hdi) -Hdi #H destruct - lapply (HL â¦ (L.âW) ?) -HL /2 width=1/ #H - elim (lsubr_inv_abbr2 â¦ H ?) -H // -Hide #K #_ #H destruct // - | #Hi #HLK #d @(nat_ind_plus â¦ d) -d - [ #e #H #_ #Hide - elim (lbotr_inv_bind â¦ H ?) -H [2: /2 width=2/ ] #HL #H destruct - @(IHL â¦ HLK â¦ HL) -IHL -HLK -HL // /2 width=1/ - | #d #_ #e #H #Hdi #Hide - lapply (lbotr_inv_skip â¦ H ?) -H // #HL - @(IHL â¦ HLK â¦ HL) -IHL -HLK -HL /2 width=1/ - ] - ] -] -qed-. - -(* Properties about local env. full refinement for substitution *************) - -(* Note: ldrop_ldrop not needed *) -lemma lbotr_ldrop: âL,d,e. - (âI,K,V,i. d â¤ i â i < d + e â â©[0, i] L â¡ K. â{I}V â I = Abbr) â - â[d, e] L. -#L elim L -L // -#L #I #V #IHL #d @(nat_ind_plus â¦ d) -d -[ #e @(nat_ind_plus â¦ 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: âL1,L2,d,e. â©[d, e] L1 â¡ L2 â âdd,ee. â[dd, ee] L1 â - dd + ee â¤ d â â[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 â¦ Hiddee Hddee) -Hddee #Hid -elim (ldrop_trans_le â¦ HL12 â¦ HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H -elim (ldrop_inv_skip2 â¦ H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct -@(lbotr_inv_ldrop â¦ HLK1 â¦ HL1) -L1 -K1 -V1 // -qed. - -lemma lbotr_ldrop_trans_be_up: âL1,L2,d,e. â©[d, e] L1 â¡ L2 â - âdd,ee. â[dd, ee] L1 â - dd â¤ d + e â d + e â¤ dd + ee â - â[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)â¦ Hdde ?) -Hdde /2 width=1/ #Hddie ->commutative_plus in Hiddee; >minus_minus_comm commutative_plus // -Hddie /2 width=1/ -qed. - -lemma lbotr_ldrop_trans_ge: âL1,L2,d,e. â©[d, e] L1 â¡ L2 â âdd,ee. â[dd, ee] L1 â - d + e â¤ dd â â[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 â¦ Hddee) -Hddee #Hdde #Hedd ->plus_minus in Hiddee; // #Hiddee -lapply (transitive_le â¦ Hdde Hddi) -Hdde #Hid -lapply (ldrop_trans_ge â¦ HL12 â¦ HLK2 ?) -L2 // -Hid #HL1K2 -@(lbotr_inv_ldrop â¦ HL1K2 â¦ HL1) -L1 >commutative_plus /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ldrop_lpx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ldrop_lpx.etc deleted file mode 100644 index d23ed28e5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ldrop_lpx.etc +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âR. t_deliftable_sn R â dropable_sn (lpx R). -#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e -[ #d #e #X #H >(lpx_inv_atom1 â¦ H) -H /2 width=3/ -| #K1 #I #V1 #X #H - elim (lpx_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ -| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H - elim (lpx_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 â¦ HL12) -L1 /3 width=3/ -| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H - elim (lpx_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (HR â¦ HV12 â¦ HWV1) -V1 - elim (IHLK1 â¦ HL12) -L1 /3 width=5/ -] -qed. - -lemma lpx_liftable_dedropable: âR. reflexive ? R â - t_liftable R â 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 â¦ H) -H /2 width=3/ -| #K1 #I #V1 #X #H - elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/ -| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 - elim (IHLK1 â¦ HK12) -K1 /3 width=5/ -| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H - elim (lpx_inv_pair1 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (lift_total W2 d e) #V2 #HWV2 - lapply (H2R â¦ HW12 â¦ HWV1 â¦ HWV2) -W1 - elim (IHLK1 â¦ HK12) -K1 /3 width=5/ -] -qed. - -fact lpx_dropable_aux: âR,L2,K2,d,e. â©[d, e] L2 â¡ K2 â âL1. lpx R L1 L2 â - d = 0 â ââK1. â©[0, e] L1 â¡ K1 & lpx R K1 K2. -#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e -[ #d #e #X #H >(lpx_inv_atom2 â¦ H) -H /2 width=3/ -| #K2 #I #V2 #X #H - elim (lpx_inv_pair2 â¦ H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/ -| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_ - elim (lpx_inv_pair2 â¦ H) -H #L1 #V1 #HL12 #HV12 #H destruct - elim (IHLK2 â¦ HL12 ?) -L2 // /3 width=3/ -| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_ - >commutative_plus normalize #H destruct -] -qed-. - -lemma lpx_dropable: âR. dropable_dx (lpx R). -/2 width=5 by lpx_dropable_aux/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lenv_px.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lenv_px.etc deleted file mode 100644 index ea916e10a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lenv_px.etc +++ /dev/null @@ -1,166 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| lpx_stom: lpx R (â) (â) -| lpx_pair: âI,K1,K2,V1,V2. - lpx R K1 K2 â R V1 V2 â lpx R (K1. â{I} V1) (K2. â{I} V2) -. - -(* Basic inversion lemmas ***************************************************) - -fact lpx_inv_atom1_aux: âR,L1,L2. lpx R L1 L2 â L1 = â â L2 = â. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_inv_atom1: âR,L2. lpx R (â) L2 â L2 = â. -/2 width=4 by lpx_inv_atom1_aux/ qed-. - -fact lpx_inv_pair1_aux: âR,L1,L2. lpx R L1 L2 â âI,K1,V1. L1 = K1. â{I} V1 â - ââK2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. â{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: âR,I,K1,V1,L2. lpx R (K1. â{I} V1) L2 â - ââK2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. â{I} V2. -/2 width=3 by lpx_inv_pair1_aux/ qed-. - -fact lpx_inv_atom2_aux: âR,L1,L2. lpx R L1 L2 â L2 = â â L1 = â. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_inv_atom2: âR,L1. lpx R L1 (â) â L1 = â. -/2 width=4 by lpx_inv_atom2_aux/ qed-. - -fact lpx_inv_pair2_aux: âR,L1,L2. lpx R L1 L2 â âI,K2,V2. L2 = K2. â{I} V2 â - ââK1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. â{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: âR,I,L1,K2,V2. lpx R L1 (K2. â{I} V2) â - ââK1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. â{I} V1. -/2 width=3 by lpx_inv_pair2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lpx_fwd_length: âR,L1,L2. lpx R L1 L2 â |L1| = |L2|. -#R #L1 #L2 #H elim H -L1 -L2 normalize // -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lpx_inv_append1: âR,L1,K1,L. lpx R (K1 @@ L1) L â - ââK2,L2. lpx R K1 K2 & lpx R L1 L2 & L = K2 @@ L2. -#R #L1 elim L1 -L1 normalize -[ #K1 #K2 #HK12 - @(ex3_2_intro â¦ K2 (â)) // (**) (* explicit constructor, /2 width=5/ does not work *) -| #L1 #I #V1 #IH #K1 #X #H - elim (lpx_inv_pair1 â¦ H) -H #L #V2 #H1 #HV12 #H destruct - elim (IH â¦ H1) -IH -H1 #K2 #L2 #HK12 #HL12 #H destruct - @(ex3_2_intro â¦ HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *) -] -qed-. - -lemma lpx_inv_append2: âR,L2,K2,L. lpx R L (K2 @@ L2) â - ââK1,L1. lpx R K1 K2 & lpx R L1 L2 & L = K1 @@ L1. -#R #L2 elim L2 -L2 normalize -[ #K2 #K1 #HK12 - @(ex3_2_intro â¦ K1 (â)) // (**) (* explicit constructor, /2 width=5/ does not work *) -| #L2 #I #V2 #IH #K2 #X #H - elim (lpx_inv_pair2 â¦ H) -H #L #V1 #H1 #HV12 #H destruct - elim (IH â¦ H1) -IH -H1 #K1 #L1 #HK12 #HL12 #H destruct - @(ex3_2_intro â¦ HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *) -] -qed-. - -(* Basic properties *********************************************************) - -lemma lpx_refl: âR. reflexive ? R â reflexive â¦ (lpx R). -#R #HR #L elim L -L // /2 width=1/ -qed. - -lemma lpx_sym: âR. symmetric ? R â symmetric â¦ (lpx R). -#R #HR #L1 #L2 #H elim H -H // /3 width=1/ -qed. - -lemma lpx_trans: âR. Transitive ? R â Transitive â¦ (lpx R). -#R #HR #L1 #L #H elim H -L // -#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H -elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ -qed. - -lemma lpx_conf: âR. confluent ? R â confluent â¦ (lpx R). -#R #HR #L0 #L1 #H elim H -L1 -[ #X #H >(lpx_inv_atom1 â¦ H) -X /2 width=3/ -| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H - elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK02 #HV02 #H destruct - elim (IHK01 â¦ HK02) -K0 #K #HK1 #HK2 - elim (HR â¦ HV01 â¦ HV02) -HR -V0 /3 width=5/ -] -qed. - -lemma lpx_TC_inj: âR,L1,L2. lpx R L1 L2 â lpx (TC â¦ R) L1 L2. -#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/ -qed. - -lemma lpx_TC_step: âR,L1,L. lpx (TC â¦ R) L1 L â - âL2. lpx R L L2 â lpx (TC â¦ 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 â¦ H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ -qed. - -lemma TC_lpx_pair_dx: âR. reflexive ? R â - âI,K,V1,V2. TC â¦ R V1 V2 â - TC â¦ (lpx R) (K.â{I}V1) (K.â{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: âR. reflexive ? R â - âI,V,K1,K2. TC â¦ (lpx R) K1 K2 â - TC â¦ (lpx R) (K1.â{I}V) (K2.â{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: âR,L1,L2. TC â¦ (lpx R) L1 L2 â lpx (TC â¦ R) L1 L2. -#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/ -qed. - -lemma lpx_inv_TC: âR. reflexive ? R â - âL1,L2. lpx (TC â¦ R) L1 L2 â TC â¦ (lpx R) L1 L2. -#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/ -qed. - -lemma lpx_append: âR,K1,K2. lpx R K1 K2 â âL1,L2. lpx R L1 L2 â - lpx R (L1 @@ K1) (L2 @@ K2). -#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lenv_top.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lenv_top.etc deleted file mode 100644 index ab90cebe7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lenv_top.etc +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 ð 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 â Î»T1,T2. True. - -definition ltop: relation lenv â lpx ttop. - -interpretation - "top reduction (environment)" - 'RTop L1 L2 = (ltop L1 L2). - -(* Basic properties *********************************************************) - -lemma ltop_refl: reflexive â¦ ltop. -/2 width=1/ qed. - -lemma ltop_sym: symmetric â¦ ltop. -/2 width=1/ qed. - -lemma ltop_trans: transitive â¦ ltop. -/2 width=3/ qed. - -lemma ltop_append: âK1,K2. K1 ð K2 â âL1,L2. L1 ð L2 â L1 @@ K1 ð L2 @@ K2. -/2 width=1/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma ltop_inv_atom1: âL2. â ð L2 â L2 = â. -/2 width=2 by lpx_inv_atom1/ qed-. - -lemma ltop_inv_pair1: âK1,I,V1,L2. K1. â{I} V1 ð L2 â - ââK2,V2. K1 ð K2 & L2 = K2. â{I} V2. -#K1 #I #V1 #L2 #H -elim (lpx_inv_pair1 â¦ H) -H /2 width=4/ -qed-. - -lemma ltop_inv_atom2: âL1. L1 ð â â L1 = â. -/2 width=2 by lpx_inv_atom2/ qed-. - -lemma ltop_inv_pair2: âL1,K2,I,V2. L1 ð K2. â{I} V2 â - ââK1,V1. K1 ð K2 & L1 = K1. â{I} V1. -#L1 #K2 #I #V2 #H -elim (lpx_inv_pair2 â¦ H) -H /2 width=4/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma ltop_fwd_length: âL1,L2. L1 ð L2 â |L1| = |L2|. -/2 width=2 by lpx_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr.etc deleted file mode 100644 index ea9f585a6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr.etc +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1 â¦ â¡ break â¦ term 46 L2 â¦ )" - 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 â - Î»L1,L2. ââL. L1 â¡ L & L â¢ â¶* [0, |L|] L2 -. - -interpretation - "focalized parallel reduction (environment)" - 'FocalizedPRed L1 L2 = (lfpr L1 L2). - -(* Basic properties *********************************************************) - -(* Note: lemma 250 *) -lemma lfpr_refl: âL. â¦Lâ¦ â¡ â¦Lâ¦. -/2 width=3/ qed. - -lemma ltpss_sn_lfpr: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â â¦L1â¦ â¡ â¦L2â¦. -/3 width=5/ qed. - -lemma ltpr_lfpr: âL1,L2. L1 â¡ L2 â â¦L1â¦ â¡ â¦L2â¦. -/3 width=3/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma lfpr_inv_atom1: âL2. â¦ââ¦ â¡ â¦L2â¦ â L2 = â. -#L2 * #L #HL >(ltpr_inv_atom1 â¦ HL) -HL #HL2 >(ltpss_sn_inv_atom1 â¦ HL2) -HL2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_aaa.etc deleted file mode 100644 index 6f6c49df3..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_aaa.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T,A. L1 â¢ T â A â âL2. â¦L1â¦ â¡ â¦L2â¦ â L2 â¢ T â A. -#L1 #T #A #HT #L2 * /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_cpr.etc deleted file mode 100644 index 8a01f263a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_cpr.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â âV1,V2. L2 â¢ V1 â¡ V2 â - âI. â¦L1. â{I} V1â¦ â¡ â¦L2. â{I} V2â¦. -#L1 #L2 * #L #HL1 #HL2 #V1 #V2 * -<(ltpss_sn_fwd_length â¦ HL2) #V #HV1 #HV2 #I -lapply (ltpss_sn_tpss_trans_eq â¦ HV2 â¦ HL2) -HV2 #V2 -@(ex2_intro â¦ (L.â{I}V)) /2 width=1/ (**) (* explicit constructor *) -qed. - -(* Properties on supclosure *************************************************) -(* -lamma fsub_cpr_trans: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. L2 â¢ T2 â¡ U2 â - ââL,U1. â¦L1â¦ â¡ â¦Lâ¦ & L â¢ T1 â¡ U1 & â¦L, U1â¦ â â¦L2, U2â¦. -#L1 #L2 #T1 #T2 #HT12 #U2 * #T #H1 #H2 -elim (fsub_tpr_trans â¦ HT12 â¦ H1) -T2 #L #U #HL1 #HT1U #HUT -elim (fsup_tpss_trans_full â¦ HUT â¦ 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 â¦ HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2 -elim (lift_total T d e) #U #HTU -elim (ldrop_ltpr_trans â¦ HLK1 â¦ HK1) -HLK1 -HK1 #L #HL1 #HLK -lapply (tpr_lift â¦ HT1 â¦ HTU1 â¦ HTU) -HT1 -HTU1 /3 width=11/ -qed-. -*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_lfpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_lfpr.etc deleted file mode 100644 index 7031e792b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfpr_lfpr.etc +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,L1,L2. â¦L0â¦ â¡ â¦L1â¦ â â¦L0â¦ â¡ â¦L2â¦ â - ââL. â¦L1â¦ â¡ â¦Lâ¦ & â¦L2â¦ â¡ â¦Lâ¦. -#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2 -lapply (ltpr_fwd_length â¦ HK01) #H ->(ltpr_fwd_length â¦ HK02) in H; #H -elim (ltpr_conf â¦ HK01 â¦ HK02) -K0 #K #HK1 #HK2 -lapply (ltpss_sn_fwd_length â¦ HKL1) #H1 -lapply (ltpss_sn_fwd_length â¦ HKL2) #H2 ->H1 in HKL1 H; -H1 #HKL1 ->H2 in HKL2; -H2 #HKL2 #H -elim (ltpr_ltpss_sn_conf â¦ HKL1 â¦ HK1) -K1 #K1 #HK1 #HLK1 -elim (ltpr_ltpss_sn_conf â¦ HKL2 â¦ HK2) -K2 #K2 #HK2 #HLK2 -elim (ltpss_sn_conf â¦ HK1 â¦ HK2) -K #K #HK1 #HK2 -lapply (ltpr_fwd_length â¦ HLK1) #H1 -lapply (ltpr_fwd_length â¦ HLK2) #H2 -/3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfprs_ltprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfprs_ltprs.etc deleted file mode 100644 index 99ae801d7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lfprs_ltprs.etc +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. L1 â¡* L2 â â¦L1â¦ â¡* â¦L2â¦. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lift.etc deleted file mode 100644 index f0d631ebb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lift.etc +++ /dev/null @@ -1,25 +0,0 @@ -definition t_liftable: relation term â Prop â - Î»R. âT1,T2. R T1 T2 â âU1,d,e. â§[d, e] T1 â¡ U1 â - âU2. â§[d, e] T2 â¡ U2 â R U1 U2. - -definition t_deliftable_sn: relation term â Prop â - Î»R. âU1,U2. R U1 U2 â âT1,d,e. â§[d, e] T1 â¡ U1 â - ââT2. â§[d, e] T2 â¡ U2 & R T1 T2. - -lemma t_liftable_TC: âR. t_liftable R â t_liftable (TC â¦ 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: âR. t_deliftable_sn R â t_deliftable_sn (TC â¦ R). -#R #HR #U1 #U2 #H elim H -U2 -[ #U2 #HU12 #T1 #d #e #HTU1 - elim (HR â¦ HU12 â¦ HTU1) -U1 /3 width=3/ -| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1 - elim (IHU1 â¦ HTU1) -U1 #T #HTU #HT1 - elim (HR â¦ HU2 â¦ HTU) -U /3 width=5/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lsubr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lsubr.etc deleted file mode 100644 index 8655a5e54..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lsubr.etc +++ /dev/null @@ -1,198 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â [ 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 â nat â relation lenv â -| lsubr_sort: âd,e. lsubr d e (â) (â) -| lsubr_OO: âL1,L2. lsubr 0 0 L1 L2 -| lsubr_abbr: âL1,L2,V,e. lsubr 0 e L1 L2 â - lsubr 0 (e + 1) (L1. âV) (L2.âV) -| lsubr_abst: âL1,L2,I,V1,V2,e. lsubr 0 e L1 L2 â - lsubr 0 (e + 1) (L1. â{I}V1) (L2. âV2) -| lsubr_skip: âL1,L2,I1,I2,V1,V2,d,e. - lsubr d e L1 L2 â lsubr (d + 1) e (L1. â{I1} V1) (L2. â{I2} V2) -. - -interpretation - "local environment refinement (substitution)" - 'SubEq L1 d e L2 = (lsubr d e L1 L2). - -definition lsubr_trans: âS. (lenv â relation S) â Prop â Î»S,R. - âL2,s1,s2. R L2 s1 s2 â - âL1,d,e. L1 â [d, e] L2 â R L1 s1 s2. - -(* Basic properties *********************************************************) - -lemma lsubr_bind_eq: âL1,L2,e. L1 â [0, e] L2 â âI,V. - L1. â{I} V â [0, e + 1] L2.â{I} V. -#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/ -qed. - -lemma lsubr_abbr_lt: âL1,L2,V,e. L1 â [0, e - 1] L2 â 0 < e â - L1. âV â [0, e] L2.âV. -#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ -qed. - -lemma lsubr_abst_lt: âL1,L2,I,V1,V2,e. L1 â [0, e - 1] L2 â 0 < e â - L1. â{I}V1 â [0, e] L2. âV2. -#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ -qed. - -lemma lsubr_skip_lt: âL1,L2,d,e. L1 â [d - 1, e] L2 â 0 < d â - âI1,I2,V1,V2. L1. â{I1} V1 â [d, e] L2. â{I2} V2. -#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/ -qed. - -lemma lsubr_bind_lt: âI,L1,L2,V,e. L1 â [0, e - 1] L2 â 0 < e â - L1. âV â [0, e] L2. â{I}V. -* /2 width=1/ qed. - -lemma lsubr_refl: âd,e,L. L â [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: âS,R. lsubr_trans S R â lsubr_trans S (Î»L. (TC â¦ (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 â¦ Hs2 â¦ HL12) -HR -Hs2 -HL12 /3 width=3/ -] -qed. - -(* Basic inversion lemmas ***************************************************) - -fact lsubr_inv_atom1_aux: âL1,L2,d,e. L1 â [d, e] L2 â L1 = â â - L2 = â â¨ (d = 0 â§ 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: âL2,d,e. â â [d, e] L2 â - L2 = â â¨ (d = 0 â§ e = 0). -/2 width=3/ qed-. - -fact lsubr_inv_skip1_aux: âL1,L2,d,e. L1 â [d, e] L2 â - âI1,K1,V1. L1 = K1.â{I1}V1 â 0 < d â - ââI2,K2,V2. K1 â [d - 1, e] K2 & L2 = K2.â{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 â¦ H) -| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false â¦ H) -| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false â¦ H) -| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ -] -qed. - -lemma lsubr_inv_skip1: âI1,K1,L2,V1,d,e. K1.â{I1}V1 â [d, e] L2 â 0 < d â - ââI2,K2,V2. K1 â [d - 1, e] K2 & L2 = K2.â{I2}V2. -/2 width=5/ qed-. - -fact lsubr_inv_atom2_aux: âL1,L2,d,e. L1 â [d, e] L2 â L2 = â â - L1 = â â¨ (d = 0 â§ 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: âL1,d,e. L1 â [d, e] â â - L1 = â â¨ (d = 0 â§ e = 0). -/2 width=3/ qed-. - -fact lsubr_inv_abbr2_aux: âL1,L2,d,e. L1 â [d, e] L2 â - âK2,V. L2 = K2.âV â d = 0 â 0 < e â - ââK1. K1 â [0, e - 1] K2 & L1 = K1.âV. -#L1 #L2 #d #e * -L1 -L2 -d -e -[ #d #e #K1 #V #H destruct -| #L1 #L2 #K1 #V #_ #_ #H - elim (lt_zero_false â¦ 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: âL1,K2,V,e. L1 â [0, e] K2.âV â 0 < e â - ââK1. K1 â [0, e - 1] K2 & L1 = K1.âV. -/2 width=5/ qed-. - -fact lsubr_inv_skip2_aux: âL1,L2,d,e. L1 â [d, e] L2 â - âI2,K2,V2. L2 = K2.â{I2}V2 â 0 < d â - ââI1,K1,V1. K1 â [d - 1, e] K2 & L1 = K1.â{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 â¦ H) -| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false â¦ H) -| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false â¦ H) -| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ -] -qed. - -lemma lsubr_inv_skip2: âI2,L1,K2,V2,d,e. L1 â [d, e] K2.â{I2}V2 â 0 < d â - ââI1,K1,V1. K1 â [d - 1, e] K2 & L1 = K1.â{I1}V1. -/2 width=5/ qed-. - -(* Basic forward lemmas *****************************************************) - -fact lsubr_fwd_length_full1_aux: âL1,L2,d,e. L1 â [d, e] L2 â - d = 0 â e = |L1| â |L1| â¤ |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: âL1,L2. L1 â [0, |L1|] L2 â |L1| â¤ |L2|. -/2 width=5/ qed-. - -fact lsubr_fwd_length_full2_aux: âL1,L2,d,e. L1 â [d, e] L2 â - d = 0 â e = |L2| â |L2| â¤ |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: âL1,L2. L1 â [0, |L2|] L2 â |L2| â¤ |L1|. -/2 width=5/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lsubr_lbotr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lsubr_lbotr.etc deleted file mode 100644 index b95a19b66..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/lsubr_lbotr.etc +++ /dev/null @@ -1,77 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â [ 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 â nat â predicate lenv â - Î»d,e. NF_sn â¦ (lsubr d e) (lsubr d e â¦). - -interpretation - "local environment full refinement (substitution)" - 'SubEqBottom d e L = (lbotr d e L). - -(* Basic properties *********************************************************) - -lemma lbotr_atom: âd,e. â[d, e] â. -#d #e #L #H -elim (lsubr_inv_atom2 â¦ H) -H -[ #H destruct // -| * #H1 #H2 destruct // -] -qed. - -lemma lbotr_OO: âL. â[0, 0] L. -// qed. - -lemma lbotr_abbr: âL,V,e. â[0, e] L â â[0, e + 1] L.âV. -#L #V #e #HL #K #H -elim (lsubr_inv_abbr2 â¦ H ?) -H // (tpr_inv_atom1 â¦ H2) -Y -|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct -] -[ >(aaa_inv_sort â¦ H1) -X // -| elim (aaa_inv_lref â¦ H1) #I #K1 #V1 #HLK1 #HA - lapply (ldrop_pair2_fwd_fw â¦ HLK1 (#i)) #HKV1 - elim (ltpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #Y #H #HLK2 - elim (ltpr_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct - lapply (IH â¦ HKV1 â¦ HA â¦ HK12 â¦ HV12) -L1 -K1 -V1 /2 width=5/ -| elim (aaa_inv_gref â¦ H1) -| elim (aaa_inv_abbr â¦ H1) -H1 #B #HB #HA - elim (tpr_inv_abbr1 â¦ H2) -H2 * - [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct - lapply (tps_lsubr_trans â¦ HT2 (L2.âV2) ?) -HT2 /2 width=1/ #HT2 - lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB /width=5/ #HB - lapply (IH â¦ HA â¦ (L2.âV2) â¦ HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/ - | -B #T #HT1 #HXT #H destruct - lapply (IH â¦ HA â¦ (L2.âV1) â¦ HT1) /width=5/ -T1 /2 width=1/ -L1 #HA - @(aaa_inv_lift â¦ HA â¦ HXT) /2 width=1/ - ] -| elim (aaa_inv_abst â¦ H1) -H1 #B #A #HB #HA #H destruct - elim (tpr_inv_abst1 â¦ H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB /width=5/ #HB - lapply (IH â¦ HA â¦ (L2.âV2) â¦ HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/ -| elim (aaa_inv_appl â¦ H1) -H1 #B #HB #HA - elim (tpr_inv_appl1 â¦ H2) -H2 * - [ #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB -HV12 /width=5/ #HB - lapply (IH â¦ HA â¦ HL12 â¦ HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/ - | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - elim (aaa_inv_abst â¦ HA) -HA #B0 #A0 #HB0 #HA0 #H destruct - lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB -HV12 /width=5/ #HB - lapply (IH â¦ HB0 â¦ HL12 W2 ?) -HB0 /width=5/ #HB0 - lapply (IH â¦ HA0 â¦ (L2.âW2) â¦ 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 â¦ HA) -HA #B0 #HW0 #HT0 - lapply (IH â¦ HW0 â¦ HL12 â¦ HW02) -HW0 /width=5/ #HW2 - lapply (IH â¦ HB â¦ HL12 â¦ HV10) -HB -HV10 /width=5/ #HV0 - lapply (IH â¦ HT0 â¦ (L2.âW2) â¦ HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2 - @(aaa_abbr â¦ HW2) -HW2 - @(aaa_appl â¦ HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) - ] -| elim (aaa_inv_cast â¦ H1) -H1 #HV1 #HT1 - elim (tpr_inv_cast1 â¦ H2) -H2 - [ * #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH â¦ HV1 â¦ HL12 â¦ HV12) -HV1 -HV12 /width=5/ #HV2 - lapply (IH â¦ HT1 â¦ HL12 â¦ HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ - | -HV1 #HT1X - lapply (IH â¦ HT1 â¦ HL12 â¦ HT1X) -IH -HT1 -HL12 -HT1X /width=5/ - ] -] -qed. - -lemma aaa_ltpr_conf: âL1,T,A. L1 â¢ T â A â âL2. L1 â¡ L2 â L2 â¢ T â A. -/2 width=5/ qed. - -lemma aaa_tpr_conf: âL,T1,A. L â¢ T1 â A â âT2. T1 â¡ T2 â L â¢ T2 â A. -/2 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ldrop.etc deleted file mode 100644 index 02404a11d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ldrop.etc +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. T2 â¡ U2 â - ââL,U1. L1 â¡ L & T1 â¡ U1 & â¦L, U1â¦ â â¦L2, U2â¦. -#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 â¦ HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2 -elim (lift_total T d e) #U #HTU -elim (ldrop_ltpr_trans â¦ HLK1 â¦ HK1) -HLK1 -HK1 #L #HL1 #HLK -lapply (tpr_lift â¦ HT1 â¦ HTU1 â¦ HTU) -HT1 -HTU1 /3 width=11/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpr.etc deleted file mode 100644 index 4a27a6e70..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpr.etc +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0:lenv. âL1. L0 â¡ L1 â âL2. L0 â¡ L2 â - ââL. L1 â¡ L & L2 â¡ L. -#L0 #L1 #H elim H -L0 -L1 /2 width=3/ -#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H -elim (ltpr_inv_pair1 â¦ H) -H #K2 #V2 #HK02 #HV02 #H destruct -elim (IHK01 â¦ HK02) -K0 #K #HK1 #HK2 -elim (tpr_conf â¦ HV01 HV02) -V0 /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpss_dx.etc deleted file mode 100644 index 00730b732..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpss_dx.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,K1,d,e. L1 â¶* [d, e] K1 â âL2. L1 â¡ L2 â - ââK2. L2 â¶* [d, e] K2 & K1 â¡ K2. -#L1 #K1 #d #e #H elim H -L1 -K1 -d -e -[ /2 width=3/ -| #L1 #I #V1 #X #H - elim (ltpr_inv_pair1 â¦ 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 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 â¦ HL12) -L1 #K2 #HLK2 #HK12 - elim (ltpr_tpr_tpss_conf â¦ HK12 â¦ HV12 â¦ HVW1) -V1 /3 width=5/ -| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H - elim (ltpr_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 â¦ HL12) -L1 #K2 #HLK2 #HK12 - elim (ltpr_tpr_tpss_conf â¦ HK12 â¦ HV12 â¦ HVW1) -V1 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpss_sn.etc deleted file mode 100644 index 79ffb7c9e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_ltpss_sn.etc +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,K1,d,e. L1 â¢ â¶* [d, e] K1 â âL2. L1 â¡ L2 â - ââK2. L2 â¢ â¶* [d, e] K2 & K1 â¡ K2. -#L1 #K1 #d #e #H -lapply (ltpss_sn_ltpssa â¦ H) -H #H -@(ltpssa_ind â¦ H) -K1 /2 width=3/ -#K #K1 #_ #HK1 #IHK #L2 #HL12 -elim (IHK â¦ HL12) -L1 #K2 #HLK2 #HK2 -elim (ltpr_ltpss_dx_conf â¦ HK1 â¦ HK2) -K /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_tps.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_tps.etc deleted file mode 100644 index 7f08be62b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_tps.etc +++ /dev/null @@ -1,57 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T1,T2,d,e. L1 â¢ T1 â¶ [d, e] T2 â âL2. L1 â¡ L2 â - ââT. L2 â¢ T1 â¶ [d, e] T & T2 â¡ 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 â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 - elim (ltpr_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1 - elim (lift_total V2 0 (i+1)) #W2 #HVW2 - lapply (tpr_lift â¦ HV12 â¦ HVW1 â¦ HVW2) -V1 /3 width=4/ -| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 - elim (IHV12 â¦ HL12) -IHV12 #V #HV1 #HV2 - elim (IHT12 (L2.â{I}V) ?) /2 width=1/ -L1 /3 width=5/ -| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 - elim (IHV12 â¦ HL12) -IHV12 - elim (IHT12 â¦ HL12) -L1 /3 width=5/ -] -qed-. - -(* Basic_1: was: pr0_subst1_back *) -lemma ltpr_tps_trans: âL2,T1,T2,d,e. L2 â¢ T1 â¶ [d, e] T2 â âL1. L1 â¡ L2 â - ââT. L1 â¢ T1 â¶ [d, e] T & T â¡ 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 â¦ HL12 â¦ HLK2) -L2 #X #HLK1 #H - elim (ltpr_inv_pair2 â¦ H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2 - elim (lift_total V1 0 (i+1)) #W1 #HVW1 - lapply (tpr_lift â¦ HV12 â¦ HVW1 â¦ HVW2) -V2 /3 width=4/ -| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 - elim (IHV12 â¦ HL12) -IHV12 #V #HV1 #HV2 - elim (IHT12 (L1.â{I}V) ?) /2 width=1/ -L2 /3 width=5/ -| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 - elim (IHV12 â¦ HL12) -IHV12 - elim (IHT12 â¦ HL12) -L2 /3 width=5/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_tpss.etc deleted file mode 100644 index 71586fa36..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpr_tpss.etc +++ /dev/null @@ -1,91 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âT1,T2. T1 â¡ T2 â - âL1,d,e,U1. L1 â¢ T1 â¶ [d, e] U1 â - âL2. L1 â¡ L2 â - ââU2. U1 â¡ U2 & L2 â¢ T2 â¶* [d, e] U2. -#T1 #T2 #H elim H -T1 -T2 -[ #I #L1 #d #e #U1 #H #L2 #HL12 - elim (ltpr_tps_conf â¦ H â¦ HL12) -L1 /3 width=3/ -| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 - elim (tps_inv_flat1 â¦ H) -H #W1 #U1 #HVW1 #HTU1 #H destruct - elim (IHV12 â¦ HVW1 â¦ HL12) -V1 - elim (IHT12 â¦ HTU1 â¦ 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 â¦ H) -H #VV1 #Y #HVV1 #HY #HX destruct - elim (tps_inv_bind1 â¦ HY) -HY #WW #TT1 #_ #HTT1 #H destruct - elim (IHV12 â¦ HVV1 â¦ HL12) -V1 #VV2 #HVV12 #HVV2 - elim (IHT12 â¦ HTT1 (L2. âWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 - lapply (tpss_lsubr_trans â¦ HTT2 (L2. âVV2) ?) -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 â¦ H) -H #W1 #U1 #HVW1 #HTU1 #H destruct - elim (IHV12 â¦ HVW1 â¦ HL12) -V1 #W2 #HW12 #HVW2 - elim (IHT1 â¦ HTU1 (L2. â{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU - elim (tpss_strip_neq â¦ HTU â¦ HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2 - lapply (tps_lsubr_trans â¦ HU2 (L2. â{I} V2) ?) -HU2 /2 width=1/ #HU2 - elim (ltpss_dx_tps_conf â¦ HU2 (L2. â{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23 - lapply (tps_lsubr_trans â¦ HU3 (â. â{I} W2) ?) -HU3 /2 width=1/ #HU3 - lapply (tpss_lsubr_trans â¦ HU23 (L2. â{I} W2) ?) -HU23 /2 width=1/ #HU23 - lapply (tpss_trans_eq â¦ HTU2 â¦ 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 â¦ H) -H #VV1 #Y #HVV1 #HY #HX destruct - elim (tps_inv_bind1 â¦ HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct - elim (IHV12 â¦ HVV1 â¦ HL12) -V1 #VV2 #HVV12 #HVV2 - elim (IHW12 â¦ HWW1 â¦ HL12) -W1 #WW2 #HWW12 #HWW2 - elim (IHT12 â¦ HTT1 (L2. âWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 - elim (lift_total VV2 0 1) #VV #H2VV - lapply (tpss_lift_ge â¦ HVV2 (L2. âWW2) â¦ HV2 â¦ 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 â¦ H) -H #W #U1 #_ #HTU1 #H destruct -V - elim (IHT1 â¦ HTU1 (L2.âW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU - elim (tpss_inv_lift1_ge â¦ HTU L2 â¦ HT2 ?) -T (ltpr_inv_atom1 â¦ HL2) -L2 // -qed-. - -lemma ltprs_inv_pair1: âI,K1,L2,V1. K1. â{I} V1 â¡* L2 â - ââK2,V2. K1 â¡* K2 & V1 â¡* V2 & L2 = K2. â{I} V2. -#I #K1 #L2 #V1 #H @(ltprs_ind â¦ H) -L2 /2 width=5/ -#L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct -elim (ltpr_inv_pair1 â¦ HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/ -qed-. - -lemma ltprs_inv_atom2: âL1. L1 â¡* â â L1 = â. -#L1 #H @(ltprs_ind_dx â¦ H) -L1 // -#L1 #L #HL1 #_ #IHL2 destruct ->(ltpr_inv_atom2 â¦ HL1) -L1 // -qed-. - -lemma ltprs_inv_pair2: âI,L1,K2,V2. L1 â¡* K2. â{I} V2 â - ââK1,V1. K1 â¡* K2 & V1 â¡* V2 & L1 = K1. â{I} V1. -#I #L1 #K2 #V2 #H @(ltprs_ind_dx â¦ H) -L1 /2 width=5/ -#L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct -elim (ltpr_inv_pair2 â¦ HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma ltprs_fwd_length: âL1,L2. L1 â¡* L2 â |L1| = |L2|. -#L1 #L2 #H @(ltprs_ind â¦ H) -L2 // -#L #L2 #_ #HL2 #IHL1 ->IHL1 -L1 >(ltpr_fwd_length â¦ HL2) -HL2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_alt.etc deleted file mode 100644 index 1d2db2daf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_alt.etc +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¡ â¡ * 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 â lpx tprs. - -interpretation - "context-free parallel computation (environment) alternative" - 'PRedStarAlt L1 L2 = (ltprsa L1 L2). - -(* Basic properties *********************************************************) - -lemma ltprs_ltprsa: âL1,L2. L1 â¡* L2 â L1 â¡â¡* L2. -/2 width=1/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma ltprsa_ltprs: âL1,L2. L1 â¡â¡* L2 â L1 â¡* L2. -/2 width=1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_ldrop.etc deleted file mode 100644 index a7c320089..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_ldrop.etc +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_ltprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_ltprs.etc deleted file mode 100644 index 7ededf2af..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltprs_ltprs.etc +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1. âL:lenv. L â¡* L1 â âL2. L â¡ L2 â - ââL0. L1 â¡ L0 & L2 â¡* L0. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem ltprs_conf: confluent â¦ ltprs. -/3 width=3/ qed. - -theorem ltprs_trans: Transitive â¦ ltprs. -/2 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx.etc deleted file mode 100644 index 753c948cc..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx.etc +++ /dev/null @@ -1,278 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¶ * [ 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 â nat â relation lenv â -| ltpss_dx_atom : âd,e. ltpss_dx d e (â) (â) -| ltpss_dx_pair : âL,I,V. ltpss_dx 0 0 (L. â{I} V) (L. â{I} V) -| ltpss_dx_tpss2: âL1,L2,I,V1,V2,e. - ltpss_dx 0 e L1 L2 â L2 â¢ V1 â¶* [0, e] V2 â - ltpss_dx 0 (e + 1) (L1. â{I} V1) (L2. â{I} V2) -| ltpss_dx_tpss1: âL1,L2,I,V1,V2,d,e. - ltpss_dx d e L1 L2 â L2 â¢ V1 â¶* [d, e] V2 â - ltpss_dx (d + 1) e (L1. â{I} V1) (L2. â{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: âd,e,L1,L2. L1 â¶* [d, e] L2 â e = 0 â 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 â¦ HV12) // -] -qed. - -lemma ltpss_dx_inv_refl_O2: âd,L1,L2. L1 â¶* [d, 0] L2 â L1 = L2. -/2 width=4/ qed-. - -fact ltpss_dx_inv_atom1_aux: âd,e,L1,L2. - L1 â¶* [d, e] L2 â L1 = â â L2 = â. -#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: âd,e,L2. â â¶* [d, e] L2 â L2 = â. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss21_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â d = 0 â 0 < e â - âK1,I,V1. L1 = K1. â{I} V1 â - ââK2,V2. K1 â¶* [0, e - 1] K2 & - K2 â¢ V1 â¶* [0, e - 1] V2 & - L2 = K2. â{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 â¦ 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: âe,K1,I,V1,L2. K1. â{I} V1 â¶* [0, e] L2 â 0 < e â - ââK2,V2. K1 â¶* [0, e - 1] K2 & - K2 â¢ V1 â¶* [0, e - 1] V2 & - L2 = K2. â{I} V2. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss11_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â 0 < d â - âI,K1,V1. L1 = K1. â{I} V1 â - ââK2,V2. K1 â¶* [d - 1, e] K2 & - K2 â¢ V1 â¶* [d - 1, e] V2 & - L2 = K2. â{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 â¦ H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_dx_inv_tpss11: âd,e,I,K1,V1,L2. K1. â{I} V1 â¶* [d, e] L2 â 0 < d â - ââK2,V2. K1 â¶* [d - 1, e] K2 & - K2 â¢ V1 â¶* [d - 1, e] V2 & - L2 = K2. â{I} V2. -/2 width=3/ qed-. - -fact ltpss_dx_inv_atom2_aux: âd,e,L1,L2. - L1 â¶* [d, e] L2 â L2 = â â L1 = â. -#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: âd,e,L1. L1 â¶* [d, e] â â L1 = â. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss22_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â d = 0 â 0 < e â - âK2,I,V2. L2 = K2. â{I} V2 â - ââK1,V1. K1 â¶* [0, e - 1] K2 & - K2 â¢ V1 â¶* [0, e - 1] V2 & - L1 = K1. â{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 â¦ 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: âe,L1,K2,I,V2. L1 â¶* [0, e] K2. â{I} V2 â 0 < e â - ââK1,V1. K1 â¶* [0, e - 1] K2 & - K2 â¢ V1 â¶* [0, e - 1] V2 & - L1 = K1. â{I} V1. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss12_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â 0 < d â - âI,K2,V2. L2 = K2. â{I} V2 â - ââK1,V1. K1 â¶* [d - 1, e] K2 & - K2 â¢ V1 â¶* [d - 1, e] V2 & - L1 = K1. â{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 â¦ H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_dx_inv_tpss12: âL1,K2,I,V2,d,e. L1 â¶* [d, e] K2. â{I} V2 â 0 < d â - ââK1,V1. K1 â¶* [d - 1, e] K2 & - K2 â¢ V1 â¶* [d - 1, e] V2 & - L1 = K1. â{I} V1. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -lemma ltpss_dx_tps2: âL1,L2,I,V1,V2,e. - L1 â¶* [0, e] L2 â L2 â¢ V1 â¶ [0, e] V2 â - L1. â{I} V1 â¶* [0, e + 1] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_dx_tps1: âL1,L2,I,V1,V2,d,e. - L1 â¶* [d, e] L2 â L2 â¢ V1 â¶ [d, e] V2 â - L1. â{I} V1 â¶* [d + 1, e] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_dx_tpss2_lt: âL1,L2,I,V1,V2,e. - L1 â¶* [0, e - 1] L2 â L2 â¢ V1 â¶* [0, e - 1] V2 â - 0 < e â L1. â{I} V1 â¶* [0, e] L2. â{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: âL1,L2,I,V1,V2,d,e. - L1 â¶* [d - 1, e] L2 â L2 â¢ V1 â¶* [d - 1, e] V2 â - 0 < d â L1. â{I} V1 â¶* [d, e] L2. â{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: âL1,L2,I,V1,V2,e. - L1 â¶* [0, e - 1] L2 â L2 â¢ V1 â¶ [0, e - 1] V2 â - 0 < e â L1. â{I} V1 â¶* [0, e] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_dx_tps1_lt: âL1,L2,I,V1,V2,d,e. - L1 â¶* [d - 1, e] L2 â L2 â¢ V1 â¶ [d - 1, e] V2 â - 0 < d â L1. â{I} V1 â¶* [d, e] L2. â{I} V2. -/3 width=1/ qed. - -(* Basic_1: was by definition: csubst1_refl *) -lemma ltpss_dx_refl: âL,d,e. L â¶* [d, e] L. -#L elim L -L // -#L #I #V #IHL * /2 width=1/ * /2 width=1/ -qed. - -lemma ltpss_dx_weak: âL1,L2,d1,e1. L1 â¶* [d1, e1] L2 â - âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â L1 â¶* [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 â¦ Hd2) #H destruct normalize in Hde2; - lapply (lt_to_le_to_lt 0 â¦ Hde2) // #He2 - lapply (le_plus_to_minus_r â¦ 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 â¦ Hde12) -Hde12 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 (ldrop_inv_atom1 â¦ H) -H // -| // -| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 - elim (le_inv_plus_l â¦ He12) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK01 â¦ 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 â¦ Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK01 â¦ HK0L2 ?) -K0 /2 width=1/ -] -qed. - -lemma ltpss_dx_ldrop_trans_ge: âL1,L0,d1,e1. L1 â¶* [d1, e1] L0 â - âL2,e2. â©[0, e2] L0 â¡ L2 â - d1 + e1 â¤ e2 â â©[0, e2] L1 â¡ L2. -#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H // -| // -| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 - elim (le_inv_plus_l â¦ He12) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK10 â¦ 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 â¦ Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK10 â¦ HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ -] -qed. - -lemma ltpss_dx_ldrop_conf_be: âL0,L1,d1,e1. L0 â¶* [d1, e1] L1 â - âL2,e2. â©[0, e2] L0 â¡ L2 â d1 â¤ e2 â e2 â¤ d1 + e1 â - ââL. L2 â¶* [0, d1 + e1 - e2] L & â©[0, e2] L1 â¡ L. -#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 - (ldrop_inv_atom1 â¦ H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 - (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK10 -He2d1 destruct (ltpss_dx_inv_atom1 â¦ H) -H /2 width=3/ -| /2 width=3/ -| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd - elim (IHLK1 â¦ 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 â¦ Hd12) -Hd12 #Hd12 #Hd2 - elim (ltpss_dx_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦ Hd12) -IHLK1 -HK12 (ltpss_dx_inv_atom1 â¦ H) -H /2 width=3/ -| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_ - lapply (le_n_O_to_eq â¦ H) -H #H destruct /2 width=3/ -| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2 - elim (IHLK1 â¦ HK12 H1 H2) -K1 -H2 - lapply (le_n_O_to_eq â¦ 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 â¦ Hd12) -Hd12 plus_minus_commutative // #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_be â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 // /2 width=1/ - >plus_minus // >commutative_plus /4 width=5/ - | elim (ltpss_dx_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_be â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 [ >plus_minus // ] /2 width=1/ - >commutative_plus /3 width=5/ - ] -] -qed-. - -lemma ldrop_ltpss_dx_trans_ge: âL1,K1,d1,e1. â©[d1, e1] L1 â¡ K1 â - âK2,d2,e2. K1 â¶* [d2, e2] K2 â d2 + e2 â¤ d1 â - ââL2. L1 â¶* [d2, e2] L2 & â©[d1, e1] L2 â¡ K2. -#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1 -[ #d1 #e1 #K2 #d2 #e2 #H #_ - >(ltpss_dx_inv_atom1 â¦ H) -H /2 width=3/ -| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H - elim (plus_le_0 â¦ H) -H #H1 #H2 destruct /2 width=3/ -| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H - elim (IHLK1 â¦ HK12 H) -K1 - elim (plus_le_0 â¦ H) -H #H1 #H2 destruct #L2 #HL12 - >(ltpss_dx_inv_refl_O2 â¦ 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 â¦ H) -X /3 width=5/ - | elim (ltpss_dx_inv_tpss21 â¦ H He2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_le â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 /2 width=1/ /3 width=5/ - | elim (ltpss_dx_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_le â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/ - ] -] -qed-. - -(* Properties on supclosure *************************************************) - -lemma fsup_tps_trans_full: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. L2 â¢ T2 â¶[0,|L2|] U2 â - ââL,U1. L1 â¶*[0,|L|] L & L â¢ T1 â¶[0,|L|] U1 & â¦L, U1â¦ â â¦L2, T2â¦. -#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 â¦ 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 â¦ HLK1 â¦ HK1 â¦ Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K - lapply (tps_lift_be â¦ HT1 â¦ HL2K â¦ HTU1 HTU â¦ Hd) // -HT1 -HTU1 #HU1 -| elim (ldrop_ltpss_dx_trans_ge â¦ HLK1 â¦ HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K - lapply (tps_lift_le â¦ HT1 â¦ HL2K â¦ HTU1 HTU Hd) -HT1 -HTU1 #HU1 -] -lapply (ltpss_dx_weak_full â¦ HL12) -HL12 #HL12 -lapply (tps_weak_full â¦ HU1) -HU1 #HU1 -@(ex3_2_intro â¦ L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *) -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_ltpss_dx.etc deleted file mode 100644 index 74b898d69..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_ltpss_dx.etc +++ /dev/null @@ -1,106 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â - âL1,d1,e1. L0 â¶* [d1, e1] L1 â - ââT. L1 â¢ T2 â¶* [d2, e2] T & - L1 â¢ U2 â¶* [d1, e1] T. -#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind â¦ H) -U2 /2 width=3/ -#U #U2 #_ #HU2 * #X2 #HTX2 #HUX2 -elim (ltpss_dx_tps_conf â¦ HU2 â¦ HL01) -L0 #X1 #HUX1 #HU2X1 -elim (tpss_strip_eq â¦ HUX2 â¦ HUX1) -U #X #HX2 #HX1 -lapply (tpss_trans_eq â¦ HU2X1 â¦ HX1) -X1 /3 width=3/ -qed. - -lemma ltpss_dx_tpss_trans_down: âL0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 â¤ d1 â - L1 â¶* [d1, e1] L0 â L0 â¢ T2 â¶* [d2, e2] U2 â - ââT. L1 â¢ T2 â¶* [d2, e2] T & L0 â¢ T â¶* [d1, e1] U2. -#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| #U #U2 #_ #HU2 * #T #HT2 #HTU - elim (tpss_strap1_down â¦ HTU â¦ HU2 ?) -U // #U #HTU #HU2 - elim (ltpss_dx_tps_trans â¦ HTU â¦ HL10) -HTU -HL10 #X #HTX #HXU - lapply (tpss_trans_eq â¦ HXU HU2) -U /3 width=3/ -] -qed. - -lemma ltpss_dx_tpss_trans_eq: âL1,T2,U2,d,e. L1 â¢ T2 â¶* [d, e] U2 â - âL0. L0 â¶* [d, e] L1 â L0 â¢ T2 â¶* [d, e] U2. -#L1 #T2 @(f2_ind â¦ fw â¦ L1 T2) -L1 -T2 #n #IH #L1 * -[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct - elim (tpss_inv_atom1 â¦ H) -H // * - #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct - lapply (ldrop_fwd_lw â¦ HLK1) #H1 normalize in H1; - elim (ltpss_dx_ldrop_trans_be â¦ HL01 â¦ HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0 - elim (ltpss_dx_inv_tpss22 â¦ H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct - lapply (tpss_fwd_tw â¦ HV01) #H2 - lapply (transitive_le (â¯{K1} + â¯{V0}) â¦ H1) -H1 /2 width=1/ -H2 #H - lapply (tpss_trans_eq â¦ HV01 HV12) -V1 #HV02 - lapply (IH â¦ HV02 â¦ 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 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct - lapply (tpss_lsubr_trans â¦ HTU2 (L1. â{I} V2) ?) -HTU2 /2 width=1/ #HTU2 - lapply (IH â¦ HVW2 â¦ HL0) -HVW2 [ /2 width=2/ ] #HVW2 - lapply (IH â¦ HTU2 (L0. â{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2 - lapply (tpss_lsubr_trans â¦ HTU2 (L0. â{I} W2) ?) -HTU2 /2 width=1/ - | elim (tpss_inv_flat1 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct - lapply (IH â¦ HVW2 â¦ HL0) -HVW2 // - lapply (IH â¦ HTU2 â¦ HL0) -IH -HTU2 // -HL0 /2 width=1/ -] -qed. - -lemma ltpss_dx_tps_trans_eq: âL0,L1,T2,U2,d,e. L0 â¶* [d, e] L1 â - L1 â¢ T2 â¶ [d, e] U2 â L0 â¢ T2 â¶* [d, e] U2. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem ltpss_dx_conf: âL0,L1,d1,e1. L0 â¶* [d1, e1] L1 â - âL2,d2,e2. L0 â¶* [d2, e2] L2 â - ââL. L1 â¶* [d2, e2] L & L2 â¶* [d1, e1] L. -#L0 @(f_ind â¦ lw â¦ L0) -L0 #n #IH * -[ #_ #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 -n - >(ltpss_dx_inv_atom1 â¦ H1) -L1 - >(ltpss_dx_inv_atom1 â¦ 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 â¦ H1) -H1 #H1 - | elim (ltpss_dx_inv_tpss21 â¦ H1 â¦ He1) -H1 #K1 #V1 #HK01 #HV01 #H1 - | elim (ltpss_dx_inv_tpss11 â¦ H1 â¦ 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 â¦ H2) -H2 #H2 - |2,4,6: elim (ltpss_dx_inv_tpss21 â¦ H2 â¦ He2) -H2 #K2 #V2 #HK02 #HV02 #H2 - |7,8,9: elim (ltpss_dx_inv_tpss11 â¦ H2 â¦ 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 â¦ HK01 â¦ HK02) // -K0 #K #HK1 #HK2 - elim (ltpss_dx_tpss_conf â¦ HV01 â¦ HK1) -HV01 #W1 #HW1 #HVW1 - elim (ltpss_dx_tpss_conf â¦ HV02 â¦ HK2) -HV02 #W2 #HW2 #HVW2 - elim (tpss_conf_eq â¦ HW1 â¦ HW2) -V0 #V #HW1 #HW2 - lapply (tpss_trans_eq â¦ HVW1 HW1) -W1 - lapply (tpss_trans_eq â¦ HVW2 HW2) -W2 /3 width=5/ - ] -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_tps.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_tps.etc deleted file mode 100644 index b10306677..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_tps.etc +++ /dev/null @@ -1,47 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â - âL1,d1,e1. L0 â¶* [d1, e1] L1 â d1 + e1 â¤ d2 â - L1 â¢ T2 â¶ [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 â¦ Hde1d2 Hdi2) -Hde1d2 #Hde1i2 - lapply (ltpss_dx_ldrop_conf_ge â¦ HL01 â¦ 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â - âL1,d1,e1. L1 â¶* [d1, e1] L0 â d1 + e1 â¤ d2 â - L1 â¢ T2 â¶ [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 â¦ Hde1d2 Hdi2) -Hde1d2 #Hde1i2 - lapply (ltpss_dx_ldrop_trans_ge â¦ HL10 â¦ 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_tpss.etc deleted file mode 100644 index 9c7875927..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_dx_tpss.etc +++ /dev/null @@ -1,101 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â - âL1,d1,e1. L0 â¶* [d1, e1] L1 â d1 + e1 â¤ d2 â - L1 â¢ T2 â¶* [d2, e2] U2. -#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind â¦ H) -U2 // -#U #U2 #_ #HU2 #IHU -lapply (ltpss_dx_tps_conf_ge â¦ HU2 â¦ HL01 ?) -L0 // -Hde1d2 /2 width=3/ -qed. - -(* Basic_1: was: subst1_subst1_back *) -lemma ltpss_dx_tps_conf: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â - âL1,d1,e1. L0 â¶* [d1, e1] L1 â - ââT. L1 â¢ T2 â¶ [d2, e2] T & - L1 â¢ U2 â¶* [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 â¦ HL01 â¦ HLK0 ?) -L0 /2 width=2/ #X #H #HLK1 - elim (ltpss_dx_inv_tpss11 â¦ H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK1) #H - elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 - lapply (tpss_lift_ge â¦ HV01 â¦ H HVW0 â¦ HVW1) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_dx_ldrop_conf_ge â¦ HL01 â¦ HLK0 ?) -L0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 - elim (IHVW2 â¦ HL01) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. â{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 â¦ HL01) -IHVW2 - elim (IHTU2 â¦ HL01) -IHTU2 -HL01 /3 width=5/ -] -qed. - -lemma ltpss_dx_tpss_trans_ge: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â - âL1,d1,e1. L1 â¶* [d1, e1] L0 â d1 + e1 â¤ d2 â - L1 â¢ T2 â¶* [d2, e2] U2. -#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind â¦ H) -U2 // -#U #U2 #_ #HU2 #IHU -lapply (ltpss_dx_tps_trans_ge â¦ HU2 â¦ HL01 ?) -L0 // -Hde1d2 /2 width=3/ -qed. - -(* Basic_1: was: subst1_subst1 *) -lemma ltpss_dx_tps_trans: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â - âL1,d1,e1. L1 â¶* [d1, e1] L0 â - ââT. L1 â¢ T2 â¶ [d2, e2] T & - L0 â¢ T â¶* [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 â¦ HL10 â¦ HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1 - elim (ltpss_dx_inv_tpss12 â¦ H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK0) -HLK0 #H - elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 - lapply (tpss_lift_ge â¦ HV01 â¦ H HVW1 â¦ HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_dx_ldrop_trans_ge â¦ HL10 â¦ HLK0 ?) -HL10 -HLK0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 - elim (IHVW2 â¦ HL10) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. â{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 â¦ HL10) -IHVW2 - elim (IHTU2 â¦ HL10) -IHTU2 -HL10 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_sn.etc deleted file mode 100644 index 577c1506d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_sn.etc +++ /dev/null @@ -1,259 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ â¶ * [ 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 â nat â relation lenv â -| ltpss_sn_atom : âd,e. ltpss_sn d e (â) (â) -| ltpss_sn_pair : âL,I,V. ltpss_sn 0 0 (L. â{I} V) (L. â{I} V) -| ltpss_sn_tpss2: âL1,L2,I,V1,V2,e. - ltpss_sn 0 e L1 L2 â L1 â¢ V1 â¶* [0, e] V2 â - ltpss_sn 0 (e + 1) (L1. â{I} V1) (L2. â{I} V2) -| ltpss_sn_tpss1: âL1,L2,I,V1,V2,d,e. - ltpss_sn d e L1 L2 â L1 â¢ V1 â¶* [d, e] V2 â - ltpss_sn (d + 1) e (L1. â{I} V1) (L2. â{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: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â e = 0 â 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 â¦ HV12) // -] -qed. - -lemma ltpss_sn_inv_refl_O2: âd,L1,L2. L1 â¢ â¶* [d, 0] L2 â L1 = L2. -/2 width=4/ qed-. - -fact ltpss_sn_inv_atom1_aux: âd,e,L1,L2. - L1 â¢ â¶* [d, e] L2 â L1 = â â L2 = â. -#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: âd,e,L2. â â¢ â¶* [d, e] L2 â L2 = â. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss21_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â d = 0 â 0 < e â - âK1,I,V1. L1 = K1. â{I} V1 â - ââK2,V2. K1 â¢ â¶* [0, e - 1] K2 & - K1 â¢ V1 â¶* [0, e - 1] V2 & - L2 = K2. â{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 â¦ 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: âe,K1,I,V1,L2. K1. â{I} V1 â¢ â¶* [0, e] L2 â 0 < e â - ââK2,V2. K1 â¢ â¶* [0, e - 1] K2 & - K1 â¢ V1 â¶* [0, e - 1] V2 & - L2 = K2. â{I} V2. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss11_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â 0 < d â - âI,K1,V1. L1 = K1. â{I} V1 â - ââK2,V2. K1 â¢ â¶* [d - 1, e] K2 & - K1 â¢ V1 â¶* [d - 1, e] V2 & - L2 = K2. â{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 â¦ H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_sn_inv_tpss11: âd,e,I,K1,V1,L2. K1. â{I} V1 â¢ â¶* [d, e] L2 â 0 < d â - ââK2,V2. K1 â¢ â¶* [d - 1, e] K2 & - K1 â¢ V1 â¶* [d - 1, e] V2 & - L2 = K2. â{I} V2. -/2 width=3/ qed-. - -fact ltpss_sn_inv_atom2_aux: âd,e,L1,L2. - L1 â¢ â¶* [d, e] L2 â L2 = â â L1 = â. -#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: âd,e,L1. L1 â¢ â¶* [d, e] â â L1 = â. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss22_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â d = 0 â 0 < e â - âK2,I,V2. L2 = K2. â{I} V2 â - ââK1,V1. K1 â¢ â¶* [0, e - 1] K2 & - K1 â¢ V1 â¶* [0, e - 1] V2 & - L1 = K1. â{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 â¦ 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: âe,L1,K2,I,V2. L1 â¢ â¶* [0, e] K2. â{I} V2 â 0 < e â - ââK1,V1. K1 â¢ â¶* [0, e - 1] K2 & - K1 â¢ V1 â¶* [0, e - 1] V2 & - L1 = K1. â{I} V1. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss12_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â 0 < d â - âI,K2,V2. L2 = K2. â{I} V2 â - ââK1,V1. K1 â¢ â¶* [d - 1, e] K2 & - K1 â¢ V1 â¶* [d - 1, e] V2 & - L1 = K1. â{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 â¦ H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_sn_inv_tpss12: âL1,K2,I,V2,d,e. L1 â¢ â¶* [d, e] K2. â{I} V2 â 0 < d â - ââK1,V1. K1 â¢ â¶* [d - 1, e] K2 & - K1 â¢ V1 â¶* [d - 1, e] V2 & - L1 = K1. â{I} V1. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -lemma ltpss_sn_tps2: âL1,L2,I,V1,V2,e. - L1 â¢ â¶* [0, e] L2 â L1 â¢ V1 â¶ [0, e] V2 â - L1. â{I} V1 â¢ â¶* [0, e + 1] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_tps1: âL1,L2,I,V1,V2,d,e. - L1 â¢ â¶* [d, e] L2 â L1 â¢ V1 â¶ [d, e] V2 â - L1. â{I} V1 â¢ â¶* [d + 1, e] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_tpss2_lt: âL1,L2,I,V1,V2,e. - L1 â¢ â¶* [0, e - 1] L2 â L1 â¢ V1 â¶* [0, e - 1] V2 â - 0 < e â L1. â{I} V1 â¢ â¶* [0, e] L2. â{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: âL1,L2,I,V1,V2,d,e. - L1 â¢ â¶* [d - 1, e] L2 â L1 â¢ V1 â¶* [d - 1, e] V2 â - 0 < d â L1. â{I} V1 â¢ â¶* [d, e] L2. â{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: âL1,L2,I,V1,V2,e. - L1 â¢ â¶* [0, e - 1] L2 â L1 â¢ V1 â¶ [0, e - 1] V2 â - 0 < e â L1. â{I} V1 â¢ â¶* [0, e] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_tps1_lt: âL1,L2,I,V1,V2,d,e. - L1 â¢ â¶* [d - 1, e] L2 â L1 â¢ V1 â¶ [d - 1, e] V2 â - 0 < d â L1. â{I} V1 â¢ â¶* [d, e] L2. â{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_refl: âL,d,e. L â¢ â¶* [d, e] L. -#L elim L -L // -#L #I #V #IHL * /2 width=1/ * /2 width=1/ -qed. - -lemma ltpss_sn_weak: âL1,L2,d1,e1. L1 â¢ â¶* [d1, e1] L2 â - âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â L1 â¢ â¶* [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 â¦ Hd2) #H destruct normalize in Hde2; - lapply (lt_to_le_to_lt 0 â¦ Hde2) // #He2 - lapply (le_plus_to_minus_r â¦ 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 â¦ Hde12) -Hde12 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 (ldrop_inv_atom1 â¦ H) -H // -| // -| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 - elim (le_inv_plus_l â¦ He12) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK01 â¦ 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 â¦ Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK01 â¦ HK0L2 ?) -K0 /2 width=1/ -] -qed. - -lemma ltpss_sn_ldrop_trans_ge: âL1,L0,d1,e1. L1 â¢ â¶* [d1, e1] L0 â - âL2,e2. â©[0, e2] L0 â¡ L2 â - d1 + e1 â¤ e2 â â©[0, e2] L1 â¡ L2. -#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H // -| // -| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 - elim (le_inv_plus_l â¦ He12) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK10 â¦ 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 â¦ Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 - lapply (IHK10 â¦ HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ -] -qed. - -lemma ltpss_sn_ldrop_conf_be: âL0,L1,d1,e1. L0 â¢ â¶* [d1, e1] L1 â - âL2,e2. â©[0, e2] L0 â¡ L2 â d1 â¤ e2 â e2 â¤ d1 + e1 â - ââL. L2 â¢ â¶* [0, d1 + e1 - e2] L & â©[0, e2] L1 â¡ L. -#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 - (ldrop_inv_atom1 â¦ H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 - (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct - lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 - [ -IHK10 -He2d1 destruct (ltpss_sn_inv_atom1 â¦ H) -H /2 width=3/ -| /2 width=3/ -| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd - elim (IHLK1 â¦ 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 â¦ Hd12) -Hd12 #Hd12 #Hd2 - elim (ltpss_sn_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦ Hd12) -IHLK1 -HK12 (ltpss_sn_inv_atom1 â¦ H) -H /2 width=3/ -| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_ - lapply (le_n_O_to_eq â¦ H) -H #H destruct /2 width=3/ -| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2 - elim (IHLK1 â¦ HK12 H1 H2) -K1 -H2 - lapply (le_n_O_to_eq â¦ 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 â¦ Hd12) -Hd12 plus_minus_commutative // #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_be â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 // /2 width=1/ - >plus_minus // >commutative_plus /4 width=5/ - | elim (ltpss_sn_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_be â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 [ >plus_minus // ] /2 width=1/ - >commutative_plus /3 width=5/ - ] -] -qed-. - -lemma ldrop_ltpss_sn_trans_ge: âL1,K1,d1,e1. â©[d1, e1] L1 â¡ K1 â - âK2,d2,e2. K1 â¢ â¶* [d2, e2] K2 â d2 + e2 â¤ d1 â - ââL2. L1 â¢ â¶* [d2, e2] L2 & â©[d1, e1] L2 â¡ K2. -#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1 -[ #d1 #e1 #K2 #d2 #e2 #H #_ - >(ltpss_sn_inv_atom1 â¦ H) -H /2 width=3/ -| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H - elim (plus_le_0 â¦ H) -H #H1 #H2 destruct /2 width=3/ -| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H - elim (IHLK1 â¦ HK12 H) -K1 - elim (plus_le_0 â¦ H) -H #H1 #H2 destruct #L2 #HL12 - >(ltpss_sn_inv_refl_O2 â¦ 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 â¦ H) -X /3 width=5/ - | elim (ltpss_sn_inv_tpss21 â¦ H He2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_le â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 /2 width=1/ /3 width=5/ - | elim (ltpss_sn_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2 - elim (lift_total W2 d1 e1) #V2 #HWV2 - lapply (tpss_lift_le â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/ - ] -] -qed-. - -(* Properties on supclosure *************************************************) - -lemma fsup_tpss_trans_full: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. L2 â¢ T2 â¶*[0,|L2|] U2 â - ââL,U1. L1 â¢ â¶*[0,|L1|] L & L â¢ T1 â¶*[0,|L|] U1 & â¦L, U1â¦ â â¦L2, T2â¦. -#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 â¦ HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 -elim (lift_total T d e) #U #HTU -lapply (ltpss_sn_fwd_length â¦ HK1) #H >H in HK1; -H #HK1 -elim (le_or_ge d (|K|)) #Hd -[ elim (ldrop_ltpss_sn_trans_be â¦ HLK1 â¦ HK1 â¦ Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K - lapply (tpss_lift_be â¦ HT1 â¦ Hd HL2K HTU1 â¦ HTU) // -HT1 -HTU1 #HU1 -| elim (ldrop_ltpss_sn_trans_ge â¦ HLK1 â¦ HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K - lapply (tpss_lift_le â¦ HT1 â¦ Hd HL2K HTU1 â¦ HTU) -HT1 -HTU1 #HU1 -] -lapply (ltpss_sn_weak_full â¦ HL12) -HL12 #HL12 -lapply (tpss_weak_full â¦ HU1) -HU1 #HU1 -@(ex3_2_intro â¦ L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *) -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_sn_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_sn_ltpss_sn.etc deleted file mode 100644 index 67c287e2b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ltpss_sn_ltpss_sn.etc +++ /dev/null @@ -1,81 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T2,U2,d,e. L1 â¢ T2 â¶* [d, e] U2 â - âL0. L0 â¢ â¶* [d, e] L1 â L0 â¢ T2 â¶* [d, e] U2. -#L1 #T2 @(f2_ind â¦ fw â¦ L1 T2) -L1 -T2 #n #IH #L1 * -[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct - elim (tpss_inv_atom1 â¦ H) -H // * - #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct - lapply (ldrop_fwd_lw â¦ HLK1) #H1 normalize in H1; - elim (ltpss_sn_ldrop_trans_be â¦ HL01 â¦ HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0 - elim (ltpss_sn_inv_tpss22 â¦ H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct - lapply (IH â¦ HV12 â¦ HK01) -IH -HV12 -HK01 [ normalize /2 width=1/ ] #HV12 - lapply (tpss_trans_eq â¦ HV01 HV12) -V1 /2 width=6/ -| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct - [ elim (tpss_inv_bind1 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct - lapply (tpss_lsubr_trans â¦ HTU2 (L1. â{I} V2) ?) -HTU2 /2 width=1/ #HTU2 - lapply (IH â¦ HVW2 â¦ HL0) -HVW2 [ /2 width=2/ ] #HVW2 - lapply (IH â¦ HTU2 (L0. â{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2 - lapply (tpss_lsubr_trans â¦ HTU2 (L0. â{I} W2) ?) -HTU2 /2 width=1/ - | elim (tpss_inv_flat1 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct - lapply (IH â¦ HVW2 â¦ HL0) -HVW2 // - lapply (IH â¦ HTU2 â¦ HL0) -IH -HTU2 // -HL0 /2 width=1/ -] -qed. - -lemma ltpss_sn_tps_trans_eq: âL0,L1,T2,U2,d,e. L0 â¢ â¶* [d, e] L1 â - L1 â¢ T2 â¶ [d, e] U2 â L0 â¢ T2 â¶* [d, e] U2. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem ltpss_sn_trans_eq: âL1,L,d,e. L1 â¢ â¶* [d, e] L â - âL2. L â¢ â¶* [d, e] L2 â L1 â¢ â¶* [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 â¦ H ?) -H // shift_append_assoc #H - elim (tps_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - elim (IH â¦ HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct - append_length minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_sn_ldrop_conf_ge â¦ HL01 â¦ HLK0 ?) -HL01 -HLK0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 - elim (IHVW2 â¦ HL01) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. â{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H - lapply (tpss_lsubr_trans â¦ H (L0.â{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 â¦ HL01) -IHVW2 - elim (IHTU2 â¦ HL01) -IHTU2 -HL01 /3 width=5/ -] -qed. - -lemma ltpss_sn_tpss_trans_ge: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â - âL1,d1,e1. L1 â¢ â¶* [d1, e1] L0 â d1 + e1 â¤ d2 â - L1 â¢ T2 â¶* [d2, e2] U2. -#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind â¦ H) -U2 // -#U #U2 #_ #HU2 #IHU -lapply (ltpss_sn_tps_trans_ge â¦ HU2 â¦ HL01 ?) -L0 // -Hde1d2 /2 width=3/ -qed. - -lemma ltpss_sn_tps_trans: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â - âL1,d1,e1. L1 â¢ â¶* [d1, e1] L0 â - ââT. L1 â¢ T2 â¶ [d2, e2] T & - L1 â¢ T â¶* [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 â¦ HL10 â¦ HLK0 ?) -L0 /2 width=2/ #X #H #HLK1 - elim (ltpss_sn_inv_tpss12 â¦ H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK1) #H - elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 - lapply (tpss_lift_ge â¦ HV01 â¦ H HVW1 â¦ HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_sn_ldrop_trans_ge â¦ HL10 â¦ HLK0 ?) -HL10 -HLK0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 - elim (IHVW2 â¦ HL10) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. â{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H - lapply (tpss_lsubr_trans â¦ H (L1.â{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 â¦ HL10) -IHVW2 - elim (IHTU2 â¦ HL10) -IHTU2 -HL10 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/snv_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/snv_ltpss_dx.etc deleted file mode 100644 index ca55239f0..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/snv_ltpss_dx.etc +++ /dev/null @@ -1,87 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,T1. â¦h, L1â¦ â¢ T1 Â¡[g] â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â â¦h, L2â¦ â¢ T2 Â¡[g]. -#h #g #L1 #T1 #H elim H -L1 -T1 -[ #L1 #k #L2 #d #e #_ #X #H - >(tpss_inv_sort1 â¦ H) -X // -| #I #L1 #K1 #V1 #i #HLK1 #HV1 #IHV1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_lref1 â¦ 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 â¦ HL12 â¦ HLK1 â¦) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss11 â¦ H â¦) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - lapply (IHV1 â¦ HK12 â¦ HV12) -IHV1 -HK12 -HV12 /2 width=5/ - | elim (ltpss_dx_ldrop_conf_be â¦ HL12 â¦ HLK1 â¦) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss21 â¦ H â¦) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - lapply (IHV1 â¦ HK12 â¦ HV12) -IHV1 -HK12 -HV12 /2 width=5/ - | lapply (ltpss_dx_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -L1 // -Hide /2 width=5/ - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 - elim (ltpss_dx_ldrop_conf_be â¦ HL12 â¦ HLK1 â¦) -L1 // /2 width=2/ #X #H1 #H2 - elim (ltpss_dx_inv_tpss21 â¦ H1 â¦) -H1 /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct - lapply (ldrop_mono â¦ H2 â¦ HLK2) -H2 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) -HLK2 #HLK2 - lapply (tpss_trans_eq â¦ HV12 HVW2) -V2 #HV1W2 - lapply (IHV1 â¦ HK12 â¦ HV1W2) -IHV1 -HK12 -HV1W2 /2 width=7/ - ] -| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IHV1 â¦ HL12 â¦ HV12) -IHV1 #HV2 - lapply (IHT1 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IHV1 â¦ HL12 â¦ HV12) -IHV1 #HV2 - lapply (IHT1 â¦ HT12) -IHT1 /2 width=1/ #HT2 - elim (ssta_ltpss_dx_tpss_conf â¦ HVW1 â¦ HL12 â¦ HV12) -V1 #W2 #HVW2 #HW12 - elim (dxprs_ltpss_dx_tpss_conf â¦ HTU1 â¦ HL12 â¦ HT12) -T1 #X #HTU2 #H - elim (tpss_inv_bind1 â¦ H) -H #W #U2 #HW0 #_ #H -U1 destruct - elim (cprs_ltpss_dx_tpss2_conf â¦ HW10 â¦ HL12 â¦ HW12 â¦ HW0) -L1 -W1 -W0 #W0 #HW20 #HW0 - lapply (dxprs_strap1 â¦ (â{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 â¦ H) -H #W2 #T2 #HW12 #HT12 #H destruct - lapply (IHW1 â¦ HL12 â¦ HW12) -IHW1 #HW2 - lapply (IHT1 â¦ HL12 â¦ HT12) -IHT1 #HT2 - elim (ssta_ltpss_dx_tpss_conf â¦ HTU1 â¦ HL12 â¦ HT12) -T1 #U2 #HTU2 #HU12 - lapply (cpcs_ltpss_dx_tpss2_conf â¦ HL12 â¦ HUW1 â¦ HU12 â¦ HW12) -L1 -W1 -U1 /2 width=4/ -] -qed-. - -lemma snv_ltpss_dx_conf: âh,g,L1,T. â¦h, L1â¦ â¢ T Â¡[g] â - âL2,d,e. L1 â¶* [d, e] L2 â â¦h, L2â¦ â¢ T Â¡[g]. -#h #g #L1 #T #HT #L2 #d #e #HL12 -@(snv_ltpss_dx_tpss_conf â¦ HT â¦ HL12) // -qed-. - -lemma snv_tpss_conf: âh,g,L,T1. â¦h, Lâ¦ â¢ T1 Â¡[g] â - âT2,d,e. L â¢ T1 â¶* [d, e] T2 â â¦h, Lâ¦ â¢ T2 Â¡[g]. -#h #g #L #T1 #HT1 #T2 #d #e #HT12 -@(snv_ltpss_dx_tpss_conf â¦ HT1 â¦ HT12) // -qed-. - -lemma snv_tps_conf: âh,g,L,T1. â¦h, Lâ¦ â¢ T1 Â¡[g] â - âT2,d,e. L â¢ T1 â¶ [d, e] T2 â â¦h, Lâ¦ â¢ T2 Â¡[g]. -#h #g #L #T1 #HT1 #T2 #d #e #HT12 -@(snv_tpss_conf â¦ HT1 T2) /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/snv_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/snv_ltpss_sn.etc deleted file mode 100644 index 3c8070249..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/snv_ltpss_sn.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,L2,d,e. L1 â¢ â¶* [d, e] L2 â - âT. â¦h, L1â¦ â¢ T Â¡[g] â â¦h, L2â¦ â¢ T Â¡[g]. -#h #g #L1 #L2 #d #e #H -lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 // -#L #L2 #_ #HL2 #IHL1 #T1 #HT1 -lapply (IHL1 â¦ HT1) -IHL1 -HT1 #HT1 -lapply (snv_ltpss_dx_conf â¦ HT1 â¦ HL2) -HT1 -HL2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ssta_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ssta_ltpss_dx.etc deleted file mode 100644 index 72eff7850..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/ssta_ltpss_dx.etc +++ /dev/null @@ -1,125 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 â¢[g] â¦l, U1â¦ â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â - ââU2. â¦h, L2â¦ â¢ T2 â¢[g] â¦l, U2â¦ & - L2 â¢ U1 â¶* [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 â¦ 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 â¦ H) -H [ | -HVW1 ] - [ #H destruct - elim (lt_or_ge i d) #Hdi [ -HVW1 | ] - [ elim (ltpss_dx_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss11 â¦ H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 â¦ HK12 â¦ HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 - lapply (ldrop_fwd_ldrop2 â¦ HLK2) #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge â¦ HW12 â¦ H â¦ HWU1 â¦ HWU2) // -HW12 -H -HWU1 - >minus_plus minus_plus #H - lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H - lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H - lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus (tpss_inv_sort1 â¦ 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 â¦ H) -H [ | -HVW1 ] - [ #H destruct - elim (lt_or_ge i d) #Hdi [ -HVW1 | ] - [ elim (ltpss_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 â¦ H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 â¦ HK12 â¦ HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 - lapply (ldrop_fwd_ldrop2 â¦ HLK2) #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge â¦ HW12 â¦ H â¦ HWU1 â¦ HWU2) // -HW12 -H -HWU1 - >minus_plus minus_plus #H - lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H - lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H - lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus shift_append_assoc normalize #H - elim (tpr_inv_bind1 â¦ H) -H * - [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct - elim (IH â¦ HT10) -IH -T1 #L #T #HL1 #H destruct - elim (tps_fwd_shift1 â¦ H0) -T #L2 #T2 #HL2 #H destruct - >append_length >HL1 >HL2 -L1 -L - @(ex2_2_intro â¦ (â.â{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 *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpr_lift.etc deleted file mode 100644 index 8d6b0363a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpr_lift.etc +++ /dev/null @@ -1,121 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ HU1 â¦ HU2) -HU1 #H destruct - [ lapply (lift_inv_sort1 â¦ HU2) -HU2 #H destruct // - | lapply (lift_inv_lref1 â¦ HU2) * * #Hid #H destruct // - | lapply (lift_inv_gref1 â¦ HU2) -HU2 #H destruct // - ] -| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 - elim (lift_inv_flat1 â¦ HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct - elim (lift_inv_flat1 â¦ 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 â¦ HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct - elim (lift_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct - elim (lift_inv_bind1 â¦ 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 â¦ HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct - elim (lift_inv_bind1 â¦ HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct - elim (lift_total T (d + 1) e) #U #HTU - @tpr_delta - [4: @(tps_lift_le â¦ HT2 â¦ 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 â¦ HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct - elim (lift_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct - elim (lift_inv_bind1 â¦ HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct - elim (lift_inv_flat1 â¦ HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct - elim (lift_trans_ge â¦ HV2 â¦ HV3 ?) -V // /3 width=4/ -| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2 - elim (lift_inv_bind1 â¦ H) -H #V3 #T3 #_ #HT13 #H destruct -V - elim (lift_conf_O1 â¦ HTU2 â¦ HT2) -T2 /3 width=4/ -| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2 - elim (lift_inv_flat1 â¦ 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 â¦ HX) -HX #H destruct /2 width=3/ - | lapply (lift_inv_lref2 â¦ HX) -HX * * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 â¦ HX) -HX #H destruct /2 width=3/ - ] -| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX - elim (lift_inv_flat2 â¦ HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct - elim (IHV12 â¦ HV01) -V1 - elim (IHT12 â¦ HT01) -T1 /3 width=5/ -| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX - elim (lift_inv_flat2 â¦ HX) -HX #V0 #Y #HV01 #HY #HX destruct - elim (lift_inv_bind2 â¦ HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct - elim (IHV12 â¦ HV01) -V1 - elim (IHT12 â¦ HT01) -T1 /3 width=5/ -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX - elim (lift_inv_bind2 â¦ HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct - elim (IHV12 â¦ HWV1) -V1 #W2 #HWV2 #HW12 - elim (IHT1 â¦ HUT1) -T1 #U #HUT #HU1 - elim (tps_inv_lift1_le â¦ HT2 â¦ 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 â¦ HX) -HX #V0 #Y #HV01 #HY #HX destruct - elim (lift_inv_bind2 â¦ HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct - elim (IHV12 â¦ HV01) -V1 #V3 #HV32 #HV03 - elim (IHW12 â¦ HW01) -W1 #W3 #HW32 #HW03 - elim (IHT12 â¦ HT01) -T1 #T3 #HT32 #HT03 - elim (lift_trans_le â¦ HV32 â¦ 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 â¦ HX) -HX #V3 #T3 #_ #HT31 #H destruct - elim (IHT1 â¦ HT31) -T1 #T1 #HT1 #HT31 - elim (lift_div_le â¦ HT2 â¦ HT1 ?) -T // /3 width=5/ -| #V #T1 #T2 #_ #IHT12 #X #d #e #HX - elim (lift_inv_flat2 â¦ HX) -HX #V0 #T0 #_ #HT01 #H destruct - elim (IHT12 â¦ HT01) -T1 /3 width=3/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was pr0_gen_abst *) -lemma tpr_inv_abst1: âa,V1,T1,U2. â{a}V1. T1 â¡ U2 â - ââV2,T2. V1 â¡ V2 & T1 â¡ T2 & U2 = â{a}V2. T2. -#a #V1 #T1 #U2 #H elim (tpr_inv_bind1 â¦ H) -H * -[ #V2 #T #T2 #HV12 #HT1 #HT2 - lapply (tps_inv_refl_SO2 â¦ HT2 ???) -HT2 // /2 width=5/ -| #T2 #_ #_ #_ #H destruct -] -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma tpr_fwd_abst1: âa,V1,T1,U2. â{a}V1.T1 â¡ U2 â âb,I,W. - ââV2,T2. â{b,I}W.T1 â¡ â{b,I}W.T2 & - U2 = â{a}V2.T2. -#a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 â¦ H) -H * -[ #V2 #T #T2 #HV12 #HT1 #HT2 - lapply (tps_inv_refl_SO2 â¦ HT2 ???) -HT2 // /3 width=4/ -| #T2 #_ #_ #_ #H destruct -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpr_tpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpr_tpr.etc deleted file mode 100644 index a733345bf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpr_tpr.etc +++ /dev/null @@ -1,261 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âI. ââX. âª{I} â¡ X & âª{I} â¡ X. -/2 width=3/ qed. - -fact tpr_conf_flat_flat: - âI,V0,V1,T0,T1,V2,T2. ( - âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â V0 â¡ V2 â T0 â¡ T1 â T0 â¡ T2 â - ââT0. â{I} V1. T1 â¡ T0 & â{I} V2. T2 â¡ T0. -#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 -elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 // #V #HV1 #HV2 -elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // /3 width=5/ -qed. - -fact tpr_conf_flat_beta: - âa,V0,V1,T1,V2,W0,U0,T2. ( - âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{U0} + 1) + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â V0 â¡ V2 â - U0 â¡ T2 â â{a}W0. U0 â¡ T1 â - ââX. âV1. T1 â¡ X & â{a}V2. T2 â¡ X. -#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H -elim (tpr_inv_abst1 â¦ H) -H #W1 #U1 #HW01 #HU01 #H destruct -elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH â¦ HT02 â¦ 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: - âa,V0,V1,T1,V2,V,W0,W2,U0,U2. ( - âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{U0} + 1) + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â V0 â¡ V2 â â§[O,1] V2 â¡ V â - W0 â¡ W2 â U0 â¡ U2 â â{a}W0. U0 â¡ T1 â - ââX. âV1. T1 â¡ X & â{a}W2. âV. U2 â¡ X. -#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H -elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2 -elim (lift_total VV 0 1) #VVV #HVV -lapply (tpr_lift â¦ HVV2 â¦ HV2 â¦ HVV) #HVVV -elim (tpr_inv_abbr1 â¦ H) -H * -(* case 1: delta *) -[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct - elim (IH â¦ HW02 â¦ HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2 - elim (IH â¦ HU02 â¦ HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2 - elim (tpr_tps_conf_bind â¦ HWW2 HUUU2 â¦ 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 â¦ HU01 â¦ HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2 - elim (tpr_inv_lift1 â¦ HU1 â¦ HTU1) -U1 #UU #HUU #HT1UU #H destruct - @(ex2_intro â¦ (âVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *) -] -qed. - -fact tpr_conf_flat_cast: - âX2,V0,V1,T0,T1. ( - âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â T0 â¡ T1 â T0 â¡ X2 â - ââX. âV1. T1 â¡ X & X2 â¡ X. -#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02 -elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // /3 width=3/ -qed. - -fact tpr_conf_beta_beta: - âa. âW0:term. âV0,V1,T0,T1,V2,T2. ( - âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{T0} + 1) + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â V0 â¡ V2 â T0 â¡ T1 â T0 â¡ T2 â - ââX. â{a}V1. T1 â¡X & â{a}V2. T2 â¡ X. -#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 -elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ -elim (IH â¦ HT01 â¦ 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: - âa,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. ( - âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â V0 â¡ V2 â T0 â¡ T1 â T0 â¡ T2 â - â. â{I1} V1 â¢ T1 â¶ [O, 1] TT1 â - â. â{I1} V2 â¢ T2 â¶ [O, 1] TT2 â - ââX. â{a,I1} V1. TT1 â¡ X & â{a,I1} V2. TT2 â¡ X. -#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2 -elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 // #V #HV1 #HV2 -elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // #T #HT1 #HT2 -elim (tpr_tps_conf_bind â¦ HV1 HT1 â¦ HTT1) -T1 #U1 #TTU1 #HTU1 -elim (tpr_tps_conf_bind â¦ HV2 HT2 â¦ HTT2) -T2 #U2 #TTU2 #HTU2 -elim (tps_conf_eq â¦ HTU1 â¦ 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: - âX2,V0,V1,T0,T1,TT1,T2. ( - âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â T0 â¡ T1 â â. âV1 â¢ T1 â¶ [O,1] TT1 â - T0 â¡ T2 â â§[O, 1] X2 â¡ T2 â - ââX. +âV1. TT1 â¡ X & X2 â¡ X. -#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2 -elim (IH â¦ HT01 â¦ HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2 -elim (tpr_tps_conf_bind ? ? V1 â¦ HT1 HTT1) -T1 // #TT #HTT1 #HTT -elim (tpr_inv_lift1 â¦ HT2 â¦ HXT2) -T2 #X #HXT #HX2 -lapply (tps_inv_lift1_eq â¦ HTT â¦ HXT) -HTT #H destruct /3 width=3/ -qed. - -(* Basic_1: was: pr0_upsilon_upsilon *) -fact tpr_conf_theta_theta: - âa,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. ( - âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{T0} + 1) + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - V0 â¡ V1 â V0 â¡ V2 â W0 â¡ W1 â W0 â¡ W2 â T0 â¡ T1 â T0 â¡ T2 â - â§[O, 1] V1 â¡ VV1 â â§[O, 1] V2 â¡ VV2 â - ââX. â{a}W1. âVV1. T1 â¡ X & â{a}W2. âVV2. T2 â¡ X. -#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2 -elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH â¦ HW01 â¦ HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2 -elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2 -elim (lift_total V 0 1) #VV #HVV -lapply (tpr_lift â¦ HV1 â¦ HVV1 â¦ HVV) -V1 #HVV1 -lapply (tpr_lift â¦ HV2 â¦ HVV2 â¦ 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: - âV0:term. âX2,TT0,T0,T1,TT2. ( - âX0:term. â¯{X0} < â¯{V0} + â¯{TT0} + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - TT0 â¡ T0 â â§[O, 1] T1 â¡ T0 â - TT0 â¡ TT2 â â§[O, 1] X2 â¡ TT2 â - ââX. T1 â¡ X & X2 â¡ X. -#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2 -elim (IH â¦ HTT0 â¦ HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2 -elim (tpr_inv_lift1 â¦ HT0 â¦ HT10) -T0 #T0 #HT0 #HT10 -elim (tpr_inv_lift1 â¦ HTT2 â¦ HXTT2) -TT2 #TT2 #HTT2 #HXTT2 -lapply (lift_inj â¦ HTT2 â¦ HT0) -HTT2 #H destruct /2 width=3/ -qed. - -fact tpr_conf_tau_tau: - âV0,T0:term. âX2,T1. ( - âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â - âX1,X2. X0 â¡ X1 â X0 â¡ X2 â - ââX. X1 â¡ X & X2 â¡ X - ) â - T0 â¡ T1 â T0 â¡ X2 â - ââX. T1 â¡ X & X2 â¡ X. -#V0 #T0 #X2 #T1 #IH #HT01 #HT02 -elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // /2 width=3/ -qed. - -(* Confluence ***************************************************************) - -(* Basic_1: was: pr0_confluence *) -theorem tpr_conf: âT0:term. âT1,T2. T0 â¡ T1 â T0 â¡ T2 â - ââT. T1 â¡ T & T2 â¡ T. -#T0 @(f_ind â¦ tw â¦ T0) -T0 #n #IH * -[ #I #_ #X1 #X2 #H1 #H2 -n - >(tpr_inv_atom1 â¦ H1) -X1 - >(tpr_inv_atom1 â¦ H2) -X2 -(* case 1: atom, atom *) - // -| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2 - [ elim (tpr_inv_bind1 â¦ H1) -H1 * - [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1 - | #T1 #HT01 #HXT1 #H11 #H12 - ] - elim (tpr_inv_bind1 â¦ 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 â¦ 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 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs.etc deleted file mode 100644 index 492b39041..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs.etc +++ /dev/null @@ -1,94 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¡ * 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 â TC â¦ tpr. - -interpretation "context-free parallel computation (term)" - 'PRedStar T1 T2 = (tprs T1 T2). - -(* Basic eliminators ********************************************************) - -lemma tprs_ind: âT1. âR:predicate term. R T1 â - (âT,T2. T1 â¡* T â T â¡ T2 â R T â R T2) â - âT2. T1 â¡* T2 â R T2. -#T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // -qed-. - -lemma tprs_ind_dx: âT2. âR:predicate term. R T2 â - (âT1,T. T1 â¡ T â T â¡* T2 â R T â R T1) â - âT1. T1 â¡* T2 â R T1. -#T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma tprs_refl: reflexive â¦ tprs. -/2 width=1/ qed. - -lemma tpr_tprs: âT1,T2. T1 â¡ T2 â T2 â¡* T2. -/2 width=1/ qed. - -lemma tprs_strap1: âT1,T,T2. T1 â¡* T â T â¡ T2 â T1 â¡* T2. -/2 width=3/ qed. - -lemma tprs_strap2: âT1,T,T2. T1 â¡ T â T â¡* T2 â T1 â¡* T2. -/2 width=3/ qed. - -(* Basic_1: was only: pr1_head_1 *) -lemma tprs_pair_sn: âI,T1,T2. T1 â¡ T2 â âV1,V2. V1 â¡* V2 â - â¡{I} V1. T1 â¡* â¡{I} V2. T2. -* [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind â¦ H) -V2 -[1,3: /3 width=1/ -|2,4: #V #V2 #_ #HV2 #IHV1 - @(tprs_strap1 â¦ IHV1) -IHV1 /2 width=1/ -] -qed. - -(* Basic_1: was only: pr1_head_2 *) -lemma tprs_pair_dx: âI,V1,V2. V1 â¡ V2 â âT1,T2. T1 â¡* T2 â - â¡{I} V1. T1 â¡* â¡{I} V2. T2. -* [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind â¦ H) -T2 -[1,3: /3 width=1/ -|2,4: #T #T2 #_ #HT2 #IHT1 - @(tprs_strap1 â¦ IHT1) -IHT1 /2 width=1/ -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma tprs_inv_atom1: âU2,k. âk â¡* U2 â U2 = âk. -#U2 #k #H @(tprs_ind â¦ H) -U2 // -#U #U2 #_ #HU2 #IHU1 destruct ->(tpr_inv_atom1 â¦ HU2) -HU2 // -qed-. - -lemma tprs_inv_cast1: âW1,T1,U2. âW1.T1 â¡* U2 â T1 â¡* U2 â¨ - ââW2,T2. W1 â¡* W2 & T1 â¡* T2 & U2 = âW2.T2. -#W1 #T1 #U2 #H @(tprs_ind â¦ H) -U2 /3 width=5/ -#U #U2 #_ #HU2 * /3 width=3/ * -#W #T #HW1 #HT1 #H destruct -elim (tpr_inv_cast1 â¦ HU2) -HU2 /3 width=3/ * -#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs_lift.etc deleted file mode 100644 index d0d173470..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs_lift.etc +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âa,V1,T1,U2. â{a}V1. T1 â¡* U2 â - ââV2,T2. V1 â¡* V2 & T1 â¡* T2 & U2 = â{a}V2. T2. -#a #V1 #T1 #U2 #H @(tprs_ind â¦ H) -U2 /2 width=5/ -#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -elim (tpr_inv_abst1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ -qed-. - -lemma tprs_inv_abst: âa,V1,V2,T1,T2. â{a}V1. T1 â¡* â{a}V2. T2 â - V1 â¡* V2 â§ T1 â¡* T2. -#a #V1 #V2 #T1 #T2 #H -elim (tprs_inv_abst1 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs_tprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs_tprs.etc deleted file mode 100644 index 87a4a71ab..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs_tprs.etc +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âT1,T. T â¡* T1 â âT2. T â¡ T2 â - ââT0. T1 â¡ T0 & T2 â¡* T0. -/3 width=3 by TC_strip1, tpr_conf/ qed. - -(* Main propertis ***********************************************************) - -(* Basic_1: was: pr1_confluence *) -theorem tprs_conf: confluent â¦ tprs. -/3 width=3/ qed. - -(* Basic_1: was: pr1_t *) -theorem tprs_trans: Transitive â¦ tprs. -/2 width=3/ qed. - -(* Basic_1: was: pr1_comp *) -lemma tprs_pair: âI,V1,V2. V1 â¡* V2 â âT1,T2. T1 â¡* T2 â - â¡{I} V1. T1 â¡* â¡{I} V2. T2. -#I #V1 #V2 #H @(tprs_ind â¦ H) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 -@(tprs_trans â¦ (â¡{I}V.T2)) /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tps.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tps.etc deleted file mode 100644 index 7d89243d5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tps.etc +++ /dev/null @@ -1,288 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 T1 break â¶ [ 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 â nat â lenv â relation term â -| tps_atom : âL,I,d,e. tps d e L (âª{I}) (âª{I}) -| tps_subst: âL,K,V,W,i,d,e. d â¤ i â i < d + e â - â©[0, i] L â¡ K. âV â â§[0, i + 1] V â¡ W â tps d e L (#i) W -| tps_bind : âL,a,I,V1,V2,T1,T2,d,e. - tps d e L V1 V2 â tps (d + 1) e (L. â{I} V2) T1 T2 â - tps d e L (â{a,I} V1. T1) (â{a,I} V2. T2) -| tps_flat : âL,I,V1,V2,T1,T2,d,e. - tps d e L V1 V2 â tps d e L T1 T2 â - tps d e L (â{I} V1. T1) (â{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: âL1,T1,T2,d,e. L1 â¢ T1 â¶ [d, e] T2 â - âL2. L2 â [d, e] L1 â L2 â¢ T1 â¶ [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 â¦ HL12 â¦ HLK1 ? ?) -HL12 -HLK1 // /2 width=4/ -| /4 width=1/ -| /3 width=1/ -] -qed. - -lemma tps_refl: âT,L,d,e. L â¢ T â¶ [d, e] T. -#T elim T -T // -#I elim I -I /2 width=1/ -qed. - -(* Basic_1: was: subst1_ex *) -lemma tps_full: âK,V,T1,L,d. â©[0, d] L â¡ (K. âV) â - ââT2,T. L â¢ T1 â¶ [d, 1] T2 & â§[d, 1] T â¡ 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 â¦ HVW i i ? ? ?) // /3 width=4/ -| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK - elim (IHW1 â¦ HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L. â{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/ - | elim (IHU1 â¦ HLK) -IHU1 -HLK /3 width=8/ - ] -] -qed. - -lemma tps_weak: âL,T1,T2,d1,e1. L â¢ T1 â¶ [d1, e1] T2 â - âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â - L â¢ T1 â¶ [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 â¦ Hd12 â¦ Hid1) -Hd12 -Hid1 #Hid2 - lapply (lt_to_le_to_lt â¦ Hide1 â¦ Hde12) -Hide1 /2 width=4/ -| /4 width=3/ -| /4 width=1/ -] -qed. - -lemma tps_weak_top: âL,T1,T2,d,e. - L â¢ T1 â¶ [d, e] T2 â L â¢ T1 â¶ [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 â¦ HLK) #Hi - lapply (le_to_lt_to_lt â¦ Hdi Hi) /3 width=4/ -| normalize /2 width=1/ -| /2 width=1/ -] -qed. - -lemma tps_weak_full: âL,T1,T2,d,e. - L â¢ T1 â¶ [d, e] T2 â L â¢ T1 â¶ [0, |L|] T2. -#L #T1 #T2 #d #e #HT12 -lapply (tps_weak â¦ HT12 0 (d + e) ? ?) -HT12 // #HT12 -lapply (tps_weak_top â¦ HT12) // -qed. - -lemma tps_split_up: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â âi. d â¤ i â i â¤ d + e â - ââT. L â¢ T1 â¶ [d, i - d] T & L â¢ T â¶ [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 â¢ (% â ?); // -Hdj /3 width=4/ - | -Hdi -Hdj #Hij - lapply (plus_minus_m_m â¦ 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 â¦ HT1 (L. â{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: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â - âi. d â¤ i â i â¤ d + e â - ââT. L â¢ T1 â¶ [i, d + e - i] T & - L â¢ T â¶ [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 â¢ (% â ?); // -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 â¦ HT1 (L. â{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: âK,T1,T2,d,e. K â¢ T1 â¶ [d, e] T2 â - âL. L @@ K â¢ T1 â¶ [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 â¦ HK0) #H -@(tps_subst â¦ (L@@K0) â¦ HVW) // (**) (* /3/ does not work *) -@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2/ -qed. - -(* Basic inversion lemmas ***************************************************) - -fact tps_inv_atom1_aux: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â âI. T1 = âª{I} â - T2 = âª{I} â¨ - ââK,V,i. d â¤ i & i < d + e & - â©[O, i] L â¡ K. âV & - â§[O, i + 1] V â¡ 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: âL,T2,I,d,e. L â¢ âª{I} â¶ [d, e] T2 â - T2 = âª{I} â¨ - ââK,V,i. d â¤ i & i < d + e & - â©[O, i] L â¡ K. âV & - â§[O, i + 1] V â¡ T2 & - I = LRef i. -/2 width=3/ qed-. - - -(* Basic_1: was: subst1_gen_sort *) -lemma tps_inv_sort1: âL,T2,k,d,e. L â¢ âk â¶ [d, e] T2 â T2 = âk. -#L #T2 #k #d #e #H -elim (tps_inv_atom1 â¦ H) -H // -* #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -(* Basic_1: was: subst1_gen_lref *) -lemma tps_inv_lref1: âL,T2,i,d,e. L â¢ #i â¶ [d, e] T2 â - T2 = #i â¨ - ââK,V. d â¤ i & i < d + e & - â©[O, i] L â¡ K. âV & - â§[O, i + 1] V â¡ T2. -#L #T2 #i #d #e #H -elim (tps_inv_atom1 â¦ H) -H /2 width=1/ -* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/ -qed-. - -lemma tps_inv_gref1: âL,T2,p,d,e. L â¢ Â§p â¶ [d, e] T2 â T2 = Â§p. -#L #T2 #p #d #e #H -elim (tps_inv_atom1 â¦ H) -H // -* #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -fact tps_inv_bind1_aux: âd,e,L,U1,U2. L â¢ U1 â¶ [d, e] U2 â - âa,I,V1,T1. U1 = â{a,I} V1. T1 â - ââV2,T2. L â¢ V1 â¶ [d, e] V2 & - L. â{I} V2 â¢ T1 â¶ [d + 1, e] T2 & - U2 = â{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: âd,e,L,a,I,V1,T1,U2. L â¢ â{a,I} V1. T1 â¶ [d, e] U2 â - ââV2,T2. L â¢ V1 â¶ [d, e] V2 & - L. â{I} V2 â¢ T1 â¶ [d + 1, e] T2 & - U2 = â{a,I} V2. T2. -/2 width=3/ qed-. - -fact tps_inv_flat1_aux: âd,e,L,U1,U2. L â¢ U1 â¶ [d, e] U2 â - âI,V1,T1. U1 = â{I} V1. T1 â - ââV2,T2. L â¢ V1 â¶ [d, e] V2 & L â¢ T1 â¶ [d, e] T2 & - U2 = â{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: âd,e,L,I,V1,T1,U2. L â¢ â{I} V1. T1 â¶ [d, e] U2 â - ââV2,T2. L â¢ V1 â¶ [d, e] V2 & L â¢ T1 â¶ [d, e] T2 & - U2 = â{I} V2. T2. -/2 width=3/ qed-. - -fact tps_inv_refl_O2_aux: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â e = 0 â 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 â¦ Hdi â¦ Hide) -Hdi -Hide shift_append_assoc normalize #H - elim (tps_inv_bind1 â¦ H) -H - #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tps_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tps_lift.etc deleted file mode 100644 index 1d65d8d63..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tps_lift.etc +++ /dev/null @@ -1,294 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,T2,d,e1. L â¢ T1 â¶ [d, e1] T2 â âe2. e1 = e2 + 1 â - âK,V. â©[0, d] L â¡ K. âV â L â¢ T1 â¶ [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 â¦ Hdi ?) /2 width=1/ #H destruct - lapply (ldrop_mono â¦ HLK0 â¦ HLK) #H destruct - | -V -Hdi /2 width=4/ - ] -| /4 width=3/ -| /3 width=3/ -] -qed. - -lemma tps_inv_S2: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e + 1] T2 â - âK,V. â©[0, d] L â¡ K. âV â L â¢ T1 â¶ [d + 1, e] T2. -/2 width=3/ qed-. - -lemma tps_inv_refl_SO2: âL,T1,T2,d. L â¢ T1 â¶ [d, 1] T2 â - âK,V. â©[0, d] L â¡ K. âV â T1 = T2. -#L #T1 #T2 #d #HT12 #K #V #HLK -lapply (tps_inv_S2 â¦ T1 T2 â¦ 0 â¦ HLK) -K // -HT12 #HT12 -lapply (tps_inv_refl_O2 â¦ HT12) -HT12 // -qed-. - -(* Relocation properties ****************************************************) - -(* Basic_1: was: subst1_lift_lt *) -lemma tps_lift_le: âK,T1,T2,dt,et. K â¢ T1 â¶ [dt, et] T2 â - âL,U1,U2,d,e. â©[d, e] L â¡ K â - â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â - dt + et â¤ d â - L â¢ U1 â¶ [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 â¦ H1 â¦ 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 â¦ Hidet â¦ Hdetd) -Hdetd #Hid - lapply (lift_inv_lref1_lt â¦ H â¦ Hid) -H #H destruct - elim (lift_trans_ge â¦ HVW â¦ HWU2 ?) -W // (lift_mono â¦ HVY â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ -] -qed. - -lemma tps_lift_be: âK,T1,T2,dt,et. K â¢ T1 â¶ [dt, et] T2 â - âL,U1,U2,d,e. â©[d, e] L â¡ K â - â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â - dt â¤ d â d â¤ dt + et â - L â¢ U1 â¶ [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 â¦ H1 â¦ 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 â¦ H) -H * #Hid #H destruct - [ -Hdtd - lapply (lt_to_le_to_lt â¦ (dt+et+e) Hidet ?) // -Hidet #Hidete - elim (lift_trans_ge â¦ HVW â¦ HWU2 ?) -W // (lift_mono â¦ HVY â¦ HVW) -V #H destruct /2 width=4/ - | -Hdti - lapply (transitive_le â¦ Hdtd Hid) -Hdtd #Hdti - lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ -] -qed. - -(* Basic_1: was: subst1_lift_ge *) -lemma tps_lift_ge: âK,T1,T2,dt,et. K â¢ T1 â¶ [dt, et] T2 â - âL,U1,U2,d,e. â©[d, e] L â¡ K â - â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â - d â¤ dt â - L â¢ U1 â¶ [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 â¦ H1 â¦ 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 â¦ Hddt â¦ Hdti) -Hddt #Hid - lapply (lift_inv_lref1_ge â¦ H â¦ Hid) -H #H destruct - lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/ -] -qed. - -(* Basic_1: was: subst1_gen_lift_lt *) -lemma tps_inv_lift1_le: âL,U1,U2,dt,et. L â¢ U1 â¶ [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt + et â¤ d â - ââT2. K â¢ T1 â¶ [dt, et] T2 & â§[d, e] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3/ - | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 â¦ 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 â¦ Hidet â¦ Hdetd) -Hdetd #Hid - lapply (lift_inv_lref2_lt â¦ H â¦ Hid) -H #H destruct - elim (ldrop_conf_lt â¦ HLK â¦ HLKV ?) -L // #L #U #HKL #_ #HUV - elim (lift_trans_le â¦ HUV â¦ HVW ?) -V // >minus_plus minus_plus plus_minus // 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 â¦ H) -H #W1 #T1 #HWV1 #HTU1 #H destruct - elim (IHV12 â¦ HLK â¦ HWV1 ? ?) -V1 // #W2 #HW12 #HWV2 - elim (IHU12 â¦ 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 â¦ H) -H #W1 #T1 #HWV1 #HTU1 #H destruct - elim (IHV12 â¦ HLK â¦ HWV1 ? ?) -V1 // - elim (IHU12 â¦ HLK â¦ HTU1 ? ?) -U1 -HLK // /3 width=5/ -] -qed. - -(* Basic_1: was: subst1_gen_lift_ge *) -lemma tps_inv_lift1_ge: âL,U1,U2,dt,et. L â¢ U1 â¶ [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - d + e â¤ dt â - ââT2. K â¢ T1 â¶ [dt - e, et] T2 & â§[d, e] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3/ - | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 â¦ 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 â¦ Hdedt â¦ Hdti) #Hdei - elim (le_inv_plus_l â¦ Hdedt) -Hdedt #_ #Hedt - elim (le_inv_plus_l â¦ Hdei) #Hdie #Hei - lapply (lift_inv_lref2_ge â¦ H â¦ Hdei) -H #H destruct - lapply (ldrop_conf_ge â¦ HLK â¦ HLKV ?) -L // #HKV - elim (lift_split â¦ HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie - #V0 #HV10 >plus_minus // 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 â¦ H) -H #W1 #T1 #HWV1 #HTU1 #H destruct - elim (le_inv_plus_l â¦ Hdetd) #_ #Hedt - elim (IHV12 â¦ HLK â¦ HWV1 ?) -V1 // #W2 #HW12 #HWV2 - elim (IHU12 â¦ HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ] - IHV12 // >IHT12 // -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_flat2 â¦ 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: âL,U1,U2,dt,et. L â¢ U1 â¶ [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - d â¤ dt â dt â¤ d + e â d + e â¤ dt + et â - ââT2. K â¢ T1 â¶ [d, dt + et - (d + e)] T2 & â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet -elim (tps_split_up â¦ HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 -lapply (tps_weak â¦ HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 -lapply (tps_inv_lift1_eq â¦ HU1 â¦ HTU1) -HU1 #HU1 destruct -elim (tps_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1 ?) -U -L // commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU -lapply (tps_weak â¦ HU2 d e ? ?) -HU2 // [ >commutative_plus (lift_mono â¦ HVT1 â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - lapply (tps_lsubr_trans â¦ HT02 (L. â{I} V1) ?) -HT02 /2 width=1/ #HT02 - elim (IHV01 â¦ HV02) -V0 #V #HV1 #HV2 - elim (IHT01 â¦ HT02) -T0 #T #HT1 #HT2 - lapply (tps_lsubr_trans â¦ HT1 (L. â{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubr_trans â¦ HT2 (L. â{I} V) ?) -HT2 /3 width=5/ -| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX - elim (tps_inv_flat1 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02) -V0 - elim (IHT01 â¦ HT02) -T0 /3 width=5/ -] -qed. - -(* Basic_1: was: subst1_confluence_neq *) -theorem tps_conf_neq: âL1,T0,T1,d1,e1. L1 â¢ T0 â¶ [d1, e1] T1 â - âL2,T2,d2,e2. L2 â¢ T0 â¶ [d2, e2] T2 â - (d1 + e1 â¤ d2 â¨ d2 + e2 â¤ d1) â - ââT. L2 â¢ T1 â¶ [d2, e2] T & L1 â¢ T2 â¶ [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 â¦ H1) -H1 - [ #H destruct /3 width=6/ - | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded - [ -Hd1 -Hde2 - lapply (transitive_le â¦ Hded Hd2) -Hded -Hd2 #H - lapply (lt_to_le_to_lt â¦ Hde1 H) -Hde1 -H #H - elim (lt_refl_false â¦ H) - | -Hd2 -Hde1 - lapply (transitive_le â¦ Hded Hd1) -Hded -Hd1 #H - lapply (lt_to_le_to_lt â¦ Hde2 H) -Hde2 -H #H - elim (lt_refl_false â¦ H) - ] - ] -| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H - elim (tps_inv_bind1 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02 H) -V0 #V #HV1 #HV2 - elim (IHT01 â¦ HT02 ?) -T0 - [ -H #T #HT1 #HT2 - lapply (tps_lsubr_trans â¦ HT1 (L2. â{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubr_trans â¦ HT2 (L1. â{I} V) ?) -HT2 /2 width=1/ /3 width=5/ - | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in â¢ (? ? %); 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02 H) -V0 - elim (IHT01 â¦ 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: âL,T1,T0,d,e. L â¢ T1 â¶ [d, e] T0 â - âT2. L â¢ T0 â¶ [d, 1] T2 â 1 â¤ e â - L â¢ T1 â¶ [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 â¦ H) -H - [ #H destruct // - | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct - lapply (lt_to_le_to_lt â¦ (d + e) Hide2 ?) /2 width=4/ - ] -| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He - lapply (tps_weak â¦ HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2 - <(tps_inv_lift1_eq â¦ HVT2 â¦ HVW) -HVT2 /2 width=4/ -| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He - elim (tps_inv_bind1 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct - lapply (tps_lsubr_trans â¦ HT02 (L. â{I} V0) ?) -HT02 /2 width=1/ #HT02 - lapply (IHT10 â¦ HT02 He) -T0 #HT12 - lapply (tps_lsubr_trans â¦ HT12 (L. â{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 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/ -] -qed. - -theorem tps_trans_down: âL,T1,T0,d1,e1. L â¢ T1 â¶ [d1, e1] T0 â - âT2,d2,e2. L â¢ T0 â¶ [d2, e2] T2 â d2 + e2 â¤ d1 â - ââT. L â¢ T1 â¶ [d2, e2] T & L â¢ T â¶ [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 â¦ Hde2d1 Hdi1) -Hde2d1 #Hde2i1 - lapply (tps_weak â¦ HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2 - <(tps_inv_lift1_eq â¦ HWT2 â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - lapply (tps_lsubr_trans â¦ HT02 (L. â{I} V0) ?) -HT02 /2 width=1/ #HT02 - elim (IHV10 â¦ HV02 ?) -IHV10 -HV02 // #V - elim (IHT10 â¦ HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2 - lapply (tps_lsubr_trans â¦ HT1 (L. â{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubr_trans â¦ HT2 (L. â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV10 â¦ HV02 ?) -V0 // - elim (IHT10 â¦ HT02 ?) -T0 // /3 width=6/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss.etc deleted file mode 100644 index 433718803..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss.etc +++ /dev/null @@ -1,188 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 T1 break â¶ * [ 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 â nat â lenv â relation term â - Î»d,e,L. TC â¦ (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: âd,e,L,T1. âR:predicate term. R T1 â - (âT,T2. L â¢ T1 â¶* [d, e] T â L â¢ T â¶ [d, e] T2 â R T â R T2) â - âT2. L â¢ T1 â¶* [d, e] T2 â R T2. -#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // -qed-. - -lemma tpss_ind_dx: âd,e,L,T2. âR:predicate term. R T2 â - (âT1,T. L â¢ T1 â¶ [d, e] T â L â¢ T â¶* [d, e] T2 â R T â R T1) â - âT1. L â¢ T1 â¶* [d, e] T2 â R T1. -#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma tps_tpss: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â L â¢ T1 â¶* [d, e] T2. -/2 width=1/ qed. - -lemma tpss_strap1: âL,T1,T,T2,d,e. - L â¢ T1 â¶* [d, e] T â L â¢ T â¶ [d, e] T2 â L â¢ T1 â¶* [d, e] T2. -/2 width=3/ qed. - -lemma tpss_strap2: âL,T1,T,T2,d,e. - L â¢ T1 â¶ [d, e] T â L â¢ T â¶* [d, e] T2 â L â¢ T1 â¶* [d, e] T2. -/2 width=3/ qed. - -lemma tpss_lsubr_trans: âL1,T1,T2,d,e. L1 â¢ T1 â¶* [d, e] T2 â - âL2. L2 â [d, e] L1 â L2 â¢ T1 â¶* [d, e] T2. -/3 width=3/ qed. - -lemma tpss_refl: âd,e,L,T. L â¢ T â¶* [d, e] T. -/2 width=1/ qed. - -lemma tpss_bind: âL,V1,V2,d,e. L â¢ V1 â¶* [d, e] V2 â - âa,I,T1,T2. L. â{I} V2 â¢ T1 â¶* [d + 1, e] T2 â - L â¢ â{a,I} V1. T1 â¶* [d, e] â{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 â¦ HT12 (L. â{I} V) ?) -HT12 /2 width=1/ #HT12 - lapply (IHV a â¦ HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) -] -qed. - -lemma tpss_flat: âL,I,V1,V2,T1,T2,d,e. - L â¢ V1 â¶* [d, e] V2 â L â¢ T1 â¶* [d, e] T2 â - L â¢ â{I} V1. T1 â¶* [d, e] â{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 â¦ HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) -] -qed. - -lemma tpss_weak: âL,T1,T2,d1,e1. L â¢ T1 â¶* [d1, e1] T2 â - âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â - L â¢ T1 â¶* [d2, e2] T2. -#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind â¦ H) -T2 -[ // -| #T #T2 #_ #HT12 #IHT - lapply (tps_weak â¦ HT12 â¦ Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/ -] -qed. - -lemma tpss_weak_top: âL,T1,T2,d,e. - L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¶* [d, |L| - d] T2. -#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 -[ // -| #T #T2 #_ #HT12 #IHT - lapply (tps_weak_top â¦ HT12) -HT12 /2 width=3/ -] -qed. - -lemma tpss_weak_full: âL,T1,T2,d,e. - L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¶* [0, |L|] T2. -#L #T1 #T2 #d #e #HT12 -lapply (tpss_weak â¦ HT12 0 (d + e) ? ?) -HT12 // #HT12 -lapply (tpss_weak_top â¦ HT12) // -qed. - -lemma tpss_append: âK,T1,T2,d,e. K â¢ T1 â¶* [d, e] T2 â - âL. L @@ K â¢ T1 â¶* [d, e] T2. -#K #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // /3 width=3/ -qed. - -(* Basic inversion lemmas ***************************************************) - -(* Note: this can be derived from tpss_inv_atom1 *) -lemma tpss_inv_sort1: âL,T2,k,d,e. L â¢ âk â¶* [d, e] T2 â T2 = âk. -#L #T2 #k #d #e #H @(tpss_ind â¦ H) -T2 -[ // -| #T #T2 #_ #HT2 #IHT destruct - >(tps_inv_sort1 â¦ HT2) -HT2 // -] -qed-. - -(* Note: this can be derived from tpss_inv_atom1 *) -lemma tpss_inv_gref1: âL,T2,p,d,e. L â¢ Â§p â¶* [d, e] T2 â T2 = Â§p. -#L #T2 #p #d #e #H @(tpss_ind â¦ H) -T2 -[ // -| #T #T2 #_ #HT2 #IHT destruct - >(tps_inv_gref1 â¦ HT2) -HT2 // -] -qed-. - -lemma tpss_inv_bind1: âd,e,L,a,I,V1,T1,U2. L â¢ â{a,I} V1. T1 â¶* [d, e] U2 â - ââV2,T2. L â¢ V1 â¶* [d, e] V2 & - L. â{I} V2 â¢ T1 â¶* [d + 1, e] T2 & - U2 = â{a,I} V2. T2. -#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind â¦ H) -U2 -[ /2 width=5/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (tps_inv_bind1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H - lapply (tpss_lsubr_trans â¦ HT1 (L. â{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ -] -qed-. - -lemma tpss_inv_flat1: âd,e,L,I,V1,T1,U2. L â¢ â{I} V1. T1 â¶* [d, e] U2 â - ââV2,T2. L â¢ V1 â¶* [d, e] V2 & L â¢ T1 â¶* [d, e] T2 & - U2 = â{I} V2. T2. -#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind â¦ H) -U2 -[ /2 width=5/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (tps_inv_flat1 â¦ HU2) -HU2 /3 width=5/ -] -qed-. - -lemma tpss_inv_refl_O2: âL,T1,T2,d. L â¢ T1 â¶* [d, 0] T2 â T1 = T2. -#L #T1 #T2 #d #H @(tpss_ind â¦ H) -T2 -[ // -| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 â¦ HT2) -HT2 // -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma tpss_fwd_tw: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â â¯{T1} â¤ â¯{T2}. -#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 -lapply (tps_fwd_tw â¦ HT2) -HT2 #HT2 -@(transitive_le â¦ IHT1) // -qed-. - -lemma tpss_fwd_shift1: âL,L1,T1,T,d,e. L â¢ L1 @@ T1 â¶*[d, e] T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#L #L1 #T1 #T #d #e #H @(tpss_ind â¦ H) -T -[ /2 width=4/ -| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct - elim (tps_fwd_shift1 â¦ H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_alt.etc deleted file mode 100644 index f7670eaa8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_alt.etc +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 T1 break â¶ â¶ * [ 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 â nat â lenv â relation term â -| tpssa_atom : âL,I,d,e. tpssa d e L (âª{I}) (âª{I}) -| tpssa_subst: âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â - â©[0, i] L â¡ K. âV1 â tpssa 0 (d + e - i - 1) K V1 V2 â - â§[0, i + 1] V2 â¡ W2 â tpssa d e L (#i) W2 -| tpssa_bind : âL,a,I,V1,V2,T1,T2,d,e. - tpssa d e L V1 V2 â tpssa (d + 1) e (L. â{I} V2) T1 T2 â - tpssa d e L (â{a,I} V1. T1) (â{a,I} V2. T2) -| tpssa_flat : âL,I,V1,V2,T1,T2,d,e. - tpssa d e L V1 V2 â tpssa d e L T1 T2 â - tpssa d e L (â{I} V1. T1) (â{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: âL1,T1,T2,d,e. L1 â¢ T1 â¶â¶* [d, e] T2 â - âL2. L2 â [d, e] L1 â L2 â¢ T1 â¶â¶* [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 â¦ HL12 â¦ HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ -| /4 width=1/ -| /3 width=1/ -] -qed. - -lemma tpssa_refl: âT,L,d,e. L â¢ T â¶â¶* [d, e] T. -#T elim T -T // -#I elim I -I /2 width=1/ -qed. - -lemma tpssa_tps_trans: âL,T1,T,d,e. L â¢ T1 â¶â¶* [d, e] T â - âT2. L â¢ T â¶ [d, e] T2 â L â¢ T1 â¶â¶* [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 â¦ H) -H // * /2 width=6/ -| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H - lapply (ldrop_fwd_ldrop2 â¦ HLK) #H0LK - lapply (tps_weak â¦ H 0 (d+e) ? ?) -H // #H - elim (tps_inv_lift1_be â¦ H â¦ H0LK â¦ HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/ -| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H - elim (tps_inv_bind1 â¦ H) -H #V2 #T2 #HV2 #HT2 #H destruct - lapply (tps_lsubr_trans â¦ HT2 (L.â{I}V) ?) -HT2 /2 width=1/ #HT2 - lapply (IHV1 â¦ HV2) -IHV1 -HV2 #HV12 - lapply (IHT1 â¦ HT2) -IHT1 -HT2 #HT12 - lapply (tpssa_lsubr_trans â¦ HT12 (L.â{I}V2) ?) -HT12 /2 width=1/ -| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H - elim (tps_inv_flat1 â¦ H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/ -] -qed. - -lemma tpss_tpssa: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¶â¶* [d, e] T2. -#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // /2 width=3/ -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma tpssa_tpss: âL,T1,T2,d,e. L â¢ T1 â¶â¶* [d, e] T2 â L â¢ T1 â¶* [d, e] T2. -#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/ -qed-. - -lemma tpss_ind_alt: âR:natânatâlenvârelation term. - (âL,I,d,e. R d e L (âª{I}) (âª{I})) â - (âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â - â©[O, i] L â¡ K.âV1 â K â¢ V1 â¶* [O, d + e - i - 1] V2 â - â§[O, i + 1] V2 â¡ W2 â R O (d+e-i-1) K V1 V2 â R d e L (#i) W2 - ) â - (âL,a,I,V1,V2,T1,T2,d,e. L â¢ V1 â¶* [d, e] V2 â - L.â{I}V2 â¢ T1 â¶* [d + 1, e] T2 â R d e L V1 V2 â - R (d+1) e (L.â{I}V2) T1 T2 â R d e L (â{a,I}V1.T1) (â{a,I}V2.T2) - ) â - (âL,I,V1,V2,T1,T2,d,e. L â¢ V1 â¶* [d, e] V2 â - L â¢ T1 â¶* [d, e] T2 â R d e L V1 V2 â - R d e L T1 T2 â R d e L (â{I}V1.T1) (â{I}V2.T2) - ) â - âd,e,L,T1,T2. L â¢ T1 â¶* [d, e] T2 â R d e L T1 T2. -#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa â¦ H) -L -T1 -T2 -d -e -// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_lift.etc deleted file mode 100644 index a68f86e32..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_lift.etc +++ /dev/null @@ -1,196 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,K,V,U1,i,d,e. - d â¤ i â i < d + e â - â©[0, i] L â¡ K. âV â K â¢ V â¶* [0, d + e - i - 1] U1 â - âU2. â§[0, i + 1] U1 â¡ U2 â L â¢ #i â¶* [d, e] U2. -#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind â¦ H) -U1 -[ /3 width=4/ -| #U #U1 #_ #HU1 #IHU #U2 #HU12 - elim (lift_total U 0 (i+1)) #U0 #HU0 - lapply (IHU â¦ HU0) -IHU #H - lapply (ldrop_fwd_ldrop2 â¦ HLK) -HLK #HLK - lapply (tps_lift_ge â¦ HU1 â¦ HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02 - lapply (tps_weak â¦ 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: âL,T2,I,d,e. L â¢ âª{I} â¶* [d, e] T2 â - T2 = âª{I} â¨ - ââK,V1,V2,i. d â¤ i & i < d + e & - â©[O, i] L â¡ K. âV1 & - K â¢ V1 â¶* [0, d + e - i - 1] V2 & - â§[O, i + 1] V2 â¡ T2 & - I = LRef i. -#L #T2 #I #d #e #H @(tpss_ind â¦ H) -T2 -[ /2 width=1/ -| #T #T2 #_ #HT2 * - [ #H destruct - elim (tps_inv_atom1 â¦ HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ] - | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI - lapply (ldrop_fwd_ldrop2 â¦ HLK) #H - elim (tps_inv_lift1_ge_up â¦ HT2 â¦ H â¦ HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 (lift_mono â¦ HTU1 â¦ H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (IHT â¦ HTU) -IHT #HU1 - lapply (tps_lift_le â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ -] -qed. - -lemma tpss_lift_be: âK,T1,T2,dt,et. K â¢ T1 â¶* [dt, et] T2 â - âL,U1,d,e. dt â¤ d â d â¤ dt + et â - â©[d, e] L â¡ K â â§[d, e] T1 â¡ U1 â - âU2. â§[d, e] T2 â¡ U2 â L â¢ U1 â¶* [dt, et + e] U2. -#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind â¦ H) -T2 -[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (IHT â¦ HTU) -IHT #HU1 - lapply (tps_lift_be â¦ HT2 â¦ HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ -] -qed. - -lemma tpss_lift_ge: âK,T1,T2,dt,et. K â¢ T1 â¶* [dt, et] T2 â - âL,U1,d,e. d â¤ dt â â©[d, e] L â¡ K â - â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â - L â¢ U1 â¶* [dt + e, et] U2. -#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind â¦ H) -T2 -[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (IHT â¦ HTU) -IHT #HU1 - lapply (tps_lift_ge â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ -] -qed. - -lemma tpss_inv_lift1_le: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt + et â¤ d â - ââT2. K â¢ T1 â¶* [dt, et] T2 & â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_le â¦ HU2 â¦ HLK â¦ HTU ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_be: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt â¤ d â d + e â¤ dt + et â - ââT2. K â¢ T1 â¶* [dt, et - e] T2 & â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_be â¦ HU2 â¦ HLK â¦ HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_ge: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - d + e â¤ dt â - ââT2. K â¢ T1 â¶* [dt - e, et] T2 & â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_eq: âL,U1,U2,d,e. - L â¢ U1 â¶* [d, e] U2 â âT1. â§[d, e] T1 â¡ U1 â U1 = U2. -#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind â¦ H) -U2 // -#U #U2 #_ #HU2 #IHU destruct -<(tps_inv_lift1_eq â¦ HU2 â¦ HTU1) -HU2 -HTU1 // -qed. - -lemma tpss_inv_lift1_ge_up: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - d â¤ dt â dt â¤ d + e â d + e â¤ dt + et â - ââT2. K â¢ T1 â¶* [d, dt + et - (d + e)] T2 & - â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_ge_up â¦ HU2 â¦ HLK â¦ HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_be_up: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt â¤ d â dt + et â¤ d + e â - ââT2. K â¢ T1 â¶* [dt, d - dt] T2 & â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_be_up â¦ HU2 â¦ HLK â¦ HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_le_up: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - dt â¤ d â d â¤ dt + et â dt + et â¤ d + e â - ââT2. K â¢ T1 â¶* [dt, d - dt] T2 & â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind â¦ H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_le_up â¦ HU2 â¦ HLK â¦ HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_tpss.etc deleted file mode 100644 index d124bb32c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tpss_tpss.etc +++ /dev/null @@ -1,96 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,T2,d. L â¢ T1 â¶* [d, 1] T2 â L â¢ T1 â¶ [d, 1] T2. -#L #T1 #T2 #d #H @(tpss_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 -lapply (tps_trans_ge â¦ IHT1 â¦ HT2 ?) // -qed-. - -(* Advanced properties ******************************************************) - -lemma tpss_strip_eq: âL,T0,T1,d1,e1. L â¢ T0 â¶* [d1, e1] T1 â - âT2,d2,e2. L â¢ T0 â¶ [d2, e2] T2 â - ââT. L â¢ T1 â¶ [d2, e2] T & L â¢ T2 â¶* [d1, e1] T. -/3 width=3/ qed. - -lemma tpss_strip_neq: âL1,T0,T1,d1,e1. L1 â¢ T0 â¶* [d1, e1] T1 â - âL2,T2,d2,e2. L2 â¢ T0 â¶ [d2, e2] T2 â - (d1 + e1 â¤ d2 â¨ d2 + e2 â¤ d1) â - ââT. L2 â¢ T1 â¶ [d2, e2] T & L1 â¢ T2 â¶* [d1, e1] T. -/3 width=3/ qed. - -lemma tpss_strap1_down: âL,T1,T0,d1,e1. L â¢ T1 â¶* [d1, e1] T0 â - âT2,d2,e2. L â¢ T0 â¶ [d2, e2] T2 â d2 + e2 â¤ d1 â - ââT. L â¢ T1 â¶ [d2, e2] T & L â¢ T â¶* [d1, e1] T2. -/3 width=3/ qed. - -lemma tpss_strap2_down: âL,T1,T0,d1,e1. L â¢ T1 â¶ [d1, e1] T0 â - âT2,d2,e2. L â¢ T0 â¶* [d2, e2] T2 â d2 + e2 â¤ d1 â - ââT. L â¢ T1 â¶* [d2, e2] T & L â¢ T â¶ [d1, e1] T2. -/3 width=3/ qed. - -lemma tpss_split_up: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â - âi. d â¤ i â i â¤ d + e â - ââT. L â¢ T1 â¶* [d, i - d] T & L â¢ T â¶* [i, d + e - i] T2. -#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind â¦ H) -T2 -[ /2 width=3/ -| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 - elim (tps_split_up â¦ HT12 â¦ Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02 - elim (tpss_strap1_down â¦ HT3 â¦ 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: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â - âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â - d â¤ dt â dt â¤ d + e â d + e â¤ dt + et â - ââT2. K â¢ T1 â¶* [d, dt + et - (d + e)] T2 & - â§[d, e] T2 â¡ U2. -#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet -elim (tpss_split_up â¦ HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 -lapply (tpss_weak â¦ HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 -lapply (tpss_inv_lift1_eq â¦ HU1 â¦ HTU1) -HU1 #HU1 destruct -elim (tpss_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1 ?) -HU2 -HLK -HTU1 // (lift_mono â¦ H1 â¦ 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 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HVW2 â¦ HWU2) -W2 // plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct - elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpys_flat/ -] -qed. - -lemma cpys_inv_lift1: âG. 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 â¦ H) -H #H destruct /2 width=3 by cpys_atom, lift_sort, ex2_intro/ - | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hid #H destruct - [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV - elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 - elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // (length_inv_zero_dx â¦ H) -L1 // -| #I1 #I2 #L1 #L2 #V #_ #IHL12 #H lapply (injective_plus_l â¦ H) -H - /3 width=1 by lsuby_pair/ -] -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact lsuby_inv_atom1_aux: âL1,L2. L1 â L2 â L1 = â â L2 = â. -#L1 #L2 * -L1 -L2 // -#I1 #I2 #L1 #L2 #V #_ #H destruct -qed-. - -lemma lsuby_inv_atom1: âL2. â â L2 â L2 = â. -/2 width=3 by lsuby_inv_atom1_aux/ qed-. - -fact lsuby_inv_pair1_aux: âL1,L2. L1 â L2 â âJ1,K1,W. L1 = K1.â{J1}W â - L2 = â â¨ ââI2,K2. K1 â K2 & L2 = K2.â{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: âI1,K1,L2,W. K1.â{I1}W â L2 â - L2 = â â¨ ââI2,K2. K1 â K2 & L2 = K2.â{I2}W. -/2 width=4 by lsuby_inv_pair1_aux/ qed-. - -fact lsuby_inv_pair2_aux: âL1,L2. L1 â L2 â âJ2,K2,W. L2 = K2.â{J2}W â - ââI1,K1. K1 â K2 & L1 = K1.â{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: âI2,L1,K2,W. L1 â K2.â{I2}W â - ââI1,K1. K1 â K2 & L1 = K1.â{I1}W. -/2 width=4 by lsuby_inv_pair2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lsuby_fwd_length: âL1,L2. L1 â L2 â |L2| â¤ |L1|. -#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/ -qed-. - -lemma lsuby_ldrop_trans: âL1,L2. L1 â L2 â - âI2,K2,W,s,i. â©[s, 0, i] L2 â¡ K2.â{I2}W â - ââI1,K1. K1 â K2 & â©[s, 0, i] L1 â¡ K1.â{I1}W. -#L1 #L2 #H elim H -L1 -L2 -[ #L #J2 #K2 #W #s #i #H - elim (ldrop_inv_atom1 â¦ H) -H #H destruct -| #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #s #i #H - elim (ldrop_inv_O1_pair1 â¦ H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ] - [ /3 width=4 by ldrop_pair, ex2_2_intro/ - | elim (IHL12 â¦ HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_drop_lt, ex2_2_intro/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/lsuby_lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/lsuby_lsuby.etc deleted file mode 100644 index 254d62928..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/lsuby_lsuby.etc +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ lsuby. -#L1 #L #H elim H -L1 -L -[ #L1 #X #H lapply (lsuby_inv_atom1 â¦ H) -H // -| #I1 #I #L1 #L #V #_ #IHL1 #X #H elim (lsuby_inv_pair1 â¦ H) -H // * - #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/psubstsnstar_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/psubstsnstar_3.etc deleted file mode 100644 index 0d6dffee6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/psubstsnstar_3.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¶ * break term 46 L2 )" - non associative with precedence 45 - for @{ 'PSubstSnStar $G $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/psubststar_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/psubststar_4.etc deleted file mode 100644 index 0e76c6f71..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/psubststar_4.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 break â¶ * break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStar $G $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpx_cpys.etc deleted file mode 100644 index 3b38ee6ab..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpx_cpys.etc +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G. lsub_trans â¦ (cpx h g G) (lsuby 0 (â)). -#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 â¦ HL12 â¦ 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: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. -#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt â¦ H) -G -L -T1 -T2 -d -e -/2 width=7 by cpx_delta, cpx_bind, cpx_flat/ -qed. - -lemma cpy_cpx: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. -/3 width=3 by cpy_cpys, cpys_cpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpx_cpys2.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpx_cpys2.etc deleted file mode 100644 index 46abd5464..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpx_cpys2.etc +++ /dev/null @@ -1,70 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. -#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt â¦ H) -G -L -T1 -T2 -d -e -/2 width=7 by cpx_delta, cpx_bind, cpx_flat/ -qed. - -lemma cpy_cpx: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. -/3 width=3 by cpy_cpys, cpys_cpx/ qed. - -lemma cpx_cpy_trans: âh,g,G,L,T1,T. â¦G, Lâ¦ â¢ T1 â¡[h, g] T â - âT2,d,e. â¦G, Lâ¦ â¢ T â¶[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[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 â¦ H) // - * /2 width=3 by cpy_cpx/ -| #G #L #k #l #Hkl #X #d #e #H >(cpy_inv_sort1 â¦ 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 â¦ HLK) #H0 - lapply (cpy_weak â¦ H 0 (â) ? ?) -H // #H - elim (cpy_inv_lift1_be â¦ H â¦ H0 â¦ 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 â¦ 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 â¦ 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 â¦ HT2 0 (â) ? ?) -HT2 // #HT2 - elim (lift_total T2 0 1) #U2 #HTU2 - lapply (cpy_lift_be â¦ HT2 (L.âV1) â¦ (â») â¦ HTU â¦ 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 â¦ HX) -HX #Y #T2 #HY #HT2 #H destruct - elim (cpy_inv_flat1 â¦ 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 â¦ HX) -HX #W2 #Y #HW2 #HY #H destruct - elim (cpy_inv_flat1 â¦ HY) -HY #U2 #T2 #HU2 #HT2 #H destruct - lapply (cpy_weak â¦ HU2 0 (â) ? ?) -HU2 // #HU2 - elim (cpy_inv_lift1_be â¦ HU2 L â¦ HVU) -U - /5 width=7 by cpx_theta, lsuby_cpy_trans, lsuby_succ, ldrop_drop/ -] -qed-. - -lemma cpx_cpys_trans: âh,g,G,L,T1,T. â¦G, Lâ¦ â¢ T1 â¡[h, g] T â - âT2,d,e. â¦G, Lâ¦ â¢ T â¶*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. -#h #g #G #L #T1 #T #HT1 #T2 #d #e #H @(cpys_ind â¦ H) -T2 -/2 width=5 by cpx_cpy_trans/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpxs_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpxs_cpys.etc deleted file mode 100644 index 881030572..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/cpxs_cpys.etc +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G. lsub_trans â¦ (cpxs h g G) (lsuby 0 (â)). -/3 width=5 by lsuby_cpx_trans, LTC_lsub_trans/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/lpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/lpx_cpys.etc deleted file mode 100644 index d3c490a89..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/lpx_cpys.etc +++ /dev/null @@ -1,69 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T1,T. â¦G, L1â¦ â¢ T1 â¡[h, g] T â - âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â - âT2,d,e. â¦G, L2â¦ â¢ T â¶[d, e] T2 â â¦G, L1â¦ â¢ T1 â¡[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 â¦ H) -H // - * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct - elim (lpx_ldrop_trans_O1 â¦ HL12 â¦ HLK2) -L2 #X #HLK1 #H - elim (lpx_inv_pair2 â¦ 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 â¦ 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 â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 - elim (lpx_inv_pair1 â¦ H) -H #K2 #V0 #HK12 #_ #H destruct - lapply (ldrop_fwd_drop2 â¦ HLK2) -V0 #HLK2 - lapply (cpy_weak â¦ HT2 0 (â) ? ?) -HT2 // #HT2 - elim (cpy_inv_lift1_be â¦ HT2 â¦ HLK2 â¦ 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 â¦ 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 â¦ 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 â¦ HT2 0 (â) ? ?) -HT2 // #HT2 - elim (lift_total T2 0 1) #U2 #HTU2 - lapply (cpy_lift_be â¦ HT2 (L2.âV1) â¦ (â») â¦ HTU â¦ 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 â¦ HX) -HX #Y #T2 #HY #HT2 #H destruct - elim (cpy_inv_flat1 â¦ 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 â¦ HX) -HX #W2 #Y #HW2 #HY #H destruct - elim (cpy_inv_flat1 â¦ HY) -HY #U2 #T2 #HU2 #HT2 #H destruct - lapply (cpy_weak â¦ HU2 0 (â) ? ?) -HU2 // #HU2 - elim (cpy_inv_lift1_be â¦ HU2 L2 â¦ HVU) -U - /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/ -] -qed-. - -lemma cpx_cpys_trans_lpx: âh,g,G,L1,T1,T. â¦G, L1â¦ â¢ T1 â¡[h, g] T â - âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â - âT2,d,e. â¦G, L2â¦ â¢ T â¶*[d, e] T2 â â¦G, L1â¦ â¢ T1 â¡[h, g] T2. -#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind â¦ H) -T2 -/2 width=7 by cpx_cpy_trans_lpx/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/lsuby.etc deleted file mode 100644 index 657ad6b6e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys2/lsuby.etc +++ /dev/null @@ -1,15 +0,0 @@ -(* -lemma lsuby_weak: âL1,L2,d1,e1. L1 âÃ[d1, e1] L2 â - âd2,e2. d1 â¤ d2 â e2 â¤ e1 â L1 âÃ[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 â¦ 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 â¦ He21) -He21 #He21 - elim (ynat_cases d2) /3 width=1 by lsuby_pair/ - * #d0 #H destruct @lsuby_succ @IHL12 // - [ destruct - -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csup.etc deleted file mode 100644 index 0a980f6a7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csup.etc +++ /dev/null @@ -1,88 +0,0 @@ -(* Basic forward lemmas *****************************************************) - -lemma csup_fwd_ldrop: âL1,L2,T1,T2. â¦L1, T1â¦ > â¦L2, T2â¦ â - âi. â©[0, i] L1 â¡ L2 â¨ â©[0, i] L2 â¡ 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 â¦ HLK1) -HLK1 /3 width=2/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lift_csup_trans_eq: âT1,U1,d,e. â§[d, e] T1 â¡ U1 â - âL,U2. â¦L, U1â¦ > â¦L, U2â¦ â - âT2. â§[d, e] T2 â¡ 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 â¦ H) -H * - [ #_ #H destruct /2 width=2/ - | #H elim (discr_lpair_x_xy â¦ H) - ] -|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H - elim (csup_inv_flat1 â¦ H) -H #_ * #H destruct /2 width=2/ -] -#i #d #e [2,3: #_ ] #L #X #H -elim (csup_inv_atom1 â¦ H) -H #I #j #HL #H destruct -lapply (ldrop_pair2_fwd_cw â¦ HL X) -HL #H -elim (lt_refl_false â¦ H) -qed-. -(* -lemma lift_csup_trans_gt: âL1,L2,U1,U2. â¦L1, U1â¦ > â¦L2, U2â¦ â - â©[0, 1] L2 â¡ L1 â âT1,d,e. â§[d, e] T1 â¡ U1 â - âT2. â§[d + 1, e] T2 â¡ U2. -#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2 -[ #I #L1 #K1 #V #i #HLK1 #HKL1 - lapply (ldrop_fwd_lw â¦ HLK1) -HLK1 #HLK1 - lapply (ldrop_fwd_lw â¦ HKL1) -HKL1 #HKL1 - lapply (transitive_le â¦ HLK1 HKL1) -L1 normalize #H - - -| #a -| #a -] -#I #L1 #W1 #U1 #HL1 - - - - #X #d #e #H - lapply (ldrop_inv_refl â¦ HL1) -HL1 -| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H - lapply (ldrop_inv_ldrop1 â¦ HL1) - - elim (lift_inv_bind2 â¦ H) -H #W2 #U2 #HW21 #HU21 #H destruct - - - /3 width=2/ /4 width=2/ - -*) - - - -(* Advanced inversion lemmas ************************************************) - -lemma csup_inv_lref2_be: âL,U,i. â¦L, Uâ¦ > â¦L, #iâ¦ â - âT,d,e. â§[d, e] T â¡ U â d â¤ i â i < d + e â â¥. -#L #U #i #H #T #d #e #HTU #Hdi #Hide -elim (lift_csup_trans_eq â¦ HTU â¦ H) -H -T #T #H -elim (lift_inv_lref2_be â¦ H ? ?) // -qed-. - - -fact csup_inv_all4_refl_aux: âL1,L2,T1,T2. â¦L1, T1â¦ > â¦L2, T2â¦ â L1 = L2 â - â¨â¨ ââa,I,U. T1 = â{a,I}T2.U - | ââI,W. T1 = â{I}W.T2 - | ââI,U. T1 = â{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 â¦ HLK V) -HLK #H - elim (lt_refl_false â¦ H) -| #a #I #L #V #T #H - elim (discr_lpair_x_xy â¦ H) -] -qed-. - -lemma csup_inv_all4_refl: âL,T1,T2. â¦L, T1â¦ > â¦L, T2â¦ â - â¨â¨ ââa,I,U. T1 = â{a,I}T2.U - | ââI,W. T1 = â{I}W.T2 - | ââI,U. T1 = â{I}T2.U. -/2 width=4 by csup_inv_all4_refl_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csup_csup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csup_csup.etc deleted file mode 100644 index 661d89697..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csup_csup.etc +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,L0,T1,T2. â¦L, T1â¦ > â¦L0, T2â¦ â - âT3. â¦L0, T2â¦ > â¦L, T3â¦ â - L = L0 â¨ â¦L, T1â¦ > â¦L, T3â¦. -#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ -[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H - lapply (ldrop_pair2_fwd_cw â¦ HLK0 T3) -HLK0 #H1 - lapply (csup_fwd_cw â¦ H) -H #H2 - lapply (transitive_lt â¦ H1 H2) -H1 -H2 #H - elim (lt_refl_false â¦ H) -| #a #I #L0 #V2 #T2 #T3 #HT23 - elim (csup_inv_ldrop â¦ HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ - qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csups_csups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csups_csups.etc deleted file mode 100644 index 9978429d7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/csup/csups_csups.etc +++ /dev/null @@ -1,57 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,L0,T1,T2. â¦L, T1â¦ > â¦L0, T2â¦ â - âT3. â¦L0, T2â¦ >+ â¦L, T3â¦ â - L = L0 â¨ â¦L, T1â¦ >+ â¦L, T3â¦. -#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ -[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H - lapply (ldrop_pair2_fwd_cw â¦ HLK0 T3) -HLK0 #H1 - lapply (csupp_fwd_cw â¦ H) -H #H2 - lapply (transitive_lt â¦ H1 H2) -H1 -H2 #H - elim (lt_refl_false â¦ H) -| #a #I #L0 #V2 #T2 #T3 #HT23 - /3 width=5/ - - elim (csup_inv_ldrop â¦ HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ - qed-. - - - - - - - - -lemma csups_strap1_fwd_refl: âL,L0,T1,T2. â¦L, T1â¦ >* â¦L0, T2â¦ â - âT3. â¦L0, T2â¦ > â¦L, T3â¦ â L = L0. -#L #L0 #T1 #T2 #H @(csups_ind_dx â¦ H) -L -T1 // -#L1 #L #T1 #T #HL1 #_ #IHL0 #T3 #HL0 -lapply (csup_trans_fwd_refl â¦ HL10) â¦ HL0) -T2 -*) -lemma lift_csups_trans_aux: âT1,U1,d,e. â§[d, e] T1 â¡ U1 â - âL1,L2,U2. â¦L1, U1â¦ >* â¦L2, U2â¦ â L1 = L2 â - âT2. â§[d, e] T2 â¡ U2. -#T1 #U1 #d #e #HTU1 #L1 #L2 #U2 #H @(csups_ind â¦ H) -L2 -U2 /2 width=2/ -T1 -#L #L2 #U #U2 #HL1 #HL2 #IHL1 #H destruct - -* -T1 -U1 -d -e diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cpcs_delift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cpcs_delift.etc deleted file mode 100644 index 7012ec11e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cpcs_delift.etc +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,V,T1,T2. L.âV â¢ â¼*[O, 1] T1 â¡ T2 â - L â¢ T2 â¬* +âV.T1. -/3 width=1/ qed. - -(* Basic_1: was only: pc3_gen_cabbr *) -lemma thin_cpcs_delift_mono: âL,U1,U2. L â¢ U1 â¬* U2 â - âK,d,e. â¼*[d, e] L â¡ K â âT1. L â¢ â¼*[d, e] U1 â¡ T1 â - âT2. L â¢ â¼*[d, e] U2 â¡ T2 â K â¢ T1 â¬* T2. -#L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2 -elim (cpcs_inv_cprs â¦ H) -H #U #HU1 #HU2 -elim (thin_cprs_delift_conf â¦ HU1 â¦ HLK â¦ HTU1) -U1 #T #HT1 #HUT -elim (thin_cprs_delift_conf â¦ HU2 â¦ HLK â¦ HTU2) -U2 -HLK #X #HT2 #H -lapply (delift_mono â¦ H â¦ HUT) -L #H destruct /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cpr_delift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cpr_delift.etc deleted file mode 100644 index 04eb4ae23..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cpr_delift.etc +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,U1,U2. L â¢ U1 â¡ U2 â - âK,d,e. â¼*[d, e] L â¡ K â âT1. L â¢ â¼*[d, e] U1 â¡ T1 â - ââT2. K â¢ T1 â¡ T2 & L â¢ â¼*[d, e] U2 â¡ T2. -#L #U1 #U2 * #U #HU1 #HU2 #K #d #e #HLK #T1 #HTU1 -elim (tpr_delift_conf â¦ HU1 â¦ HTU1) -U1 #T #HT1 #HUT -elim (le_or_ge (|L|) d) #Hd -[ elim (thin_delift_tpss_conf_le â¦ HU2 â¦ HUT â¦ HLK ?) -| elim (le_or_ge (|L|) (d+e)) #Hde - [ elim (thin_delift_tpss_conf_le_up â¦ HU2 â¦ HUT â¦ HLK ? ? ?) - | elim (thin_delift_tpss_conf_be â¦ HU2 â¦ HUT â¦ HLK ? ?) - ] -] -U -HLK // -Hd [2,3: -Hde] #T2 #HT2 -lapply (cpr_intro â¦ HT1 HT2) -T /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cprs_delift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cprs_delift.etc deleted file mode 100644 index 6b7892611..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/cprs_delift.etc +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,V,T1,T2. L.âV â¢ â¼*[O, 1] T1 â¡ T2 â L â¢ +âV.T1 â¡* T2. -#L #V #T1 #T2 * #T #HT1 #HT2 -@(cprs_strap2 â¦ (+âV.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: âL,U1,U2. L â¢ U1 â¡* U2 â - âK,d,e. â¼*[d, e] L â¡ K â âT1. L â¢ â¼*[d, e] U1 â¡ T1 â - ââT2. K â¢ T1 â¡* T2 & L â¢ â¼*[d, e] U2 â¡ T2. -#L #U1 #U2 #H @(cprs_ind â¦ H) -U2 /2 width=3/ -#U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1 -elim (IHU1 â¦ HLK â¦ HTU1) -U1 #T #HT1 #HUT -elim (thin_cpr_delift_conf â¦ HU2 â¦ HLK â¦ HUT) -U -HLK /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift.etc deleted file mode 100644 index eda839607..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift.etc +++ /dev/null @@ -1,112 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break â¼ * [ term 46 d , break term 46 e ] break term 46 T1 â¡ 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 â nat â lenv â relation term â - Î»d,e,L,T1,T2. ââT. L â¢ T1 â¶* [d, e] T & â§[d, e] T2 â¡ T. - -interpretation "inverse basic relocation (term)" - 'TSubst L T1 d e T2 = (delift d e L T1 T2). - -(* Basic properties *********************************************************) - -lemma lift_delift: âT1,T2,d,e. â§[d, e] T1 â¡ T2 â - âL. L â¢ â¼*[d, e] T2 â¡ T1. -/2 width=3/ qed. - -lemma delift_refl_O2: âL,T,d. L â¢ â¼*[d, 0] T â¡ T. -/2 width=3/ qed. - -lemma delift_lsubr_trans: âL1,T1,T2,d,e. L1 â¢ â¼*[d, e] T1 â¡ T2 â - âL2. L2 â [d, e] L1 â L2 â¢ â¼*[d, e] T1 â¡ T2. -#L1 #T1 #T2 #d #e * /3 width=3/ -qed. - -lemma delift_sort: âL,d,e,k. L â¢ â¼*[d, e] âk â¡ âk. -/2 width=3/ qed. - -lemma delift_lref_lt: âL,d,e,i. i < d â L â¢ â¼*[d, e] #i â¡ #i. -/3 width=3/ qed. - -lemma delift_lref_ge: âL,d,e,i. d + e â¤ i â L â¢ â¼*[d, e] #i â¡ #(i - e). -/3 width=3/ qed. - -lemma delift_gref: âL,d,e,p. L â¢ â¼*[d, e] Â§p â¡ Â§p. -/2 width=3/ qed. - -lemma delift_bind: âa,I,L,V1,V2,T1,T2,d,e. - L â¢ â¼*[d, e] V1 â¡ V2 â L. â{I} V2 â¢ â¼*[d+1, e] T1 â¡ T2 â - L â¢ â¼*[d, e] â{a,I} V1. T1 â¡ â{a,I} V2. T2. -#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2 -lapply (tpss_lsubr_trans â¦ HT1 (L. â{I} V) ?) -HT1 /2 width=1/ /3 width=5/ -qed. - -lemma delift_flat: âI,L,V1,V2,T1,T2,d,e. - L â¢ â¼*[d, e] V1 â¡ V2 â L â¢ â¼*[d, e] T1 â¡ T2 â - L â¢ â¼*[d, e] â{I} V1. T1 â¡ â{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: âL,U2,d,e,k. L â¢ â¼*[d, e] âk â¡ U2 â U2 = âk. -#L #U2 #d #e #k * #U #HU ->(tpss_inv_sort1 â¦ HU) -HU #HU2 ->(lift_inv_sort2 â¦ HU2) -HU2 // -qed-. - -lemma delift_inv_gref1: âL,U2,d,e,p. L â¢ â¼*[d, e] Â§p â¡ U2 â U2 = Â§p. -#L #U #d #e #p * #U #HU ->(tpss_inv_gref1 â¦ HU) -HU #HU2 ->(lift_inv_gref2 â¦ HU2) -HU2 // -qed-. - -lemma delift_inv_bind1: âa,I,L,V1,T1,U2,d,e. L â¢ â¼*[d, e] â{a,I} V1. T1 â¡ U2 â - ââV2,T2. L â¢ â¼*[d, e] V1 â¡ V2 & - L. â{I} V2 â¢ â¼*[d+1, e] T1 â¡ T2 & - U2 = â{a,I} V2. T2. -#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 -elim (tpss_inv_bind1 â¦ HU) -HU #V #T #HV1 #HT1 #X destruct -elim (lift_inv_bind2 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 -lapply (tpss_lsubr_trans â¦ HT1 (L. â{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ -qed-. - -lemma delift_inv_flat1: âI,L,V1,T1,U2,d,e. L â¢ â¼*[d, e] â{I} V1. T1 â¡ U2 â - ââV2,T2. L â¢ â¼*[d, e] V1 â¡ V2 & - L â¢ â¼*[d, e] T1 â¡ T2 & - U2 = â{I} V2. T2. -#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 -elim (tpss_inv_flat1 â¦ HU) -HU #V #T #HV1 #HT1 #X destruct -elim (lift_inv_flat2 â¦ HU2) -HU2 /3 width=5/ -qed-. - -lemma delift_inv_refl_O2: âL,T1,T2,d. L â¢ â¼*[d, 0] T1 â¡ T2 â T1 = T2. -#L #T1 #T2 #d * #T #HT1 ->(tpss_inv_refl_O2 â¦ HT1) -HT1 #HT2 ->(lift_inv_refl_O2 â¦ HT2) -HT2 // -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma delift_fwd_tw: âL,T1,T2,d,e. L â¢ â¼*[d, e] T1 â¡ T2 â â¯{T1} â¤ â¯{T2}. -#L #T1 #T2 #d #e * #T #HT1 #HT2 ->(lift_fwd_tw â¦ HT2) -T2 /2 width=4 by tpss_fwd_tw/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_alt.etc deleted file mode 100644 index e22911032..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_alt.etc +++ /dev/null @@ -1,104 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break â¼ â¼ * [ term 46 d , break term 46 e ] break term 46 T1 â¡ 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 â nat â lenv â relation term â -| delifta_sort : âL,d,e,k. delifta d e L (âk) (âk) -| delifta_lref_lt: âL,d,e,i. i < d â delifta d e L (#i) (#i) -| delifta_lref_be: âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â - â©[0, i] L â¡ K. âV1 â delifta 0 (d + e - i - 1) K V1 V2 â - â§[0, d] V2 â¡ W2 â delifta d e L (#i) W2 -| delifta_lref_ge: âL,d,e,i. d + e â¤ i â delifta d e L (#i) (#(i - e)) -| delifta_gref : âL,d,e,p. delifta d e L (Â§p) (Â§p) -| delifta_bind : âL,a,I,V1,V2,T1,T2,d,e. - delifta d e L V1 V2 â delifta (d + 1) e (L. â{I} V2) T1 T2 â - delifta d e L (â{a,I} V1. T1) (â{a,I} V2. T2) -| delifta_flat : âL,I,V1,V2,T1,T2,d,e. - delifta d e L V1 V2 â delifta d e L T1 T2 â - delifta d e L (â{I} V1. T1) (â{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: âL1,T1,T2,d,e. L1 â¢ â¼â¼*[d, e] T1 â¡ T2 â - âL2. L2 â [d, e] L1 â L2 â¢ â¼â¼*[d, e] T1 â¡ 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 â¦ HL12 â¦ HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ -| /4 width=1/ -| /3 width=1/ -] -qed. - -lemma delift_delifta: âL,T1,T2,d,e. L â¢ â¼*[d, e] T1 â¡ T2 â L â¢ â¼â¼*[d, e] T1 â¡ T2. -#L #T1 @(f2_ind â¦ fw â¦ L T1) -L -T1 #n #IH #L * -[ * #i #Hn #T2 #d #e #H destruct - [ >(delift_inv_sort1 â¦ H) -H // - | elim (delift_inv_lref1 â¦ H) -H * /2 width=1/ - #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2 - lapply (ldrop_pair2_fwd_fw â¦ HLK) #H - lapply (IH â¦ HV12) // -H /2 width=6/ - | >(delift_inv_gref1 â¦ H) -H // - ] -| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H - [ elim (delift_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (delift_lsubr_trans â¦ HT12 (L.â{I}V1) ?) -HT12 /2 width=1/ #HT12 - lapply (IH â¦ HV12) -HV12 // #HV12 - lapply (IH â¦ HT12) -IH -HT12 /2 width=1/ #HT12 - lapply (delifta_lsubr_trans â¦ HT12 (L.â{I}V2) ?) -HT12 /2 width=1/ - | elim (delift_inv_flat1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH â¦ HV12) -HV12 // - lapply (IH â¦ HT12) -IH -HT12 // /2 width=1/ - ] -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma delifta_delift: âL,T1,T2,d,e. L â¢ â¼â¼*[d, e] T1 â¡ T2 â L â¢ â¼*[d, e] T1 â¡ 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: âR:ââââlenvârelation term. - (âL,d,e,k. R d e L (âk) (âk)) â - (âL,d,e,i. i < d â R d e L (#i) (#i)) â - (âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â - â©[O, i] L â¡ K.âV1 â K â¢ â¼*[O, d + e - i - 1] V1 â¡ V2 â - â§[O, d] V2 â¡ W2 â R O (d+e-i-1) K V1 V2 â R d e L (#i) W2 - ) â - (âL,d,e,i. d + e â¤ i â R d e L (#i) (#(i - e))) â - (âL,d,e,p. R d e L (Â§p) (Â§p)) â - (âL,a,I,V1,V2,T1,T2,d,e. L â¢ â¼*[d, e] V1 â¡ V2 â - L.â{I}V2 â¢ â¼*[d + 1, e] T1 â¡ T2 â R d e L V1 V2 â - R (d+1) e (L.â{I}V2) T1 T2 â R d e L (â{a,I}V1.T1) (â{a,I}V2.T2) - ) â - (âL,I,V1,V2,T1,T2,d,e. L â¢ â¼*[d, e] V1 â¡ V2 â - Lâ¢ â¼*[d, e] T1 â¡ T2 â R d e L V1 V2 â - R d e L T1 T2 â R d e L (â{I}V1.T1) (â{I}V2.T2) - ) â - âd,e,L,T1,T2. L â¢ â¼*[d, e] T1 â¡ T2 â R d e L T1 T2. -#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_delift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_delift.etc deleted file mode 100644 index a5c563565..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_delift.etc +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T,T1,T2,d,e. - L â¢ â¼*[d, e] T â¡ T1 â L â¢ â¼*[d, e] T â¡ T2 â T1 = T2. -#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2 -elim (tpss_conf_eq â¦ H1TU1 â¦ H1TU2) -T #U #HU1 #HU2 -lapply (tpss_inv_lift1_eq â¦ HU1 â¦ H2TU1) -HU1 #H destruct -lapply (tpss_inv_lift1_eq â¦ HU2 â¦ H2TU2) -HU2 #H destruct -lapply (lift_inj â¦ H2TU1 â¦ H2TU2) // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_lift.etc deleted file mode 100644 index 8abaeb9f9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/delift_lift.etc +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,K,V1,V2,U2,i,d,e. d â¤ i â i < d + e â - â©[0, i] L â¡ K. âV1 â K â¢ â¼*[0, d + e - i - 1] V1 â¡ V2 â - â§[0, d] V2 â¡ U2 â L â¢ â¼*[d, e] #i â¡ 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 â¦ HV2 â¦ HVU ? ?) -HV2 // >minus_plus commutative_plus in â¢ (??%??â?); 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 â¦ V1 â¦ Hde HL) // #V2 #HV12 - elim (IH (L.â{I}V1) T1 â¦ (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12 - lapply (delift_lsubr_trans â¦ HT12 (L.â{I}V2) ?) -HT12 /2 width=1/ /3 width=4/ -| #I #V1 #T1 #H #d #e #Hde #HL destruct - elim (IH â¦ V1 â¦ Hde HL) // #V2 #HV12 - elim (IH â¦ T1 â¦ Hde HL) -IH -Hde -HL // /3 width=2/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma delift_inv_lref1_lt: âL,U2,i,d,e. L â¢ â¼*[d, e] #i â¡ U2 â i < d â U2 = #i. -#L #U2 #i #d #e * #U #HU #HU2 #Hid -elim (tpss_inv_lref1 â¦ HU) -HU -[ #H destruct >(lift_inv_lref2_lt â¦ HU2) // -| * #K #V1 #V2 #Hdi - lapply (lt_to_le_to_lt â¦ Hid Hdi) -Hid -Hdi #Hi - elim (lt_refl_false â¦ Hi) -] -qed-. - -lemma delift_inv_lref1_be: âL,U2,d,e,i. L â¢ â¼*[d, e] #i â¡ U2 â - d â¤ i â i < d + e â - ââK,V1,V2. â©[0, i] L â¡ K. âV1 & - K â¢ â¼*[0, d + e - i - 1] V1 â¡ V2 & - â§[0, d] V2 â¡ U2. -#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide -elim (tpss_inv_lref1 â¦ HU) -HU -[ #H destruct elim (lift_inv_lref2_be â¦ HU2 ? ?) // -| * #K #V1 #V #_ #_ #HLK #HV1 #HVU - elim (lift_div_be â¦ HVU â¦ HU2 ? ?) -U // /2 width=1/ /3 width=6/ -] -qed-. - -lemma delift_inv_lref1_ge: âL,U2,i,d,e. L â¢ â¼*[d, e] #i â¡ U2 â - d + e â¤ i â U2 = #(i - e). -#L #U2 #i #d #e * #U #HU #HU2 #Hdei -elim (tpss_inv_lref1 â¦ HU) -HU -[ #H destruct >(lift_inv_lref2_ge â¦ HU2) // -| * #K #V1 #V2 #_ #Hide - lapply (lt_to_le_to_lt â¦ Hide Hdei) -Hide -Hdei #Hi - elim (lt_refl_false â¦ Hi) -] -qed-. - -lemma delift_inv_lref1: âL,U2,i,d,e. L â¢ â¼*[d, e] #i â¡ U2 â - â¨â¨ (i < d â§ U2 = #i) - | (ââK,V1,V2. d â¤ i & i < d + e & - â©[0, i] L â¡ K. âV1 & - K â¢ â¼*[0, d + e - i - 1] V1 â¡ V2 & - â§[0, d] V2 â¡ U2 - ) - | (d + e â¤ i â§ U2 = #(i - e)). -#L #U2 #i #d #e #H -elim (lt_or_ge i d) #Hdi -[ elim (delift_inv_lref1_lt â¦ H Hdi) -H /3 width=1/ -| elim (lt_or_ge i (d+e)) #Hide - [ elim (delift_inv_lref1_be â¦ H Hdi Hide) -H /3 width=6/ - | elim (delift_inv_lref1_ge â¦ H Hide) -H /3 width=1/ - ] -] -qed-. - -(* Properties on basic term relocation **************************************) - -lemma delift_lift_le: âK,T1,T2,dt,et. K â¢ â¼*[dt, et] T1 â¡ T2 â - âL,U1,d,e. dt + et â¤ d â â©[d, e] L â¡ K â - â§[d, e] T1 â¡ U1 â âU2. â§[d - et, e] T2 â¡ U2 â - L â¢ â¼*[dt, et] U1 â¡ 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 â¦ HT1 â¦ HLK HTU1 â¦ HTU) -T1 -HLK // #HU1 -elim (lift_trans_ge â¦ HT2 â¦ HTU ?) -T // -Hdetd #T #HT2 #HTU ->(lift_mono â¦ HTU2 â¦ HT2) -T2 /2 width=3/ -qed. - -lemma delift_lift_be: âK,T1,T2,dt,et. K â¢ â¼*[dt, et] T1 â¡ T2 â - âL,U1,d,e. dt â¤ d â d â¤ dt + et â - â©[d, e] L â¡ K â â§[d, e] T1 â¡ U1 â - L â¢ â¼*[dt, et + e] U1 â¡ 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 â¦ HT1 â¦ HLK HTU1 â¦ HTU) -T1 -HLK // #HU1 -lapply (lift_trans_be â¦ HT2 â¦ HTU ? ?) -T // -Hdtd -Hddet /2 width=3/ -qed. - -lemma delift_lift_ge: âK,T1,T2,dt,et. K â¢ â¼*[dt, et] T1 â¡ T2 â - âL,U1,d,e. d â¤ dt â â©[d, e] L â¡ K â - â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â - L â¢ â¼*[dt + e, et] U1 â¡ 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 â¦ HT1 â¦ HLK HTU1 â¦ HTU) -T1 -HLK // #HU1 -elim (lift_trans_le â¦ HT2 â¦ HTU ?) -T // -Hddt #T #HT2 #HTU ->(lift_mono â¦ HTU2 â¦ HT2) -T2 /2 width=3/ -qed. - -lemma delift_inv_lift1_eq: âL,U1,T2,d,e. L â¢ â¼*[d, e] U1 â¡ T2 â - âK. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â T1 = T2. -#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1 -lapply (tpss_inv_lift1_eq â¦ HU12 â¦ HTU1) -L -K #H destruct -lapply (lift_inj â¦ HTU1 â¦ HTU2) -U2 // -qed-. - -lemma delift_lift_div_be: âL,T1,T,d,e,i. L â¢ â¼*[i, d + e - i] T1 â¡ T â - âT2. â§[d, i - d] T2 â¡ T â d â¤ i â i â¤ d + e â - L â¢ â¼*[d, e] T1 â¡ T2. -#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide -lapply (tpss_weak â¦ HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10 -lapply (lift_trans_be â¦ HT2 â¦ HT0 ? ?) -T // ->commutative_plus >commutative_plus in â¢ (? ? (? % ?) ? ? â ?); -commutative_plus in H; >plus_plus_comm_23 #H - elim (le_plus_xySz_x_false â¦ H) - ] - ] -| #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2 - elim (cpr_inv_bind1 â¦ H1) -H1 * - [ #V #T #HV1 #HT1 #H destruct - elim (cpr_inv_bind1 â¦ H2) -H2 * - [ #V2 #T2 #HV2 #HT2 #H destruct - elim (IH â¦ HV1 â¦ HV2) // #HV12 destruct - [ @or_introl #H destruct /2 width=1/ - | elim (IH â¦ HT1 â¦ HT2) // /2 width=1/ -L1 -L2 #HT12 destruct - @or_introl #H destruct /2 width=1/ - ] - | #T2 #HT2 #HXT2 #H1 #H2 destruct - elim (IH â¦ HT1 â¦ 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 (+âV1.T1) 0 1) #Y2 #HXY2 - lapply (cpr_lift â¦ H2 (L2.âV1) â¦ HXY1 â¦ HXY2) /2 width=1/ -X1 /4 width=5/ - ] -| #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2 - elim (cpr_inv_flat1 â¦ H1) -H1 * - [ #V #T #HV1 #HT1 #H destruct - elim (cpr_inv_flat1 â¦ H2) -H2 * - [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ - | #HX2 #H destruct /3 width=5/ - ] - | #HX1 #H destruct /3 width=5/ -] -qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/etc/ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/etc/ldrop.etc deleted file mode 100644 index e4caef0c8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/etc/ldrop.etc +++ /dev/null @@ -1,11 +0,0 @@ -lemma ldrop_bor1: âL1,L2,s1,s2,d,e. â©[s1, d, e] L1 â¡ L2 â â©[s1 â¨ s2, d, e] L1 â¡ L2. -#L1 #L2 * /2 width=1 by ldrop_gen/ -qed. - -lemma ldrop_bor2: âL1,L2,s1,s2,d,e. â©[s2, d, e] L1 â¡ L2 â â©[s1 â¨ s2, d, e] L1 â¡ L2. -#L1 #L2 #s1 #s2 >commutative_orb /2 width=1 by ldrop_bor1/ -qed. - -(* Basic_1: was: drop_conf_rev *) -axiom ldrop_div: âe1,L1,L. â©[0, e1] L1 â¡ L â âe2,L2. â©[0, e2] L2 â¡ L â - ââL0. â©[0, e1] L0 â¡ L2 & â©[e1, e2] L0 â¡ L1. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fleq/fpbc_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fleq/fpbc_lift.etc deleted file mode 100644 index 286290135..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fleq/fpbc_lift.etc +++ /dev/null @@ -1,9 +0,0 @@ -include "basic_2/reduction/lpx_ldrop.ma". - -lemma lpx_fpbc_trans: âh,g,G1,G2,K1,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â - â¦G1, K1â¦ â¢ â¡[h, g] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡[h, g] T & â¦G1, K1, Tâ¦ â»[h, g] â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g] L2. -#h #g #G1 #G2 #K1 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 -[ #G2 #L2 #T2 #H12 #HKL1 elim (lpx_fqu_trans â¦ H12 â¦ HKL1) -L1 - /3 width=5 by fpbc_fqu, ex3_2_intro/ -| #T2 #HT12 #H #HKL1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fleq/fpbs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fleq/fpbs.etc deleted file mode 100644 index 017421feb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fleq/fpbs.etc +++ /dev/null @@ -1,49 +0,0 @@ -lemma fpbs_fpbu_sn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â¡[0] â¦G2, L2, T2â¦ â¨ - ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦. -(* ALTERNATIVE PROOF -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx â¦ 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 â¦ H1) -H1 #H1 - [ /3 width=1 by -*) -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt â¦ H) -H -#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct -[ -HT1 elim (fqus_inv_gen â¦ HT2) -HT2 - [ #H elim (fqup_inv_step_sn â¦ 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 â¦ 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 â¦ 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: âh,g,G1,G2,L1,L,L2,T1,T2. â¦G1, L1â¦ â¢â¡ [h, g] L â - â¦G1, L, T1â¦ â[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â[h, g] â¦G2, L2, T2â¦. - -lemma fpbs_fwd_fpb_sn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â[h, g] â¦G2, L2, T2â¦ â¨ - ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx â¦ 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 â¦ 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 - diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpa.etc deleted file mode 100644 index 8f72937c4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpa.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| fpa_fwd: âL,K,T,U,d,e. â©[s, d, e] L â¡ K â â§[d, e] T â¡ U â fpa s K T L U -| fpa_bwd: âL,K,T,U,d,e. â©[s, d, e] L â¡ K â â§[d, e] T â¡ U â 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: âs. bi_reflexive â¦ (fpa s). -/2 width=4 by fpa_fwd/ qed. - -lemma fpa_sym: âs. bi_symmetric â¦ (fpa s). -#s #L1 #L2 #T1 #T2 * /2 width=4 by fpa_fwd, fpa_bwd/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpa_fpa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpa_fpa.etc deleted file mode 100644 index c62b8bfb9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpa_fpa.etc +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âs. bi_confluent â¦ (fpa s). -/3 width=4 by fpa_sym, ex2_2_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas.etc deleted file mode 100644 index 6817d8c90..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas.etc +++ /dev/null @@ -1,59 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â bi_relation lenv term â Î»s. bi_TC â¦ (fpa s). - -interpretation - "multiple ajustment (restricted closure)" - 'RAjustStar L1 T1 s L2 T2 = (fpas s L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma fpas_ind: âL1,T1,s. âR:relation2 â¦. R L1 T1 â - (âL,L2,T,T2. â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³[s] â¦L2, T2â¦ â R L T â R L2 T2) â - âL2,T2. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â R L2 T2. -#L1 #T1 #s #R #IH1 #IH2 #L2 #T2 #H -@(bi_TC_star_ind â¦ IH1 IH2 L2 T2 H) // -qed-. - -lemma fpas_ind_dx: âL2,T2,s. âR:relation2 â¦. R L2 T2 â - (âL1,L,T1,T. â¦L1, T1â¦ â³[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³*[s] â¦L2, T2â¦ â R L T â R L1 T1) â - âL1,T1. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â R L1 T1. -#L2 #T2 #s #R #IH1 #IH2 #L1 #T1 #H -@(bi_TC_star_ind_dx â¦ IH1 IH2 L1 T1 H) // -qed-. - -(* Basic properties *********************************************************) - -lemma fpas_refl: âs. bi_reflexive â¦ (fpas s). -/2 width=1 by bi_inj/ qed. - -lemma fpas_sym: âs. bi_symmetric â¦ (fpas s). -/3 width=1 by fpa_sym, bi_TC_symmetric/ qed-. - -lemma fpa_fpas: âL1,L2,T1,T2,s. â¦L1, T1â¦ â³[s] â¦L2, T2â¦ â â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. -/2 width=1 by bi_inj/ qed. - -lemma fpas_strap1: âL1,L,L2,T1,T,T2,s. â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³[s] â¦L2, T2â¦ â - â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. -/2 width=4 by bi_step/ qed-. - -lemma fpas_strap2: âL1,L,L2,T1,T,T2,s. â¦L1, T1â¦ â³[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³*[s] â¦L2, T2â¦ â - â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. -/2 width=4 by bi_TC_strap/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas_fpas.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas_fpas.etc deleted file mode 100644 index 28d648e7b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas_fpas.etc +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,L1,L2,T0,T1,T2,s. â¦L0, T0â¦ â³[s] â¦L1, T1â¦ â â¦L0, T0â¦ â³*[s] â¦L2, T2â¦ â - ââL,T. â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ & â¦L2, T2â¦ â³[s] â¦L, Tâ¦. -/3 width=4 by fpa_conf, bi_TC_strip/ qed-. - -(* Main properties **********************************************************) - -theorem fpas_conf: âs. bi_confluent â¦ (fpas s). -/3 width=4 by fpa_conf, bi_TC_confluent/ qed-. - -theorem fpas_trans: âs. bi_transitive â¦ (fpas s). -/2 width=4 by bi_TC_transitive/ qed-. - -theorem fpas_canc_sn: âL,L1,L2,T,T1,T2,s. - â¦L, Tâ¦ â³*[s] â¦L1, T1â¦â â¦L, Tâ¦ â³*[s] â¦L2, T2â¦ â â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. -/3 width=4 by fpas_trans, fpas_sym/ qed-. - -theorem fpas_canc_dx: âL1,L2,L,T1,T2,T,s. - â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ â â¦L2, T2â¦ â³*[s] â¦L, Tâ¦ â â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. -/3 width=4 by fpas_trans, fpas_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas_vector.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas_vector.etc deleted file mode 100644 index 65e8190e4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/fpas_vector.etc +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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) â -| fpasv_nil : âL1,L2,T1,T2. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â fpasv s L1 (â) L2 (â) -| fpasv_cons: âL1,L2,T1s,T2s,T1,T2. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â - fpasv s L1 T1s L2 T2s â - 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: âL1,L2,V1s,T1,X,s. â¦L1, â¶ V1s.T1â¦ â³*[s] â¦L2, Xâ¦ â - ââV2s,T2. â¦L1, V1sâ¦ â³*[s] â¦L2, V2sâ¦ & â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ & - X = â¶ V2s.T2. -#L1 #L2 #V1s elim V1s -V1s -[ #T1 #X #s #H - @(ex3_2_intro â¦ (â) X) /2 width=3 by fpasv_nil/ (**) (* explicit constructor *) -| #V1 #V1s #IHV1s #T1 #X #s #H - elim (lifts_inv_flat1 â¦ H) -H #V2 #Y #HV12 #HY #H destruct - elim (IHV1s â¦ 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: âV1s,V2s,des. â§*[des] V1s â¡ V2s â - âT1,T2. â§*[des] T1 â¡ T2 â - â§*[des] â¶ V1s. T1 â¡ â¶ V2s. T2. -#V1s #V2s #des #H elim H -V1s -V2s /3 width=1 by lifts_flat/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/gcp.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/gcp.etc deleted file mode 100644 index b9b2db820..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/gcp.etc +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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] â relation3 genv lenv term. - -definition CP0 â Î»RR:relation4 genv lenv term term. Î»RS:relation term. - âG,L1,L2,T1,T2,s. â¦L1, T1â¦ â³[s] â¦L2, T2â¦ â - NF â¦ (RR G L1) RS T1 â NF â¦ (RR G L2) RS T2. - -definition CP0s â Î»RR:relation4 genv lenv term term. Î»RS:relation term. - âG,L1,L2,T1,T2,s. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â - NF â¦ (RR G L1) RS T1 â NF â¦ (RR G L2) RS T2. - -definition CP1 â Î»RR:relation4 genv lenv term term. Î»RS:relation term. - âG,L. âk. NF â¦ (RR G L) RS (âk). - -definition CP2 â Î»RP:candidate. - âG,L,T,k. RP G L (ââk.T) â RP G L T. - -(* requirements for generic computation properties *) -record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop â -{ cp0: CP0 RR RS; - cp1: CP1 RR RS; - cp2: CP2 RP -}. - -(* Basic properties *********************************************************) - -(* Basic_1: was just: nf2_lift1 *) -lemma gcp_fpas: âRR,RS. CP0 RR RS â CP0s RR RS. -#RR #RS #HRR #G #L1 #L2 #T1 #T2 #s #H @(fpas_ind â¦ H) -L2 -T2 /3 width=5 by/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/gcp_cr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/gcp_cr.etc deleted file mode 100644 index 76490987f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/gcp_cr.etc +++ /dev/null @@ -1,185 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â Î»C:candidate. âG,L1,L2,T1,T2. â¦L1, T1â¦ â³[â»] â¦L2, T2â¦ â - C G L1 T1 â C G L2 T2. - -definition S0s â Î»C:candidate. âG,L1,L2,T1,T2. â¦L1, T1â¦ â³*[â»] â¦L2, T2â¦ â - C G L1 T1 â C G L2 T2. - -(* Note: this is Girard's CR1 *) -definition S1 â Î»RP,C:candidate. - âG,L,T. C G L T â RP G L T. - -(* Note: this is Tait's iii, or Girard's CR4 *) -definition S2 â Î»RR:relation4 genv lenv term term. Î»RS:relation term. Î»RP,C:candidate. - âG,L,Vs. all â¦ (RP G L) Vs â - âT. ðâ¦Tâ¦ â NF â¦ (RR G L) RS T â C G L (â¶Vs.T). - -(* Note: this generalizes Tait's ii *) -definition S3 â Î»C:candidate. - âa,G,L,Vs,V,T,W. - C G L (â¶Vs.â{a}âW.V.T) â C G L (â¶Vs.âV.â{a}W.T). - -definition S4 â Î»RP,C:candidate. - âG,L,Vs. all â¦ (RP G L) Vs â âk. C G L (â¶Vs.âk). - -definition S5 â Î»C:candidate. âI,G,L,K,Vs,V1,V2,i. - C G L (â¶Vs.V2) â â§[0, i+1] V1 â¡ V2 â - â©[i] L â¡ K.â{I}V1 â C G L (â¶Vs.#i). - -definition S6 â Î»RP,C:candidate. - âG,L,V1s,V2s. â§[0, 1] V1s â¡ V2s â - âa,V,T. C G (L.âV) (â¶V2s.T) â RP G L V â C G L (â¶V1s.â{a}V.T). - -definition S7 â Î»C:candidate. - âG,L,Vs,T,W. C G L (â¶Vs.T) â C G L (â¶Vs.W) â C G L (â¶Vs.âW.T). - -(* requirements for the generic reducibility candidate *) -record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop â -{ 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 â candidate â candidate â - Î»C1,C2,G,K,T. âL,V,U. â¦K, Tâ¦ â³*[â»] â¦L, Uâ¦ â - C1 G L V â C2 G L (âV.U). - -(* the reducibility candidate associated to an atomic arity *) -let rec acr (RP:candidate) (A:aarity) on A: candidate â -match A with -[ AAtom â RP -| APair B A â 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: âC. S0 C â S0s C. -#C #HC #G #L1 #L2 #T1 #T2 #H @(fpas_ind â¦ H) -L2 -T2 /3 width=5 by/ -qed. -(* -lemma rp_lifts: âRR,RS,RP. gcr RR RS RP RP â - âdes,G,L0,L,V,V0. â©*[â», des] L0 â¡ L â â§*[des] V â¡ V0 â - RP G L V â RP G L0 V0. -#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV -@gcr_lifts /width=7 by/ -@(s0 â¦ HRP) -qed. - -(* Basic_1: was only: sns3_lifts1 *) -lemma rp_liftsv_all: âRR,RS,RP. gcr RR RS RP RP â - âdes,G,L0,L,Vs,V0s. â©*[â», des] L0 â¡ L â â§*[des] Vs â¡ V0s â - all â¦ (RP G L) Vs â all â¦ (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: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â - âA. 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 â¦ H1RP G L) #k #HK - lapply (H L (âk) T ? ?) -H // - [ @(s2 â¦ IHB â¦ (â)) // - | #H @(cp2 â¦ H1RP â¦ k) @(s1 â¦ IHA) // - ] -| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #H #HB - elim (lifts_inv_applv1 â¦ H) -H #V0s #T0 #HV0s #HT0 #H destruct - lapply (s1 â¦ IHB â¦ HB) #HV0 - @(s2 â¦ IHA â¦ (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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct - elim (lifts_inv_flat1 â¦ HY) -HY #U0 #X #HU0 #HX #H destruct - elim (lifts_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT0 #H destruct - @(s3 â¦ IHA â¦ (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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct - >(lifts_inv_sort1 â¦ HY) -Y - lapply (s1 â¦ IHB â¦ HB) #HV0 - @(s4 â¦ IHA â¦ (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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct - elim (lifts_inv_lref1 â¦ HY) -HY #i0 #Hi0 #H destruct - elim (drops_drop_trans â¦ HL0 â¦ HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0 - >(at_mono â¦ Hi1 â¦ Hi0) in HL02; -i1 #HL02 - elim (drops_inv_skip2 â¦ Hdes0 â¦ H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct - elim (lift_total W1 0 (i0 + 1)) #W2 #HW12 - elim (lifts_lift_trans â¦ Hdes0 â¦ HVW1 â¦ HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2 - >(lift_mono â¦ HV13 â¦ HV12) in HVW2; -V3 #HVW2 - @(s5 â¦ IHA â¦ (V0 @ V0s) â¦ 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 â¦ H) -H #V10s #Y #HV10s #HY #H destruct - elim (lifts_inv_bind1 â¦ 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 â¦ IHA â¦ (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/ - @(HA â¦ (des + 1)) /2 width=2 by drops_skip/ - [ @lifts_applv // - elim (liftsv_liftv_trans_le â¦ HV10s â¦ HV120s) -V10s #V10s #HV10s #HV120s - >(liftv_mono â¦ HV12s â¦ HV10s) -V1s // - | @(s0 â¦ IHB â¦ HB â¦ 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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct - elim (lifts_inv_flat1 â¦ HY) -HY #W0 #T0 #HW0 #HT0 #H destruct - @(s7 â¦ IHA â¦ (V0 @ V0s)) /3 width=5 by lifts_applv/ -] -qed. - -lemma acr_abst: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â - âa,G,L,W,T,A,B. â¦G, L, Wâ¦ Ïµ[RP] ãBã â ( - âL0,V0,W0,T0,des. â©*[â», des] L0 â¡ L â â§*[des] W â¡ W0 â â§*[des + 1] T â¡ T0 â - â¦G, L0, V0â¦ Ïµ[RP] ãBã â â¦G, L0, W0â¦ Ïµ[RP] ãBã â â¦G, L0.ââW0.V0, T0â¦ Ïµ[RP] ãAã - ) â - â¦G, L, â{a}W.Tâ¦ Ïµ[RP] ãâ¡B.Aã. -#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HL0 #H #HB -lapply (acr_gcr â¦ H1RP H2RP A) #HCA -lapply (acr_gcr â¦ H1RP H2RP B) #HCB -elim (lifts_inv_bind1 â¦ H) -H #W0 #T0 #HW0 #HT0 #H destruct -lapply (gcr_lifts â¦ HL0 â¦ HW0 HW) -HW [ @(s0 â¦ HCB) ] #HW0 -@(s3 â¦ HCA â¦ (â)) -@(s6 â¦ HCA â¦ (â) (â)) // -[ @(HA â¦ HL0) // -| lapply (s1 â¦ HCB) -HCB #HCB - @(s7 â¦ H2RP â¦ (â)) /2 width=1 by/ -] -qed. - -(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *) -(* Basic_1: removed local theorems 1: sc3_sn3_abst *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/rajust_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/rajust_5.etc deleted file mode 100644 index ef0fbc49b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/rajust_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 L1, break term 46 T1 â¦ â³ [ break term 46 s ] break â¦ term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'RAjust $L1 $T1 $s $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/rajuststar_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/rajuststar_5.etc deleted file mode 100644 index 7cf9191d6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpa/rajuststar_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 L1, break term 46 T1 â¦ â³ * [ break term 46 s ] break â¦ term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'RAjustStar $L1 $T1 $s $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbc.etc deleted file mode 100644 index 9c803f041..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbc.etc +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â tri_relation genv lenv term â - Î»h,g,G1,L1,T1,G2,L2,T2. - ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¡[0] â¦G2, L2, T2â¦. - -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: âh,g,G1,G2,L1,L2,T1,T2. - â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦. -/2 width=5 by ex2_3_intro/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbc_fleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbc_fleq.etc deleted file mode 100644 index aa5cd7f2b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbc_fleq.etc +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G, L, Tâ¦ â - â¦G, L, Tâ¦ â¡[0] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦. -#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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¡[0] â¦G, L, Tâ¦ â - â¦G, L, Tâ¦ â»â¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 * -#G0 #L0 #T0 #H0 #H02 elim (fleq_fpbu_trans â¦ H1 â¦ H0) -G -L -T -/3 width=9 by fleq_trans, ex2_3_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg.etc deleted file mode 100644 index 574c35e10..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg.etc +++ /dev/null @@ -1,61 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â tri_relation genv lenv term â - Î»h,g. tri_TC â¦ (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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -/2 width=1 by tri_inj/ qed. - -lemma fpbg_strap1: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. - â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â»â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -/2 width=5 by tri_step/ qed. - -lemma fpbg_strap2: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. - â¦G1, L1, T1â¦ â»â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -/2 width=5 by tri_TC_strap/ qed. - -lemma fpbu_fpbg: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»[h,g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -/3 width=1 by fpbu_fpbc, fpbc_fpbg/ qed. - -(* Basic eliminators ********************************************************) - -lemma fpbg_ind: âh,g,G1,L1,T1. âR:relation3 â¦. - (âG2,L2,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â R G2 L2 T2) â - (âG,G2,L,L2,T,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â»â¡[h, g] â¦G2, L2, T2â¦ â R G L T â R G2 L2 T2) â - âG2,L2,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â R G2 L2 T2. -#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H -@(tri_TC_ind â¦ IH1 IH2 G2 L2 T2 H) -qed-. - -lemma fpbg_ind_dx: âh,g,G2,L2,T2. âR:relation3 â¦. - (âG1,L1,T1. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â R G1 L1 T1) â - (âG1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â R G L T â R G1 L1 T1) â - âG1,L1,T1. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â R G1 L1 T1. -#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H -@(tri_TC_ind_dx â¦ IH1 IH2 G1 L1 T1 H) -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fleq.etc deleted file mode 100644 index 754811ec2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fleq.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G1,G,L1,L,T1,T. â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â - âG2,L2,T2. â¦G, L, Tâ¦ â¡[0] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind â¦ 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: âh,g,G,G2,L,L2,T,T2. â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â - âG1,L1,T1. â¦G1, L1, T1â¦ â¡[0] â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx â¦ H) -G -L -T -[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/ -| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fpbg.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fpbg.etc deleted file mode 100644 index 57dd5ebb4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fpbg.etc +++ /dev/null @@ -1,53 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â - ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx â¦ 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind â¦ H) -G2 -L2 -T2 -[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans â¦ IH1) -IH1 ] (**) (* full auto fails *) -/2 width=1 by fpbc_fpbs/ -qed-. - -(* Advanced properties ******************************************************) - -lemma fpbs_fpbu_trans: âh,g,F1,F2,K1,K2,T1,T2. â¦F1, K1, T1â¦ â¥[h, g] â¦F2, K2, T2â¦ â - âG2,L2,U2. â¦F2, K2, T2â¦ â»[h, g] â¦G2, L2, U2â¦ â - ââG1,L1,U1. â¦F1, K1, T1â¦ â»[h, g] â¦G1, L1, U1â¦ & â¦G1, L1, U1â¦ â¥[h, g] â¦G2, L2, U2â¦. -/5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-. - -(* Man properties ***********************************************************) - -theorem fpbg_trans: âh,g. tri_transitive â¦ (fpbg h g). -/2 width=5 by tri_TC_transitive/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fpbs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fpbs.etc deleted file mode 100644 index 4b8662295..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_fpbs.etc +++ /dev/null @@ -1,88 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. - â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â½[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu â¦ H2) -H2 -/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/ -qed-. - -lemma fpb_fpbg_trans: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. - â¦G1, L1, T1â¦ â½[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu â¦ 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: âh,g,G1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â¥[h, g] â¦G, L, Tâ¦ â - âG2,L2,T2. â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind â¦ H) -G -L -T /3 width=5 by fpb_fpbg_trans/ -qed-. - -(* Note: this is used in the closure proof *) -lemma fpbg_fpbs_trans: âh,g,G,G2,L,L2,T,T2. â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦ â - âG1,L1,T1. â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx â¦ H) -G -L -T /3 width=5 by fpbg_fpb_trans/ -qed-. - -lemma fpbu_fpbs_fpbg: âh,g,G1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ â - âG2,L2,T2. â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed. - -(* Note: this is used in the closure proof *) -lemma fqup_fpbg: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn â¦ H) -H -/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/ -qed. - -lemma cpxs_fpbg: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡*[h, g] T2 â - (T1 = T2 â â¥) â â¦G, L, T1â¦ >â¡[h, g] â¦G, L, T2â¦. -#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn â¦ H â¦ H0) -H -H0 -/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/ -qed. - -lemma lstas_fpbg: âh,g,G,L,T1,T2,l2. â¦G, Lâ¦ â¢ T1 â¢*[h, l2] T2 â (T1 = T2 â â¥) â - âl1. l2 â¤ l1 â â¦G, Lâ¦ â¢ T1 âª[h, g] l1 â â¦G, L, T1â¦ >â¡[h, g] â¦G, L, T2â¦. -/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed. - -lemma lpxs_fpbg: âh,g,G,L1,L2,T. â¦G, L1â¦ â¢ â¡*[h, g] L2 â - (L1 â¡[T, 0] L2 â â¥) â â¦G, L1, Tâ¦ >â¡[h, g] â¦G, L2, Tâ¦. -#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn â¦ H â¦ H0) -H -H0 -/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/ -qed. - -lemma fpbs_fpbg: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â¡[0] â¦G2, L2, T2â¦ â¨ - â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind â¦ H) -G2 -L2 -T2 -[ /2 width=1 by or_introl/ -| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_lift.etc deleted file mode 100644 index 58ecd1200..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbg_lift.etc +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 âª[h, g] l+1 â - â¦G, Lâ¦ â¢ T1 â¢*[h, 1] T2 â â¦G, L, T1â¦ >â¡[h, g] â¦G, L, T2â¦. -/4 width=2 by fpbu_fpbg, sta_fpbu/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbs_fpbc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbs_fpbc.etc deleted file mode 100644 index a88d51925..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/fpbs_fpbc.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -/3 width=5 by fpbu_fwd_fpb, fpbs_strap2, fleq_fpbs/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/lazybtpredproper_8.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/lazybtpredproper_8.etc deleted file mode 100644 index 6abdbeb59..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpbc/lazybtpredproper_8.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â»â¡ break [ term 46 h, break term 46 g ] break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'LazyBTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq.etc deleted file mode 100644 index c909e2ed6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq.etc +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â - Î»G1,L1,T1,G2,L2,T2. - â§â§ 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 â¦ bteq. -/2 width=1 by and3_intro/ qed. - -lemma bteq_sym: tri_symmetric â¦ bteq. -#G1 #G2 #L1 #L2 #T1 #T2 * // -qed-. - -lemma bteq_dec: âG1,G2,L1,L2,T1,T2. Decidable (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦). -#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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq_6.etc deleted file mode 100644 index 5ee962fe5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'BTEq $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq_bteq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq_bteq.etc deleted file mode 100644 index 7870b2644..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/bteq_bteq.etc +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ bteq. -#G1 #G #L1 #L #T1 #T * // -qed-. - -theorem bteq_canc_sn: âG,G1,G2,L,L1,L2,T,T1,T2. â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ â - â¦G, L, Tâ¦ â â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦. -/3 width=5 by bteq_trans, bteq_sym/ qed-. - -theorem bteq_canc_dx: âG1,G2,G,L1,L2,L,T1,T2,T. â¦G1, L1, T1â¦ â â¦G, L, Tâ¦ â - â¦G2, L2, T2â¦ â â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦. -/3 width=5 by bteq_trans, bteq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/btpredsn_8.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/btpredsn_8.etc deleted file mode 100644 index 0350d1495..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/btpredsn_8.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â¢ â â¡ break [ term 46 h , break term 46 g ] break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'BTPRedSn $h $g $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpb.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpb.etc deleted file mode 100644 index e6698349d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpb.etc +++ /dev/null @@ -1,14 +0,0 @@ -include "basic_2/reduction/fpn.ma". - -(* Basic forward lemmas *****************************************************) - -lemma fpb_bteq_fwd_fpn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G2, L2, T2â¦. -#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 â¦ H) -H - [ #H1 #H2 elim (fqu_fwd_bteq â¦ H1 H2) - | * #HG #HL #HT #_ destruct // - ] -| #T2 #HT12 * // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpbc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpbc.etc deleted file mode 100644 index 1ed366cdd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpbc.etc +++ /dev/null @@ -1,24 +0,0 @@ -lemma fpb_fpbc_or_fpn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â¨ - â¦G1, L1, T1â¦ â¢ â¡[h,g] â¦G2, L2, T2â¦. -#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 â¦ 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â - (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â - â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 elim (fpb_fpbc_or_fpn â¦ H) -H // -#H elim H0 -H0 /2 width=3 by fpn_fwd_bteq/ -qed. - -lemma fpbc_fwd_bteq: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥. -#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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpbs_conj.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpbs_conj.etc deleted file mode 100644 index 8d0f137df..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpbs_conj.etc +++ /dev/null @@ -1,34 +0,0 @@ -(* -inclade "basic_2/computation/fpns.ma". -inclade "basic_2/computation/fpbs.ma". -inclade "basic_2/reduction/fpbc.ma". - -lemma fpn_dec: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â - Decidable (â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G2, L2, T2â¦). -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fpb_fpbc_or_fpn â¦ H) -H /2 width=1 by or_introl/ -#H12 @or_intror -#H @(fpbc_fwd_bteq â¦ H12) -H12 @(fpn_fwd_bteq â¦ H) -qed-. -*) -(* -lemma fpns_dec: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - Decidable (â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦). -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind â¦ H) -G2 -L2 -T2 /2 width=1 by or_introl/ -#G #G2 #L #L2 #T #T2 #H #H2 * -#H1 elim (fpn_dec â¦ 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind â¦ 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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpn.etc deleted file mode 100644 index 406e6eb8e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpn.etc +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| fpn_intro: âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â L1 â[T] L2 â 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: âh,g. tri_reflexive â¦ (fpn h g). -/2 width=1 by fpn_intro/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma fpn_inv_gen: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â - â§â§ G1 = G2 & â¦G1, L1â¦ â¢ â¡[h, g] L2 & L1 â[T1] L2 & T1 = T2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and4_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpn_fpn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpn_fpn.etc deleted file mode 100644 index adda4b538..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpn_fpn.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â - âG2,L2,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â - ââF2,K2,V2. â¦F1, K1, V1â¦ â â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. -#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 -#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqu_trans â¦ H12 â¦ HKL1 HV1) -L1 -/3 width=5 by fpn_intro, ex2_3_intro/ -qed-. - -lemma fpn_fquq_trans: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â - âG2,L2,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â - ââF2,K2,V2. â¦F1, K1, V1â¦ ââ¸® â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. -#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 -#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fquq_trans â¦ H12 â¦ HKL1 HV1) -L1 -/3 width=5 by fpn_intro, ex2_3_intro/ -qed-. - -lemma fpn_fqup_trans: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â - âG2,L2,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â - ââF2,K2,V2. â¦F1, K1, V1â¦ â+ â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. -#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 -#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqup_trans â¦ H12 â¦ HKL1 HV1) -L1 -/3 width=5 by fpn_intro, ex2_3_intro/ -qed-. - -lemma fpn_fqus_trans: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â - âG2,L2,T2. â¦G1, L1, T1â¦ â* â¦G2, L2, T2â¦ â - ââF2,K2,V2. â¦F1, K1, V1â¦ â* â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. -#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 -#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqus_trans â¦ H12 â¦ HKL1 HV1) -L1 -/3 width=5 by fpn_intro, ex2_3_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpns.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpns.etc deleted file mode 100644 index 0a208fedf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpns.etc +++ /dev/null @@ -1,58 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â tri_relation genv lenv term â - Î»h,g. tri_TC â¦ (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: âh,g,G1,L1,T1. âR:relation3 â¦. R G1 L1 T1 â - (âG,G2,L,L2,T,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â R G L T â R G2 L2 T2) â - âG2,L2,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â R G2 L2 T2. -#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H -lapply (tri_TC_star_ind â¦ IH1 IH2 G2 L2 T2 H) // -qed-. - -lemma fpns_ind_dx: âh,g,G2,L2,T2. âR:relation3 â¦. R G2 L2 T2 â - (âG1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â R G L T â R G1 L1 T1) â - âG1,L1,T1. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â R G1 L1 T1. -#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H -@(tri_TC_star_ind_dx â¦ IH1 IH2 G1 L1 T1 H) // -qed-. - -(* Basic_properties *********************************************************) - -lemma fpns_refl: âh,g. tri_reflexive â¦ (fpns h g). -/2 width=1 by tri_inj/ qed. - -lemma fpn_fpns: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦. -/2 width=1 by tri_inj/ qed. - -lemma fpns_strap1: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G, L, Tâ¦ â - â¦G, L, Tâ¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦. -/2 width=5 by tri_step/ qed-. - -lemma fpns_strap2: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G, L, Tâ¦ â - â¦G, L, Tâ¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦. -/2 width=5 by tri_TC_strap/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpns_fpns.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpns_fpns.etc deleted file mode 100644 index 5aaea6a9e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fpns_fpns.etc +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â - â§â§ G1 = G2 & â¦G1, L1â¦ â¢ â¡*[h, g] L2 & L1 â[T1] L2 & T1 = T2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpns_ind â¦ 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 â¦ H2) -H2 #HG #HL2 #HT #HT2 destruct -/3 width=3 by lpxs_strap1, lleq_trans, and4_intro/ -qed-. - -(* Advanced properties ******************************************************) - -lemma lpxs_lleq_fpns: âh,g,G,L1,L2,T. â¦G, L1â¦ â¢ â¡*[h, g] L2 â L1 â[T] L2 â - â¦G, L1, Tâ¦ â¢ ââ¡*[h, g] â¦G, L2, Tâ¦. -#h #g #G #L1 #L2 #T #H @(lpxs_ind â¦ 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 â¦ fqus. -/2 width=5 by tri_TC_transitive/ qed-. - - -(* -lemma fpns_refl: âh,g. tri_reflexive â¦ (fpns h g). -/2 width=1 by and3_intro/ qed. - -lemma fpn_fpns: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=1 by lpx_lpxs, and3_intro/ -qed. - -lemma fpns_strap1: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G, L, Tâ¦ â - â¦G, L, Tâ¦ â¢ â¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. -#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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G, L, Tâ¦ â - â¦G, L, Tâ¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. -#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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=4 by lpxs_fwd_length, and3_intro/ -qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fqu.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fqu.etc deleted file mode 100644 index 1e4defb3c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fqu.etc +++ /dev/null @@ -1,14 +0,0 @@ -include "basic_2/grammar/bteq.ma". - -lemma fqu_fwd_bteq: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥. -#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 â¦ H) -| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_x â¦ H) -| #a #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y â¦ H) -| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y â¦ H) -| #G #L #K #T #U #e #HLK #_ * #_ #H - lapply (ldrop_fwd_length_lt4 â¦ HLK ?) // >H -L #H - elim (lt_refl_false â¦ H) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb.etc deleted file mode 100644 index 3cd6861ad..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb.etc +++ /dev/null @@ -1,12 +0,0 @@ -(* Basic eliminators ********************************************************) - -theorem fsb_ind_alt: âh,g. âR: relation3 â¦. ( - âG1,L1,T1. â¦G1, L1â¦ â¢ â¦¥[h,g] T1 â ( - âG2,L2,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â - (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â R G2 L2 T2 - ) â R G1 L1 T1 - ) â - âG,L,T. â¦G, Lâ¦ â¢ â¦¥[h, g] T â 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb_alt.etc deleted file mode 100644 index 8c69a001d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb_alt.etc +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| fsba_intro: âG1,L1,T1. ( - âG2,L2,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â fsba h g G2 L2 T2 - ) â 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: âh,g. âR: relation3 â¦. ( - âG1,L1,T1. â¦G1, L1â¦ â¢ â¦¥â¦¥[h,g] T1 â ( - âG2,L2,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â - (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â R G2 L2 T2 - ) â R G1 L1 T1 - ) â - âG,L,T. â¦G, Lâ¦ â¢ â¦¥â¦¥[h, g] T â 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: âh,g,G1,L1,T1. ( - âG,G2,L,L2,T,T2. â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦ â - â¦G1, L1, T1â¦ â â¦G, L, Tâ¦ â - (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â fsba h g G2 L2 T2 - ) â fsba h g G1 L1 T1. -/4 width=5 by fsba_intro/ qed-. - -lemma fsba_fpbs_trans: âh,g,G1,L1,T1. â¦G1, L1â¦ â¢ â¦¥â¦¥[h, g] T1 â - âG2,L2,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â â¦G2, L2â¦ â¢ â¦¥â¦¥[h, g] T2. -#h #g #G1 #L1 #T1 #H @(fsba_ind_alt â¦ H) -G1 -L1 -T1 -#G1 #L1 #T1 #H0 #IH0 #G #L #T #H1 @fsba_intro -#G2 #L2 #T2 #H2 #_ lapply (fpbs_trans â¦ H1 â¦ 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 â¦ H) -IH0 -H // @(fpbs_trans â¦ H1 â¦ H2) - -lemma fsba_intro_fpb: âh,g,G1,L1,T1. ( - âG2,L2,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â - (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â â¦G2, L2â¦ â¢ â¦¥â¦¥[h, g] T2 - ) â â¦G1, L1â¦ â¢ â¦¥â¦¥[h, g] T1. -#h #g #G1 #L1 #T1 #IH1 @fsba_intro_aux -#G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx â¦ 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 â¦ H21) -IH1 -H21 - [ - | -H10 -H210 #H -(* -(* Main inversion lemmas ****************************************************) - -theorem fsba_inv_fsb: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¦¥â¦¥[h, g] T â â¦G, Lâ¦ â¢ â¦¥[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: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¦¥[h, g] T â â¦G, Lâ¦ â¢ â¦¥â¦¥[h, g] T. -#h #g #G #L #T #H @(fsb_ind_alt â¦ H) -G -L -T -/4 width=1 by fsba_intro_fpb/ -qed. -(* -| fsba_intro: âG1,L1,T1. ( - âG2,L2,T2. â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â fsb h g G2 L2 T2 - ) â fsb h g G1 L1 T1 -. - - - -(****************************************************************************) - -include "basic_2/substitution/fqup.ma". - -lemma fsb_csx: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¬*[h, g] T â â¦G, Lâ¦ â¢ â¦¥[h, g] T. -#h #g #G #L #T #H @(csx_ind â¦ H) -T -*)*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/lazyeq_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/lazyeq_6.etc deleted file mode 100644 index 105a47363..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/lazyeq_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'LazyEq $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr.etc deleted file mode 100644 index f58a52c37..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr.etc +++ /dev/null @@ -1,59 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break â¦ term 46 L1, break term 46 T1 â¦ â¡ break â¦ term 46 L2 , break term 46 T2 â¦ )" - 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 â bi_relation lenv term â - Î»L,L1,T1,L2,T2. |L1| = |L2| â§ L â¢ L1 @@ T1 â¡ 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: âL. bi_reflexive â¦ (cfpr L). -/2 width=1/ qed. - -lemma fpr_cfpr: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â â¢ â¦L1, T1â¦ â¡ â¦L2, T2â¦. -#L1 #L2 #T1 #T2 * /3 width=1/ -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma cfpr_inv_atom1: âL,L2,T1,T2. L â¢ â¦â, T1â¦ â¡ â¦L2, T2â¦ â L â¢ T1 â¡ T2 â§ L2 = â. -#L #L2 #T1 #T2 * #H >(length_inv_zero_sn â¦ H) /2 width=1/ -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma fpr_inv_pair1_sn: âI,K1,L2,V1,T1,T2. â¦â.â{I}V1@@K1, T1â¦ â¡ â¦L2, T2â¦ â - ââK2,V2. V1 â¡ V2 & - â.â{I}V2 â¢ â¦K1, T1â¦ â¡ â¦K2, T2â¦ & - L2 = â.â{I}V2@@K2. -#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H -elim (length_inv_pos_sn_append â¦ H) -H #I2 #K2 #V2 #HK12 #H destruct ->shift_append_assoc >shift_append_assoc normalize in â¢ (%â?); #H -elim (tpr_inv_bind1 â¦ H) -H * -[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/ -| #T0 #_ #_ #H destruct -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_aaa.etc deleted file mode 100644 index 72802e499..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_aaa.etc +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T1,A. L1 â¢ T1 â A â - âL2,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â L2 â¢ T2 â A. -#L1 #T1 #A #HT1 #L2 #T2 #H -elim (fpr_inv_all â¦ H) -H -/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_cfpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_cfpr.etc deleted file mode 100644 index f442be28b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_cfpr.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL. bi_confluent â¦ (cfpr L). -#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02 -elim (cpr_conf â¦ HT01 HT02) -L0 -T0 #X #H1 #H2 -elim (cpr_fwd_shift1 â¦ H1) #L0 #T0 #HL10 #H destruct /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_cpr.etc deleted file mode 100644 index a1e20a025..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_cpr.etc +++ /dev/null @@ -1,64 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L. L1 â¡ L â âL2,T1,T2. L â¢ T1 â¡ T2 â - L â¢ â¶* [0, |L|] L2 â â¦L1, T1â¦ â¡ â¦L2, T2â¦. -#L1 #L #H elim H -L1 -L -[ #L2 #T1 #T2 #HT12 #HL2 - lapply (ltpss_sn_inv_atom1 â¦ HL2) -HL2 #H destruct - lapply (cpr_inv_atom â¦ HT12) -HT12 /2 width=1/ -| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H - elim (ltpss_sn_inv_tpss21 â¦ H ?) -H // append_length >append_length #H - lapply (injective_plus_r â¦ H) -H #H - @(ex3_intro â¦ (â.â{I}V@@Y)) append_length H -H >commutative_plus /3 width=1/ -] -qed-. - -lemma fpr_inv_all: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â - ââL. L1 â¡ L & L â¢ T1 â¡ T2 & L â¢ â¶* [0, |L|] L2. -#L1 #L2 #T1 #T2 #H -lapply (fpr_cfpr â¦ H) -H #H -elim (cfpr_inv_all â¦ H) -H /2 width=4/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_ltpss.etc deleted file mode 100644 index 66e102e63..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/cfpr_ltpss.etc +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âI,L,K1,L2,V1,T1,T2. L â¢ â¦â.â{I}V1@@K1, T1â¦ â¡ â¦L2, T2â¦ â - ââK2,V,V2. V1 â¡ V & L â¢ V â¶* [0, |L|] V2 & - L.â{I}V â¢ â¦K1, T1â¦ â¡ â¦K2, T2â¦ & - L2 = â.â{I}V2@@K2. -* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H -elim (length_inv_pos_sn_append â¦ H) -H #I2 #K2 #V2 #HK12 #H destruct ->shift_append_assoc >shift_append_assoc normalize in â¢ (??%%â?); #H -[ elim (cpr_inv_abbr1 â¦ H) -H * - [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/ - | #T0 #_ #_ #H destruct - ] -| elim (cpr_inv_abst1 â¦ H Abst V2) -H - #V #T * #V0 #HV10 #HV0 #HT1 #H destruct - lapply (ltpss_sn_cpr_trans (L.âV0) â¦ 0 (|L|+1) â¦ HT1) -HT1 /2 width=1/ #HT12 - /3 width=7/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpc.etc deleted file mode 100644 index ee362471b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpc.etc +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1 , break term 46 T1 â¦ â¬ break â¦ term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }. - -notation "hvbox( â¦ term 46 L1 , break term 46 T1 â¦ â¬ â¬ break â¦ term 46 L2 , break term 46 T2 â¦ )" - 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 â - Î»L1,T1,L2,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â¨ â¦L2, T2â¦ â¡ â¦L1, T1â¦. - -interpretation - "context-free parallel conversion (closure)" - 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2). - -(* Basic properties *********************************************************) - -lemma fpc_refl: bi_reflexive â¦ fpc. -/2 width=1/ qed. - -lemma fpc_sym: bi_symmetric â¦ fpc. -#L1 #L2 #T1 #T2 * /2 width=1/ -qed. - -(* Basic forward lemmas *****************************************************) - -lemma fpc_fwd_fpr: âL1,L2,T1,T2. â¦L1, T1â¦ â¬ â¦L2, T2â¦ â - ââL,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ & â¦L2, T2â¦ â¡ â¦L, Tâ¦. -#L1 #L2 #T1 #T2 * /2 width=4/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpc_fpc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpc_fpc.etc deleted file mode 100644 index 22fc16f37..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpc_fpc.etc +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,L1,T0,T1. â¦L0, T0â¦ â¬ â¦L1, T1â¦ â - âL2,T2. â¦L0, T0â¦ â¬ â¦L2, T2â¦ â - ââL,T. â¦L1, T1â¦ â¬ â¦L, Tâ¦ & â¦L2, T2â¦ â¬ â¦L, Tâ¦. -/3 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs.etc deleted file mode 100644 index 68c946ce8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs.etc +++ /dev/null @@ -1,84 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1 , break term 46 T1 â¦ â¬ * break â¦ term 46 L2 , break term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }. - -notation "hvbox( â¦ term 46 L1 , break term 46 T1 â¦ â¬ â¬ * break â¦ term 46 L2 , break term 46 T2 â¦ )" - 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 â bi_TC â¦ fpc. - -interpretation "context-free parallel equivalence (closure)" - 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma fpcs_ind: âL1,T1. âR:relation2 lenv term. R L1 T1 â - (âL,L2,T,T2. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â â¦L, Tâ¦ â¬ â¦L2, T2â¦ â R L T â R L2 T2) â - âL2,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â R L2 T2. -/3 width=7 by bi_TC_star_ind/ qed-. - -lemma fpcs_ind_dx: âL2,T2. âR:relation2 lenv term. R L2 T2 â - (âL1,L,T1,T. â¦L1, T1â¦ â¬ â¦L, Tâ¦ â â¦L, Tâ¦ â¬* â¦L2, T2â¦ â R L T â R L1 T1) â - âL1,T1. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â R L1 T1. -/3 width=7 by bi_TC_star_ind_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma fpcs_refl: bi_reflexive â¦ fpcs. -/2 width=1/ qed. - -lemma fpcs_sym: bi_symmetric â¦ fpcs. -/3 width=1/ qed. - -lemma fpc_fpcs: âL1,L2,T1,T2. â¦L1, T1â¦ â¬ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/2 width=1/ qed. - -lemma fpcs_strap1: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â â¦L, Tâ¦ â¬ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/2 width=4/ qed. - -lemma fpcs_strap2: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¬ â¦L, Tâ¦ â â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/2 width=4/ qed. - -lemma fpcs_fpr_dx: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=1/ qed. - -lemma fpcs_fpr_sn: âL1,L2,T1,T2. â¦L2, T2â¦ â¡ â¦L1, T1â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=1/ qed. - -lemma fpcs_fpr_strap1: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â - âL2,T2. â¦L, Tâ¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4/ qed. - -lemma fpcs_fpr_strap2: âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â - âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4/ qed. - -lemma fpcs_fpr_div: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â - âL2,T2. â¦L2, T2â¦ â¡ â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4/ qed. - -lemma fpr_div: âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡ â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4/ qed-. - -lemma fpcs_fpr_conf: âL1,L,T1,T. â¦L, Tâ¦ â¡ â¦L1, T1â¦ â - âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_aaa.etc deleted file mode 100644 index 9f4327bff..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_aaa.etc +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â - âA1. L1 â¢ T1 â A1 â âA2. L2 â¢ T2 â A2 â - A1 = A2. -#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2 -elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 -lapply (aaa_fprs_conf â¦ HT1 â¦ H1) -L1 -T1 #HT1 -lapply (aaa_fprs_conf â¦ HT2 â¦ H2) -L2 -T2 #HT2 -lapply (aaa_mono â¦ HT1 â¦ HT2) -L -T // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_cpcs.etc deleted file mode 100644 index b1288f751..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_cpcs.etc +++ /dev/null @@ -1,86 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â âV1,V2. V1 â¡ V2 â - âI. â¦L1, â{I}V1.T1â¦ â¬* â¦L2, â{I}V2.T2â¦. -#L1 #L2 #T1 #T2 #HT12 -elim (fpcs_inv_fprs â¦ HT12) -HT12 -/3 width=6 by fprs_flat_dx_tpr, fprs_div/ (**) (* auto too slow without trace *) -qed. - -lemma fpcs_shift: âI,L1,L2,V1,V2,T1,T2. â¦L1, -â{I}V1.T1â¦ â¬* â¦L2, -â{I}V2.T2â¦ â - â¦L1.â{I}V1, T1â¦ â¬* â¦L2.â{I}V2, T2â¦. -#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 -elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 -elim (fprs_bind2_minus â¦ H1) -H1 #W1 #U1 #HTU1 #H destruct -elim (fprs_bind2_minus â¦ H2) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/ -qed. - -(* Advanced inversion lemmas ************************************************) - -lemma fpcs_inv_shift: âI,L1,L2,V1,V2,T1,T2. â¦L1.â{I}V1, T1â¦ â¬* â¦L2.â{I}V2, T2â¦ â - â¦L1, -â{I}V1.T1â¦ â¬* â¦L2, -â{I}V2.T2â¦. -#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 -elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 -elim (fprs_inv_pair1 â¦ H1) -H1 #K1 #U1 #_ #HTU1 #H destruct -elim (fprs_inv_pair1 â¦ H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma fpcs_fwd_bind_minus: âI,L1,L2,V1,V2,T1,T2. â¦L1, -â{I}V1.T1â¦ â¬* â¦L2, -â{I}V2.T2â¦ â - âb. â¦L1, â{b,I}V1.T1â¦ â¬* â¦L2, â{b,I}V2.T2â¦. -#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b -elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 -elim (fprs_fwd_bind2_minus â¦ H1 b) -H1 #W1 #U1 #HTU1 #H destruct -elim (fprs_fwd_bind2_minus â¦ H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/ -qed-. - -lemma fpcs_fwd_shift: âI,L1,L2,V1,V2,T1,T2. â¦L1.â{I}V1, T1â¦ â¬* â¦L2.â{I}V2, T2â¦ â - âb. â¦L1, â{b,I}V1.T1â¦ â¬* â¦L2, â{b,I}V2.T2â¦. -/3 width=1 by fpcs_inv_shift, fpcs_fwd_bind_minus/ qed-. - -lemma fpcs_fwd_abst24: âa,L1,L2,V1,V2,T1,T2. â¦L1, â{a}V1.T1â¦ â¬* â¦L2, â{a}V2.T2â¦ â - âb,I,W. â¦L1, â{b,I}W.T1â¦ â¬* â¦L2, â{b,I}W.T2â¦. -#a #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b #I #W -elim (fpcs_inv_fprs â¦ H12) -H12 #L #U #H1 #H2 -elim (fprs_fwd_abst2 â¦ H1 b I W) -H1 #W1 #U1 #HTU1 #H destruct -elim (fprs_fwd_abst2 â¦ H2 b I W) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/ -qed-. - -lemma fpcs_fwd_abst13: âL1,L2,V1,V2,T1,T2. â¦L1.âV1, T1â¦ â¬* â¦L2.âV2, T2â¦ â - âI,W. â¦L1.â{I}W, T1â¦ â¬* â¦L2.â{I}W, T2â¦. -/4 width=4 by fpcs_fwd_shift, fpcs_fwd_abst24, fpcs_shift/ qed-. - -(* Properties on context-sensitive parallel equivalence for terms ***********) - -lemma cpcs_fpcs: âL,T1,T2. L â¢ T1 â¬* T2 â â¦L, T1â¦ â¬* â¦L, T2â¦. -#L #T1 #T2 #H -elim (cpcs_inv_cprs â¦ 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: âL,T1,T2. â¦L, T1â¦ â¬* â¦L, T2â¦ â L â¢ T1 â¬* T2. -#L #T1 #T2 #H -elim (fpcs_inv_fprs â¦ H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_fpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_fpcs.etc deleted file mode 100644 index 270e8dc40..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_fpcs.etc +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â - ââL,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ & â¦L2, T2â¦ â¡* â¦L, Tâ¦. -#L1 #L2 #T1 #T2 #H @(fpcs_ind â¦ H) -L2 -T2 -[ /3 width=4/ -| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 - [ elim (fprs_strip â¦ HT2 â¦ HT0) -L -T #L #T #HT2 #HT0 - lapply (fprs_strap1 â¦ HT10 â¦ HT0) -L0 -T0 /2 width=4/ - | lapply (fprs_strap2 â¦ HT2 â¦ HT0) -L -T /2 width=4/ - ] -] -qed-. - -(* Advanced properties ******************************************************) - -lemma fpr_fprs_conf: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 -elim (fprs_strip â¦ HT2 â¦ HT1) /2 width=4 by fpr_fprs_div/ -qed-. - -lemma fprs_fpr_conf: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â â¦L2, T2â¦ â¬* â¦L1, T1â¦. -#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 -elim (fprs_strip â¦ HT2 â¦ HT1) /2 width=4 by fprs_fpr_div/ -qed-. - -lemma fprs_conf: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â â¦L, Tâ¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 -elim (fprs_conf â¦ HT1 â¦ HT2) /2 width=4/ -qed-. - -lemma fpcs_strip: âL0,L1,T0,T1. â¦L0, T0â¦ â¬ â¦L1, T1â¦ â - âL2,T2. â¦L0, T0â¦ â¬* â¦L2, T2â¦ â - ââL,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ & â¦L2, T2â¦ â¬ â¦L, Tâ¦. -/3 width=4/ qed. - -(* Main properties **********************************************************) - -theorem fpcs_trans: bi_transitive â¦ fpcs. -/2 width=4/ qed. - -theorem fpcs_canc_sn: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¬* â¦L1, T1â¦ â â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) - -theorem fpcs_canc_dx: âL1,L2,L,T1,T2,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â â¦L2, T2â¦ â¬* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_fprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_fprs.etc deleted file mode 100644 index 43239eadd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpcs_fprs.etc +++ /dev/null @@ -1,55 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L2 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 /width=1/ /3 width=4/ -qed. - -lemma fpcs_fprs_sn: âL1,L2,T1,T2. â¦L2, T2â¦ â¡* â¦L1, T1â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L2 #T1 #T2 #H @(fprs_ind_dx â¦ H) -L2 -T2 /width=1/ /3 width=4/ -qed. - -lemma fpcs_fprs_strap1: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â âL2,T2. â¦L, Tâ¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind â¦ H) -L2 -T2 /width=1/ /2 width=4/ -qed. - -lemma fpcs_fprs_strap2: âL1,L,T1,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx â¦ H) -L1 -T1 /width=1/ /2 width=4/ -qed. - -lemma fpcs_fprs_div: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx â¦ H) -L2 -T2 /width=1/ /2 width=4/ -qed. - -lemma fpcs_fprs_conf: âL1,L,T1,T. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind â¦ H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *) -qed. - -lemma fprs_div: âL1,L,T1,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx â¦ H) -L2 -T2 /2 width=1/ /2 width=4/ -qed. - -lemma fprs_fpr_div: âL1,L,T1,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡ â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=7 by bi_step, fprs_div/ qed-. - -lemma fpr_fprs_div: âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. -/3 width=4 by bi_step, fprs_div/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr.etc deleted file mode 100644 index 3bb168f9e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr.etc +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1, break term 46 T1 â¦ â¡ break â¦ term 46 L2 , break term 46 T2 â¦ )" - 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 â - Î»L1,T1,L2,T2. |L1| = |L2| â§ L1 @@ T1 â¡ 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 â¦ fpr. -/2 width=1/ qed. - -lemma fpr_shift: âI1,I2,L1,L2,V1,V2,T1,T2. - â¦L1, -â{I1}V1.T1â¦ â¡ â¦L2, -â{I2}V2.T2â¦ â - â¦L1.â{I1}V1, T1â¦ â¡ â¦L2.â{I2}V2, T2â¦. -#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12 -@conj // normalize // (**) (* explicit constructor *) -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma fpr_inv_atom1: âL2,T1,T2. â¦â, T1â¦ â¡ â¦L2, T2â¦ â T1 â¡ T2 â§ L2 = â. -#L2 #T1 #T2 * #H -lapply (length_inv_zero_sn â¦ H) -H #H destruct /2 width=1/ -qed-. - -lemma fpr_inv_atom3: âL1,T1,T2. â¦L1,T1â¦ â¡ â¦â,T2â¦ â T1 â¡ T2 â§ L1 = â. -#L1 #T1 #T2 * #H -lapply (length_inv_zero_dx â¦ H) -H #H destruct /2 width=1/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma fpr_fwd_pair1: âI1,K1,L2,V1,T1,T2. â¦K1.â{I1}V1, T1â¦ â¡ â¦L2, T2â¦ â - ââI2,K2,V2. â¦K1, -â{I1}V1.T1â¦ â¡ â¦K2, -â{I2}V2.T2â¦ & - L2 = K2.â{I2}V2. -#I1 #K1 #L2 #V1 #T1 #T2 * #H -elim (length_inv_pos_sn â¦ H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/ -qed-. - -lemma fpr_fwd_pair3: âI2,L1,K2,V2,T1,T2. â¦L1, T1â¦ â¡ â¦K2.â{I2}V2, T2â¦ â - ââI1,K1,V1. â¦K1, -â{I1}V1.T1â¦ â¡ â¦K2, -â{I2}V2.T2â¦ & - L1 = K1.â{I1}V1. -#I2 #L1 #K2 #V2 #T1 #T2 * #H -elim (length_inv_pos_dx â¦ H) -H #I1 #K1 #V1 #HK12 #H destruct /3 width=5/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr_cpr.etc deleted file mode 100644 index baa630f5b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr_cpr.etc +++ /dev/null @@ -1,125 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. L1 â¡ L2 â âT1,T2. T1 â¡ T2 â â¦L1, T1â¦ â¡ â¦L2, T2â¦. -/3 width=4/ qed. - -lemma cpr_fpr: âL,T1,T2. L â¢ T1 â¡ T2 â â¦L, T1â¦ â¡ â¦L, T2â¦. -/2 width=4/ qed. - -lemma fpr_lift: âK1,K2,T1,T2. â¦K1, T1â¦ â¡ â¦K2, T2â¦ â - âd,e,L1. â©[d, e] L1 â¡ K1 â - âU1,U2. â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â - ââL2. â¦L1, U1â¦ â¡ â¦L2, U2â¦ & â©[d, e] L2 â¡ K2. -#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2 -elim (fpr_inv_all â¦ HT12) -HT12 #K #HK1 #HT12 #HK2 -elim (ldrop_ltpr_trans â¦ HLK1 â¦ HK1) -K1 #L #HL1 #HLK -lapply (cpr_lift â¦ HLK â¦ HTU1 â¦ HTU2 HT12) -T1 -T2 #HU12 -elim (le_or_ge (|K|) d) #Hd -[ elim (ldrop_ltpss_sn_trans_ge â¦ HLK â¦ HK2 â¦) -| elim (ldrop_ltpss_sn_trans_be â¦ HLK â¦ HK2 â¦) -] // -Hd #L2 #HL2 #HLK2 -lapply (ltpss_sn_weak_full â¦ HL2) -K /3 width=4/ -qed-. - -(* Advanced properties ******************************************************) - -lemma fpr_flat_dx: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â âV1,V2. V1 â¡ V2 â - âI. â¦L1, â{I}V1.T1â¦ â¡ â¦L2, â{I}V2.T2â¦. -#L1 #L2 #T1 #T2 #HT12 -elim (fpr_inv_all â¦ HT12) -HT12 /4 width=4/ -qed. - -lemma fpr_bind_sn: âL1,L2,V1,V2. â¦L1, V1â¦ â¡ â¦L2, V2â¦ â âT1,T2. T1 â¡ T2 â - âa,I. â¦L1, â{a,I}V1.T1â¦ â¡ â¦L2, â{a,I}V2.T2â¦. -#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I -elim (fpr_inv_all â¦ H) /3 width=4/ -qed. - -lemma fpr_bind2_minus: âI,L1,L2,V1,T1,U2. â¦L1, -â{I}V1.T1â¦ â¡ â¦L2, U2â¦ â - ââV2,T2. â¦L1.â{I}V1, T1â¦ â¡ â¦L2.â{I}V2, T2â¦ & - U2 = -â{I}V2.T2. -#I1 #L1 #L2 #V1 #T1 #U2 #H -elim (fpr_inv_all â¦ H) -H #L #HL1 #H #HL2 -elim (cpr_fwd_bind1_minus â¦ H false) -H /4 width=4/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma fpr_fwd_bind2_minus: âI,L1,L,V1,T1,T. â¦L1, -â{I}V1.T1â¦ â¡ â¦L, Tâ¦ â âb. - ââV2,T2. â¦L1, â{b,I}V1.T1â¦ â¡ â¦L, â{b,I}V2.T2â¦ & - T = -â{I}V2.T2. -#I #L1 #L #V1 #T1 #T #H1 #b -elim (fpr_inv_all â¦ H1) -H1 #L0 #HL10 #HT1 #HL0 -elim (cpr_fwd_bind1_minus â¦ HT1 b) -HT1 /3 width=4/ -qed-. - -lemma fpr_fwd_shift_bind_minus: âI1,I2,L1,L2,V1,V2,T1,T2. - â¦L1, -â{I1}V1.T1â¦ â¡ â¦L2, -â{I2}V2.T2â¦ â - â¦L1, V1â¦ â¡ â¦L2, V2â¦ â§ I1 = I2. -* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H -elim (fpr_inv_all â¦ H) -H #L #HL1 #H #HL2 -[ elim (cpr_inv_abbr1 â¦ H) -H * - [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/ - | #T #_ #_ #H destruct - ] -| elim (cpr_inv_abst1 â¦ H Abst V2) -H - #V #T #HV1 #_ #H destruct /3 width=4/ -] -qed-. - -lemma fpr_fwd_abst2: âa,L1,L2,V1,T1,U2. â¦L1, â{a}V1.T1â¦ â¡ â¦L2, U2â¦ â âb,I,W. - ââV2,T2. â¦L1, â{b,I}W.T1â¦ â¡ â¦L2, â{b,I}W.T2â¦ & - U2 = â{a}V2.T2. -#a #L1 #L2 #V1 #T1 #U2 #H -elim (fpr_inv_all â¦ H) #L #HL1 #H #HL2 #b #I #W -elim (cpr_fwd_abst1 â¦ H b I W) -H /3 width=4/ -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma fpr_inv_pair1: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡ â¦L2, T2â¦ â - ââK2,V2. â¦K1, V1â¦ â¡ â¦K2, V2â¦ & - â¦K1, -â{I}V1.T1â¦ â¡ â¦K2, -â{I}V2.T2â¦ & - L2 = K2.â{I}V2. -#I1 #K1 #X #V1 #T1 #T2 #H -elim (fpr_fwd_pair1 â¦ H) -H #I2 #K2 #V2 #HT12 #H destruct -elim (fpr_fwd_shift_bind_minus â¦ HT12) #HV12 #H destruct /2 width=5/ -qed-. - -lemma fpr_inv_pair3: âI,L1,K2,V2,T1,T2. â¦L1, T1â¦ â¡ â¦K2.â{I}V2, T2â¦ â - ââK1,V1. â¦K1, V1â¦ â¡ â¦K2, V2â¦ & - â¦K1, -â{I}V1.T1â¦ â¡ â¦K2, -â{I}V2.T2â¦ & - L1 = K1.â{I}V1. -#I2 #X #K2 #V2 #T1 #T2 #H -elim (fpr_fwd_pair3 â¦ H) -H #I1 #K1 #V1 #HT12 #H destruct -elim (fpr_fwd_shift_bind_minus â¦ HT12) #HV12 #H destruct /2 width=5/ -qed-. - -(* More advanced forward lemmas *********************************************) - -lemma fpr_fwd_pair1_full: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡ â¦L2, T2â¦ â - âb. ââK2,V2. â¦K1, V1â¦ â¡ â¦K2, V2â¦ & - â¦K1, â{b,I}V1.T1â¦ â¡ â¦K2, â{b,I}V2.T2â¦ & - L2 = K2.â{I}V2. -#I #K1 #L2 #V1 #T1 #T2 #H #b -elim (fpr_inv_pair1 â¦ H) -H #K2 #V2 #HV12 #HT12 #H destruct -elim (fpr_fwd_bind2_minus â¦ HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr_fpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr_fpr.etc deleted file mode 100644 index 3f7ac2ceb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fpr_fpr.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ fpr. -#L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02 -elim (tpr_conf â¦ HT01 HT02) -L0 -T0 #X #H1 #H2 -elim (tpr_fwd_shift1 â¦ H1) #L #T #HL1 #H destruct /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs.etc deleted file mode 100644 index f42bdf7e2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs.etc +++ /dev/null @@ -1,58 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1 , term 46 T1 â¦ â¡ * break â¦ term 46 L2 , term 46 T2 â¦ )" - non associative with precedence 45 - for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }. - -notation "hvbox( â¦ term 46 L1 , term 46 T1 â¦ â¡ â¡ * break â¦ term 46 L2 , term 46 T2 â¦ )" - 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 â bi_TC â¦ fpr. - -interpretation "context-free parallel computation (closure)" - 'FocalizedPRedStar L1 T1 L2 T2 = (fprs L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma fprs_ind: âL1,T1. âR:relation2 lenv term. R L1 T1 â - (âL,L2,T,T2. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â R L T â R L2 T2) â - âL2,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â R L2 T2. -/3 width=7 by bi_TC_star_ind/ qed-. - -lemma fprs_ind_dx: âL2,T2. âR:relation2 lenv term. R L2 T2 â - (âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â â¦L, Tâ¦ â¡* â¦L2, T2â¦ â R L T â R L1 T1) â - âL1,T1. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â R L1 T1. -/3 width=7 by bi_TC_star_ind_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma fpr_fprs: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¡* â¦L2, T2â¦. -/2 width=1/ qed. - -lemma fprs_refl: bi_reflexive â¦ fprs. -/2 width=1/ qed. - -lemma fprs_strap1: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â - â¦L1, T1â¦ â¡* â¦L2, T2â¦. -/2 width=4/ qed. - -lemma fprs_strap2: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â â¦L, Tâ¦ â¡* â¦L2, T2â¦ â - â¦L1, T1â¦ â¡* â¦L2, T2â¦. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_aaa.etc deleted file mode 100644 index b76637ff7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_aaa.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T1,A. L1 â¢ T1 â A â - âL2,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â L2 â¢ T2 â A. -#L1 #T1 #A #HT1 #L2 #T2 #HLT12 -@(bi_TC_Conf3 â¦ HT1 ?? HLT12) /2 width=4/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_cprs.etc deleted file mode 100644 index 9d4d954b8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_cprs.etc +++ /dev/null @@ -1,138 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â âV1,V2. V1 â¡ V2 â - âI. â¦L1, â{I}V1.T1â¦ â¡* â¦L2, â{I}V2.T2â¦. -#L1 #L2 #T1 #T2 #HT12 @(fprs_ind â¦ HT12) -L2 -T2 /3 width=1/ -#L #L2 #T #T2 #_ #HT2 #IHT2 #V1 #V2 #HV12 #I -lapply (IHT2 â¦ HV12 I) -IHT2 -HV12 /3 width=6/ -qed. - -lemma fprs_bind2_minus: âI,L1,L2,V1,T1,U2. â¦L1, -â{I}V1.T1â¦ â¡* â¦L2, U2â¦ â - ââV2,T2. â¦L1.â{I}V1, T1â¦ â¡* â¦L2.â{I}V2, T2â¦ & - U2 = -â{I}V2.T2. -#I #L1 #L2 #V1 #T1 #U2 #H @(fprs_ind â¦ H) -L2 -U2 /2 width=4/ -#L #L2 #U #U2 #_ #HU2 * #V #T #HT1 #H destruct -elim (fpr_bind2_minus â¦ HU2) -HU2 /3 width=4/ -qed-. - -lemma fprs_lift: âK1,K2,T1,T2. â¦K1, T1â¦ â¡* â¦K2, T2â¦ â - âd,e,L1. â©[d, e] L1 â¡ K1 â - âU1,U2. â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â - ââL2. â¦L1, U1â¦ â¡* â¦L2, U2â¦ & â©[d, e] L2 â¡ K2. -#K1 #K2 #T1 #T2 #HT12 @(fprs_ind â¦ HT12) -K2 -T2 -[ #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2 - >(lift_mono â¦ HTU2 â¦ 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 â¦ HLK1 â¦ HTU1 HTU) -K1 -T1 #L #HU1 #HKL - elim (fpr_lift â¦ HT2 â¦ HKL â¦ HTU HTU2) -K -T -T2 /3 width=4/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma fprs_inv_pair1: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡* â¦L2, T2â¦ â - ââK2,V2. â¦K1, V1â¦ â¡* â¦K2, V2â¦ & - â¦K1, -â{I}V1.T1â¦ â¡* â¦K2, -â{I}V2.T2â¦ & - L2 = K2.â{I}V2. -#I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 /2 width=5/ -#L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct -elim (fpr_inv_pair1 â¦ HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/ -qed-. - -lemma fprs_inv_pair3: âI,L1,K2,V2,T1,T2. â¦L1, T1â¦ â¡* â¦K2.â{I}V2, T2â¦ â - ââK1,V1. â¦K1, V1â¦ â¡* â¦K2, V2â¦ & - â¦K1, -â{I}V1.T1â¦ â¡* â¦K2, -â{I}V2.T2â¦ & - L1 = K1.â{I}V1. -#I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx â¦ H) -L1 -T1 /2 width=5/ -#L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct -elim (fpr_inv_pair3 â¦ HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma fprs_fwd_bind2_minus: âI,L1,L,V1,T1,T. â¦L1, -â{I}V1.T1â¦ â¡* â¦L, Tâ¦ â âb. - ââV2,T2. â¦L1, â{b,I}V1.T1â¦ â¡* â¦L, â{b,I}V2.T2â¦ & - T = -â{I}V2.T2. -#I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind â¦ H1) -L -T /2 width=4/ -#L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct -elim (fpr_fwd_bind2_minus â¦ H0 b) -H0 /3 width=4/ -qed-. - -lemma fprs_fwd_pair1_full: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡* â¦L2, T2â¦ â - âb. ââK2,V2. â¦K1, V1â¦ â¡* â¦K2, V2â¦ & - â¦K1, â{b,I}V1.T1â¦ â¡* â¦K2, â{b,I}V2.T2â¦ & - L2 = K2.â{I}V2. -#I #K1 #L2 #V1 #T1 #T2 #H #b -elim (fprs_inv_pair1 â¦ H) -H #K2 #V2 #HV12 #HT12 #H destruct -elim (fprs_fwd_bind2_minus â¦ HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/ -qed-. - -lemma fprs_fwd_abst2: âa,L1,L2,V1,T1,U2. â¦L1, â{a}V1.T1â¦ â¡* â¦L2, U2â¦ â âb,I,W. - ââV2,T2. â¦L1, â{b,I}W.T1â¦ â¡* â¦L2, â{b,I}W.T2â¦ & - U2 = â{a}V2.T2. -#a #L1 #L2 #V1 #T1 #U2 #H #b #I #W @(fprs_ind â¦ H) -L2 -U2 /2 width=4/ -#L #L2 #U #U2 #_ #H2 * #V #T #HT1 #H destruct -elim (fpr_fwd_abst2 â¦ H2 b I W) -H2 /3 width=4/ -qed-. - -(* Properties on context-sensitive parallel computation for terms ***********) - -lemma cprs_fprs: âL,T1,T2. L â¢ T1 â¡* T2 â â¦L, T1â¦ â¡* â¦L, T2â¦. -#L #T1 #T2 #H @(cprs_ind â¦ H) -T2 // /3 width=4/ -qed. - -(* Forward lemmas on context-sensitive parallel computation for terms *******) - -lemma fprs_fwd_cprs: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â L1 â¢ T1 â¡* T2. -#L1 #L2 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 // -#L #L2 #T #T2 #H1 #H2 #IH1 -elim (fpr_inv_all â¦ H2) -H2 #L0 #HL0 #HT2 #_ -L2 -lapply (lfprs_cpr_trans L1 â¦ HT2) -HT2 /3 width=3/ -qed-. -(* -(* Advanced properties ******************************************************) - -lamma fpr_bind_sn: âL1,L2,V1,V2. â¦L1, V1â¦ â¡ â¦L2, V2â¦ â âT1,T2. T1 â¡ T2 â - âa,I. â¦L1, â{a,I}V1.T1â¦ â¡ â¦L2, â{a,I}V2.T2â¦. -#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I -elim (fpr_inv_all â¦ H) /3 width=4/ -qed. - -(* Advanced forward lemmas **************************************************) - -lamma fpr_fwd_shift_bind_minus: âI1,I2,L1,L2,V1,V2,T1,T2. - â¦L1, -â{I1}V1.T1â¦ â¡ â¦L2, -â{I2}V2.T2â¦ â - â¦L1, V1â¦ â¡ â¦L2, V2â¦ â§ I1 = I2. -* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H -elim (fpr_inv_all â¦ H) -H #L #HL1 #H #HL2 -[ elim (cpr_inv_abbr1 â¦ H) -H * - [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/ - | #T #_ #_ #H destruct - ] -| elim (cpr_inv_abst1 â¦ H Abst V2) -H - #V #T #HV1 #_ #H destruct /3 width=4/ -] -qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_fprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_fprs.etc deleted file mode 100644 index e0c1b3058..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/fprs_fprs.etc +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,L1,T0,T1. â¦L0, T0â¦ â¡ â¦L1, T1â¦ â - âL2,T2. â¦L0, T0â¦ â¡* â¦L2, T2â¦ â - ââL,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ & â¦L2, T2â¦ â¡ â¦L, Tâ¦. -#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 -/2 width=4/ qed. - -(* Main propertis ***********************************************************) - -theorem fprs_conf: bi_confluent â¦ fprs. -/2 width=4/ qed. - -theorem fprs_trans: bi_transitive â¦ fprs. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lenv_px_bi.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lenv_px_bi.etc deleted file mode 100644 index 931d075ab..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lenv_px_bi.etc +++ /dev/null @@ -1,88 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| lpx_bi_stom: lpx_bi R (â) (â) -| lpx_bi_pair: âI,K1,K2,V1,V2. - lpx_bi R K1 K2 â R K1 V1 K2 V2 â - lpx_bi R (K1. â{I} V1) (K2. â{I} V2) -. - -(* Basic inversion lemmas ***************************************************) - -fact lpx_bi_inv_atom1_aux: âR,L1,L2. lpx_bi R L1 L2 â L1 = â â L2 = â. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_bi_inv_atom1: âR,L2. lpx_bi R (â) L2 â L2 = â. -/2 width=4 by lpx_bi_inv_atom1_aux/ qed-. - -fact lpx_bi_inv_pair1_aux: âR,L1,L2. lpx_bi R L1 L2 â - âI,K1,V1. L1 = K1. â{I} V1 â - ââK2,V2. lpx_bi R K1 K2 & - R K1 V1 K2 V2 & L2 = K2. â{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: âR,I,K1,V1,L2. lpx_bi R (K1. â{I} V1) L2 â - ââK2,V2. lpx_bi R K1 K2 & R K1 V1 K2 V2 & - L2 = K2. â{I} V2. -/2 width=3 by lpx_bi_inv_pair1_aux/ qed-. - -fact lpx_bi_inv_atom2_aux: âR,L1,L2. lpx_bi R L1 L2 â L2 = â â L1 = â. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_bi_inv_atom2: âR,L1. lpx_bi R L1 (â) â L1 = â. -/2 width=4 by lpx_bi_inv_atom2_aux/ qed-. - -fact lpx_bi_inv_pair2_aux: âR,L1,L2. lpx_bi R L1 L2 â - âI,K2,V2. L2 = K2. â{I} V2 â - ââK1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & - L1 = K1. â{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: âR,I,L1,K2,V2. lpx_bi R L1 (K2. â{I} V2) â - ââK1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & - L1 = K1. â{I} V1. -/2 width=3 by lpx_bi_inv_pair2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lpx_bi_fwd_length: âR,L1,L2. lpx_bi R L1 L2 â |L1| = |L2|. -#R #L1 #L2 #H elim H -L1 -L2 normalize // -qed-. - -(* Basic properties *********************************************************) - -lemma lpx_bi_refl: âR. bi_reflexive ? ? R â reflexive â¦ (lpx_bi R). -#R #HR #L elim L -L // /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfpr_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfpr_alt.etc deleted file mode 100644 index 95ec60ba2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfpr_alt.etc +++ /dev/null @@ -1,87 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ â¡ â¡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'PRedSnAlt $L1 $L2 }. - -notation "hvbox( â¦ term 46 L1 â¦ â¡ â¡ break â¦ term 46 L2 â¦ )" - 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 â lpx_bi fpr. - -interpretation - "focalized parallel reduction (environment) alternative" - 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2). - -(* Basic properties *********************************************************) - -lemma lfpra_refl: reflexive â¦ lfpra. -/2 width=1/ qed. - -lemma fpr_lfpra: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1â¦ â¡â¡ â¦L2â¦. -#L1 elim L1 -L1 -[ #L2 #T1 #T2 #H - elim (fpr_inv_atom1 â¦ H) -H #_ #H destruct // -| #L1 #I #V1 #IH #L2 #T1 #T2 #H - elim (fpr_inv_pair1 â¦ H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/ -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma lfpra_inv_atom1: âL2. â¦ââ¦ â¡â¡ â¦L2â¦ â L2 = â. -/2 width=2 by lpx_bi_inv_atom1/ qed-. - -lemma lfpra_inv_pair1: âK1,I,V1,L2. â¦K1. â{I} V1â¦ â¡â¡ â¦L2â¦ â - ââK2,V2. â¦K1â¦ â¡â¡ â¦K2â¦ & â¦K1, V1â¦ â¡ â¦K2, V2â¦ & - L2 = K2. â{I} V2. -/2 width=1 by lpx_bi_inv_pair1/ qed-. - -lemma lfpra_inv_atom2: âL1. â¦L1â¦ â¡â¡ â¦ââ¦ â L1 = â. -/2 width=2 by lpx_bi_inv_atom2/ qed-. - -lemma lfpra_inv_pair2: âL1,K2,I,V2. â¦L1â¦ â¡â¡ â¦K2. â{I} V2â¦ â - ââK1,V1. â¦K1â¦ â¡â¡ â¦K2â¦ & â¦K1, V1â¦ â¡ â¦K2, V2â¦ & - L1 = K1. â{I} V1. -/2 width=1 by lpx_bi_inv_pair2/ qed-. - -lemma lfpra_inv_fpr: âL1,L2. â¦L1â¦ â¡â¡ â¦L2â¦ â âT.â¦L1, Tâ¦ â¡ â¦L2, Tâ¦. -#L1 #L2 * -L1 -L2 // /3 width=1/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lfpra_fwd_length: âL1,L2. â¦L1â¦ â¡â¡ â¦L2â¦ â |L1| = |L2|. -/2 width=2 by lpx_bi_fwd_length/ qed-. - -(* Main properties **********************************************************) - -theorem lfpr_lfpra: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â â¦L1â¦ â¡â¡ â¦L2â¦. -#L1 #L2 #H -lapply (lfpr_inv_fpr â¦ H (â0)) -H /2 width=3/ -qed. - -theorem lfpra_lfpr: âL1,L2. â¦L1â¦ â¡â¡ â¦L2â¦ â â¦L1â¦ â¡ â¦L2â¦. -#L1 #L2 #H -lapply (lfpra_inv_fpr â¦ H (â0)) -H /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfpr_fpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfpr_fpr.etc deleted file mode 100644 index f798d566b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfpr_fpr.etc +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1â¦ â¡ â¦L2â¦. -#L1 #L2 #T1 #T2 #H -elim (fpr_inv_all â¦ H) -H /2 width=3/ -qed. - -(* Inversion lemmas on context-free parallel reduction for closures *********) - -lemma lfpr_inv_fpr: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â âT. â¦L1, Tâ¦ â¡ â¦L2, Tâ¦. -#L1 #L2 * /2 width=4/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfprs_fprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfprs_fprs.etc deleted file mode 100644 index a5c42ba65..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpr/lfprs_fprs.etc +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. â¦L1â¦ â¡* â¦L2â¦ â âT. â¦L1, Tâ¦ â¡* â¦L2, Tâ¦. -#L1 #L2 #H @(lfprs_ind â¦ H) -L2 // -#L #L2 #_ #HL2 #IHL1 #T -lapply (lfpr_inv_fpr â¦ HL2 T) -HL2 /3 width=4/ -qed-. - -(* Properties on context-free parallel computation for closures *************) - -lemma fprs_lfprs: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â â¦L1â¦ â¡* â¦L2â¦. -#L1 #L2 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 // /3 width=5/ -qed. - -lemma lfprs_fprs_trans: âL1,L,L2,T1,T2. â¦L1â¦ â¡* â¦Lâ¦ â â¦L, T1â¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¡* â¦L2, T2â¦. -#L1 #L #L2 #T1 #T2 #HL1 #HL2 -lapply (lfprs_inv_fprs â¦ HL1 T1) -HL1 /2 width=4/ -qed. -(* -lamma lfprs_cprs_conf: âL1,L,L2,T1,T2. â¦L1â¦ â¡* â¦L2â¦ â L1 â¢ T1 â¡* T2 â â¦L1, T1â¦ â¡* â¦L2, T2â¦. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsup.etc deleted file mode 100644 index 077fd7422..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsup.etc +++ /dev/null @@ -1,123 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1, break term 46 T1 â¦ â§ break â¦ term 46 L2 , break term 46 T2 â¦ )" - 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 â -| frsup_bind_sn: âa,I,L,V,T. frsup L (â{a,I}V.T) L V -| frsup_bind_dx: âa,I,L,V,T. frsup L (â{a,I}V.T) (L.â{I}V) T -| frsup_flat_sn: âI,L,V,T. frsup L (â{I}V.T) L V -| frsup_flat_dx: âI,L,V,T. frsup L (â{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: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â - âJ. T1 = âª{J} â â¥. -#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: âJ,L1,L2,T2. â¦L1, âª{J}â¦ â§ â¦L2, T2â¦ â â¥. -/2 width=7 by frsup_inv_atom1_aux/ qed-. - -fact frsup_inv_bind1_aux: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â - âb,J,W,U. T1 = â{b,J}W.U â - (L2 = L1 â§ T2 = W) â¨ - (L2 = L1.â{J}W â§ 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: âb,J,L1,L2,W,U,T2. â¦L1, â{b,J}W.Uâ¦ â§ â¦L2, T2â¦ â - (L2 = L1 â§ T2 = W) â¨ - (L2 = L1.â{J}W â§ T2 = U). -/2 width=4 by frsup_inv_bind1_aux/ qed-. - -fact frsup_inv_flat1_aux: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â - âJ,W,U. T1 = â{J}W.U â - L2 = L1 â§ (T2 = W â¨ 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: âJ,L1,L2,W,U,T2. â¦L1, â{J}W.Uâ¦ â§ â¦L2, T2â¦ â - L2 = L1 â§ (T2 = W â¨ T2 = U). -/2 width=4 by frsup_inv_flat1_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma frsup_fwd_fw: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¯{L2, T2} < â¯{L1, T1}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ -qed-. - -lemma frsup_fwd_lw: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¯{L1} â¤ â¯{L2}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ -qed-. - -lemma frsup_fwd_tw: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¯{T2} < â¯{T1}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/ -qed-. - -lemma frsup_fwd_append: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â âL. L2 = L1 @@ L. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #a -| #a #I #L #V #_ @(ex_intro â¦ (â.â{I}V)) // -] -#I #L #V #T @(ex_intro â¦ (â)) // -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lift_frsup_trans: âT1,U1,d,e. â§[d, e] T1 â¡ U1 â - âL,K,U2. â¦L, U1â¦ â§ â¦L @@ K, U2â¦ â - âT2. â§[d + |K|, e] T2 â¡ 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 â¦ H) -H * - [ -HTU1 #H1 #H2 destruct - >(append_inv_refl_dx â¦ H1) -L -K normalize /2 width=2/ - | -HVW1 #H1 #H2 destruct - >(append_inv_pair_dx â¦ 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 â¦ H) -H #H1 * #H2 destruct - >(append_inv_refl_dx â¦ H1) -L -K normalize /2 width=2/ -] -#i #d #e [2,3: #_ ] #L #K #X #H -elim (frsup_inv_atom1 â¦ H) -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsupp.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsupp.etc deleted file mode 100644 index 9f7a8dc9d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsupp.etc +++ /dev/null @@ -1,110 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1, break term 46 T1 â¦ â§ + break â¦ term 46 L2 , break term 46 T2 â¦ )" - 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 â bi_TC â¦ frsup. - -interpretation "plus-iterated restricted structural predecessor (closure)" - 'RestSupTermPlus L1 T1 L2 T2 = (frsupp L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma frsupp_ind: âL1,T1. âR:relation2 lenv term. - (âL2,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â R L2 T2) â - (âL,T,L2,T2. â¦L1, T1â¦ â§+ â¦L, Tâ¦ â â¦L, Tâ¦ â§ â¦L2, T2â¦ â R L T â R L2 T2) â - âL2,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â R L2 T2. -#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H -@(bi_TC_ind â¦ IH1 IH2 ? ? H) -qed-. - -lemma frsupp_ind_dx: âL2,T2. âR:relation2 lenv term. - (âL1,T1. â¦L1, T1â¦ â§ â¦L2, T2â¦ â R L1 T1) â - (âL1,L,T1,T. â¦L1, T1â¦ â§ â¦L, Tâ¦ â â¦L, Tâ¦ â§+ â¦L2, T2â¦ â R L T â R L1 T1) â - âL1,T1. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â R L1 T1. -#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H -@(bi_TC_ind_dx â¦ IH1 IH2 ? ? H) -qed-. - -(* Baic inversion lemmas ****************************************************) - -lemma frsupp_inv_dx: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¦L1, T1â¦ â§ â¦L2, T2â¦ â¨ - ââL,T. â¦L1, T1â¦ â§+ â¦L, Tâ¦ & â¦L, Tâ¦ â§ â¦L2, T2â¦. -/2 width=1 by bi_TC_decomp_r/ qed-. - -lemma frsupp_inv_sn: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¦L1, T1â¦ â§ â¦L2, T2â¦ â¨ - ââL,T. â¦L1, T1â¦ â§ â¦L, Tâ¦ & â¦L, Tâ¦ â§+ â¦L2, T2â¦. -/2 width=1 by bi_TC_decomp_l/ qed-. - -(* Basic properties *********************************************************) - -lemma frsup_frsupp: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¦L1, T1â¦ â§+ â¦L2, T2â¦. -/2 width=1/ qed. - -lemma frsupp_strap1: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â§+ â¦L, Tâ¦ â â¦L, Tâ¦ â§ â¦L2, T2â¦ â - â¦L1, T1â¦ â§+ â¦L2, T2â¦. -/2 width=4/ qed. - -lemma frsupp_strap2: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â§ â¦L, Tâ¦ â â¦L, Tâ¦ â§+ â¦L2, T2â¦ â - â¦L1, T1â¦ â§+ â¦L2, T2â¦. -/2 width=4/ qed. - -(* Basic forward lemmas *****************************************************) - -lemma frsupp_fwd_fw: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¯{L2, T2} < â¯{L1, T1}. -#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 -/3 width=3 by frsup_fwd_fw, transitive_lt/ -qed-. - -lemma frsupp_fwd_lw: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¯{L1} â¤ â¯{L2}. -#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ 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 â¦ HL2) -HL2 /2 width=3 by transitive_le/ -qed-. - -lemma frsupp_fwd_tw: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¯{T2} < â¯{T1}. -#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 -/3 width=3 by frsup_fwd_tw, transitive_lt/ -qed-. - -lemma frsupp_fwd_append: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â âL. L2 = L1 @@ L. -#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 /2 width=3 by frsup_fwd_append/ -#L #T #L2 #T2 #_ #HL2 * #K1 #H destruct -elim (frsup_fwd_append â¦ HL2) -HL2 #K2 #H destruct /2 width=2/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lift_frsupp_trans: âL,U1,K,U2. â¦L, U1â¦ â§+ â¦L @@ K, U2â¦ â - âT1,d,e. â§[d, e] T1 â¡ U1 â - âT2. â§[d + |K|, e] T2 â¡ U2. -#L #U1 @(f2_ind â¦ fw â¦ L U1) -L -U1 #n #IH -#L #U1 #Hn #K #U2 #H #T1 #d #e #HTU1 destruct -elim (frsupp_inv_sn â¦ H) -H /2 width=5 by lift_frsup_trans/ * -#L0 #U0 #HL0 #HL -elim (frsup_fwd_append â¦ HL0) #K0 #H destruct -elim (frsupp_fwd_append â¦ HL) #L0 >append_assoc #H -elim (append_inj_dx â¦ H ?) -H // #_ #H destruct -(append_inv_refl_dx â¦ (sym_eq â¦ H1)) -H1 normalize /2 width=2/ -| /2 width=5 by lift_frsupp_trans/ -] -qed-. - -(* Advanced inversion lemmas for frsupp **************************************) - -lemma frsupp_inv_atom1_frsups: âJ,L1,L2,T2. â¦L1, âª{J}â¦ â§+ â¦L2, T2â¦ â â¥. -#J #L1 #L2 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 // -#L2 #T2 #H elim (frsup_inv_atom1 â¦ H) -qed-. - -lemma frsupp_inv_bind1_frsups: âb,J,L1,L2,W,U,T2. â¦L1, â{b,J}W.Uâ¦ â§+ â¦L2, T2â¦ â - â¦L1, Wâ¦ â§* â¦L2, T2â¦ â¨ â¦L1.â{J}W, Uâ¦ â§* â¦L2, T2â¦. -#b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind â¦ H) -L2 -T2 -[ #L2 #T2 #H - elim (frsup_inv_bind1 â¦ H) -H * #H1 #H2 destruct /2 width=1/ -| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ -] -qed-. - -lemma frsupp_inv_flat1_frsups: âJ,L1,L2,W,U,T2. â¦L1, â{J}W.Uâ¦ â§+ â¦L2, T2â¦ â - â¦L1, Wâ¦ â§* â¦L2, T2â¦ â¨ â¦L1, Uâ¦ â§* â¦L2, T2â¦. -#J #L1 #L2 #W #U #T2 #H @(frsupp_ind â¦ H) -L2 -T2 -[ #L2 #T2 #H - elim (frsup_inv_flat1 â¦ H) -H #H1 * #H2 destruct /2 width=1/ -| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsups_frsups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsups_frsups.etc deleted file mode 100644 index e7b7de26e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/frsups_frsups.etc +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ frsups. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/ssta_frsups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/ssta_frsups.etc deleted file mode 100644 index 9eb1fbd8b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/frsup/ssta_frsups.etc +++ /dev/null @@ -1,70 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L,T,U,l. â¦h, Lâ¦ â¢ T â¢[g] â¦l, Uâ¦ â â¦L, Uâ¦ â§+ â¦L, Tâ¦ â â¥. -#h #g #L #T #U #l #H elim H -L -T -U -l -[ #L #k #l #_ #H - elim (frsupp_inv_atom1_frsups â¦ H) -| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H - elim (lift_frsupp_trans â¦ (â) â¦ H â¦ HWU) -U #X #H - elim (lift_inv_lref2_be â¦ H ? ?) -H // -| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H - elim (lift_frsupp_trans â¦ (â) â¦ H â¦ HWU) -U #X #H - elim (lift_inv_lref2_be â¦ H ? ?) -H // -| #a #I #L #V #T #U #l #_ #IHTU #H - elim (frsupp_inv_bind1_frsups â¦ H) -H #H [2: /4 width=4/ ] -IHTU - lapply (frsups_fwd_fw â¦ H) -H normalize - (lcoeq_inv_O2 â¦ HL1) -HL1 // -| #I1 #I #L1 #L #V1 #V #e #_ #IHL1 #X #H elim (lcoeq_inv_pair1 â¦ 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 â¦ H) -H // - #L2 #HL2 #H destruct /3 width=1 by lcoeq_succ/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lcoeq/ldrop_lcoeq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lcoeq/ldrop_lcoeq.etc deleted file mode 100644 index 4891949c2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lcoeq/ldrop_lcoeq.etc +++ /dev/null @@ -1,51 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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: âL1,L2,d,e. L1 â[d, e] L2 â - âI,K2,W,s,i. â©[s, 0, i] L2 â¡ K2.â{I}W â i < d â - ââK1. K1 â[â«°(d-i), e] K2 & â©[s, 0, i] L1 â¡ K1.â{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 â¦ H) -H #H destruct -| #I #L1 #L2 #V #_ #_ #J #K2 #W #s #i #_ #H - elim (ylt_yle_false â¦ H) // -| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #J #K2 #W #s #i #_ #H - elim (ylt_yle_false â¦ H) // -| #I #L1 #L2 #V #d #e #HL12 #IHL12 #J #K2 #W #s #i #H - elim (ldrop_inv_O1_pair1 â¦ 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 yminus_succ (cpys_inv_sort1 â¦ H) -H // -qed. - -lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â[Â§p, d] L2. -#L1 #L2 #d #k #HL12 @conj // -HL12 -#U @conj #H >(cpys_inv_gref1 â¦ H) -H // -qed. - -lemma lleq_bind: âa,I,L1,L2,V,T,d. - L1 â[V, d] L2 â L1.â{I}V â[T, â«¯d] L2.â{I}V â - L1 â[â{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 â¦ 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: âI,L1,L2,V,T,d. - L1 â[V, d] L2 â L1 â[T, d] L2 â L1 â[â{I}V.T, d] L2. -#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12 -#X @conj #H elim (cpys_inv_flat1 â¦ 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: âL1,L2,U,dt. L1 â[U, dt] L2 â âT,d,e. â§[d, e] T â¡ U â - d â¤ dt â dt â¤ d + e â L1 â[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 â¦ HU0 â¦ HTU) // -HU0 /4 width=5 by cpys_weak/ -qed-. - -lemma lsuby_lleq_trans: âL2,L,T,d. L2 â[T, d] L â - âL1. L1 âÃ[d, â] L2 â |L1| = |L2| â L1 â[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 â¦ 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: âL,L1,T,d. L â[T, d] L1 â - âL2. L1 âÃ[d, â] L2 â |L1| = |L2| â L â[T, d] L2. -/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-. - -lemma lleq_lsuby_repl: âL1,L2,T,d. L1 â[T, d] L2 â - âK1. K1 âÃ[d, â] L1 â |K1| = |L1| â - âK2. L2 âÃ[d, â] K2 â |L2| = |K2| â - K1 â[T, d] K2. -/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lleq_fwd_length: âL1,L2,T,d. L1 â[T, d] L2 â |L1| = |L2|. -#L1 #L2 #T #d * // -qed-. - -lemma lleq_fwd_ldrop_sn: âL1,L2,T,d. L1 â[d, T] L2 â âK1,i. â©[i] L1 â¡ K1 â - âK2. â©[i] L2 â¡ K2. -#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length â¦ H) -H -#HL12 lapply (ldrop_fwd_length_le2 â¦ HLK1) -HLK1 /2 width=1 by ldrop_O1_le/ -qed-. - -lemma lleq_fwd_ldrop_dx: âL1,L2,T,d. L1 â[d, T] L2 â âK2,i. â©[i] L2 â¡ K2 â - âK1. â©[i] L1 â¡ K1. -/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-. - -lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,d. - L1 â[â{a,I}V.T, d] L2 â L1 â[V, d] L2. -#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12 -#U elim (H (â{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 â¦ H) -H -#X #Y #H1 #H2 #H destruct // -qed-. - -lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,d. - L1 â[â{a,I}V.T, d] L2 â L1.â{I}V â[T, â«¯d] L2.â{I}V. -#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12 -#U elim (H (â{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 â¦ H) -H -#X #Y #H1 #H2 #H destruct // -qed-. - -lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,d. - L1 â[â{I}V.T, d] L2 â L1 â[V, d] L2. -#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12 -#U elim (H (â{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 â¦ H) -H -#X #Y #H1 #H2 #H destruct // -qed-. - -lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,d. - L1 â[â{I}V.T, d] L2 â L1 â[T, d] L2. -#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12 -#U elim (H (â{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 â¦ H) -H -#X #Y #H1 #H2 #H destruct // -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma lleq_inv_bind: âa,I,L1,L2,V,T,d. L1 â[â{a,I}V.T, d] L2 â - L1 â[V, d] L2 â§ L1.â{I}V â[T, â«¯d] L2.â{I}V. -/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-. - -lemma lleq_inv_flat: âI,L1,L2,V,T,d. L1 â[â{I}V.T, d] L2 â - L1 â[V, d] L2 â§ L1 â[T, d] L2. -/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_alt.etc deleted file mode 100644 index a8fb526e5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_alt.etc +++ /dev/null @@ -1,85 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| lleqa_sort: âL1,L2,d,k. |L1| = |L2| â lleqa d (âk) L1 L2 -| lleqa_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â lleqa d (#i) L1 L2 -| lleqa_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â - lleqa (yinj 0) V K1 K2 â lleqa d (#i) L1 L2 -| lleqa_free: âL1,L2,d,i. |L1| = |L2| â |L1| â¤ i â |L2| â¤ i â lleqa d (#i) L1 L2 -| lleqa_gref: âL1,L2,d,p. |L1| = |L2| â lleqa d (Â§p) L1 L2 -| lleqa_bind: âa,I,L1,L2,V,T,d. - lleqa d V L1 L2 â lleqa (â«¯d) T (L1.â{I}V) (L2.â{I}V) â - lleqa d (â{a,I}V.T) L1 L2 -| lleqa_flat: âI,L1,L2,V,T,d. - lleqa d V L1 L2 â lleqa d T L1 L2 â lleqa d (â{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: âL1,L2,T,d. L1 ââ[T, d] L2 â L1 â[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: âL1,T,L2,d. L1 â[T, d] L2 â L1 ââ[T, d] L2. -#L1 #T @(f2_ind â¦ rfw â¦ 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 â¦ 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 â¦ H) -H /3 width=1 by lleqa_bind/ -| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat â¦ H) -H /3 width=1 by lleqa_flat/ -] -qed. - -(* Advanced eliminators *****************************************************) - -lemma lleq_ind_alt: âR:relation4 ynat term lenv lenv. ( - âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2 - ) â ( - âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2 - ) â ( - âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â - K1 â[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2 - ) â ( - âL1,L2,d,i. |L1| = |L2| â |L1| â¤ i â |L2| â¤ i â R d (#i) L1 L2 - ) â ( - âL1,L2,d,p. |L1| = |L2| â R d (Â§p) L1 L2 - ) â ( - âa,I,L1,L2,V,T,d. - L1 â[V, d]L2 â L1.â{I}V â[T, â«¯d] L2.â{I}V â - R d V L1 L2 â R (â«¯d) T (L1.â{I}V) (L2.â{I}V) â R d (â{a,I}V.T) L1 L2 - ) â ( - âI,L1,L2,V,T,d. - L1 â[V, d]L2 â L1 â[T, d] L2 â - R d V L1 L2 â R d T L1 L2 â R d (â{I}V.T) L1 L2 - ) â - âd,T,L1,L2. L1 â[T, d] L2 â R d T L1 L2. -#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa â¦ H) -H -/3 width=9 by lleqa_inv_lleq/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_ext.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_ext.etc deleted file mode 100644 index d6bb03fa7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_ext.etc +++ /dev/null @@ -1,91 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T,d0. L1 â[T, d0] L2 â âd. d0 = d + 1 â - âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â - K1 â[V, 0] K2 â L1 â[T, d] L2. -#L1 #L2 #T #d0 #H @(lleq_ind_alt â¦ 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: âT,L1,L2,d. L1 â[T, d+1] L2 â - âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â - K1 â[V, 0] K2 â L1 â[T, d] L2. -/2 width=7 by lleq_inv_S_aux/ qed-. - -lemma lleq_inv_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â - L1 â[V, 0] L2 â§ L1.â{I}V â[T, 0] L2.â{I}V. -#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind â¦ H) -H -/3 width=7 by ldrop_pair, conj, lleq_inv_S/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lleq_fwd_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â - L1.â{I}V â[T, 0] L2.â{I}V. -#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O â¦ H) -H // -qed-. - -(* Advanced properties ******************************************************) - -lemma lleq_ge: âL1,L2,T,d1. L1 â[T, d1] L2 â âd2. d1 â¤ d2 â L1 â[T, d2] L2. -#L1 #L2 #T #d1 #H @(lleq_ind_alt â¦ 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 â¦ HV) #HK12 #Hid2 - lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) - normalize in â¢ (%â%â?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/ - | /3 width=8 by lleq_lref, yle_trans/ - ] -] -qed-. - -lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â[V, 0] L2 â L1.â{I}V â[T, 0] L2.â{I}V â - L1 â[â{a,I}V.T, 0] L2. -/3 width=3 by lleq_ge, lleq_bind/ qed. - -lemma lleq_bind_repl_SO: âI1,I2,L1,L2,V1,V2,T. L1.â{I1}V1 â[T, 0] L2.â{I2}V2 â - âJ1,J2,W1,W2. L1.â{J1}W1 â[T, 1] L2.â{J2}W2. -#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge â¦ HT 1 ?) // -HT -#HT @(lleq_lsuby_repl â¦ HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *) -qed-. - -lemma lleq_bind_repl_O: âI,L1,L2,V,T. L1.â{I}V â[T, 0] L2.â{I}V â - âJ,W. L1 â[W, 0] L2 â L1.â{J}W â[T, 0] L2.â{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: âa,I,L1,L2,V,T. (L1 â[â{a,I}V.T, 0] L2 â â¥) â - (L1 â[V, 0] L2 â â¥) â¨ (L1.â{I}V â[T, 0] L2.â{I}V â â¥). -#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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_fqus.etc deleted file mode 100644 index 18744a539..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_fqus.etc +++ /dev/null @@ -1,75 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ â â¦G2, K2, Uâ¦ â - âL1. L1 â[T, 0] L2 â - ââK1. â¦G1, L1, Tâ¦ â â¦G2, K1, Uâ¦ & K1 â[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 â¦ H â¦ I L2 V) -H // - #I1 #K1 #H1 #H2 lapply (ldrop_inv_O2 â¦ 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 â¦ H) - | elim (lleq_inv_flat â¦ H) - ] -H - /2 width=3 by fqu_pair_sn, ex2_intro/ -| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O â¦ H) -H - #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/ -| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat â¦ 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 â¦ HLK2) -K2 - lapply (lleq_fwd_length â¦ HL12) -T -U // - ] -] -qed-. - -lemma lleq_fquq_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ ââ¸® â¦G2, K2, Uâ¦ â - âL1. L1 â[T, 0] L2 â - ââK1. â¦G1, L1, Tâ¦ ââ¸® â¦G2, K1, Uâ¦ & K1 â[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen â¦ H) -H -[ #H elim (lleq_fqu_trans â¦ H â¦ 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: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ â+ â¦G2, K2, Uâ¦ â - âL1. L1 â[T, 0] L2 â - ââK1. â¦G1, L1, Tâ¦ â+ â¦G2, K1, Uâ¦ & K1 â[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind â¦ H) -G2 -K2 -U -[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans â¦ HTU â¦ HL12) -L2 - /3 width=3 by fqu_fqup, ex2_intro/ -| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU â¦ HL12) -L2 - #K1 #HTU #HK1 elim (lleq_fqu_trans â¦ HU2 â¦ HK1) -K - /3 width=5 by fqup_strap1, ex2_intro/ -] -qed-. - -lemma lleq_fqus_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ â* â¦G2, K2, Uâ¦ â - âL1. L1 â[T, 0] L2 â - ââK1. â¦G1, L1, Tâ¦ â* â¦G2, K1, Uâ¦ & K1 â[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen â¦ H) -H -[ #H elim (lleq_fqup_trans â¦ H â¦ HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/ -| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_ldrop.etc deleted file mode 100644 index 9cb597c6f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_ldrop.etc +++ /dev/null @@ -1,123 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â[#i, d] L2. -#L1 #L2 #d #i #Hid #HL12 @conj // -HL12 -#U @conj #H elim (cpys_inv_lref1 â¦ H) -H // * -#I #Z #Y #X #H elim (ylt_yle_false â¦ Hid â¦ H) -qed. - -lemma lleq_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â - K1 â[V, 0] K2 â L1 â[#i, d] L2. -#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ] -[ lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H1 - lapply (ldrop_fwd_length â¦ HLK2) -HLK2 #H2 - >H1 >H2 -H1 -H2 normalize // -| #U @conj #H elim (cpys_inv_lref1 â¦ H) -H // * - >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU - [ letin HLK â HLK1 | letin HLK â HLK2 ] - lapply (ldrop_mono â¦ H â¦ HLK) -H #H destruct elim (IH W) - /3 width=7 by cpys_subst_Y2/ -] -qed. - -lemma lleq_free: âL1,L2,d,i. |L1| â¤ i â |L2| â¤ i â |L1| = |L2| â L1 â[#i, d] L2. -#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12 -#U @conj #H elim (cpys_inv_lref1 â¦ H) -H // * -#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 â¦ 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: âK1,K2,T,dt. K1 â[T, dt] K2 â - âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âU. â§[d, e] T â¡ U â dt â¤ d â L1 â[U, dt] L2. -#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd -lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) -#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0 -[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ] -elim (cpys_inv_lift1_be â¦ HU0 â¦ HLKA â¦ 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: âK1,K2,T,dt. K1 â[T, dt] K2 â - âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âU. â§[d, e] T â¡ U â d â¤ dt â L1 â[U, dt+e] L2. -#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt -lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) -#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0 -[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ] -elim (cpys_inv_lift1_ge â¦ HU0 â¦ HLKA â¦ 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: âL1,L2,U,dt. L1 â[U, dt] L2 â - âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âT. â§[d, e] T â¡ U â dt â¤ d â K1 â[T, dt] K2. -#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd -lapply (ldrop_fwd_length_minus2 â¦ HLK1) lapply (ldrop_fwd_length_minus2 â¦ 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 â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ] -lapply (cpys_lift_be â¦ HT0 â¦ HLKA â¦ HTU â¦ HTU0) // -HT0 ->yplus_Y1 #HU0 elim (cpys_inv_lift1_be â¦ (H0 HU0) â¦ HLKB â¦ HTU) // -L1 -L2 -U -Hdtd -#X #HT0 #HX lapply (lift_inj â¦ HX â¦ HTU0) -U0 // -qed-. - -lemma lleq_inv_lift_ge: âL1,L2,U,dt. L1 â[U, dt] L2 â - âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âT. â§[d, e] T â¡ U â yinj d + e â¤ dt â K1 â[T, dt-e] K2. -#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt -lapply (ldrop_fwd_length_minus2 â¦ HLK1) lapply (ldrop_fwd_length_minus2 â¦ HLK2) -#H2 #H1 @conj // -HL12 -H1 -H2 -elim (yle_inv_plus_inj2 â¦ Hdedt) #Hddt #Hedt -#T0 elim (lift_total T0 d e) -#U0 #HTU0 elim (IH U0) -IH -#H12 #H21 @conj #HT0 -[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ] -lapply (cpys_lift_ge â¦ HT0 â¦ HLKA â¦ HTU â¦ HTU0) // -HT0 -Hddt ->ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge â¦ (H0 HU0) â¦ HLKB â¦ HTU) // -L1 -L2 -U -Hdedt -Hedt -#X #HT0 #HX lapply (lift_inj â¦ HX â¦ HTU0) -U0 // -qed-. - -lemma lleq_inv_lift_be: âL1,L2,U,dt. L1 â[U, dt] L2 â - âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âT. â§[d, e] T â¡ U â d â¤ dt â dt â¤ yinj d + e â K1 â[T, d] K2. -#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde -lapply (ldrop_fwd_length_minus2 â¦ HLK1) lapply (ldrop_fwd_length_minus2 â¦ 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 â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ] -lapply (cpys_lift_ge â¦ HT0 â¦ HLKA â¦ HTU â¦ HTU0) // -HT0 -#HU0 lapply (cpys_weak â¦ HU0 dt (â) ? ?) // -HU0 -#HU0 lapply (H0 HU0) -#HU0 lapply (cpys_weak â¦ HU0 d (â) ? ?) // -HU0 -#HU0 elim (cpys_inv_lift1_ge_up â¦ HU0 â¦ HLKB â¦ HTU) // -L1 -L2 -U -Hddt -Hdtde -#X #HT0 #HX lapply (lift_inj â¦ HX â¦ HTU0) -U0 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_lleq.etc deleted file mode 100644 index 2a8873d9b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lleq_alt/lleq_lleq.etc +++ /dev/null @@ -1,175 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. âd:ynat. âi:nat. L1 â[#i, d] L2 â - â¨â¨ |L1| â¤ i â§ |L2| â¤ i - | yinj i < d - | ââI1,I2,K1,K2,V. â©[i] L1 â¡ K1.â{I1}V & - â©[i] L2 â¡ K2.â{I2}V & - K1 â[V, yinj 0] K2 & d â¤ 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 â¦ Hi) #I1 #K1 #V1 #HLK1 -elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2 -lapply (ldrop_fwd_length_minus2 â¦ HLK2) #H -lapply (ldrop_fwd_length_minus2 â¦ HLK1) >HL12 yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/ -qed-. - -lemma lleq_fwd_lref_dx: âL1,L2,d,i. L1 â[#i, d] L2 â - âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â - i < d â¨ - ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2 & d â¤ i. -#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref â¦ H) -H [ * || * ] -[ #_ #H elim (lt_refl_false i) - lapply (ldrop_fwd_length_lt2 â¦ 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 â¦ HLK22 â¦ HLK2) -L2 - #H destruct /3 width=5 by ex3_2_intro, or_intror/ -] -qed-. - -lemma lleq_fwd_lref_sn: âL1,L2,d,i. L1 â[#i, d] L2 â - âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â - i < d â¨ - ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2 & d â¤ i. -#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 â¦ d â¦ 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: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â - âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â - ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2. -#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx â¦ H â¦ HLK2) -L2 -[ #H elim (ylt_yle_false â¦ H Hdi) -| * /2 width=4 by ex2_2_intro/ -] -qed-. - -lemma lleq_inv_lref_ge_sn: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â - âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â - ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2. -#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 â¦ Hdi â¦ HLK1) -Hdi -HLK1 -/3 width=4 by lleq_sym, ex2_2_intro/ -qed-. - -lemma lleq_inv_lref_ge_gen: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â - âI1,I2,K1,K2,V1,V2. - â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â - V1 = V2 â§ K1 â[V2, 0] K2. -#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 -elim (lleq_inv_lref_ge_sn â¦ HL12 â¦ HLK1) // -L1 -d -#J #Y #HY lapply (ldrop_mono â¦ HY â¦ HLK2) -L2 -i #H destruct /2 width=1 by conj/ -qed-. - -lemma lleq_inv_lref_ge: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â - âI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â - K1 â[V, 0] K2. -#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2 -elim (lleq_inv_lref_ge_gen â¦ HL12 â¦ HLK1 HLK2) // -qed-. - -(* Advanced properties ******************************************************) - -lemma lleq_dec: âT,L1,L2,d. Decidable (L1 â[T, d] L2). -#T #L1 @(f2_ind â¦ rfw â¦ 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 â¦ HiL2) #I2 #K2 #V2 #HLK2 - elim (ldrop_O1_lt â¦ HiL1) #I1 #K1 #V1 #HLK1 - elim (eq_term_dec V2 V1) - [ #H3 elim (IH K1 V1 â¦ K2 0) destruct - /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/ - ] - -IH #H3 @or_intror - #H elim (lleq_fwd_lref â¦ 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 â¦ HLY1 â¦ HLK1) -HLY1 -HLK1 - lapply (ldrop_mono â¦ HLY2 â¦ 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 â¦ L2 d) /2 width=1 by/ - elim (IH (L1.â{I}V) T â¦ (L2.â{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/ - #H1 #H2 @or_intror - #H elim (lleq_inv_bind â¦ H) -H /2 width=1 by/ -| #I #V #T #Hn #L2 #d destruct - elim (IH L1 V â¦ L2 d) /2 width=1 by/ - elim (IH L1 T â¦ L2 d) -IH /3 width=1 by or_introl, lleq_flat/ - #H1 #H2 @or_intror - #H elim (lleq_inv_flat â¦ H) -H /2 width=1 by/ -] --n /4 width=3 by lleq_fwd_length, or_intror/ -qed-. - -(* Main properties **********************************************************) - -theorem lleq_trans: âd,T. Transitive â¦ (lleq d T). -#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/ -qed-. - -theorem lleq_canc_sn: âL,L1,L2,T,d. L â[d, T] L1â L â[d, T] L2 â L1 â[d, T] L2. -/3 width=3 by lleq_trans, lleq_sym/ qed-. - -theorem lleq_canc_dx: âL1,L2,L,T,d. L1 â[d, T] L â L2 â[d, T] L â L1 â[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: âa,I,L1,L2,V,T,d. (L1 â[â{a,I}V.T, d] L2 â â¥) â - (L1 â[V, d] L2 â â¥) â¨ (L1.â{I}V â[T, â«¯d] L2.â{I}V â â¥). -#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: âI,L1,L2,V,T,d. (L1 â[â{I}V.T, d] L2 â â¥) â - (L1 â[V, d] L2 â â¥) â¨ (L1 â[T, d] L2 â â¥). -#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: âd,T,L1,L. L1â[T, d] L â - âL2. (L â[T, d] L2 â â¥) â (L1 â[T, d] L2 â â¥). -/3 width=3 by lleq_canc_sn/ qed-. -works with /4 width=8/ so lleq_canc_sn is more convenient -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/lazynegatedeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/lazynegatedeq_4.etc deleted file mode 100644 index 6b0cb85e5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/lazynegatedeq_4.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( L1 â§£ break [ term 46 T , break term 46 d ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'LazyNegatedEq $T $d $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/lazynegatedeqalt_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/lazynegatedeqalt_4.etc deleted file mode 100644 index b150f63ff..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/lazynegatedeqalt_4.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( L1 â§£ â§£ break [ term 46 T , break term 46 d ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'LazyNegatedEqAlt $T $d $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq.etc deleted file mode 100644 index d41a4e83c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â - Î»d,T,L1,L2. |L1| = |L2| â§ (L1 â[T, d] L2 â â¥). - -interpretation - "negated lazy equivalence (local environment)" - 'LazyNegatedEq T d L1 L2 = (llneq d T L1 L2). diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq_alt.etc deleted file mode 100644 index 00eb2c34e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq_alt.etc +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| llneqa_neq: âI1,I2,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â - â©[i]L1 â¡ K1.â{I1}V1 â â©[i]L2 â¡ K2.â{I2}V2 â - |K1| = |K2| â (V1 = V2 â â¥) â llneqa d (#i) L1 L2 -| llneqa_eq : âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â - â©[i]L1 â¡ K1.â{I1}V â â©[i]L2 â¡ K2.â{I2}V â - llneqa 0 (V) K1 K2 â llneqa d (#i) L1 L2 -| llneqa_bind_sn: âa,I,L1,L2,V,T,d. - llneqa d V L1 L2 â llneqa d (â{a,I}V.T) L1 L2 -| llneqa_bind_dx: âa,I,L1,L2,V,T,d. - llneqa (â«¯d) T (L1.â{I}V) (L2.â{I}V) â llneqa d (â{a,I}V.T) L1 L2 -| llneqa_flat_sn: âI,L1,L2,V,T,d. - llneqa d V L1 L2 â llneqa d (â{I}V.T) L1 L2 -| llneqa_flat_dx: âI,L1,L2,V,T,d. - llneqa d T L1 L2 â llneqa d (â{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: âT,L1,L2,d. L1 â§£[T, d] L2 â L1 â§£â§£[T, d] L2. -#T #L1 @(f2_ind â¦ rfw â¦ 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 â¦ HiL1) #I1 #K1 #V1 #HLK1 - elim (ldrop_O1_lt L2 i) /2 width=1 by/ #I2 #K2 #V2 #HLK2 - lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) normalize - elim (eq_term_dec V1 V2) #HnV12 destruct - [2: #H @(llneqa_neq â¦ HLK1 â¦ 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 â¦ HLK1 â¦ 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 â¦ 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 â¦ H) -H - /5 width=1 by llneqa_flat_dx, llneqa_flat_sn, conj/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq_ext.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq_ext.etc deleted file mode 100644 index 235d869c2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llneq/llneq_ext.etc +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2,T,d. L1 â§£â§£[T, d] L2 â |L1| â¤ d â â¥. -#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 â¦ HLK1) -HLK1 normalize - #H lapply (yle_trans â¦ H â¦ 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 â¦ HLK1) -HLK1 normalize - #H lapply (yle_trans â¦ H â¦ Hdi) -d - /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/ -| #a #I #L1 (cprs_inv_sort1 â¦ H1) -T #H2 -lapply (cprs_inv_sort1 â¦ H2) -L #H destruct // -qed-. - -lemma cpcs_inv_abst1: âa,G,L,W1,T1,T. â¦G, Lâ¦ â¢ â{a}W1.T1 â¬* T â - ââW2,T2. â¦G, Lâ¦ â¢ T â¡* â{a}W2.T2 & â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* â{a}W2.T2. -#a #G #L #W1 #T1 #T #H -elim (cpcs_inv_cprs â¦ H) -H #X #H1 #H2 -elim (cprs_inv_abst1 â¦ H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct -/3 width=6 by cprs_bind, ex2_2_intro/ -qed-. - -lemma cpcs_inv_abst2: âa,G,L,W1,T1,T. â¦G, Lâ¦ â¢ T â¬* â{a}W1.T1 â - ââW2,T2. â¦G, Lâ¦ â¢ T â¡* â{a}W2.T2 & â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* â{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: âa,G,L,W,T,k. â¦G, Lâ¦ â¢ âk â¬* â{a}W.T â â¥. -#a #G #L #W #T #k #H -elim (cpcs_inv_cprs â¦ H) -H #X #H1 ->(cprs_inv_sort1 â¦ H1) -X #H2 -elim (cprs_inv_abst1 â¦ H2) -H2 #W0 #T0 #_ #_ #H destruct -qed-. - -(* Basic_1: was: pc3_gen_lift *) -lemma cpcs_inv_lift: âG,L,K,s,d,e. â©[s, d, e] L â¡ K â - âT1,U1. â§[d, e] T1 â¡ U1 â âT2,U2. â§[d, e] T2 â¡ U2 â - â¦G, Lâ¦ â¢ U1 â¬* U2 â â¦G, Kâ¦ â¢ T1 â¬* T2. -#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12 -elim (cpcs_inv_cprs â¦ HU12) -HU12 #U #HU1 #HU2 -elim (cprs_inv_lift1 â¦ HU1 â¦ HLK â¦ HTU1) -U1 #T #HTU #HT1 -elim (cprs_inv_lift1 â¦ HU2 â¦ HLK â¦ HTU2) -L -U2 #X #HXU ->(lift_inj â¦ HXU â¦ HTU) -X -U -d -e /2 width=3 by cprs_div/ -qed-. - -(* Advanced properties ******************************************************) - -lemma llpr_cpcs_trans: âG,L1,L2,T1,T2. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â â¦G, L1â¦ â¢ â¡[T2, 0] L2 â - â¦G, L2â¦ â¢ T1 â¬* T2 â â¦G, L1â¦ â¢ T1 â¬* T2. -#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs â¦ H) -H -/4 width=5 by cprs_div, cprs_llpr_trans/ -qed-. - -lemma llprs_cpcs_trans: âG,L1,L2,T1,T2. â¦G, L1â¦ â¢ â¡*[T1, 0] L2 â â¦G, L1â¦ â¢ â¡*[T2, 0] L2 â - â¦G, L2â¦ â¢ T1 â¬* T2 â â¦G, L1â¦ â¢ T1 â¬* T2. -#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs â¦ H) -H -/4 width=5 by cprs_div, llprs_cprs_trans/ -qed-. - -lemma cpr_cprs_conf_cpcs: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¡* T1 â â¦G, Lâ¦ â¢ T â¡ T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. -#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip â¦ HT1 â¦ HT2) -HT1 -HT2 -/2 width=3 by cpr_cprs_div/ -qed-. - -lemma cprs_cpr_conf_cpcs: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¡* T1 â â¦G, Lâ¦ â¢ T â¡ T2 â â¦G, Lâ¦ â¢ T2 â¬* T1. -#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip â¦ HT1 â¦ HT2) -HT1 -HT2 -/2 width=3 by cprs_cpr_div/ -qed-. - -lemma cprs_conf_cpcs: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¡* T1 â â¦G, Lâ¦ â¢ T â¡* T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. -#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf â¦ HT1 â¦ 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: âG,L1,L2,T1. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â - âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â â¦G, L2â¦ â¢ T1 â¬* T2. -#G #L1 #L2 #T1 #HL12 #T2 #HT12 elim (cprs_llpr_conf_dx â¦ HT12 â¦ 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: âG,L1,L2,T1. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â - âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â â¦G, L2â¦ â¢ T1 â¬* T2. -/3 width=5 by llpr_cprs_conf, cpr_cprs/ qed-. - -(* Basic_1: was only: pc3_thin_dx *) -lemma cpcs_flat: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¬* V2 â âT1,T2. â¦G, Lâ¦ â¢ T1 â¬* T2 â - âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¬* â{I}V2.T2. -#G #L #V1 #V2 #HV12 #T1 #T2 #HT12 -elim (cpcs_inv_cprs â¦ HV12) -HV12 -elim (cpcs_inv_cprs â¦ HT12) -HT12 -/3 width=5 by cprs_flat, cprs_div/ -qed. - -lemma cpcs_flat_dx_cpr_rev: âG,L,V1,V2. â¦G, Lâ¦ â¢ V2 â¡ V1 â âT1,T2. â¦G, Lâ¦ â¢ T1 â¬* T2 â - âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¬* â{I}V2.T2. -/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed. - -lemma cpcs_bind_dx: âa,I,G,L,V,T1,T2. â¦G, L.â{I}Vâ¦ â¢ T1 â¬* T2 â - â¦G, Lâ¦ â¢ â{a,I}V.T1 â¬* â{a,I}V.T2. -#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs â¦ HT12) -HT12 -/3 width=5 by cprs_div, cprs_bind/ -qed. - -lemma cpcs_bind_sn: âa,I,G,L,V1,V2,T. â¦G, Lâ¦ â¢ V1 â¬* V2 â â¦G, Lâ¦ â¢ â{a,I}V1. T â¬* â{a,I}V2. T. -#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs â¦ HV12) -HV12 -/3 width=5 by cprs_div, cprs_bind/ -qed. - -lemma lsubr_cpcs_trans: âG,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¬* T2 â - âL2. L2 â L1 â â¦G, L2â¦ â¢ T1 â¬* T2. -#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs â¦ HT12) -HT12 -/3 width=5 by cprs_div, lsubr_cprs_trans/ -qed-. - -(* Basic_1: was: pc3_lift *) -lemma cpcs_lift: âG,L,K,s,d,e. â©[s, d, e] L â¡ K â - âT1,U1. â§[d, e] T1 â¡ U1 â âT2,U2. â§[d, e] T2 â¡ U2 â - â¦G, Kâ¦ â¢ T1 â¬* T2 â â¦G, Lâ¦ â¢ U1 â¬* U2. -#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12 -elim (cpcs_inv_cprs â¦ HT12) -HT12 #T #HT1 #HT2 -elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/ -qed. - -lemma cpcs_strip: âG,L,T1,T. â¦G, Lâ¦ â¢ T â¬* T1 â âT2. â¦G, Lâ¦ â¢ T â¬ T2 â - ââT0. â¦G, Lâ¦ â¢ T1 â¬ T0 & â¦G, Lâ¦ â¢ T2 â¬* T0. -#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-. - -(* More inversion lemmas ****************************************************) - -axiom cpcs_inv_abst_sn: âa1,a2,G,L,W1,W2,T1,T2. â¦G, Lâ¦ â¢ â{a1}W1.T1 â¬* â{a2}W2.T2 â - â§â§ â¦G, Lâ¦ â¢ W1 â¬* W2 & â¦G, L.âW1â¦ â¢ T1 â¬* T2 & a1 = a2. -(* -#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H -elim (cpcs_inv_cprs â¦ H) -H #T #H1 #H2 -elim (cprs_inv_abst1 â¦ H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct -elim (cprs_inv_abst1 â¦ H2) -H2 #W #T #HW2 #HT2 #H destruct -lapply (llprs_cprs_conf â¦ (L.âW) â¦ HT2) /2 width=1 by llprs_pair/ -HT2 #HT2 -lapply (llprs_cpcs_trans â¦ (L.âW1) â¦ 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: âa1,a2,G,L,W1,W2,T1,T2. â¦G, Lâ¦ â¢ â{a1}W1.T1 â¬* â{a2}W2.T2 â - â§â§ â¦G, Lâ¦ â¢ W1 â¬* W2 & â¦G, L.âW2â¦ â¢ T1 â¬* T2 & a1 = a2. -#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym â¦ HT12) -HT12 -#HT12 elim (cpcs_inv_abst_sn â¦ HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/ -qed-. - -(* Main properties **********************************************************) - -(* Basic_1: was pc3_t *) -theorem cpcs_trans: âG,L,T1,T. â¦G, Lâ¦ â¢ T1 â¬* T â âT2. â¦G, Lâ¦ â¢ T â¬* T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. -#G #L #T1 #T #HT1 #T2 @(trans_TC â¦ HT1) qed-. - -theorem cpcs_canc_sn: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¬* T1 â â¦G, Lâ¦ â¢ T â¬* T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. -/3 width=3 by cpcs_trans, cpcs_sym/ qed-. - -theorem cpcs_canc_dx: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T1 â¬* T â â¦G, Lâ¦ â¢ T2 â¬* T â â¦G, Lâ¦ â¢ T1 â¬* T2. -/3 width=3 by cpcs_trans, cpcs_sym/ qed-. - -lemma cpcs_bind1: âa,I,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¬* V2 â - âT1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¬* T2 â - â¦G, Lâ¦ â¢ â{a,I}V1. T1 â¬* â{a,I}V2. T2. -/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed. - -lemma cpcs_bind2: âa,I,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¬* V2 â - âT1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¬* T2 â - â¦G, Lâ¦ â¢ â{a,I}V1. T1 â¬* â{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: âG,L1,L2,T1,T2. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â â¦G, L1â¦ â¢ â¡[T2, 0] L2 â - â¦G, L1â¦ â¢ T1 â¬* T2 â â¦G, L2â¦ â¢ T1 â¬* T2. -#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs â¦ H) -H -/3 width=5 by cpcs_canc_dx, llpr_cprs_conf/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/cprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/cprs_cprs.etc deleted file mode 100644 index a85488cdf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/cprs_cprs.etc +++ /dev/null @@ -1,166 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L. Transitive â¦ (cprs G L). -normalize /2 width=3 by trans_TC/ qed-. - -(* Basic_1: was: pr3_confluence *) -(* Basic_1: includes: pr1_confluence *) -theorem cprs_conf: âG,L. confluent2 â¦ (cprs G L) (cprs G L). -normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-. - -theorem cprs_bind: âa,I,G,L,V1,V2,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ V1 â¡* V2 â - â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{a,I}V2.T2. -#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind â¦ H) -V2 -/3 width=5 by cprs_trans, cprs_bind_dx/ -qed. - -(* Basic_1: was: pr3_flat *) -theorem cprs_flat: âI,G,L,V1,V2,T1,T2. â¦G, Lâ¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ V1 â¡* V2 â - â¦G, Lâ¦ â¢ â{I}V1.T1 â¡* â{I}V2.T2. -#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind â¦ H) -V2 -/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/ -qed. - -theorem cprs_beta_rc: âa,G,L,V1,V2,W1,W2,T1,T2. - â¦G, Lâ¦ â¢ V1 â¡ V2 â â¦G, L.âW1â¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ W1 â¡* W2 â - â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}âW2.V2.T2. -#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind â¦ H) -W2 /2 width=1 by cprs_beta_dx/ -#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *) -@(cprs_trans â¦ IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/ -qed. - -theorem cprs_beta: âa,G,L,V1,V2,W1,W2,T1,T2. - â¦G, L.âW1â¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ W1 â¡* W2 â â¦G, Lâ¦ â¢ V1 â¡* V2 â - â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}âW2.V2.T2. -#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind â¦ H) -V2 /2 width=1 by cprs_beta_rc/ -#V #V2 #_ #HV2 #IHV1 -@(cprs_trans â¦ IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/ -qed. - -theorem cprs_theta_rc: âa,G,L,V1,V,V2,W1,W2,T1,T2. - â¦G, Lâ¦ â¢ V1 â¡ V â â§[0, 1] V â¡ V2 â â¦G, L.âW1â¦ â¢ T1 â¡* T2 â - â¦G, Lâ¦ â¢ W1 â¡* W2 â â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}W2.âV2.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: âa,G,L,V1,V,V2,W1,W2,T1,T2. - â§[0, 1] V â¡ V2 â â¦G, Lâ¦ â¢ W1 â¡* W2 â â¦G, L.âW1â¦ â¢ T1 â¡* T2 â - â¦G, Lâ¦ â¢ V1 â¡* V â â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}W2.âV2.T2. -#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx â¦ 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: âG,L,V1,T1,U2. â¦G, Lâ¦ â¢ âV1.T1 â¡* U2 â - â¨â¨ ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡* V2 & â¦G, Lâ¦ â¢ T1 â¡* T2 & - U2 = âV2. T2 - | ââa,W,T. â¦G, Lâ¦ â¢ T1 â¡* â{a}W.T & - â¦G, Lâ¦ â¢ â{a}âW.V1.T â¡* U2 - | ââa,V0,V2,V,T. â¦G, Lâ¦ â¢ V1 â¡* V0 & â§[0,1] V0 â¡ V2 & - â¦G, Lâ¦ â¢ T1 â¡* â{a}V.T & - â¦G, Lâ¦ â¢ â{a}V.âV2.T â¡* U2. -#G #L #V1 #T1 #U2 #H @(cprs_ind â¦ H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/ -#U #U2 #_ #HU2 * * -[ #V0 #T0 #HV10 #HT10 #H destruct - elim (cpr_inv_appl1 â¦ 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 â¦ HV10 â¦ HV02) -V0 #HV12 - lapply (lsubr_cpr_trans â¦ HT2 (L.ââW.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 â¦ 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: âG. s_r_transitive â¦ (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 â¦ HL12 â¦ 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 â¦ HL12) -HL12 /4 width=1 by cprs_bind/ -| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12 - elim (llpr_inv_flat â¦ HL12) -HL12 /3 width=1 by cprs_flat/ -| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12 - elim (llpr_inv_bind_O â¦ HL12) /3 width=3 by cprs_zeta/ -| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12 - elim (llpr_inv_flat â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 - elim (llpr_inv_bind_O â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 - elim (llpr_inv_bind_O â¦ HL12) /3 width=3 by cprs_theta/ -] -qed-. - -lemma cpr_bind2: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡ V2 â âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡ T2 â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{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: âG. s_rs_transitive â¦ (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: âG,L. confluent2 â¦ (cprs G L) (cpr G L). -normalize /4 width=3 by cpr_conf, TC_strip1/ qed-. - -lemma cprs_llpr_conf_dx: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡* T1 â âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â - ââT. â¦G, L1â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. -#G #L0 #T0 #T1 #H @(cprs_ind_dx â¦ T0 H) -T0 /2 width=3 by ex2_intro/ -#T0 #T #HT0 #_ #IHT1 #L1 #HL01 -elim (IHT1 â¦ L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2 -elim (llpr_cpr_conf_dx â¦ HT0 â¦ HL01) -L0 #T3 #HT03 #HT3 -elim (cprs_strip â¦ HT2 â¦ HT3) -T -/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/ -qed-. - -lemma cprs_llpr_conf_sn: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡* T1 â - âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â - ââT. â¦G, L0â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. -#G #L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cprs_llpr_conf_dx â¦ HT01 â¦ HL01) -/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/ -qed-. - -lemma cprs_bind2_dx: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡ V2 â - âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡* T2 â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{a,I}V2.T2. -/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/cprs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/cprs_lift.etc deleted file mode 100644 index 27f0bb6e5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/cprs_lift.etc +++ /dev/null @@ -1,63 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,K,V,V2,i. - â©[i] L â¡ K.âV â â¦G, Kâ¦ â¢ V â¡* V2 â - âW2. â§[0, i + 1] V2 â¡ W2 â â¦G, Lâ¦ â¢ #i â¡* 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 â¦ HLK) -HLK #HLK -elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/ -qed. - -lemma cprs_llpr_conf: âG. s_r_confluent1 â¦ (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: âG,L,T2,i. â¦G, Lâ¦ â¢ #i â¡* T2 â - T2 = #i â¨ - ââK,V1,T1. â©[i] L â¡ K.âV1 & â¦G, Kâ¦ â¢ V1 â¡* T1 & - â§[0, i + 1] T1 â¡ T2. -#G #L #T2 #i #H @(cprs_ind â¦ H) -T2 /2 width=1 by or_introl/ -#T #T2 #_ #HT2 * -[ #H destruct - elim (cpr_inv_lref1 â¦ 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 â¦ HLK) #H0LK - elim (cpr_inv_lift1 â¦ HT2 â¦ H0LK â¦ 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: âG. l_liftable (cprs G). -/3 width=10 by l_liftable_LTC, cpr_lift/ qed. - -(* Basic_1: was: pr3_gen_lift *) -lemma cprs_inv_lift1: âG. l_deliftable_sn (cprs G). -/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/lazypredsn_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/lazypredsn_5.etc deleted file mode 100644 index 01ce586f8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/lazypredsn_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡ break [ term 46 T , break term 46 d ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'LazyPRedSn $G $L1 $L2 $T $d }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/lazypredsnstar_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/lazypredsnstar_5.etc deleted file mode 100644 index 824933fba..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/lazypredsnstar_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡* break [ term 46 T , break term 46 d ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'LazyPRedSnStar $G $L1 $L2 $T $d }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr.etc deleted file mode 100644 index e4c1a01be..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr.etc +++ /dev/null @@ -1,51 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â relation4 ynat term lenv lenv â Î»G. 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: âI,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[â{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡[V, d] L2 â§ â¦G, L1â¦ â¢ â¡[T, d] L2. -/2 width=2 by llpx_sn_inv_flat/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma llpr_fwd_length: âG,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[T, d] L2 â |L1| = |L2|. -/2 width=4 by llpx_sn_fwd_length/ qed-. - -(* Basic properties *********************************************************) - -lemma llpr_lref: âI,G,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I}V1 â â©[i] L2 â¡ K2.â{I}V2 â - â¦G, K1â¦ â¢ â¡[V1, 0] K2 â â¦G, K1â¦ â¢ V1 â¡ V2 â â¦G, L1â¦ â¢ â¡[#i, d] L2. -/2 width=9 by llpx_sn_lref/ qed. - -(* Note: lemma 250 *) -lemma llpr_refl: âG,T,d. reflexive â¦ (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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr_ldrop.etc deleted file mode 100644 index 74f7cc6b3..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr_ldrop.etc +++ /dev/null @@ -1,97 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[#i, d] L2 â d â¤ i â - âI,K2,V2. â©[i] L2 â¡ K2.â{I}V2 â - ââK1,V1. â©[i] L1 â¡ K1.â{I}V1 & - â¦G, K1â¦ â¢ â¡[V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡ V2. -/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-. - -lemma llpr_inv_lref_ge_sn: âG,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[#i, d] L2 â d â¤ i â - âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â - ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & - â¦G, K1â¦ â¢ â¡[V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡ V2. -/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-. - -lemma llpr_inv_lref_ge_bi: âG,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[#i, d] L2 â d â¤ i â - âI1,I2,K1,K2,V1,V2. - â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â - â§â§ I1 = I2 & â¦G, K1â¦ â¢ â¡[V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡ V2. -/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. - -lemma llpr_inv_bind_O: âa,I,G,L1,L2,V,T. â¦G, L1â¦ â¢ â¡ [â{a,I}V.T, 0] L2 â - â¦G, L1â¦ â¢ â¡[V, 0] L2 â§ â¦G, L1.â{I}Vâ¦ â¢ â¡[T, 0] L2.â{I}V. -/2 width=2 by llpx_sn_inv_bind_O/ qed-. - -lemma llpr_bind_repl_O: âI,G,L1,L2,V1,V2,T. â¦G, L1.â{I}V1â¦ â¢ â¡[T, 0] L2.â{I}V2 â - âJ,W1,W2. â¦G, L1â¦ â¢ â¡[W1, 0] L2 â â¦G, L1â¦ â¢ W1 â¡ W2 â â¦G, L1.â{J}W1â¦ â¢ â¡[T, 0] L2.â{J}W2. -/2 width=4 by llpx_sn_bind_repl_O/ qed-. - -(* Advanced properties ******************************************************) - -lemma llpr_ssta_conf: âh,g,G. s_r_confluent1 â¦ (ssta h g G) (llpr G 0). -/3 width=10 by ssta_llpx_sn_conf, cpr_lift/ qed-. - -lemma llpr_cpr_conf: âG. s_r_confluent1 â¦ (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: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â - âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â - ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, Lâ¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ â â¦G2, L2, U2â¦. -#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 â¦ HU2 â¦ HLK â¦ HUT â¦ HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/ -qed-. - -lemma fquq_cpr_trans_dx: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â - âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â - ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, Lâ¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ ââ¸® â¦G2, L2, U2â¦. -#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen â¦ H) -H -[ #HT12 elim (fqu_cpr_trans_dx â¦ HT12 â¦ 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: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â - âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â - ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, L1â¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ â â¦G2, L2, U2â¦. -#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 â¦ HU2 â¦ HLK â¦ HUT â¦ HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/ -qed-. - -lemma fquq_cpr_trans_sn: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â - âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â - ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, L1â¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ ââ¸® â¦G2, L2, U2â¦. -#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen â¦ H) -H -[ #HT12 elim (fqu_cpr_trans_sn â¦ HT12 â¦ HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/ -| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr_llpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr_llpr.etc deleted file mode 100644 index 4e5b9e2be..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpr_llpr.etc +++ /dev/null @@ -1,375 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: - âI,G,L1,L2. ââT. â¦G, L1â¦ â¢ âª{I} â¡ T & â¦G, L2â¦ â¢ âª{I} â¡ T. -/2 width=3 by cpr_atom, ex2_intro/ qed-. - -fact cpr_conf_llpr_atom_delta: - âG,L0,i. ( - âL,T. â¦G, L0, #iâ¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âK0,V0. â©[i] L0 â¡ K0.âV0 â - âV2. â¦G, K0â¦ â¢ V0 â¡ V2 â âT2. â§[O, i + 1] V2 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[#i, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[#i, 0] L2 â - ââT. â¦G, L1â¦ â¢ #i â¡ T & â¦G, L2â¦ â¢ T2 â¡ T. -#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -elim (llpr_inv_lref_ge_sn â¦ HL01 â¦ HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01 -elim (llpr_inv_lref_ge_sn â¦ HL02 â¦ HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_ -lapply (ldrop_fwd_drop2 â¦ HLK2) -W2 #HLK2 -lapply (fqup_lref â¦ G â¦ HLK0) -HLK0 #HLK0 -elim (IH â¦ HLK0 â¦ HV01 â¦ HV02 â¦ HK01 â¦ 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: - âG,L0,i. ( - âL,T. â¦G, L0, #iâ¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âK0,V0. â©[i] L0 â¡ K0.âV0 â - âV1. â¦G, K0â¦ â¢ V0 â¡ V1 â âT1. â§[O, i + 1] V1 â¡ T1 â - âKX,VX. â©[i] L0 â¡ KX.âVX â - âV2. â¦G, KXâ¦ â¢ VX â¡ V2 â âT2. â§[O, i + 1] V2 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[#i, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[#i, 0] L2 â - ââT. â¦G, L1â¦ â¢ T1 â¡ T & â¦G, L2â¦ â¢ T2 â¡ 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 â¦ H â¦ HLK0) -H #H destruct -elim (llpr_inv_lref_ge_sn â¦ HL01 â¦ HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_ -lapply (ldrop_fwd_drop2 â¦ HLK1) -W1 #HLK1 -elim (llpr_inv_lref_ge_sn â¦ HL02 â¦ HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_ -lapply (ldrop_fwd_drop2 â¦ HLK2) -W2 #HLK2 -lapply (fqup_lref â¦ G â¦ HLK0) -HLK0 #HLK0 -elim (IH â¦ HLK0 â¦ HV01 â¦ HV02 â¦ HK01 â¦ 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: - âa,I,G,L0,V0,T0. ( - âL,T. â¦G, L0, â{a,I}V0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0.â{I}V0â¦ â¢ T0 â¡ T1 â - âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âT2. â¦G, L0.â{I}V0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[â{a,I}V0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[â{a,I}V0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ â{a,I}V1.T1 â¡ T & â¦G, L2â¦ â¢ â{a,I}V2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #H1T0 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2V0 #H2T0 -elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) // -elim (IH â¦ HT01 â¦ HT02 (L1.â{I}V1) â¦ (L2.â{I}V2)) -IH -/3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/ -qed-. - -fact cpr_conf_llpr_bind_zeta: - âG,L0,V0,T0. ( - âL,T. â¦G, L0, +âV0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0.âV0â¦ â¢ T0 â¡ T1 â - âT2. â¦G, L0.âV0â¦ â¢ T0 â¡ T2 â âX2. â§[O, 1] X2 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ +âV1.T1 â¡ T & â¦G, L2â¦ â¢ X2 â¡ T. -#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 -elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1V0 #H1T0 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2V0 #H2T0 -elim (IH â¦ HT01 â¦ HT02 (L1.âV1) â¦ (L2.âV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2 -elim (cpr_inv_lift1 â¦ HT2 L2 â¦ HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/ -qed-. - -fact cpr_conf_llpr_zeta_zeta: - âG,L0,V0,T0. ( - âL,T. â¦G, L0, +âV0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âT1. â¦G, L0.âV0â¦ â¢ T0 â¡ T1 â âX1. â§[O, 1] X1 â¡ T1 â - âT2. â¦G, L0.âV0â¦ â¢ T0 â¡ T2 â âX2. â§[O, 1] X2 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ X1 â¡ T & â¦G, L2â¦ â¢ X2 â¡ T. -#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 -#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 -elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1V0 #H1T0 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2V0 #H2T0 -elim (IH â¦ HT01 â¦ HT02 (L1.âV0) â¦ (L2.âV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2 -elim (cpr_inv_lift1 â¦ HT1 L1 â¦ HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1 -elim (cpr_inv_lift1 â¦ HT2 L2 â¦ HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2 -lapply (lift_inj â¦ HT2 â¦ HT1) -T #H destruct /2 width=3 by ex2_intro/ -qed-. - -fact cpr_conf_llpr_flat_flat: - âI,G,L0,V0,T0. ( - âL,T. â¦G, L0, â{I}V0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0â¦ â¢ T0 â¡ T1 â - âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âT2. â¦G, L0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[â{I}V0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[â{I}V0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ â{I}V1.T1 â¡ T & â¦G, L2â¦ â¢ â{I}V2.T2 â¡ T. -#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (llpr_inv_flat â¦ HL01) -HL01 #H1V0 #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #H2T0 -elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) // -elim (IH â¦ HT01 â¦ HT02 â¦ H1T0 â¦ H2T0) /3 width=5 by cpr_flat, ex2_intro/ -qed-. - -fact cpr_conf_llpr_flat_tau: - âG,L0,V0,T0. ( - âL,T. â¦G, L0, âV0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â âT2. â¦G, L0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ âV1.T1 â¡ T & â¦G, L2â¦ â¢ T2 â¡ T. -#G #L0 #V0 #T0 #IH #V1 #T1 #HT01 -#T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (llpr_inv_flat â¦ HL01) -HL01 #_ #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #_ #H2T0 -elim (IH â¦ HT01 â¦ HT02 â¦ H1T0 â¦ H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/ -qed-. - -fact cpr_conf_llpr_tau_tau: - âG,L0,V0,T0. ( - âL,T. â¦G, L0, âV0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âT1. â¦G, L0â¦ â¢ T0 â¡ T1 â âT2. â¦G, L0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ T1 â¡ T & â¦G, L2â¦ â¢ T2 â¡ T. -#G #L0 #V0 #T0 #IH #T1 #HT01 -#T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (llpr_inv_flat â¦ HL01) -HL01 #_ #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #_ #H2T0 -elim (IH â¦ HT01 â¦ HT02 â¦ H1T0 â¦ H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/ -qed-. - -fact cpr_conf_llpr_flat_beta: - âa,G,L0,V0,W0,T0. ( - âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0â¦ â¢ â{a}W0.T0 â¡ T1 â - âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ âV1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}âW2.V2.T2 â¡ 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 â¦ H) -H #W1 #T1 #HW01 #HT01 #H destruct -elim (llpr_inv_flat â¦ HL01) -HL01 #H1V0 #HL01 -elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 -elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 -elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1 by/ #W #HW1 #HW2 -elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 -lapply (lsubr_cpr_trans â¦ HT2 (L2.ââW2.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: - âa,G,L0,V0,W0,T0. ( - âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0â¦ â¢ â{a}W0.T0 â¡ T1 â - âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âU2. â§[O, 1] V2 â¡ U2 â - âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ âV1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}W2.âU2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #HL01 -elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 -elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 -elim (lift_total V 0 1) #U #HVU -lapply (cpr_lift â¦ HV2 (L2.âW2) â¦ HVU2 â¦ HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2 -elim (cpr_inv_abbr1 â¦ H) -H * -[ #W1 #T1 #HW01 #HT01 #H destruct - elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1 by/ - elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /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 â¦ HT01 â¦ HT02 (L1.âW2) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 - elim (cpr_inv_lift1 â¦ HT1 L1 â¦ HXT1) -HXT1 - /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/ -] -qed-. - -fact cpr_conf_llpr_beta_beta: - âa,G,L0,V0,W0,T0. ( - âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âW1. â¦G, L0â¦ â¢ W0 â¡ W1 â âT1. â¦G, L0.âW0â¦ â¢ T0 â¡ T1 â - âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ â{a}âW1.V1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}âW2.V2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #HL01 -elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 -elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 -elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1/ #W #HW1 #HW2 -elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 -lapply (lsubr_cpr_trans â¦ HT1 (L1.ââW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/ -lapply (lsubr_cpr_trans â¦ HT2 (L2.ââW2.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: - âa,G,L0,V0,W0,T0. ( - âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â - âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â - âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â - ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 - ) â - âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âU1. â§[O, 1] V1 â¡ U1 â - âW1. â¦G, L0â¦ â¢ W0 â¡ W1 â âT1. â¦G, L0.âW0â¦ â¢ T0 â¡ T1 â - âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âU2. â§[O, 1] V2 â¡ U2 â - âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â - âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â - ââT. â¦G, L1â¦ â¢ â{a}W1.âU1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}W2.âU2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #HL01 -elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 -elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 -elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 -elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 -elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1 by/ -elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 -elim (lift_total V 0 1) #U #HVU -lapply (cpr_lift â¦ HV1 (L1.âW1) â¦ HVU1 â¦ HVU) -HVU1 /2 width=2 by ldrop_drop/ -lapply (cpr_lift â¦ HV2 (L2.âW2) â¦ HVU2 â¦ 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: âG. llpx_sn_confluent2 (cpr G) (cpr G). -#G #L0 #T0 @(fqup_wf_ind_eq â¦ 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 â¦ H1) -H1 - elim (cpr_inv_atom1 â¦ 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 â¦ H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #T1 #HT01 #HXT1 #H11 #H12 - ] - elim (cpr_inv_bind1 â¦ 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 â¦ 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 â¦ 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: âG,L. confluent â¦ (cpr G L). -/2 width=6 by cpr_conf_llpr/ qed-. - -(* Properties on context-sensitive parallel reduction for terms *************) - -lemma llpr_cpr_conf_dx: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â - ââT. â¦G, L1â¦ â¢ T0 â¡ T & â¦G, L1â¦ â¢ T1 â¡ T. -#G #L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_conf_llpr â¦ HT01 T0 â¦ HL01 â¦ HL01) /2 width=3 by ex2_intro/ -qed-. - -lemma llpr_cpr_conf_sn: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â - ââT. â¦G, L1â¦ â¢ T0 â¡ T & â¦G, L0â¦ â¢ T1 â¡ T. -#G #L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_conf_llpr â¦ HT01 T0 â¦ L0 â¦ HL01) /2 width=3 by ex2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs.etc deleted file mode 100644 index d3c2a6f27..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs.etc +++ /dev/null @@ -1,61 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â relation4 ynat term lenv lenv â - Î»G,d. LTC â¦ (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: âG,L1,T,d. âR:predicate lenv. R L1 â - (âL,L2. â¦G, L1â¦ â¢ â¡*[T, d] L â â¦G, Lâ¦ â¢ â¡[T, d] L2 â R L â R L2) â - âL2. â¦G, L1â¦ â¢ â¡*[T, d] L2 â R L2. -#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12 -@(TC_star_ind â¦ HL1 IHL1 â¦ HL12) // -qed-. - -lemma llprs_ind_dx: âG,L2,T,d. âR:predicate lenv. R L2 â - (âL1,L. â¦G, L1â¦ â¢ â¡[T, d] L â â¦G, Lâ¦ â¢ â¡*[T, d] L2 â R L â R L1) â - âL1. â¦G, L1â¦ â¢ â¡*[T, d] L2 â R L1. -#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12 -@(TC_star_ind_dx â¦ HL2 IHL2 â¦ HL12) // -qed-. - -(* Basic properties *********************************************************) - -lemma lpr_llprs: âG,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[T, d] L2 â â¦G, L1â¦ â¢ â¡*[T, d] L2. -/2 width=1 by inj/ qed. - -lemma llprs_refl: âG,L,T,d. â¦G, Lâ¦ â¢ â¡*[T, d] L. -/2 width=1 by lpr_llprs/ qed. - -lemma llprs_strap1: âG,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡*[T, d] L â â¦G, Lâ¦ â¢ â¡[T, d] L2 â â¦G, L1â¦ â¢ â¡*[T, d] L2. -normalize /2 width=3 by step/ qed-. - -lemma llprs_strap2: âG,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡[T, d] L â â¦G, Lâ¦ â¢ â¡*[T, d] L2 â â¦G, L1â¦ â¢ â¡*[T, d] L2. -normalize /2 width=3 by TC_strap/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma llprs_fwd_length: âG,L1,L2,T,d. â¦G, L1â¦ â¢ â¡*[T, d] L2 â |L1| = |L2|. -#G #L1 #L2 #T #d #H @(llprs_ind â¦ H) -L2 -/3 width=6 by llpr_fwd_length, trans_eq/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs_cprs.etc deleted file mode 100644 index b9bf0f963..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs_cprs.etc +++ /dev/null @@ -1,122 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡* V2 â - âI,T. â¦G, L.â{I}V1â¦ â¢ â¡*[T, 0] L.â{I}V2. -/2 width=1 by llpx_sn_TC_pair_dx/ qed. - -(* Properties on context-sensitive parallel computation for terms ***********) - -lemma llprs_cpr_trans: âG. s_r_transitive â¦ (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: âG. s_rs_transitive â¦ (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: âG2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡* U2 â - âL0. â¦G2, L0â¦ â¢ â¡*[T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[U2, O] L2. -#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx â¦ H) -L0 // -#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 â¦ IHL2) -IHL2 -lapply (llprs_cprs_trans â¦ HTU2 â¦ 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: âG2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡ U2 â - âL0. â¦G2, L0â¦ â¢ â¡*[T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[U2, O] L2. -#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx â¦ H) -L0 // -#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 â¦ IHL2) -IHL2 -lapply (llprs_cpr_trans â¦ HTU2 â¦ HL2) -L2 #HTU2 -/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/ -qed-. - -lemma llprs_cprs_conf_sn: âG,L0,L1,T0. â¦G, L0â¦ â¢ â¡*[T0, 0] L1 â - âT1. â¦G, L0â¦ â¢ T0 â¡* T1 â - ââT. â¦G, L0â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. -#G #L0 #L1 #T0 #H @(llprs_ind_dx â¦ H) -L0 /2 width=3 by ex2_intro/ -#L0 #L #HL0 #HL1 #IHL1 #T1 #HT01 elim (cprs_llpr_conf_sn â¦ HT01 â¦ HL0) -#T2 #HT12 #HT02 elim (IHL1 â¦ HT02) -IHL1 -HT02 -lapply (cprs_trans â¦ HT01 â¦ HT12) #HT02 -lapply (cprs_llpr_conf â¦ HT02 â¦ HL0) -HT02 -HL0 -/4 width=5 by cprs_llpr_trans, cprs_trans, ex2_intro/ -qed-. - -lemma llprs_cpr_conf_sn: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â - âL1. â¦G, L0â¦ â¢ â¡*[T0, 0] L1 â - ââT. â¦G, L0â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. -/3 width=3 by llprs_cprs_conf_sn, cpr_cprs/ qed-. - -lemma cprs_bind2: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡* V2 â - âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡* T2 â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{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: âa,G,L,W1,T1,U2. â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* U2 â - ââW2,T2. â¦G, Lâ¦ â¢ W1 â¡* W2 & â¦G, L.âW1â¦ â¢ T1 â¡* T2 & - U2 = â{a}W2.T2. -#a #G #L #V1 #T1 #U2 #H @(cprs_ind â¦ H) -U2 /2 width=5 by ex3_2_intro/ -#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct -elim (cpr_inv_abst1 â¦ HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct -lapply (llprs_cpr_trans â¦ HT02 (L.âV1) ?) -/3 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro/ -qed-. - -lemma cprs_inv_abst: âa,G,L,W1,W2,T1,T2. â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* â{a}W2.T2 â - â¦G, Lâ¦ â¢ W1 â¡* W2 â§ â¦G, L.âW1â¦ â¢ T1 â¡* T2. -#a #G #L #W1 #W2 #T1 #T2 #H -elim (cprs_inv_abst1 â¦ H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/ -qed-. - -(* Basic_1: was pr3_gen_abbr *) -lemma cprs_inv_abbr1: âa,G,L,V1,T1,U2. â¦G, Lâ¦ â¢ â{a}V1.T1 â¡* U2 â ( - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡* V2 & â¦G, L.âV1â¦ â¢ T1 â¡* T2 & - U2 = â{a}V2.T2 - ) â¨ - ââT2. â¦G, L.âV1â¦ â¢ T1 â¡* T2 & â§[0, 1] U2 â¡ T2 & a = true. -#a #G #L #V1 #T1 #U2 #H @(cprs_ind â¦ H) -U2 /3 width=5 by ex3_2_intro, or_introl/ -#U0 #U2 #_ #HU02 * * -[ #V0 #T0 #HV10 #HT10 #H destruct - elim (cpr_inv_abbr1 â¦ HU02) -HU02 * - [ #V2 #T2 #HV02 #HT02 #H destruct - lapply (llprs_cpr_trans â¦ HT02 (L.âV1) ?) - /4 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/ - | #T2 #HT02 #HUT2 - lapply (llprs_cpr_trans â¦ HT02 (L.âV1) ?) -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 â¦ HU02 (L.âV1) â¦ HU01 â¦ 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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs_llprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs_llprs.etc deleted file mode 100644 index 4f97672f3..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llprs_llprs.etc +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âG,T,d. Transitive â¦ (llprs G d T). -normalize /2 width=3 by trans_TC/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpx_sn_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpx_sn_llpx_sn.etc deleted file mode 100644 index 0df9690ee..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpr/llpx_sn_llpx_sn.etc +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 (lenvârelation term) â Î»R1,R2. - âL0,T0,T1. R1 L0 T0 T1 â âT2. R2 L0 T0 T2 â - âL1. llpx_sn R1 0 T0 L0 L1 â âL2. llpx_sn R2 0 T0 L0 L2 â - ââT. R2 L1 T1 T & R1 L2 T2 T. - -(* Note: we miss llpx_sn_conf and derivatives: lpr_conf lprs_conf *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/cpxs_llpx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/cpxs_llpx.etc deleted file mode 100644 index bec908ef7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/cpxs_llpx.etc +++ /dev/null @@ -1,53 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G. s_r_confluent1 â¦ (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: âh,g,G. s_r_transitive â¦ (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 â¦ HL12 â¦ 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 â¦ HL12) -HL12 /4 width=1 by cpxs_bind/ -| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12 - elim (llpx_inv_flat â¦ HL12) -HL12 /3 width=1 by cpxs_flat/ -| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12 - elim (llpx_inv_bind_O â¦ HL12) /3 width=3 by cpxs_zeta/ -| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12 - elim (llpx_inv_flat â¦ HL12) /3 width=1 by cpxs_tau/ -| #G #L2 #V1 #V2 #T2 #HV12 #IHV12 #L1 #HL12 - elim (llpx_inv_flat â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 - elim (llpx_inv_bind_O â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 - elim (llpx_inv_bind_O â¦ HL12) /3 width=3 by cpxs_theta/ -] -qed-. - -lemma llpx_cpxs_trans: âh,g,G. s_rs_transitive â¦ (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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/csx_llpx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/csx_llpx.etc deleted file mode 100644 index c14492f21..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/csx_llpx.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T. â¦G, L1â¦ â¢ â¬*[h, g] T â - âL2. â¦G, L1â¦ â¢ â¡[h, g, T, 0] L2 â â¦G, L2â¦ â¢ â¬*[h, g] T. -#h #g #G #L1 #T #H @(csx_ind_alt â¦ H) -T -/5 width=3 by csx_intro_cpxs, llpx_cpxs_trans, cpxs_llpx_conf/ (* 2 cpxs_llpx_trans *) -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/csx_llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/csx_llpxs.etc deleted file mode 100644 index aad3e8bf0..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/csx_llpxs.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T. â¦G, L1â¦ â¢ â¬*[h, g] T â - âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, 0] L2 â â¦G, L2â¦ â¢ â¬*[h, g] T. -#h #g #G #L1 #T #HT #L2 #H @(llpxs_ind â¦ H) -L2 /3 by llpxs_strap1, csx_llpx_conf/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/fpbs_lpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/fpbs_lpr.etc deleted file mode 100644 index 34b60ad1d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/fpbs_lpr.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2,T1,T2. â¦G, L1â¦ â¢ T1 â¡ T2 â â¦G, L1â¦ â¢ â¡ L2 â - â¦G, L1, T1â¦ â¥[h, g] â¦G, L2, T2â¦. -/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: âh,g,G,L1,L2,T1,T2,U2,l2. - â¦G, L1â¦ â¢ T1 â¡ T2 â â¦G, L1â¦ â¢ â¡ L2 â - â¦G, L2â¦ â¢ T2 âª[h, g] l2+1 â â¦G, L2â¦ â¢ T2 â¢[h, g] U2 â - â¦G, L1, T1â¦ â¥[h, g] â¦G, L2, U2â¦. -/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, ssta_cpx, fpb_cpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazypredsn_7.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazypredsn_7.etc deleted file mode 100644 index 2350f5efc..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazypredsn_7.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡ 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazypredsnstar_7.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazypredsnstar_7.etc deleted file mode 100644 index f8209e2f8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazypredsnstar_7.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡* 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazysn_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazysn_6.etc deleted file mode 100644 index 0c52a6ea5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazysn_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( G â¢ â â¬ * 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazysnalt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazysnalt_6.etc deleted file mode 100644 index 0976a3136..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/lazysnalt_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( G â¢ â â¬ â¬ * 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx.etc deleted file mode 100644 index 3276008b4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx.etc +++ /dev/null @@ -1,69 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â genv â relation4 ynat term lenv lenv â - Î»h,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: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡[h, g, V, d] L2 â§ â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. -/2 width=2 by llpx_sn_inv_flat/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma llpx_fwd_length: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â |L1| = |L2|. -/2 width=4 by llpx_sn_fwd_length/ qed-. - -lemma llpx_fwd_flat_dx: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. -/2 width=3 by llpx_sn_fwd_flat_dx/ qed-. - -lemma llpx_fwd_pair_sn: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â¡{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡[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: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{a,I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡[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: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{a,I}V.T, d] L2 â - â¦G, L1.â{I}Vâ¦ â¢ â¡[h, g, T, â«¯d] L2.â{I}V. -/2 width=2 by llpx_sn_fwd_bind_dx/ qed-. - -(* Note: this might be removed *) -lemma llpx_fwd_flat_sn: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡[h, g, V, d] L2. -/2 width=3 by llpx_sn_fwd_flat_sn/ qed-. - -(* Basic properties *********************************************************) - -lemma llpx_lref: âh,g,I,G,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I}V1 â â©[i] L2 â¡ K2.â{I}V2 â - â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 â â¦G, K1â¦ â¢ V1 â¡[h, g] V2 â â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2. -/2 width=9 by llpx_sn_lref/ qed. - -lemma llpx_refl: âh,g,G,T,d. reflexive â¦ (llpx h g G d T). -/2 width=1 by llpx_sn_refl/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_aaa.etc deleted file mode 100644 index 99eef8d55..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_aaa.etc +++ /dev/null @@ -1,82 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T1,A. â¦G, L1â¦ â¢ T1 â A â - âT2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â - âL2. â¦G, L1â¦ â¢ â¡[h, g, T1, 0] L2 â â¦G, L2â¦ â¢ T2 â A. -#h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A -[ #g #L1 #k #X #H elim (cpx_inv_sort1 â¦ H) -H // * // -| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12 - elim (cpx_inv_lref1 â¦ H) -H - [ #H destruct - elim (llpx_inv_lref_ge_sn â¦ HL12 â¦ HLK1) -L1 /3 width=6 by aaa_lref/ - | * #J #Y #Z #V2 #H #HV12 #HV2 - lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct - elim (llpx_inv_lref_ge_sn â¦ HL12 â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 - elim (cpx_inv_abbr1 â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 - elim (cpx_inv_abst1 â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 - elim (cpx_inv_appl1 â¦ 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 â¦ HV12 â¦ HV1) -IHV1 -HV12 #HV2 - lapply (IHT1 (â{b}W2.U2) â¦ HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H - elim (aaa_inv_abst â¦ 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 â¦ B â¦ (L2.âW2) â¦ HV2) -HV2 /2 width=2 by ldrop_drop/ #HV2 - lapply (IHT1 (â{b}W2.U2) â¦ HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H - elim (aaa_inv_abbr â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 - elim (cpx_inv_cast1 â¦ 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: âh,g,G,L,T1,A. â¦G, Lâ¦ â¢ T1 â A â âT2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â â¦G, Lâ¦ â¢ T2 â A. -/2 width=7 by aaa_cpx_llpx_conf/ qed-. - -lemma aaa_llpx_conf: âh,g,G,L1,T,A. â¦G, L1â¦ â¢ T â A â âL2. â¦G, L1â¦ â¢ â¡[h, g, T, 0] L2 â â¦G, L2â¦ â¢ T â A. -/2 width=7 by aaa_cpx_llpx_conf/ qed-. - -lemma aaa_cpr_conf: âG,L,T1,A. â¦G, Lâ¦ â¢ T1 â A â âT2. â¦G, Lâ¦ â¢ T1 â¡ T2 â â¦G, Lâ¦ â¢ T2 â A. -/3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-. -(* -lamma aaa_llpr_conf: âG,L1,T,A. â¦G, L1â¦ â¢ T â A â âL2. â¦G, L1â¦ â¢ â¡[T, 0] L2 â â¦G, L2â¦ â¢ T â A. -/3 width=5 by aaa_llpx_conf, llpr_llpx/ qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_ldrop.etc deleted file mode 100644 index 3e4bcab8f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_ldrop.etc +++ /dev/null @@ -1,90 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2 â d â¤ i â - âI,K2,V2. â©[i] L2 â¡ K2.â{I}V2 â - ââK1,V1. â©[i] L1 â¡ K1.â{I}V1 & - â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡[h, g] V2. -/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-. - -lemma llpx_inv_lref_ge_sn: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2 â d â¤ i â - âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â - ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & - â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡[h, g] V2. -/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-. - -lemma llpx_inv_lref_ge_bi: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2 â d â¤ i â - âI1,I2,K1,K2,V1,V2. - â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â - â§â§ I1 = I2 & â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡[h, g] V2. -/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. - -lemma llpx_inv_bind_O: âh,g,a,I,G,L1,L2,V,T. â¦G, L1â¦ â¢ â¡ [h, g, â{a,I}V.T, 0] L2 â - â¦G, L1â¦ â¢ â¡[h, g, V, 0] L2 â§ â¦G, L1.â{I}Vâ¦ â¢ â¡[h, g, T, 0] L2.â{I}V. -/2 width=2 by llpx_sn_inv_bind_O/ qed-. - -lemma llpx_bind_repl_O: âh,g,I,G,L1,L2,V1,V2,T. â¦G, L1.â{I}V1â¦ â¢ â¡[h, g, T, 0] L2.â{I}V2 â - âJ,W1,W2. â¦G, L1â¦ â¢ â¡[h, g, W1, 0] L2 â â¦G, L1â¦ â¢ W1 â¡[h, g] W2 â â¦G, L1.â{J}W1â¦ â¢ â¡[h, g, T, 0] L2.â{J}W2. -/2 width=4 by llpx_sn_bind_repl_O/ qed-. - -(* Advanced forward lemmas **************************************************) - -lemma llpx_fwd_bind_O_dx: âh,g,a,I,G,L1,L2,V,T. â¦G, L1â¦ â¢ â¡[h, g, â{a,I}V.T, 0] L2 â - â¦G, L1.â{I}Vâ¦ â¢ â¡[h, g, T, 0] L2.â{I}V. -/2 width=3 by llpx_sn_fwd_bind_O_dx/ qed-. - -(* Advanced properties ******************************************************) - -lemma llpx_cpx_conf: âh,g,G. s_r_confluent1 â¦ (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: âh,g,G,L1,L2,U. â¦G, L1â¦ â¢ â¡[h, g, U, 0] L2 â - âK2,e. â©[e] L2 â¡ K2 â âT. â§[0, e] T â¡ U â - ââK1. â©[e] L1 â¡ K1 & â¦G, K1â¦ â¢ â¡[h, g, T, 0] K2. -/2 width=5 by llpx_sn_ldrop_trans_O/ qed-. - -(* Properties on supclosure *************************************************) - -lemma llpx_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡[h, g] T & â¦G1, K1, Tâ¦ â â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[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 â¦ 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 â¦ HU1 â¦ HLK1) -L1 - /3 width=5 by fqu_drop, ex3_2_intro/ -] -qed-. - -lemma llpx_fquq_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡[h, g] T & â¦G1, K1, Tâ¦ ââ¸® â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g, T2, 0] L2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen â¦ H) -H -[ #HT12 elim (llpx_fqu_trans â¦ HT12 â¦ HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/ -| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_lleq.etc deleted file mode 100644 index e936cc531..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_lleq.etc +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. -/2 width=1 by llpx_sn_lrefl/ qed-. - -lemma lleq_llpx_trans: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â - âL.â¦G, L2â¦ â¢ â¡[h, g, T, d] L â â¦G, L1â¦ â¢ â¡[h, g, T, d] L. -/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_trans/ qed-. - -lemma lleq_llpx_conf: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â - âL.â¦G, L1â¦ â¢ â¡[h, g, T, d] L â â¦G, L2â¦ â¢ â¡[h, g, T, d] L. -/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_lpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_lpr.etc deleted file mode 100644 index 6ceb64b51..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpx_lpr.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. -/3 width=4 by cpr_cpx, lpx_sn_llpx_sn, llpx_sn_co/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs.etc deleted file mode 100644 index 750d0b5b1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs.etc +++ /dev/null @@ -1,89 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â genv â relation4 ynat term lenv lenv â - Î»h,g,G,d. LTC â¦ (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: âh,g,G,L1,T,d. âR:predicate lenv. R L1 â - (âL,L2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡[h, g, T, d] L2 â R L â R L2) â - âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â R L2. -#h #g #G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12 -@(TC_star_ind â¦ HL1 IHL1 â¦ HL12) // -qed-. - -lemma llpxs_ind_dx: âh,g,G,L2,T,d. âR:predicate lenv. R L2 â - (âL1,L. â¦G, L1â¦ â¢ â¡[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L2 â R L â R L1) â - âL1. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â R L1. -#h #g #G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12 -@(TC_star_ind_dx â¦ HL2 IHL2 â¦ HL12) // -qed-. - -(* Basic properties *********************************************************) - -lemma llpx_llpxs: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. -normalize /2 width=1 by inj/ qed. - -lemma llpxs_refl: âh,g,G,L,T,d. â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L. -/2 width=1 by llpx_llpxs/ qed. - -lemma llpxs_strap1: âh,g,G,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. -normalize /2 width=3 by step/ qed. - -lemma llpxs_strap2: âh,g,G,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. -normalize /2 width=3 by TC_strap/ qed. - -(* Basic forward lemmas *****************************************************) - -lemma llpxs_fwd_length: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â |L1| = |L2|. -#h #g #G #L1 #L2 #T #d #H @(llpxs_ind â¦ H) -L2 -/3 width=8 by llpx_fwd_length, trans_eq/ -qed-. - -(* Note: this might be moved *) -lemma llpxs_fwd_bind_sn: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{a,I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡*[h, g, V, d] L2. -#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 -/3 width=6 by llpx_fwd_bind_sn, llpxs_strap1/ -qed-. - -(* Note: this might be moved *) -lemma llpxs_fwd_bind_dx: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{a,I}V.T, d] L2 â - â¦G, L1.â{I}Vâ¦ â¢ â¡*[h, g, T, â«¯d] L2.â{I}V. -#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 -/3 width=6 by llpx_fwd_bind_dx, llpxs_strap1/ -qed-. - -(* Note: this might be moved *) -lemma llpxs_fwd_flat_sn: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡*[h, g, V, d] L2. -#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 -/3 width=6 by llpx_fwd_flat_sn, llpxs_strap1/ -qed-. - -(* Note: this might be moved *) -lemma llpxs_fwd_flat_dx: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{I}V.T, d] L2 â - â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. -#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 -/3 width=6 by llpx_fwd_flat_dx, llpxs_strap1/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_aaa.etc deleted file mode 100644 index ededb03a2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_aaa.etc +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T,A. â¦G, L1â¦ â¢ T â A â - âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, 0] L2 â â¦G, L2â¦ â¢ T â A. -#h #g #G #L1 #T #A #HT #L2 #HL12 -@(TC_Conf3 â¦ (Î»L,A. â¦G, Lâ¦ â¢ T â A) â¦ HT ? HL12) /2 width=5 by aaa_llpx_conf/ -qed-. -(* -lamma aaa_llprs_conf: âG,L1,T,A. â¦G, L1â¦ â¢ T â A â - âL2. â¦G, L1â¦ â¢ â¡*[T, 0] L2 â â¦G, L2â¦ â¢ T â A. -/3 width=5 by aaa_llpxs_conf, llprs_llpxs/ qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_cpxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_cpxs.etc deleted file mode 100644 index 35472d7e5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_cpxs.etc +++ /dev/null @@ -1,172 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 â - âI,T. â¦G, L.â{I}V1â¦ â¢ â¡*[h, g, T, 0] L.â{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: âh,g,G. s_r_transitive â¦ (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: âh,g,G. s_rs_transitive â¦ (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: âh,g,G2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡*[h, g] U2 â - âL0. â¦G2, L0â¦ â¢ â¡*[h, g, T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[h, g, U2, O] L2. -#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx â¦ H) -L0 // -#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 â¦ IHL2) -IHL2 -lapply (llpxs_cpxs_trans â¦ HTU2 â¦ 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: âh,g,G2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡[h, g] U2 â - âL0. â¦G2, L0â¦ â¢ â¡*[h, g, T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[h, g, U2, O] L2. -#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx â¦ H) -L0 // -#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 â¦ IHL2) -IHL2 -lapply (llpxs_cpx_trans â¦ HTU2 â¦ HL2) -L2 #HTU2 -/3 width=3 by llpx_cpxs_trans, cpxs_llpx_conf/ -qed-. - -lemma cpxs_bind2: âh,g,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 â - âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡*[h, g] T2 â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡*[h, g] â{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: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡*[h, g, #i, d] L2 â d â¤ i â - âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â - ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & - â¦G, K1â¦ â¢ â¡*[h, g, V2, 0] K2 & â¦G, K1â¦ â¢ V1 â¡*[h, g] V2. -#h #g #G #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 @(llpxs_ind â¦ H) -L2 /2 width=5 by ex3_2_intro/ -HLK1 -#L #L2 #_ #HL2 * #K #V #HLK #HK1 #HV1 elim (llpx_inv_lref_ge_sn â¦ HL2 â¦ HLK) // -HL2 -HLK -Hdi -#K2 #V2 #HLK2 #HK2 #HV2 -@(ex3_2_intro â¦ 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: âh,g,a,G,L,V1,T1,U2. â¦G, Lâ¦ â¢ â{a}V1.T1 â¡*[h, g] U2 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 & â¦G, L.âV1â¦ â¢ T1 â¡*[h, g] T2 & - U2 = â{a}V2.T2. -#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind â¦ H) -U2 /2 width=5 by ex3_2_intro/ -#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct -elim (cpx_inv_abst1 â¦ HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct -lapply (llpxs_cpx_trans â¦ HT02 (L.âV1) ?) -/3 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro/ -qed-. - -lemma cpxs_inv_abbr1: âh,g,a,G,L,V1,T1,U2. â¦G, Lâ¦ â¢ â{a}V1.T1 â¡*[h, g] U2 â ( - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 & â¦G, L.âV1â¦ â¢ T1 â¡*[h, g] T2 & - U2 = â{a}V2.T2 - ) â¨ - ââT2. â¦G, L.âV1â¦ â¢ T1 â¡*[h, g] T2 & â§[0, 1] U2 â¡ T2 & a = true. -#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind â¦ H) -U2 /3 width=5 by ex3_2_intro, or_introl/ -#U0 #U2 #_ #HU02 * * -[ #V0 #T0 #HV10 #HT10 #H destruct - elim (cpx_inv_abbr1 â¦ HU02) -HU02 * - [ #V2 #T2 #HV02 #HT02 #H destruct - lapply (llpxs_cpx_trans â¦ HT02 (L.âV1) ?) - /4 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/ - | #T2 #HT02 #HUT2 - lapply (llpxs_cpx_trans â¦ HT02 (L.âV1) ?) -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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â+ â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g, T2, 0] L2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind â¦ H) -G2 -L2 -T2 -[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (llpx_fqu_trans â¦ H12 â¦ 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 â¦ HLK1) -L1 - #L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fqu_trans â¦ H2 â¦ HL0) -L - #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans â¦ HT0 â¦ HT3) -T - /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/ -] -qed-. - -lemma llpx_fqus_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â* â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â* â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g, T2, 0] L2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind â¦ H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/ -#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 â¦ HLK1) -L1 -#L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fquq_trans â¦ H2 â¦ HL0) -L -#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans â¦ HT0 â¦ HT3) -T -/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/ -qed-. - -lemma llpxs_fquq_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡*[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ ââ¸® â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡*[h, g, T2, 0] L2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx â¦ H) -K1 -[ /2 width=5 by ex3_2_intro/ -| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12 - lapply (llpx_cpxs_trans â¦ HT1 â¦ HK1) -HT1 #HT1 - lapply (cpxs_llpx_conf â¦ HT1 â¦ HK1) -HK1 #HK1 - elim (llpx_fquq_trans â¦ HT2 â¦ HK1) -K - /3 width=7 by llpxs_strap2, cpxs_strap1, ex3_2_intro/ -] -qed-. - -lemma llpxs_fqup_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡*[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â+ â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡*[h, g, T2, 0] L2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx â¦ H) -K1 -[ /2 width=5 by ex3_2_intro/ -| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12 - lapply (llpx_cpxs_trans â¦ HT1 â¦ HK1) -HT1 #HT1 - lapply (cpxs_llpx_conf â¦ HT1 â¦ HK1) -HK1 #HK1 - elim (llpx_fqup_trans â¦ HT2 â¦ HK1) -K - /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/ -] -qed-. - -lemma llpxs_fqus_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â* â¦G2, L2, T2â¦ â - âK1. â¦G1, K1â¦ â¢ â¡*[h, g, T1, 0] L1 â - ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â* â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡*[h, g, T2, 0] L2. -#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx â¦ H) -K1 -[ /2 width=5 by ex3_2_intro/ -| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12 - lapply (llpx_cpxs_trans â¦ HT1 â¦ HK1) -HT1 #HT1 - lapply (cpxs_llpx_conf â¦ HT1 â¦ HK1) -HK1 #HK1 - elim (llpx_fqus_trans â¦ HT2 â¦ HK1) -K - /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lleq.etc deleted file mode 100644 index a38ba395a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lleq.etc +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. -/3 width=1 by llpx_lrefl, llpx_llpxs/ qed-. - -lemma lleq_llpxs_trans: âh,g,G,L2,L,T,d. â¦G, L2â¦ â¢ â¡*[h, g, T, d] L â - âL1. L1 â[T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L. -#h #g #G #L2 #L #T #d #H @(llpxs_ind â¦ H) -L -/3 width=3 by llpxs_strap1, llpxs_lrefl/ -qed-. - -lemma lleq_llpxs_conf: âh,g,G,L1,L,T,d. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L â - âL2. L1 â[T, d] L2 â â¦G, L2â¦ â¢ â¡*[h, g, T, d] L. -/3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-. -(* -foct leq_lpxs_trans_lleq_aux: âh,g,G,L1,L0,d,e. L1 â[d, e] L0 â e = â â - âL2. â¦G, L0â¦ â¢ â¡*[h, g] L2 â - ââL. L â[d, e] L2 & â¦G, L1â¦ â¢ â¡*[h, g] L & - (âT. L0 â[T, d] L2 â L1 â[T, d] L). -#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e -[ #d #e #_ #L2 #H >(lpxs_inv_atom1 â¦ 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 â¦ H) -H #L2 #V2 #HL02 #HV02 #H destruct - lapply (ysucc_inv_Y_dx â¦ He) -He #He - elim (IHL10 â¦ HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH - @(ex3_intro â¦ (L.â{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/ - #T elim (IH T) #HL0dx #HL0sn - @conj #H @(lleq_leq_repl â¦ 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 â¦ H) -H #L2 #V2 #HL02 #HV02 #H destruct - elim (IHL10 â¦ HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH - @(ex3_intro â¦ (L.â{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/ - #T elim (IH T) #HL0dx #HL0sn - @conj #H @(lleq_leq_repl â¦ H) -H /3 width=1 by leq_sym, leq_succ/ -] -qed-. - -lamma leq_lpxs_trans_lleq: âh,g,G,L1,L0,d. L1 â[d, â] L0 â - âL2. â¦G, L0â¦ â¢ â¡*[h, g] L2 â - ââL. L â[d, â] L2 & â¦G, L1â¦ â¢ â¡*[h, g] L & - (âT. L0 â[T, d] L2 â L1 â[T, d] L). -/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_llpxs.etc deleted file mode 100644 index d316c5207..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_llpxs.etc +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,T,d. Transitive â¦ (llpxs h g G d T). -normalize /2 width=3 by trans_TC/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lprs.etc deleted file mode 100644 index 32d7bbc38..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lprs.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡* L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. -normalize /3 width=3 by lpr_llpx, monotonic_TC/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx.etc deleted file mode 100644 index 286d0645f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx.etc +++ /dev/null @@ -1,57 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â relation4 ynat term genv lenv â - Î»h,g,d,T,G. SN â¦ (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: âh,g,G,T,d. âR:predicate lenv. - (âL1. G â¢ ââ¬*[h, g, T, d] L1 â - (âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â R L2) â - R L1 - ) â - âL. G â¢ ââ¬*[h, g, T, d] L â 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: âh,g,G,L1,T,d. - (âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬*[h, g, T, d] L2) â - G â¢ ââ¬*[h, g, T, d] L1. -/5 width=1 by lleq_sym, SN_intro/ qed. - -lemma llsx_sort: âh,g,G,L,d,k. G â¢ ââ¬*[h, g, âk, 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: âh,g,G,L,d,p. G â¢ ââ¬*[h, g, Â§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. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_alt.etc deleted file mode 100644 index 0d23439b9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_alt.etc +++ /dev/null @@ -1,107 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh. sd h â relation4 ynat term genv lenv â - Î»h,g,d,T,G. SN â¦ (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: âh,g,G,T,d. âR:predicate lenv. - (âL1. G â¢ ââ¬â¬*[h, g, T, d] L1 â - (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â R L2) â - R L1 - ) â - âL. G â¢ ââ¬â¬*[h, g, T, d] L â 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: âh,g,G,L1,T,d. - (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬â¬*[h, g, T, d] L2) â - G â¢ ââ¬â¬*[h, g, T, d] L1. -/5 width=1 by lleq_sym, SN_intro/ qed. - -fact llsxa_intro_aux: âh,g,G,L1,T,d. - (âL,L2. â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L2 â L1 â[T, d] L â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬â¬*[h, g, T, d] L2) â - G â¢ ââ¬â¬*[h, g, T, d] L1. -/4 width=3 by llsxa_intro/ qed-. - -lemma llsxa_llpxs_trans: âh,g,G,L1,T,d. G â¢ ââ¬â¬*[h, g, T, d] L1 â - âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â G â¢ ââ¬â¬*[h, g, T, d] L2. -#h #g #G #L1 #T #d #H @(llsxa_ind â¦ 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: âh,g,G,L1,T,d. - (âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬â¬*[h, g, T, d] L2) â - G â¢ ââ¬â¬*[h, g, T, d] L1. -#h #g #G #L1 #T #d #IH @llsxa_intro_aux -#L #L2 #H @(llpxs_ind_dx â¦ 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: âh,g,G,L,T,d. G â¢ ââ¬*[h, g, T, d] L â G â¢ ââ¬â¬*[h, g, T, d] L. -#h #g #G #L #T #d #H @(llsx_ind â¦ H) -L -/4 width=1 by llsxa_intro_llpx/ -qed. - -(* Main inversion lemmas ****************************************************) - -theorem llsxa_inv_llsx: âh,g,G,L,T,d. G â¢ ââ¬â¬*[h, g, T, d] L â G â¢ ââ¬*[h, g, T, d] L. -#h #g #G #L #T #d #H @(llsxa_ind â¦ H) -L -/4 width=1 by llsx_intro, llpx_llpxs/ -qed-. - -(* Advanced properties ******************************************************) - -lemma llsx_intro_alt: âh,g,G,L1,T,d. - (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬*[h, g, T, d] L2) â - G â¢ ââ¬*[h, g, T, d] L1. -/6 width=1 by llsxa_inv_llsx, llsx_llsxa, llsxa_intro/ qed. - -lemma llsx_llpxs_trans: âh,g,G,L1,T,d. G â¢ ââ¬*[h, g, T, d] L1 â - âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â G â¢ ââ¬*[h, g, T, d] L2. -/4 width=3 by llsxa_inv_llsx, llsx_llsxa, llsxa_llpxs_trans/ -qed-. - -(* Advanced eliminators *****************************************************) - -lemma llsx_ind_alt: âh,g,G,T,d. âR:predicate lenv. - (âL1. G â¢ ââ¬*[h, g, T, d] L1 â - (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â R L2) â - R L1 - ) â - âL. G â¢ ââ¬*[h, g, T, d] L â R L. -#h #g #G #T #d #R #IH #L #H @(llsxa_ind h g G T d â¦ L) -/4 width=1 by llsxa_inv_llsx, llsx_llsxa/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_csx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_csx.etc deleted file mode 100644 index 9c284732c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_csx.etc +++ /dev/null @@ -1,96 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â - âL2. â¦G, L1â¦â¢ â¡[h, g, T2, O] L2 â - ââL. â¦G, L1â¦â¢ â¡[h, g, T1, O] L & L â[T2, 0] L2. -(* -fact llsx_cpx_trans_aux: âh,g,G,L0,T1,T2. â¦G, L0â¦ â¢ T1 â¡[h, g] T2 â - âL1,d. G â¢ ââ¬*[h, g, T1, d] L1 â d = 0 â - L0 â[T1, d] L1 â âL2. L1 â[T2, d] L2 â - G â¢ ââ¬*[h, g, T2, d] L2. -#h #g #G #L0 #T1 #T2 #HT12 #L1 #d #H @(llsx_ind â¦ H) -L1 -#L1 #_ #IHL1 #Hd #He011 #L2 #He122 @llsx_intro -#L3 #Hx223 #Hn223 destruct -lapply (lleq_cpx_conf_sn â¦ HT12 â¦ He011) #He021 -lapply (lleq_cpx_conf â¦ HT12 â¦ He011) -HT12 #HT12 -lapply (lleq_llpx_trans â¦ He122 â¦ Hx223) -Hx223 #Hx123 -elim (cpx_llpx_trans â¦ HT12 â¦ Hx123) -Hx123 #L4 #Hx114 #He423 -(* lapply (lleq_cpx_conf â¦ Hx114 â¦ He011) #He120 *) -@(IHL1 â¦ Hx114) // -IHL1 -[ #HL13 @HnL2 -HnL2 -*) - -fact llsx_cpx_trans_aux: âh,g,G,L1,T1,d. G â¢ ââ¬*[h, g, T1, d] L1 â d = 0 â - âT2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â - âL2. L1 â[T1, d] L2 â G â¢ ââ¬*[h, g, T2, 0] L2. -#h #g #G #L1 #T1 #d #H @(llsx_ind â¦ H) -L1 -#L1 #_ #IHL1 #Hd #T2 #HT12 #L2 #He112 @llsx_intro -#L3 #Hx223 #Hn223 destruct -lapply (lleq_cpx_conf_sn â¦ HT12 â¦ He112) #He122 -lapply (lleq_cpx_conf â¦ HT12 â¦ He112) -HT12 #HT12 -elim (cpx_llpx_trans â¦ HT12 â¦ Hx223) #L4 #Hx214 #He423 -@(IHL1 â¦ L4) // -*) -axiom llsx_cpx_trans_O: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â - G â¢ ââ¬*[h, g, T1, 0] L â G â¢ ââ¬*[h, g, T2, 0] L. - -(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************) - -(* Advanced properties ******************************************************) - -lemma llsx_lref_be_lpxs: âh,g,I,G,K1,V,i,d. d â¤ yinj i â â¦G, K1â¦ â¢ â¬*[h, g] V â - âK2. G â¢ ââ¬*[h, g, V, 0] K2 â â¦G, K1â¦ â¢ â¡*[h, g, V, 0] K2 â - âL2. â©[i] L2 â¡ K2.â{I}V â G â¢ ââ¬*[h, g, #i, d] L2. -#h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt â¦ H) -V -#V0 #_ #IHV0 #K2 #H @(llsx_ind â¦ H) -K2 -#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @llsx_intro -#L2 #HL02 #HnL02 elim (llpx_inv_lref_ge_sn â¦ HL02 â¦ 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 â¦ HV02 â¦ HK02) -HK02 #HK02 - @(IHV0 â¦ HnV02 â¦ 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: âh,g,I,G,K,V,i,d. d â¤ yinj i â â¦G, Kâ¦ â¢ â¬*[h, g] V â - G â¢ ââ¬*[h, g, V, 0] K â - âL. â©[i] L â¡ K.â{I}V â G â¢ ââ¬*[h, g, #i, d] L. -/2 width=8 by llsx_lref_be_lpxs/ qed. - -(* Main properties **********************************************************) - -theorem csx_llsx: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¬*[h, g] T â âd. G â¢ ââ¬*[h, g, T, d] L. -#h #g #G #L #T @(fqup_wf_ind_eq â¦ 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 â¦ Hi) -Hi - #I #K #V #HLK lapply (csx_inv_lref_bind â¦ HLK â¦ 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 â¦ H) -H /3 width=1 by llsx_bind/ -| #I #V #T #HG #HL #HT #H #d destruct - elim (csx_fwd_flat â¦ H) -H /3 width=1 by llsx_flat/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_ldrop.etc deleted file mode 100644 index 2d16e3240..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_ldrop.etc +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L,d,i. |L| â¤ i â G â¢ ââ¬*[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: âh,g,G,L,d,i. yinj i < d â G â¢ ââ¬*[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. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_llpx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_llpx.etc deleted file mode 100644 index 9392afba2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_llpx.etc +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L1,T,d. G â¢ ââ¬*[h, g, T, d] L1 â - âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â G â¢ ââ¬*[h, g, T, d] L2. -#h #g #G #L1 #T #d #H @(llsx_ind â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_llpxs.etc deleted file mode 100644 index af37a372f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx_llpxs.etc +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,a,I,G,L1,V,d. G â¢ ââ¬*[h, g, V, d] L1 â - âY,T. G â¢ ââ¬*[h, g, T, â«¯d] Y â - âL2. Y = L2.â{I}V â â¦G, L1â¦ â¢ â¡*[h, g, â{a,I}V.T, d] L2 â - G â¢ ââ¬*[h, g, â{a,I}V.T, d] L2. -#h #g #a #I #G #L1 #V #d #H @(llsx_ind_alt â¦ H) -L1 -#L1 #HL1 #IHL1 #Y #T #H @(llsx_ind_alt â¦ H) -Y -#Y #HY #IHY #L2 #H #HL12 destruct @llsx_intro_alt -#L0 #HL20 lapply (llpxs_fwd_bind_dx â¦ HL20) -lapply (llpxs_trans â¦ HL12 â¦ HL20) -#HL10 #HT #H elim (nlleq_inv_bind â¦ H) -H [ -HL1 -IHY | -HY -IHL1 ] -[ #HnV elim (lleq_dec V L1 L2 d) - [ -HL20 #HV @(IHL1 â¦ 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: âh,g,a,I,G,L,V,d. G â¢ ââ¬*[h, g, V, d] L â - âT. G â¢ ââ¬*[h, g, T, â«¯d] L.â{I}V â - G â¢ ââ¬*[h, g, â{a,I}V.T, d] L. -/2 width=3 by llsx_bind_llpxs_aux/ qed. - -lemma llsx_flat_llpxs: âh,g,I,G,L1,V,d. G â¢ ââ¬*[h, g, V, d] L1 â - âL2,T. G â¢ ââ¬*[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, â{I}V.T, d] L2 â - G â¢ ââ¬*[h, g, â{I}V.T, d] L2. -#h #g #I #G #L1 #V #d #H @(llsx_ind_alt â¦ H) -L1 -#L1 #HL1 #IHL1 #L2 #T #H @(llsx_ind_alt â¦ H) -L2 -#L2 #HL2 #IHL2 #HL12 @llsx_intro_alt -#L0 #HL20 lapply (llpxs_fwd_flat_dx â¦ HL20) -lapply (llpxs_trans â¦ HL12 â¦ HL20) -#HL10 #HT #H elim (nlleq_inv_flat â¦ H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ] -[ #HnV elim (lleq_dec V L1 L2 d) - [ #HV @(IHL1 â¦ 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: âh,g,I,G,L,V,d. G â¢ ââ¬*[h, g, V, d] L â - âT. G â¢ ââ¬*[h, g, T, d] L â G â¢ ââ¬*[h, g, â{I}V.T, d] L. -/2 width=3 by llsx_flat_llpxs/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx_sn/llpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx_sn/llpx_sn_tc.etc deleted file mode 100644 index 682b48c68..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx_sn/llpx_sn_tc.etc +++ /dev/null @@ -1,160 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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) â Î»S,R. - âLs,T1,T2. S Ls T1 T2 â - âLd. TC â¦ (llpx_sn R 0 T1) Ls Ld â TC â¦ (llpx_sn R 0 T2) Ls Ld. - -lemma TC_llpx_sn_s_confluent: âS,R. (llpx_sn_confluent1 S R) â TC_llpx_sn_confluent1 S R. -#S #R #HSR #Ls #T1 #T2 #HT12 #Ld #H -generalize in match HT12; -HT12 -@(TC_ind_dx â¦ Ls H) -Ls -[ /3 width=3 by inj/ -| #Ls #L #HLs #_ #IHLd #HT12 - @(TC_strap â¦ L) /2 width=3 by/ @IHLd -IHLd - -lemma TC_llpx_sn_lref_refl: âR. (âL.reflexive â¦ (R L)) â - âI,L1,K1,K2,V,d,i. d â¤ yinj i â â©[i] L1 â¡ K1.â{I}V â - TC lenv (llpx_sn R 0 V) K1 K2 â - âL2. â©[i] L2 â¡ K2.â{I}V â TC â¦ (llpx_sn R d (#i)) L1 L2. -#R #HR #I #L1 #K1 #K2 #V #d #i #Hdi #HLK1 #H @(TC_star_ind â¦ 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 â¦ HLK2) - #H elim (ldrop_O1_ex (K.â{I}V) i L2) [2: normalize in H â¢ %; >(llpx_sn_fwd_length â¦ HV) ] - /4 width=11 by llpx_sn_lref, step/ -] -qed-. - -lemma TC_llpx_sn_lref: âR. (âL.reflexive â¦ (R L)) â (llpx_sn_confluent1 R R) â - âI,K1,V1,V2,d,i. d â¤ yinj i â LTC â¦ R K1 V1 V2 â - âK2. TC lenv (llpx_sn R 0 V1) K1 K2 â âL1. â©[i] L1 â¡ K1.â{I}V1 â - âL2. â©[i] L2 â¡ K2.â{I}V2 â TC â¦ (llpx_sn R d (#i)) L1 L2. -#R #H1R #H2R #I #K1 #V1 #V2 #d #i #Hdi #H @(TC_star_ind_dx â¦ 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 â¦ HLK1) - #H elim (ldrop_O1_ex (K1.â{I}V) i L1) [2: normalize in H â¢ %; // ] -H - #L #_ #HLK @(TC_strap â¦ L) - [ @(llpx_sn_lref â¦ HLK1 â¦ HLK) /2 width=1 by llpx_sn_refl/ - | @(IHV2 â¦ HLK â¦ HLK2) - -HLK1 -HLK2 -HLK -IHV2 -Hdi @(TC_llpx_sn_s_confluent R R â¦ HK12) // - ] -] - - -lemma llpx_sn_LTC_TC_llpx_sn: âR. (âL. reflexive â¦ (R L)) â - âL1,L2,T,d. llpx_sn (LTC â¦ R) d T L1 L2 â - TC â¦ (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: âR. (âL. reflexive â¦ (R L)) â - âI,L1,L2. TC â¦ (lpx_sn R) L1 L2 â - âV1,V2. LTC â¦ R L1 V1 V2 â - TC â¦ (lpx_sn R) (L1. â{I} V1) (L2. â{I} V2). -#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx â¦ 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: âR. (âL. reflexive â¦ (R L)) â - âL1,L2. lpx_sn (LTC â¦ R) L1 L2 â - TC â¦ (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: âR,L1. TC â¦ (lpx_sn R) L1 (â) â L1 = â. -#R #L1 #H @(TC_ind_dx â¦ 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: âR. s_rs_trans â¦ R (lpx_sn R) â - âI,L1,K2,V2. TC â¦ (lpx_sn R) L1 (K2.â{I}V2) â - ââK1,V1. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ R K1 V1 V2 & L1 = K1. â{I} V1. -#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx â¦ L1 H) -L1 -[ #L1 #H elim (lpx_sn_inv_pair2 â¦ H) -H /3 width=5 by inj, ex3_2_intro/ -| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct - elim (lpx_sn_inv_pair2 â¦ HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct - lapply (HR â¦ HV2 â¦ HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/ -] -qed-. - -lemma TC_lpx_sn_ind: âR. s_rs_trans â¦ R (lpx_sn R) â - âS:relation lenv. - S (â) (â) â ( - âI,K1,K2,V1,V2. - TC â¦ (lpx_sn R) K1 K2 â LTC â¦ R K1 V1 V2 â - S K1 K2 â S (K1.â{I}V1) (K2.â{I}V2) - ) â - âL2,L1. TC â¦ (lpx_sn R) L1 L2 â S L1 L2. -#R #HR #S #IH1 #IH2 #L2 elim L2 -L2 -[ #X #H >(TC_lpx_sn_inv_atom2 â¦ H) -X // -| #L2 #I #V2 #IHL2 #X #H - elim (TC_lpx_sn_inv_pair2 â¦ H) // -H -HR - #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/ -] -qed-. - -lemma TC_lpx_sn_inv_atom1: âR,L2. TC â¦ (lpx_sn R) (â) L2 â L2 = â. -#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: âR. s_rs_trans â¦ R (lpx_sn R) â - âL1,L2. TC â¦ (lpx_sn R) L1 L2 â - âI,K1,V1. L1 = K1.â{I}V1 â - ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ R K1 V1 V2 & L2 = K2. â{I} V2. -#R #HR #L1 #L2 #H @(TC_lpx_sn_ind â¦ 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: âR. s_rs_trans â¦ R (lpx_sn R) â - âI,K1,L2,V1. TC â¦ (lpx_sn R) (K1.â{I}V1) L2 â - ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ R K1 V1 V2 & L2 = K2. â{I} V2. -/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-. - -lemma TC_lpx_sn_inv_lpx_sn_LTC: âR. s_rs_trans â¦ R (lpx_sn R) â - âL1,L2. TC â¦ (lpx_sn R) L1 L2 â - lpx_sn (LTC â¦ 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: âR,L1,L2. TC â¦ (lpx_sn R) L1 L2 â |L1| = |L2|. -#R #L1 #L2 #H elim H -L2 -[ #L2 #HL12 >(lpx_sn_fwd_length â¦ HL12) -HL12 // -| #L #L2 #_ #HL2 #IHL1 - >IHL1 -L1 >(lpx_sn_fwd_length â¦ HL2) -HL2 // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx_sn/lpx_conj.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx_sn/lpx_conj.etc deleted file mode 100644 index 5950f1b92..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx_sn/lpx_conj.etc +++ /dev/null @@ -1,120 +0,0 @@ -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: âA:Type[0]. âR:predicate A. - ((âa. R a)ââ¥) â (âa. R a â â¥). -/3 width=2 by ex_intro/ qed-. - -lemma lt_repl_sn_trans_tw: âL1,L2,T1,T2. â¯{L1, T1} < â¯{L2, T2} â - âU1. â¯{U1} = â¯{T1} â â¯{L1, U1} < â¯{L2, T2}. -normalize in â¢ (?â?â?â?â?%%â?â?â?%%); // -qed-. - -axiom cpx_fwd_lift1: âh,g,G,L,U1,U2. â¦G, Lâ¦ â¢ U1 â¡[h, g] U2 â - âT1,d,e. â§[d, e] T1 â¡ U1 â âT2. â§[d, e] T2 â¡ 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 â¦ H) -H #H destruct /2 width=2 by ex_intro/ - | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=2 by lift_lref_ge_minus, lift_lref_lt, ex_intro/ - | lapply (lift_inv_gref2 â¦ H) -H #H destruct /2 width=2 by ex_intro/ - ] -| #G #L #k #l #Hkl #T1 #d #e #H - lapply (lift_inv_sort2 â¦ 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 â¦ H) -H * #Hid #H destruct - [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV - elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 - elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // (length_inv_zero_dx â¦ 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 â¦ H) -H #Is #L1s #W1s #_ #H destruct - elim (is_lift_dec U1 0 1) [ -IHU1 -HW12d | -HU1 ] - [ * #T1 #HTU1 lapply (lift_fwd_tw â¦ HTU1) #H - lapply (lleq_inv_lift_le â¦ HU1 L1s L1d â¦ HTU1 ?) -HU1 /2 width=1 by ldrop_drop/ - #HT1 elim (cpx_inv_lift1 â¦ HU12 L1d â¦ HTU1) -HU12 -HTU1 /2 width=4 by ldrop_drop/ - #T2 #HTU2 #HT12 elim (IH â¦ HT1 â¦ HL12d â¦ HT12) /2 width=3 by lt_repl_sn_trans_tw/ -IH -HT1 -HT12 -H - #L2s #HL12s #HT2 @(ex2_intro â¦ (L2s.â{Is}W1s)) - /3 width=10 by lleq_lift_le, lpx_pair, ldrop_drop/ (**) (* full auto too slow *) - | #HnU1 lapply (not_ex_to_all_not â¦ HnU1) -HnU1 #HnU1 - elim (IHU1 â¦ HnU1) [2,3,4: // |5,6,7,8,9,10: skip ] -HnU1 #H1 #H2 #HW1s destruct - elim (IH â¦ HW1s â¦ HL12d â¦ HW12d) // #L2s #HL12s #HW2d - @(ex2_intro â¦ (L2s.â{Id}W2d)) /3 width=3 by lleq_cpx_trans, lpx_pair/ - lapply (lleq_fwd_length â¦ HW2d) #HL2sd -HW12d -HW1s - @lleq_intro_alt [ normalize // ] -HL2sd - #I2s #I2d #K2s #K2d #V2s #V2d #i @(nat_ind_plus â¦ i) -i - [ #_ #_ #HLK2s #HLK2d -IH -IHU1 -HU12 -HL12s -HL12d - lapply (ldrop_inv_O2 â¦ HLK2s) -HLK2s #H destruct - lapply (ldrop_inv_O2 â¦ HLK2d) -HLK2d #H destruct /2 width=1 by and3_intro/ - | #i #_ #_ #HnU2 #HLK2s #HLK2d - lapply (cpx_fwd_nlift2 â¦ HU12 â¦ HnU2) -HU12 -HnU2 #HnU1 - lapply (ldrop_inv_drop1 â¦ HLK2s) -HLK2s #HLK2s - lapply (ldrop_inv_drop1 â¦ HLK2d) -HLK2d #HLK2d - elim (lpx_ldrop_trans_O1 â¦ HL12s â¦ HLK2s) -L2s #Y #HLK1s #H - elim (lpx_inv_pair2 â¦ H) -H #K1s #V1s #HK12s #HV12s #H destruct - elim (lpx_ldrop_trans_O1 â¦ HL12d â¦ HLK2d) -L2d #Y #HLK1d #H - elim (lpx_inv_pair2 â¦ H) -H #K1d #V1d #HK12d #HV12d #H destruct - elim (IHU1 â¦ 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 â¦ HLK1s) -HLK1s #H - elim (IH â¦ HV1d â¦ HK12d â¦ HV12d) // -IH -HV1d -HK12d -HV12d - [ #Y #H1Y #H2Y - - - diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpc.etc deleted file mode 100644 index b80b290b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpc.etc +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1 â¦ â¬ â¬ break â¦ term 46 L2 â¦ )" - 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 â - Î»L1,L2. â¦L1â¦ â¡ â¦L2â¦ â¨ â¦L2â¦ â¡ â¦L1â¦. - -interpretation - "focalized parallel conversion (local environment)" - 'FocalizedPConv L1 L2 = (lfpc L1 L2). - -(* Basic properties *********************************************************) - -lemma lfpc_refl: âL. â¦Lâ¦ â¬ â¦Lâ¦. -/2 width=1/ qed. - -lemma lfpc_sym: âL1,L2. â¦L1â¦ â¬ â¦L2â¦ â â¦L2â¦ â¬ â¦L1â¦. -#L1 #L2 * /2 width=1/ -qed. - -lemma lfpc_lfpr: âL1,L2. â¦L1â¦ â¬ â¦L2â¦ â ââL. â¦L1â¦ â¡ â¦Lâ¦ & â¦L2â¦ â¡ â¦Lâ¦. -#L1 #L2 * /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpc_lfpc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpc_lfpc.etc deleted file mode 100644 index 69e444adb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpc_lfpc.etc +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL0,L1,L2. â¦L0â¦ â¬ â¦L1â¦ â â¦L0â¦ â¬ â¦L2â¦ â - ââL. â¦L1â¦ â¬ â¦Lâ¦ & â¦L2â¦ â¬ â¦Lâ¦. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs.etc deleted file mode 100644 index 3674637dd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs.etc +++ /dev/null @@ -1,76 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 L1 â¦ â¬ â¬ * break â¦ term 46 L2 â¦ )" - 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 â TC â¦ lfpc. - -interpretation "focalized parallel equivalence (local environment)" - 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2). - -(* Basic eliminators ********************************************************) - -lemma lfpcs_ind: âL1. âR:predicate lenv. R L1 â - (âL,L2. â¦L1â¦ â¬* â¦Lâ¦ â â¦Lâ¦ â¬ â¦L2â¦ â R L â R L2) â - âL2. â¦L1â¦ â¬* â¦L2â¦ â R L2. -#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind â¦ HL1 IHL1 â¦ HL12) // -qed-. - -lemma lfpcs_ind_dx: âL2. âR:predicate lenv. R L2 â - (âL1,L. â¦L1â¦ â¬ â¦Lâ¦ â â¦Lâ¦ â¬* â¦L2â¦ â R L â R L1) â - âL1. â¦L1â¦ â¬* â¦L2â¦ â R L1. -#L2 #R #HL2 #IHL2 #L1 #HL12 -@(TC_star_ind_dx â¦ HL2 IHL2 â¦ HL12) // -qed-. - -(* Basic properties *********************************************************) - -lemma lfpcs_refl: reflexive â¦ lfpcs. -/2 width=1/ qed. - -lemma lfpcs_sym: symmetric â¦ lfpcs. -/3 width=1/ qed. - -lemma lfpc_lfpcs: âL1,L2. â¦L1â¦ â¬ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/2 width=1/ qed. - -lemma lfpcs_strap1: âL1,L,L2. â¦L1â¦ â¬* â¦Lâ¦ â â¦Lâ¦ â¬ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/2 width=3/ qed. - -lemma lfpcs_strap2: âL1,L,L2. â¦L1â¦ â¬ â¦Lâ¦ â â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/2 width=3/ qed. - -lemma lfpcs_lfpr_dx: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=1/ qed. - -lemma lfpcs_lfpr_sn: âL1,L2. â¦L2â¦ â¡ â¦L1â¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=1/ qed. - -lemma lfpcs_lfpr_strap1: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦Lâ¦ â¡ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=3/ qed. - -lemma lfpcs_lfpr_strap2: âL1,L. â¦L1â¦ â¡ â¦Lâ¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=3/ qed. - -lemma lfpcs_lfpr_div: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦L2â¦ â¡ â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=3/ qed. - -lemma lfpcs_lfpr_conf: âL1,L. â¦Lâ¦ â¡ â¦L1â¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_aaa.etc deleted file mode 100644 index b7cea0b7a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_aaa.etc +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â - âT,A1. L1 â¢ T â A1 â âA2. L2 â¢ T â A2 â - A1 = A2. -#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2 -elim (lfpcs_inv_lfprs â¦ HL12) -HL12 #L #HL1 #HL2 -lapply (aaa_lfprs_conf â¦ HT1 â¦ HL1) -L1 #HT1 -lapply (aaa_lfprs_conf â¦ HT2 â¦ HL2) -L2 #HT2 -lapply (aaa_mono â¦ HT1 â¦ HT2) -L -T // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_fpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_fpcs.etc deleted file mode 100644 index 8f3688890..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_fpcs.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â âT. â¦L1, Tâ¦ â¬* â¦L2, Tâ¦. -#L1 #L2 #HL12 #T -elim (lfpcs_inv_lfprs â¦ HL12) -HL12 #L #HL1 #HL2 -lapply (lfprs_inv_fprs â¦ HL1 T) -HL1 -lapply (lfprs_inv_fprs â¦ HL2 T) -HL2 /2 width=4/ -qed-. - -(* Properties on context-free parallel equivalence for closures *************) - -lemma fpcs_lfpcs: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L2 #T1 #T2 #HT12 -elim (fpcs_inv_fprs â¦ HT12) -HT12 /3 width=5 by fprs_lfprs, lfprs_div/ (**) (* auto too slow without trace *) -qed. - -lemma fpcs_lift: âK1,K2,T1,T2. â¦K1, T1â¦ â¬* â¦K2, T2â¦ â - âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â - âd,e. â©[d, e] L1 â¡ K1 â â©[d, e] L2 â¡ K2 â - âU1,U2. â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â - â¦L1, U1â¦ â¬* â¦L2, U2â¦. -#K1 #K2 #T1 #T2 #HT12 #L1 #L2 #HL12 #d #e #HLK1 #HLK2 #U1 #U2 #HTU1 #HTU2 -elim (fpcs_inv_fprs â¦ HT12) -HT12 #K #T #HT1 #HT2 -elim (lift_total T d e) #U #HTU -elim (fprs_lift â¦ HT1 â¦ HLK1 â¦ HTU1 HTU) -K1 -T1 #K1 #HU1 #_ -elim (fprs_lift â¦ HT2 â¦ HLK2 â¦ HTU2 HTU) -K2 -T2 -T #K2 #HU2 #_ -K -d -e -lapply (lfpcs_lfprs_conf K1 â¦ HL12) -HL12 /2 width=3/ #H -lapply (lfpcs_lfprs_strap1 â¦ H K2 ?) -H /2 width=3/ #HK12 -lapply (lfpcs_inv_fpcs â¦ HK12 U) -HK12 #HU -/3 width=4 by fpcs_fprs_strap2, fpcs_fprs_div/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfpcs.etc deleted file mode 100644 index 1e9521736..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfpcs.etc +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â - ââL. â¦L1â¦ â¡* â¦Lâ¦ & â¦L2â¦ â¡* â¦Lâ¦. -#L1 #L2 #H @(lfpcs_ind â¦ H) -L2 -[ /3 width=3/ -| #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0 - [ elim (lfprs_strip â¦ HL0 â¦ HL2) -L #L #HL0 #HL2 - lapply (lfprs_strap1 â¦ HL10 â¦ HL0) -L0 /2 width=3/ - | lapply (lfprs_strap2 â¦ HL2 â¦ HL0) -L /2 width=3/ - ] -] -qed-. - -(* Advanced properties ******************************************************) - -lemma lfpcs_strip: âL,L1. â¦Lâ¦ â¬* â¦L1â¦ â âL2. â¦Lâ¦ â¬ â¦L2â¦ â - ââL0. â¦L1â¦ â¬ â¦L0â¦ & â¦L2â¦ â¬* â¦L0â¦. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem lfpcs_trans: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/2 width=3/ qed. - -theorem lfpcs_canc_sn: âL,L1,L2. â¦Lâ¦ â¬* â¦L1â¦ â â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=3 by lfpcs_trans, lfpcs_sym/ qed. - -theorem lfpcs_canc_dx: âL,L1,L2. â¦L1â¦ â¬* â¦Lâ¦ â â¦L2â¦ â¬* â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. -/3 width=3 by lfpcs_trans, lfpcs_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfprs.etc deleted file mode 100644 index baf2caf27..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfprs.etc +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,L2. â¦L1â¦ â¡* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L2 #H @(lfprs_ind â¦ H) -L2 /width=1/ /3 width=3/ -qed. - -lemma lfpcs_lfprs_sn: âL1,L2. â¦L2â¦ â¡* â¦L1â¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L2 #H @(lfprs_ind_dx â¦ H) -L2 /width=1/ /3 width=3/ -qed. - -lemma lfpcs_lfprs_strap1: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦Lâ¦ â¡* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L #HL1 #L2 #H @(lfprs_ind â¦ H) -L2 /width=1/ /2 width=3/ -qed. - -lemma lfpcs_lfprs_strap2: âL1,L. â¦L1â¦ â¡* â¦Lâ¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L #H #L2 #HL2 @(lfprs_ind_dx â¦ H) -L1 /width=1/ /2 width=3/ -qed. - -lemma lfpcs_lfprs_div: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦L2â¦ â¡* â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L #HL1 #L2 #H @(lfprs_ind_dx â¦ H) -L2 /width=1/ /2 width=3/ -qed. - -lemma lfpcs_lfprs_conf: âL1,L. â¦Lâ¦ â¡* â¦L1â¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L #H #L2 #HL2 @(lfprs_ind â¦ H) -L1 /width=1/ /2 width=3/ -qed. - -lemma lfprs_div: âL1,L. â¦L1â¦ â¡* â¦Lâ¦ â âL2. â¦L2â¦ â¡* â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. -#L1 #L #HL1 #L2 #H @(lfprs_ind_dx â¦ H) -L2 /2 width=1/ /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/cpr_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/cpr_llpx_sn.etc deleted file mode 100644 index 4db858798..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/cpr_llpx_sn.etc +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âR. (âI,L.reflexive â¦ (R I L)) â - (âI.l_liftable (R I)) â - (âI.l_deliftable_sn (R I)) â - âG. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H - #Kd #V1d #HLKd #HV1s #HV1sd - lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs - lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd - @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H - #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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 â¦ H) -H - #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/cpx_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/cpx_llpx_sn.etc deleted file mode 100644 index 418106494..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/cpx_llpx_sn.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âR. (âI,L.reflexive â¦ (R I L)) â - (âI.l_liftable (R I)) â - (âI.l_deliftable_sn (R I)) â - âh,g,G. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H - #Kd #V1d #HLKd #HV1s #HV1sd - lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs - lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd - @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H /2 width=1 by/ -| #G #Ls #V1 #V2 #T #_ #IHV12 #Ld #H elim (llpx_sn_inv_flat â¦ 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 â¦ H) -H - #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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 â¦ H) -H - #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/lleq.etc deleted file mode 100644 index dc138a492..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/lleq.etc +++ /dev/null @@ -1,160 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â Î»I,L,T1,T2. T1 = T2. - -definition lleq: relation4 ynat term lenv lenv â 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) â - Î»R. âI,L2,T1,T2. R I L2 T1 T2 â âL1. L1 â¡[T1, 0] L2 â R I L1 T1 T2. - -(* Basic inversion lemmas ***************************************************) - -lemma lleq_ind: âR:relation4 ynat term lenv lenv. ( - âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2 - ) â ( - âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2 - ) â ( - âI,L1,L2,K1,K2,V,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â - K1 â¡[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2 - ) â ( - âL1,L2,d,i. |L1| = |L2| â |L1| â¤ i â |L2| â¤ i â R d (#i) L1 L2 - ) â ( - âL1,L2,d,p. |L1| = |L2| â R d (Â§p) L1 L2 - ) â ( - âa,I,L1,L2,V,T,d. - L1 â¡[V, d]L2 â L1.â{I}V â¡[T, â«¯d] L2.â{I}V â - R d V L1 L2 â R (â«¯d) T (L1.â{I}V) (L2.â{I}V) â R d (â{a,I}V.T) L1 L2 - ) â ( - âI,L1,L2,V,T,d. - L1 â¡[V, d]L2 â L1 â¡[T, d] L2 â - R d V L1 L2 â R d T L1 L2 â R d (â{I}V.T) L1 L2 - ) â - âd,T,L1,L2. L1 â¡[T, d] L2 â 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: âa,I,L1,L2,V,T,d. L1 â¡[â{a,I}V.T, d] L2 â - L1 â¡[V, d] L2 â§ L1.â{I}V â¡[T, â«¯d] L2.â{I}V. -/2 width=2 by llpx_sn_inv_bind/ qed-. - -lemma lleq_inv_flat: âI,L1,L2,V,T,d. L1 â¡[â{I}V.T, d] L2 â - L1 â¡[V, d] L2 â§ L1 â¡[T, d] L2. -/2 width=2 by llpx_sn_inv_flat/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lleq_fwd_length: âL1,L2,T,d. L1 â¡[T, d] L2 â |L1| = |L2|. -/2 width=4 by llpx_sn_fwd_length/ qed-. - -lemma lleq_fwd_lref: âL1,L2,d,i. L1 â¡[#i, d] L2 â - â¨â¨ |L1| â¤ i â§ |L2| â¤ i - | yinj i < d - | ââI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V & - â©[i] L2 â¡ K2.â{I}V & - K1 â¡[V, yinj 0] K2 & d â¤ yinj i. -#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref â¦ H) /2 width=1/ -* /3 width=7 by or3_intro2, ex4_4_intro/ -qed-. - -lemma lleq_fwd_ldrop_sn: âL1,L2,T,d. L1 â¡[d, T] L2 â âK1,i. â©[i] L1 â¡ K1 â - âK2. â©[i] L2 â¡ K2. -/2 width=7 by llpx_sn_fwd_ldrop_sn/ qed-. - -lemma lleq_fwd_ldrop_dx: âL1,L2,T,d. L1 â¡[d, T] L2 â âK2,i. â©[i] L2 â¡ K2 â - âK1. â©[i] L1 â¡ K1. -/2 width=7 by llpx_sn_fwd_ldrop_dx/ qed-. - -lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,d. - L1 â¡[â{a,I}V.T, d] L2 â L1 â¡[V, d] L2. -/2 width=4 by llpx_sn_fwd_bind_sn/ qed-. - -lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,d. - L1 â¡[â{a,I}V.T, d] L2 â L1.â{I}V â¡[T, â«¯d] L2.â{I}V. -/2 width=2 by llpx_sn_fwd_bind_dx/ qed-. - -lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,d. - L1 â¡[â{I}V.T, d] L2 â L1 â¡[V, d] L2. -/2 width=3 by llpx_sn_fwd_flat_sn/ qed-. - -lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,d. - L1 â¡[â{I}V.T, d] L2 â L1 â¡[T, d] L2. -/2 width=3 by llpx_sn_fwd_flat_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma lleq_sort: âL1,L2,d,k. |L1| = |L2| â L1 â¡[âk, d] L2. -/2 width=1 by llpx_sn_sort/ qed. - -lemma lleq_skip: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â¡[#i, d] L2. -/2 width=1 by llpx_sn_skip/ qed. - -lemma lleq_lref: âI,L1,L2,K1,K2,V,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â - K1 â¡[V, 0] K2 â L1 â¡[#i, d] L2. -/2 width=9 by llpx_sn_lref/ qed. - -lemma lleq_free: âL1,L2,d,i. |L1| â¤ i â |L2| â¤ i â |L1| = |L2| â L1 â¡[#i, d] L2. -/2 width=1 by llpx_sn_free/ qed. - -lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â¡[Â§p, d] L2. -/2 width=1 by llpx_sn_gref/ qed. - -lemma lleq_bind: âa,I,L1,L2,V,T,d. - L1 â¡[V, d] L2 â L1.â{I}V â¡[T, â«¯d] L2.â{I}V â - L1 â¡[â{a,I}V.T, d] L2. -/2 width=1 by llpx_sn_bind/ qed. - -lemma lleq_flat: âI,L1,L2,V,T,d. - L1 â¡[V, d] L2 â L1 â¡[T, d] L2 â L1 â¡[â{I}V.T, d] L2. -/2 width=1 by llpx_sn_flat/ qed. - -lemma lleq_refl: âd,T. reflexive â¦ (lleq d T). -/2 width=1 by llpx_sn_refl/ qed. - -lemma lleq_Y: âL1,L2,T. |L1| = |L2| â L1 â¡[T, â] L2. -/2 width=1 by llpx_sn_Y/ qed. - -lemma lleq_sym: âd,T. symmetric â¦ (lleq d T). -#d #T #L1 #L2 #H @(lleq_ind â¦ 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: âL1,L2,U,dt. L1 â¡[U, dt] L2 â - âT,d,e. â§[d, e] T â¡ U â - dt â¤ d + e â L1 â¡[U, d] L2. -/2 width=6 by llpx_sn_ge_up/ qed-. - -lemma lleq_ge: âL1,L2,T,d1. L1 â¡[T, d1] L2 â âd2. d1 â¤ d2 â L1 â¡[T, d2] L2. -/2 width=3 by llpx_sn_ge/ qed-. - -lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â¡[V, 0] L2 â L1.â{I}V â¡[T, 0] L2.â{I}V â - L1 â¡[â{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: âR. (âI,L. reflexive â¦ (R I L)) â - âL1,L2,T,d. L1 â¡[T, d] L2 â llpx_sn R d T L1 L2. -/2 width=3 by llpx_sn_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/llpx_sn.etc deleted file mode 100644 index 918b42605..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/llpx_sn.etc +++ /dev/null @@ -1,209 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â -| llpx_sn_sort: âL1,L2,d,k. |L1| = |L2| â llpx_sn R d (âk) L1 L2 -| llpx_sn_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â llpx_sn R d (#i) L1 L2 -| llpx_sn_lref: âI,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â - â©[i] L1 â¡ K1.â{I}V1 â â©[i] L2 â¡ K2.â{I}V2 â - llpx_sn R (yinj 0) V1 K1 K2 â R I K1 V1 V2 â llpx_sn R d (#i) L1 L2 -| llpx_sn_free: âL1,L2,d,i. |L1| â¤ i â |L2| â¤ i â |L1| = |L2| â llpx_sn R d (#i) L1 L2 -| llpx_sn_gref: âL1,L2,d,p. |L1| = |L2| â llpx_sn R d (Â§p) L1 L2 -| llpx_sn_bind: âa,I,L1,L2,V,T,d. - llpx_sn R d V L1 L2 â llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{I}V) â - llpx_sn R d (â{a,I}V.T) L1 L2 -| llpx_sn_flat: âI,L1,L2,V,T,d. - llpx_sn R d V L1 L2 â llpx_sn R d T L1 L2 â llpx_sn R d (â{I}V.T) L1 L2 -. - -(* Basic inversion lemmas ***************************************************) - -fact llpx_sn_inv_bind_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â - âa,I,V,T. X = â{a,I}V.T â - llpx_sn R d V L1 L2 â§ llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{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: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â - llpx_sn R d V L1 L2 â§ llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{I}V). -/2 width=4 by llpx_sn_inv_bind_aux/ qed-. - -fact llpx_sn_inv_flat_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â - âI,V,T. X = â{I}V.T â - llpx_sn R d V L1 L2 â§ 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: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â - llpx_sn R d V L1 L2 â§ 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: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â |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 â¦ HLK1) -HLK1 -lapply (ldrop_fwd_length â¦ HLK2) -HLK2 -normalize // -qed-. - -lemma llpx_sn_fwd_ldrop_sn: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â - âK1,i. â©[i] L1 â¡ K1 â âK2. â©[i] L2 â¡ K2. -#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length â¦ H) -H -#HL12 lapply (ldrop_fwd_length_le2 â¦ HLK1) -HLK1 /2 width=1 by ldrop_O1_le/ -qed-. - -lemma llpx_sn_fwd_ldrop_dx: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â - âK2,i. â©[i] L2 â¡ K2 â âK1. â©[i] L1 â¡ K1. -#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length â¦ H) -H -#HL12 lapply (ldrop_fwd_length_le2 â¦ HLK2) -HLK2 /2 width=1 by ldrop_O1_le/ -qed-. - -fact llpx_sn_fwd_lref_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â âi. X = #i â - â¨â¨ |L1| â¤ i â§ |L2| â¤ i - | yinj i < d - | ââI,K1,K2,V1,V2. â©[i] L1 â¡ K1.â{I}V1 & - â©[i] L2 â¡ K2.â{I}V2 & - llpx_sn R (yinj 0) V1 K1 K2 & - R I K1 V1 V2 & d â¤ 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: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â - â¨â¨ |L1| â¤ i â§ |L2| â¤ i - | yinj i < d - | ââI,K1,K2,V1,V2. â©[i] L1 â¡ K1.â{I}V1 & - â©[i] L2 â¡ K2.â{I}V2 & - llpx_sn R (yinj 0) V1 K1 K2 & - R I K1 V1 V2 & d â¤ yinj i. -/2 width=3 by llpx_sn_fwd_lref_aux/ qed-. - -lemma llpx_sn_fwd_bind_sn: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â - llpx_sn R d V L1 L2. -#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind â¦ H) -H // -qed-. - -lemma llpx_sn_fwd_bind_dx: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â - llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{I}V). -#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind â¦ H) -H // -qed-. - -lemma llpx_sn_fwd_flat_sn: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â - llpx_sn R d V L1 L2. -#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat â¦ H) -H // -qed-. - -lemma llpx_sn_fwd_flat_dx: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â - llpx_sn R d T L1 L2. -#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat â¦ H) -H // -qed-. - -lemma llpx_sn_fwd_pair_sn: âR,I,L1,L2,V,T,d. llpx_sn R d (â¡{I}V.T) L1 L2 â - 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: âR. (âI,L. reflexive â¦ (R I L)) â âT,L,d. llpx_sn R d T L L. -#R #HR #T #L @(f2_ind â¦ rfw â¦ 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 â¦ HiL) -HiL destruct /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/ -qed-. - -lemma llpx_sn_Y: âR,T,L1,L2. |L1| = |L2| â llpx_sn R (â) T L1 L2. -#R #T #L1 @(f2_ind â¦ rfw â¦ 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: âR,L1,L2,U,dt. llpx_sn R dt U L1 L2 â âT,d,e. â§[d, e] T â¡ U â - dt â¤ d + e â 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 â¦ H) -H /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde - elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12 - elim (ylt_yle_false â¦ Hidt) -Hidt - @(yle_trans â¦ 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 â¦ H) -H * #Hid #H destruct - [ lapply (llpx_sn_fwd_length â¦ HW1) -HW1 #HK12 - lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) - normalize in â¢ (%â%â?); -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 â¦ 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 â¦ H) -H #V #T #HVW >commutative_plus #HTU #H destruct - @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *) - @(IHT â¦ 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 â¦ H) -H #HVW #HTU #H destruct - /3 width=4 by llpx_sn_flat/ -] -qed-. - -(**) (* the minor premise comes first *) -lemma llpx_sn_ge: âR,L1,L2,T,d1,d2. d1 â¤ d2 â - llpx_sn R d1 T L1 L2 â 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: âR,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 â - llpx_sn R 0 T (L1.â{I}V) (L2.â{I}V) â - llpx_sn R 0 (â{a,I}V.T) L1 L2. -/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-. - -lemma llpx_sn_co: âR1,R2. (âI,L,T1,T2. R1 I L T1 T2 â R2 I L T1 T2) â - âL1,L2,T,d. llpx_sn R1 d T L1 L2 â 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/llpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/llpx_sn_alt.etc deleted file mode 100644 index 0916edb67..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/llpx_sn_alt.etc +++ /dev/null @@ -1,62 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â relation4 ynat term lenv lenv â - Î»R,d,T,L1,L2. |L1| = |L2| â§ - (âI1,I2,K1,K2,V1,V2,i. d â¤ yinj i â (L1 â¢ i ~Ïµ ð*[d]â¦Tâ¦ â â¥) â - â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â - I1 = I2 â§ R I1 K1 V1 V2 - ). - -(* Main properties **********************************************************) - -theorem llpx_sn_llpx_sn_alt: âR,T,L1,L2,d. llpx_sn R d T L1 L2 â llpx_sn_alt R d T L1 L2. -#R #U #L1 @(f2_ind â¦ rfw â¦ L1 U) -L1 -U -#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r â¦ H) -H -#HL12 #IHU @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv_ge â¦ H) -H // -[ -n #HnU elim (IHU â¦ 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 â¦ HLK10) #H - lapply (ldrop_conf_ge â¦ H â¦ HLK1 ?) -H /2 width=1 by lt_to_le/ (minus_plus_m_m j (i+1)) in â¢ (%â?); >commutative_plus (lift_inv_sort1 â¦ H) -X - lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ H) -H - * #Hdi #H destruct - [ lapply (ldrop_fwd_length_eq2 â¦ HLK1 HLK2 HK12) -K1 -K2 -d - /2 width=1 by llpx_sn_skip/ - | elim (ylt_yle_false â¦ 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 â¦ H) -H - * #Hdi #H destruct [ -HK12 | -IHK12 ] - [ elim (ldrop_trans_lt â¦ HLK1 â¦ HK11) // -K1 - elim (ldrop_trans_lt â¦ HLK2 â¦ HK22) // -Hdi -K2 - /3 width=18 by llpx_sn_lref/ - | lapply (ldrop_trans_ge_comm â¦ HLK1 â¦ HK11 ?) // -K1 - lapply (ldrop_trans_ge_comm â¦ HLK2 â¦ 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 â¦ H) -H - * #Hid #H destruct - lapply (ldrop_fwd_length_eq2 â¦ HLK1 HLK2 HK12) -HK12 - [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/ - | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #HLK1 - lapply (ldrop_fwd_length â¦ 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 â¦ H) -X - lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ 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 â¦ H) -H - #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ -] -qed-. - -lemma llpx_sn_lift_ge: âR,K1,K2,T,d0. llpx_sn R d0 T K1 K2 â - âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âU. â§[d, e] T â¡ U â d â¤ d0 â 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 â¦ H) -X - lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ H) -H - * #_ #H destruct - lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ H) -H - * #Hid #H destruct - [ elim (ylt_yle_false â¦ 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 â¦ HLK1 â¦ HK11 ?) // -K1 - lapply (ldrop_trans_ge_comm â¦ HLK2 â¦ 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 â¦ H) -H - * #Hid #H destruct - lapply (ldrop_fwd_length_eq2 â¦ HLK1 HLK2 HK12) -HK12 - [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/ - | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #HLK1 - lapply (ldrop_fwd_length â¦ 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 â¦ H) -X - lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ 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 â¦ 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: âR. (âI. l_deliftable_sn (R I)) â - âL1,L2,U,d0. llpx_sn R d0 U L1 L2 â - âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âT. â§[d, e] T â¡ U â d0 â¤ d â 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 â¦ H) -X - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H - * #_ #H destruct - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H - * #Hid #H destruct [ -HK12 | -IHK12 ] - [ elim (ldrop_conf_lt â¦ HLK1 â¦ HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1 - elim (ldrop_conf_lt â¦ HLK2 â¦ HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2 - elim (HR â¦ HW12 â¦ HKL11 â¦ HVW1) -HR #V0 #HV0 #HV12 - lapply (lift_inj â¦ HV0 â¦ HVW2) -HV0 -HVW2 #H destruct - /3 width=10 by llpx_sn_lref/ - | lapply (ldrop_conf_ge â¦ HLK1 â¦ HLK11 ?) // -L1 - lapply (ldrop_conf_ge â¦ HLK2 â¦ HLK22 ?) // -L2 -Hid0 - elim (le_inv_plus_l â¦ 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 â¦ H) -H - * #_ #H destruct - lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) - [ lapply (ldrop_fwd_length_le4 â¦ HLK1) -HLK1 - lapply (ldrop_fwd_length_le4 â¦ HLK2) -HLK2 - #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) - | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H >H in HL1; -H - lapply (ldrop_fwd_length â¦ 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 â¦ H) -X - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ 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 â¦ H) -H - #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ -] -qed-. - -lemma llpx_sn_inv_lift_be: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â - âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âT. â§[d, e] T â¡ U â d â¤ d0 â d0 â¤ yinj d + e â 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 â¦ H) -X - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H - * #Hid #H destruct - [ lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) -L1 -L2 - -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/ - | elim (ylt_yle_false â¦ 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 â¦ H) -H - * #Hid #H destruct - [ elim (ylt_yle_false â¦ Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0 - /3 width=3 by ylt_yle_trans, ylt_inj/ - | lapply (ldrop_conf_ge â¦ HLK1 â¦ HLK11 ?) // -L1 - lapply (ldrop_conf_ge â¦ HLK2 â¦ HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e - elim (le_inv_plus_l â¦ 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 â¦ H) -H - * #_ #H destruct - lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) - [ lapply (ldrop_fwd_length_le4 â¦ HLK1) -HLK1 - lapply (ldrop_fwd_length_le4 â¦ HLK2) -HLK2 - #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) - | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H >H in HL1; -H - lapply (ldrop_fwd_length â¦ 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 â¦ H) -X - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H - >commutative_plus #V #T #HVW #HTU #H destruct - @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) - @(IHU â¦ 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 â¦ H) -H - #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ -] -qed-. - -lemma llpx_sn_inv_lift_ge: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â - âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â - âT. â§[d, e] T â¡ U â yinj d + e â¤ d0 â 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 â¦ H) -X - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H - * #Hid #H destruct [ -Hid0 | -Hded0 ] - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ 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 â¦ H) -H - * #Hid #H destruct - [ elim (ylt_yle_false â¦ 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 â¦ HLK1 â¦ HLK11 ?) // -L1 - lapply (ldrop_conf_ge â¦ HLK2 â¦ 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 â¦ H) -H - * #_ #H destruct - lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) - [ lapply (ldrop_fwd_length_le4 â¦ HLK1) -HLK1 - lapply (ldrop_fwd_length_le4 â¦ HLK2) -HLK2 - #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) - | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H >H in HL1; -H - lapply (ldrop_fwd_length â¦ 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 â¦ H) -X - lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H - #V #T #HVW #HTU #H destruct - @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) - (lpx_sn_inv_atom1 â¦ H) -H - /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/ -| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 â¦ 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 â¦ H) -H - #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 â¦ 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 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (HR â¦ HV12 â¦ HLK1 â¦ HWV1) -V1 - elim (IHLK1 â¦ HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/ -] -qed-. - -lemma lpx_sn_liftable_dedropable: âR. (âI,L. reflexive ? (R I L)) â - (âI. l_liftable (R I)) â 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 â¦ H) -H - /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/ -| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 â¦ H) -H - #K2 #V2 #HK12 #HV12 #H destruct - lapply (lpx_sn_fwd_length â¦ HK12) - #H @(ex3_intro â¦ (K2.â{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 â¦ 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 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (lift_total W2 d e) #V2 #HWV2 - lapply (H2R â¦ HW12 â¦ HLK1 â¦ HWV1 â¦ HWV2) -W1 - elim (IHLK1 â¦ HK12) -K1 - /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/ -] -qed-. - -fact lpx_sn_dropable_aux: âR,L2,K2,s,d,e. â©[s, d, e] L2 â¡ K2 â âL1. lpx_sn R L1 L2 â - d = 0 â ââK1. â©[s, 0, e] L1 â¡ 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 â¦ H) -H - /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/ -| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 â¦ 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 â¦ H) -H - #L1 #V1 #HL12 #HV12 #H destruct - elim (IHLK2 â¦ HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/ -| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_ - (lpx_sn_inv_atom1 â¦ H1) -X1 - >(lpx_sn_inv_atom1 â¦ 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 â¦ H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct - elim (lpx_sn_inv_pair1 â¦ H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct - elim (IH â¦ HL01 â¦ HL02) -IH normalize // #L #HL1 #HL2 - elim (HR12 â¦ HV01 â¦ HV02 â¦ HL01 â¦ HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/lpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/lpx_sn_tc.etc deleted file mode 100644 index 2a3a87cf4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/lpx_sn_tc.etc +++ /dev/null @@ -1,119 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âR. (âI,L. reflexive â¦ (R I L)) â - âL1,L2. TC â¦ (lpx_sn R) L1 L2 â - âI,V. TC â¦ (lpx_sn R) (L1.â{I}V) (L2. â{I}V). -#R #HR #L1 #L2 #H @(TC_star_ind â¦ 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: âR. (âI,L. reflexive â¦ (R I L)) â - âI,L1,L2. TC â¦ (lpx_sn R) L1 L2 â - âV1,V2. LTC â¦ (R I) L1 V1 V2 â - TC â¦ (lpx_sn R) (L1.â{I}V1) (L2. â{I}V2). -#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx â¦ 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: âR. (âI,L. reflexive â¦ (R I L)) â - âL1,L2. lpx_sn (Î»I.LTC â¦ (R I)) L1 L2 â - TC â¦ (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: âR,L1. TC â¦ (lpx_sn R) L1 (â) â L1 = â. -#R #L1 #H @(TC_ind_dx â¦ 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: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â - âI,L1,K2,V2. TC â¦ (lpx_sn R) L1 (K2.â{I}V2) â - ââK1,V1. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ (R I) K1 V1 V2 & L1 = K1.â{I}V1. -#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx â¦ L1 H) -L1 -[ #L1 #H elim (lpx_sn_inv_pair2 â¦ H) -H /3 width=5 by inj, ex3_2_intro/ -| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct - elim (lpx_sn_inv_pair2 â¦ HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct - lapply (HR â¦ HV2 â¦ HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/ -] -qed-. - -lemma TC_lpx_sn_ind: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â - âS:relation lenv. - S (â) (â) â ( - âI,K1,K2,V1,V2. - TC â¦ (lpx_sn R) K1 K2 â LTC â¦ (R I) K1 V1 V2 â - S K1 K2 â S (K1.â{I}V1) (K2.â{I}V2) - ) â - âL2,L1. TC â¦ (lpx_sn R) L1 L2 â S L1 L2. -#R #HR #S #IH1 #IH2 #L2 elim L2 -L2 -[ #X #H >(TC_lpx_sn_inv_atom2 â¦ H) -X // -| #L2 #I #V2 #IHL2 #X #H - elim (TC_lpx_sn_inv_pair2 â¦ H) // -H -HR - #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/ -] -qed-. - -lemma TC_lpx_sn_inv_atom1: âR,L2. TC â¦ (lpx_sn R) (â) L2 â L2 = â. -#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: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â - âL1,L2. TC â¦ (lpx_sn R) L1 L2 â - âI,K1,V1. L1 = K1.â{I}V1 â - ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ (R I) K1 V1 V2 & L2 = K2. â{I} V2. -#R #HR #L1 #L2 #H @(TC_lpx_sn_ind â¦ 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: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â - âI,K1,L2,V1. TC â¦ (lpx_sn R) (K1.â{I}V1) L2 â - ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ (R I) K1 V1 V2 & L2 = K2. â{I} V2. -/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-. - -lemma TC_lpx_sn_inv_lpx_sn_LTC: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â - âL1,L2. TC â¦ (lpx_sn R) L1 L2 â - lpx_sn (Î»I.LTC â¦ (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: âR,L1,L2. TC â¦ (lpx_sn R) L1 L2 â |L1| = |L2|. -#R #L1 #L2 #H elim H -L2 -[ #L2 #HL12 >(lpx_sn_fwd_length â¦ HL12) -HL12 // -| #L #L2 #_ #HL2 #IHL1 - >IHL1 -L1 >(lpx_sn_fwd_length â¦ HL2) -HL2 // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/ssta_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/ssta_llpx_sn.etc deleted file mode 100644 index 3b1db6a9a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpx_sn/ssta_llpx_sn.etc +++ /dev/null @@ -1,44 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âR. (âI,L.reflexive â¦ (R I L)) â - (âI.l_liftable (R I)) â - âh,g,G. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H - #Kd #V1d #HLKd #HV1s #HV1sd - lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs - lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd - @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ H â¦ HLKs) // -H - #Kd #V1d #HLKd #HV1s #HV1sd - lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs - lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd - @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ H) -H - /3 width=1 by llpx_sn_flat/ -| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat â¦ H) -H - /3 width=1 by llpx_sn_flat/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn.etc deleted file mode 100644 index c4359c35f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn.etc +++ /dev/null @@ -1,118 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 L1 : â break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqN $h $L1 $L2 }. - -notation "hvbox( h â¢ break term 46 L1 : : â 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 â -| lsubn_atom: lsubn h (â) (â) -| lsubn_pair: âI,L1,L2,W. lsubn h L1 L2 â lsubn h (L1. â{I} W) (L2. â{I} W) -| lsubn_abbr: âL1,L2,V,W. â¦h, L1â¦ â¢ V : W â â¦h, L2â¦ â¢ V : W â - lsubn h L1 L2 â lsubn h (L1. âV) (L2. âW) -. - -interpretation - "local environment refinement (native type assigment)" - 'CrSubEqN h L1 L2 = (lsubn h L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubn_inv_atom1_aux: âh,L1,L2. h â¢ L1 :â L2 â L1 = â â L2 = â. -#h #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #_ #_ #_ #H destruct -] -qed. - -lemma lsubn_inv_atom1: âh,L2. h â¢ â :â L2 â L2 = â. -/2 width=4/ qed-. - -fact lsubn_inv_pair1_aux: âh,L1,L2. h â¢ L1 :â L2 â âI,K1,V. L1 = K1. â{I} V â - (ââK2. h â¢ K1 :â K2 & L2 = K2. â{I} V) â¨ - ââK2,W. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & - h â¢ K1 :â K2 & L2 = K2. âW & 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: âh,I,K1,L2,V. h â¢ K1. â{I} V :â L2 â - (ââK2. h â¢ K1 :â K2 & L2 = K2. â{I} V) â¨ - ââK2,W. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & - h â¢ K1 :â K2 & L2 = K2. âW & I = Abbr. -/2 width=3/ qed-. - -fact lsubn_inv_atom2_aux: âh,L1,L2. h â¢ L1 :â L2 â L2 = â â L1 = â. -#h #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #_ #_ #_ #H destruct -] -qed. - -lemma lsubc_inv_atom2: âh,L1. h â¢ L1 :â â â L1 = â. -/2 width=4/ qed-. - -fact lsubn_inv_pair2_aux: âh,L1,L2. h â¢ L1 :â L2 â âI,K2,W. L2 = K2. â{I} W â - (ââK1. h â¢ K1 :â K2 & L1 = K1. â{I} W) â¨ - ââK1,V. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & - h â¢ K1 :â K2 & L1 = K1. âV & 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: âh,I,L1,K2,W. h â¢ L1 :â K2. â{I} W â - (ââK1. h â¢ K1 :â K2 & L1 = K1. â{I} W) â¨ - ââK1,V. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & - h â¢ K1 :â K2 & L1 = K1. âV & I = Abst. -/2 width=3/ qed-. - -(* Basic_forward lemmas *****************************************************) - -lemma lsubn_fwd_lsubs1: âh,L1,L2. h â¢ L1 :â L2 â L1 â¼[0, |L1|] L2. -#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -lemma lsubn_fwd_lsubs2: âh,L1,L2. h â¢ L1 :â L2 â L1 â¼[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: âh,L. h â¢ L :â 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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_cpcs.etc deleted file mode 100644 index 5f610bc96..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_cpcs.etc +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,L2. h â¢ L1 :â L2 â - âT1,T2. L2 â¢ T1 â¡ T2 â L1 â¢ T1 â¡ T2. -/3 width=4 by lsubn_fwd_lsubs2, cpr_lsubs_trans/ qed. - -lemma cprs_lsubn_trans: âh,L1,L2. h â¢ L1 :â L2 â - âT1,T2. L2 â¢ T1 â¡* T2 â L1 â¢ T1 â¡* T2. -/3 width=4 by lsubn_fwd_lsubs2, cprs_lsubs_trans/ qed. - -(* Basic_1: was: csubt_pc3 *) -lemma cpcs_lsubn_trans: âh,L1,L2. h â¢ L1 :â L2 â - âT1,T2. L2 â¢ T1 â¬* T2 â L1 â¢ T1 â¬* T2. -/3 width=4 by lsubn_fwd_lsubs2, cpcs_lsubs_trans/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_ldrop.etc deleted file mode 100644 index a16fff618..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_ldrop.etc +++ /dev/null @@ -1,64 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,L2. h â¢ L1 :â L2 â âK1,e. â©[0, e] L1 â¡ K1 â - ââK2. h â¢ K1 :â K2 & â©[0, e] L2 â¡ K2. -#h #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ 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: âh,L1,L2. h â¢ L1 :â L2 â âK2,e. â©[0, e] L2 â¡ K2 â - ââK1. h â¢ K1 :â K2 & â©[0, e] L1 â¡ K1. -#h #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ - ] -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_lsubn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_lsubn.etc deleted file mode 100644 index 9ef3dda88..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_lsubn.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,L. h â¢ L1 :â L â âL2. h â¢ L :â L2 â h â¢ L1 :â L2. -#h #L1 #L #H elim H -L1 -L -[ #X #H >(lsubn_inv_atom1 â¦ H) -H // -| #I #L1 #L #V #HL1 #H1W #IHL1 #X #H - elim (lsubn_inv_pair1 â¦ 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 â¦ H) -H * #L2 - [ #HL2 #H #HW destruct /3 width=1/ - | #V #_ #_ #_ #_ #H destruct - ] -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_nta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_nta.etc deleted file mode 100644 index 5832b00b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubn/lsubn_nta.etc +++ /dev/null @@ -1,47 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L2,T,U. â¦h, L2â¦ â¢ T : U â âL1. h â¢ L1 :â L2 â - â¦h, L1â¦ â¢ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 - elim (lsubn_inv_pair2 â¦ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 - elim (lsubn_inv_pair2 â¦ 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/aaa_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/aaa_lpss.etc deleted file mode 100644 index e930823ad..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/aaa_lpss.etc +++ /dev/null @@ -1,54 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T1,A. L1 â¢ T1 â A â âT2. L1 â¢ T1 â¶* T2 â - âL2. L1 â¢ â¶* L2 â L2 â¢ T2 â A. -#L1 #T1 #A #H elim H -L1 -T1 -A -[ #L1 #k #X #H - >(cpss_inv_sort1 â¦ H) -H // -| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12 - elim (cpss_inv_lref1 â¦ H) -H - [ #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/ - | * #Y #Z #V2 #H #HV12 #HV2 - lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #Z #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #V0 #HK12 #_ #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) -V0 /3 width=7/ - ] -| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12 - elim (cpss_inv_bind1 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/ -] -qed-. - -lemma aaa_cpss_conf: âL,T1,A. L â¢ T1 â A â âT2. L â¢ T1 â¶* T2 â L â¢ T2 â A. -/2 width=5 by aaa_cpss_lpss_conf/ qed-. - -lemma aaa_lpss_conf: âL1,T,A. L1 â¢ T â A â âL2. L1 â¢ â¶* L2 â L2 â¢ T â A. -/2 width=5 by aaa_cpss_lpss_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpcs_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpcs_lpss.etc deleted file mode 100644 index 09d29754f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpcs_lpss.etc +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL1,T1,T2. L1 â¢ T1 â¬* T2 â - âL2. L1 â¢ â¶* L2 â L2 â¢ T1 â¬* T2. -#L1 #T1 #T2 #H #L2 #HL12 -elim (cpcs_inv_cprs â¦ H) -H #T #HT1 #HT2 -elim (cprs_lpss_conf_dx â¦ HT1 â¦ HL12) -HT1 #U1 #H1 #HTU1 -elim (cprs_lpss_conf_dx â¦ HT2 â¦ HL12) -L1 #U2 #H2 #HTU2 -elim (cpss_conf â¦ H1 â¦ H2) -T #U #HU1 #HU2 -lapply (cprs_cpss_trans â¦ HTU1 â¦ HU1) -U1 -lapply (cprs_cpss_trans â¦ HTU2 â¦ HU2) -U2 /2 width=3/ -qed-. - -lemma cpcs_cpss_lpss_conf: âL1,T,T2. L1 â¢ T â¬* T2 â âT1. L1 â¢ T â¶* T1 â - âL2. L1 â¢ â¶* L2 â L2 â¢ T1 â¬* T2. -#L1 #T #T2 #HT2 #T1 #HT1 #L2 #HL12 -lapply (cpcs_lpss_conf â¦ HT2 â¦ HL12) -HT2 #HT2 -elim (lpss_cpss_conf_dx â¦ HT1 â¦ HL12) -L1 #T0 #HT0 #HT10 -lapply (cpcs_cpss_conf â¦ HT0 â¦ HT2) -T #HT02 -lapply (cpcs_cpss_strap2 â¦ HT10 â¦ HT02) -T0 // -qed-. - -lemma cpcs_cpss2_lpss_conf: âL1,T1,T2. L1 â¢ T1 â¬* T2 â - âT3. L1 â¢ T1 â¶* T3 â âT4. L1 â¢ T2 â¶* T4 â - âL2. L1 â¢ â¶* L2 â L2 â¢ T3 â¬* T4. -#L1 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 #L2 #HL12 -lapply (cpcs_cpss_lpss_conf â¦ HT12 â¦ HT13 â¦ HL12) -T1 #HT32 -elim (lpss_cpss_conf_dx â¦ HT24 â¦ HL12) -L1 #T #HT2 #HT4 -lapply (cpcs_cpss_strap1 â¦ HT32 â¦ HT2) -T2 #HT3 -lapply (cpcs_cpss_div â¦ HT3 â¦ HT4) -T // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpqs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpqs.etc deleted file mode 100644 index 41120361a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpqs.etc +++ /dev/null @@ -1,227 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 T1 â¤ * 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 â relation term â -| cpqs_atom : âI,L. cpqs L (âª{I}) (âª{I}) -| cpqs_delta: âL,K,V,V2,W2,i. - â©[0, i] L â¡ K. âV â cpqs K V V2 â - â§[0, i + 1] V2 â¡ W2 â cpqs L (#i) W2 -| cpqs_bind : âa,I,L,V1,V2,T1,T2. - cpqs L V1 V2 â cpqs (L. â{I} V1) T1 T2 â - cpqs L (â{a,I} V1. T1) (â{a,I} V2. T2) -| cpqs_flat : âI,L,V1,V2,T1,T2. - cpqs L V1 V2 â cpqs L T1 T2 â - cpqs L (â{I} V1. T1) (â{I} V2. T2) -| cpqs_zeta : âL,V,T1,T,T2. cpqs (L.âV) T1 T â - â§[0, 1] T2 â¡ T â cpqs L (+âV. T1) T2 -| cpqs_tau : âL,V,T1,T2. cpqs L T1 T2 â cpqs L (âV. 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 â¦ 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 â¦ HL12 â¦ HLK1) -HL12 -HLK1 /3 width=6/ -| /4 width=1/ -|4,6: /3 width=1/ -| /4 width=3/ -] -qed-. - -lemma cpss_cpqs: âL,T1,T2. L â¢ T1 â¶* T2 â L â¢ T1 â¤* T2. -#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/ -qed. - -lemma cpqs_refl: âT,L. L â¢ T â¤* T. -/2 width=1/ qed. - -lemma cpqs_delift: âL,K,V,T1,d. â©[0, d] L â¡ (K. âV) â - ââT2,T. L â¢ T1 â¤* T2 & â§[d, 1] T â¡ T2. -#L #K #V #T1 #d #HLK -elim (cpss_delift â¦ T1 â¦ HLK) -HLK /3 width=4/ -qed-. - -lemma cpqs_append: l_appendable_sn â¦ 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 â¦ HK0) #H -@(cpqs_delta â¦ (L@@K0) V1 â¦ HVW2) // -@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2/ (**) (* /3/ does not work *) -qed. - -(* Basic inversion lemmas ***************************************************) - -fact cpqs_inv_atom1_aux: âL,T1,T2. L â¢ T1 â¤* T2 â âI. T1 = âª{I} â - T2 = âª{I} â¨ - ââK,V,V2,i. â©[O, i] L â¡ K. âV & - K â¢ V â¤* V2 & - â§[O, i + 1] V2 â¡ 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: âI,L,T2. L â¢ âª{I} â¤* T2 â - T2 = âª{I} â¨ - ââK,V,V2,i. â©[O, i] L â¡ K. âV & - K â¢ V â¤* V2 & - â§[O, i + 1] V2 â¡ T2 & - I = LRef i. -/2 width=3 by cpqs_inv_atom1_aux/ qed-. - -lemma cpqs_inv_sort1: âL,T2,k. L â¢ âk â¤* T2 â T2 = âk. -#L #T2 #k #H -elim (cpqs_inv_atom1 â¦ H) -H // -* #K #V #V2 #i #_ #_ #_ #H destruct -qed-. - -lemma cpqs_inv_lref1: âL,T2,i. L â¢ #i â¤* T2 â - T2 = #i â¨ - ââK,V,V2. â©[O, i] L â¡ K. âV & - K â¢ V â¤* V2 & - â§[O, i + 1] V2 â¡ T2. -#L #T2 #i #H -elim (cpqs_inv_atom1 â¦ H) -H /2 width=1/ -* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/ -qed-. - -lemma cpqs_inv_gref1: âL,T2,p. L â¢ Â§p â¤* T2 â T2 = Â§p. -#L #T2 #p #H -elim (cpqs_inv_atom1 â¦ H) -H // -* #K #V #V2 #i #_ #_ #_ #H destruct -qed-. - -fact cpqs_inv_bind1_aux: âL,U1,U2. L â¢ U1 â¤* U2 â - âa,I,V1,T1. U1 = â{a,I} V1. T1 â ( - ââV2,T2. L â¢ V1 â¤* V2 & - L. â{I} V1 â¢ T1 â¤* T2 & - U2 = â{a,I} V2. T2 - ) â¨ - ââT. L.âV1 â¢ T1 â¤* T & â§[0, 1] U2 â¡ 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: âa,I,L,V1,T1,U2. L â¢ â{a,I} V1. T1 â¤* U2 â ( - ââV2,T2. L â¢ V1 â¤* V2 & - L. â{I} V1 â¢ T1 â¤* T2 & - U2 = â{a,I} V2. T2 - ) â¨ - ââT. L.âV1 â¢ T1 â¤* T & â§[0, 1] U2 â¡ T & a = true & I = Abbr. -/2 width=3 by cpqs_inv_bind1_aux/ qed-. - -lemma cpqs_inv_abbr1: âa,L,V1,T1,U2. L â¢ â{a} V1. T1 â¤* U2 â ( - ââV2,T2. L â¢ V1 â¤* V2 & - L. â V1 â¢ T1 â¤* T2 & - U2 = â{a} V2. T2 - ) â¨ - ââT. L.âV1 â¢ T1 â¤* T & â§[0, 1] U2 â¡ T & a = true. -#a #L #V1 #T1 #U2 #H -elim (cpqs_inv_bind1 â¦ H) -H * /3 width=3/ /3 width=5/ -qed-. - -lemma cpqs_inv_abst1: âa,L,V1,T1,U2. L â¢ â{a} V1. T1 â¤* U2 â - ââV2,T2. L â¢ V1 â¤* V2 & - L. â V1 â¢ T1 â¤* T2 & - U2 = â{a} V2. T2. -#a #L #V1 #T1 #U2 #H -elim (cpqs_inv_bind1 â¦ H) -H * -[ /3 width=5/ -| #T #_ #_ #_ #H destruct -] -qed-. - -fact cpqs_inv_flat1_aux: âL,U1,U2. L â¢ U1 â¤* U2 â - âI,V1,T1. U1 = â{I} V1. T1 â ( - ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & - U2 = â{I} V2. T2 - ) â¨ - (L â¢ T1 â¤* U2 â§ 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: âI,L,V1,T1,U2. L â¢ â{I} V1. T1 â¤* U2 â ( - ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & - U2 = â{I} V2. T2 - ) â¨ - (L â¢ T1 â¤* U2 â§ I = Cast). -/2 width=3 by cpqs_inv_flat1_aux/ qed-. - -lemma cpqs_inv_appl1: âL,V1,T1,U2. L â¢ â V1. T1 â¤* U2 â - ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & - U2 = â V2. T2. -#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 â¦ H) -H * -[ /3 width=5/ -| #_ #H destruct -] -qed-. - -lemma cpqs_inv_cast1: âL,V1,T1,U2. L â¢ â V1. T1 â¤* U2 â ( - ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & - U2 = â V2. T2 - ) â¨ - L â¢ T1 â¤* U2. -#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 â¦ H) -H * /2 width=1/ /3 width=5/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cpqs_fwd_shift1: âL1,L,T1,T. L â¢ L1 @@ T1 â¤* T â - ââL2,T2. |L1| = |L2| & T = L2 @@ T2. -#L1 @(lenv_ind_dx â¦ L1) -L1 normalize -[ #L #T1 #T #HT1 - @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) -| #I #L1 #V1 #IH #L #T1 #X - >shift_append_assoc normalize #H - elim (cpqs_inv_bind1 â¦ H) -H * - [ #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) - | #T #_ #_ #H destruct - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpqs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpqs_lift.etc deleted file mode 100644 index 78529b6b1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpqs_lift.etc +++ /dev/null @@ -1,81 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ H1 â¦ 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 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HVW2 â¦ HWU2) -W2 // plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct - elim (lift_inv_flat1 â¦ 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 â¦ H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct - elim (lift_conf_O1 â¦ HTU2 â¦ HT2) -T2 /4 width=5/ -| #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2 - elim (lift_inv_flat1 â¦ 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 â¦ H) -H #H destruct /2 width=3/ - | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hid #H destruct - [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV - elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 - elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // shift_append_assoc normalize #H - elim (cpss_inv_bind1 â¦ H) -H - #V0 #T0 #_ #HT10 #H destruct - elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro â¦ (â.â{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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpss_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpss_lift.etc deleted file mode 100644 index 9745b6cde..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpss_lift.etc +++ /dev/null @@ -1,71 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â¦ H1 â¦ 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 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HVW2 â¦ HWU2) -W2 // plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct - elim (lift_inv_flat1 â¦ 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 â¦ H) -H #H destruct /2 width=3/ - | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hid #H destruct - [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV - elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 - elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // shift_append_assoc #H shift_append_assoc #H (cpss_inv_sort1 â¦ H) -X // -| #I #L1 #K1 #V1 #i #HLK1 #_ #IHV1 #W2 #H #L2 #HL12 - elim (cpss_inv_lref1 â¦ H) -H - [ #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct - lapply (IHV1 â¦ HV12 â¦ HK12) -IHV1 -HV12 -HK12 /2 width=5/ - | * #K0 #V0 #V2 #HLK0 #HV12 #HVW2 - lapply (ldrop_mono â¦ HLK0 â¦ HLK1) -HLK0 #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #V #HK12 #_ #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) -V #HLK2 - lapply (IHV1 â¦ HV12 â¦ HK12) -IHV1 -HV12 -HK12 /2 width=7/ - ] -| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12 - elim (cpss_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IHV1 â¦ HV12 â¦ HL12) -IHV1 #HV2 - lapply (IHT1 â¦ HT12 (L2.â{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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IHV1 â¦ HV12 â¦ HL12) -IHV1 #HV2 - lapply (IHT1 â¦ HT12 â¦ HL12) -IHT1 #HT2 - elim (ssta_cpss_lpss_conf â¦ HVW1 â¦ HV12 â¦ HL12) -V1 #W2 #HVW2 #HW12 - elim (dxprs_cpss_lpss_conf â¦ HTU1 â¦ HT12 â¦ HL12) -T1 #X #HTU2 #H - elim (cpss_inv_bind1 â¦ H) -H #W #U2 #HW0 #_ #H -U1 destruct - elim (cprs_cpss2_lpss_conf_dx â¦ HW10 â¦ HW12 â¦ HW0 â¦ HL12) -L1 -W1 -W0 #W0 #HW20 #HW0 - lapply (dxprs_strap1 â¦ (â{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 â¦ H) -H #W2 #T2 #HW12 #HT12 #H destruct - lapply (IHW1 â¦ HW12 â¦ HL12) -IHW1 #HW2 - lapply (IHT1 â¦ HT12 â¦ HL12) -IHT1 #HT2 - elim (ssta_cpss_lpss_conf â¦ HTU1 â¦ HT12 â¦ HL12) -T1 #U2 #HTU2 #HU12 - lapply (cpcs_cpss2_lpss_conf â¦ HUW1 â¦ HU12 â¦ HW12 â¦ HL12) -L1 -W1 -U1 /2 width=4/ -] -qed-. - -lemma snv_lpss_conf: âh,g,L1,T. â¦h, L1â¦ â¢ T Â¡[g] â - âL2. L1 â¢ â¶* L2 â â¦h, L2â¦ â¢ T Â¡[g]. -#h #g #L1 #T #HT #L2 #HL12 -@(snv_cpss_lpss_conf â¦ HT â¦ HL12) // -qed-. - -lemma snv_cpss_conf: âh,g,L,T1. â¦h, Lâ¦ â¢ T1 Â¡[g] â - âT2. L â¢ T1 â¶* T2 â â¦h, Lâ¦ â¢ T2 Â¡[g]. -#h #g #L #T1 #HT1 #T2 #HT12 -@(snv_cpss_lpss_conf â¦ HT1 â¦ HT12) // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/ssta_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/ssta_lpss.etc deleted file mode 100644 index 7471b2304..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/ssta_lpss.etc +++ /dev/null @@ -1,81 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 â¢[g] â¦l, U1â¦ â - âT2. L1 â¢ T1 â¶* T2 â âL2. L1 â¢ â¶* L2 â - ââU2. â¦h, L2â¦ â¢ T2 â¢[g] â¦l, U2â¦ & L1 â¢ U1 â¶* U2. -#h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l -[ #L1 #k1 #l1 #Hkl1 #X #H - >(cpss_inv_sort1 â¦ H) -H /3 width=3/ -| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #_ #HWU1 #IHVW1 #X #H #L2 #HL12 - elim (cpss_inv_lref1 â¦ H) -H - [ #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK1) -HLK1 #HLK1 - elim (IHVW1 â¦ HV12 â¦ HK12) -IHVW1 -HV12 -HK12 #W2 #HVW2 #HW12 - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (cpss_lift â¦ HW12 â¦ HLK1 â¦ HWU1 â¦ HWU2) -HW12 -HLK1 -HWU1 /3 width=6/ - | * #Y #Z #V2 #H #HV12 #HV2 - lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #Z #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #V0 #HK12 #_ #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) -V0 #HLK2 - lapply (ldrop_fwd_ldrop2 â¦ HLK1) -HLK1 #HLK1 - elim (IHVW1 â¦ HV12 â¦ HK12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (ssta_lift â¦ HVW2 â¦ HLK2 â¦ HV2 â¦ HWU2) -HVW2 -HLK2 -HV2 - lapply (cpss_lift â¦ HW12 â¦ HLK1 â¦ HWU1 â¦ 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 â¦ H) -H [ | -IHWV1 -HWU1 -HL12 ] - [ #H destruct - elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 - elim (lpss_inv_pair1 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK1) -HLK1 #HLK1 - elim (IHWV1 â¦ HW12 â¦ HK12) -IHWV1 -HK12 #V2 #HWV2 #_ - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (cpss_lift â¦ HW12 â¦ HLK1 â¦ HWU1 â¦ HWU2) -HW12 -HLK1 -HWU1 /3 width=6/ - | * #K2 #V2 #W2 #HLK2 #_ #_ - lapply (ldrop_mono â¦ HLK2 â¦ HLK1) -HLK1 -HLK2 #H destruct - ] -| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12 - elim (cpss_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - elim (IHTU1 â¦ HT12 (L2.â{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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - elim (IHTU1 â¦ HT12 â¦ HL12) -IHTU1 -HT12 -HL12 /3 width=5/ -| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12 - elim (cpss_inv_flat1 â¦ H) -H #W2 #T2 #HW12 #HT12 #H destruct - elim (IHTU1 â¦ HT12 â¦ HL12) -IHTU1 -HT12 -HL12 /3 width=3/ -] -qed-. - -lemma ssta_cpss_conf: âh,g,L,T1,U1,l. â¦h, Lâ¦ â¢ T1 â¢[g] â¦l, U1â¦ â - âT2. L â¢ T1 â¶* T2 â - ââU2. â¦h, Lâ¦ â¢ T2 â¢[g] â¦l, U2â¦ & L â¢ U1 â¶* U2. -/2 width=3 by ssta_cpss_lpss_conf/ qed-. - -lemma ssta_lpss_conf: âh,g,L1,T,U1,l. â¦h, L1â¦ â¢ T â¢[g] â¦l, U1â¦ â - âL2. L1 â¢ â¶* L2 â - ââU2. â¦h, L2â¦ â¢ T â¢[g] â¦l, U2â¦ & L1 â¢ U1 â¶* U2. -/2 width=3 by ssta_cpss_lpss_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/sstas_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/sstas_lpss.etc deleted file mode 100644 index 3ecbc2256..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/sstas_lpss.etc +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,T1,U1. â¦h, L1â¦ â¢ T1 â¢*[g] U1 â - âT2. L1 â¢ T1 â¶* T2 â âL2. L1 â¢ â¶* L2 â - ââU2. â¦h, L2â¦ â¢ T2 â¢*[g] U2 & L1 â¢ U1 â¶* U2. -#h #g #L1 #T1 #U1 #H @(sstas_ind_dx â¦ H) -T1 /2 width=3/ -#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12 -elim (ssta_cpss_lpss_conf â¦ HTU0 â¦ HT0 â¦ HL12) -HTU0 -HT0 #U #HTU #HU0 -elim (IHU01 â¦ HU0 â¦ HL12) -IHU01 -U0 -HL12 /3 width=4/ -qed-. - -lemma sstas_cpss_conf: âh,g,L,T1,U1. â¦h, Lâ¦ â¢ T1 â¢*[g] U1 â - âT2. L â¢ T1 â¶* T2 â - ââU2. â¦h, Lâ¦ â¢ T2 â¢*[g] U2 & L â¢ U1 â¶* U2. -/2 width=3 by sstas_cpss_lpss_conf/ qed-. - -lemma sstas_lpss_conf: âh,g,L1,T,U1. â¦h, L1â¦ â¢ T â¢*[g] U1 â - âL2. L1 â¢ â¶* L2 â - ââU2. â¦h, L2â¦ â¢ T â¢*[g] U2 & L1 â¢ U1 â¶* U2. -/2 width=3 by sstas_cpss_lpss_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn.etc deleted file mode 100644 index f6b54500e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn.etc +++ /dev/null @@ -1,110 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 L1 : â [ ] 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 â -| lsubsn_atom: lsubsn h (â) (â) -| lsubsn_pair: âI,L1,L2,W. lsubsn h L1 L2 â - lsubsn h (L1. â{I} W) (L2. â{I} W) -| lsubsn_abbr: âL1,L2,V,W,l. â¦h, L1â¦ â¢ V :[l+1] W â â¦h, L2â¦ â¢ V :[l+1] W â - lsubsn h L1 L2 â lsubsn h (L1. âV) (L2. âW) -. - -interpretation - "local environment refinement (stratified native type assigment)" - 'StratifiedCrSubEqN h L1 L2 = (lsubsn h L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubsn_inv_atom1_aux: âh,L1,L2. h â¢ L1 :â[] L2 â L1 = â â L2 = â. -#h #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #l #_ #_ #_ #H destruct -] -qed. - -lemma lsubsn_inv_atom1: âh,L2. h â¢ â :â[] L2 â L2 = â. -/2 width=5/ qed-. - -fact lsubsn_inv_pair1_aux: âh,L1,L2. h â¢ L1 :â[] L2 â - âI,K1,V. L1 = K1. â{I} V â - (ââK2. h â¢ K1 :â[] K2 & L2 = K2. â{I} V) â¨ - ââK2,W,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & - h â¢ K1 :â[] K2 & L2 = K2. âW & 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: âh,I,K1,L2,V. h â¢ K1. â{I} V :â[] L2 â - (ââK2. h â¢ K1 :â[] K2 & L2 = K2. â{I} V) â¨ - ââK2,W,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & - h â¢ K1 :â[] K2 & L2 = K2. âW & I = Abbr. -/2 width=3/ qed-. - -fact lsubsn_inv_atom2_aux: âh,L1,L2. h â¢ L1 :â[] L2 â L2 = â â L1 = â. -#h #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #l #_ #_ #_ #H destruct -] -qed. - -lemma lsubsn_inv_atom2: âh,L1. h â¢ L1 :â[] â â L1 = â. -/2 width=5/ qed-. - -fact lsubsn_inv_pair2_aux: âh,L1,L2. h â¢ L1 :â[] L2 â - âI,K2,W. L2 = K2. â{I} W â - (ââK1. h â¢ K1 :â[] K2 & L1 = K1. â{I} W) â¨ - ââK1,V,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & - h â¢ K1 :â[] K2 & L1 = K1. âV & 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: âh,I,L1,K2,W. h â¢ L1 :â[] K2. â{I} W â - (ââK1. h â¢ K1 :â[] K2 & L1 = K1. â{I} W) â¨ - ââK1,V,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & - h â¢ K1 :â[] K2 & L1 = K1. âV & I = Abst. -/2 width=3/ qed-. - -(* Basic_forward lemmas *****************************************************) - -lemma lsubsn_fwd_lsubs1: âh,L1,L2. h â¢ L1 :â[] L2 â L1 â¼[0, |L1|] L2. -#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -lemma lsubsn_fwd_lsubs2: âh,L1,L2. h â¢ L1 :â[] L2 â L1 â¼[0, |L2|] L2. -#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -lemma lsubsn_refl: âh,L. h â¢ L :â[] L. -#h #L elim L -L // /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_cpcs.etc deleted file mode 100644 index 38f7e17bd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_cpcs.etc +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,L2. h â¢ L1 :â[] L2 â - âT1,T2. L2 â¢ T1 â¡ T2 â L1 â¢ T1 â¡ T2. -/3 width=5 by lsubsn_fwd_lsubs2, cpr_lsubs_trans/ qed-. - -lemma cprs_lsubsn_trans: âh,L1,L2. h â¢ L1 :â[] L2 â - âT1,T2. L2 â¢ T1 â¡* T2 â L1 â¢ T1 â¡* T2. -/3 width=5 by lsubsn_fwd_lsubs2, cprs_lsubs_trans/ qed-. - -lemma cpcs_lsubsn_trans: âh,L1,L2. h â¢ L1 :â[] L2 â - âT1,T2. L2 â¢ T1 â¬* T2 â L1 â¢ T1 â¬* T2. -/3 width=5 by lsubsn_fwd_lsubs2, cpcs_lsubs_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_ldrop.etc deleted file mode 100644 index 092883783..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_ldrop.etc +++ /dev/null @@ -1,65 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,L2. h â¢ L1 :â[] L2 â - âK1,e. â©[0, e] L1 â¡ K1 â - ââK2. h â¢ K1 :â[] K2 & â©[0, e] L2 â¡ K2. -#h #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK1) -L1 /3 width=3/ - ] -] -qed. - -(* Note: the constant 0 cannot be generalized *) -lemma lsubsn_ldrop_O1_trans: âh,L1,L2. h â¢ L1 :â[] L2 â - âK2,e. â©[0, e] L2 â¡ K2 â - ââK1. h â¢ K1 :â[] K2 & â©[0, e] L1 â¡ K1. -#h #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ - | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ - ] -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_snta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_snta.etc deleted file mode 100644 index c6215302c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/lsubsn_snta.etc +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L2,T,U,l. â¦h, L2â¦ â¢ T :[l] U â - âL1. h â¢ L1 :â[] L2 â â¦h, L1â¦ â¢ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 - elim (lsubsn_inv_pair2 â¦ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 - elim (lsubsn_inv_pair2 â¦ H) -H * #K1 [ -HWV2 | -IHWV2 ] - [ #HK12 #H destruct /3 width=6/ - | #V1 #l0 #H1 #H2 #_ #H #_ destruct - elim (snta_fwd_correct â¦ H2) -H2 #V #H - elim (snta_mono â¦ HWV2 â¦ 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 â¦ HL12 â¦ HU12) -HU12 /3 width=3/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta.etc deleted file mode 100644 index 71ed42179..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta.etc +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ h , break L â¦ â¢ break term 46 T1 : * break [ l ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeTypeStar $h $l $L $T1 $T2 }. - -notation "hvbox( â¦ h , break L â¦ â¢ 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 â lenv â relation term â -| snta_sort: âL,k. snta h 0 L (âk) (â(next h k)) -| snta_ldef: âL,K,V,W,U,i,l. â©[0, i] L â¡ K. âV â snta h l K V W â - â§[0, i + 1] W â¡ U â snta h l L (#i) U -| snta_ldec: âL,K,W,V,U,i,l. â©[0, i] L â¡ K. âW â snta h l K W V â - â§[0, i + 1] W â¡ U â snta h (l+1) L (#i) U -| snta_bind: âI,L,V,W,T,U,l1,l2. snta h l1 L V W â snta h l2 (L. â{I} V) T U â - snta h l2 L (â{I}V.T) (â{I}V.U) -| snta_appl: âL,V,W1,W2,T,U,l1,l2. snta h (l1+1) L V W2 â - snta h l2 L (âW1.T) (âW2.U) â - snta h l2 L (âV.âW1.T) (âV.âW2.U) -| snta_pure: âL,V,T,U,W,l. snta h (l+1) L T U â snta h l L (âV.U) W â - snta h (l+1) L (âV.T) (âV.U) -| snta_cast: âL,T,U,W,l1,l2. snta h l2 L T U â snta h l1 L U W â - snta h l2 L (âU.T) U -| snta_conv: âL,T,U1,U2,V2,l. snta h l L T U1 â L â¢ U1 â¬* U2 â - snta h (l-1) L U2 V2 â snta h l L T U2 -. - -interpretation "stratified native type assignment (term)" - 'StratifiedNativeType h l L T U = (snta h l L T U). diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_lift.etc deleted file mode 100644 index 424bfe89f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_lift.etc +++ /dev/null @@ -1,238 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âk0. T = âk0 â - l = 0 â§ L â¢ â(next h k0) â¬* 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 â¦ Hk0 â¦ HU12) -U1 /2 width=1/ -] -qed. - -lemma snta_inv_sort1: âh,L,U,k,l. â¦h, Lâ¦ â¢ âk :[l] U â - l = 0 â§ L â¢ â(next h k) â¬* U. -/2 width=4/ qed-. - -fact snta_inv_lref1_aux: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âj. T = #j â - (ââK,V,W,U0. â©[0, j] L â¡ K. âV & â¦h, Kâ¦ â¢ V :[l] W & - â§[0, j + 1] W â¡ U0 & L â¢ U0 â¬* U - ) â¨ - (ââK,W,V,U0. â©[0, j] L â¡ K. âW & â¦h, Kâ¦ â¢ W :[l-1] V & - â§[0, j + 1] W â¡ U0 & l > 0 & L â¢ U0 â¬* 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 â¦ HU01 â¦ HU12) -U1 /3 width=8/ -] -qed. - -lemma snta_inv_lref1: âh,L,U,i,l. â¦h, Lâ¦ â¢ #i :[l] U â - (ââK,V,W,U0. â©[0, i] L â¡ K. âV & â¦h, Kâ¦ â¢ V :[l] W & - â§[0, i + 1] W â¡ U0 & L â¢ U0 â¬* U - ) â¨ - (ââK,W,V,U0. â©[0, i] L â¡ K. âW & â¦h, Kâ¦ â¢ W :[l-1] V & - â§[0, i + 1] W â¡ U0 & l > 0 & L â¢ U0 â¬* U - ). -/2 width=3/ qed-. - -fact snta_inv_bind1_aux: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âJ,X,Y. T = â{J}Y.X â - ââZ1,Z2,l0. â¦h, Lâ¦ â¢ Y :[l0] Z1 & - â¦h, L.â{J}Yâ¦ â¢ X :[l] Z2 & - L â¢ â{J}Y.Z2 â¬* 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 â¦ HU1 â¦ HU12) -U1 /2 width=3/ -] -qed. - -lemma snta_inv_bind1: âh,J,L,Y,X,U,l. â¦h, Lâ¦ â¢ â{J}Y.X :[l] U â - ââZ1,Z2,l0. â¦h, Lâ¦ â¢ Y :[l0] Z1 & â¦h, L.â{J}Yâ¦ â¢ X :[l] Z2 & - L â¢ â{J}Y.Z2 â¬* U. -/2 width=3/ qed-. - -fact snta_inv_cast1_aux: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âX,Y. T = âY.X â - â¦h, Lâ¦ â¢ X :[l] Y â§ L â¢ Y â¬* 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 â¦ HU1 â¦ HU12) -U1 /2 width=1/ -] -qed. - -lemma snta_inv_cast1: âh,L,X,Y,U,l. â¦h, Lâ¦ â¢ âY.X :[l] U â - â¦h, Lâ¦ â¢ X :[l] Y â§ L â¢ Y â¬* U. -/2 width=3/ qed-. - -(* Properties on relocation *************************************************) - -lemma snta_lift: âh,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 :[l] U1 â - âL2,d,e. â©[d, e] L2 â¡ L1 â - âT2. â§[d, e] T1 â¡ T2 â âU2. â§[d, e] U1 â¡ U2 â - â¦h, L2â¦ â¢ 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 â¦ H1) -X1 - >(lift_inv_sort1 â¦ H2) -X2 // -| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HW1 â¦ HWU2 ?) -W // #W2 #HW12 #HWU2 - elim (ldrop_trans_le â¦ HL21 â¦ HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H - elim (ldrop_inv_skip2 â¦ H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct - /3 width=8/ - | lapply (lift_trans_be â¦ HW1 â¦ HWU2 ? ?) -W // /2 width=1/ #HW1U2 - lapply (ldrop_trans_ge â¦ HL21 â¦ 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 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HW1 â¦ HWU2 ?) -W // (tpr_inv_atom1 â¦ H) -H // -| #L1 #K1 #V1 #W #U #i1 #l #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H #Hl -IH - >(tpr_inv_atom1 â¦ H) -T2 - elim (ltpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #HLK2 #H - elim (ltpr_inv_pair1 â¦ 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 â¦ H) -T2 - elim (ltpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #HLK2 #H - elim (ltpr_inv_pair1 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) #HLK - elim (lift_total V1 0 (i+1)) #W #HW - lapply (snta_lift h â¦ HLK â¦ HWU1 â¦ HW) /2 width=1/ -HLK -HW - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpr_lift â¦ HW12 â¦ HWU1 â¦ HWU2) -HWU1 #HU12 - @(snta_conv â¦ 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 â¦ H) -H * - [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct - lapply (IHVW1 â¦ HL12 â¦ HV12) #HV2W1 - lapply (IHVW1 L2 â¦ V1 ?) // -IHVW1 #HWV1 - lapply (IHTU1 (L2.â{I}V2) â¦ HT1) -HT1 /2 width=1/ #HTU1 - lapply (IHTU1 (L2.â{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H - lapply (tps_lsubs_trans â¦ HT2 (L2.â{I}V2) ?) -HT2 /2 width=1/ #HT2 - lapply (snta_tps_conf â¦ HTU1 â¦ HT2) -T #HT2U1 - elim (snta_fwd_correct â¦ H) -H #U2 #HU12 - @(snta_conv â¦ (â{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | #T #HT1 #HTX #H destruct - lapply (IHVW1 â¦ HL12 V1 ?) -IHVW1 // #HVW1 - lapply (IHTU1 (L2.âV1) â¦ HT1) -T1 /2 width=1/ -L1 #H - elim (snta_fwd_correct â¦ H) #T1 #HUT1 - elim (snta_ldrop_conf â¦ H L2 0 1 ? ?) -H // /2 width=1/ #T0 #U0 #HTU0 #H #HU10 - lapply (delift_inv_lift1_eq â¦ H L2 â¦ HTX) -H -HTX /2 width=1/ #H destruct - @(snta_conv â¦ HTU0) /2 width=2/ - ] -*) -| #L1 #V1 #W11 #W2 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH -(* - elim (tpr_inv_appl1 â¦ H) -H * - [ #V2 #Y #HV12 #HY #H destruct - elim (tpr_inv_abst1 â¦ HY) -HY #W2 #T2 #HW12 #HT12 #H destruct - lapply (IHTU1 L2 ? (âW1.T1) ?) // #H - elim (snta_fwd_correct â¦ H) -H #X #H - elim (snta_inv_bind1 â¦ H) -H #W #U #HW #HU #_ - @(snta_conv â¦ (âV2.âW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) - | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - lapply (IHVW1 â¦ HL12 â¦ HV12) #HVW2 - lapply (IHVW1 â¦ HL12 V1 ?) -IHVW1 // #HV1W2 - lapply (IHTU1 â¦ HL12 (âW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1 - elim (snta_fwd_correct â¦ H1) #T #H2 - elim (snta_inv_bind1 â¦ H1) -H1 #W #U2 #HW2 #HTU2 #H - elim (cpcs_inv_abst â¦ H Abst W2) -H #_ #HU21 - elim (snta_inv_bind1 â¦ H2) -H2 #W0 #U0 #_ #H #_ -T -W0 - lapply (lsubsn_snta_trans â¦ HTU2 (L2.âV2) ?) -HTU2 /2 width=1/ #HTU2 - @(snta_conv â¦ (âV2.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 â¦ H) -H * - [ #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpr â¦ HV12 L2) #HV - elim (snta_fwd_correct h L2 (âV1.T1) (âV1.U1) (l+1) ?) [2: /3 width=6/ ] #U - @(snta_conv â¦ (âV2.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 â¦ HL12 (âW0.T2) ? ?) -IHTU1 // /2 width=1/ -T0 #H1 - lapply (IH â¦ (âV2.U1) â¦ HUW1 HL12 ?) // /3 width=1/ #H2 - lapply (snta_pure â¦ H1 H2) -H2 #H - elim (snta_inv_bind1 â¦ H1) -H1 #V0 #U2 #l1 #HWV0 #HTU2 #HU21 - @(snta_conv â¦ (âV2.U2)) (**) (* explicit constructor *) - [2: -(* - @snta_bind /3 width=2/ /3 width=6/ (**) (* /4 width=6/ is a bit slow *) -*) - |3: @(cpcs_cpr_conf â¦ (âV1.âW0.U2)) /2 width=1/ - |4: /2 width=5/ - | skip - ] -(* - elim (snta_fwd_pure1 â¦ H) -H #T1 #W2 #HVW2 #HUT1 #HTW1 - - elim (cpcs_inv_abst1 â¦ HU21) #W3 #U3 #HU13 #H - elim (cprs_inv_abst â¦ H Abst W0) -H #HW03 #_ - elim (pippo â¦ IH â¦ 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 â¦ HV1 â¦ HL12 â¦ HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH â¦ HB0 â¦ HL12 W2 ?) -HB0 /width=5/ #HB0 - lapply (IH â¦ HA0 â¦ (L2.âW2) â¦ HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ - -axiom pippo: â¦h, Lâ¦ â¢ âV.X : Y â - ââW,T. L â¢ X â¡* âW.T & â¦h, Lâ¦ â¢ âV : W. - -*) -(* SEGMENT 2 -| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 â¦ H) -H #U2 #T2 #HU12 #HT12 #H destruct - lapply (cpr_tpss â¦ HU12) /4 width=4/ -| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 - @(snta_conv â¦ U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) -] -qed. -*) - -(* SEGMENT 3 -fact snta_ltpr_tpr_conf_aux: âh,L,T,L1,T1,U. â¦h, L1â¦ â¢ T1 : U â L = L1 â T = T1 â - âL2. L1 â¡ L2 â âT2. T1 â¡ T2 â â¦h, L2â¦ â¢ T2 : U. - - - | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct - elim (snta_inv_abbr â¦ HT1) -HT1 #B0 #HW0 #HT0 - lapply (IH â¦ HW0 â¦ HL12 â¦ HW02) -HW0 /width=5/ #HW2 - lapply (IH â¦ HV1 â¦ HL12 â¦ HV10) -HV1 -HV10 /width=5/ #HV0 - lapply (IH â¦ HT0 â¦ (L2.âW2) â¦ HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2 - @(snta_abbr â¦ HW2) -HW2 - @(snta_appl â¦ 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 â¦ H) -H - [ * #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH â¦ HV1 â¦ HL12 â¦ HV12) -HV1 -HV12 /width=5/ #HV2 - lapply (IH â¦ HT1 â¦ HL12 â¦ HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ - | -HV1 #HT1X - lapply (IH â¦ HT1 â¦ HL12 â¦ HT1X) -IH -HT1 -HL12 -HT1X /width=5/ - ] -] -qed. - -lemma snta_ltpr_tpr_conf: âh,L1,T1,U. â¦h, L1â¦ â¢ T1 : U â âL2. L1 â¡ L2 â - âT2. T1 â¡ T2 â â¦h, L2â¦ â¢ T2 : U. - -/2 width=9/ qed. - -axiom snta_ltpr_conf: âL1,T,A. L1 â¢ T : A â âL2. L1 â¡ L2 â L2 â¢ T : A. -/2 width=5/ qed. - -axiom snta_tpr_conf: âL,T1,A. L â¢ T1 : A â âT2. T1 â¡ T2 â L â¢ T2 : A. -/2 width=5/ qed. -*) -*)*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_ltpss.etc deleted file mode 100644 index 0e5f3930e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_ltpss.etc +++ /dev/null @@ -1,123 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,T1,U,l. â¦h, L1â¦ â¢ T1 :[l] U â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶* [d, e] T2 â â¦h, L2â¦ â¢ 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 â¦ H) -H // -| #L1 #K1 #V1 #W #U #i #l #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 â¦ H) -H - [ #H destruct - elim (lt_or_ge i d) #Hdi - [ elim (ltpss_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct - /3 width=7/ - | lapply (ltpss_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -L1 // -Hide /3 width=7/ - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 - elim (ltpss_ldrop_conf_be â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct - lapply (ldrop_mono â¦ HL2K0 â¦ HLK2) -HL2K0 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) -HLK2 #HLK2 - lapply (tpss_trans_eq â¦ 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 â¦ 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 â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 â¦ H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) #HLK - lapply (snta_lift h â¦ HLK â¦ HWU1 â¦ HW) [ /2 width=4/ | skip ] -HW #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge â¦ HW12 â¦ HLK â¦ HWU1 â¦ HWU2) -HLK -HWU1 // #HU12 - lapply (cpr_tpss â¦ HU12) -HU12 #HU12 - @(snta_conv â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK2) #HLK - lapply (snta_lift h â¦ HLK â¦ HWU1 â¦ HW) [ /2 width=4/ | skip ] -HW #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge â¦ HW12 â¦ HLK â¦ HWU1 â¦ HWU2) -HLK -HWU1 // #HU12 - lapply (cpr_tpss â¦ HU12) -HU12 #HU12 - @(snta_conv â¦ U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | lapply (ltpss_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -L1 // -Hide /2 width=6/ - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_ - elim (ltpss_ldrop_conf_be â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct - lapply (ldrop_mono â¦ HL2K0 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss â¦ HV12) #HV - lapply (IHTU1 (L2.â{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H - elim (snta_fwd_correct â¦ H) -H #U2 #HU12 - @(snta_conv â¦ (â{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 â¦ H) -H #V2 #Y #HV12 #HY #H destruct - elim (tpss_inv_bind1 â¦ HY) -HY #W21 #T2 #HW121 #HT12 #H destruct - lapply (cpr_tpss â¦ HV12) #HVV12 - lapply (IHTU1 L2 d e ? (âW21.T2) ?) -IHTU1 // /2 width=1/ -HW121 -HT12 #H0 - elim (snta_fwd_correct â¦ H0) #X #H - elim (snta_inv_bind1 â¦ H) -H #W #U #l0 #HW #HU #_ - @(snta_conv â¦ (âV2.âW12.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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss â¦ HV12) #HV - elim (snta_fwd_correct h L2 (âV1.T1) (âV1.U1) (l+1) ?) [2: /3 width=4/ ] #U - @(snta_conv â¦ (âV2.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 â¦ HTU1) -HTU1 #U #H - elim (snta_mono â¦ HUW1 â¦ H) -HUW1 -H #H #_ -U destruct - elim (tpss_inv_flat1 â¦ H) -H #U2 #T2 #HU12 #HT12 #H destruct - lapply (cpr_tpss â¦ HU12) #HU /4 width=4/ -| #L1 #T1 #U11 #U12 #U #l #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 - @(snta_conv â¦ U11) /2 width=5/ (**) (* explicit constructor, /3 width=7/ is too slow *) -] -qed. - -lemma snta_ltpss_tps_conf: âh,L1,T1,U,l. â¦h, L1â¦ â¢ T1 :[l] U â - âL2,d,e. L1 â¶* [d, e] L2 â - âT2. L2 â¢ T1 â¶ [d, e] T2 â â¦h, L2â¦ â¢ T2 :[l] U. -/3 width=7/ qed. - -lemma snta_ltpss_conf: âh,L1,T,U,l. â¦h, L1â¦ â¢ T :[l] U â - âL2,d,e. L1 â¶* [d, e] L2 â â¦h, L2â¦ â¢ T :[l] U. -/2 width=7/ qed. - -lemma snta_tpss_conf: âh,L,T1,U,l. â¦h, Lâ¦ â¢ T1 :[l] U â - âT2,d,e. L â¢ T1 â¶* [d, e] T2 â â¦h, Lâ¦ â¢ T2 :[l] U. -/2 width=7/ qed. - -lemma snta_tps_conf: âh,L,T1,U,l. â¦h, Lâ¦ â¢ T1 :[l] U â - âT2,d,e. L â¢ T1 â¶ [d, e] T2 â â¦h, Lâ¦ â¢ T2 :[l] U. -/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_snta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_snta.etc deleted file mode 100644 index db71e1192..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_snta.etc +++ /dev/null @@ -1,65 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L,T,U1,l1. â¦h, Lâ¦ â¢ T :[l1] U1 â - âU2,l2. â¦h, Lâ¦ â¢ T :[l2] U2 â l1 = l2 â§ L â¢ U1 â¬* U2. -#h #L #T #U1 #l1 #H elim H -L -T -U1 -l1 -[ #L #k #X #l2 #H - lapply (snta_inv_sort1 â¦ H) -H * /2 width=1/ -| #L #K #V #W11 #W12 #i #l1 #HLK #_ #HW112 #IHVW11 #X #l2 #H - elim (snta_inv_lref1 â¦ H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX - lapply (ldrop_mono â¦ HLK0 â¦ HLK) -HLK0 #H destruct - lapply (ldrop_fwd_ldrop2 â¦ HLK) -HLK #HLK - elim (IHVW11 â¦ HVW21) -IHVW11 -HVW21 #Hl12 #HW121 - lapply (cpcs_lift â¦ HLK â¦ HW112 â¦ HW212 ?) // -K -W11 -W21 /3 width=3/ -| #L #K #W #V1 #V #i #l1 #HLK #_ #HWV #IHWV1 #X #l2 #H - elim (snta_inv_lref1 â¦ H) -H * #K0 #W0 #V2 #V0 #HLK0 #HW0V2 #HWV0 [2: #HL2 ] #HX - lapply (ldrop_mono â¦ HLK0 â¦ HLK) -HLK0 -HLK #H destruct - lapply (lift_mono â¦ HWV0 â¦ HWV) -HWV0 -HWV #H destruct - elim (IHWV1 â¦ HW0V2) -IHWV1 -HW0V2 /3 width=1/ -| #I #L #V #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H - elim (snta_inv_bind1 â¦ H) -H #W2 #U2 #l20 #_ #HTU2 #H - elim (IHTU1 â¦ HTU2) -IHTU1 -HTU2 #Hl12 #HU12 - lapply (cpcs_trans â¦ (â{I}V.U1) â¦ H) -H /2 width=1/ -| #L #V #W #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H - elim (snta_fwd_pure1 â¦ H) -H #U2 #W2 #l20 #_ #HTU2 #H - elim (IHTU1 â¦ HTU2) -IHTU1 -HTU2 #Hl12 #HU12 - lapply (cpcs_trans â¦ (âV.âW1.U1) â¦ H) -H /2 width=1/ -| #L #V #T #U1 #W1 #l1 #_ #_ #IHTU1 #_ #X #l2 #H - elim (snta_fwd_pure1 â¦ H) -H #U2 #W2 #l20 #_ #HTU2 #H - elim (IHTU1 â¦ HTU2) -IHTU1 -HTU2 #Hl12 #HU12 - lapply (cpcs_trans â¦ (âV.U1) â¦ H) -H /2 width=1/ -| #L #T #U1 #W1 #l10 #l1 #_ #_ #IHTU1 #_ #X #l2 #H - elim (snta_inv_cast1 â¦ H) -H #HTU1 - elim (IHTU1 â¦ HTU1) -IHTU1 -HTU1 /2 width=1/ -| #L #T #U11 #U12 #V12 #l1 #_ #HU112 #_ #IHTU11 #_ #U2 #l2 #HTU2 - elim (IHTU11 â¦ HTU2) -IHTU11 -HTU2 #Hl12 #H - lapply (cpcs_canc_sn â¦ HU112 â¦ H) -U11 /2 width=1/ -] -qed-. - -(* Advanced properties ******************************************************) - -lemma snta_cast_alt: âh,L,T,W,U,l. â¦h, Lâ¦ â¢ T :[l] W â â¦h, Lâ¦ â¢ T :[l] U â - â¦h, Lâ¦ â¢ âW.T :[l] U. -#h #L #T #W #U #l #HTW #HTU -elim (snta_mono â¦ HTW â¦ HTU) #_ #HWU -elim (snta_fwd_correct â¦ HTU) -HTU /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_thin.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_thin.etc deleted file mode 100644 index ceb5375bf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snta/snta_thin.etc +++ /dev/null @@ -1,116 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 :[l] U1 â - âL2,d,e. â½ [d, e] L1 â L1 â¼*[d, e] â¡ L2 â - ââT2,U2. â¦h, L2â¦ â¢ T2 :[l] U2 & - L1 â¢ T1 â¼*[d, e] â¡ T2 & L1 â¢ U1 â¼*[d, e] â¡ 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 â¦ HLK1 â¦ HL1 ?) -HL1 /2 width=2/ #H - lapply (sfr_inv_skip â¦ H ?) -H /2 width=1/ #HK1 - elim (thin_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2 - elim (thin_inv_delift1 â¦ H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 â¦ HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12 - lapply (delift_mono â¦ H â¦ HV12) -H -HV12 #H destruct - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (ldrop_fwd_ldrop2 â¦ HLK1) -V1 #HLK1 - lapply (delift_lift_ge â¦ HW12 â¦ HLK1 HWU1 â¦ HWU2) -HW12 -HLK1 -HWU1 // - >minus_plus minus_plus #HU1 - lapply (lift_conf_be â¦ HWU2 â¦ HW2U ?) -W2 /2 width=1/ #HU2 - lapply (delift_lift_div_be â¦ HU1 â¦ HU2 ? ?) -U // /2 width=1/ /3 width=8/ - | lapply (transitive_le â¦ (i+1) Hide ?) /2 width=1/ #Hdei - lapply (thin_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -HL12 -HLK1 // #HL2K1 - elim (lift_split â¦ HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W - commutative_plus minus_plus commutative_plus â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ T2 Â¡[h, g]. -/3 width=6 by lpr_lpx, cpr_cpx/ qed-. - -fact snv_sta_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - âG,L,T. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, Lâ¦ â¢ T Â¡[h, g] â - âl. â¦G, Lâ¦ â¢ T âª[h, g] l+1 â - âU. â¦G, Lâ¦ â¢ T â¢[h] U â â¦G, Lâ¦ â¢ U Â¡[h, g]. -/3 width=6 by sta_cpx/ qed-. - -fact snv_cpxs_lpx_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âT2. â¦G, L1â¦ â¢ T1 â¡*[h, g] T2 â âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â â¦G, L2â¦ â¢ T2 Â¡[h, g]. -#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H -@(cpxs_ind â¦ H) -T2 /4 width=6 by fpbg_fpbs_trans, cpxs_fpbs/ -qed-. - -fact snv_cprs_lpr_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ T2 Â¡[h, g]. -/3 width=10 by snv_cpxs_lpx_aux, cprs_cpxs, lpr_lpx/ qed-. - -fact snv_lstas_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âl1,l2. l2 â¤ l1 â â¦G, L1â¦ â¢ T1 âª[h, g] l1 â - âU1. â¦G, L1â¦ â¢ T1 â¢*[h, l2] U1 â â¦G, L1â¦ â¢ U1 Â¡[h, g]. -/3 width=12 by snv_cpxs_lpx_aux, lstas_cpxs/ qed-. - -fact da_cprs_lpr_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âl. â¦G, L1â¦ â¢ T1 âª[h, g] l â - âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ T2 âª[h, g] l. -#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H -@(cprs_ind â¦ H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/ -qed-. - -fact da_cpcs_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - âG,L,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, T1â¦ â â¦G, Lâ¦ â¢ T1 Â¡[h, g] â - âT2. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, T2â¦ â â¦G, Lâ¦ â¢ T2 Â¡[h, g] â - âl1. â¦G, Lâ¦ â¢ T1 âª[h, g] l1 â âl2. â¦G, Lâ¦ â¢ T2 âª[h, g] l2 â - â¦G, Lâ¦ â¢ T1 â¬* T2 â 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 â¦ H) -H /4 width=18 by da_cprs_lpr_aux, da_mono/ -qed-. - -fact sta_cpr_lpr_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âl. â¦G, L1â¦ â¢ T1 âª[h, g] l+1 â - âU1. â¦G, L1â¦ â¢ T1 â¢[h] U1 â - âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â - ââU2. â¦G, L2â¦ â¢ T2 â¢[h] U2 & â¦G, L2â¦ â¢ U1 â¬* U2. -#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #H01 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12 -elim (IH â¦ H01 â¦ 1 â¦ Hl U1 â¦ HT12 â¦ HL12) -H01 -Hl -HT12 -HL12 -/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/ -qed-. - -fact lstas_cprs_lpr_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âl1,l2. l2 â¤ l1 â â¦G, L1â¦ â¢ T1 âª[h, g] l1 â - âU1. â¦G, L1â¦ â¢ T1 â¢*[h, l2] U1 â - âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â - ââU2. â¦G, L2â¦ â¢ T2 â¢*[h, l2] U2 & â¦G, L2â¦ â¢ U1 â¬* U2. -#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H -@(cprs_ind â¦ H) -T2 [ /2 width=10 by/ ] -#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12 -elim (IHT1 L1) // -IHT1 #U #HTU #HU1 -elim (IH1 â¦ Hl21 â¦ HTU â¦ HTT2 â¦ 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: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âl,l1. l â¤ l1 â â¦G, L1â¦ â¢ T1 âª[h, g] l1 â âU1. â¦G, L1â¦ â¢ T1 â¢*[h, l] U1 â - âT2. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T2â¦ â â¦G, L1â¦ â¢ T2 Â¡[h, g] â - âl2. l â¤ l2 â â¦G, L1â¦ â¢ T2 âª[h, g] l2 â âU2. â¦G, L1â¦ â¢ T2 â¢*[h, l] U2 â - â¦G, L1â¦ â¢ T1 â¬* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ U1 â¬* 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 â¦ H) -H #T #H1 #H2 -elim (lstas_cprs_lpr_aux â¦ H01 HT1 â¦ Hl1 HTl1 â¦ HTU1 â¦ H1 â¦ HL12) -T1 /2 width=1 by/ #W1 #H1 #HUW1 -elim (lstas_cprs_lpr_aux â¦ H02 HT2 â¦ Hl2 HTl2 â¦ HTU2 â¦ H2 â¦ HL12) -T2 /2 width=1 by/ #W2 #H2 #HUW2 -L0 -T0 -lapply (lstas_mono â¦ H1 â¦ H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/ -qed-. - -fact lstas_cpds_aux: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - âG,L,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, T1â¦ â â¦G, Lâ¦ â¢ T1 Â¡[h, g] â - âl1,l2. l2 â¤ l1 â â¦G, Lâ¦ â¢ T1 âª[h, g] l1 â - âU1. â¦G, Lâ¦ â¢ T1 â¢*[h, l2] U1 â âT2. â¦G, Lâ¦ â¢ T1 â¢*â¡*[h, g] T2 â - ââU2,l. l â¤ l2 & â¦G, Lâ¦ â¢ T2 â¢*[h, l] U2 & â¦G, Lâ¦ â¢ U1 â¢*â¬*[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 â¦ H â¦ Hl1) -H #H destruct -lapply (lstas_da_conf â¦ HTU1 â¦ Hl1) #Hl12 -elim (le_or_ge l2 l) #Hl2 -[ lapply (lstas_conf_le â¦ HTU1 â¦ HT1T) -HT1T - /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/ -| lapply (lstas_da_conf â¦ HT1T â¦ Hl1) #Hl1l - lapply (lstas_conf_le â¦ HT1T â¦ HTU1) -HTU1 // #HTU1 - elim (lstas_cprs_lpr_aux â¦ IH3 IH2 IH1 â¦ Hl1l â¦ HTU1 â¦ 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: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â - âU1. â¦G, L1â¦ â¢ T1 â¢*â¡*[h, g] U1 â - âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â - ââU2. â¦G, L2â¦ â¢ T2 â¢*â¡*[h, g] U2 & â¦G, L2â¦ â¢ U1 â¡* 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 â¦ H01 â¦ HTW1 â¦ HT12 â¦ HL12) -IH1 // #W2 #HTW2 #HW12 -lapply (IH2 â¦ H01 â¦ Hl1 â¦ HT12 â¦ HL12) -L0 -T0 // -T1 -lapply (lpr_cprs_conf â¦ HL12 â¦ HWU1) -L1 #HWU1 -lapply (cpcs_canc_sn â¦ HW12 HWU1) -W1 #H -elim (cpcs_inv_cprs â¦ H) -H /3 width=7 by ex4_3_intro, ex2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snv/snv_da_lpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snv/snv_da_lpr.etc deleted file mode 100644 index 119a40243..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/snv/snv_da_lpr.etc +++ /dev/null @@ -1,93 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G0,L0,T0. - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â 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 â¦ H2) -H2 - lapply (cpr_inv_sort1 â¦ 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 â¦ H1) -H1 #I0 #K0 #X0 #H #HX0 - elim (da_inv_lref â¦ H2) -H2 * #K1 [ #V1 | #W1 #l1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct - lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct - elim (cpr_inv_lref1 â¦ H3) -H3 - [1,3: #H destruct - lapply (fqup_lref â¦ G1 â¦ HLK1) - elim (lpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #H #HLK2 - elim (lpr_inv_pair1 â¦ 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 â¦ H â¦ HLK1) -H #H destruct - lapply (fqup_lref â¦ G1 â¦ HLK1) - elim (lpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #H #HLK2 - elim (lpr_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #_ #H destruct - lapply (ldrop_fwd_drop2 â¦ HLK2) -V2 - /4 width=8 by da_lift, fqup_fpbg/ - ] -| #p #_ #_ #HT0 #H1 destruct -IH2 -IH1 - elim (snv_inv_gref â¦ H1) -| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2 - elim (snv_inv_bind â¦ H1) -H1 #_ #HT1 - lapply (da_inv_bind â¦ H2) -H2 - elim (cpr_inv_bind1 â¦ 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 â¦ H1) -H1 #b0 #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10 - lapply (da_inv_flat â¦ H2) -H2 #Hl - elim (cpr_inv_appl1 â¦ 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 â¦ HT1) -HT1 #HW #HU1 - lapply (da_inv_bind â¦ Hl) -Hl #Hl - elim (cpds_inv_abst1 â¦ HT10) -HT10 #W3 #U3 #HW3 #_ #H destruct -U3 - lapply (cprs_div â¦ HW3 â¦ HW10) -W3 #HWW1 - lapply (da_sta_conf â¦ HVW1 â¦ Hl0) â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â 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 â¦ H2) -H2 // * // -| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2 - elim (snv_inv_lref â¦ H1) -H1 #I #K1 #V1 #HLK1 #HV1 - elim (lpx_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 - elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct - lapply (fqup_lref â¦ G1 â¦ HLK1) #HKL - elim (cpx_inv_lref1 â¦ H2) -H2 - [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/ - | * #I0 #K0 #V0 #W0 #H #HVW0 #W0 -HV12 - lapply (ldrop_mono â¦ H â¦ HLK1) -HLK1 -H #H destruct - lapply (ldrop_fwd_drop2 â¦ 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 â¦ H1) -| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2 - elim (snv_inv_bind â¦ H1) -H1 #HV1 #HT1 - elim (cpx_inv_bind1 â¦ 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 â¦ H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1 - elim (cpx_inv_appl1 â¦ H2) -H2 * - [ #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH1 â¦ HV12 â¦ HL12) /2 width=1 by fqup_fpbg/ #HV2 - lapply (IH1 â¦ HT12 â¦ HL12) /2 width=1 by fqup_fpbg/ #HT2 -(* lapply (IH2 â¦ Hl0 â¦ HV12 â¦ HL12) /2 width=1 by fqup_fpbg/ #H2l0 - elim (sta_cpr_lpr_aux â¦ IH3 â¦ Hl0 â¦ HVW1 â¦ HV12 â¦ HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fqup_fpbg/ -HV1 #W2 #HVW2 #HW12 - elim (cpds_cpr_lpr_aux â¦ IH2 IH3 â¦ HTU1 â¦ HT12 â¦ HL12) /2 width=1 by fqup_fpbg/ -HT12 -HTU1 #X #HTU2 #H - elim (cprs_inv_abst1 â¦ H) -H #W20 #U2 #HW120 #_ #H destruct - lapply (lpr_cprs_conf â¦ HL12 â¦ HW10) -L1 #HW10 - lapply (cpcs_cprs_strap1 â¦ HW10 â¦ HW120) -W1 #HW120 - lapply (cpcs_canc_sn â¦ HW12 HW120) -W10 #HW20 - elim (cpcs_inv_cprs â¦ HW20) -HW20 #W0 #HW20 #HW200 - lapply (cpds_cprs_trans â¦ (â{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 â¦ HT1) -HT1 #HW20 #HT20 - elim (cpds_inv_abst1 â¦ HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30 - lapply (cprs_div â¦ HW10 â¦ HW230) -W30 #HW120 - lapply (snv_sta_aux â¦ IH1 â¦ Hl0 â¦ HVW1) /2 width=1 by fqup_fpbg/ #HW10 - lapply (da_sta_conf â¦ HVW1 â¦ Hl0) â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â - (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â - âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â 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 â¦ H2) -X2 - >(cpr_inv_sort1 â¦ H3) -X3 /2 width=3 by ex2_intro/ -| #i #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus â¦ l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3 - [ lapply (lstas_inv_O â¦ H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ] - elim (snv_inv_lref â¦ H1) -H1 #I0 #K0 #X0 #HK0 #HX0 - elim (da_inv_lref â¦ Hl1) -Hl1 * #K1 [ #V1 | #W1 #l0 ] #HLK1 [ #HVl1 | #HWl1 #H destruct ] - lapply (ldrop_mono â¦ HK0 â¦ HLK1) -HK0 #H destruct - elim (lstas_inv_lref1 â¦ H2) -H2 * #K0 #V0 #W0 [2,4: #X0 ] #HK0 [1,2: #_ -X0 ] #HVW0 #HX2 - lapply (ldrop_mono â¦ HK0 â¦ HLK1) -HK0 #H destruct - [ lapply (le_plus_to_le_r â¦ Hl21) -Hl21 #Hl21 ] - lapply (fqup_lref â¦ G1 â¦ HLK1) #HKV1 - elim (lpr_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 - elim (lpr_inv_pair1 â¦ H) -H #K2 [ #W2 | #V2 ] #HK12 [ #HW12 | #HV12 ] #H destruct - lapply (ldrop_fwd_drop2 â¦ HLK2) #H2 - elim (cpr_inv_lref1 â¦ H3) -H3 - [1,3: #H destruct -HLK1 - |2,4: * #K0 #V0 #X0 #H #HVX0 #HX0 - lapply (ldrop_mono â¦ H â¦ HLK1) -H -HLK1 #H destruct - ] - [ lapply (IH2 â¦ HWl1 â¦ HW12 â¦ HK12) /2 width=1 by fqup_fpbg/ -IH2 #H - elim (da_inv_sta â¦ H) -H - elim (IH1 â¦ HWl1 â¦ HVW0 â¦ HW12 â¦ 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 â¦ HVl1 â¦ HVW0 â¦ HV12 â¦ 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 â¦ HVl1 â¦ HVW0 â¦ HVX0 â¦ 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 â¦ 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 â¦ H1) -H1 #_ #HT1 - lapply (da_inv_bind â¦ Hl1) -Hl1 #Hl1 - elim (lstas_inv_bind1 â¦ H2) -H2 #U1 #HTU1 #H destruct - elim (cpr_inv_bind1 â¦ H3) -H3 * - [ #V2 #T2 #HV12 #HT12 #H destruct - elim (IH1 â¦ Hl1 â¦ HTU1 â¦ HT12 (L2.â{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 â¦ Hl1 â¦ HTU1 â¦ HT13 (L2.âV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13 - elim (lstas_inv_lift1 â¦ HTU3 L2 â¦ 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 â¦ H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU10 - lapply (da_inv_flat â¦ Hl1) -Hl1 #Hl1 - elim (lstas_inv_appl1 â¦ H2) -H2 #U1 #HTU1 #H destruct - elim (cpr_inv_appl1 â¦ H3) -H3 * - [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2 - elim (IH1 â¦ Hl1 â¦ HTU1 â¦ HT12 â¦ 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 â¦ HT1) -HT1 #HW2 #HT2 - lapply (da_inv_bind â¦ Hl1) -Hl1 #Hl1 - elim (lstas_inv_bind1 â¦ HTU1) -HTU1 #U2 #HTU2 #H destruct - elim (cpds_inv_abst1 â¦ HTU10) -HTU10 #W0 #U0 #HW20 #_ #H destruct - lapply (cprs_div â¦ HW10 â¦ HW20) -W0 #HW12 - lapply (da_sta_conf â¦ HVW1 â¦ Hl0) iter_SO // -qed-. - -lemma lstas_inv_gref1: âh,G,L,X,p,l. â¦G, Lâ¦ â¢ Â§p â¢*[h, l+1] X â â¥. -#h #G #L #X #p #l #H elim (lstas_inv_step_sn â¦ H) -H -#U #H #HUX elim (sta_inv_gref1 â¦ H) -qed-. - -lemma lstas_inv_bind1: âh,a,I,G,L,V,T,X,l. â¦G, Lâ¦ â¢ â{a,I}V.T â¢*[h, l] X â - ââU. â¦G, L.â{I}Vâ¦ â¢ T â¢*[h, l] U & X = â{a,I}V.U. -#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx â¦ H) -X -l /2 width=3 by ex2_intro/ -#l #X #X0 #_ #HX0 * #U #HTU #H destruct -elim (sta_inv_bind1 â¦ HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/ -qed-. - -lemma lstas_inv_appl1: âh,G,L,V,T,X,l. â¦G, Lâ¦ â¢ âV.T â¢*[h, l] X â - ââU. â¦G, Lâ¦ â¢ T â¢*[h, l] U & X = âV.U. -#h #G #L #V #T #X #l #H @(lstas_ind_dx â¦ H) -X -l /2 width=3 by ex2_intro/ -#l #X #X0 #_ #HX0 * #U #HTU #H destruct -elim (sta_inv_appl1 â¦ HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/ -qed-. - -lemma lstas_inv_cast1: âh,G,L,W,T,U,l. â¦G, Lâ¦ â¢ âW.T â¢*[h, l+1] U â â¦G, Lâ¦ â¢ T â¢*[h, l+1] U. -#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn â¦ H) -H -#U #H #HUX lapply (sta_inv_cast1 â¦ H) -H /2 width=3 by lstar_S/ -qed-. - -(* Basic properties *********************************************************) - -lemma lstas_refl: âh,G,L. reflexive â¦ (lstas h G L 0). -// qed. - -lemma sta_lstas: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â â¦G, Lâ¦ â¢ T â¢*[h, 1] U. -/2 width=1 by lstar_step/ qed. - -lemma lstas_step_sn: âh,G,L,T1,U1,U2,l. â¦G, Lâ¦ â¢ T1 â¢[h] U1 â â¦G, Lâ¦ â¢ U1 â¢*[h, l] U2 â - â¦G, Lâ¦ â¢ T1 â¢*[h, l+1] U2. -/2 width=3 by lstar_S/ qed. - -lemma lstas_step_dx: âh,G,L,T1,T2,U2,l. â¦G, Lâ¦ â¢ T1 â¢*[h, l] T2 â â¦G, Lâ¦ â¢ T2 â¢[h] U2 â - â¦G, Lâ¦ â¢ T1 â¢*[h, l+1] U2. -/2 width=3 by lstar_dx/ qed. - -lemma lstas_split: âh,G,L. inv_ltransitive â¦ (lstas h G L). -/2 width=1 by lstar_inv_ltransitive/ qed-. - -lemma lstas_sort: âh,G,L,l,k. â¦G, Lâ¦ â¢ âk â¢*[h, l] â((next h)^l k). -#h #G #L #l @(nat_ind_plus â¦ l) -l // -#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/ -qed. - -lemma lstas_bind: âh,I,G,L,V,T,U,l. â¦G, L.â{I}Vâ¦ â¢ T â¢*[h, l] U â - âa. â¦G, Lâ¦ â¢ â{a,I}V.T â¢*[h, l] â{a,I}V.U. -#h #I #G #L #V #T #U #l #H @(lstas_ind_dx â¦ H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/ -qed. - -lemma lstas_appl: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l] U â - âV.â¦G, Lâ¦ â¢ âV.T â¢*[h, l] âV.U. -#h #G #L #T #U #l #H @(lstas_ind_dx â¦ H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/ -qed. - -lemma lstas_cast: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l+1] U â - âW. â¦G, Lâ¦ â¢ âW.T â¢*[h, l+1] U. -#h #G #L #T #U #l #H elim (lstas_inv_step_sn â¦ 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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_aaa.etc deleted file mode 100644 index 7497f44b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_aaa.etc +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,G,L,l. Conf3 â¦ (aaa G L) (lstas h G L l). -/3 width=6 by sta_aaa_conf, lstar_Conf3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_alt.etc deleted file mode 100644 index 0ad7b9bbc..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_alt.etc +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â relation4 lenv nat term term â -| lstasa_O : âG,L,T. lstasa h G L 0 T T -| lstasa_sort: âG,L,l,k. lstasa h G L l (âk) (â((next h)^l k)) -| lstasa_ldef: âG,L,K,V,W,U,i,l. â©[i] L â¡ K.âV â lstasa h G K (l+1) V W â - â§[0, i+1] W â¡ U â lstasa h G L (l+1) (#i) U -| lstasa_ldec: âG,L,K,W,V,V0,U,i,l. â©[i] L â¡ K.âW â â¦G, Kâ¦ â¢ W â¢[h] V0 â - lstasa h G K l W V â â§[0, i+1] V â¡ U â lstasa h G L (l+1) (#i) U -| lstasa_bind: âa,I,G,L,V,T,U,l. lstasa h G (L.â{I}V) l T U â - lstasa h G L l (â{a,I}V.T) (â{a,I}V.U) -| lstasa_appl: âG,L,V,T,U,l. lstasa h G L l T U â lstasa h G L l (âV.T) (âV.U) -| lstasa_cast: âG,L,W,T,U,l. lstasa h G L (l+1) T U â lstasa h G L (l+1) (âW.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: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â â¦G, Lâ¦ â¢ T â¢â¢*[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: âh,G,L,T1,T,l. â¦G, Lâ¦ â¢ T1 â¢â¢*[h, l] T â - âT2. â¦G, Lâ¦ â¢ T â¢[h] T2 â â¦G, Lâ¦ â¢ T1 â¢â¢*[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 â¦ H) -X >commutative_plus // -| #G #L #K #V #W #U #i #l #HLK #_ #HWU #IHVW #U2 #HU2 - lapply (drop_fwd_drop2 â¦ HLK) #H - elim (sta_inv_lift1 â¦ HU2 â¦ H â¦ 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 â¦ HLK) #H - elim (sta_inv_lift1 â¦ HU2 â¦ H â¦ HVU) -H -U /3 width=8 by lstasa_ldec/ -| #a #I #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H - elim (sta_inv_bind1 â¦ 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 â¦ H) -H #U #HU1 #H destruct /3 width=1 by lstasa_appl/ -| /3 width=1 by lstasa_cast/ -] -qed. - -(* Main properties **********************************************************) - -theorem lstas_lstasa: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l] U â â¦G, Lâ¦ â¢ T â¢â¢*[h, l] U. -#h #G #L #T #U #l #H @(lstas_ind_dx â¦ H) -U -l /2 width=3 by lstasa_step_dx, lstasa_O/ -qed. - -(* Main inversion lemmas ****************************************************) - -theorem lstasa_inv_lstas: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢â¢*[h, l] U â â¦G, Lâ¦ â¢ T â¢*[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: âh. âR:genvârelation4 lenv nat term term. - (âG,L,T. R G L O T T) â - (âG,L,l,k. R G L l (âk) (â((next h)^l k))) â ( - âG,L,K,V,W,U,i,l. - â©[i] L â¡ K.âV â â¦G, Kâ¦ â¢ V â¢*[h, l+1] W â â§[O, i+1] W â¡ U â - R G K (l+1) V W â R G L (l+1) (#i) U - ) â ( - âG,L,K,W,V,V0,U,i,l. - â©[i] L â¡ K.âW â â¦G, Kâ¦ â¢ W â¢[h] V0 â - â¦G, Kâ¦ â¢ W â¢*[h, l]V â â§[O, i+1] V â¡ U â - R G K l W V â R G L (l+1) (#i) U - ) â ( - âa,I,G,L,V,T,U,l. â¦G, L.â{I}Vâ¦ â¢ T â¢*[h, l] U â - R G (L.â{I}V) l T U â R G L l (â{a,I}V.T) (â{a,I}V.U) - ) â ( - âG,L,V,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l] U â - R G L l T U â R G L l (âV.T) (âV.U) - ) â ( - âG,L,W,T,U,l. â¦G, Lâ¦â¢ T â¢*[h, l+1] U â - R G L (l+1) T U â R G L (l+1) (âW.T) U - ) â - âG,L,l,T,U. â¦G, Lâ¦ â¢ T â¢*[h, l] U â R G L l T U. -#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #G #L #l #T #U #H -elim (lstas_lstasa â¦ H) /3 width=10 by lstasa_inv_lstas/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_da.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_da.etc deleted file mode 100644 index 8b11d1dc0..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_da.etc +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,G,L,T,U,l1. â¦G, Lâ¦ â¢ T â¢*[h, l1] U â - âl2. â¦G, Lâ¦ â¢ T âª[h, g] l2 â â¦G, Lâ¦ â¢ U âª[h, g] l2-l1. -#h #g #G #L #T #U #l1 #H @(lstas_ind_dx â¦ H) -U -l1 // -#l1 #U #U0 #_ #HU0 #IHTU #l2 #HT -(plus_minus_m_m â¦ Hl12) in â¢ (%â?); -Hl12 >commutative_plus #H -elim (lstas_split â¦ H) -H #U #HTU ->(lstas_mono â¦ HTU â¦ HTU1) -T // -qed-. - -(* Advanced properties ******************************************************) - -lemma lstas_sta_conf_pos: âh,G,L,T,U1. â¦G, Lâ¦ â¢ T â¢[h] U1 â - âU2,l. â¦G, Lâ¦ â¢ T â¢*[h, l+1] U2 â â¦G, Lâ¦ â¢ U1 â¢*[h, l] U2. -#h #G #L #T #U1 #HTU1 #U2 #l #HTU2 -lapply (lstas_conf_le â¦ T U1 1 â¦ HTU2) -HTU2 /2 width=1 by sta_lstas/ -qed-. - -lemma lstas_strip_pos: âh,G,L,T1,U1. â¦G, Lâ¦ â¢ T1 â¢[h] U1 â - âT2,l. â¦G, Lâ¦ â¢ T1 â¢*[h, l+1] T2 â - ââU2. â¦G, Lâ¦ â¢ T2 â¢[h] U2 & â¦G, Lâ¦ â¢ U1 â¢*[h, l+1] U2. -#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12 -elim (lstas_fwd_correct â¦ HTU1 â¦ HT12) -lapply (lstas_sta_conf_pos â¦ HTU1 â¦ HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta.etc deleted file mode 100644 index fc6bf6a2a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta.etc +++ /dev/null @@ -1,143 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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 â -| sta_sort: âG,L,k. sta h G L (âk) (â(next h k)) -| sta_ldef: âG,L,K,V,W,U,i. â©[i] L â¡ K.âV â sta h G K V W â - â§[0, i + 1] W â¡ U â sta h G L (#i) U -| sta_ldec: âG,L,K,W,V,U,i. â©[i] L â¡ K.âW â sta h G K W V â - â§[0, i + 1] W â¡ U â sta h G L (#i) U -| sta_bind: âa,I,G,L,V,T,U. sta h G (L.â{I}V) T U â - sta h G L (â{a,I}V.T) (â{a,I}V.U) -| sta_appl: âG,L,V,T,U. sta h G L T U â sta h G L (âV.T) (âV.U) -| sta_cast: âG,L,W,T,U. sta h G L T U â sta h G L (âW.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: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âk0. T = âk0 â - U = â(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: âh,G,L,U,k. â¦G, Lâ¦ â¢ âk â¢[h] U â U = â(next h k). -/2 width=5 by sta_inv_sort1_aux/ qed-. - -fact sta_inv_lref1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âj. T = #j â - (ââK,V,W. â©[j] L â¡ K.âV & â¦G, Kâ¦ â¢ V â¢[h] W & - â§[0, j+1] W â¡ U - ) â¨ - (ââK,W,V. â©[j] L â¡ K.âW & â¦G, Kâ¦ â¢ W â¢[h] V & - â§[0, j+1] W â¡ 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: âh,G,L,U,i. â¦G, Lâ¦ â¢ #i â¢[h] U â - (ââK,V,W. â©[i] L â¡ K.âV & â¦G, Kâ¦ â¢ V â¢[h] W & - â§[0, i+1] W â¡ U - ) â¨ - (ââK,W,V. â©[i] L â¡ K.âW & â¦G, Kâ¦ â¢ W â¢[h] V & - â§[0, i+1] W â¡ U - ). -/2 width=3 by sta_inv_lref1_aux/ qed-. - -fact sta_inv_gref1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âp0. T = Â§p0 â â¥. -#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: âh,G,L,U,p. â¦G, Lâ¦ â¢ Â§p â¢[h] U â â¥. -/2 width=8 by sta_inv_gref1_aux/ qed-. - -fact sta_inv_bind1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âb,J,X,Y. T = â{b,J}Y.X â - ââZ. â¦G, L.â{J}Yâ¦ â¢ X â¢[h] Z & U = â{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: âh,b,J,G,L,Y,X,U. â¦G, Lâ¦ â¢ â{b,J}Y.X â¢[h] U â - ââZ. â¦G, L.â{J}Yâ¦ â¢ X â¢[h] Z & U = â{b,J}Y.Z. -/2 width=3 by sta_inv_bind1_aux/ qed-. - -fact sta_inv_appl1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âX,Y. T = âY.X â - ââZ. â¦G, Lâ¦ â¢ X â¢[h] Z & U = âY.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: âh,G,L,Y,X,U. â¦G, Lâ¦ â¢ âY.X â¢[h] U â - ââZ. â¦G, Lâ¦ â¢ X â¢[h] Z & U = âY.Z. -/2 width=3 by sta_inv_appl1_aux/ qed-. - -fact sta_inv_cast1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âX,Y. T = âY.X â - â¦G, Lâ¦ â¢ X â¢[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: âh,G,L,X,Y,U. â¦G, Lâ¦ â¢ âY.X â¢[h] U â â¦G, Lâ¦ â¢ X â¢[h] U. -/2 width=4 by sta_inv_cast1_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_aaa.etc deleted file mode 100644 index affd2eded..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_aaa.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,G,L,T,A. â¦G, Lâ¦ â¢ T â A â âU. â¦G, Lâ¦ â¢ T â¢[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: âh,G,L. Conf3 â¦ (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 â¦ H) -H #H destruct // -| #I #G #L #K #V #B #i #HLK #HV #IHV #U #H - elim (sta_inv_lref1 â¦ H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HU - lapply (drop_mono â¦ HLK0 â¦ HLK) -HLK0 #H0 destruct - lapply (drop_fwd_drop2 â¦ HLK) -HLK #HLK - @(aaa_lift â¦ HLK â¦ HU) -HU -L /2 width=2 by/ -| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H - elim (sta_inv_bind1 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H /2 width=2 by/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_lift.etc deleted file mode 100644 index 4d342878f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_lift.etc +++ /dev/null @@ -1,113 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,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 â¦ H1) -X1 - >(lift_inv_sort1 â¦ 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 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HW1 â¦ HWU2) -W // #W2 #HW12 #HWU2 - elim (drop_trans_le â¦ HL21 â¦ HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H - elim (drop_inv_skip2 â¦ 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 â¦ HW1 â¦ HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2 - lapply (drop_trans_ge â¦ HL21 â¦ 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 â¦ H) * #Hid #H destruct - [ elim (lift_trans_ge â¦ HW1 â¦ HWU2) -W // (lift_inv_sort2 â¦ 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 â¦ H) * #Hid #H destruct [ -HVW2 | -IHVW2 ] - [ elim (drop_conf_lt â¦ HL21 â¦ HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 - elim (IHVW2 â¦ HK21 â¦ HV12) -K2 -V2 #W1 #HW12 #HVW1 - elim (lift_trans_le â¦ HW12 â¦ HW2) -W2 // >minus_plus 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 â¦ H) * #Hid #H destruct [ -HWV2 | -IHWV2 ] - [ elim (drop_conf_lt â¦ HL21 â¦ HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12 - elim (IHWV2 â¦ HK21 â¦ HW12) -K2 #V1 #_ #HWV1 - elim (lift_trans_le â¦ HW12 â¦ HW2) -W2 // >minus_plus 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 â¦ H) -H #V1 #T1 #HV12 #HT12 #H destruct - elim (IHTU2 (L1.â{I}V1) â¦ 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 â¦ H) -H #V1 #T1 #HV12 #HT12 #H destruct - elim (IHTU2 â¦ HL21 â¦ 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 â¦ H) -H #W1 #T1 #_ #HT12 #H destruct - elim (IHTU2 â¦ HL21 â¦ HT12) -L2 -HT12 /3 width=3 by sta_cast, ex2_intro/ -] -qed-. - -(* Advanced forward lemmas **************************************************) - -(* Basic_1: was: sty0_correct *) -lemma sta_fwd_correct: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âT0. â¦G, Lâ¦ â¢ U â¢[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 â¦ 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 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_llpx_sn.etc deleted file mode 100644 index 3adf28fc4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_llpx_sn.etc +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âR. (âL. reflexive â¦ (R L)) â l_liftable R â - âh,G. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H - #Kd #V1d #HLKd #HV1s #HV1sd - lapply (drop_fwd_drop2 â¦ HLKs) -HLKs #HLKs - lapply (drop_fwd_drop2 â¦ HLKd) -HLKd #HLKd - @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ H â¦ HLKs) // -H - #Kd #V1d #HLKd #HV1s #HV1sd - lapply (drop_fwd_drop2 â¦ HLKs) -HLKs #HLKs - lapply (drop_fwd_drop2 â¦ HLKd) -HLKd #HLKd - @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ H) -H - /3 width=1 by llpx_sn_flat/ -| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat â¦ H) -H - /3 width=1 by llpx_sn_flat/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_sta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_sta.etc deleted file mode 100644 index 6d54d56d7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/sta_sta.etc +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,G,L. singlevalued â¦ (sta h G L). -#h #G #L #T #U1 #H elim H -G -L -T -U1 -[ #G #L #k #X #H >(sta_inv_sort1 â¦ H) -X // -| #G #L #K #V #W #U1 #i #HLK #_ #HWU1 #IHVW #U2 #H - elim (sta_inv_lref1 â¦ H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HW0U2 - lapply (drop_mono â¦ HLK0 â¦ HLK) -HLK -HLK0 #H destruct - lapply (IHVW â¦ HVW0) -IHVW -HVW0 #H destruct - >(lift_mono â¦ HWU1 â¦ HW0U2) -W0 -U1 // -| #G #L #K #W #V #U1 #i #HLK #_ #HWU1 #IHWV #U2 #H - elim (sta_inv_lref1 â¦ H) -H * #K0 #W0 #V0 #HLK0 #HWV0 #HV0U2 - lapply (drop_mono â¦ HLK0 â¦ HLK) -HLK -HLK0 #H destruct - lapply (IHWV â¦ HWV0) -IHWV -HWV0 #H destruct - >(lift_mono â¦ HWU1 â¦ HV0U2) -W -U1 // -| #a #I #G #L #V #T #U1 #_ #IHTU1 #X #H - elim (sta_inv_bind1 â¦ H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ -| #G #L #V #T #U1 #_ #IHTU1 #X #H - elim (sta_inv_appl1 â¦ H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ -| #G #L #W #T #U1 #_ #IHTU1 #U2 #H - lapply (sta_inv_cast1 â¦ H) -H /2 width=1 by/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/statictype_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/statictype_5.etc deleted file mode 100644 index 9249be4e0..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/statictype_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 â¢ break [ term 46 h ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'StaticType $h $G $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/statictypestaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/statictypestaralt_6.etc deleted file mode 100644 index a6e46be62..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/statictypestaralt_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************************) - -notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 â¢ â¢ * 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/chnf.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/chnf.etc deleted file mode 100644 index f6106d38d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/chnf.etc +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break ðð â¦ term 46 T â¦ )" - 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 â predicate term â Î»L. NF â¦ (cpr L) tshf. - -interpretation - "context-sensitive head normality (term)" - 'HdNormal L T = (chnf L T). - -(* Basic inversion lemmas ***************************************************) - -lemma chnf_inv_tshf: âL,T. L â¢ ððâ¦Tâ¦ â T â T. -normalize /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -lemma tshf_thnf: âT. T â T â â â¢ ððâ¦Tâ¦. -#T #HT #T2 #H elim (cpr_fwd_tshf1 â¦ H) -H // -#H elim H // -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/cpr_tshf.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/cpr_tshf.etc deleted file mode 100644 index 0fc5e1d0c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/cpr_tshf.etc +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âL,T1,T2. L â¢ T1 â¡ T2 â T1 â T1 â - T2 â T1 â¨ (L = â â â¥). -#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 â¦ HLK) -HLK #H destruct -| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H - elim (tshf_inv_bind1 â¦ H) -H #W2 #U2 #H1 * #H2 destruct /2 width=1/ -| #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H - elim (tshf_inv_flat1 â¦ H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct - lapply (IHT12 HT1U2) -IHT12 -HT1U2 * #HUT2 /3 width=1/ - lapply (simple_tshf_repl_sn â¦ HUT2 HT1) /3 width=1/ -| #L #V #T #T1 #T2 #_ #_ #_ #H - elim (tshf_inv_bind1 â¦ H) -H #W2 #U2 #H1 * #H2 destruct -| #L #V #T1 #T2 #_ #_ #H - elim (tshf_inv_flat1 â¦ H) -H #W2 #U2 #_ #_ #_ #H destruct -| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H - elim (tshf_inv_flat1 â¦ H) -H #W2 #U2 #_ #H - elim (simple_inv_bind â¦ H) -| #a #L #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H - elim (tshf_inv_flat1 â¦ H) -H #U1 #U2 #_ #H - elim (simple_inv_bind â¦ H) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/tshf.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/tshf.etc deleted file mode 100644 index 583c3f46f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/tshf/tshf.etc +++ /dev/null @@ -1,98 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 â¢ break term 46 T1 â break term 46 T2 )" - non associative with precedence 45 - for @{ 'Hom $L $T1 $T2 }. - -notation "hvbox( L â¢ break ðð â¦ term 46 T â¦ )" - non associative with precedence 45 - for @{ 'HdReducible $L $T }. - -notation "hvbox( L â¢ break ðð â¦ term 46 T â¦ )" - non associative with precedence 45 - for @{ 'NotHdReducible $L $T }. - -include "basic_2/grammar/term_simple.ma". - -(* SAME HEAD TERM FORMS *****************************************************) - -inductive tshf: relation term â - | tshf_atom: âI. tshf (âª{I}) (âª{I}) - | tshf_abbr: âV1,V2,T1,T2. tshf (-âV1. T1) (-âV2. T2) - | tshf_abst: âa,V1,V2,T1,T2. tshf (â{a}V1. T1) (â{a}V2. T2) - | tshf_appl: âV1,V2,T1,T2. tshf T1 T2 â ðâ¦T1â¦ â ðâ¦T2â¦ â - tshf (âV1. T1) (âV2. T2) -. - -interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2). - -(* Basic properties *********************************************************) - -lemma tshf_sym: âT1,T2. T1 â T2 â T2 â T1. -#T1 #T2 #H elim H -T1 -T2 /2 width=1/ -qed. - -lemma tshf_refl2: âT1,T2. T1 â T2 â T2 â T2. -#T1 #T2 #H elim H -T1 -T2 // /2 width=1/ -qed. - -lemma tshf_refl1: âT1,T2. T1 â T2 â T1 â T1. -/3 width=2/ qed. - -lemma simple_tshf_repl_dx: âT1,T2. T1 â T2 â ðâ¦T1â¦ â ðâ¦T2â¦. -#T1 #T2 #H elim H -T1 -T2 // -[ #V1 #V2 #T1 #T2 #H - elim (simple_inv_bind â¦ H) -| #a #V1 #V2 #T1 #T2 #H - elim (simple_inv_bind â¦ H) -] -qed. (**) (* remove from index *) - -lemma simple_tshf_repl_sn: âT1,T2. T1 â T2 â ðâ¦T2â¦ â ðâ¦T1â¦. -/3 width=3/ qed-. - -(* Basic inversion lemmas ***************************************************) - -fact tshf_inv_bind1_aux: âT1,T2. T1 â T2 â âa,I,W1,U1. T1 = â{a,I}W1.U1 â - ââW2,U2. T2 = â{a,I}W2. U2 & - (Bind2 a I = Bind2 false Abbr â¨ 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: âa,I,W1,U1,T2. â{a,I}W1.U1 â T2 â - ââW2,U2. T2 = â{a,I}W2. U2 & - (Bind2 a I = Bind2 false Abbr â¨ I = Abst). -/2 width=5/ qed-. - -fact tshf_inv_flat1_aux: âT1,T2. T1 â T2 â âI,W1,U1. T1 = â{I}W1.U1 â - ââW2,U2. U1 â U2 & ðâ¦U1â¦ & ðâ¦U2â¦ & - I = Appl & T2 = âW2. 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: âI,W1,U1,T2. â{I}W1.U1 â T2 â - ââW2,U2. U1 â U2 & ðâ¦U1â¦ & ðâ¦U2â¦ & - I = Appl & T2 = âW2. U2. -/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_aaa.etc deleted file mode 100644 index 98fe01347..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_aaa.etc +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L,T,A. L â¢ T â A â âU. â¦h, Lâ¦ â¢ T â¢â¡[g] U â L â¢ U â A. -#h #g #L #T #A #HT #U * /2 width=3/ /2 width=6/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_lift.etc deleted file mode 100644 index 8f5deca51..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_lift.etc +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,a,L,V1,T1,U2. â¦h, Lâ¦ â¢ â{a}V1.T1 â¢â¡[g] U2 â - ââV2,T2. L â¢ V1 â¡ V2 & â¦h, L. âV1â¦ â¢ T1 â¢â¡[g] T2 & - U2 = â{a}V2. T2. -#h #g #a #L #V1 #T1 #U2 * -[ #H elim (cpr_inv_abst1 â¦ H Abst V1) /3 width=5/ -| #l #H elim (ssta_inv_bind1 â¦ H) /3 width=5/ -] -qed-. - -(* Relocation properties ****************************************************) - -lemma xpr_lift: âL,K,d,e. â©[d, e] L â¡ K â - âT1,U1. â§[d, e] T1 â¡ U1 â âT2,U2. â§[d, e] T2 â¡ U2 â - âh,g. â¦h, Kâ¦ â¢ T1 â¢â¡[g] T2 â â¦h, Lâ¦ â¢ U1 â¢â¡[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: âL,K,d,e. â©[d, e] L â¡ K â - âT1,U1. â§[d, e] T1 â¡ U1 â âh,g,U2. â¦h, Lâ¦ â¢ U1 â¢â¡[g] U2 â - ââT2. â§[d, e] T2 â¡ U2 & â¦h, Kâ¦ â¢ T1 â¢â¡[g] T2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 * [ #HU12 | #l #HU12 ] -[ elim (cpr_inv_lift1 â¦ HLK â¦ HTU1 â¦ HU12) -L -U1 /3 width=3/ -| elim (ssta_inv_lift1 â¦ HU12 â¦ HLK â¦ HTU1) -L -U1 /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_lsubss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_lsubss.etc deleted file mode 100644 index bc66cd2cd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xpr_lsubss.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,L2. h â¢ L1 â¢â[g] L2 â - âT1,T2. â¦h, L2â¦ â¢ T1 â¢â¡[g] T2 â â¦h, L1â¦ â¢ T1 â¢â¡[g] T2. -#h #g #L1 #L2 #HL12 #T1 #T2 * [ | #l ] #HT12 -[ lapply (lsubss_fwd_lsubs2 â¦ HL12) -HL12 /3 width=3/ -| /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs.etc deleted file mode 100644 index bffc4e735..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs.etc +++ /dev/null @@ -1,72 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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( â¦ term 46 h , break term 46 L â¦ â¢ break term 46 T1 â¢ â¡ * break [ term 46 g ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'XPRedStar $h $g $L $T1 $T2 }. - -notation "hvbox( â¦ term 46 h , break term 46 L â¦ â¢ â¢ â¬ * 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: âh. sd h â lenv â relation term â - Î»h,g,L. TC â¦ (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: âh,g,L,T1. âR:predicate term. R T1 â - (âT,T2. â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T â â¦h, Lâ¦ â¢ T â¢â¡[g] T2 â R T â R T2) â - âT2. â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2 â R T2. -#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // -qed-. - -lemma xprs_ind_dx: âh,g,L,T2. âR:predicate term. R T2 â - (âT1,T. â¦h, Lâ¦ â¢ T1 â¢â¡[g] T â â¦h, Lâ¦ â¢ T â¢â¡*[g] T2 â R T â R T1) â - âT1. â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2 â R T1. -#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma xprs_refl: âh,g,L. reflexive â¦ (xprs h g L). -/2 width=1/ qed. - -lemma xprs_strap1: âh,g,L,T1,T,T2. - â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T â â¦h, Lâ¦ â¢ T â¢â¡[g] T2 â â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2. -/2 width=3/ qed. - -lemma xprs_strap2: âh,g,L,T1,T,T2. - â¦h, Lâ¦ â¢ T1 â¢â¡[g] T â â¦h, Lâ¦ â¢ T â¢â¡*[g] T2 â â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2. -/2 width=3/ qed. - -(* Basic inversion lemmas ***************************************************) -(* -axiom xprs_inv_cnf1: âL,T,U. L â¢ T â¡* U â L â¢ ðâ¦Tâ¦ â T = U. -#L #T #U #H @(xprs_ind_dx â¦ H) -T // -#T0 #T #H1T0 #_ #IHT #H2T0 -lapply (H2T0 â¦ H1T0) -H1T0 #H destruct /2 width=1/ -qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_aaa.etc deleted file mode 100644 index 5beb8fe19..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_aaa.etc +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L,T,A. L â¢ T â A â âU. â¦h, Lâ¦ â¢ T â¢â¡*[g] U â L â¢ U â A. -#h #g #L #T #A #HT #U #H @(xprs_ind â¦ H) -U // /2 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_cprs.etc deleted file mode 100644 index 13a4f8889..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_cprs.etc +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L,T1,T2. L â¢ T1 â¡* T2 â â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2. -#h #g #L #T1 #T2 #H @(cprs_ind â¦ H) -T2 // /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_lift.etc deleted file mode 100644 index cb151a194..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_lift.etc +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,a,L,V1,T1,U2. â¦h, Lâ¦ â¢ â{a}V1. T1 â¢â¡*[g] U2 â - ââV2,T2. L â¢ V1 â¡* V2 & U2 = â{a}V2. T2. -#h #g #a #L #V1 #T1 #U2 #H @(xprs_ind â¦ H) -U2 /2 width=4/ -#U #U2 #_ #HU2 * #V #T #HV1 #H destruct -elim (xpr_inv_abst1 â¦ HU2) -HU2 #V2 #T2 #HV2 #_ #H destruct /3 width=4/ -qed-. - -(* Relocation properties ****************************************************) - -lemma xprs_lift: âL,K,d,e. â©[d, e] L â¡ K â âT1,U1. â§[d, e] T1 â¡ U1 â - âh,g,T2. â¦h, Kâ¦ â¢ T1 â¢â¡*[g] T2 â âU2. â§[d, e] T2 â¡ U2 â - â¦h, Lâ¦ â¢ U1 â¢â¡*[g] U2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 #HT12 @(xprs_ind â¦ HT12) -T2 -[ -HLK #T2 #HT12 - <(lift_mono â¦ HTU1 â¦ HT12) -T1 // -| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (xpr_lift â¦ HLK â¦ HTU â¦ HTU2 â¦ HT2) -T2 -HLK /3 width=3/ -] -qed. - -lemma xprs_inv_lift1: âL,K,d,e. â©[d, e] L â¡ K â - âT1,U1. â§[d, e] T1 â¡ U1 â âh,g,U2. â¦h, Lâ¦ â¢ U1 â¢â¡*[g] U2 â - ââT2. â§[d, e] T2 â¡ U2 & â¦h, Kâ¦ â¢ T1 â¢â¡*[g] T2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 #HU12 @(xprs_ind â¦ HU12) -U2 /2 width=3/ --HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 -elim (xpr_inv_lift1 â¦ HLK â¦ HTU â¦ HU2) -U -HLK /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_lsubss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_lsubss.etc deleted file mode 100644 index c883c14f3..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_lsubss.etc +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L1,L2. h â¢ L1 â¢â[g] L2 â - âT1,T2. â¦h, L2â¦ â¢ T1 â¢â¡*[g] T2 â â¦h, L1â¦ â¢ T1 â¢â¡*[g] T2. -#h #g #L1 #L2 #HL12 #T1 #T2 #H @(xprs_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 -lapply (lsubss_xpr_trans â¦ HL12 â¦ HT2) -L2 /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_xprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_xprs.etc deleted file mode 100644 index 9593f0550..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/xpr/xprs_xprs.etc +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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: âh,g,L. transitive â¦ (xprs h g L). -/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml index e40771d8e..33d86efda 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml +++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml @@ -26,159 +26,82 @@ - Stage "B" - + + Î»Î´-2B is repackaged without Î»Î´-ground-2. + + Î»Î´-2B is released. - + Applicability condition is now parametrized with a generic subset of numbers. - + Applicability condition parametrized with an initial interval of numbers allows Î»Î´-2B to generalize both Î»Î´-2A and Î»Î´-1B. - + Extended (Î»Î´-2A) and restricted (Î»Î´-1B) validity is decidable (anniversary milestone). - + Preservation of validity for rt-computation does not need the sort degree parameter (i.e. no induction on the degree). - + Extended (Î»Î´-2A) and restricted (Î»Î´-1B) validity rules justified. - + Î»Î´-2A completed with confluence of rt-computation and preservation of validity for rt-computation. - + Behavioral component rt_computation completed. - + "Big tree" theorem (anniversary milestone). - + Support for rt-computation completed. - + Exclusion binder in local environments. Syntactic component updated: syntax, relocation, s_transition, s_computation, static, i_static. - + Strong rt-normalization for simply typed terms (anniversary milestone). - + Behavioral component rt_transition completed. - + Generic candidates of reducibility. - + Confluence for parallel r-transition on referred entries of local environments. - + Confluence for context-sensitive parallel r-transition on terms. - + Syntactic component completed: syntax, relocation, s_transition, s_computation, static (anniversary milestone). - + Relocation with reference transforming maps (rtmap). - + Î»Î´-2B is started. - Stage "A" - - Î»Î´-2A is repackaged (was Î»Î´-2A1). - - - Î»Î´-2A appears too complex and is dismissed. - - - Î»Î´-2A is released. - - - Iterated static type assignment defined (more elegantly) - as a primitive notion. - - - Preservation of stratified native validity - for context-sensitive computation on terms. - - - Strong qrst-normalization - for simply typed terms. - - - Lazy equivalence on local environments - added as q-step to rst-computation on closures - (anniversary milestone). - - - Parametrized slicing on local environments - comprises both versions of this operation - (one from basic_1, the other used in basic_2 till now). - - - Passive support for global environments. - - - Reaxiomatized Î²-reductum as in rt-reduction. - - - Context-sensitive strong rt-normalization - for simply typed terms. - - - Reaxiomatized substitution and reduction - commute with respect to subclosure - (anniversary milestone). - - - Mutual recursive preservation of stratified native validity - for rst-computation on closures. - - - Confluence for context-free parallel reduction on closures. - - - Term binders polarized to control Î¶-reduction (not released). - - - Context-sensitive subject equivalence - for atomic arity assignment - (anniversary milestone). - - - Context-sensitive strong normalization - for simply typed terms. - - - Generic candidates of reducibility. - - - Confluence for context-sensitive parallel reduction on terms. - - - Confluence for context-free parallel reduction on terms. - - - Î»Î´-2A is started. - - Logical Structure of the Specification This table reports the specification's components and their planes. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma index 8d71a75c4..42b5b3621 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma @@ -19,7 +19,7 @@ include "basic_2A/reduction/cnr.ma". (* Basic_1: includes: pr1_pr0 *) definition cprs: relation4 genv lenv term term â - Î»G. LTC â¦ (cpr G). + Î»G. CTC â¦ (cpr G). interpretation "context-sensitive parallel computation (term)" 'PRedStar G L T1 T2 = (cprs G L T1 T2). @@ -60,7 +60,7 @@ lemma cprs_strap2: âG,L,T1,T,T2. normalize /2 width=3 by TC_strap/ qed-. lemma lsubr_cprs_trans: âG. lsub_trans â¦ (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 *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma index 79500576b..c62335d55 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "ground_2/xoa/ex_4_5.ma". include "basic_2A/reduction/lpr_lpr.ma". include "basic_2A/computation/cprs_lift.ma". @@ -125,7 +126,7 @@ lemma cpr_bind2: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡ V2 â âI,T1,T2. â¦G, L. (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *) lemma lpr_cprs_trans: âG. s_rs_transitive â¦ (cpr G) (Î»_. 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 *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma index 173808baa..ab1b48c56 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma @@ -19,7 +19,7 @@ include "basic_2A/computation/cprs.ma". (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************) definition cpxs: âh. sd h â relation4 genv lenv term term â - Î»h,g,G. LTC â¦ (cpx h g G). + Î»h,g,G. CTC â¦ (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). @@ -57,7 +57,7 @@ lemma cpxs_strap2: âh,g,G,L,T1,T. â¦G, Lâ¦ â¢ T1 â¡[h, g] T â normalize /2 width=3 by TC_strap/ qed. lemma lsubr_cpxs_trans: âh,g,G. lsub_trans â¦ (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: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ T1 â¡*[h, g] T2. @@ -145,7 +145,10 @@ lemma cpxs_inv_sort1: âh,g,G,L,U2,k. â¦G, Lâ¦ â¢ âk â¡*[h, g] U2 â elim (cpx_inv_sort1 â¦ HU2) -HU2 [ #H destruct /2 width=4 by ex2_2_intro/ | * #d0 #Hkd0 #H destruct -d - @(ex2_2_intro â¦ (n+1) d0) /2 width=1 by deg_inv_prec/ >iter_SO // + @(ex2_2_intro â¦ (n+1) d0) + [ iter_SO // + ] ] ] qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma index 61002527e..075ecfa55 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma @@ -116,7 +116,7 @@ lemma cpx_bind2: âh,g,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡[h, g] V2 â (* Advanced properties ******************************************************) lemma lpx_cpxs_trans: âh,g,G. s_rs_transitive â¦ (cpx h g G) (Î»_.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: âh,g,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡[h, g] V2 â diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma index 025cfdc2f..b110276dc 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma @@ -20,5 +20,5 @@ include "basic_2A/computation/cpxs.ma". (* Properties on equivalence for local environments *************************) lemma lreq_cpxs_trans: âh,g,G. lsub_trans â¦ (cpxs h g G) (lreq 0 (â)). -/3 width=5 by lreq_cpx_trans, LTC_lsub_trans/ +/3 width=5 by lreq_cpx_trans, CTC_lsub_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma index 6bda54375..fc3b016e6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma @@ -27,7 +27,7 @@ lemma csx_lift: âh,g,G,L2,L1,T1,s,l,m. â¦G, L1â¦ â¢ â¬*[h, g] T1 â @csx_intro #T #HLT2 #HT2 elim (cpx_inv_lift1 â¦ HLT2 â¦ HL21 â¦ HT12) -HLT2 #T0 #HT0 #HLT10 @(IHT1 â¦ HLT10) // -L1 -L2 #H destruct ->(lift_mono â¦ HT0 â¦ HT12) in HT2; -T1 /2 width=1 by/ +>(lift_mono â¦ HT0 â¦ HT12) in HT2; -T0 /2 width=1 by/ qed. (* Basic_1: was just: sn3_gen_lift *) @@ -38,7 +38,7 @@ lemma csx_inv_lift: âh,g,G,L2,L1,T1,s,l,m. â¦G, L1â¦ â¢ â¬*[h, g] T1 â elim (lift_total T l m) #T0 #HT0 lapply (cpx_lift â¦ HLT2 â¦ HL12 â¦ HT21 â¦ HT0) -HLT2 #HLT10 @(IHT1 â¦ HLT10) // -L1 -L2 #H destruct ->(lift_inj â¦ HT0 â¦ HT21) in HT2; -T1 /2 width=1 by/ +>(lift_inj â¦ HT0 â¦ HT21) in HT2; -T0 /2 width=1 by/ qed. (* Advanced inversion lemmas ************************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma index 51345fb0f..bc309b3b1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma @@ -44,7 +44,7 @@ elim (csxv_inv_cons â¦ HVVs) #HV #HVs #X #H #H0 elim (cpxs_fwd_sort_vector â¦ H) -H #H [ elim H0 -H0 // -| -H0 @(csx_cpxs_trans â¦ (â¶(V@Vs).â(next h k))) /2 width=1 by cpxs_flat_dx/ +| -H0 @(csx_cpxs_trans â¦ (â¶(Vâ¨®Vs).â(next h k))) /2 width=1 by cpxs_flat_dx/ ] qed. @@ -93,7 +93,7 @@ lapply (csx_appl_theta â¦ HW12 â¦ H) -H -HW12 #H lapply (csx_fwd_pair_sn â¦ H) #HW1 lapply (csx_fwd_flat_dx â¦ H) #H1 @csx_appl_simple_tsts /2 width=3 by simple_flat/ -IHV12s -HW1 -H1 #X #H1 #H2 -elim (cpxs_fwd_theta_vector â¦ (V2@V2s) â¦ H1) -H1 /2 width=1 by liftv_cons/ -HV12s -HV12 +elim (cpxs_fwd_theta_vector â¦ (V2â¨®V2s) â¦ 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/ ] diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma index b45bc8b5e..43391ecfa 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma @@ -26,7 +26,7 @@ interpretation (* Basic inversion lemmas ***************************************************) -lemma csxv_inv_cons: âh,g,G,L,T,Ts. â¦G, Lâ¦ â¢ â¬*[h, g] T @ Ts â +lemma csxv_inv_cons: âh,g,G,L,T,Ts. â¦G, Lâ¦ â¢ â¬*[h, g] T â¨® Ts â â¦G, Lâ¦ â¢ â¬*[h, g] T â§ â¦G, Lâ¦ â¢ â¬*[h, g] Ts. normalize // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma index 1bbe0d510..7c7d65359 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma @@ -32,7 +32,7 @@ theorem acr_aaa_csubc_lifts: âRR,RS,RP. [ #G #L #k #L0 #cs #HL0 #X #H #L2 #HL20 >(lifts_inv_sort1 â¦ H) -H lapply (acr_gcr â¦ H1RP H2RP (âª)) #HAtom - lapply (s4 â¦ HAtom G L2 (â)) /2 width=1 by/ + lapply (s4 â¦ HAtom G L2 (âº)) /2 width=1 by/ | #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #cs #HL01 #X #H #L2 #HL20 lapply (acr_gcr â¦ H1RP H2RP B) #HB elim (lifts_inv_lref1 â¦ H) -H #i1 #Hi1 #H destruct @@ -45,7 +45,7 @@ theorem acr_aaa_csubc_lifts: âRR,RS,RP. [ #K2 #HK20 #H destruct elim (lift_total V0 0 (i0 +1)) #V #HV0 elim (lifts_lift_trans â¦ Hi0 â¦ Hcs0 â¦ HV10 â¦ HV0) -HV10 #V2 #HV12 #HV2 - lapply (s5 â¦ HB ? G ? ? (â) â¦ HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *) + lapply (s5 â¦ HB ? G ? ? (âº) â¦ 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 â¦ HLK2) #HLK2b @@ -53,15 +53,15 @@ theorem acr_aaa_csubc_lifts: âRR,RS,RP. lapply (aaa_mono â¦ H2KV0A â¦ 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¦ HB ? G ? ? (â) â¦ (âV3.V) â¦ HLK2) /2 width=1 by lift_flat/ - lapply (s7 â¦ HB G L2 (â)) /3 width=7 by gcr_lift/ + lapply (s5 â¦ HB ? G ? ? (âº) â¦ (âV3.V) â¦ HLK2) /2 width=1 by lift_flat/ + lapply (s7 â¦ HB G L2 (âº)) /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 â¦ H) -H #V0 #T0 #HV0 #HT0 #H destruct lapply (acr_gcr â¦ H1RP H2RP A) #HA lapply (acr_gcr â¦ H1RP H2RP B) #HB lapply (s1 â¦ HB) -HB #HB - lapply (s6 â¦ HA G L2 (â) (â)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/ + lapply (s6 â¦ HA G L2 (âº) (âº)) /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 â¦ H) -H #W0 #T0 #HW0 #HT0 #H destruct @(acr_abst â¦ H1RP H2RP) /2 width=5 by/ @@ -76,7 +76,7 @@ theorem acr_aaa_csubc_lifts: âRR,RS,RP. | #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #cs #HL0 #X #H #L2 #HL20 elim (lifts_inv_flat1 â¦ H) -H #V0 #T0 #HV0 #HT0 #H destruct lapply (acr_gcr â¦ H1RP H2RP A) #HA - lapply (s7 â¦ HA G L2 (â)) /3 width=5 by/ + lapply (s7 â¦ HA G L2 (âº)) /3 width=5 by/ ] qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma index a35bc63d2..bb48bab78 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma @@ -14,7 +14,6 @@ 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". @@ -99,24 +98,24 @@ lemma acr_gcr: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â [ #G #L #T #H elim (cp1 â¦ H1RP G L) #k #HK lapply (H L (âk) T (â) ? ? ?) -H // - [ lapply (s2 â¦ IHB G L (â) â¦ HK) // + [ lapply (s2 â¦ IHB G L (âº) â¦ 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 â¦ H) -H #V0s #T0 #HV0s #HT0 #H destruct lapply (s1 â¦ IHB â¦ HB) #HV0 - @(s2 â¦ IHA â¦ (V0 @ V0s)) + @(s2 â¦ IHA â¦ (V0 â¨® 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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct elim (lifts_inv_flat1 â¦ HY) -HY #U0 #X #HU0 #HX #H destruct elim (lifts_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT0 #H destruct - @(s3 â¦ IHA â¦ (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/ + @(s3 â¦ IHA â¦ (V0 â¨® 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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct >(lifts_inv_sort1 â¦ HY) -Y lapply (s1 â¦ IHB â¦ HB) #HV0 - @(s4 â¦ IHA â¦ (V0 @ V0s)) /3 width=7 by gcp2_lifts_all, conj/ + @(s4 â¦ IHA â¦ (V0 â¨® 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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct elim (lifts_inv_lref1 â¦ HY) -HY #i0 #Hi0 #H destruct @@ -126,13 +125,13 @@ lemma acr_gcr: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â elim (lift_total W1 0 (i0 + 1)) #W2 #HW12 elim (lifts_lift_trans â¦ Hcs0 â¦ HVW1 â¦ HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2 >(lift_mono â¦ HV13 â¦ HV12) in HVW2; -V3 #HVW2 - @(s5 â¦ IHA â¦ (V0 @ V0s) â¦ HW12 HL02) /3 width=5 by lifts_applv/ + @(s5 â¦ IHA â¦ (V0 â¨® V0s) â¦ 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 â¦ H) -H #V10s #Y #HV10s #HY #H destruct elim (lifts_inv_bind1 â¦ 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 â¦ IHA â¦ (V10 @ V10s) (V20 @ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/ + @(s6 â¦ IHA â¦ (V10 â¨® V10s) (V20 â¨® V20s)) /3 width=7 by gcp2_lifts, liftv_cons/ @(HA â¦ (cs + 1)) /2 width=2 by drops_skip/ [ @lifts_applv // elim (liftsv_liftv_trans_le â¦ HV10s â¦ HV120s) -V10s #V10s #HV10s #HV120s @@ -142,7 +141,7 @@ lemma acr_gcr: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â | #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB elim (lifts_inv_applv1 â¦ H) -H #V0s #Y #HV0s #HY #H destruct elim (lifts_inv_flat1 â¦ HY) -HY #W0 #T0 #HW0 #HT0 #H destruct - @(s7 â¦ IHA â¦ (V0 @ V0s)) /3 width=5 by lifts_applv/ + @(s7 â¦ IHA â¦ (V0 â¨® V0s)) /3 width=5 by lifts_applv/ ] qed. @@ -157,11 +156,11 @@ lapply (acr_gcr â¦ H1RP H2RP A) #HCA lapply (acr_gcr â¦ H1RP H2RP B) #HCB elim (lifts_inv_bind1 â¦ H) -H #W0 #T0 #HW0 #HT0 #H destruct lapply (gcr_lifts â¦ H1RP â¦ HL0 â¦ HW0 HW) -HW #HW0 -lapply (s3 â¦ HCA â¦ a G L0 (â)) #H @H -H -lapply (s6 â¦ HCA G L0 (â) (â) ?) // #H @H -H +lapply (s3 â¦ HCA â¦ a G L0 (âº)) #H @H -H +lapply (s6 â¦ HCA G L0 (âº) (âº) ?) // #H @H -H [ @(HA â¦ HL0) // | lapply (s1 â¦ HCB) -HCB #HCB - lapply (s7 â¦ H2RP G L0 (â)) /3 width=1 by/ + lapply (s7 â¦ H2RP G L0 (âº)) /3 width=1 by/ ] qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma index c77910134..465a42cd0 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "ground_2/ynat/ynat_minus_sn.ma". include "basic_2A/notation/relations/cosn_5.ma". include "basic_2A/computation/lsx.ma". @@ -21,7 +22,7 @@ inductive lcosx (h) (g) (G): relation2 ynat lenv â | lcosx_sort: âl. lcosx h g G l (â) | lcosx_skip: âI,L,T. lcosx h g G 0 L â lcosx h g G 0 (L.â{I}T) | lcosx_pair: âI,L,T,l. G â¢ â¬*[h, g, T, l] L â - lcosx h g G l L â lcosx h g G (â«¯l) (L.â{I}T) + lcosx h g G l L â lcosx h g G (âl) (L.â{I}T) . interpretation @@ -36,7 +37,7 @@ qed. lemma lcosx_drop_trans_lt: âh,g,G,L,l. G â¢ ~â¬*[h, g, l] L â âI,K,V,i. â¬[i] L â¡ K.â{I}V â i < l â - G â¢ ~â¬*[h, g, â«°(l-i)] K â§ G â¢ â¬*[h, g, V, â«°(l-i)] K. + G â¢ ~â¬*[h, g, â(l-i)] K â§ G â¢ â¬*[h, g, V, â(l-i)] K. #h #g #G #L #l #H elim H -L -l [ #l #J #K #V #i #H elim (drop_inv_atom1 â¦ H) -H #H destruct | #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false â¦ H) -H // @@ -44,7 +45,7 @@ lemma lcosx_drop_trans_lt: âh,g,G,L,l. G â¢ ~â¬*[h, g, l] L â elim (drop_inv_O1_pair1 â¦ H) -H * #Hi #HLK destruct [ >ypred_succ /2 width=1 by conj/ | lapply (ylt_pred â¦ Hil ?) -Hil /2 width=1 by ylt_inj/ >ypred_succ #Hil - elim (IHL â¦ HLK ?) -IHL -HLK yminus_SO2 // + elim (IHL â¦ HLK ?) -IHL -HLK >minus_SO_dx // <(ypred_succ l) in â¢ (%â%â?); >yminus_pred /2 width=1 by ylt_inj, conj/ ] ] @@ -52,23 +53,23 @@ qed-. (* Basic inversion lemmas ***************************************************) -fact lcosx_inv_succ_aux: âh,g,G,L,x. G â¢ ~â¬*[h, g, x] L â âl. x = â«¯l â +fact lcosx_inv_succ_aux: âh,g,G,L,x. G â¢ ~â¬*[h, g, x] L â âl. x = âl â L = â â¨ ââI,K,V. L = K.â{I}V & G â¢ ~â¬*[h, g, l] K & G â¢ â¬*[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 â¦ H) -| #I #L #T #l #HT #HL #x #H <(ysucc_inj â¦ H) -x +| #I #L #T #l #HT #HL #x #H <(ysucc_inv_inj â¦ H) -x /3 width=6 by ex3_3_intro, or_intror/ ] qed-. -lemma lcosx_inv_succ: âh,g,G,L,l. G â¢ ~â¬*[h, g, â«¯l] L â L = â â¨ +lemma lcosx_inv_succ: âh,g,G,L,l. G â¢ ~â¬*[h, g, âl] L â L = â â¨ ââI,K,V. L = K.â{I}V & G â¢ ~â¬*[h, g, l] K & G â¢ â¬*[h, g, V, l] K. /2 width=3 by lcosx_inv_succ_aux/ qed-. -lemma lcosx_inv_pair: âh,g,I,G,L,T,l. G â¢ ~â¬*[h, g, â«¯l] L.â{I}T â +lemma lcosx_inv_pair: âh,g,I,G,L,T,l. G â¢ ~â¬*[h, g, âl] L.â{I}T â G â¢ ~â¬*[h, g, l] L â§ G â¢ â¬*[h, g, T, l] L. #h #g #I #G #L #T #l #H elim (lcosx_inv_succ â¦ H) -H [ #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma index 47de7c8bf..4850a771f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma @@ -12,7 +12,6 @@ (* *) (**************************************************************************) -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". @@ -28,9 +27,11 @@ lemma lsx_cpx_trans_lcosx: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â #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 (â«¯i)) /2 width=1 by ylt_fwd_le_succ/ - elim (lcosx_drop_trans_lt â¦ HL â¦ HLK) // -HL -Hli - lapply (drop_fwd_drop2 â¦ HLK) -HLK /3 width=7 by lsx_lift_ge/ + [ elim (lcosx_drop_trans_lt â¦ HL â¦ HLK) // -HL + (yminus_plus l (âi)) + /4 width=7 by lsx_lift_ge, drop_fwd_drop2/ | lapply (lsx_fwd_lref_be â¦ H â¦ HLK) // -H -Hli lapply (drop_fwd_drop2 â¦ HLK) -HLK /4 width=10 by lsx_ge, lsx_lift_le/ @@ -42,7 +43,7 @@ lemma lsx_cpx_trans_lcosx: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #l #HL #H elim (lsx_inv_flat â¦ H) -H /3 width=1 by lsx_flat/ | #G #L #V #U1 #U2 #T2 #_ #HTU2 #IHU12 #l #HL #H - elim (lsx_inv_bind â¦ H) -H + elim (lsx_inv_bind â¦ 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 â¦ H) -H /2 width=1 by/ @@ -57,7 +58,7 @@ lemma lsx_cpx_trans_lcosx: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â elim (lsx_inv_flat â¦ H) -H #HV1 #H elim (lsx_inv_bind â¦ 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 â¦ (L.âW1)) /3 width=1 by lcosx_pair, lreq_succ/ ] qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma index a96eaa23a..bc5beaf2c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma @@ -49,12 +49,12 @@ lemma lprs_ind_alt: âG. âR:relation lenv. (* Properties on context-sensitive parallel computation for terms ***********) lemma lprs_cpr_trans: âG. s_r_transitive â¦ (cpr G) (Î»_. 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: âG. s_rs_transitive â¦ (cpr G) (Î»_. 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma index c4ba7ccf0..5bef4a5ec 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma @@ -48,11 +48,11 @@ lemma lpxs_ind_alt: âh,g,G. âR:relation lenv. (* Properties on context-sensitive extended parallel computation for terms **) lemma lpxs_cpx_trans: âh,g,G. s_r_transitive â¦ (cpx h g G) (Î»_.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: âh,g,G. s_rs_transitive â¦ (cpx h g G) (Î»_.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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma index a0bb73d7a..07f244d3a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma @@ -12,6 +12,8 @@ (* *) (**************************************************************************) +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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma index a50714b20..c897a9333 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma @@ -45,7 +45,7 @@ qed-. (* Advanced forward lemmas **************************************************) lemma lsx_fwd_bind_dx: âh,g,a,I,G,L,V,T,l. G â¢ â¬*[h, g, â{a,I}V.T, l] L â - G â¢ â¬*[h, g, T, â«¯l] L.â{I}V. + G â¢ â¬*[h, g, T, âl] L.â{I}V. #h #g #a #I #G #L #V1 #T #l #H @(lsx_ind â¦ H) -L #L1 #_ #IHL1 @lsx_intro #Y #H #HT elim (lpx_inv_pair1 â¦ H) -H @@ -59,5 +59,5 @@ qed-. (* Advanced inversion lemmas ************************************************) lemma lsx_inv_bind: âh,g,a,I,G,L,V,T,l. G â¢ â¬*[h, g, â{a, I}V.T, l] L â - G â¢ â¬*[h, g, V, l] L â§ G â¢ â¬*[h, g, T, â«¯l] L.â{I}V. + G â¢ â¬*[h, g, V, l] L â§ G â¢ â¬*[h, g, T, âl] L.â{I}V. /3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma index d77771eec..74f255161 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma @@ -20,7 +20,7 @@ include "basic_2A/computation/lsx_alt.ma". (* Advanced properties ******************************************************) fact lsx_bind_lpxs_aux: âh,g,a,I,G,L1,V,l. G â¢ â¬*[h, g, V, l] L1 â - âY,T. G â¢ â¬*[h, g, T, â«¯l] Y â + âY,T. G â¢ â¬*[h, g, T, âl] Y â âL2. Y = L2.â{I}V â â¦G, L1â¦ â¢ â¡*[h, g] L2 â G â¢ â¬*[h, g, â{a,I}V.T, l] L2. #h #g #a #I #G #L1 #V #l #H @(lsx_ind_alt â¦ H) -L1 @@ -37,7 +37,7 @@ fact lsx_bind_lpxs_aux: âh,g,a,I,G,L1,V,l. G â¢ â¬*[h, g, V, l] L1 â qed-. lemma lsx_bind: âh,g,a,I,G,L,V,l. G â¢ â¬*[h, g, V, l] L â - âT. G â¢ â¬*[h, g, T, â«¯l] L.â{I}V â + âT. G â¢ â¬*[h, g, T, âl] L.â{I}V â G â¢ â¬*[h, g, â{a,I}V.T, l] L. /2 width=3 by lsx_bind_lpxs_aux/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma index f26ee0989..00c805ea2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma index b2fb883ff..3519800c6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma @@ -12,6 +12,8 @@ (* *) (**************************************************************************) +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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma index ffae53ad7..55bae3317 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma @@ -18,7 +18,7 @@ include "basic_2A/conversion/cpc.ma". (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) definition cpcs: relation4 genv lenv term term â - Î»G. LTC â¦ (cpc G). + Î»G. CTC â¦ (cpc G). interpretation "context-sensitive parallel equivalence (term)" 'PConvStar G L T1 T2 = (cpcs G L T1 T2). diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cir_append.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cir_append.etc new file mode 100644 index 000000000..efd097f84 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cir_append.etc @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,T,W. â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, â.âW @@ Lâ¦ â¢ â¡ ðâ¦Tâ¦. +/3 width=2 by crr_inv_labst_last/ qed. + +lemma cir_tif: âG,T,W. â¦G, ââ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, â.âWâ¦ â¢ â¡ ðâ¦Tâ¦. +/3 width=2 by crr_inv_trr/ qed. + +(* Advanced inversion lemmas ************************************************) + +lemma cir_inv_append_sn: âG,L,K,T. â¦G, K @@ Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦. +/3 width=1/ qed-. + +lemma cir_inv_tir: âG,T,W. â¦G, â.âWâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, ââ¦ â¢ â¡ ðâ¦Tâ¦. +/3 width=1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cix_append.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cix_append.etc new file mode 100644 index 000000000..7b5522ebd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cix_append.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L,K,T. â¦G, K @@ Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦. +/3 width=1 by crx_append_sn/ qed-. + +lemma cix_inv_tix: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â â¦G, ââ¦ â¢ â¡[h, g] ðâ¦Tâ¦. +/3 width=1 by trx_crx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cl_shift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cl_shift.etc new file mode 100644 index 000000000..5fdab7148 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cl_shift.etc @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â match L with +[ LAtom â T +| LPair L I V â shift L (-â{I} V. T) +]. + +interpretation "shift (closure)" 'Append L T = (shift L T). + +(* Basic properties *********************************************************) + +lemma shift_append_assoc: âL,K. âT:term. (L @@ K) @@ T = L @@ K @@ T. +#L #K elim K -K // normalize // +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma shift_inj: âL1,L2. âT1,T2:term. L1 @@ T1 = L2 @@ T2 â |L1| = |L2| â + L1 = L2 â§ T1 = T2. +#L1 elim L1 -L1 +[ * normalize /2 width=1/ + #L2 #I2 #V2 #T1 #T2 #_ shift_append_assoc normalize #H + elim (cpr_inv_bind1 â¦ H) -H * + [ #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *) + | #T #_ #_ #H destruct + ] +] +qed-. + diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpx.etc new file mode 100644 index 000000000..5f8bf520c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpx.etc @@ -0,0 +1,25 @@ +lemma cpx_append: âh,g,G. l_appendable_sn â¦ (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 â¦ HK0) #H +@(cpx_delta â¦ I â¦ (L@@K0) V1 â¦ HVW2) // +@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *) +qed. + +lemma cpx_fwd_shift1: âh,g,G,L1,L,T1,T. â¦G, Lâ¦ â¢ L1 @@ T1 â¡[h, g] T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#h #g #G #L1 @(lenv_ind_dx â¦ L1) -L1 normalize +[ #L #T1 #T #HT1 + @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) +| #I #L1 #V1 #IH #L #T1 #X + >shift_append_assoc normalize #H + elim (cpx_inv_bind1 â¦ H) -H * + [ #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *) + | #T #_ #_ #H destruct + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpy2.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpy2.etc new file mode 100644 index 000000000..e941c9653 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpy2.etc @@ -0,0 +1,27 @@ +include "basic_2/grammar/cl_shift.ma". +include "basic_2/relocation/ldrop_append.ma". + +lemma cpy_append: âG,d,e. l_appendable_sn â¦ (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 â¦ HK0) #H +@(cpy_subst I â¦ (L@@K0) â¦ HVW) // (**) (* /4/ does not work *) +@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2 by lt_to_le/ +qed-. + +lemma cpy_fwd_shift1: âG,L1,L,T1,T,d,e. â¦G, Lâ¦ â¢ L1 @@ T1 â¶[d, e] T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#G #L1 @(lenv_ind_dx â¦ L1) -L1 normalize +[ #L #T1 #T #d #e #HT1 + @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) +| #I #L1 #V1 #IH #L #T1 #X #d #e + >shift_append_assoc normalize #H + elim (cpy_inv_bind1 â¦ H) -H + #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *) + /2 width=3 by trans_eq/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpys0.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpys0.etc new file mode 100644 index 000000000..fa9bcf6de --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpys0.etc @@ -0,0 +1,24 @@ +include "basic_2/grammar/cl_shift.ma". +include "basic_2/relocation/ldrop_append.ma". + +lemma cpys_append: âG. l_appendable_sn â¦ (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 â¦ HK0) #H +@(cpys_delta â¦ I â¦ (L@@K0) V1 â¦ HVW2) // +@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *) +qed. + +lemma cpys_fwd_shift1: âG,L1,L,T1,T. â¦G, Lâ¦ â¢ L1 @@ T1 â¶*Ã T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#G #L1 @(lenv_ind_dx â¦ L1) -L1 normalize +[ #L #T1 #T #HT1 @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) +| #I #L1 #V1 #IH #L #T1 #X >shift_append_assoc normalize + #H elim (cpys_inv_bind1 â¦ H) -H + #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpys2.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpys2.etc new file mode 100644 index 000000000..22f74d0a8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/cpys2.etc @@ -0,0 +1,14 @@ +lemma cpys_append: âG,d,e. l_appendable_sn â¦ (cpys d e G). +#G #d #e #K #T1 #T2 #H @(cpys_ind â¦ H) -T2 +/3 width=3 by cpys_strap1, cpy_append/ +qed-. + +lemma cpys_fwd_shift1: âG,L,L1,T1,T,d,e. â¦G, Lâ¦ â¢ L1 @@ T1 â¶*[d, e] T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#G #L #L1 #T1 #T #d #e #H @(cpys_ind â¦ H) -T +[ /2 width=4 by ex2_2_intro/ +| #T #X #_ #HX * #L0 #T0 #HL10 #H destruct + elim (cpy_fwd_shift1 â¦ HX) -HX #L2 #T2 #HL02 #H destruct + /2 width=4 by ex2_2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/append/crr_append.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/crr_append.etc new file mode 100644 index 000000000..dbf794d99 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/append/crr_append.etc @@ -0,0 +1,56 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,K,T. â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, K @@ Lâ¦ â¢ â¡ ðâ¦Tâ¦. +#G #L #K0 #T #H elim H -L -T /2 width=1/ +#L #K #V #i #HLK +lapply (ldrop_fwd_length_lt2 â¦ HLK) #Hi +lapply (ldrop_O1_append_sn_le â¦ HLK â¦ K0) -HLK /2 width=2/ -Hi /2 width=3/ +qed. + +lemma trr_crr: âG,L,T. â¦G, ââ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦. +#G #L #T #H lapply (crr_append_sn â¦ H) // +qed. + +(* Advanced inversion lemmas ************************************************) + +fact crr_inv_labst_last_aux: âG,L1,T,W. â¦G, L1â¦ â¢ â¡ ðâ¦Tâ¦ â + âL2. L1 = â.âW @@ L2 â â¦G, L2â¦ â¢ â¡ ðâ¦Tâ¦. +#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 â¦ HLK1) + >append_length >commutative_plus normalize in â¢ (??% â ?); #H + elim (le_to_or_lt_eq i (|L2|)) /2 width=1/ -H #Hi destruct + [ elim (ldrop_O1_lt â¦ Hi) #I2 #K2 #V2 #HLK2 + lapply (ldrop_O1_inv_append1_le â¦ HLK1 â¦ HLK2) -HLK1 /2 width=2/ -Hi + normalize #H destruct /2 width=3/ + | lapply (ldrop_O1_inv_append1_ge â¦ HLK1 ?) -HLK1 // shift_append_assoc #H (ldrop_fwd_length â¦ HL) -HL // +qed. + +lemma cny_lref_top: âG,L,d,e,i. d+e â¤ yinj i â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦#iâ¦. +#G #L #d #e #i #Hdei #X #H elim (cpy_inv_lref1 â¦ H) -H // * +#I #K #V #_ #H elim (ylt_yle_false â¦ H) // +qed. + +lemma cny_lref_skip: âG,L,d,e,i. yinj i < d â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦#iâ¦. +#G #L #d #e #i #Hid #X #H elim (cpy_inv_lref1 â¦ H) -H // * +#I #K #V #H elim (ylt_yle_false â¦ H) // +qed. + +lemma cny_gref: âG,L,d,e,p. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Â§pâ¦. +#G #L #d #e #p #X #H elim (cpy_inv_gref1 â¦ H) -H // +qed. + +lemma cny_bind: âG,L,V,d,e. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Vâ¦ â + âI,T. â¦G, L.â{I}Vâ¦ â¢ â¶[â«¯d, e] ðâ¦Tâ¦ â + âa. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦â{a,I}V.Tâ¦. +#G #L #V1 #d #e #HV1 #I #T1 #HT1 #a #X #H +elim (cpy_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct +>(HV1 â¦ HV12) -V2 >(HT1 â¦ HT12) -T2 // +qed. + +lemma cny_flat: âG,L,V,d,e. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Vâ¦ â + âT. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Tâ¦ â + âI. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦â{I}V.Tâ¦. +#G #L #V1 #d #e #HV1 #T1 #HT1 #I #X #H +elim (cpy_inv_flat1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct +>(HV1 â¦ HV12) -V2 >(HT1 â¦ HT12) -T2 // +qed. + +lemma cny_narrow: âG,L,T,d1,e1. â¦G, Lâ¦ â¢ â¶[d1, e1] ðâ¦Tâ¦ â + âd2,e2. d1 â¤ d2 â d2 + e2 â¤ d1 + e1 â â¦G, Lâ¦ â¢ â¶[d2, e2] ðâ¦Tâ¦. +#G #L #T1 #d1 #e1 #HT1 #d2 #e2 #Hd12 #Hde21 #T2 #HT12 +@HT1 /2 width=5 by cpy_weak/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cny_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cny_lift.etc new file mode 100644 index 000000000..213fa6abb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cny_lift.etc @@ -0,0 +1,118 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â dt + et â¤ d â â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdetd #U2 #HU12 +elim (cpy_inv_lift1_le â¦ HU12 â¦ HLK â¦ HTU1) // -L -Hdetd #T2 #HT12 +>(HT1 â¦ HT12) -K /2 width=5 by lift_mono/ +qed-. + +lemma cny_lift_be: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â dt â¤ d â yinj d â¤ dt + et â â¦G, Lâ¦ â¢ â¶[dt, et+e] ðâ¦Uâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hdtd #Hddet #U2 #HU12 +elim (cpy_inv_lift1_be â¦ HU12 â¦ HLK â¦ HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hdtd -Hddet #T2 +>yplus_minus_inj #HT12 >(HT1 â¦ HT12) -K /2 width=5 by lift_mono/ +qed-. + +lemma cny_lift_ge: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â d â¤ dt â â¦G, Lâ¦ â¢ â¶[dt+e, et] ðâ¦Uâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #U2 #HU12 +elim (cpy_inv_lift1_ge â¦ HU12 â¦ HLK â¦ HTU1) /2 width=1 by monotonic_yle_plus_dx/ -L -Hddt #T2 +>yplus_minus_inj #HT12 >(HT1 â¦ HT12) -K /2 width=5 by lift_mono/ +qed-. + +(* Inversion lemmas on relocation *******************************************) + +lemma cny_inv_lift_le: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â dt + et â¤ d â â¦G, Kâ¦ â¢ â¶[dt, et] ðâ¦Tâ¦. +#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 â¦ HT12 â¦ HLK â¦ HTU1 â¦ HTU2 ?) // -K -Hdetd #HU12 +lapply (HU1 â¦ HU12) -L /2 width=5 by lift_inj/ +qed-. + +lemma cny_inv_lift_be: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â dt â¤ d â yinj d + e â¤ dt + et â â¦G, Kâ¦ â¢ â¶[dt, et-e] ðâ¦Tâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdtd #Hdedet #T2 #HT12 +lapply (yle_fwd_plus_ge_inj â¦ Hdedet) // #Heet +elim (yle_inv_plus_inj2 â¦ Hdedet) -Hdedet #Hddete #Hedet +elim (lift_total T2 d e) #U2 #HTU2 +lapply (cpy_lift_be â¦ HT12 â¦ HLK â¦ HTU1 â¦ HTU2 ? ?) // [ >yplus_minus_assoc_inj // ] -K -Hdtd -Hddete +>ymax_pre_sn // -Heet #HU12 +lapply (HU1 â¦ HU12) -L /2 width=5 by lift_inj/ +qed-. + +lemma cny_inv_lift_ge: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â yinj d + e â¤ dt â â¦G, Kâ¦ â¢ â¶[dt-e, et] ðâ¦Tâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdedt #T2 #HT12 +elim (yle_inv_plus_inj2 â¦ Hdedt) -Hdedt #Hddte #Hedt +elim (lift_total T2 d e) #U2 #HTU2 +lapply (cpy_lift_ge â¦ HT12 â¦ HLK â¦ HTU1 â¦ HTU2 ?) // -K -Hddte +>ymax_pre_sn // -Hedt #HU12 +lapply (HU1 â¦ HU12) -L /2 width=5 by lift_inj/ +qed-. + +(* Advanced inversion lemmas on relocation **********************************) + +lemma cny_inv_lift_ge_up: âG,L,K,T,U,s,d,dt,e,et. â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦ â â©[s, d, e] L â¡ K â + â§[d, e] T â¡ U â d â¤ dt â dt â¤ yinj d + e â yinj d + e â¤ dt + et â + â¦G, Kâ¦ â¢ â¶[d, dt + et - (yinj d + e)] ðâ¦Tâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hddt #Hdtde #Hdedet +lapply (cny_narrow â¦ HU1 (d+e) (dt+et-(d+e)) ? ?) -HU1 [ >ymax_pre_sn_comm ] // #HU1 +lapply (cny_inv_lift_ge â¦ HU1 â¦ HLK â¦ HTU1 ?) // -L -U1 +>yplus_minus_inj // +qed-. + +lemma cny_inv_lift_subst: âG,L,K,V,W,i,d,e. d â¤ yinj i â i < d + e â + â©[i+1] L â¡ K â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Wâ¦ â + â§[O, i+1] V â¡ W â â¦G, Kâ¦ â¢ â¶[O, â«°(d+e-i)] ðâ¦Vâ¦. +#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW +lapply (cny_inv_lift_ge_up â¦ HW â¦ HLK â¦ HVW ? ? ?) // +>yplus_O1 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: âG,L,K,T,U,s,d,dt,e,et. â¦G, Kâ¦ â¢ â¶[yinj d, dt + et - (yinj d + yinj e)] ðâ¦Tâ¦ â + â©[s, d, e] L â¡ K â â§[d, e] T â¡ U â + yinj d â¤ dt â dt â¤ yinj d + yinj e â yinj d + yinj e â¤ dt + et â + â¦G, Lâ¦ â¢ â¶[dt, et] ðâ¦Uâ¦. +#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #Hdtde #Hdedet +lapply (cny_lift_be â¦ HT1 â¦ HLK â¦ HTU1 ? ?) // -K -T1 +#HU1 @(cny_narrow â¦ HU1) -HU1 // (**) (* auto fails *) +qed-. + +lemma cny_lift_subst: âG,L,K,V,W,i,d,e. d â¤ yinj i â i < d + e â + â©[i+1] L â¡ K â â¦G, Kâ¦ â¢ â¶[O, â«°(d+e-i)] ðâ¦Vâ¦ â + â§[O, i+1] V â¡ W â â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦Wâ¦. +#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW +@(cny_lift_ge_up â¦ HLK â¦ HVW) // >yplus_O1 yplus_SO2 +[ >yminus_succ2 // +| /2 width=3 by yle_trans/ +| /2 width=1 by ylt_fwd_le_succ1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpx_cpzs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpx_cpzs.etc new file mode 100644 index 000000000..6c208d51b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpx_cpzs.etc @@ -0,0 +1,73 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âk1,k2. âk1 = âk2 â k1 = k2. +#k1 #k2 #H destruct // +qed-. + +axiom cpzs_inv_subst: âI,G,L,K,V1,V2,W2,i. + â©[i] L â¡ K.â{I}V1 â â§[O, i+1] V2 â¡ W2 â + â¦G, Lâ¦ â¢ #i â*[O, â] W2 â â¦G, Kâ¦â¢ V1 â*[O, â] V2. + +axiom cpzs_subst: âI,G,L,K,V1,V2,W2,i. + â©[i] L â¡ K.â{I}V1 â â§[O, i+1] V2 â¡ W2 â + â¦G, Kâ¦â¢ V1 â*[O, â] V2 â â¦G, Lâ¦ â¢ #i â*[O, â] W2. + +(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************) + +(* Forward lemmas on delta-equivalence for terms ****************************) + +lemma cpx_fwd_cpys_cpzs: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â + âd,e. â¦G, Lâ¦ â¢ T1 â*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¶*[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 â¦ H) + | lapply (cpys_inv_sort1 â¦ H) -H #H >(destruct_tsort_tsort â¦ H) + ] -H #H elim (lt_refl_false â¦ H) +| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #d #e @conj #H +(* + [ @(cpys_subst â¦ HLK â¦ HVW2) // >yminus_Y_inj /3 width=7 by cpzs_inv_subst/ + | elim (cpys_inv_lref1_ldrop â¦ H â¦ HLK â¦ 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 (â«¯d) e) -IHT12 + #IHTdx #IHTsn #IHVdx #IHVsn @conj #H + [ elim (cpzs_inv_bind â¦ H) -H /3 width=1 by cpys_bind/ + | elim (cpys_inv_bind1 â¦ 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 â¦ H) -H /3 width=1 by cpys_flat/ + | elim (cpys_inv_flat1 â¦ 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 â¦ H) + | elim (cpys_inv_flat1 â¦ 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 â¦ H) + | elim (cpys_inv_flat1 â¦ H) -H #X1 #X2 #H1 #H2 #H destruct + ] +] + + \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye.etc new file mode 100644 index 000000000..71e06ea72 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye.etc @@ -0,0 +1,89 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â ynat â relation4 genv lenv term term â + Î»d,e,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â§ â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦T2â¦. + +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: âG,L2,T1,T2,d,e. â¦G, L2â¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â + âL1. L1 âÃ[d, e] L2 â L2 âÃ[d, e] L1 â â¦G, L1â¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦. +#G #L2 #T1 #T2 #d #e * +/4 width=8 by lsuby_cpys_trans, lsuby_cny_conf, conj/ +qed-. + +lemma cpye_sort: âG,L,d,e,k. â¦G, Lâ¦ â¢ âk â¶*[d, e] ðâ¦âkâ¦. +/3 width=5 by cny_sort, conj/ qed. + +lemma cpye_free: âG,L,d,e,i. |L| â¤ i â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦#iâ¦. +/3 width=6 by cny_lref_free, conj/ qed. + +lemma cpye_top: âG,L,d,e,i. d + e â¤ yinj i â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦#iâ¦. +/3 width=6 by cny_lref_top, conj/ qed. + +lemma cpye_skip: âG,L,d,e,i. yinj i < d â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦#iâ¦. +/3 width=6 by cny_lref_skip, conj/ qed. + +lemma cpye_gref: âG,L,d,e,p. â¦G, Lâ¦ â¢ Â§p â¶*[d, e] ðâ¦Â§pâ¦. +/3 width=5 by cny_gref, conj/ qed. + +lemma cpye_bind: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â + âI,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯d, e] ðâ¦T2â¦ â + âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[d, e] ðâ¦â{a,I}V2.T2â¦. +#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: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â + âT1,T2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â + âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[d, e] ðâ¦â{I}V2.T2â¦. +#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: âG,L,X,d,e,k. â¦G, Lâ¦ â¢ âk â¶*[d, e] ðâ¦Xâ¦ â X = âk. +#G #L #X #d #e #k * /2 width=5 by cpys_inv_sort1/ +qed-. + +lemma cpye_inv_gref1: âG,L,X,d,e,p. â¦G, Lâ¦ â¢ Â§p â¶*[d, e] ðâ¦Xâ¦ â X = Â§p. +#G #L #X #d #e #p * /2 width=5 by cpys_inv_gref1/ +qed-. + +lemma cpye_inv_bind1: âa,I,G,L,V1,T1,X,d,e. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[d, e] ðâ¦Xâ¦ â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ & â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯d, e] ðâ¦T2â¦ & + X = â{a,I}V2.T2. +#a #I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_bind1 â¦ H1) -H1 +#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_bind â¦ H2) -H2 +/5 width=8 by lsuby_cny_conf, lsuby_succ, ex3_2_intro, conj/ +qed-. + +lemma cpye_inv_flat1: âI,G,L,V1,T1,X,d,e. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[d, e] ðâ¦Xâ¦ â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ & â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ & + X = â{I}V2.T2. +#I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_flat1 â¦ H1) -H1 +#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_flat â¦ H2) -H2 +/3 width=5 by ex3_2_intro, conj/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_alt.etc new file mode 100644 index 000000000..a9dc53cf5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_alt.etc @@ -0,0 +1,89 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â ynat â relation4 genv lenv term term â +| cpyea_sort : âG,L,d,e,k. cpyea d e G L (âk) (âk) +| cpyea_free : âG,L,d,e,i. |L| â¤ i â cpyea d e G L (#i) (#i) +| cpyea_top : âG,L,d,e,i. d + e â¤ yinj i â cpyea d e G L (#i) (#i) +| cpyea_skip : âG,L,d,e,i. yinj i < d â cpyea d e G L (#i) (#i) +| cpyea_subst: âI,G,L,K,V1,V2,W2,i,d,e. d â¤ yinj i â yinj i < d+e â + â©[i] L â¡ K.â{I}V1 â cpyea (yinj 0) (â«°(d+e-yinj i)) G K V1 V2 â + â§[0, i+1] V2 â¡ W2 â cpyea d e G L (#i) W2 +| cpyea_gref : âG,L,d,e,p. cpyea d e G L (Â§p) (Â§p) +| cpyea_bind : âa,I,G,L,V1,V2,T1,T2,d,e. + cpyea d e G L V1 V2 â cpyea (â«¯d) e G (L.â{I}V1) T1 T2 â + cpyea d e G L (â{a,I}V1.T1) (â{a,I}V2.T2) +| cpyea_flat : âI,G,L,V1,V2,T1,T2,d,e. + cpyea d e G L V1 V2 â cpyea d e G L T1 T2 â + cpyea d e G L (â{I}V1.T1) (â{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: âG,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â â¦G, Lâ¦ â¢ T1 â¶â¶*[d, e] ðâ¦T2â¦. +#G #L #T1 @(fqup_wf_ind_eq â¦ G L T1) -G -L -T1 +#Z #Y #X #IH #G #L * * +[ #k #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_sort1 â¦ H) -H // +| #i #HG #HL #HT #T2 #d #e #H destruct + elim (cpye_inv_lref1 â¦ 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 â¦ H) -H // +| #a #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct + elim (cpye_inv_bind1 â¦ H) -H /3 width=1 by cpyea_bind/ +| #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct + elim (cpye_inv_flat1 â¦ H) -H /3 width=1 by cpyea_flat/ +] +qed. + +(* Main inversion properties ************************************************) + +theorem cpyea_inv_cpye: âG,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶â¶*[d, e] ðâ¦T2â¦ â â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦. +#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: âR:ynatâynatârelation4 genv lenv term term. + (âG,L,d,e,k. R d e G L (âk) (âk)) â + (âG,L,d,e,i. |L| â¤ i â R d e G L (#i) (#i)) â + (âG,L,d,e,i. d + e â¤ yinj i â R d e G L (#i) (#i)) â + (âG,L,d,e,i. yinj i < d â R d e G L (#i) (#i)) â + (âI,G,L,K,V1,V2,W2,i,d,e. d â¤ yinj i â yinj i < d + e â + â©[i] L â¡ K.â{I}V1 â â¦G, Kâ¦ â¢ V1 â¶*[yinj O, â«°(d+e-yinj i)] ðâ¦V2â¦ â + â§[O, i+1] V2 â¡ W2 â R (yinj O) (â«°(d+e-yinj i)) G K V1 V2 â R d e G L (#i) W2 + ) â + (âG,L,d,e,p. R d e G L (Â§p) (Â§p)) â + (âa,I,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â + â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯d, e] ðâ¦T2â¦ â R d e G L V1 V2 â + R (â«¯d) e G (L.â{I}V1) T1 T2 â R d e G L (â{a,I}V1.T1) (â{a,I}V2.T2) + ) â + (âI,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ V1 â¶*[d, e] ðâ¦V2â¦ â + â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â R d e G L V1 V2 â + R d e G L T1 T2 â R d e G L (â{I}V1.T1) (â{I}V2.T2) + ) â + âd,e,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦ â 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 â¦ H) -G -L -T1 -T2 -d -e +/3 width=8 by cpyea_inv_cpye/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_cpye.etc new file mode 100644 index 000000000..42bd19b8e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_cpye.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,T,T2,d,e. â¦G, Lâ¦ â¢ T â¶*[d, e] ðâ¦T2â¦ â + âT1. â¦G, Lâ¦ â¢ T â¶*[d, e] T1 â â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2. +#G #L #T #T2 #d #e * #H2 #HT2 #T1 #H1 elim (cpys_conf_eq â¦ H1 â¦ H2) -T +#T0 #HT10 #HT20 >(cpys_inv_cny1 â¦ HT2 â¦ HT20) -T2 // +qed-. + \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_lift.etc new file mode 100644 index 000000000..3c0abe617 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpye_lift.etc @@ -0,0 +1,169 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âI,G,L,K,V1,V2,W2,i,d,e. d â¤ yinj i â i < d + e â + â©[i] L â¡ K.â{I}V1 â â¦G, Kâ¦ â¢ V1 â¶*[O, â«°(d+e-i)] ðâ¦V2â¦ â + â§[O, i+1] V2 â¡ W2 â â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦W2â¦. +#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: âG,L,T1,d,e. âT2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦T2â¦. +#G #L #T1 @(fqup_wf_ind_eq â¦ 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 â¦ 0 (â«°(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 â¦ d e) // elim (IH G (L.â{I}V1) T1 â¦ (â«¯d) e) // + /3 width=2 by cpye_bind, ex_intro/ +| #I #V1 #T1 #HG #HL #HT #d #e destruct + elim (IH G L V1 â¦ d e) // elim (IH G L T1 â¦ d e) // + /3 width=2 by cpye_flat, ex_intro/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma cpye_inv_lref1: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â + â¨â¨ |L| â¤ i â§ T2 = #i + | d + e â¤ yinj i â§ T2 = #i + | yinj i < d â§ T2 = #i + | ââI,K,V1,V2. d â¤ yinj i & yinj i < d + e & + â©[i] L â¡ K.â{I}V1 & + â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦ & + â§[O, i+1] V2 â¡ T2. +#G #L #T2 #i #d #e * #H1 #H2 elim (cpys_inv_lref1 â¦ H1) -H1 +[ #H destruct elim (cny_inv_lref â¦ 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 â¦ HLK â¦ HVT2) (**) (* explicit constructor *) + /4 width=13 by cny_inv_lift_subst, ldrop_fwd_drop2, conj/ +] +qed-. + +lemma cpye_inv_lref1_free: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â + (â¨â¨ |L| â¤ i | d + e â¤ yinj i | yinj i < d) â T2 = #i. +#G #L #T2 #d #e #i #H * elim (cpye_inv_lref1 â¦ H) -H * // +#I #K #V1 #V2 #Hdi #Hide #HLK #_ #_ #H +[ elim (lt_refl_false i) -d + @(lt_to_le_to_lt â¦ H) -H /2 width=5 by ldrop_fwd_length_lt2/ (**) (* full auto slow: 19s *) +] +elim (ylt_yle_false â¦ H) // +qed-. + +lemma cpye_inv_lref1_lget: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â + âI,K,V1. â©[i] L â¡ K.â{I}V1 â + â¨â¨ d + e â¤ yinj i â§ T2 = #i + | yinj i < d â§ T2 = #i + | ââV2. d â¤ yinj i & yinj i < d + e & + â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦ & + â§[O, i+1] V2 â¡ T2. +#G #L #T2 #d #e #i #H #I #K #V1 #HLK elim (cpye_inv_lref1 â¦ H) -H * +[ #H elim (lt_refl_false i) -T2 -d + @(lt_to_le_to_lt â¦ 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 â¦ HLY â¦ HLK) -HLY -HLK #H destruct + /3 width=3 by or3_intro2, ex4_intro/ +] +qed-. + +lemma cpye_inv_lref1_subst_ex: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â + âI,K,V1. d â¤ yinj i â yinj i < d + e â + â©[i] L â¡ K.â{I}V1 â + ââV2. â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦ & + â§[O, i+1] V2 â¡ T2. +#G #L #T2 #d #e #i #H #I #K #V1 #Hdi #Hide #HLK +elim (cpye_inv_lref1_lget â¦ H â¦ HLK) -H * /2 width=3 by ex2_intro/ +#H elim (ylt_yle_false â¦ H) // +qed-. + +lemma cpye_inv_lref1_subst: âG,L,T2,d,e,i. â¦G, Lâ¦ â¢ #i â¶*[d, e] ðâ¦T2â¦ â + âI,K,V1,V2. d â¤ yinj i â yinj i < d + e â + â©[i] L â¡ K.â{I}V1 â â§[O, i+1] V2 â¡ T2 â + â¦G, Kâ¦ â¢ V1 â¶*[yinj 0, â«°(d+e-yinj i)] ðâ¦V2â¦. +#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2 +elim (cpye_inv_lref1_subst_ex â¦ H â¦ HLK) -H -HLK // +#X2 #H0 #HXT2 lapply (lift_inj â¦ HXT2 â¦ HVT2) -HXT2 -HVT2 #H destruct // +qed-. + +(* Inversion lemmas on relocation *******************************************) + +lemma cpye_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â + âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt + et â¤ d â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[dt, et] ðâ¦T2â¦ & â§[d, e] T2 â¡ U2. +#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdetd +elim (cpys_inv_lift1_le â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 +lapply (cny_inv_lift_le â¦ HU2 â¦ HLK â¦ HTU2 ?) -L +/3 width=3 by ex2_intro, conj/ +qed-. + +lemma cpye_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â + âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt â¤ d â yinj d + e â¤ dt + et â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[dt, et - e] ðâ¦T2â¦ & â§[d, e] T2 â¡ U2. +#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet +elim (cpys_inv_lift1_be â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 +lapply (cny_inv_lift_be â¦ HU2 â¦ HLK â¦ HTU2 ? ?) -L +/3 width=3 by ex2_intro, conj/ +qed-. + +lemma cpye_inv_lift1_ge: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â + âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + yinj d + e â¤ dt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[dt - e, et] ðâ¦T2â¦ & â§[d, e] T2 â¡ U2. +#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdedt +elim (cpys_inv_lift1_ge â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 +lapply (cny_inv_lift_ge â¦ HU2 â¦ HLK â¦ HTU2 ?) -L +/3 width=3 by ex2_intro, conj/ +qed-. + +lemma cpye_inv_lift1_ge_up: âG,L,U1,U2,dt,et. â¦G, Lâ¦ â¢ U1 â¶*[dt, et] ðâ¦U2â¦ â + âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + d â¤ dt â dt â¤ yinj d + e â yinj d + e â¤ dt + et â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[d, dt + et - (yinj d + e)] ðâ¦T2â¦ & + â§[d, e] T2 â¡ 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 â¦ HU12 â¦ HLK â¦ HTU1) -U1 // #T2 #HT12 #HTU2 +lapply (cny_inv_lift_ge_up â¦ HU2 â¦ HLK â¦ HTU2 ? ? ?) -L +/3 width=3 by ex2_intro, conj/ +qed-. + +lemma cpye_inv_lift1_subst: âG,L,W1,W2,d,e. â¦G, Lâ¦ â¢ W1 â¶*[d, e] ðâ¦W2â¦ â + âK,V1,i. â©[i+1] L â¡ K â â§[O, i+1] V1 â¡ W1 â + d â¤ yinj i â i < d + e â + ââV2. â¦G, Kâ¦ â¢ V1 â¶*[O, â«°(d+e-i)] ðâ¦V2â¦ & â§[O, i+1] V2 â¡ W2. +#G #L #W1 #W2 #d #e * #HW12 #HW2 #K #V1 #i #HLK #HVW1 #Hdi #Hide +elim (cpys_inv_lift1_subst â¦ HW12 â¦ HLK â¦ HVW1) -W1 // #V2 #HV12 #HVW2 +lapply (cny_inv_lift_subst â¦ HLK HW2 HVW2) -L +/3 width=3 by ex2_intro, conj/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpys_cny.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpys_cny.etc new file mode 100644 index 000000000..1499ceea5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpys_cny.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,T1,d,e. â¦G, Lâ¦ â¢ â¶[d, e] ðâ¦T1â¦ â + âT2. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â T1 = T2. +#G #L #T1 #d #e #HT1 #T2 #H @(cpys_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 destruct <(HT1 â¦ HT2) -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpzs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpzs.etc new file mode 100644 index 000000000..0fada97a0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/cpzs.etc @@ -0,0 +1,83 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â ynat â relation4 genv lenv term term â + Î»d,e,G,L,T1,T2. + ââT. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦Tâ¦ & â¦G, Lâ¦ â¢ T2 â¶*[d, e] ðâ¦Tâ¦. + +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: âG,L,T1,T,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] ðâ¦Tâ¦ â + âT2. â¦G, Lâ¦ â¢ T2 â¶*[d, e] ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ T1 â*[d, e] T2. +/2 width=3 by ex2_intro/ qed. + +lemma cpzs_refl: âG,L,d,e. reflexive â¦ (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: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â*[d, e] V2 â + âI,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â*[â«¯d, e] T2 â + âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â*[d, e] â{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: âG,L,V1,V2,d,e. â¦G, Lâ¦ â¢ V1 â*[d, e] V2 â + âT1,T2. â¦G, Lâ¦ â¢ T1 â*[d, e] T2 â + âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â*[d, e] â{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: âG,L,d,e,k1,k2. â¦G, Lâ¦ â¢ âk1 â*[d, e] âk2 â k1 = k2. +#G #L #d #e #k1 #k2 * #X #H1 #H2 +lapply (cpye_inv_sort1 â¦ H1) -H1 #H1 +lapply (cpye_inv_sort1 â¦ H2) -H2 #H2 +destruct // +qed-. + +lemma cpzs_inv_bind: âa1,a2,I1,I2,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ â{a1,I1}V1.T1 â*[d, e] â{a2,I2}V2.T2 â + â§â§ a1 = a2 & I1 = I2 + & â¦G, Lâ¦ â¢ V1 â*[d, e] V2 & â¦G, L.â{I1}V1â¦ â¢ T1 â*[â«¯d, e] T2. +#a1 #a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2 +elim (cpye_inv_bind1 â¦ H1) -H1 #V2 #T2 #HV12 #HT12 #H1 +elim (cpye_inv_bind1 â¦ 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: âI1,I2,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ â{I1}V1.T1 â*[d, e] â{I2}V2.T2 â + â§â§ I1 = I2 + & â¦G, Lâ¦ â¢ V1 â*[d, e] V2 & â¦G, Lâ¦ â¢ T1 â*[d, e] T2. +#I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2 +elim (cpye_inv_flat1 â¦ H1) -H1 #V2 #T2 #HV12 #HT12 #H1 +elim (cpye_inv_flat1 â¦ H2) -H2 #W2 #U2 #HW12 #HU12 #H2 +destruct /3 width=3 by cpye_div, and3_intro/ +qed-. + +lemma cpzs_inv_flat_bind: âa2,I1,I2,G,L,V1,V2,T1,T2,d,e. â¦G, Lâ¦ â¢ â{I1}V1.T1 â*[d, e] â{a2,I2}V2.T2 â â¥. +#a2 #I1 #I2 #G #L #V1 #V2 #T1 #T2 #d #e * #X #H1 #H2 +elim (cpye_inv_flat1 â¦ H1) -H1 #V2 #T2 #HV12 #HT12 #H1 +elim (cpye_inv_bind1 â¦ H2) -H2 #W2 #U2 #HW12 #HU12 #H2 +destruct +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lleq_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lleq_cpye.etc new file mode 100644 index 000000000..e6890e637 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lleq_cpye.etc @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T,d. L1 â[T, d] L2 â âG,T1. â¦G, L1â¦ â¢ T â¶*[d, â] ðâ¦T1â¦ â + âT2. â¦G, L2â¦ â¢ T â¶*[d, â] ðâ¦T2â¦ â T1 = T2. +#L1 #L2 #T #d #H @(lleq_ind_alt â¦ H) -L1 -L2 -T -d +[ #L1 #L2 #d #k #_ #G #T1 #H1 #T2 #H2 + >(cpye_inv_sort1 â¦ H1) -H1 >(cpye_inv_sort1 â¦ H2) -H2 // +| #L1 #L2 #d #i #_ #Hid #G #T1 #H1 #T2 #H2 + >(cpye_inv_lref1_free â¦ H1) -H1 [ >(cpye_inv_lref1_free â¦ 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 â¦ H1 â¦ HLK1) -H1 -HLK1 // + elim (cpye_inv_lref1_subst_ex â¦ H2 â¦ HLK2) -H2 -HLK2 // + >yminus_Y_inj #V2 #HV2 #HVT2 #V1 #HV1 #HVT1 + lapply (IHV â¦ HV1 â¦ 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 â¦ H1) -H1 [ >(cpye_inv_lref1_free â¦ H2) -H2 ] + /2 width=1 by or3_intro0/ +| #L1 #L2 #d #p #_ #G #T1 #H1 #T2 #H2 + >(cpye_inv_gref1 â¦ H1) -H1 >(cpye_inv_gref1 â¦ H2) -H2 // +| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2 + elim (cpye_inv_bind1 â¦ H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct + elim (cpye_inv_bind1 â¦ 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 â¦ H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct + elim (cpye_inv_flat1 â¦ H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct + /3 width=3 by eq_f2/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lpx_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lpx_cpye.etc new file mode 100644 index 000000000..48816c85d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lpx_cpye.etc @@ -0,0 +1,44 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,d,e. + âL0,T0,T1. â¦G, L0â¦ â¢ T0 â¡[h, g] T1 â âL1. â¦G, L0â¦ â¢ â¡[h, g] L1 â + âT2. â¦G, L0â¦ â¢ T0 â¶*[d, e] T2 â + ââT. â¦G, L1â¦ â¢ T1 â¶*[d, e] T & â¦G, L0â¦ â¢ T2 â¡[h, g] T. + +(* SN EXTENDED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *********************) + +(* Forward lemmas on evaluation for extended substitution *******************) + +lemma cpx_cpys_cpye_fwd_lpx: âh,g,G,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â + âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â + âU1,d,e. â¦G, L1â¦ â¢ T1 â¶*[d, e] U1 â + âU2. â¦G, L2â¦ â¢ T2 â¶*[d, e] ðâ¦U2â¦ â + â¦G, L1â¦ â¢ U1 â¡[h, g] U2. +#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e #HTU1 +elim (cpx_cpys_conf_lpx â¦ HT12 â¦ HL12 â¦ HTU1) -T1 +/3 width=9 by cpx_cpys_trans_lpx, cpye_cpys_conf/ +qed-. + +lemma cpx_cpye_fwd_lpx: âh,g,G,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â + âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â + âU1,d,e. â¦G, L1â¦ â¢ T1 â¶*[d, e] ðâ¦U1â¦ â + âU2. â¦G, L2â¦ â¢ T2 â¶*[d, e] ðâ¦U2â¦ â + â¦G, L1â¦ â¢ U1 â¡[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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lpxs_cpye.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lpxs_cpye.etc new file mode 100644 index 000000000..a6f0f86bd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/lpxs_cpye.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2. â¦G, L1â¦ â¢ â¡*[h, g] L2 â + âT1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â + âU1,d,e. â¦G, L1â¦ â¢ T1 â¶*[d, e] ðâ¦U1â¦ â + âU2. â¦G, L2â¦ â¢ T2 â¶*[d, e] ðâ¦U2â¦ â + â¦G, L1â¦ â¢ U1 â¡*[h, g] U2. +#h #g #G #L1 #L2 #H @(lpxs_ind_dx â¦ 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 â¦ HT12 â¦ HL1 â¦ HTU1 â¦ HTX2) -T1 + /4 width=9 by lpx_cpxs_trans, cpxs_strap2/ (**) (* full auto too long: 41s *) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/pdeltaconvstar_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/pdeltaconvstar_6.etc new file mode 100644 index 000000000..b7de5f543 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/pdeltaconvstar_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 break â * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PDeltaConvStar $G $L $T1 $d $e $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubsteval_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubsteval_6.etc new file mode 100644 index 000000000..360f43d29 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubsteval_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G, break term 46 L â¦ â¢ break term 46 T1 â¶ * break [ term 46 d , break term 46 e ] break ð â¦ term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'PSubstEval $G $L $T1 $T2 $d $e }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubstevalalt_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubstevalalt_6.etc new file mode 100644 index 000000000..18cdf608f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubstevalalt_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G, break term 46 L â¦ â¢ break term 46 T1 â¶ â¶ * break [ term 46 d , break term 46 e ] break ð â¦ term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'PSubstEvalAlt $G $L $T1 $T2 $d $e }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubstnormal_5.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubstnormal_5.etc new file mode 100644 index 000000000..4a7a78afc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cny/psubstnormal_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G, break term 46 L â¦ â¢ â¶ break [ term 46 d , break term 46 e ] ð break â¦ term 46 T â¦ )" + non associative with precedence 45 + for @{ 'PSubstNormal $G $L $T $d $e }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees.etc new file mode 100644 index 000000000..e1b9b06b2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees.etc @@ -0,0 +1,130 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â + Î»d,i,L,U1. âU2. â¦â, Lâ¦ â¢ U1 â¶*[d, â] U2 â âT2. â§[i, 1] T2 â¡ 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: âL,U,d,i. L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â âT. â§[i, 1] T â¡ U. +/2 width=1 by/ qed-. + +lemma cofrees_fwd_bind_sn: âa,I,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{a,I}W.Uâ¦ â + L â¢ i ~Ïµ ð*[d]â¦Wâ¦. +#a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (â{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1 +#X #H elim (lift_inv_bind2 â¦ H) -H /2 width=2 by ex_intro/ +qed-. + +lemma cofrees_fwd_bind_dx: âa,I,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{a,I}W.Uâ¦ â + L.â{I}W â¢ i+1 ~Ïµ ð*[â«¯d]â¦Uâ¦. +#a #I #L #W #U1 #i #d #H #U2 #HU12 elim (H (â{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1 +#X #H elim (lift_inv_bind2 â¦ H) -H /2 width=2 by ex_intro/ +qed-. + +lemma cofrees_fwd_flat_sn: âI,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{I}W.Uâ¦ â + L â¢ i ~Ïµ ð*[d]â¦Wâ¦. +#I #L #W1 #U #i #d #H #W2 #HW12 elim (H (â{I}W2.U)) /2 width=1 by cpys_flat/ -W1 +#X #H elim (lift_inv_flat2 â¦ H) -H /2 width=2 by ex_intro/ +qed-. + +lemma cofrees_fwd_flat_dx: âI,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{I}W.Uâ¦ â + L â¢ i ~Ïµ ð*[d]â¦Uâ¦. +#I #L #W #U1 #i #d #H #U2 #HU12 elim (H (â{I}W.U2)) /2 width=1 by cpys_flat/ -U1 +#X #H elim (lift_inv_flat2 â¦ H) -H /2 width=2 by ex_intro/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma cofrees_inv_gen: âL,U,U0,d,i. â¦â, Lâ¦ â¢ U â¶*[d, â] U0 â (âT. â§[i, 1] T â¡ U0 â â¥) â + L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥. +#L #U #U0 #d #i #HU0 #HnU0 #HU elim (HU â¦ HU0) -L -U -d /2 width=2 by/ +qed-. + +lemma cofrees_inv_lref_eq: âL,d,i. L â¢ i ~Ïµ ð*[d]â¦#iâ¦ â â¥. +#L #d #i #H elim (H (#i)) -H // +#X #H elim (lift_inv_lref2_be â¦ H) -H // +qed-. + +lemma cofrees_inv_bind: âa,I,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{a,I}W.Uâ¦ â + L â¢ i ~Ïµ ð*[d]â¦Wâ¦ â§ L.â{I}W â¢ i+1 ~Ïµ ð*[â«¯d]â¦Uâ¦. +/3 width=8 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-. + +lemma cofrees_inv_flat: âI,L,W,U,i,d. L â¢ i ~Ïµ ð*[d]â¦â{I}W.Uâ¦ â + L â¢ i ~Ïµ ð*[d]â¦Wâ¦ â§ L â¢ i ~Ïµ ð*[d]â¦Uâ¦. +/3 width=7 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-. + +(* Basic Properties *********************************************************) + +lemma cofrees_lsuby_conf: âL1,U,d,i. L1 â¢ i ~Ïµ ð*[d]â¦Uâ¦ â + âL2. L1 â[d, â] L2 â L2 â¢ i ~Ïµ ð*[d]â¦Uâ¦. +/3 width=3 by lsuby_cpys_trans/ qed-. + +lemma cofrees_sort: âL,d,i,k. L â¢ i ~Ïµ ð*[d]â¦âkâ¦. +#L #d #i #k #X #H >(cpys_inv_sort1 â¦ H) -X /2 width=2 by ex_intro/ +qed. + +lemma cofrees_gref: âL,d,i,p. L â¢ i ~Ïµ ð*[d]â¦Â§pâ¦. +#L #d #i #p #X #H >(cpys_inv_gref1 â¦ H) -X /2 width=2 by ex_intro/ +qed. + +lemma cofrees_bind: âL,V,d,i. L â¢ i ~Ïµ ð*[d] â¦Vâ¦ â + âI,T. L.â{I}V â¢ i+1 ~Ïµ ð*[â«¯d]â¦Tâ¦ â + âa. L â¢ i ~Ïµ ð*[d]â¦â{a,I}V.Tâ¦. +#L #W1 #d #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 â¦ H) -H +#W2 #U2 #HW12 #HU12 #H destruct +elim (HW1 â¦ HW12) elim (HU1 â¦ HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/ +qed. + +lemma cofrees_flat: âL,V,d,i. L â¢ i ~Ïµ ð*[d]â¦Vâ¦ â âT. L â¢ i ~Ïµ ð*[d]â¦Tâ¦ â + âI. L â¢ i ~Ïµ ð*[d]â¦â{I}V.Tâ¦. +#L #W1 #d #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 â¦ H) -H +#W2 #U2 #HW12 #HU12 #H destruct +elim (HW1 â¦ HW12) elim (HU1 â¦ HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/ +qed. + +lemma cofrees_cpy_trans: âL,U1,U2,d. â¦â, Lâ¦ â¢ U1 â¶[d, â] U2 â + âi. L â¢ i ~Ïµ ð*[d]â¦U1â¦ â L â¢ i ~Ïµ ð*[d]â¦U2â¦. +/3 width=3 by cpys_strap2/ qed-. + +axiom cofrees_dec: âL,T,d,i. Decidable (L â¢ i ~Ïµ ð*[d]â¦Tâ¦). + +(* Basic negated properties *************************************************) + +lemma frees_cpy_div: âL,U1,U2,d. â¦â, Lâ¦ â¢ U1 â¶[d, â] U2 â + âi. (L â¢ i ~Ïµ ð*[d]â¦U2â¦ â â¥) â (L â¢ i ~Ïµ ð*[d]â¦U1â¦ â â¥). +/3 width=7 by cofrees_cpy_trans/ qed-. + +(* Basic negated inversion lemmas *******************************************) + +lemma frees_inv_bind: âa,I,L,V,T,d,i. (L â¢ i ~Ïµ ð*[d]â¦â{a,I}V.Tâ¦ â â¥) â + (L â¢ i ~Ïµ ð*[d]â¦Vâ¦ â â¥) â¨ (L.â{I}V â¢ i+1 ~Ïµ ð*[â«¯d]â¦Tâ¦ â â¥). +#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: âI,L,V,T,d,i. (L â¢ i ~Ïµ ð*[d]â¦â{I}V.Tâ¦ â â¥) â + (L â¢ i ~Ïµ ð*[d]â¦Vâ¦ â â¥) â¨ (L â¢ i ~Ïµ ð*[d]â¦Tâ¦ â â¥). +#I #L #W #U #d #H elim (cofrees_dec L W d) +/4 width=8 by cofrees_flat, or_intror, or_introl/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees_alt.etc new file mode 100644 index 000000000..7bb9c9ca7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees_alt.etc @@ -0,0 +1,54 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,U,d,i. (âT. â§[i, 1] T â¡ U â â¥) â (L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥). +#L #U #d #i #HnTU #H elim (cofrees_fwd_lift â¦ H) -H /2 width=2 by/ +qed-. + +lemma frees_inv_ge: âL,U,d,i. d â¤ yinj i â (L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥) â + (âT. â§[i, 1] T â¡ U â â¥) â¨ + ââI,K,W,j. d â¤ yinj j & j < i & â©[j]L â¡ K.â{I}W & + (K â¢ i-j-1 ~Ïµ ð*[yinj 0]â¦Wâ¦ â â¥) & (âT. â§[j, 1] T â¡ U â â¥). +#L #U #d #i #Hdi #H @(frees_ind â¦ H) -U /3 width=2 by or_introl/ +#U1 #U2 #HU12 #HU2 * +[ #HnU2 elim (cpy_fwd_nlift2_ge â¦ HU12 â¦ 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 â¦ HU12 â¦ 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 â¦ HLK1 â¦ HLK2 ?) -HLK2 /2 width=1 by lt_to_le/ + #HK12 lapply (ldrop_inv_drop1_lt â¦ HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12 + #HK12 + @or_intror @(ex5_4_intro â¦ HLK1 â¦ HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/ + @(frees_be â¦ HK12 â¦ HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1 + >minus_plus in â¢ (??(?(?%?)?)??â?); >minus_plus in â¢ (??(?(??%)?)??â?); >arith_b1 /2 width=1 by/ +] +qed-. + +lemma frees_ind_ge: âR:relation4 ynat nat lenv term. + (âd,i,L,U. d â¤ yinj i â (âT. â§[i, 1] T â¡ U â â¥) â R d i L U) â + (âd,i,j,I,L,K,W,U. d â¤ yinj j â j < i â â©[j]L â¡ K.â{I}W â (K â¢ i-j-1 ~Ïµ ð*[0]â¦Wâ¦ â â¥) â (âT. â§[j, 1] T â¡ U â â¥) â R 0 (i-j-1) K W â R d i L U) â + âd,i,L,U. d â¤ yinj i â (L â¢ i ~Ïµ ð*[d]â¦Uâ¦ â â¥) â R d i L U. +#R #IH1 #IH2 #d #i #L #U +generalize in match d; -d generalize in match i; -i +@(f2_ind â¦ rfw â¦ L U) -L -U +#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees_lift.etc new file mode 100644 index 000000000..f93f72216 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cofrees/cofrees_lift.etc @@ -0,0 +1,180 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,d,i,j. L â¢ i ~Ïµ ð*[d]â¦#jâ¦ â d â¤ yinj j â j < i â + âI,K,W. â©[j]L â¡ K.â{I}W â K â¢ i-j-1 ~Ïµ ð*[yinj 0]â¦Wâ¦. +#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 â¦ HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/ +>minus_plus minus_plus_plus_l // +| #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct + elim (cofrees_inv_flat â¦ H1) -H1 #HW #HU + elim (nlift_inv_flat â¦ H2) -H2 [ /3 width=9 by/ ] + #HnU @(IH â¦ HU â¦ HnU â¦ HLK) // (**) (* full auto fails *) +] +qed-. + +(* Advanced properties ******************************************************) + +lemma cofrees_lref_skip: âL,d,i,j. j < i â yinj j < d â L â¢ i ~Ïµ ð*[d]â¦#jâ¦. +#L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 â¦ H) -H +[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/ +| * #I #K #W1 #W2 #Hdj elim (ylt_yle_false â¦ Hdj) -i -I -L -K -W1 -W2 -X // +] +qed. + +lemma cofrees_lref_lt: âL,d,i,j. i < j â L â¢ i ~Ïµ ð*[d]â¦#jâ¦. +#L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 â¦ 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 â¦ H i j) /2 width=2 by lt_to_le, ex_intro/ +] +qed. + +lemma cofrees_lref_gt: âI,L,K,W,d,i,j. j < i â â©[j] L â¡ K.â{I}W â + K â¢ (i-j-1) ~Ïµ ð*[O]â¦Wâ¦ â L â¢ i ~Ïµ ð*[d]â¦#jâ¦. +#I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 â¦ H) -H +[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/ +| * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono â¦ HLK0 â¦ HLK) -L + #H destruct elim (HW1 â¦ HW12) -I -K -W1 -d + #V2 #HVW2 elim (lift_trans_le â¦ HVW2 â¦ HW2) -W2 // + >minus_plus minus_plus yplus_inj >yminus_Y_inj #T2 #HT12 + lapply (cpys_weak â¦ HT12 (d-yinj e0) (â) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12 + | elim (cpys_inv_lift1_ge â¦ HU12 â¦ HLK â¦ HTU1) // #T2 + ] +| elim (cpys_inv_lift1_be â¦ HU12 â¦ HLK â¦ HTU1) // >yminus_Y_inj #T2 #HT12 + lapply (cpys_weak â¦ HT12 (d-yinj e0) (â) ? ?) // -HT12 +] +-s -L #HT12 #HTU2 +elim (HT1 â¦ HT12) -T1 #V2 #HVT2 +elim (lift_trans_le â¦ HVT2 â¦ HTU2 ?) // (tpss_inv_sort1 â¦ H) -H // +| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 â¦ H) -H + [ #H destruct + elim (lt_or_ge i d) #Hdi + [ elim (ltpss_dx_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss11 â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss21 â¦ 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 â¦ HL12 â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_dx_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct + lapply (ldrop_mono â¦ HL2K0 â¦ HLK2) -HL2K0 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) -HLK2 #HLK2 + lapply (tpss_trans_eq â¦ HV12 HVW2) -V2 /3 width=7/ + ] +| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_bind1 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ +] +qed. + +lemma aaa_ltpss_dx_tps_conf: âL1,T1,A. L1 â¢ T1 â A â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶ [d, e] T2 â L2 â¢ T2 â A. +/3 width=7/ qed. + +lemma aaa_ltpss_dx_conf: âL1,T,A. L1 â¢ T â A â + âL2,d,e. L1 â¶* [d, e] L2 â L2 â¢ T â A. +/2 width=7/ qed. + +lemma aaa_tpss_conf: âL,T1,A. L â¢ T1 â A â + âT2,d,e. L â¢ T1 â¶* [d, e] T2 â L â¢ T2 â A. +/2 width=7/ qed. + +lemma aaa_tps_conf: âL,T1,A. L â¢ T1 â A â + âT2,d,e. L â¢ T1 â¶ [d, e] T2 â L â¢ T2 â A. +/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/aaa_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/aaa_ltpss_sn.etc new file mode 100644 index 000000000..4f2a44827 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/aaa_ltpss_sn.etc @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T,A. L1 â¢ T â A â + âL2,d,e. L1 â¢ â¶* [d, e] L2 â L2 â¢ T â A. +#L1 #T #A #HT #L2 #d #e #HL12 +lapply (ltpss_sn_ltpssa â¦ HL12) -HL12 #HL12 +@(TC_Conf3 â¦ (Î»L,A. L â¢ T â A) â¦ HT ? HL12) /2 width=5/ +qed. + +lemma aaa_ltpss_sn_tpss_conf: âL1,T1,A. L1 â¢ T1 â A â + âL2,d,e. L1 â¢ â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â L2 â¢ T2 â A. +/3 width=5/ qed. + +lemma aaa_ltpss_sn_tps_conf: âL1,T1,A. L1 â¢ T1 â A â + âL2,d,e. L1 â¢ â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶ [d, e] T2 â L2 â¢ T2 â A. +/3 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cnf_cif.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cnf_cif.etc new file mode 100644 index 000000000..5a1ae5a5b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cnf_cif.etc @@ -0,0 +1,106 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,T2,d,e. L â¢ T1 â¶[d, e] T2 â L â¢ ðâ¦T1â¦ â 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 â¦ HLK ?) // +| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H + elim (cif_inv_bind â¦ H) -H #HV1 #HT1 * #H destruct + lapply (IHV12 â¦ HV1) -IHV12 -HV1 #H destruct /3 width=1/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H + elim (cif_inv_flat â¦ H) -H #HV1 #HT1 #_ #_ /3 width=1/ +] +qed. + +lemma tpss_cif_eq: âL,T1,T2,d,e. L â¢ T1 â¶*[d, e] T2 â L â¢ ðâ¦T1â¦ â T1 = T2. +#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 #HT1 +lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/ +qed. + +lemma tpr_cif_eq: âT1,T2. T1 â¡ T2 â âL. L â¢ ðâ¦T1â¦ â T1 = T2. +#T1 #T2 #H elim H -T1 -T2 +[ // +| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H + [ elim (cif_inv_appl â¦ H) -H #HV1 #HT1 #_ + >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // + | elim (cif_inv_ri2 â¦ H) /2 width=1/ + ] +| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H + elim (cif_inv_appl â¦ H) -H #_ #_ #H + elim (simple_inv_bind â¦ H) +| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H + [ lapply (tps_lsubr_trans â¦ HT2 (L.âV2) ?) -HT2 /2 width=1/ #HT2 + elim (cif_inv_bind â¦ H) -H #HV1 #HT1 * #H destruct + lapply (IHV1 â¦ HV1) -IHV1 -HV1 #H destruct + lapply (IHT1 â¦ HT1) -IHT1 #H destruct + lapply (tps_cif_eq â¦ HT2 ?) -HT2 // + | <(tps_inv_refl_SO2 â¦ HT2 ?) -HT2 // + elim (cif_inv_ib2 â¦ H) -H /2 width=1/ /3 width=2/ + ] +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H + elim (cif_inv_appl â¦ H) -H #_ #_ #H + elim (simple_inv_bind â¦ H) +| #V1 #T1 #T #T2 #_ #_ #_ #L #H + elim (cif_inv_ri2 â¦ H) /2 width=1/ +| #V1 #T1 #T2 #_ #_ #L #H + elim (cif_inv_ri2 â¦ H) /2 width=1/ +] +qed. + +lemma cpr_cif_eq: âL,T1,T2. L â¢ T1 â¡ T2 â L â¢ ðâ¦T1â¦ â T1 = T2. +#L #T1 #T2 * #T0 #HT10 #HT02 #HT1 +lapply (tpr_cif_eq â¦ HT10 â¦ HT1) -HT10 #H destruct /2 width=5/ +qed. + +theorem cif_cnf: âL,T. L â¢ ðâ¦Tâ¦ â L â¢ ðâ¦Tâ¦. +/3 width=3/ qed. + +(* Note: this property is unusual *) +lemma cnf_crf_false: âL,T. L â¢ ðâ¦Tâ¦ â L â¢ ðâ¦Tâ¦ â â¥. +#L #T #H elim H -L -T +[ #L #K #V #i #HLK #H + elim (cnf_inv_delta â¦ HLK H) +| #L #V #T #_ #IHV #H + elim (cnf_inv_appl â¦ H) -H /2 width=1/ +| #L #V #T #_ #IHT #H + elim (cnf_inv_appl â¦ H) -H /2 width=1/ +| #I #L #V #T * #H1 #H2 destruct + [ elim (cnf_inv_zeta â¦ H2) + | elim (cnf_inv_tau â¦ H2) + ] +|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct + [1,3: elim (cnf_inv_abbr â¦ H2) -H2 /2 width=1/ + |*: elim (cnf_inv_abst â¦ H2) -H2 /2 width=1/ + ] +| #a #L #V #W #T #H + elim (cnf_inv_appl â¦ H) -H #_ #_ #H + elim (simple_inv_bind â¦ H) +| #a #L #V #W #T #H + elim (cnf_inv_appl â¦ H) -H #_ #_ #H + elim (simple_inv_bind â¦ H) +] +qed. + +theorem cnf_cif: âL,T. L â¢ ðâ¦Tâ¦ â L â¢ ðâ¦Tâ¦. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpcs_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpcs_ltpss_dx.etc new file mode 100644 index 000000000..80f418b08 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpcs_ltpss_dx.etc @@ -0,0 +1,50 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,d,e. L1 â¶* [d, e] L2 â + âT1,T2. L1 â¢ T1 â¬* T2 â L2 â¢ T1 â¬* T2. +#L1 #L2 #d #e #HL12 #T1 #T2 #H +elim (cpcs_inv_cprs â¦ H) -H #T #HT1 #HT2 +elim (cprs_ltpss_dx_conf â¦ HT1 â¦ HL12) -HT1 #U1 #HTU1 #H1 +elim (cprs_ltpss_dx_conf â¦ HT2 â¦ HL12) -L1 #U2 #HTU2 #H2 +elim (tpss_conf_eq â¦ H1 â¦ H2) -T #U #HU1 #HU2 +lapply (cprs_tpss_trans â¦ HTU1 â¦ HU1) -U1 +lapply (cprs_tpss_trans â¦ HTU2 â¦ HU2) -U2 /2 width=3/ +qed-. + +lemma cpcs_ltpss_dx_tpss_conf: âL1,L2,d,e. L1 â¶* [d, e] L2 â + âT,T2. L1 â¢ T â¬* T2 â + âT1. L2 â¢ T â¶* [d, e] T1 â + L2 â¢ T1 â¬* T2. +#L1 #L2 #d #e #HL12 #T #T2 #HT2 #T1 #HT1 +lapply (cpcs_ltpss_dx_conf â¦ HL12 â¦ HT2) -L1 #HT2 +lapply (cpcs_tpss_conf â¦ HT1 â¦ HT2) -T // +qed-. + +lemma cpcs_ltpss_dx_tpss2_conf: âL1,L2,d,e. L1 â¶* [d, e] L2 â + âT1,T2. L1 â¢ T1 â¬* T2 â + âT3. L2 â¢ T1 â¶* [d, e] T3 â + âT4. L2 â¢ T2 â¶* [d, e] T4 â + L2 â¢ T3 â¬* T4. +#L1 #L2 #d #e #HL12 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 +lapply (cpcs_ltpss_dx_tpss_conf â¦ HL12 â¦ HT12 â¦ HT13) -L1 -T1 #HT32 +lapply (cpcs_tpss_strap1 â¦ HT32 â¦ HT24) -T2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpcs_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpcs_ltpss_sn.etc new file mode 100644 index 000000000..5bf1ed57f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpcs_ltpss_sn.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â + âT1,T2. L1 â¢ T1 â¬* T2 â L2 â¢ T1 â¬* T2. +#L1 #L2 #d #e #H +lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 // +#L #L2 #_ #HL2 #IHL1 #T1 #U1 #HTU1 +lapply (IHL1 â¦ HTU1) -IHL1 -HTU1 #HTU1 +lapply (cpcs_ltpss_dx_conf â¦ HL2 â¦ HTU1) -HTU1 -HL2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr.etc new file mode 100644 index 000000000..d82299c38 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr.etc @@ -0,0 +1,111 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â relation term â + Î»L,T1,T2. ââT. T1 â¡ T & L â¢ T â¶* [0, |L|] T2. + +interpretation + "context-sensitive parallel reduction (term)" + 'PRed L T1 T2 = (cpr L T1 T2). + +(* Basic properties *********************************************************) + +lemma cpr_intro: âL,T1,T,T2,d,e. T1 â¡ T â L â¢ T â¶* [d, e] T2 â L â¢ T1 â¡ T2. +/3 width=5/ qed-. + +(* Basic_1: was by definition: pr2_free *) +lemma cpr_tpr: âT1,T2. T1 â¡ T2 â âL. L â¢ T1 â¡ T2. +/2 width=3/ qed. + +lemma cpr_tpss: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¡ T2. +/3 width=5/ qed. + +lemma cpr_refl: âL,T. L â¢ T â¡ T. +/2 width=1/ qed. + +(* Note: new property *) +(* Basic_1: was only: pr2_thin_dx *) +lemma cpr_flat: âI,L,V1,V2,T1,T2. + L â¢ V1 â¡ V2 â L â¢ T1 â¡ T2 â L â¢ â{I} V1. T1 â¡ â{I} V2. T2. +#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/ +qed. + +lemma cpr_cast: âL,V,T1,T2. + L â¢ T1 â¡ T2 â L â¢ âV. T1 â¡ 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: âL1,T1,T2. L1 â¢ T1 â¡ T2 â + âL2. L2 â [0, |L1|] L1 â L2 â¢ T1 â¡ T2. +#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12 +lapply (tpss_lsubr_trans â¦ HT2 â¦ HL12) -HT2 -HL12 /3 width=4/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Basic_1: was: pr2_gen_csort *) +lemma cpr_inv_atom: âT1,T2. â â¢ T1 â¡ T2 â T1 â¡ T2. +#T1 #T2 * #T #HT normalize #HT2 +<(tpss_inv_refl_O2 â¦ HT2) -HT2 // +qed-. + +(* Basic_1: was: pr2_gen_sort *) +lemma cpr_inv_sort1: âL,T2,k. L â¢ âk â¡ T2 â T2 = âk. +#L #T2 #k * #X #H +>(tpr_inv_atom1 â¦ H) -H #H +>(tpss_inv_sort1 â¦ H) -H // +qed-. + +(* Basic_1: was: pr2_gen_cast *) +lemma cpr_inv_cast1: âL,V1,T1,U2. L â¢ âV1. T1 â¡ U2 â ( + ââV2,T2. L â¢ V1 â¡ V2 & L â¢ T1 â¡ T2 & + U2 = âV2. T2 + ) â¨ L â¢ T1 â¡ U2. +#L #V1 #T1 #U2 * #X #H #HU2 +elim (tpr_inv_cast1 â¦ H) -H /3 width=3/ +* #V #T #HV1 #HT1 #H destruct +elim (tpss_inv_flat1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cpr_fwd_bind1_minus: âI,L,V1,T1,T. L â¢ -â{I}V1.T1 â¡ T â âb. + ââV2,T2. L â¢ â{b,I}V1.T1 â¡ â{b,I}V2.T2 & + T = -â{I}V2.T2. +#I #L #V1 #T1 #T * #X #H1 #H2 #b +elim (tpr_fwd_bind1_minus â¦ H1 b) -H1 #V0 #T0 #HT10 #H destruct +elim (tpss_inv_bind1 â¦ H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ +qed-. + +lemma cpr_fwd_shift1: âL,L1,T1,T. L â¢ L1 @@ T1 â¡ T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#L #L1 #T1 #T * #X #H1 #H2 +elim (tpr_fwd_shift1 â¦ H1) -H1 #L0 #T0 #HL10 #H destruct +elim (tpss_fwd_shift1 â¦ 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_aaa.etc new file mode 100644 index 000000000..7c1273334 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_aaa.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,A. L â¢ T1 â A â âT2. L â¢ T1 â¡ T2 â L â¢ T2 â A. +#L #T1 #A #HT1 #T2 * /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_cpr.etc new file mode 100644 index 000000000..d231c849f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_cpr.etc @@ -0,0 +1,63 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âa,I,L,V1,V2,T1,T2. L â¢ V1 â¡ V2 â T1 â¡ T2 â + L â¢ â{a,I} V1. T1 â¡ â{a,I} V2. T2. +#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12 +@ex2_intro [2: @(tpr_delta â¦ HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *) +qed. + +(* Basic_1: was only: pr2_gen_cbind *) +lemma cpr_bind_dx: âa,I,L,V1,V2,T1,T2. V1 â¡ V2 â L. â{I} V2 â¢ T1 â¡ T2 â + L â¢ â{a,I} V1. T1 â¡ â{a,I} V2. T2. +#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2 +elim (tpss_split_up â¦ HT2 1 ? ?) -HT2 // #T0 (tpr_inv_atom1 â¦ H) -H #H +elim (tpss_inv_lref1 â¦ H) -H /2 width=1/ +* /3 width=6/ +qed-. + +(* Basic_1: was pr2_gen_abbr *) +lemma cpr_inv_abbr1: âa,L,V1,T1,U2. L â¢ â{a}V1. T1 â¡ U2 â + (ââV,V2,T2. V1 â¡ V & L â¢ V â¶* [O, |L|] V2 & + L. âV â¢ T1 â¡ T2 & + U2 = â{a}V2. T2 + ) â¨ + ââT2. L.âV1 â¢ T1 â¡ T2 & â§[0,1] U2 â¡ T2 & a = true. +#a #L #V1 #T1 #Y * #X #H1 #H2 +elim (tpr_inv_abbr1 â¦ H1) -H1 * +[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct + elim (tpss_inv_bind1 â¦ H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct + lapply (tps_lsubr_trans â¦ HT0 (L. âV) ?) -HT0 /2 width=1/ #HT0 + lapply (tps_weak_full â¦ HT0) -HT0 #HT0 + lapply (tpss_lsubr_trans â¦ HT02 (L. âV) ?) -HT02 /2 width=1/ #HT02 + lapply (tpss_weak_full â¦ HT02) -HT02 #HT02 + lapply (tpss_strap2 â¦ HT0 HT02) -T0 /4 width=7/ +| #T2 #HT12 #HXT2 #H destruct + elim (lift_total Y 0 1) #Z #HYZ + lapply (tpss_lift_ge â¦ H2 (L.âV1) â¦ HXT2 â¦ HYZ) -X // /2 width=1/ #H + lapply (cpr_intro â¦ HT12 â¦ H) -T2 /3 width=3/ +] +qed-. + +(* Basic_1: was: pr2_gen_abst *) +lemma cpr_inv_abst1: âa,L,V1,T1,U2. L â¢ â{a}V1. T1 â¡ U2 â âI,W. + ââV2,T2. L â¢ V1 â¡ V2 & L. â{I} W â¢ T1 â¡ T2 & U2 = â{a}V2. T2. +#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W +elim (tpr_inv_abst1 â¦ H1) -H1 #V #T #HV1 #HT1 #H destruct +elim (tpss_inv_bind1 â¦ H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct +lapply (tpss_lsubr_trans â¦ HT2 (L. â{I} W) ?) -HT2 /2 width=1/ /4 width=5/ +qed-. + +(* Basic_1: was pr2_gen_appl *) +lemma cpr_inv_appl1: âL,V1,U0,U2. L â¢ âV1. U0 â¡ U2 â + â¨â¨ ââV2,T2. L â¢ V1 â¡ V2 & L â¢ U0 â¡ T2 & + U2 = âV2. T2 + | ââa,V2,W,T1,T2. L â¢ V1 â¡ V2 & L. âV2 â¢ T1 â¡ T2 & + U0 = â{a}W. T1 & + U2 = â{a}V2. T2 + | ââa,V2,V,W1,W2,T1,T2. L â¢ V1 â¡ V2 & L â¢ W1 â¡ W2 & L. âW2 â¢ T1 â¡ T2 & + â§[0,1] V2 â¡ V & + U0 = â{a}W1. T1 & + U2 = â{a}W2. âV. T2. +#L #V1 #U0 #Y * #X #H1 #H2 +elim (tpr_inv_appl1 â¦ H1) -H1 * +[ #V #U #HV1 #HU0 #H destruct + elim (tpss_inv_flat1 â¦ 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 â¦ H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct + lapply (tpss_weak â¦ 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 â¦ H2) -H2 #W2 #X #HW02 #HX #HY destruct + elim (tpss_inv_flat1 â¦ HX) -HX #V2 #T2 #HV2 #HT2 #H destruct + elim (tpss_inv_lift1_ge â¦ HV2 â¦ HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V (ltpr_fwd_length â¦ HL12) in HT2; #HT2 +elim (ltpr_tpr_tpss_conf â¦ HL12 â¦ HT2) -L1 /3 width=3/ +qed. + +lemma cpr_ltpr_conf_tpss: âL1,L2. L1 â¡ L2 â âT1,T2. L1 â¢ T1 â¡ T2 â + âd,e,U1. L1 â¢ T1 â¶* [d, e] U1 â + ââU2. L2 â¢ U1 â¡ U2 & L2 â¢ T2 â¡ U2. +#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 +elim (cpr_ltpr_conf_eq â¦ HT12 â¦ HL12) -HT12 #T #HT1 #HT2 +elim (cpr_tpss_ltpr â¦ HL12 â¦ HT1 â¦ HTU1) -L1 -HT1 #U2 #HU12 #HTU2 +lapply (tpss_weak_full â¦ HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_ltpss_dx.etc new file mode 100644 index 000000000..5d90e6865 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_ltpss_dx.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T,U1. L1 â¢ T â¡ U1 â + âL2,d,e. L1 â¶* [d, e] L2 â + ââU2. L2 â¢ T â¡ U2 & L2 â¢ U1 â¶* [d, e] U2. +#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12 +lapply (ltpss_dx_fwd_length â¦ HL12) #H >H in HU1; -H #HU1 +elim (ltpss_dx_tpss_conf â¦ HU1 â¦ HL12) -L1 /3 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_ltpss_sn.etc new file mode 100644 index 000000000..f946e4cfb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_ltpss_sn.etc @@ -0,0 +1,35 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â + âT1,T2. L2 â¢ T1 â¡ T2 â L1 â¢ T1 â¡ T2. +#L1 #L2 #d #e #HL12 #T1 #T2 * +lapply (ltpss_sn_weak_full â¦ HL12) +<(ltpss_sn_fwd_length â¦ HL12) -HL12 /3 width=5/ +qed. + +lemma ltpss_sn_cpr_conf: âL1,T,U1. L1 â¢ T â¡ U1 â + âL2,d,e. L1 â¢ â¶* [d, e] L2 â + ââU2. L2 â¢ T â¡ U2 & L1 â¢ U1 â¶* [d, e] U2. +#L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12 +lapply (ltpss_sn_fwd_length â¦ HL12) #H >H in HU1; -H #HU1 +elim (ltpss_sn_tpss_conf â¦ HU1 â¦ HL12) -HU1 -HL12 /3 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_tpss.etc new file mode 100644 index 000000000..d8c7225d9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cpr_tpss.etc @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,T. L â¢ T1 â¡ T â + âT2,d,e. L â¢ T â¶* [d, e] T2 â L â¢ T1 â¡ T2. +#L #T1 #T * #T0 #HT10 #HT0 #T2 #d #e #HT2 +lapply (tpss_weak_full â¦ HT2) -HT2 #HT2 +lapply (tpss_trans_eq â¦ HT0 HT2) -T /2 width=3/ +qed. + +lemma cpr_tps_trans: âL,T1,T. L â¢ T1 â¡ T â + âT2,d,e. L â¢ T â¶ [d, e] T2 â L â¢ T1 â¡ T2. +/3 width=5/ qed. + +lemma cpr_tpss_conf: âL,T0,T1. L â¢ T0 â¡ T1 â + âT2,d,e. L â¢ T0 â¶* [d, e] T2 â + ââT. L â¢ T1 â¶* [d, e] T & L â¢ T2 â¡ T. +#L #T0 #T1 * #U0 #HTU0 #HU0T1 #T2 #d #e #HT02 +elim (tpr_tpss_conf â¦ HTU0 â¦ HT02) -T0 #T0 #HT20 #HUT0 +elim (tpss_conf_eq â¦ HU0T1 â¦ HUT0) -U0 /3 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpr.etc new file mode 100644 index 000000000..2682a7609 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpr.etc @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T1,U1,d,e. L1 â¢ T1 â¶* [d, e] U1 â + âL2. L1 â¡ L2 â âT2. L2 â¢ T1 â¡* T2 â + ââU2. L2 â¢ U1 â¡* U2 & L2 â¢ T2 â¶* [d, e] U2. +#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2 +[ #T2 #HT12 + elim (cpr_tpss_ltpr â¦ HL12 â¦ HT12 â¦ HTU1) -L1 -T1 /3 width=3/ +| #T #T2 #_ #HT2 * #U #HU1 #HTU + elim (cpr_tpss_ltpr â¦ HT2 â¦ HTU) -L1 -T // /3 width=3/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpss_dx.etc new file mode 100644 index 000000000..7f8618007 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpss_dx.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T,U1. L1 â¢ T â¡* U1 â + âL2,d,e. L1 â¶* [d, e] L2 â + ââU2. L2 â¢ T â¡* U2 & L2 â¢ U1 â¶* [d, e] U2. +#L1 #T #U1 #H @(cprs_ind â¦ H) -U1 /2 width=3/ +#T1 #U1 #_ #HTU1 #IHT1 #L2 #d #e #HL12 +elim (IHT1 â¦ HL12) -IHT1 #U #HTU #HT1U +elim (ltpss_dx_cpr_conf â¦ HTU1 â¦ HL12) -L1 #U0 #HT1U0 #HU10 +elim (cpr_tpss_conf â¦ HT1U0 â¦ HT1U) -T1 #U2 #HU02 #HU2 +lapply (tpss_trans_eq â¦ HU10 HU02) -U0 /3 width=3/ +qed-. + +lemma cprs_ltpss_dx_tpss_conf: âL1,T1,U1. L1 â¢ T1 â¡* U1 â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â + ââU2. L2 â¢ T2 â¡* U2 & L2 â¢ U1 â¶* [d, e] U2. +#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 +elim (cprs_ltpss_dx_conf â¦ HTU1 â¦ HL12) -L1 #U #HT1U #HU1 +elim (cprs_tpss_conf â¦ HT1U â¦ HT12) -T1 #T #HUT #HT2 +lapply (tpss_trans_eq â¦ HU1 HUT) -U /2 width=3/ +qed-. + +lemma cprs_ltpss_dx_tpss2_conf: âL1,T1,U1. L1 â¢ T1 â¡* U1 â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â + âU2. L2 â¢ U1 â¶* [d, e] U2 â + ââU. L2 â¢ T2 â¡* U & L2 â¢ U2 â¶* [d, e] U. +#L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 #U2 #HU12 +elim (cprs_ltpss_dx_tpss_conf â¦ HTU1 â¦ HL12 â¦ HT12) -L1 -T1 #U #HT2U #HU1 +elim (tpss_conf_eq â¦ HU12 â¦ HU1) -U1 #U0 #HU20 #HU0 +lapply (cprs_tpss_trans â¦ HT2U â¦ HU0) -U /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpss_sn.etc new file mode 100644 index 000000000..32009d11d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_ltpss_sn.etc @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â + âT,U1. L1 â¢ T â¡* U1 â + ââU2. L2 â¢ T â¡* U2 & L1 â¢ U1 â¶* [d, e] U2. +#L1 #L2 #d #e #H +lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 /2 width=3/ +#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1 +lapply (ltpssa_ltpss_sn â¦ HL1) -HL1 #HL1 +lapply (ltpss_sn_dx_trans_eq â¦ HL1 â¦ HL2) -HL1 #HL12 +elim (IHL1 â¦ HTU1) -IHL1 -HTU1 #U #HTU #HU1 +elim (cprs_ltpss_dx_conf â¦ HTU â¦ HL2) -HTU -HL2 #U2 #HTU2 #HU2 +lapply (ltpss_sn_tpss_trans_eq â¦ HU2 â¦ HL12) -HU2 -HL12 #HU2 +lapply (tpss_trans_eq â¦ HU1 HU2) -U /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_tpss.etc new file mode 100644 index 000000000..0c575ae02 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cprs_tpss.etc @@ -0,0 +1,38 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,T. L â¢ T1 â¡* T â + âT2,d,e. L â¢ T â¶* [d, e] T2 â L â¢ T1 â¡* T2. +#L #T1 #T #H @(cprs_ind â¦ H) -T /2 width=3/ /3 width=5/ +qed. + +lemma cprs_tps_trans: âL,T1,T. L â¢ T1 â¡* T â + âT2,d,e. L â¢ T â¶ [d, e] T2 â L â¢ T1 â¡* T2. +/3 width=5 by inj, cprs_tpss_trans/ qed. (**) (* auto too slow without trace *) + +lemma cprs_tpss_conf: âL,T0,T1. L â¢ T0 â¡* T1 â + âT2,d,e. L â¢ T0 â¶* [d, e] T2 â + ââT. L â¢ T1 â¶* [d, e] T & L â¢ T2 â¡* T. +#L #T0 #T1 #H @(cprs_ind â¦ H) -T1 /2 width=3/ +#T #T1 #_ #HT1 #IHT0 #T2 #d #e #HT02 +elim (IHT0 â¦ HT02) -T0 #T0 #HT0 #HT20 +elim (cpr_tpss_conf â¦ HT1 â¦ HT0) -T /3 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/dxprs_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/dxprs_ltpss_dx.etc new file mode 100644 index 000000000..8e3260dd4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/dxprs_ltpss_dx.etc @@ -0,0 +1,50 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,T,U1. â¦h, L1â¦ â¢ T â¢*â¡*[g] U1 â + âL2,d,e. L1 â¶* [d, e] L2 â + ââU2. â¦h, L2â¦ â¢ T â¢*â¡*[g] U2 & L2 â¢ U1 â¶* [d, e] U2. +#h #g #L1 #T #U1 * #U #HTU #HU1 #L2 #d #e #HL12 +elim (sstas_ltpss_dx_conf â¦ HTU â¦ HL12) -HTU #U0 #HTU0 #HU0 +elim (cprs_ltpss_dx_conf â¦ HU1 â¦ HL12) -L1 #U2 #HU2 #HU12 +elim (cprs_tpss_conf â¦ HU2 â¦ HU0) -U #U #HU2 #HU0 +lapply (tpss_trans_eq â¦ HU12 HU2) -U2 /3 width=3/ +qed-. + +lemma dxprs_tpss_conf: âh,g,L,T1,U1. â¦h, Lâ¦ â¢ T1 â¢*â¡*[g] U1 â + âT2,d,e. L â¢ T1 â¶* [d, e] T2 â + ââU2. â¦h, Lâ¦ â¢ T2 â¢*â¡*[g] U2 & L â¢ U1 â¶* [d, e] U2. +#h #g #L #T1 #U1 * #W1 #HTW1 #HWU1 #T2 #d #e #HT12 +elim (sstas_tpss_conf â¦ HTW1 â¦ HT12) -T1 #W2 #HTW2 #HW12 +elim (cprs_tpss_conf â¦ HWU1 â¦ HW12) -W1 /3 width=3/ +qed-. + +lemma dxprs_ltpss_dx_tpss_conf: âh,g,L1,T1,U1. â¦h, L1â¦ â¢ T1 â¢*â¡*[g] U1 â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â + ââU2. â¦h, L2â¦ â¢ T2 â¢*â¡*[g] U2 & + L2 â¢ U1 â¶* [d, e] U2. +#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12 +elim (dxprs_ltpss_dx_conf â¦ HTU1 â¦ HL12) -L1 #U #HT1U #HU1 +elim (dxprs_tpss_conf â¦ HT1U â¦ HT12) -T1 #U2 #HTU2 #HU2 +lapply (tpss_trans_eq â¦ HU1 HU2) -U /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/dxprs_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/dxprs_ltpss_sn.etc new file mode 100644 index 000000000..553e4658b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/dxprs_ltpss_sn.etc @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,L2,d,e. L1 â¢ â¶* [d, e] L2 â + âT,U1. â¦h, L1â¦ â¢ T â¢*â¡*[g] U1 â + ââU2. â¦h, L2â¦ â¢ T â¢*â¡*[g] U2 & L1 â¢ U1 â¶* [d, e] U2. +#h #g #L1 #L2 #d #e #H +lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 [ /2 width=3/ ] +#L #L2 #HL1 #HL2 #IHL1 #T #U1 #HTU1 +lapply (ltpssa_ltpss_sn â¦ HL1) -HL1 #HL1 +lapply (ltpss_sn_dx_trans_eq â¦ HL1 â¦ HL2) -HL1 #HL12 +elim (IHL1 â¦ HTU1) -IHL1 -HTU1 #U #HTU #HU1 +elim (dxprs_ltpss_dx_conf â¦ HTU â¦ HL2) -HTU -HL2 #U2 #HTU2 #HU2 +lapply (ltpss_sn_tpss_trans_eq â¦ HU2 â¦ HL12) -HU2 -HL12 #HU2 +lapply (tpss_trans_eq â¦ HU1 HU2) -U /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lbotr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lbotr.etc new file mode 100644 index 000000000..df6b2c11e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lbotr.etc @@ -0,0 +1,94 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âI,L,K,V,i. â©[0, i] L â¡ K. â{I}V â âd,e. â[d, e] L â + d â¤ i â i < d + e â I = Abbr. +#I #L elim L -L +[ #K #V #i #H + lapply (ldrop_inv_atom1 â¦ H) -H #H destruct +| #L #J #W #IHL #K #V #i #H + elim (ldrop_inv_O1 â¦ H) -H * + [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct + lapply (le_n_O_to_eq â¦ Hdi) -Hdi #H destruct + lapply (HL â¦ (L.âW) ?) -HL /2 width=1/ #H + elim (lsubr_inv_abbr2 â¦ H ?) -H // -Hide #K #_ #H destruct // + | #Hi #HLK #d @(nat_ind_plus â¦ d) -d + [ #e #H #_ #Hide + elim (lbotr_inv_bind â¦ H ?) -H [2: /2 width=2/ ] #HL #H destruct + @(IHL â¦ HLK â¦ HL) -IHL -HLK -HL // /2 width=1/ + | #d #_ #e #H #Hdi #Hide + lapply (lbotr_inv_skip â¦ H ?) -H // #HL + @(IHL â¦ HLK â¦ HL) -IHL -HLK -HL /2 width=1/ + ] + ] +] +qed-. + +(* Properties about local env. full refinement for substitution *************) + +(* Note: ldrop_ldrop not needed *) +lemma lbotr_ldrop: âL,d,e. + (âI,K,V,i. d â¤ i â i < d + e â â©[0, i] L â¡ K. â{I}V â I = Abbr) â + â[d, e] L. +#L elim L -L // +#L #I #V #IHL #d @(nat_ind_plus â¦ d) -d +[ #e @(nat_ind_plus â¦ 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: âL1,L2,d,e. â©[d, e] L1 â¡ L2 â âdd,ee. â[dd, ee] L1 â + dd + ee â¤ d â â[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 â¦ Hiddee Hddee) -Hddee #Hid +elim (ldrop_trans_le â¦ HL12 â¦ HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H +elim (ldrop_inv_skip2 â¦ H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct +@(lbotr_inv_ldrop â¦ HLK1 â¦ HL1) -L1 -K1 -V1 // +qed. + +lemma lbotr_ldrop_trans_be_up: âL1,L2,d,e. â©[d, e] L1 â¡ L2 â + âdd,ee. â[dd, ee] L1 â + dd â¤ d + e â d + e â¤ dd + ee â + â[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)â¦ Hdde ?) -Hdde /2 width=1/ #Hddie +>commutative_plus in Hiddee; >minus_minus_comm commutative_plus // -Hddie /2 width=1/ +qed. + +lemma lbotr_ldrop_trans_ge: âL1,L2,d,e. â©[d, e] L1 â¡ L2 â âdd,ee. â[dd, ee] L1 â + d + e â¤ dd â â[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 â¦ Hddee) -Hddee #Hdde #Hedd +>plus_minus in Hiddee; // #Hiddee +lapply (transitive_le â¦ Hdde Hddi) -Hdde #Hid +lapply (ldrop_trans_ge â¦ HL12 â¦ HLK2 ?) -L2 // -Hid #HL1K2 +@(lbotr_inv_ldrop â¦ HL1K2 â¦ HL1) -L1 >commutative_plus /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lpx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lpx.etc new file mode 100644 index 000000000..d23ed28e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lpx.etc @@ -0,0 +1,68 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âR. t_deliftable_sn R â dropable_sn (lpx R). +#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e +[ #d #e #X #H >(lpx_inv_atom1 â¦ H) -H /2 width=3/ +| #K1 #I #V1 #X #H + elim (lpx_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H + elim (lpx_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 â¦ HL12) -L1 /3 width=3/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H + elim (lpx_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (HR â¦ HV12 â¦ HWV1) -V1 + elim (IHLK1 â¦ HL12) -L1 /3 width=5/ +] +qed. + +lemma lpx_liftable_dedropable: âR. reflexive ? R â + t_liftable R â 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 â¦ H) -H /2 width=3/ +| #K1 #I #V1 #X #H + elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 + elim (IHLK1 â¦ HK12) -K1 /3 width=5/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H + elim (lpx_inv_pair1 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (lift_total W2 d e) #V2 #HWV2 + lapply (H2R â¦ HW12 â¦ HWV1 â¦ HWV2) -W1 + elim (IHLK1 â¦ HK12) -K1 /3 width=5/ +] +qed. + +fact lpx_dropable_aux: âR,L2,K2,d,e. â©[d, e] L2 â¡ K2 â âL1. lpx R L1 L2 â + d = 0 â ââK1. â©[0, e] L1 â¡ K1 & lpx R K1 K2. +#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e +[ #d #e #X #H >(lpx_inv_atom2 â¦ H) -H /2 width=3/ +| #K2 #I #V2 #X #H + elim (lpx_inv_pair2 â¦ H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/ +| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_ + elim (lpx_inv_pair2 â¦ H) -H #L1 #V1 #HL12 #HV12 #H destruct + elim (IHLK2 â¦ HL12 ?) -L2 // /3 width=3/ +| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_ + >commutative_plus normalize #H destruct +] +qed-. + +lemma lpx_dropable: âR. dropable_dx (lpx R). +/2 width=5 by lpx_dropable_aux/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lenv_px.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lenv_px.etc new file mode 100644 index 000000000..ea916e10a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lenv_px.etc @@ -0,0 +1,166 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| lpx_stom: lpx R (â) (â) +| lpx_pair: âI,K1,K2,V1,V2. + lpx R K1 K2 â R V1 V2 â lpx R (K1. â{I} V1) (K2. â{I} V2) +. + +(* Basic inversion lemmas ***************************************************) + +fact lpx_inv_atom1_aux: âR,L1,L2. lpx R L1 L2 â L1 = â â L2 = â. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_inv_atom1: âR,L2. lpx R (â) L2 â L2 = â. +/2 width=4 by lpx_inv_atom1_aux/ qed-. + +fact lpx_inv_pair1_aux: âR,L1,L2. lpx R L1 L2 â âI,K1,V1. L1 = K1. â{I} V1 â + ââK2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. â{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: âR,I,K1,V1,L2. lpx R (K1. â{I} V1) L2 â + ââK2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. â{I} V2. +/2 width=3 by lpx_inv_pair1_aux/ qed-. + +fact lpx_inv_atom2_aux: âR,L1,L2. lpx R L1 L2 â L2 = â â L1 = â. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_inv_atom2: âR,L1. lpx R L1 (â) â L1 = â. +/2 width=4 by lpx_inv_atom2_aux/ qed-. + +fact lpx_inv_pair2_aux: âR,L1,L2. lpx R L1 L2 â âI,K2,V2. L2 = K2. â{I} V2 â + ââK1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. â{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: âR,I,L1,K2,V2. lpx R L1 (K2. â{I} V2) â + ââK1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. â{I} V1. +/2 width=3 by lpx_inv_pair2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lpx_fwd_length: âR,L1,L2. lpx R L1 L2 â |L1| = |L2|. +#R #L1 #L2 #H elim H -L1 -L2 normalize // +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lpx_inv_append1: âR,L1,K1,L. lpx R (K1 @@ L1) L â + ââK2,L2. lpx R K1 K2 & lpx R L1 L2 & L = K2 @@ L2. +#R #L1 elim L1 -L1 normalize +[ #K1 #K2 #HK12 + @(ex3_2_intro â¦ K2 (â)) // (**) (* explicit constructor, /2 width=5/ does not work *) +| #L1 #I #V1 #IH #K1 #X #H + elim (lpx_inv_pair1 â¦ H) -H #L #V2 #H1 #HV12 #H destruct + elim (IH â¦ H1) -IH -H1 #K2 #L2 #HK12 #HL12 #H destruct + @(ex3_2_intro â¦ HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *) +] +qed-. + +lemma lpx_inv_append2: âR,L2,K2,L. lpx R L (K2 @@ L2) â + ââK1,L1. lpx R K1 K2 & lpx R L1 L2 & L = K1 @@ L1. +#R #L2 elim L2 -L2 normalize +[ #K2 #K1 #HK12 + @(ex3_2_intro â¦ K1 (â)) // (**) (* explicit constructor, /2 width=5/ does not work *) +| #L2 #I #V2 #IH #K2 #X #H + elim (lpx_inv_pair2 â¦ H) -H #L #V1 #H1 #HV12 #H destruct + elim (IH â¦ H1) -IH -H1 #K1 #L1 #HK12 #HL12 #H destruct + @(ex3_2_intro â¦ HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *) +] +qed-. + +(* Basic properties *********************************************************) + +lemma lpx_refl: âR. reflexive ? R â reflexive â¦ (lpx R). +#R #HR #L elim L -L // /2 width=1/ +qed. + +lemma lpx_sym: âR. symmetric ? R â symmetric â¦ (lpx R). +#R #HR #L1 #L2 #H elim H -H // /3 width=1/ +qed. + +lemma lpx_trans: âR. Transitive ? R â Transitive â¦ (lpx R). +#R #HR #L1 #L #H elim H -L // +#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H +elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ +qed. + +lemma lpx_conf: âR. confluent ? R â confluent â¦ (lpx R). +#R #HR #L0 #L1 #H elim H -L1 +[ #X #H >(lpx_inv_atom1 â¦ H) -X /2 width=3/ +| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H + elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK02 #HV02 #H destruct + elim (IHK01 â¦ HK02) -K0 #K #HK1 #HK2 + elim (HR â¦ HV01 â¦ HV02) -HR -V0 /3 width=5/ +] +qed. + +lemma lpx_TC_inj: âR,L1,L2. lpx R L1 L2 â lpx (TC â¦ R) L1 L2. +#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/ +qed. + +lemma lpx_TC_step: âR,L1,L. lpx (TC â¦ R) L1 L â + âL2. lpx R L L2 â lpx (TC â¦ 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 â¦ H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ +qed. + +lemma TC_lpx_pair_dx: âR. reflexive ? R â + âI,K,V1,V2. TC â¦ R V1 V2 â + TC â¦ (lpx R) (K.â{I}V1) (K.â{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: âR. reflexive ? R â + âI,V,K1,K2. TC â¦ (lpx R) K1 K2 â + TC â¦ (lpx R) (K1.â{I}V) (K2.â{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: âR,L1,L2. TC â¦ (lpx R) L1 L2 â lpx (TC â¦ R) L1 L2. +#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/ +qed. + +lemma lpx_inv_TC: âR. reflexive ? R â + âL1,L2. lpx (TC â¦ R) L1 L2 â TC â¦ (lpx R) L1 L2. +#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/ +qed. + +lemma lpx_append: âR,K1,K2. lpx R K1 K2 â âL1,L2. lpx R L1 L2 â + lpx R (L1 @@ K1) (L2 @@ K2). +#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lenv_top.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lenv_top.etc new file mode 100644 index 000000000..ab90cebe7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lenv_top.etc @@ -0,0 +1,68 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 ð 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 â Î»T1,T2. True. + +definition ltop: relation lenv â lpx ttop. + +interpretation + "top reduction (environment)" + 'RTop L1 L2 = (ltop L1 L2). + +(* Basic properties *********************************************************) + +lemma ltop_refl: reflexive â¦ ltop. +/2 width=1/ qed. + +lemma ltop_sym: symmetric â¦ ltop. +/2 width=1/ qed. + +lemma ltop_trans: transitive â¦ ltop. +/2 width=3/ qed. + +lemma ltop_append: âK1,K2. K1 ð K2 â âL1,L2. L1 ð L2 â L1 @@ K1 ð L2 @@ K2. +/2 width=1/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma ltop_inv_atom1: âL2. â ð L2 â L2 = â. +/2 width=2 by lpx_inv_atom1/ qed-. + +lemma ltop_inv_pair1: âK1,I,V1,L2. K1. â{I} V1 ð L2 â + ââK2,V2. K1 ð K2 & L2 = K2. â{I} V2. +#K1 #I #V1 #L2 #H +elim (lpx_inv_pair1 â¦ H) -H /2 width=4/ +qed-. + +lemma ltop_inv_atom2: âL1. L1 ð â â L1 = â. +/2 width=2 by lpx_inv_atom2/ qed-. + +lemma ltop_inv_pair2: âL1,K2,I,V2. L1 ð K2. â{I} V2 â + ââK1,V1. K1 ð K2 & L1 = K1. â{I} V1. +#L1 #K2 #I #V2 #H +elim (lpx_inv_pair2 â¦ H) -H /2 width=4/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma ltop_fwd_length: âL1,L2. L1 ð L2 â |L1| = |L2|. +/2 width=2 by lpx_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr.etc new file mode 100644 index 000000000..ea9f585a6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr.etc @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1 â¦ â¡ break â¦ term 46 L2 â¦ )" + 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 â + Î»L1,L2. ââL. L1 â¡ L & L â¢ â¶* [0, |L|] L2 +. + +interpretation + "focalized parallel reduction (environment)" + 'FocalizedPRed L1 L2 = (lfpr L1 L2). + +(* Basic properties *********************************************************) + +(* Note: lemma 250 *) +lemma lfpr_refl: âL. â¦Lâ¦ â¡ â¦Lâ¦. +/2 width=3/ qed. + +lemma ltpss_sn_lfpr: âL1,L2,d,e. L1 â¢ â¶* [d, e] L2 â â¦L1â¦ â¡ â¦L2â¦. +/3 width=5/ qed. + +lemma ltpr_lfpr: âL1,L2. L1 â¡ L2 â â¦L1â¦ â¡ â¦L2â¦. +/3 width=3/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma lfpr_inv_atom1: âL2. â¦ââ¦ â¡ â¦L2â¦ â L2 = â. +#L2 * #L #HL >(ltpr_inv_atom1 â¦ HL) -HL #HL2 >(ltpss_sn_inv_atom1 â¦ HL2) -HL2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_aaa.etc new file mode 100644 index 000000000..6f6c49df3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_aaa.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T,A. L1 â¢ T â A â âL2. â¦L1â¦ â¡ â¦L2â¦ â L2 â¢ T â A. +#L1 #T #A #HT #L2 * /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_cpr.etc new file mode 100644 index 000000000..8a01f263a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_cpr.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â âV1,V2. L2 â¢ V1 â¡ V2 â + âI. â¦L1. â{I} V1â¦ â¡ â¦L2. â{I} V2â¦. +#L1 #L2 * #L #HL1 #HL2 #V1 #V2 * +<(ltpss_sn_fwd_length â¦ HL2) #V #HV1 #HV2 #I +lapply (ltpss_sn_tpss_trans_eq â¦ HV2 â¦ HL2) -HV2 #V2 +@(ex2_intro â¦ (L.â{I}V)) /2 width=1/ (**) (* explicit constructor *) +qed. + +(* Properties on supclosure *************************************************) +(* +lamma fsub_cpr_trans: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. L2 â¢ T2 â¡ U2 â + ââL,U1. â¦L1â¦ â¡ â¦Lâ¦ & L â¢ T1 â¡ U1 & â¦L, U1â¦ â â¦L2, U2â¦. +#L1 #L2 #T1 #T2 #HT12 #U2 * #T #H1 #H2 +elim (fsub_tpr_trans â¦ HT12 â¦ H1) -T2 #L #U #HL1 #HT1U #HUT +elim (fsup_tpss_trans_full â¦ HUT â¦ 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 â¦ HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2 +elim (lift_total T d e) #U #HTU +elim (ldrop_ltpr_trans â¦ HLK1 â¦ HK1) -HLK1 -HK1 #L #HL1 #HLK +lapply (tpr_lift â¦ HT1 â¦ HTU1 â¦ HTU) -HT1 -HTU1 /3 width=11/ +qed-. +*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_lfpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_lfpr.etc new file mode 100644 index 000000000..7031e792b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfpr_lfpr.etc @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,L1,L2. â¦L0â¦ â¡ â¦L1â¦ â â¦L0â¦ â¡ â¦L2â¦ â + ââL. â¦L1â¦ â¡ â¦Lâ¦ & â¦L2â¦ â¡ â¦Lâ¦. +#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2 +lapply (ltpr_fwd_length â¦ HK01) #H +>(ltpr_fwd_length â¦ HK02) in H; #H +elim (ltpr_conf â¦ HK01 â¦ HK02) -K0 #K #HK1 #HK2 +lapply (ltpss_sn_fwd_length â¦ HKL1) #H1 +lapply (ltpss_sn_fwd_length â¦ HKL2) #H2 +>H1 in HKL1 H; -H1 #HKL1 +>H2 in HKL2; -H2 #HKL2 #H +elim (ltpr_ltpss_sn_conf â¦ HKL1 â¦ HK1) -K1 #K1 #HK1 #HLK1 +elim (ltpr_ltpss_sn_conf â¦ HKL2 â¦ HK2) -K2 #K2 #HK2 #HLK2 +elim (ltpss_sn_conf â¦ HK1 â¦ HK2) -K #K #HK1 #HK2 +lapply (ltpr_fwd_length â¦ HLK1) #H1 +lapply (ltpr_fwd_length â¦ HLK2) #H2 +/3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfprs_ltprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfprs_ltprs.etc new file mode 100644 index 000000000..99ae801d7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lfprs_ltprs.etc @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. L1 â¡* L2 â â¦L1â¦ â¡* â¦L2â¦. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lift.etc new file mode 100644 index 000000000..f0d631ebb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lift.etc @@ -0,0 +1,25 @@ +definition t_liftable: relation term â Prop â + Î»R. âT1,T2. R T1 T2 â âU1,d,e. â§[d, e] T1 â¡ U1 â + âU2. â§[d, e] T2 â¡ U2 â R U1 U2. + +definition t_deliftable_sn: relation term â Prop â + Î»R. âU1,U2. R U1 U2 â âT1,d,e. â§[d, e] T1 â¡ U1 â + ââT2. â§[d, e] T2 â¡ U2 & R T1 T2. + +lemma t_liftable_TC: âR. t_liftable R â t_liftable (TC â¦ 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: âR. t_deliftable_sn R â t_deliftable_sn (TC â¦ R). +#R #HR #U1 #U2 #H elim H -U2 +[ #U2 #HU12 #T1 #d #e #HTU1 + elim (HR â¦ HU12 â¦ HTU1) -U1 /3 width=3/ +| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1 + elim (IHU1 â¦ HTU1) -U1 #T #HTU #HT1 + elim (HR â¦ HU2 â¦ HTU) -U /3 width=5/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lsubr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lsubr.etc new file mode 100644 index 000000000..8655a5e54 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lsubr.etc @@ -0,0 +1,198 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â [ 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 â nat â relation lenv â +| lsubr_sort: âd,e. lsubr d e (â) (â) +| lsubr_OO: âL1,L2. lsubr 0 0 L1 L2 +| lsubr_abbr: âL1,L2,V,e. lsubr 0 e L1 L2 â + lsubr 0 (e + 1) (L1. âV) (L2.âV) +| lsubr_abst: âL1,L2,I,V1,V2,e. lsubr 0 e L1 L2 â + lsubr 0 (e + 1) (L1. â{I}V1) (L2. âV2) +| lsubr_skip: âL1,L2,I1,I2,V1,V2,d,e. + lsubr d e L1 L2 â lsubr (d + 1) e (L1. â{I1} V1) (L2. â{I2} V2) +. + +interpretation + "local environment refinement (substitution)" + 'SubEq L1 d e L2 = (lsubr d e L1 L2). + +definition lsubr_trans: âS. (lenv â relation S) â Prop â Î»S,R. + âL2,s1,s2. R L2 s1 s2 â + âL1,d,e. L1 â [d, e] L2 â R L1 s1 s2. + +(* Basic properties *********************************************************) + +lemma lsubr_bind_eq: âL1,L2,e. L1 â [0, e] L2 â âI,V. + L1. â{I} V â [0, e + 1] L2.â{I} V. +#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/ +qed. + +lemma lsubr_abbr_lt: âL1,L2,V,e. L1 â [0, e - 1] L2 â 0 < e â + L1. âV â [0, e] L2.âV. +#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ +qed. + +lemma lsubr_abst_lt: âL1,L2,I,V1,V2,e. L1 â [0, e - 1] L2 â 0 < e â + L1. â{I}V1 â [0, e] L2. âV2. +#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ +qed. + +lemma lsubr_skip_lt: âL1,L2,d,e. L1 â [d - 1, e] L2 â 0 < d â + âI1,I2,V1,V2. L1. â{I1} V1 â [d, e] L2. â{I2} V2. +#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/ +qed. + +lemma lsubr_bind_lt: âI,L1,L2,V,e. L1 â [0, e - 1] L2 â 0 < e â + L1. âV â [0, e] L2. â{I}V. +* /2 width=1/ qed. + +lemma lsubr_refl: âd,e,L. L â [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: âS,R. lsubr_trans S R â lsubr_trans S (Î»L. (TC â¦ (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 â¦ Hs2 â¦ HL12) -HR -Hs2 -HL12 /3 width=3/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +fact lsubr_inv_atom1_aux: âL1,L2,d,e. L1 â [d, e] L2 â L1 = â â + L2 = â â¨ (d = 0 â§ 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: âL2,d,e. â â [d, e] L2 â + L2 = â â¨ (d = 0 â§ e = 0). +/2 width=3/ qed-. + +fact lsubr_inv_skip1_aux: âL1,L2,d,e. L1 â [d, e] L2 â + âI1,K1,V1. L1 = K1.â{I1}V1 â 0 < d â + ââI2,K2,V2. K1 â [d - 1, e] K2 & L2 = K2.â{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 â¦ H) +| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false â¦ H) +| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false â¦ H) +| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ +] +qed. + +lemma lsubr_inv_skip1: âI1,K1,L2,V1,d,e. K1.â{I1}V1 â [d, e] L2 â 0 < d â + ââI2,K2,V2. K1 â [d - 1, e] K2 & L2 = K2.â{I2}V2. +/2 width=5/ qed-. + +fact lsubr_inv_atom2_aux: âL1,L2,d,e. L1 â [d, e] L2 â L2 = â â + L1 = â â¨ (d = 0 â§ 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: âL1,d,e. L1 â [d, e] â â + L1 = â â¨ (d = 0 â§ e = 0). +/2 width=3/ qed-. + +fact lsubr_inv_abbr2_aux: âL1,L2,d,e. L1 â [d, e] L2 â + âK2,V. L2 = K2.âV â d = 0 â 0 < e â + ââK1. K1 â [0, e - 1] K2 & L1 = K1.âV. +#L1 #L2 #d #e * -L1 -L2 -d -e +[ #d #e #K1 #V #H destruct +| #L1 #L2 #K1 #V #_ #_ #H + elim (lt_zero_false â¦ 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: âL1,K2,V,e. L1 â [0, e] K2.âV â 0 < e â + ââK1. K1 â [0, e - 1] K2 & L1 = K1.âV. +/2 width=5/ qed-. + +fact lsubr_inv_skip2_aux: âL1,L2,d,e. L1 â [d, e] L2 â + âI2,K2,V2. L2 = K2.â{I2}V2 â 0 < d â + ââI1,K1,V1. K1 â [d - 1, e] K2 & L1 = K1.â{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 â¦ H) +| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false â¦ H) +| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false â¦ H) +| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ +] +qed. + +lemma lsubr_inv_skip2: âI2,L1,K2,V2,d,e. L1 â [d, e] K2.â{I2}V2 â 0 < d â + ââI1,K1,V1. K1 â [d - 1, e] K2 & L1 = K1.â{I1}V1. +/2 width=5/ qed-. + +(* Basic forward lemmas *****************************************************) + +fact lsubr_fwd_length_full1_aux: âL1,L2,d,e. L1 â [d, e] L2 â + d = 0 â e = |L1| â |L1| â¤ |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: âL1,L2. L1 â [0, |L1|] L2 â |L1| â¤ |L2|. +/2 width=5/ qed-. + +fact lsubr_fwd_length_full2_aux: âL1,L2,d,e. L1 â [d, e] L2 â + d = 0 â e = |L2| â |L2| â¤ |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: âL1,L2. L1 â [0, |L2|] L2 â |L2| â¤ |L1|. +/2 width=5/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lsubr_lbotr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lsubr_lbotr.etc new file mode 100644 index 000000000..b95a19b66 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/lsubr_lbotr.etc @@ -0,0 +1,77 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â [ 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 â nat â predicate lenv â + Î»d,e. NF_sn â¦ (lsubr d e) (lsubr d e â¦). + +interpretation + "local environment full refinement (substitution)" + 'SubEqBottom d e L = (lbotr d e L). + +(* Basic properties *********************************************************) + +lemma lbotr_atom: âd,e. â[d, e] â. +#d #e #L #H +elim (lsubr_inv_atom2 â¦ H) -H +[ #H destruct // +| * #H1 #H2 destruct // +] +qed. + +lemma lbotr_OO: âL. â[0, 0] L. +// qed. + +lemma lbotr_abbr: âL,V,e. â[0, e] L â â[0, e + 1] L.âV. +#L #V #e #HL #K #H +elim (lsubr_inv_abbr2 â¦ H ?) -H // (tpr_inv_atom1 â¦ H2) -Y +|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct +] +[ >(aaa_inv_sort â¦ H1) -X // +| elim (aaa_inv_lref â¦ H1) #I #K1 #V1 #HLK1 #HA + lapply (ldrop_pair2_fwd_fw â¦ HLK1 (#i)) #HKV1 + elim (ltpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #Y #H #HLK2 + elim (ltpr_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct + lapply (IH â¦ HKV1 â¦ HA â¦ HK12 â¦ HV12) -L1 -K1 -V1 /2 width=5/ +| elim (aaa_inv_gref â¦ H1) +| elim (aaa_inv_abbr â¦ H1) -H1 #B #HB #HA + elim (tpr_inv_abbr1 â¦ H2) -H2 * + [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct + lapply (tps_lsubr_trans â¦ HT2 (L2.âV2) ?) -HT2 /2 width=1/ #HT2 + lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB /width=5/ #HB + lapply (IH â¦ HA â¦ (L2.âV2) â¦ HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/ + | -B #T #HT1 #HXT #H destruct + lapply (IH â¦ HA â¦ (L2.âV1) â¦ HT1) /width=5/ -T1 /2 width=1/ -L1 #HA + @(aaa_inv_lift â¦ HA â¦ HXT) /2 width=1/ + ] +| elim (aaa_inv_abst â¦ H1) -H1 #B #A #HB #HA #H destruct + elim (tpr_inv_abst1 â¦ H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB /width=5/ #HB + lapply (IH â¦ HA â¦ (L2.âV2) â¦ HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/ +| elim (aaa_inv_appl â¦ H1) -H1 #B #HB #HA + elim (tpr_inv_appl1 â¦ H2) -H2 * + [ #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB -HV12 /width=5/ #HB + lapply (IH â¦ HA â¦ HL12 â¦ HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/ + | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + elim (aaa_inv_abst â¦ HA) -HA #B0 #A0 #HB0 #HA0 #H destruct + lapply (IH â¦ HB â¦ HL12 â¦ HV12) -HB -HV12 /width=5/ #HB + lapply (IH â¦ HB0 â¦ HL12 W2 ?) -HB0 /width=5/ #HB0 + lapply (IH â¦ HA0 â¦ (L2.âW2) â¦ 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 â¦ HA) -HA #B0 #HW0 #HT0 + lapply (IH â¦ HW0 â¦ HL12 â¦ HW02) -HW0 /width=5/ #HW2 + lapply (IH â¦ HB â¦ HL12 â¦ HV10) -HB -HV10 /width=5/ #HV0 + lapply (IH â¦ HT0 â¦ (L2.âW2) â¦ HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2 + @(aaa_abbr â¦ HW2) -HW2 + @(aaa_appl â¦ HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) + ] +| elim (aaa_inv_cast â¦ H1) -H1 #HV1 #HT1 + elim (tpr_inv_cast1 â¦ H2) -H2 + [ * #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH â¦ HV1 â¦ HL12 â¦ HV12) -HV1 -HV12 /width=5/ #HV2 + lapply (IH â¦ HT1 â¦ HL12 â¦ HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ + | -HV1 #HT1X + lapply (IH â¦ HT1 â¦ HL12 â¦ HT1X) -IH -HT1 -HL12 -HT1X /width=5/ + ] +] +qed. + +lemma aaa_ltpr_conf: âL1,T,A. L1 â¢ T â A â âL2. L1 â¡ L2 â L2 â¢ T â A. +/2 width=5/ qed. + +lemma aaa_tpr_conf: âL,T1,A. L â¢ T1 â A â âT2. T1 â¡ T2 â L â¢ T2 â A. +/2 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ldrop.etc new file mode 100644 index 000000000..02404a11d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ldrop.etc @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. T2 â¡ U2 â + ââL,U1. L1 â¡ L & T1 â¡ U1 & â¦L, U1â¦ â â¦L2, U2â¦. +#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 â¦ HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2 +elim (lift_total T d e) #U #HTU +elim (ldrop_ltpr_trans â¦ HLK1 â¦ HK1) -HLK1 -HK1 #L #HL1 #HLK +lapply (tpr_lift â¦ HT1 â¦ HTU1 â¦ HTU) -HT1 -HTU1 /3 width=11/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpr.etc new file mode 100644 index 000000000..4a27a6e70 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpr.etc @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0:lenv. âL1. L0 â¡ L1 â âL2. L0 â¡ L2 â + ââL. L1 â¡ L & L2 â¡ L. +#L0 #L1 #H elim H -L0 -L1 /2 width=3/ +#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H +elim (ltpr_inv_pair1 â¦ H) -H #K2 #V2 #HK02 #HV02 #H destruct +elim (IHK01 â¦ HK02) -K0 #K #HK1 #HK2 +elim (tpr_conf â¦ HV01 HV02) -V0 /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpss_dx.etc new file mode 100644 index 000000000..00730b732 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpss_dx.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,K1,d,e. L1 â¶* [d, e] K1 â âL2. L1 â¡ L2 â + ââK2. L2 â¶* [d, e] K2 & K1 â¡ K2. +#L1 #K1 #d #e #H elim H -L1 -K1 -d -e +[ /2 width=3/ +| #L1 #I #V1 #X #H + elim (ltpr_inv_pair1 â¦ 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 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 â¦ HL12) -L1 #K2 #HLK2 #HK12 + elim (ltpr_tpr_tpss_conf â¦ HK12 â¦ HV12 â¦ HVW1) -V1 /3 width=5/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H + elim (ltpr_inv_pair1 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 â¦ HL12) -L1 #K2 #HLK2 #HK12 + elim (ltpr_tpr_tpss_conf â¦ HK12 â¦ HV12 â¦ HVW1) -V1 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpss_sn.etc new file mode 100644 index 000000000..79ffb7c9e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_ltpss_sn.etc @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,K1,d,e. L1 â¢ â¶* [d, e] K1 â âL2. L1 â¡ L2 â + ââK2. L2 â¢ â¶* [d, e] K2 & K1 â¡ K2. +#L1 #K1 #d #e #H +lapply (ltpss_sn_ltpssa â¦ H) -H #H +@(ltpssa_ind â¦ H) -K1 /2 width=3/ +#K #K1 #_ #HK1 #IHK #L2 #HL12 +elim (IHK â¦ HL12) -L1 #K2 #HLK2 #HK2 +elim (ltpr_ltpss_dx_conf â¦ HK1 â¦ HK2) -K /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tps.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tps.etc new file mode 100644 index 000000000..7f08be62b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tps.etc @@ -0,0 +1,57 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T1,T2,d,e. L1 â¢ T1 â¶ [d, e] T2 â âL2. L1 â¡ L2 â + ââT. L2 â¢ T1 â¶ [d, e] T & T2 â¡ 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 â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 + elim (ltpr_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1 + elim (lift_total V2 0 (i+1)) #W2 #HVW2 + lapply (tpr_lift â¦ HV12 â¦ HVW1 â¦ HVW2) -V1 /3 width=4/ +| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 + elim (IHV12 â¦ HL12) -IHV12 #V #HV1 #HV2 + elim (IHT12 (L2.â{I}V) ?) /2 width=1/ -L1 /3 width=5/ +| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 + elim (IHV12 â¦ HL12) -IHV12 + elim (IHT12 â¦ HL12) -L1 /3 width=5/ +] +qed-. + +(* Basic_1: was: pr0_subst1_back *) +lemma ltpr_tps_trans: âL2,T1,T2,d,e. L2 â¢ T1 â¶ [d, e] T2 â âL1. L1 â¡ L2 â + ââT. L1 â¢ T1 â¶ [d, e] T & T â¡ 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 â¦ HL12 â¦ HLK2) -L2 #X #HLK1 #H + elim (ltpr_inv_pair2 â¦ H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2 + elim (lift_total V1 0 (i+1)) #W1 #HVW1 + lapply (tpr_lift â¦ HV12 â¦ HVW1 â¦ HVW2) -V2 /3 width=4/ +| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 + elim (IHV12 â¦ HL12) -IHV12 #V #HV1 #HV2 + elim (IHT12 (L1.â{I}V) ?) /2 width=1/ -L2 /3 width=5/ +| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 + elim (IHV12 â¦ HL12) -IHV12 + elim (IHT12 â¦ HL12) -L2 /3 width=5/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tpss.etc new file mode 100644 index 000000000..71586fa36 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tpss.etc @@ -0,0 +1,91 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âT1,T2. T1 â¡ T2 â + âL1,d,e,U1. L1 â¢ T1 â¶ [d, e] U1 â + âL2. L1 â¡ L2 â + ââU2. U1 â¡ U2 & L2 â¢ T2 â¶* [d, e] U2. +#T1 #T2 #H elim H -T1 -T2 +[ #I #L1 #d #e #U1 #H #L2 #HL12 + elim (ltpr_tps_conf â¦ H â¦ HL12) -L1 /3 width=3/ +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 â¦ H) -H #W1 #U1 #HVW1 #HTU1 #H destruct + elim (IHV12 â¦ HVW1 â¦ HL12) -V1 + elim (IHT12 â¦ HTU1 â¦ 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 â¦ H) -H #VV1 #Y #HVV1 #HY #HX destruct + elim (tps_inv_bind1 â¦ HY) -HY #WW #TT1 #_ #HTT1 #H destruct + elim (IHV12 â¦ HVV1 â¦ HL12) -V1 #VV2 #HVV12 #HVV2 + elim (IHT12 â¦ HTT1 (L2. âWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 + lapply (tpss_lsubr_trans â¦ HTT2 (L2. âVV2) ?) -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 â¦ H) -H #W1 #U1 #HVW1 #HTU1 #H destruct + elim (IHV12 â¦ HVW1 â¦ HL12) -V1 #W2 #HW12 #HVW2 + elim (IHT1 â¦ HTU1 (L2. â{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU + elim (tpss_strip_neq â¦ HTU â¦ HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2 + lapply (tps_lsubr_trans â¦ HU2 (L2. â{I} V2) ?) -HU2 /2 width=1/ #HU2 + elim (ltpss_dx_tps_conf â¦ HU2 (L2. â{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23 + lapply (tps_lsubr_trans â¦ HU3 (â. â{I} W2) ?) -HU3 /2 width=1/ #HU3 + lapply (tpss_lsubr_trans â¦ HU23 (L2. â{I} W2) ?) -HU23 /2 width=1/ #HU23 + lapply (tpss_trans_eq â¦ HTU2 â¦ 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 â¦ H) -H #VV1 #Y #HVV1 #HY #HX destruct + elim (tps_inv_bind1 â¦ HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct + elim (IHV12 â¦ HVV1 â¦ HL12) -V1 #VV2 #HVV12 #HVV2 + elim (IHW12 â¦ HWW1 â¦ HL12) -W1 #WW2 #HWW12 #HWW2 + elim (IHT12 â¦ HTT1 (L2. âWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 + elim (lift_total VV2 0 1) #VV #H2VV + lapply (tpss_lift_ge â¦ HVV2 (L2. âWW2) â¦ HV2 â¦ 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 â¦ H) -H #W #U1 #_ #HTU1 #H destruct -V + elim (IHT1 â¦ HTU1 (L2.âW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU + elim (tpss_inv_lift1_ge â¦ HTU L2 â¦ HT2 ?) -T (ltpr_inv_atom1 â¦ HL2) -L2 // +qed-. + +lemma ltprs_inv_pair1: âI,K1,L2,V1. K1. â{I} V1 â¡* L2 â + ââK2,V2. K1 â¡* K2 & V1 â¡* V2 & L2 = K2. â{I} V2. +#I #K1 #L2 #V1 #H @(ltprs_ind â¦ H) -L2 /2 width=5/ +#L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct +elim (ltpr_inv_pair1 â¦ HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/ +qed-. + +lemma ltprs_inv_atom2: âL1. L1 â¡* â â L1 = â. +#L1 #H @(ltprs_ind_dx â¦ H) -L1 // +#L1 #L #HL1 #_ #IHL2 destruct +>(ltpr_inv_atom2 â¦ HL1) -L1 // +qed-. + +lemma ltprs_inv_pair2: âI,L1,K2,V2. L1 â¡* K2. â{I} V2 â + ââK1,V1. K1 â¡* K2 & V1 â¡* V2 & L1 = K1. â{I} V1. +#I #L1 #K2 #V2 #H @(ltprs_ind_dx â¦ H) -L1 /2 width=5/ +#L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct +elim (ltpr_inv_pair2 â¦ HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma ltprs_fwd_length: âL1,L2. L1 â¡* L2 â |L1| = |L2|. +#L1 #L2 #H @(ltprs_ind â¦ H) -L2 // +#L #L2 #_ #HL2 #IHL1 +>IHL1 -L1 >(ltpr_fwd_length â¦ HL2) -HL2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_alt.etc new file mode 100644 index 000000000..1d2db2daf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_alt.etc @@ -0,0 +1,38 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¡ â¡ * 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 â lpx tprs. + +interpretation + "context-free parallel computation (environment) alternative" + 'PRedStarAlt L1 L2 = (ltprsa L1 L2). + +(* Basic properties *********************************************************) + +lemma ltprs_ltprsa: âL1,L2. L1 â¡* L2 â L1 â¡â¡* L2. +/2 width=1/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma ltprsa_ltprs: âL1,L2. L1 â¡â¡* L2 â L1 â¡* L2. +/2 width=1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_ldrop.etc new file mode 100644 index 000000000..a7c320089 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_ldrop.etc @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_ltprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_ltprs.etc new file mode 100644 index 000000000..7ededf2af --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltprs_ltprs.etc @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1. âL:lenv. L â¡* L1 â âL2. L â¡ L2 â + ââL0. L1 â¡ L0 & L2 â¡* L0. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem ltprs_conf: confluent â¦ ltprs. +/3 width=3/ qed. + +theorem ltprs_trans: Transitive â¦ ltprs. +/2 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx.etc new file mode 100644 index 000000000..753c948cc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx.etc @@ -0,0 +1,278 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¶ * [ 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 â nat â relation lenv â +| ltpss_dx_atom : âd,e. ltpss_dx d e (â) (â) +| ltpss_dx_pair : âL,I,V. ltpss_dx 0 0 (L. â{I} V) (L. â{I} V) +| ltpss_dx_tpss2: âL1,L2,I,V1,V2,e. + ltpss_dx 0 e L1 L2 â L2 â¢ V1 â¶* [0, e] V2 â + ltpss_dx 0 (e + 1) (L1. â{I} V1) (L2. â{I} V2) +| ltpss_dx_tpss1: âL1,L2,I,V1,V2,d,e. + ltpss_dx d e L1 L2 â L2 â¢ V1 â¶* [d, e] V2 â + ltpss_dx (d + 1) e (L1. â{I} V1) (L2. â{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: âd,e,L1,L2. L1 â¶* [d, e] L2 â e = 0 â 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 â¦ HV12) // +] +qed. + +lemma ltpss_dx_inv_refl_O2: âd,L1,L2. L1 â¶* [d, 0] L2 â L1 = L2. +/2 width=4/ qed-. + +fact ltpss_dx_inv_atom1_aux: âd,e,L1,L2. + L1 â¶* [d, e] L2 â L1 = â â L2 = â. +#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: âd,e,L2. â â¶* [d, e] L2 â L2 = â. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss21_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â d = 0 â 0 < e â + âK1,I,V1. L1 = K1. â{I} V1 â + ââK2,V2. K1 â¶* [0, e - 1] K2 & + K2 â¢ V1 â¶* [0, e - 1] V2 & + L2 = K2. â{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 â¦ 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: âe,K1,I,V1,L2. K1. â{I} V1 â¶* [0, e] L2 â 0 < e â + ââK2,V2. K1 â¶* [0, e - 1] K2 & + K2 â¢ V1 â¶* [0, e - 1] V2 & + L2 = K2. â{I} V2. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss11_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â 0 < d â + âI,K1,V1. L1 = K1. â{I} V1 â + ââK2,V2. K1 â¶* [d - 1, e] K2 & + K2 â¢ V1 â¶* [d - 1, e] V2 & + L2 = K2. â{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 â¦ H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_dx_inv_tpss11: âd,e,I,K1,V1,L2. K1. â{I} V1 â¶* [d, e] L2 â 0 < d â + ââK2,V2. K1 â¶* [d - 1, e] K2 & + K2 â¢ V1 â¶* [d - 1, e] V2 & + L2 = K2. â{I} V2. +/2 width=3/ qed-. + +fact ltpss_dx_inv_atom2_aux: âd,e,L1,L2. + L1 â¶* [d, e] L2 â L2 = â â L1 = â. +#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: âd,e,L1. L1 â¶* [d, e] â â L1 = â. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss22_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â d = 0 â 0 < e â + âK2,I,V2. L2 = K2. â{I} V2 â + ââK1,V1. K1 â¶* [0, e - 1] K2 & + K2 â¢ V1 â¶* [0, e - 1] V2 & + L1 = K1. â{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 â¦ 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: âe,L1,K2,I,V2. L1 â¶* [0, e] K2. â{I} V2 â 0 < e â + ââK1,V1. K1 â¶* [0, e - 1] K2 & + K2 â¢ V1 â¶* [0, e - 1] V2 & + L1 = K1. â{I} V1. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss12_aux: âd,e,L1,L2. L1 â¶* [d, e] L2 â 0 < d â + âI,K2,V2. L2 = K2. â{I} V2 â + ââK1,V1. K1 â¶* [d - 1, e] K2 & + K2 â¢ V1 â¶* [d - 1, e] V2 & + L1 = K1. â{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 â¦ H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_dx_inv_tpss12: âL1,K2,I,V2,d,e. L1 â¶* [d, e] K2. â{I} V2 â 0 < d â + ââK1,V1. K1 â¶* [d - 1, e] K2 & + K2 â¢ V1 â¶* [d - 1, e] V2 & + L1 = K1. â{I} V1. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +lemma ltpss_dx_tps2: âL1,L2,I,V1,V2,e. + L1 â¶* [0, e] L2 â L2 â¢ V1 â¶ [0, e] V2 â + L1. â{I} V1 â¶* [0, e + 1] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_dx_tps1: âL1,L2,I,V1,V2,d,e. + L1 â¶* [d, e] L2 â L2 â¢ V1 â¶ [d, e] V2 â + L1. â{I} V1 â¶* [d + 1, e] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_dx_tpss2_lt: âL1,L2,I,V1,V2,e. + L1 â¶* [0, e - 1] L2 â L2 â¢ V1 â¶* [0, e - 1] V2 â + 0 < e â L1. â{I} V1 â¶* [0, e] L2. â{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: âL1,L2,I,V1,V2,d,e. + L1 â¶* [d - 1, e] L2 â L2 â¢ V1 â¶* [d - 1, e] V2 â + 0 < d â L1. â{I} V1 â¶* [d, e] L2. â{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: âL1,L2,I,V1,V2,e. + L1 â¶* [0, e - 1] L2 â L2 â¢ V1 â¶ [0, e - 1] V2 â + 0 < e â L1. â{I} V1 â¶* [0, e] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_dx_tps1_lt: âL1,L2,I,V1,V2,d,e. + L1 â¶* [d - 1, e] L2 â L2 â¢ V1 â¶ [d - 1, e] V2 â + 0 < d â L1. â{I} V1 â¶* [d, e] L2. â{I} V2. +/3 width=1/ qed. + +(* Basic_1: was by definition: csubst1_refl *) +lemma ltpss_dx_refl: âL,d,e. L â¶* [d, e] L. +#L elim L -L // +#L #I #V #IHL * /2 width=1/ * /2 width=1/ +qed. + +lemma ltpss_dx_weak: âL1,L2,d1,e1. L1 â¶* [d1, e1] L2 â + âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â L1 â¶* [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 â¦ Hd2) #H destruct normalize in Hde2; + lapply (lt_to_le_to_lt 0 â¦ Hde2) // #He2 + lapply (le_plus_to_minus_r â¦ 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 â¦ Hde12) -Hde12 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 (ldrop_inv_atom1 â¦ H) -H // +| // +| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 + elim (le_inv_plus_l â¦ He12) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK01 â¦ 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 â¦ Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK01 â¦ HK0L2 ?) -K0 /2 width=1/ +] +qed. + +lemma ltpss_dx_ldrop_trans_ge: âL1,L0,d1,e1. L1 â¶* [d1, e1] L0 â + âL2,e2. â©[0, e2] L0 â¡ L2 â + d1 + e1 â¤ e2 â â©[0, e2] L1 â¡ L2. +#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H // +| // +| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 + elim (le_inv_plus_l â¦ He12) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK10 â¦ 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 â¦ Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK10 â¦ HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ +] +qed. + +lemma ltpss_dx_ldrop_conf_be: âL0,L1,d1,e1. L0 â¶* [d1, e1] L1 â + âL2,e2. â©[0, e2] L0 â¡ L2 â d1 â¤ e2 â e2 â¤ d1 + e1 â + ââL. L2 â¶* [0, d1 + e1 - e2] L & â©[0, e2] L1 â¡ L. +#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 + (ldrop_inv_atom1 â¦ H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 + (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK10 -He2d1 destruct (ltpss_dx_inv_atom1 â¦ H) -H /2 width=3/ +| /2 width=3/ +| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd + elim (IHLK1 â¦ 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 â¦ Hd12) -Hd12 #Hd12 #Hd2 + elim (ltpss_dx_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦ Hd12) -IHLK1 -HK12 (ltpss_dx_inv_atom1 â¦ H) -H /2 width=3/ +| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_ + lapply (le_n_O_to_eq â¦ H) -H #H destruct /2 width=3/ +| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2 + elim (IHLK1 â¦ HK12 H1 H2) -K1 -H2 + lapply (le_n_O_to_eq â¦ 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 â¦ Hd12) -Hd12 plus_minus_commutative // #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_be â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 // /2 width=1/ + >plus_minus // >commutative_plus /4 width=5/ + | elim (ltpss_dx_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_be â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 [ >plus_minus // ] /2 width=1/ + >commutative_plus /3 width=5/ + ] +] +qed-. + +lemma ldrop_ltpss_dx_trans_ge: âL1,K1,d1,e1. â©[d1, e1] L1 â¡ K1 â + âK2,d2,e2. K1 â¶* [d2, e2] K2 â d2 + e2 â¤ d1 â + ââL2. L1 â¶* [d2, e2] L2 & â©[d1, e1] L2 â¡ K2. +#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1 +[ #d1 #e1 #K2 #d2 #e2 #H #_ + >(ltpss_dx_inv_atom1 â¦ H) -H /2 width=3/ +| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H + elim (plus_le_0 â¦ H) -H #H1 #H2 destruct /2 width=3/ +| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H + elim (IHLK1 â¦ HK12 H) -K1 + elim (plus_le_0 â¦ H) -H #H1 #H2 destruct #L2 #HL12 + >(ltpss_dx_inv_refl_O2 â¦ 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 â¦ H) -X /3 width=5/ + | elim (ltpss_dx_inv_tpss21 â¦ H He2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_le â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 /2 width=1/ /3 width=5/ + | elim (ltpss_dx_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_le â¦ HW12 â¦ HLK2 HWV1 â¦ HWV2) -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/ + ] +] +qed-. + +(* Properties on supclosure *************************************************) + +lemma fsup_tps_trans_full: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. L2 â¢ T2 â¶[0,|L2|] U2 â + ââL,U1. L1 â¶*[0,|L|] L & L â¢ T1 â¶[0,|L|] U1 & â¦L, U1â¦ â â¦L2, T2â¦. +#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 â¦ 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 â¦ HLK1 â¦ HK1 â¦ Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K + lapply (tps_lift_be â¦ HT1 â¦ HL2K â¦ HTU1 HTU â¦ Hd) // -HT1 -HTU1 #HU1 +| elim (ldrop_ltpss_dx_trans_ge â¦ HLK1 â¦ HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K + lapply (tps_lift_le â¦ HT1 â¦ HL2K â¦ HTU1 HTU Hd) -HT1 -HTU1 #HU1 +] +lapply (ltpss_dx_weak_full â¦ HL12) -HL12 #HL12 +lapply (tps_weak_full â¦ HU1) -HU1 #HU1 +@(ex3_2_intro â¦ L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_ltpss_dx.etc new file mode 100644 index 000000000..74b898d69 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_ltpss_dx.etc @@ -0,0 +1,106 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â + âL1,d1,e1. L0 â¶* [d1, e1] L1 â + ââT. L1 â¢ T2 â¶* [d2, e2] T & + L1 â¢ U2 â¶* [d1, e1] T. +#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind â¦ H) -U2 /2 width=3/ +#U #U2 #_ #HU2 * #X2 #HTX2 #HUX2 +elim (ltpss_dx_tps_conf â¦ HU2 â¦ HL01) -L0 #X1 #HUX1 #HU2X1 +elim (tpss_strip_eq â¦ HUX2 â¦ HUX1) -U #X #HX2 #HX1 +lapply (tpss_trans_eq â¦ HU2X1 â¦ HX1) -X1 /3 width=3/ +qed. + +lemma ltpss_dx_tpss_trans_down: âL0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 â¤ d1 â + L1 â¶* [d1, e1] L0 â L0 â¢ T2 â¶* [d2, e2] U2 â + ââT. L1 â¢ T2 â¶* [d2, e2] T & L0 â¢ T â¶* [d1, e1] U2. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| #U #U2 #_ #HU2 * #T #HT2 #HTU + elim (tpss_strap1_down â¦ HTU â¦ HU2 ?) -U // #U #HTU #HU2 + elim (ltpss_dx_tps_trans â¦ HTU â¦ HL10) -HTU -HL10 #X #HTX #HXU + lapply (tpss_trans_eq â¦ HXU HU2) -U /3 width=3/ +] +qed. + +lemma ltpss_dx_tpss_trans_eq: âL1,T2,U2,d,e. L1 â¢ T2 â¶* [d, e] U2 â + âL0. L0 â¶* [d, e] L1 â L0 â¢ T2 â¶* [d, e] U2. +#L1 #T2 @(f2_ind â¦ fw â¦ L1 T2) -L1 -T2 #n #IH #L1 * +[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct + elim (tpss_inv_atom1 â¦ H) -H // * + #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct + lapply (ldrop_fwd_lw â¦ HLK1) #H1 normalize in H1; + elim (ltpss_dx_ldrop_trans_be â¦ HL01 â¦ HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0 + elim (ltpss_dx_inv_tpss22 â¦ H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct + lapply (tpss_fwd_tw â¦ HV01) #H2 + lapply (transitive_le (â¯{K1} + â¯{V0}) â¦ H1) -H1 /2 width=1/ -H2 #H + lapply (tpss_trans_eq â¦ HV01 HV12) -V1 #HV02 + lapply (IH â¦ HV02 â¦ 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 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct + lapply (tpss_lsubr_trans â¦ HTU2 (L1. â{I} V2) ?) -HTU2 /2 width=1/ #HTU2 + lapply (IH â¦ HVW2 â¦ HL0) -HVW2 [ /2 width=2/ ] #HVW2 + lapply (IH â¦ HTU2 (L0. â{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2 + lapply (tpss_lsubr_trans â¦ HTU2 (L0. â{I} W2) ?) -HTU2 /2 width=1/ + | elim (tpss_inv_flat1 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct + lapply (IH â¦ HVW2 â¦ HL0) -HVW2 // + lapply (IH â¦ HTU2 â¦ HL0) -IH -HTU2 // -HL0 /2 width=1/ +] +qed. + +lemma ltpss_dx_tps_trans_eq: âL0,L1,T2,U2,d,e. L0 â¶* [d, e] L1 â + L1 â¢ T2 â¶ [d, e] U2 â L0 â¢ T2 â¶* [d, e] U2. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem ltpss_dx_conf: âL0,L1,d1,e1. L0 â¶* [d1, e1] L1 â + âL2,d2,e2. L0 â¶* [d2, e2] L2 â + ââL. L1 â¶* [d2, e2] L & L2 â¶* [d1, e1] L. +#L0 @(f_ind â¦ lw â¦ L0) -L0 #n #IH * +[ #_ #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 -n + >(ltpss_dx_inv_atom1 â¦ H1) -L1 + >(ltpss_dx_inv_atom1 â¦ 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 â¦ H1) -H1 #H1 + | elim (ltpss_dx_inv_tpss21 â¦ H1 â¦ He1) -H1 #K1 #V1 #HK01 #HV01 #H1 + | elim (ltpss_dx_inv_tpss11 â¦ H1 â¦ 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 â¦ H2) -H2 #H2 + |2,4,6: elim (ltpss_dx_inv_tpss21 â¦ H2 â¦ He2) -H2 #K2 #V2 #HK02 #HV02 #H2 + |7,8,9: elim (ltpss_dx_inv_tpss11 â¦ H2 â¦ 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 â¦ HK01 â¦ HK02) // -K0 #K #HK1 #HK2 + elim (ltpss_dx_tpss_conf â¦ HV01 â¦ HK1) -HV01 #W1 #HW1 #HVW1 + elim (ltpss_dx_tpss_conf â¦ HV02 â¦ HK2) -HV02 #W2 #HW2 #HVW2 + elim (tpss_conf_eq â¦ HW1 â¦ HW2) -V0 #V #HW1 #HW2 + lapply (tpss_trans_eq â¦ HVW1 HW1) -W1 + lapply (tpss_trans_eq â¦ HVW2 HW2) -W2 /3 width=5/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_tps.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_tps.etc new file mode 100644 index 000000000..b10306677 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_tps.etc @@ -0,0 +1,47 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â + âL1,d1,e1. L0 â¶* [d1, e1] L1 â d1 + e1 â¤ d2 â + L1 â¢ T2 â¶ [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 â¦ Hde1d2 Hdi2) -Hde1d2 #Hde1i2 + lapply (ltpss_dx_ldrop_conf_ge â¦ HL01 â¦ 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â + âL1,d1,e1. L1 â¶* [d1, e1] L0 â d1 + e1 â¤ d2 â + L1 â¢ T2 â¶ [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 â¦ Hde1d2 Hdi2) -Hde1d2 #Hde1i2 + lapply (ltpss_dx_ldrop_trans_ge â¦ HL10 â¦ 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_tpss.etc new file mode 100644 index 000000000..9c7875927 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_dx_tpss.etc @@ -0,0 +1,101 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â + âL1,d1,e1. L0 â¶* [d1, e1] L1 â d1 + e1 â¤ d2 â + L1 â¢ T2 â¶* [d2, e2] U2. +#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind â¦ H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltpss_dx_tps_conf_ge â¦ HU2 â¦ HL01 ?) -L0 // -Hde1d2 /2 width=3/ +qed. + +(* Basic_1: was: subst1_subst1_back *) +lemma ltpss_dx_tps_conf: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â + âL1,d1,e1. L0 â¶* [d1, e1] L1 â + ââT. L1 â¢ T2 â¶ [d2, e2] T & + L1 â¢ U2 â¶* [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 â¦ HL01 â¦ HLK0 ?) -L0 /2 width=2/ #X #H #HLK1 + elim (ltpss_dx_inv_tpss11 â¦ H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK1) #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tpss_lift_ge â¦ HV01 â¦ H HVW0 â¦ HVW1) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_dx_ldrop_conf_ge â¦ HL01 â¦ HLK0 ?) -L0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 â¦ HL01) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. â{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 â¦ HL01) -IHVW2 + elim (IHTU2 â¦ HL01) -IHTU2 -HL01 /3 width=5/ +] +qed. + +lemma ltpss_dx_tpss_trans_ge: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â + âL1,d1,e1. L1 â¶* [d1, e1] L0 â d1 + e1 â¤ d2 â + L1 â¢ T2 â¶* [d2, e2] U2. +#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind â¦ H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltpss_dx_tps_trans_ge â¦ HU2 â¦ HL01 ?) -L0 // -Hde1d2 /2 width=3/ +qed. + +(* Basic_1: was: subst1_subst1 *) +lemma ltpss_dx_tps_trans: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â + âL1,d1,e1. L1 â¶* [d1, e1] L0 â + ââT. L1 â¢ T2 â¶ [d2, e2] T & + L0 â¢ T â¶* [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 â¦ HL10 â¦ HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1 + elim (ltpss_dx_inv_tpss12 â¦ H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK0) -HLK0 #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tpss_lift_ge â¦ HV01 â¦ H HVW1 â¦ HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_dx_ldrop_trans_ge â¦ HL10 â¦ HLK0 ?) -HL10 -HLK0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 â¦ HL10) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. â{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 â¦ HL10) -IHVW2 + elim (IHTU2 â¦ HL10) -IHTU2 -HL10 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn.etc new file mode 100644 index 000000000..577c1506d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn.etc @@ -0,0 +1,259 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ â¶ * [ 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 â nat â relation lenv â +| ltpss_sn_atom : âd,e. ltpss_sn d e (â) (â) +| ltpss_sn_pair : âL,I,V. ltpss_sn 0 0 (L. â{I} V) (L. â{I} V) +| ltpss_sn_tpss2: âL1,L2,I,V1,V2,e. + ltpss_sn 0 e L1 L2 â L1 â¢ V1 â¶* [0, e] V2 â + ltpss_sn 0 (e + 1) (L1. â{I} V1) (L2. â{I} V2) +| ltpss_sn_tpss1: âL1,L2,I,V1,V2,d,e. + ltpss_sn d e L1 L2 â L1 â¢ V1 â¶* [d, e] V2 â + ltpss_sn (d + 1) e (L1. â{I} V1) (L2. â{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: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â e = 0 â 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 â¦ HV12) // +] +qed. + +lemma ltpss_sn_inv_refl_O2: âd,L1,L2. L1 â¢ â¶* [d, 0] L2 â L1 = L2. +/2 width=4/ qed-. + +fact ltpss_sn_inv_atom1_aux: âd,e,L1,L2. + L1 â¢ â¶* [d, e] L2 â L1 = â â L2 = â. +#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: âd,e,L2. â â¢ â¶* [d, e] L2 â L2 = â. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss21_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â d = 0 â 0 < e â + âK1,I,V1. L1 = K1. â{I} V1 â + ââK2,V2. K1 â¢ â¶* [0, e - 1] K2 & + K1 â¢ V1 â¶* [0, e - 1] V2 & + L2 = K2. â{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 â¦ 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: âe,K1,I,V1,L2. K1. â{I} V1 â¢ â¶* [0, e] L2 â 0 < e â + ââK2,V2. K1 â¢ â¶* [0, e - 1] K2 & + K1 â¢ V1 â¶* [0, e - 1] V2 & + L2 = K2. â{I} V2. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss11_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â 0 < d â + âI,K1,V1. L1 = K1. â{I} V1 â + ââK2,V2. K1 â¢ â¶* [d - 1, e] K2 & + K1 â¢ V1 â¶* [d - 1, e] V2 & + L2 = K2. â{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 â¦ H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_sn_inv_tpss11: âd,e,I,K1,V1,L2. K1. â{I} V1 â¢ â¶* [d, e] L2 â 0 < d â + ââK2,V2. K1 â¢ â¶* [d - 1, e] K2 & + K1 â¢ V1 â¶* [d - 1, e] V2 & + L2 = K2. â{I} V2. +/2 width=3/ qed-. + +fact ltpss_sn_inv_atom2_aux: âd,e,L1,L2. + L1 â¢ â¶* [d, e] L2 â L2 = â â L1 = â. +#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: âd,e,L1. L1 â¢ â¶* [d, e] â â L1 = â. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss22_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â d = 0 â 0 < e â + âK2,I,V2. L2 = K2. â{I} V2 â + ââK1,V1. K1 â¢ â¶* [0, e - 1] K2 & + K1 â¢ V1 â¶* [0, e - 1] V2 & + L1 = K1. â{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 â¦ 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: âe,L1,K2,I,V2. L1 â¢ â¶* [0, e] K2. â{I} V2 â 0 < e â + ââK1,V1. K1 â¢ â¶* [0, e - 1] K2 & + K1 â¢ V1 â¶* [0, e - 1] V2 & + L1 = K1. â{I} V1. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss12_aux: âd,e,L1,L2. L1 â¢ â¶* [d, e] L2 â 0 < d â + âI,K2,V2. L2 = K2. â{I} V2 â + ââK1,V1. K1 â¢ â¶* [d - 1, e] K2 & + K1 â¢ V1 â¶* [d - 1, e] V2 & + L1 = K1. â{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 â¦ H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false â¦ H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_sn_inv_tpss12: âL1,K2,I,V2,d,e. L1 â¢ â¶* [d, e] K2. â{I} V2 â 0 < d â + ââK1,V1. K1 â¢ â¶* [d - 1, e] K2 & + K1 â¢ V1 â¶* [d - 1, e] V2 & + L1 = K1. â{I} V1. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +lemma ltpss_sn_tps2: âL1,L2,I,V1,V2,e. + L1 â¢ â¶* [0, e] L2 â L1 â¢ V1 â¶ [0, e] V2 â + L1. â{I} V1 â¢ â¶* [0, e + 1] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_tps1: âL1,L2,I,V1,V2,d,e. + L1 â¢ â¶* [d, e] L2 â L1 â¢ V1 â¶ [d, e] V2 â + L1. â{I} V1 â¢ â¶* [d + 1, e] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_tpss2_lt: âL1,L2,I,V1,V2,e. + L1 â¢ â¶* [0, e - 1] L2 â L1 â¢ V1 â¶* [0, e - 1] V2 â + 0 < e â L1. â{I} V1 â¢ â¶* [0, e] L2. â{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: âL1,L2,I,V1,V2,d,e. + L1 â¢ â¶* [d - 1, e] L2 â L1 â¢ V1 â¶* [d - 1, e] V2 â + 0 < d â L1. â{I} V1 â¢ â¶* [d, e] L2. â{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: âL1,L2,I,V1,V2,e. + L1 â¢ â¶* [0, e - 1] L2 â L1 â¢ V1 â¶ [0, e - 1] V2 â + 0 < e â L1. â{I} V1 â¢ â¶* [0, e] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_tps1_lt: âL1,L2,I,V1,V2,d,e. + L1 â¢ â¶* [d - 1, e] L2 â L1 â¢ V1 â¶ [d - 1, e] V2 â + 0 < d â L1. â{I} V1 â¢ â¶* [d, e] L2. â{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_refl: âL,d,e. L â¢ â¶* [d, e] L. +#L elim L -L // +#L #I #V #IHL * /2 width=1/ * /2 width=1/ +qed. + +lemma ltpss_sn_weak: âL1,L2,d1,e1. L1 â¢ â¶* [d1, e1] L2 â + âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â L1 â¢ â¶* [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 â¦ Hd2) #H destruct normalize in Hde2; + lapply (lt_to_le_to_lt 0 â¦ Hde2) // #He2 + lapply (le_plus_to_minus_r â¦ 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 â¦ Hde12) -Hde12 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 (ldrop_inv_atom1 â¦ H) -H // +| // +| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 + elim (le_inv_plus_l â¦ He12) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK01 â¦ 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 â¦ Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK01 â¦ HK0L2 ?) -K0 /2 width=1/ +] +qed. + +lemma ltpss_sn_ldrop_trans_ge: âL1,L0,d1,e1. L1 â¢ â¶* [d1, e1] L0 â + âL2,e2. â©[0, e2] L0 â¡ L2 â + d1 + e1 â¤ e2 â â©[0, e2] L1 â¡ L2. +#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H // +| // +| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 + elim (le_inv_plus_l â¦ He12) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK10 â¦ 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 â¦ Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 â¦ H ?) -H // #HK0L2 + lapply (IHK10 â¦ HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ +] +qed. + +lemma ltpss_sn_ldrop_conf_be: âL0,L1,d1,e1. L0 â¢ â¶* [d1, e1] L1 â + âL2,e2. â©[0, e2] L0 â¡ L2 â d1 â¤ e2 â e2 â¤ d1 + e1 â + ââL. L2 â¢ â¶* [0, d1 + e1 - e2] L & â©[0, e2] L1 â¡ L. +#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 â¦ H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 + (ldrop_inv_atom1 â¦ H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq â¦ He2) -He2 #H destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l â¦ Hd1e2) #_ #He2 + (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 â¦ 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 â¦ He2) -He2 #He2 destruct + lapply (ldrop_inv_refl â¦ 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 â¦ H) -H * * #He2 #HK0L2 + [ -IHK10 -He2d1 destruct (ltpss_sn_inv_atom1 â¦ H) -H /2 width=3/ +| /2 width=3/ +| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd + elim (IHLK1 â¦ 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 â¦ Hd12) -Hd12 #Hd12 #Hd2 + elim (ltpss_sn_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦ Hd12) -IHLK1 -HK12 (ltpss_sn_inv_atom1 â¦ H) -H /2 width=3/ +| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_ + lapply (le_n_O_to_eq â¦ H) -H #H destruct /2 width=3/ +| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2 + elim (IHLK1 â¦ HK12 H1 H2) -K1 -H2 + lapply (le_n_O_to_eq â¦ 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 â¦ Hd12) -Hd12 plus_minus_commutative // #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_be â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 // /2 width=1/ + >plus_minus // >commutative_plus /4 width=5/ + | elim (ltpss_sn_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_be â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 [ >plus_minus // ] /2 width=1/ + >commutative_plus /3 width=5/ + ] +] +qed-. + +lemma ldrop_ltpss_sn_trans_ge: âL1,K1,d1,e1. â©[d1, e1] L1 â¡ K1 â + âK2,d2,e2. K1 â¢ â¶* [d2, e2] K2 â d2 + e2 â¤ d1 â + ââL2. L1 â¢ â¶* [d2, e2] L2 & â©[d1, e1] L2 â¡ K2. +#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1 +[ #d1 #e1 #K2 #d2 #e2 #H #_ + >(ltpss_sn_inv_atom1 â¦ H) -H /2 width=3/ +| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H + elim (plus_le_0 â¦ H) -H #H1 #H2 destruct /2 width=3/ +| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H + elim (IHLK1 â¦ HK12 H) -K1 + elim (plus_le_0 â¦ H) -H #H1 #H2 destruct #L2 #HL12 + >(ltpss_sn_inv_refl_O2 â¦ 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 â¦ H) -X /3 width=5/ + | elim (ltpss_sn_inv_tpss21 â¦ H He2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_le â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 /2 width=1/ /3 width=5/ + | elim (ltpss_sn_inv_tpss11 â¦ H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (IHLK1 â¦ HK12 â¦) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2 + elim (lift_total W2 d1 e1) #V2 #HWV2 + lapply (tpss_lift_le â¦ HW12 â¦ HLK1 HWV1 â¦ HWV2) -HLK1 -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/ + ] +] +qed-. + +(* Properties on supclosure *************************************************) + +lemma fsup_tpss_trans_full: âL1,L2,T1,T2. â¦L1, T1â¦ â â¦L2, T2â¦ â âU2. L2 â¢ T2 â¶*[0,|L2|] U2 â + ââL,U1. L1 â¢ â¶*[0,|L1|] L & L â¢ T1 â¶*[0,|L|] U1 & â¦L, U1â¦ â â¦L2, T2â¦. +#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 â¦ HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 +elim (lift_total T d e) #U #HTU +lapply (ltpss_sn_fwd_length â¦ HK1) #H >H in HK1; -H #HK1 +elim (le_or_ge d (|K|)) #Hd +[ elim (ldrop_ltpss_sn_trans_be â¦ HLK1 â¦ HK1 â¦ Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K + lapply (tpss_lift_be â¦ HT1 â¦ Hd HL2K HTU1 â¦ HTU) // -HT1 -HTU1 #HU1 +| elim (ldrop_ltpss_sn_trans_ge â¦ HLK1 â¦ HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K + lapply (tpss_lift_le â¦ HT1 â¦ Hd HL2K HTU1 â¦ HTU) -HT1 -HTU1 #HU1 +] +lapply (ltpss_sn_weak_full â¦ HL12) -HL12 #HL12 +lapply (tpss_weak_full â¦ HU1) -HU1 #HU1 +@(ex3_2_intro â¦ L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn_ltpss_sn.etc new file mode 100644 index 000000000..67c287e2b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn_ltpss_sn.etc @@ -0,0 +1,81 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T2,U2,d,e. L1 â¢ T2 â¶* [d, e] U2 â + âL0. L0 â¢ â¶* [d, e] L1 â L0 â¢ T2 â¶* [d, e] U2. +#L1 #T2 @(f2_ind â¦ fw â¦ L1 T2) -L1 -T2 #n #IH #L1 * +[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct + elim (tpss_inv_atom1 â¦ H) -H // * + #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct + lapply (ldrop_fwd_lw â¦ HLK1) #H1 normalize in H1; + elim (ltpss_sn_ldrop_trans_be â¦ HL01 â¦ HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0 + elim (ltpss_sn_inv_tpss22 â¦ H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct + lapply (IH â¦ HV12 â¦ HK01) -IH -HV12 -HK01 [ normalize /2 width=1/ ] #HV12 + lapply (tpss_trans_eq â¦ HV01 HV12) -V1 /2 width=6/ +| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct + [ elim (tpss_inv_bind1 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct + lapply (tpss_lsubr_trans â¦ HTU2 (L1. â{I} V2) ?) -HTU2 /2 width=1/ #HTU2 + lapply (IH â¦ HVW2 â¦ HL0) -HVW2 [ /2 width=2/ ] #HVW2 + lapply (IH â¦ HTU2 (L0. â{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2 + lapply (tpss_lsubr_trans â¦ HTU2 (L0. â{I} W2) ?) -HTU2 /2 width=1/ + | elim (tpss_inv_flat1 â¦ H) -H #W2 #U2 #HVW2 #HTU2 #H destruct + lapply (IH â¦ HVW2 â¦ HL0) -HVW2 // + lapply (IH â¦ HTU2 â¦ HL0) -IH -HTU2 // -HL0 /2 width=1/ +] +qed. + +lemma ltpss_sn_tps_trans_eq: âL0,L1,T2,U2,d,e. L0 â¢ â¶* [d, e] L1 â + L1 â¢ T2 â¶ [d, e] U2 â L0 â¢ T2 â¶* [d, e] U2. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem ltpss_sn_trans_eq: âL1,L,d,e. L1 â¢ â¶* [d, e] L â + âL2. L â¢ â¶* [d, e] L2 â L1 â¢ â¶* [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 â¦ H ?) -H // shift_append_assoc #H + elim (tps_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + elim (IH â¦ HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct + append_length minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_sn_ldrop_conf_ge â¦ HL01 â¦ HLK0 ?) -HL01 -HLK0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 â¦ HL01) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. â{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H + lapply (tpss_lsubr_trans â¦ H (L0.â{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 â¦ HL01) -IHVW2 + elim (IHTU2 â¦ HL01) -IHTU2 -HL01 /3 width=5/ +] +qed. + +lemma ltpss_sn_tpss_trans_ge: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶* [d2, e2] U2 â + âL1,d1,e1. L1 â¢ â¶* [d1, e1] L0 â d1 + e1 â¤ d2 â + L1 â¢ T2 â¶* [d2, e2] U2. +#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind â¦ H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltpss_sn_tps_trans_ge â¦ HU2 â¦ HL01 ?) -L0 // -Hde1d2 /2 width=3/ +qed. + +lemma ltpss_sn_tps_trans: âL0,T2,U2,d2,e2. L0 â¢ T2 â¶ [d2, e2] U2 â + âL1,d1,e1. L1 â¢ â¶* [d1, e1] L0 â + ââT. L1 â¢ T2 â¶ [d2, e2] T & + L1 â¢ T â¶* [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 â¦ HL10 â¦ HLK0 ?) -L0 /2 width=2/ #X #H #HLK1 + elim (ltpss_sn_inv_tpss12 â¦ H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK1) #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tpss_lift_ge â¦ HV01 â¦ H HVW1 â¦ HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_sn_ldrop_trans_ge â¦ HL10 â¦ HLK0 ?) -HL10 -HLK0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 â¦ HL10) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. â{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H + lapply (tpss_lsubr_trans â¦ H (L1.â{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 â¦ HL10) -IHVW2 + elim (IHTU2 â¦ HL10) -IHTU2 -HL10 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/snv_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/snv_ltpss_dx.etc new file mode 100644 index 000000000..ca55239f0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/snv_ltpss_dx.etc @@ -0,0 +1,87 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,T1. â¦h, L1â¦ â¢ T1 Â¡[g] â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â â¦h, L2â¦ â¢ T2 Â¡[g]. +#h #g #L1 #T1 #H elim H -L1 -T1 +[ #L1 #k #L2 #d #e #_ #X #H + >(tpss_inv_sort1 â¦ H) -X // +| #I #L1 #K1 #V1 #i #HLK1 #HV1 #IHV1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_lref1 â¦ 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 â¦ HL12 â¦ HLK1 â¦) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss11 â¦ H â¦) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + lapply (IHV1 â¦ HK12 â¦ HV12) -IHV1 -HK12 -HV12 /2 width=5/ + | elim (ltpss_dx_ldrop_conf_be â¦ HL12 â¦ HLK1 â¦) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss21 â¦ H â¦) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + lapply (IHV1 â¦ HK12 â¦ HV12) -IHV1 -HK12 -HV12 /2 width=5/ + | lapply (ltpss_dx_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -L1 // -Hide /2 width=5/ + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 + elim (ltpss_dx_ldrop_conf_be â¦ HL12 â¦ HLK1 â¦) -L1 // /2 width=2/ #X #H1 #H2 + elim (ltpss_dx_inv_tpss21 â¦ H1 â¦) -H1 /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct + lapply (ldrop_mono â¦ H2 â¦ HLK2) -H2 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) -HLK2 #HLK2 + lapply (tpss_trans_eq â¦ HV12 HVW2) -V2 #HV1W2 + lapply (IHV1 â¦ HK12 â¦ HV1W2) -IHV1 -HK12 -HV1W2 /2 width=7/ + ] +| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IHV1 â¦ HL12 â¦ HV12) -IHV1 #HV2 + lapply (IHT1 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IHV1 â¦ HL12 â¦ HV12) -IHV1 #HV2 + lapply (IHT1 â¦ HT12) -IHT1 /2 width=1/ #HT2 + elim (ssta_ltpss_dx_tpss_conf â¦ HVW1 â¦ HL12 â¦ HV12) -V1 #W2 #HVW2 #HW12 + elim (dxprs_ltpss_dx_tpss_conf â¦ HTU1 â¦ HL12 â¦ HT12) -T1 #X #HTU2 #H + elim (tpss_inv_bind1 â¦ H) -H #W #U2 #HW0 #_ #H -U1 destruct + elim (cprs_ltpss_dx_tpss2_conf â¦ HW10 â¦ HL12 â¦ HW12 â¦ HW0) -L1 -W1 -W0 #W0 #HW20 #HW0 + lapply (dxprs_strap1 â¦ (â{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 â¦ H) -H #W2 #T2 #HW12 #HT12 #H destruct + lapply (IHW1 â¦ HL12 â¦ HW12) -IHW1 #HW2 + lapply (IHT1 â¦ HL12 â¦ HT12) -IHT1 #HT2 + elim (ssta_ltpss_dx_tpss_conf â¦ HTU1 â¦ HL12 â¦ HT12) -T1 #U2 #HTU2 #HU12 + lapply (cpcs_ltpss_dx_tpss2_conf â¦ HL12 â¦ HUW1 â¦ HU12 â¦ HW12) -L1 -W1 -U1 /2 width=4/ +] +qed-. + +lemma snv_ltpss_dx_conf: âh,g,L1,T. â¦h, L1â¦ â¢ T Â¡[g] â + âL2,d,e. L1 â¶* [d, e] L2 â â¦h, L2â¦ â¢ T Â¡[g]. +#h #g #L1 #T #HT #L2 #d #e #HL12 +@(snv_ltpss_dx_tpss_conf â¦ HT â¦ HL12) // +qed-. + +lemma snv_tpss_conf: âh,g,L,T1. â¦h, Lâ¦ â¢ T1 Â¡[g] â + âT2,d,e. L â¢ T1 â¶* [d, e] T2 â â¦h, Lâ¦ â¢ T2 Â¡[g]. +#h #g #L #T1 #HT1 #T2 #d #e #HT12 +@(snv_ltpss_dx_tpss_conf â¦ HT1 â¦ HT12) // +qed-. + +lemma snv_tps_conf: âh,g,L,T1. â¦h, Lâ¦ â¢ T1 Â¡[g] â + âT2,d,e. L â¢ T1 â¶ [d, e] T2 â â¦h, Lâ¦ â¢ T2 Â¡[g]. +#h #g #L #T1 #HT1 #T2 #d #e #HT12 +@(snv_tpss_conf â¦ HT1 T2) /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/snv_ltpss_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/snv_ltpss_sn.etc new file mode 100644 index 000000000..3c8070249 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/snv_ltpss_sn.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,L2,d,e. L1 â¢ â¶* [d, e] L2 â + âT. â¦h, L1â¦ â¢ T Â¡[g] â â¦h, L2â¦ â¢ T Â¡[g]. +#h #g #L1 #L2 #d #e #H +lapply (ltpss_sn_ltpssa â¦ H) -H #H @(ltpssa_ind â¦ H) -L2 // +#L #L2 #_ #HL2 #IHL1 #T1 #HT1 +lapply (IHL1 â¦ HT1) -IHL1 -HT1 #HT1 +lapply (snv_ltpss_dx_conf â¦ HT1 â¦ HL2) -HT1 -HL2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ssta_ltpss_dx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ssta_ltpss_dx.etc new file mode 100644 index 000000000..72eff7850 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ssta_ltpss_dx.etc @@ -0,0 +1,125 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 â¢[g] â¦l, U1â¦ â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â + ââU2. â¦h, L2â¦ â¢ T2 â¢[g] â¦l, U2â¦ & + L2 â¢ U1 â¶* [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 â¦ 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 â¦ H) -H [ | -HVW1 ] + [ #H destruct + elim (lt_or_ge i d) #Hdi [ -HVW1 | ] + [ elim (ltpss_dx_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss11 â¦ H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 â¦ HK12 â¦ HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 + lapply (ldrop_fwd_ldrop2 â¦ HLK2) #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge â¦ HW12 â¦ H â¦ HWU1 â¦ HWU2) // -HW12 -H -HWU1 + >minus_plus minus_plus #H + lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H + lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H + lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus (tpss_inv_sort1 â¦ 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 â¦ H) -H [ | -HVW1 ] + [ #H destruct + elim (lt_or_ge i d) #Hdi [ -HVW1 | ] + [ elim (ltpss_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 â¦ H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 â¦ HK12 â¦ HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 + lapply (ldrop_fwd_ldrop2 â¦ HLK2) #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge â¦ HW12 â¦ H â¦ HWU1 â¦ HWU2) // -HW12 -H -HWU1 + >minus_plus minus_plus #H + lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H + lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H + lapply (tpss_weak â¦ H d e ? ?) [1,2: normalize [ >commutative_plus shift_append_assoc normalize #H + elim (tpr_inv_bind1 â¦ H) -H * + [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct + elim (IH â¦ HT10) -IH -T1 #L #T #HL1 #H destruct + elim (tps_fwd_shift1 â¦ H0) -T #L2 #T2 #HL2 #H destruct + >append_length >HL1 >HL2 -L1 -L + @(ex2_2_intro â¦ (â.â{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 *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpr_lift.etc new file mode 100644 index 000000000..8d6b0363a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpr_lift.etc @@ -0,0 +1,121 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ HU1 â¦ HU2) -HU1 #H destruct + [ lapply (lift_inv_sort1 â¦ HU2) -HU2 #H destruct // + | lapply (lift_inv_lref1 â¦ HU2) * * #Hid #H destruct // + | lapply (lift_inv_gref1 â¦ HU2) -HU2 #H destruct // + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 + elim (lift_inv_flat1 â¦ HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct + elim (lift_inv_flat1 â¦ 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 â¦ HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct + elim (lift_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct + elim (lift_inv_bind1 â¦ 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 â¦ HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct + elim (lift_inv_bind1 â¦ HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct + elim (lift_total T (d + 1) e) #U #HTU + @tpr_delta + [4: @(tps_lift_le â¦ HT2 â¦ 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 â¦ HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct + elim (lift_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct + elim (lift_inv_bind1 â¦ HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct + elim (lift_inv_flat1 â¦ HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct + elim (lift_trans_ge â¦ HV2 â¦ HV3 ?) -V // /3 width=4/ +| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2 + elim (lift_inv_bind1 â¦ H) -H #V3 #T3 #_ #HT13 #H destruct -V + elim (lift_conf_O1 â¦ HTU2 â¦ HT2) -T2 /3 width=4/ +| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2 + elim (lift_inv_flat1 â¦ 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 â¦ HX) -HX #H destruct /2 width=3/ + | lapply (lift_inv_lref2 â¦ HX) -HX * * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 â¦ HX) -HX #H destruct /2 width=3/ + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX + elim (lift_inv_flat2 â¦ HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct + elim (IHV12 â¦ HV01) -V1 + elim (IHT12 â¦ HT01) -T1 /3 width=5/ +| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX + elim (lift_inv_flat2 â¦ HX) -HX #V0 #Y #HV01 #HY #HX destruct + elim (lift_inv_bind2 â¦ HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct + elim (IHV12 â¦ HV01) -V1 + elim (IHT12 â¦ HT01) -T1 /3 width=5/ +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX + elim (lift_inv_bind2 â¦ HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct + elim (IHV12 â¦ HWV1) -V1 #W2 #HWV2 #HW12 + elim (IHT1 â¦ HUT1) -T1 #U #HUT #HU1 + elim (tps_inv_lift1_le â¦ HT2 â¦ 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 â¦ HX) -HX #V0 #Y #HV01 #HY #HX destruct + elim (lift_inv_bind2 â¦ HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct + elim (IHV12 â¦ HV01) -V1 #V3 #HV32 #HV03 + elim (IHW12 â¦ HW01) -W1 #W3 #HW32 #HW03 + elim (IHT12 â¦ HT01) -T1 #T3 #HT32 #HT03 + elim (lift_trans_le â¦ HV32 â¦ 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 â¦ HX) -HX #V3 #T3 #_ #HT31 #H destruct + elim (IHT1 â¦ HT31) -T1 #T1 #HT1 #HT31 + elim (lift_div_le â¦ HT2 â¦ HT1 ?) -T // /3 width=5/ +| #V #T1 #T2 #_ #IHT12 #X #d #e #HX + elim (lift_inv_flat2 â¦ HX) -HX #V0 #T0 #_ #HT01 #H destruct + elim (IHT12 â¦ HT01) -T1 /3 width=3/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was pr0_gen_abst *) +lemma tpr_inv_abst1: âa,V1,T1,U2. â{a}V1. T1 â¡ U2 â + ââV2,T2. V1 â¡ V2 & T1 â¡ T2 & U2 = â{a}V2. T2. +#a #V1 #T1 #U2 #H elim (tpr_inv_bind1 â¦ H) -H * +[ #V2 #T #T2 #HV12 #HT1 #HT2 + lapply (tps_inv_refl_SO2 â¦ HT2 ???) -HT2 // /2 width=5/ +| #T2 #_ #_ #_ #H destruct +] +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma tpr_fwd_abst1: âa,V1,T1,U2. â{a}V1.T1 â¡ U2 â âb,I,W. + ââV2,T2. â{b,I}W.T1 â¡ â{b,I}W.T2 & + U2 = â{a}V2.T2. +#a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 â¦ H) -H * +[ #V2 #T #T2 #HV12 #HT1 #HT2 + lapply (tps_inv_refl_SO2 â¦ HT2 ???) -HT2 // /3 width=4/ +| #T2 #_ #_ #_ #H destruct +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpr_tpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpr_tpr.etc new file mode 100644 index 000000000..a733345bf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpr_tpr.etc @@ -0,0 +1,261 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âI. ââX. âª{I} â¡ X & âª{I} â¡ X. +/2 width=3/ qed. + +fact tpr_conf_flat_flat: + âI,V0,V1,T0,T1,V2,T2. ( + âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â V0 â¡ V2 â T0 â¡ T1 â T0 â¡ T2 â + ââT0. â{I} V1. T1 â¡ T0 & â{I} V2. T2 â¡ T0. +#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 +elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 // #V #HV1 #HV2 +elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // /3 width=5/ +qed. + +fact tpr_conf_flat_beta: + âa,V0,V1,T1,V2,W0,U0,T2. ( + âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{U0} + 1) + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â V0 â¡ V2 â + U0 â¡ T2 â â{a}W0. U0 â¡ T1 â + ââX. âV1. T1 â¡ X & â{a}V2. T2 â¡ X. +#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H +elim (tpr_inv_abst1 â¦ H) -H #W1 #U1 #HW01 #HU01 #H destruct +elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH â¦ HT02 â¦ 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: + âa,V0,V1,T1,V2,V,W0,W2,U0,U2. ( + âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{U0} + 1) + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â V0 â¡ V2 â â§[O,1] V2 â¡ V â + W0 â¡ W2 â U0 â¡ U2 â â{a}W0. U0 â¡ T1 â + ââX. âV1. T1 â¡ X & â{a}W2. âV. U2 â¡ X. +#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H +elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2 +elim (lift_total VV 0 1) #VVV #HVV +lapply (tpr_lift â¦ HVV2 â¦ HV2 â¦ HVV) #HVVV +elim (tpr_inv_abbr1 â¦ H) -H * +(* case 1: delta *) +[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct + elim (IH â¦ HW02 â¦ HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2 + elim (IH â¦ HU02 â¦ HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2 + elim (tpr_tps_conf_bind â¦ HWW2 HUUU2 â¦ 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 â¦ HU01 â¦ HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2 + elim (tpr_inv_lift1 â¦ HU1 â¦ HTU1) -U1 #UU #HUU #HT1UU #H destruct + @(ex2_intro â¦ (âVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *) +] +qed. + +fact tpr_conf_flat_cast: + âX2,V0,V1,T0,T1. ( + âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â T0 â¡ T1 â T0 â¡ X2 â + ââX. âV1. T1 â¡ X & X2 â¡ X. +#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02 +elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // /3 width=3/ +qed. + +fact tpr_conf_beta_beta: + âa. âW0:term. âV0,V1,T0,T1,V2,T2. ( + âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{T0} + 1) + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â V0 â¡ V2 â T0 â¡ T1 â T0 â¡ T2 â + ââX. â{a}V1. T1 â¡X & â{a}V2. T2 â¡ X. +#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 +elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ +elim (IH â¦ HT01 â¦ 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: + âa,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. ( + âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â V0 â¡ V2 â T0 â¡ T1 â T0 â¡ T2 â + â. â{I1} V1 â¢ T1 â¶ [O, 1] TT1 â + â. â{I1} V2 â¢ T2 â¶ [O, 1] TT2 â + ââX. â{a,I1} V1. TT1 â¡ X & â{a,I1} V2. TT2 â¡ X. +#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2 +elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 // #V #HV1 #HV2 +elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // #T #HT1 #HT2 +elim (tpr_tps_conf_bind â¦ HV1 HT1 â¦ HTT1) -T1 #U1 #TTU1 #HTU1 +elim (tpr_tps_conf_bind â¦ HV2 HT2 â¦ HTT2) -T2 #U2 #TTU2 #HTU2 +elim (tps_conf_eq â¦ HTU1 â¦ 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: + âX2,V0,V1,T0,T1,TT1,T2. ( + âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â T0 â¡ T1 â â. âV1 â¢ T1 â¶ [O,1] TT1 â + T0 â¡ T2 â â§[O, 1] X2 â¡ T2 â + ââX. +âV1. TT1 â¡ X & X2 â¡ X. +#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2 +elim (IH â¦ HT01 â¦ HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2 +elim (tpr_tps_conf_bind ? ? V1 â¦ HT1 HTT1) -T1 // #TT #HTT1 #HTT +elim (tpr_inv_lift1 â¦ HT2 â¦ HXT2) -T2 #X #HXT #HX2 +lapply (tps_inv_lift1_eq â¦ HTT â¦ HXT) -HTT #H destruct /3 width=3/ +qed. + +(* Basic_1: was: pr0_upsilon_upsilon *) +fact tpr_conf_theta_theta: + âa,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. ( + âX0:term. â¯{X0} < â¯{V0} + (â¯{W0} + â¯{T0} + 1) + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + V0 â¡ V1 â V0 â¡ V2 â W0 â¡ W1 â W0 â¡ W2 â T0 â¡ T1 â T0 â¡ T2 â + â§[O, 1] V1 â¡ VV1 â â§[O, 1] V2 â¡ VV2 â + ââX. â{a}W1. âVV1. T1 â¡ X & â{a}W2. âVV2. T2 â¡ X. +#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2 +elim (IH â¦ HV01 â¦ HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH â¦ HW01 â¦ HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2 +elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2 +elim (lift_total V 0 1) #VV #HVV +lapply (tpr_lift â¦ HV1 â¦ HVV1 â¦ HVV) -V1 #HVV1 +lapply (tpr_lift â¦ HV2 â¦ HVV2 â¦ 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: + âV0:term. âX2,TT0,T0,T1,TT2. ( + âX0:term. â¯{X0} < â¯{V0} + â¯{TT0} + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + TT0 â¡ T0 â â§[O, 1] T1 â¡ T0 â + TT0 â¡ TT2 â â§[O, 1] X2 â¡ TT2 â + ââX. T1 â¡ X & X2 â¡ X. +#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2 +elim (IH â¦ HTT0 â¦ HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2 +elim (tpr_inv_lift1 â¦ HT0 â¦ HT10) -T0 #T0 #HT0 #HT10 +elim (tpr_inv_lift1 â¦ HTT2 â¦ HXTT2) -TT2 #TT2 #HTT2 #HXTT2 +lapply (lift_inj â¦ HTT2 â¦ HT0) -HTT2 #H destruct /2 width=3/ +qed. + +fact tpr_conf_tau_tau: + âV0,T0:term. âX2,T1. ( + âX0:term. â¯{X0} < â¯{V0} + â¯{T0} + 1 â + âX1,X2. X0 â¡ X1 â X0 â¡ X2 â + ââX. X1 â¡ X & X2 â¡ X + ) â + T0 â¡ T1 â T0 â¡ X2 â + ââX. T1 â¡ X & X2 â¡ X. +#V0 #T0 #X2 #T1 #IH #HT01 #HT02 +elim (IH â¦ HT01 â¦ HT02) -HT01 -HT02 -IH // /2 width=3/ +qed. + +(* Confluence ***************************************************************) + +(* Basic_1: was: pr0_confluence *) +theorem tpr_conf: âT0:term. âT1,T2. T0 â¡ T1 â T0 â¡ T2 â + ââT. T1 â¡ T & T2 â¡ T. +#T0 @(f_ind â¦ tw â¦ T0) -T0 #n #IH * +[ #I #_ #X1 #X2 #H1 #H2 -n + >(tpr_inv_atom1 â¦ H1) -X1 + >(tpr_inv_atom1 â¦ H2) -X2 +(* case 1: atom, atom *) + // +| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2 + [ elim (tpr_inv_bind1 â¦ H1) -H1 * + [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1 + | #T1 #HT01 #HXT1 #H11 #H12 + ] + elim (tpr_inv_bind1 â¦ 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 â¦ 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 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs.etc new file mode 100644 index 000000000..492b39041 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs.etc @@ -0,0 +1,94 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¡ * 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 â TC â¦ tpr. + +interpretation "context-free parallel computation (term)" + 'PRedStar T1 T2 = (tprs T1 T2). + +(* Basic eliminators ********************************************************) + +lemma tprs_ind: âT1. âR:predicate term. R T1 â + (âT,T2. T1 â¡* T â T â¡ T2 â R T â R T2) â + âT2. T1 â¡* T2 â R T2. +#T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // +qed-. + +lemma tprs_ind_dx: âT2. âR:predicate term. R T2 â + (âT1,T. T1 â¡ T â T â¡* T2 â R T â R T1) â + âT1. T1 â¡* T2 â R T1. +#T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma tprs_refl: reflexive â¦ tprs. +/2 width=1/ qed. + +lemma tpr_tprs: âT1,T2. T1 â¡ T2 â T2 â¡* T2. +/2 width=1/ qed. + +lemma tprs_strap1: âT1,T,T2. T1 â¡* T â T â¡ T2 â T1 â¡* T2. +/2 width=3/ qed. + +lemma tprs_strap2: âT1,T,T2. T1 â¡ T â T â¡* T2 â T1 â¡* T2. +/2 width=3/ qed. + +(* Basic_1: was only: pr1_head_1 *) +lemma tprs_pair_sn: âI,T1,T2. T1 â¡ T2 â âV1,V2. V1 â¡* V2 â + â¡{I} V1. T1 â¡* â¡{I} V2. T2. +* [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind â¦ H) -V2 +[1,3: /3 width=1/ +|2,4: #V #V2 #_ #HV2 #IHV1 + @(tprs_strap1 â¦ IHV1) -IHV1 /2 width=1/ +] +qed. + +(* Basic_1: was only: pr1_head_2 *) +lemma tprs_pair_dx: âI,V1,V2. V1 â¡ V2 â âT1,T2. T1 â¡* T2 â + â¡{I} V1. T1 â¡* â¡{I} V2. T2. +* [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind â¦ H) -T2 +[1,3: /3 width=1/ +|2,4: #T #T2 #_ #HT2 #IHT1 + @(tprs_strap1 â¦ IHT1) -IHT1 /2 width=1/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma tprs_inv_atom1: âU2,k. âk â¡* U2 â U2 = âk. +#U2 #k #H @(tprs_ind â¦ H) -U2 // +#U #U2 #_ #HU2 #IHU1 destruct +>(tpr_inv_atom1 â¦ HU2) -HU2 // +qed-. + +lemma tprs_inv_cast1: âW1,T1,U2. âW1.T1 â¡* U2 â T1 â¡* U2 â¨ + ââW2,T2. W1 â¡* W2 & T1 â¡* T2 & U2 = âW2.T2. +#W1 #T1 #U2 #H @(tprs_ind â¦ H) -U2 /3 width=5/ +#U #U2 #_ #HU2 * /3 width=3/ * +#W #T #HW1 #HT1 #H destruct +elim (tpr_inv_cast1 â¦ HU2) -HU2 /3 width=3/ * +#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs_lift.etc new file mode 100644 index 000000000..d0d173470 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs_lift.etc @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âa,V1,T1,U2. â{a}V1. T1 â¡* U2 â + ââV2,T2. V1 â¡* V2 & T1 â¡* T2 & U2 = â{a}V2. T2. +#a #V1 #T1 #U2 #H @(tprs_ind â¦ H) -U2 /2 width=5/ +#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct +elim (tpr_inv_abst1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ +qed-. + +lemma tprs_inv_abst: âa,V1,V2,T1,T2. â{a}V1. T1 â¡* â{a}V2. T2 â + V1 â¡* V2 â§ T1 â¡* T2. +#a #V1 #V2 #T1 #T2 #H +elim (tprs_inv_abst1 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs_tprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs_tprs.etc new file mode 100644 index 000000000..87a4a71ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tprs_tprs.etc @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âT1,T. T â¡* T1 â âT2. T â¡ T2 â + ââT0. T1 â¡ T0 & T2 â¡* T0. +/3 width=3 by TC_strip1, tpr_conf/ qed. + +(* Main propertis ***********************************************************) + +(* Basic_1: was: pr1_confluence *) +theorem tprs_conf: confluent â¦ tprs. +/3 width=3/ qed. + +(* Basic_1: was: pr1_t *) +theorem tprs_trans: Transitive â¦ tprs. +/2 width=3/ qed. + +(* Basic_1: was: pr1_comp *) +lemma tprs_pair: âI,V1,V2. V1 â¡* V2 â âT1,T2. T1 â¡* T2 â + â¡{I} V1. T1 â¡* â¡{I} V2. T2. +#I #V1 #V2 #H @(tprs_ind â¦ H) -V2 /2 width=1/ +#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 +@(tprs_trans â¦ (â¡{I}V.T2)) /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tps.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tps.etc new file mode 100644 index 000000000..7d89243d5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tps.etc @@ -0,0 +1,288 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 T1 break â¶ [ 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 â nat â lenv â relation term â +| tps_atom : âL,I,d,e. tps d e L (âª{I}) (âª{I}) +| tps_subst: âL,K,V,W,i,d,e. d â¤ i â i < d + e â + â©[0, i] L â¡ K. âV â â§[0, i + 1] V â¡ W â tps d e L (#i) W +| tps_bind : âL,a,I,V1,V2,T1,T2,d,e. + tps d e L V1 V2 â tps (d + 1) e (L. â{I} V2) T1 T2 â + tps d e L (â{a,I} V1. T1) (â{a,I} V2. T2) +| tps_flat : âL,I,V1,V2,T1,T2,d,e. + tps d e L V1 V2 â tps d e L T1 T2 â + tps d e L (â{I} V1. T1) (â{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: âL1,T1,T2,d,e. L1 â¢ T1 â¶ [d, e] T2 â + âL2. L2 â [d, e] L1 â L2 â¢ T1 â¶ [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 â¦ HL12 â¦ HLK1 ? ?) -HL12 -HLK1 // /2 width=4/ +| /4 width=1/ +| /3 width=1/ +] +qed. + +lemma tps_refl: âT,L,d,e. L â¢ T â¶ [d, e] T. +#T elim T -T // +#I elim I -I /2 width=1/ +qed. + +(* Basic_1: was: subst1_ex *) +lemma tps_full: âK,V,T1,L,d. â©[0, d] L â¡ (K. âV) â + ââT2,T. L â¢ T1 â¶ [d, 1] T2 & â§[d, 1] T â¡ 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 â¦ HVW i i ? ? ?) // /3 width=4/ +| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK + elim (IHW1 â¦ HLK) -IHW1 #W2 #W #HW12 #HW2 + [ elim (IHU1 (L. â{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/ + | elim (IHU1 â¦ HLK) -IHU1 -HLK /3 width=8/ + ] +] +qed. + +lemma tps_weak: âL,T1,T2,d1,e1. L â¢ T1 â¶ [d1, e1] T2 â + âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â + L â¢ T1 â¶ [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 â¦ Hd12 â¦ Hid1) -Hd12 -Hid1 #Hid2 + lapply (lt_to_le_to_lt â¦ Hide1 â¦ Hde12) -Hide1 /2 width=4/ +| /4 width=3/ +| /4 width=1/ +] +qed. + +lemma tps_weak_top: âL,T1,T2,d,e. + L â¢ T1 â¶ [d, e] T2 â L â¢ T1 â¶ [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 â¦ HLK) #Hi + lapply (le_to_lt_to_lt â¦ Hdi Hi) /3 width=4/ +| normalize /2 width=1/ +| /2 width=1/ +] +qed. + +lemma tps_weak_full: âL,T1,T2,d,e. + L â¢ T1 â¶ [d, e] T2 â L â¢ T1 â¶ [0, |L|] T2. +#L #T1 #T2 #d #e #HT12 +lapply (tps_weak â¦ HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tps_weak_top â¦ HT12) // +qed. + +lemma tps_split_up: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â âi. d â¤ i â i â¤ d + e â + ââT. L â¢ T1 â¶ [d, i - d] T & L â¢ T â¶ [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 â¢ (% â ?); // -Hdj /3 width=4/ + | -Hdi -Hdj #Hij + lapply (plus_minus_m_m â¦ 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 â¦ HT1 (L. â{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: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â + âi. d â¤ i â i â¤ d + e â + ââT. L â¢ T1 â¶ [i, d + e - i] T & + L â¢ T â¶ [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 â¢ (% â ?); // -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 â¦ HT1 (L. â{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: âK,T1,T2,d,e. K â¢ T1 â¶ [d, e] T2 â + âL. L @@ K â¢ T1 â¶ [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 â¦ HK0) #H +@(tps_subst â¦ (L@@K0) â¦ HVW) // (**) (* /3/ does not work *) +@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact tps_inv_atom1_aux: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â âI. T1 = âª{I} â + T2 = âª{I} â¨ + ââK,V,i. d â¤ i & i < d + e & + â©[O, i] L â¡ K. âV & + â§[O, i + 1] V â¡ 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: âL,T2,I,d,e. L â¢ âª{I} â¶ [d, e] T2 â + T2 = âª{I} â¨ + ââK,V,i. d â¤ i & i < d + e & + â©[O, i] L â¡ K. âV & + â§[O, i + 1] V â¡ T2 & + I = LRef i. +/2 width=3/ qed-. + + +(* Basic_1: was: subst1_gen_sort *) +lemma tps_inv_sort1: âL,T2,k,d,e. L â¢ âk â¶ [d, e] T2 â T2 = âk. +#L #T2 #k #d #e #H +elim (tps_inv_atom1 â¦ H) -H // +* #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +(* Basic_1: was: subst1_gen_lref *) +lemma tps_inv_lref1: âL,T2,i,d,e. L â¢ #i â¶ [d, e] T2 â + T2 = #i â¨ + ââK,V. d â¤ i & i < d + e & + â©[O, i] L â¡ K. âV & + â§[O, i + 1] V â¡ T2. +#L #T2 #i #d #e #H +elim (tps_inv_atom1 â¦ H) -H /2 width=1/ +* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/ +qed-. + +lemma tps_inv_gref1: âL,T2,p,d,e. L â¢ Â§p â¶ [d, e] T2 â T2 = Â§p. +#L #T2 #p #d #e #H +elim (tps_inv_atom1 â¦ H) -H // +* #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +fact tps_inv_bind1_aux: âd,e,L,U1,U2. L â¢ U1 â¶ [d, e] U2 â + âa,I,V1,T1. U1 = â{a,I} V1. T1 â + ââV2,T2. L â¢ V1 â¶ [d, e] V2 & + L. â{I} V2 â¢ T1 â¶ [d + 1, e] T2 & + U2 = â{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: âd,e,L,a,I,V1,T1,U2. L â¢ â{a,I} V1. T1 â¶ [d, e] U2 â + ââV2,T2. L â¢ V1 â¶ [d, e] V2 & + L. â{I} V2 â¢ T1 â¶ [d + 1, e] T2 & + U2 = â{a,I} V2. T2. +/2 width=3/ qed-. + +fact tps_inv_flat1_aux: âd,e,L,U1,U2. L â¢ U1 â¶ [d, e] U2 â + âI,V1,T1. U1 = â{I} V1. T1 â + ââV2,T2. L â¢ V1 â¶ [d, e] V2 & L â¢ T1 â¶ [d, e] T2 & + U2 = â{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: âd,e,L,I,V1,T1,U2. L â¢ â{I} V1. T1 â¶ [d, e] U2 â + ââV2,T2. L â¢ V1 â¶ [d, e] V2 & L â¢ T1 â¶ [d, e] T2 & + U2 = â{I} V2. T2. +/2 width=3/ qed-. + +fact tps_inv_refl_O2_aux: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â e = 0 â 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 â¦ Hdi â¦ Hide) -Hdi -Hide shift_append_assoc normalize #H + elim (tps_inv_bind1 â¦ H) -H + #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tps_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tps_lift.etc new file mode 100644 index 000000000..1d65d8d63 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tps_lift.etc @@ -0,0 +1,294 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,T2,d,e1. L â¢ T1 â¶ [d, e1] T2 â âe2. e1 = e2 + 1 â + âK,V. â©[0, d] L â¡ K. âV â L â¢ T1 â¶ [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 â¦ Hdi ?) /2 width=1/ #H destruct + lapply (ldrop_mono â¦ HLK0 â¦ HLK) #H destruct + | -V -Hdi /2 width=4/ + ] +| /4 width=3/ +| /3 width=3/ +] +qed. + +lemma tps_inv_S2: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e + 1] T2 â + âK,V. â©[0, d] L â¡ K. âV â L â¢ T1 â¶ [d + 1, e] T2. +/2 width=3/ qed-. + +lemma tps_inv_refl_SO2: âL,T1,T2,d. L â¢ T1 â¶ [d, 1] T2 â + âK,V. â©[0, d] L â¡ K. âV â T1 = T2. +#L #T1 #T2 #d #HT12 #K #V #HLK +lapply (tps_inv_S2 â¦ T1 T2 â¦ 0 â¦ HLK) -K // -HT12 #HT12 +lapply (tps_inv_refl_O2 â¦ HT12) -HT12 // +qed-. + +(* Relocation properties ****************************************************) + +(* Basic_1: was: subst1_lift_lt *) +lemma tps_lift_le: âK,T1,T2,dt,et. K â¢ T1 â¶ [dt, et] T2 â + âL,U1,U2,d,e. â©[d, e] L â¡ K â + â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â + dt + et â¤ d â + L â¢ U1 â¶ [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 â¦ H1 â¦ 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 â¦ Hidet â¦ Hdetd) -Hdetd #Hid + lapply (lift_inv_lref1_lt â¦ H â¦ Hid) -H #H destruct + elim (lift_trans_ge â¦ HVW â¦ HWU2 ?) -W // (lift_mono â¦ HVY â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ +] +qed. + +lemma tps_lift_be: âK,T1,T2,dt,et. K â¢ T1 â¶ [dt, et] T2 â + âL,U1,U2,d,e. â©[d, e] L â¡ K â + â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â + dt â¤ d â d â¤ dt + et â + L â¢ U1 â¶ [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 â¦ H1 â¦ 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 â¦ H) -H * #Hid #H destruct + [ -Hdtd + lapply (lt_to_le_to_lt â¦ (dt+et+e) Hidet ?) // -Hidet #Hidete + elim (lift_trans_ge â¦ HVW â¦ HWU2 ?) -W // (lift_mono â¦ HVY â¦ HVW) -V #H destruct /2 width=4/ + | -Hdti + lapply (transitive_le â¦ Hdtd Hid) -Hdtd #Hdti + lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ +] +qed. + +(* Basic_1: was: subst1_lift_ge *) +lemma tps_lift_ge: âK,T1,T2,dt,et. K â¢ T1 â¶ [dt, et] T2 â + âL,U1,U2,d,e. â©[d, e] L â¡ K â + â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â + d â¤ dt â + L â¢ U1 â¶ [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 â¦ H1 â¦ 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 â¦ Hddt â¦ Hdti) -Hddt #Hid + lapply (lift_inv_lref1_ge â¦ H â¦ Hid) -H #H destruct + lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/ +] +qed. + +(* Basic_1: was: subst1_gen_lift_lt *) +lemma tps_inv_lift1_le: âL,U1,U2,dt,et. L â¢ U1 â¶ [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt + et â¤ d â + ââT2. K â¢ T1 â¶ [dt, et] T2 & â§[d, e] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3/ + | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 â¦ 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 â¦ Hidet â¦ Hdetd) -Hdetd #Hid + lapply (lift_inv_lref2_lt â¦ H â¦ Hid) -H #H destruct + elim (ldrop_conf_lt â¦ HLK â¦ HLKV ?) -L // #L #U #HKL #_ #HUV + elim (lift_trans_le â¦ HUV â¦ HVW ?) -V // >minus_plus minus_plus plus_minus // 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 â¦ H) -H #W1 #T1 #HWV1 #HTU1 #H destruct + elim (IHV12 â¦ HLK â¦ HWV1 ? ?) -V1 // #W2 #HW12 #HWV2 + elim (IHU12 â¦ 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 â¦ H) -H #W1 #T1 #HWV1 #HTU1 #H destruct + elim (IHV12 â¦ HLK â¦ HWV1 ? ?) -V1 // + elim (IHU12 â¦ HLK â¦ HTU1 ? ?) -U1 -HLK // /3 width=5/ +] +qed. + +(* Basic_1: was: subst1_gen_lift_ge *) +lemma tps_inv_lift1_ge: âL,U1,U2,dt,et. L â¢ U1 â¶ [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + d + e â¤ dt â + ââT2. K â¢ T1 â¶ [dt - e, et] T2 & â§[d, e] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3/ + | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 â¦ 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 â¦ Hdedt â¦ Hdti) #Hdei + elim (le_inv_plus_l â¦ Hdedt) -Hdedt #_ #Hedt + elim (le_inv_plus_l â¦ Hdei) #Hdie #Hei + lapply (lift_inv_lref2_ge â¦ H â¦ Hdei) -H #H destruct + lapply (ldrop_conf_ge â¦ HLK â¦ HLKV ?) -L // #HKV + elim (lift_split â¦ HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie + #V0 #HV10 >plus_minus // 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 â¦ H) -H #W1 #T1 #HWV1 #HTU1 #H destruct + elim (le_inv_plus_l â¦ Hdetd) #_ #Hedt + elim (IHV12 â¦ HLK â¦ HWV1 ?) -V1 // #W2 #HW12 #HWV2 + elim (IHU12 â¦ HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ] + IHV12 // >IHT12 // +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_flat2 â¦ 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: âL,U1,U2,dt,et. L â¢ U1 â¶ [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + d â¤ dt â dt â¤ d + e â d + e â¤ dt + et â + ââT2. K â¢ T1 â¶ [d, dt + et - (d + e)] T2 & â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet +elim (tps_split_up â¦ HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 +lapply (tps_weak â¦ HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 +lapply (tps_inv_lift1_eq â¦ HU1 â¦ HTU1) -HU1 #HU1 destruct +elim (tps_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1 ?) -U -L // commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU +lapply (tps_weak â¦ HU2 d e ? ?) -HU2 // [ >commutative_plus (lift_mono â¦ HVT1 â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (tps_lsubr_trans â¦ HT02 (L. â{I} V1) ?) -HT02 /2 width=1/ #HT02 + elim (IHV01 â¦ HV02) -V0 #V #HV1 #HV2 + elim (IHT01 â¦ HT02) -T0 #T #HT1 #HT2 + lapply (tps_lsubr_trans â¦ HT1 (L. â{I} V) ?) -HT1 /2 width=1/ + lapply (tps_lsubr_trans â¦ HT2 (L. â{I} V) ?) -HT2 /3 width=5/ +| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (tps_inv_flat1 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02) -V0 + elim (IHT01 â¦ HT02) -T0 /3 width=5/ +] +qed. + +(* Basic_1: was: subst1_confluence_neq *) +theorem tps_conf_neq: âL1,T0,T1,d1,e1. L1 â¢ T0 â¶ [d1, e1] T1 â + âL2,T2,d2,e2. L2 â¢ T0 â¶ [d2, e2] T2 â + (d1 + e1 â¤ d2 â¨ d2 + e2 â¤ d1) â + ââT. L2 â¢ T1 â¶ [d2, e2] T & L1 â¢ T2 â¶ [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 â¦ H1) -H1 + [ #H destruct /3 width=6/ + | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded + [ -Hd1 -Hde2 + lapply (transitive_le â¦ Hded Hd2) -Hded -Hd2 #H + lapply (lt_to_le_to_lt â¦ Hde1 H) -Hde1 -H #H + elim (lt_refl_false â¦ H) + | -Hd2 -Hde1 + lapply (transitive_le â¦ Hded Hd1) -Hded -Hd1 #H + lapply (lt_to_le_to_lt â¦ Hde2 H) -Hde2 -H #H + elim (lt_refl_false â¦ H) + ] + ] +| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (tps_inv_bind1 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02 H) -V0 #V #HV1 #HV2 + elim (IHT01 â¦ HT02 ?) -T0 + [ -H #T #HT1 #HT2 + lapply (tps_lsubr_trans â¦ HT1 (L2. â{I} V) ?) -HT1 /2 width=1/ + lapply (tps_lsubr_trans â¦ HT2 (L1. â{I} V) ?) -HT2 /2 width=1/ /3 width=5/ + | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in â¢ (? ? %); 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02 H) -V0 + elim (IHT01 â¦ 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: âL,T1,T0,d,e. L â¢ T1 â¶ [d, e] T0 â + âT2. L â¢ T0 â¶ [d, 1] T2 â 1 â¤ e â + L â¢ T1 â¶ [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 â¦ H) -H + [ #H destruct // + | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct + lapply (lt_to_le_to_lt â¦ (d + e) Hide2 ?) /2 width=4/ + ] +| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He + lapply (tps_weak â¦ HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2 + <(tps_inv_lift1_eq â¦ HVT2 â¦ HVW) -HVT2 /2 width=4/ +| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (tps_inv_bind1 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct + lapply (tps_lsubr_trans â¦ HT02 (L. â{I} V0) ?) -HT02 /2 width=1/ #HT02 + lapply (IHT10 â¦ HT02 He) -T0 #HT12 + lapply (tps_lsubr_trans â¦ HT12 (L. â{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 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/ +] +qed. + +theorem tps_trans_down: âL,T1,T0,d1,e1. L â¢ T1 â¶ [d1, e1] T0 â + âT2,d2,e2. L â¢ T0 â¶ [d2, e2] T2 â d2 + e2 â¤ d1 â + ââT. L â¢ T1 â¶ [d2, e2] T & L â¢ T â¶ [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 â¦ Hde2d1 Hdi1) -Hde2d1 #Hde2i1 + lapply (tps_weak â¦ HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2 + <(tps_inv_lift1_eq â¦ HWT2 â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (tps_lsubr_trans â¦ HT02 (L. â{I} V0) ?) -HT02 /2 width=1/ #HT02 + elim (IHV10 â¦ HV02 ?) -IHV10 -HV02 // #V + elim (IHT10 â¦ HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2 + lapply (tps_lsubr_trans â¦ HT1 (L. â{I} V) ?) -HT1 /2 width=1/ + lapply (tps_lsubr_trans â¦ HT2 (L. â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV10 â¦ HV02 ?) -V0 // + elim (IHT10 â¦ HT02 ?) -T0 // /3 width=6/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss.etc new file mode 100644 index 000000000..433718803 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss.etc @@ -0,0 +1,188 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 T1 break â¶ * [ 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 â nat â lenv â relation term â + Î»d,e,L. TC â¦ (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: âd,e,L,T1. âR:predicate term. R T1 â + (âT,T2. L â¢ T1 â¶* [d, e] T â L â¢ T â¶ [d, e] T2 â R T â R T2) â + âT2. L â¢ T1 â¶* [d, e] T2 â R T2. +#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // +qed-. + +lemma tpss_ind_dx: âd,e,L,T2. âR:predicate term. R T2 â + (âT1,T. L â¢ T1 â¶ [d, e] T â L â¢ T â¶* [d, e] T2 â R T â R T1) â + âT1. L â¢ T1 â¶* [d, e] T2 â R T1. +#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma tps_tpss: âL,T1,T2,d,e. L â¢ T1 â¶ [d, e] T2 â L â¢ T1 â¶* [d, e] T2. +/2 width=1/ qed. + +lemma tpss_strap1: âL,T1,T,T2,d,e. + L â¢ T1 â¶* [d, e] T â L â¢ T â¶ [d, e] T2 â L â¢ T1 â¶* [d, e] T2. +/2 width=3/ qed. + +lemma tpss_strap2: âL,T1,T,T2,d,e. + L â¢ T1 â¶ [d, e] T â L â¢ T â¶* [d, e] T2 â L â¢ T1 â¶* [d, e] T2. +/2 width=3/ qed. + +lemma tpss_lsubr_trans: âL1,T1,T2,d,e. L1 â¢ T1 â¶* [d, e] T2 â + âL2. L2 â [d, e] L1 â L2 â¢ T1 â¶* [d, e] T2. +/3 width=3/ qed. + +lemma tpss_refl: âd,e,L,T. L â¢ T â¶* [d, e] T. +/2 width=1/ qed. + +lemma tpss_bind: âL,V1,V2,d,e. L â¢ V1 â¶* [d, e] V2 â + âa,I,T1,T2. L. â{I} V2 â¢ T1 â¶* [d + 1, e] T2 â + L â¢ â{a,I} V1. T1 â¶* [d, e] â{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 â¦ HT12 (L. â{I} V) ?) -HT12 /2 width=1/ #HT12 + lapply (IHV a â¦ HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_flat: âL,I,V1,V2,T1,T2,d,e. + L â¢ V1 â¶* [d, e] V2 â L â¢ T1 â¶* [d, e] T2 â + L â¢ â{I} V1. T1 â¶* [d, e] â{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 â¦ HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_weak: âL,T1,T2,d1,e1. L â¢ T1 â¶* [d1, e1] T2 â + âd2,e2. d2 â¤ d1 â d1 + e1 â¤ d2 + e2 â + L â¢ T1 â¶* [d2, e2] T2. +#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind â¦ H) -T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak â¦ HT12 â¦ Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/ +] +qed. + +lemma tpss_weak_top: âL,T1,T2,d,e. + L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¶* [d, |L| - d] T2. +#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak_top â¦ HT12) -HT12 /2 width=3/ +] +qed. + +lemma tpss_weak_full: âL,T1,T2,d,e. + L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¶* [0, |L|] T2. +#L #T1 #T2 #d #e #HT12 +lapply (tpss_weak â¦ HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tpss_weak_top â¦ HT12) // +qed. + +lemma tpss_append: âK,T1,T2,d,e. K â¢ T1 â¶* [d, e] T2 â + âL. L @@ K â¢ T1 â¶* [d, e] T2. +#K #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // /3 width=3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_sort1: âL,T2,k,d,e. L â¢ âk â¶* [d, e] T2 â T2 = âk. +#L #T2 #k #d #e #H @(tpss_ind â¦ H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct + >(tps_inv_sort1 â¦ HT2) -HT2 // +] +qed-. + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_gref1: âL,T2,p,d,e. L â¢ Â§p â¶* [d, e] T2 â T2 = Â§p. +#L #T2 #p #d #e #H @(tpss_ind â¦ H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct + >(tps_inv_gref1 â¦ HT2) -HT2 // +] +qed-. + +lemma tpss_inv_bind1: âd,e,L,a,I,V1,T1,U2. L â¢ â{a,I} V1. T1 â¶* [d, e] U2 â + ââV2,T2. L â¢ V1 â¶* [d, e] V2 & + L. â{I} V2 â¢ T1 â¶* [d + 1, e] T2 & + U2 = â{a,I} V2. T2. +#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind â¦ H) -U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (tps_inv_bind1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (tpss_lsubr_trans â¦ HT1 (L. â{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ +] +qed-. + +lemma tpss_inv_flat1: âd,e,L,I,V1,T1,U2. L â¢ â{I} V1. T1 â¶* [d, e] U2 â + ââV2,T2. L â¢ V1 â¶* [d, e] V2 & L â¢ T1 â¶* [d, e] T2 & + U2 = â{I} V2. T2. +#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind â¦ H) -U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (tps_inv_flat1 â¦ HU2) -HU2 /3 width=5/ +] +qed-. + +lemma tpss_inv_refl_O2: âL,T1,T2,d. L â¢ T1 â¶* [d, 0] T2 â T1 = T2. +#L #T1 #T2 #d #H @(tpss_ind â¦ H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 â¦ HT2) -HT2 // +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma tpss_fwd_tw: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â â¯{T1} â¤ â¯{T2}. +#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (tps_fwd_tw â¦ HT2) -HT2 #HT2 +@(transitive_le â¦ IHT1) // +qed-. + +lemma tpss_fwd_shift1: âL,L1,T1,T,d,e. L â¢ L1 @@ T1 â¶*[d, e] T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#L #L1 #T1 #T #d #e #H @(tpss_ind â¦ H) -T +[ /2 width=4/ +| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct + elim (tps_fwd_shift1 â¦ H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_alt.etc new file mode 100644 index 000000000..f7670eaa8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_alt.etc @@ -0,0 +1,105 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 T1 break â¶ â¶ * [ 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 â nat â lenv â relation term â +| tpssa_atom : âL,I,d,e. tpssa d e L (âª{I}) (âª{I}) +| tpssa_subst: âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â + â©[0, i] L â¡ K. âV1 â tpssa 0 (d + e - i - 1) K V1 V2 â + â§[0, i + 1] V2 â¡ W2 â tpssa d e L (#i) W2 +| tpssa_bind : âL,a,I,V1,V2,T1,T2,d,e. + tpssa d e L V1 V2 â tpssa (d + 1) e (L. â{I} V2) T1 T2 â + tpssa d e L (â{a,I} V1. T1) (â{a,I} V2. T2) +| tpssa_flat : âL,I,V1,V2,T1,T2,d,e. + tpssa d e L V1 V2 â tpssa d e L T1 T2 â + tpssa d e L (â{I} V1. T1) (â{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: âL1,T1,T2,d,e. L1 â¢ T1 â¶â¶* [d, e] T2 â + âL2. L2 â [d, e] L1 â L2 â¢ T1 â¶â¶* [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 â¦ HL12 â¦ HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ +| /4 width=1/ +| /3 width=1/ +] +qed. + +lemma tpssa_refl: âT,L,d,e. L â¢ T â¶â¶* [d, e] T. +#T elim T -T // +#I elim I -I /2 width=1/ +qed. + +lemma tpssa_tps_trans: âL,T1,T,d,e. L â¢ T1 â¶â¶* [d, e] T â + âT2. L â¢ T â¶ [d, e] T2 â L â¢ T1 â¶â¶* [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 â¦ H) -H // * /2 width=6/ +| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H + lapply (ldrop_fwd_ldrop2 â¦ HLK) #H0LK + lapply (tps_weak â¦ H 0 (d+e) ? ?) -H // #H + elim (tps_inv_lift1_be â¦ H â¦ H0LK â¦ HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/ +| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H + elim (tps_inv_bind1 â¦ H) -H #V2 #T2 #HV2 #HT2 #H destruct + lapply (tps_lsubr_trans â¦ HT2 (L.â{I}V) ?) -HT2 /2 width=1/ #HT2 + lapply (IHV1 â¦ HV2) -IHV1 -HV2 #HV12 + lapply (IHT1 â¦ HT2) -IHT1 -HT2 #HT12 + lapply (tpssa_lsubr_trans â¦ HT12 (L.â{I}V2) ?) -HT12 /2 width=1/ +| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H + elim (tps_inv_flat1 â¦ H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/ +] +qed. + +lemma tpss_tpssa: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â L â¢ T1 â¶â¶* [d, e] T2. +#L #T1 #T2 #d #e #H @(tpss_ind â¦ H) -T2 // /2 width=3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma tpssa_tpss: âL,T1,T2,d,e. L â¢ T1 â¶â¶* [d, e] T2 â L â¢ T1 â¶* [d, e] T2. +#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/ +qed-. + +lemma tpss_ind_alt: âR:natânatâlenvârelation term. + (âL,I,d,e. R d e L (âª{I}) (âª{I})) â + (âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â + â©[O, i] L â¡ K.âV1 â K â¢ V1 â¶* [O, d + e - i - 1] V2 â + â§[O, i + 1] V2 â¡ W2 â R O (d+e-i-1) K V1 V2 â R d e L (#i) W2 + ) â + (âL,a,I,V1,V2,T1,T2,d,e. L â¢ V1 â¶* [d, e] V2 â + L.â{I}V2 â¢ T1 â¶* [d + 1, e] T2 â R d e L V1 V2 â + R (d+1) e (L.â{I}V2) T1 T2 â R d e L (â{a,I}V1.T1) (â{a,I}V2.T2) + ) â + (âL,I,V1,V2,T1,T2,d,e. L â¢ V1 â¶* [d, e] V2 â + L â¢ T1 â¶* [d, e] T2 â R d e L V1 V2 â + R d e L T1 T2 â R d e L (â{I}V1.T1) (â{I}V2.T2) + ) â + âd,e,L,T1,T2. L â¢ T1 â¶* [d, e] T2 â R d e L T1 T2. +#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa â¦ H) -L -T1 -T2 -d -e +// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_lift.etc new file mode 100644 index 000000000..a68f86e32 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_lift.etc @@ -0,0 +1,196 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,K,V,U1,i,d,e. + d â¤ i â i < d + e â + â©[0, i] L â¡ K. âV â K â¢ V â¶* [0, d + e - i - 1] U1 â + âU2. â§[0, i + 1] U1 â¡ U2 â L â¢ #i â¶* [d, e] U2. +#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind â¦ H) -U1 +[ /3 width=4/ +| #U #U1 #_ #HU1 #IHU #U2 #HU12 + elim (lift_total U 0 (i+1)) #U0 #HU0 + lapply (IHU â¦ HU0) -IHU #H + lapply (ldrop_fwd_ldrop2 â¦ HLK) -HLK #HLK + lapply (tps_lift_ge â¦ HU1 â¦ HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02 + lapply (tps_weak â¦ 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: âL,T2,I,d,e. L â¢ âª{I} â¶* [d, e] T2 â + T2 = âª{I} â¨ + ââK,V1,V2,i. d â¤ i & i < d + e & + â©[O, i] L â¡ K. âV1 & + K â¢ V1 â¶* [0, d + e - i - 1] V2 & + â§[O, i + 1] V2 â¡ T2 & + I = LRef i. +#L #T2 #I #d #e #H @(tpss_ind â¦ H) -T2 +[ /2 width=1/ +| #T #T2 #_ #HT2 * + [ #H destruct + elim (tps_inv_atom1 â¦ HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ] + | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI + lapply (ldrop_fwd_ldrop2 â¦ HLK) #H + elim (tps_inv_lift1_ge_up â¦ HT2 â¦ H â¦ HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 (lift_mono â¦ HTU1 â¦ H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT â¦ HTU) -IHT #HU1 + lapply (tps_lift_le â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ +] +qed. + +lemma tpss_lift_be: âK,T1,T2,dt,et. K â¢ T1 â¶* [dt, et] T2 â + âL,U1,d,e. dt â¤ d â d â¤ dt + et â + â©[d, e] L â¡ K â â§[d, e] T1 â¡ U1 â + âU2. â§[d, e] T2 â¡ U2 â L â¢ U1 â¶* [dt, et + e] U2. +#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind â¦ H) -T2 +[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT â¦ HTU) -IHT #HU1 + lapply (tps_lift_be â¦ HT2 â¦ HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ +] +qed. + +lemma tpss_lift_ge: âK,T1,T2,dt,et. K â¢ T1 â¶* [dt, et] T2 â + âL,U1,d,e. d â¤ dt â â©[d, e] L â¡ K â + â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â + L â¢ U1 â¶* [dt + e, et] U2. +#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind â¦ H) -T2 +[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT â¦ HTU) -IHT #HU1 + lapply (tps_lift_ge â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ +] +qed. + +lemma tpss_inv_lift1_le: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt + et â¤ d â + ââT2. K â¢ T1 â¶* [dt, et] T2 & â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_le â¦ HU2 â¦ HLK â¦ HTU ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_be: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt â¤ d â d + e â¤ dt + et â + ââT2. K â¢ T1 â¶* [dt, et - e] T2 & â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_be â¦ HU2 â¦ HLK â¦ HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_ge: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + d + e â¤ dt â + ââT2. K â¢ T1 â¶* [dt - e, et] T2 & â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_eq: âL,U1,U2,d,e. + L â¢ U1 â¶* [d, e] U2 â âT1. â§[d, e] T1 â¡ U1 â U1 = U2. +#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind â¦ H) -U2 // +#U #U2 #_ #HU2 #IHU destruct +<(tps_inv_lift1_eq â¦ HU2 â¦ HTU1) -HU2 -HTU1 // +qed. + +lemma tpss_inv_lift1_ge_up: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + d â¤ dt â dt â¤ d + e â d + e â¤ dt + et â + ââT2. K â¢ T1 â¶* [d, dt + et - (d + e)] T2 & + â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_ge_up â¦ HU2 â¦ HLK â¦ HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_be_up: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt â¤ d â dt + et â¤ d + e â + ââT2. K â¢ T1 â¶* [dt, d - dt] T2 & â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_be_up â¦ HU2 â¦ HLK â¦ HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_le_up: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + dt â¤ d â d â¤ dt + et â dt + et â¤ d + e â + ââT2. K â¢ T1 â¶* [dt, d - dt] T2 & â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind â¦ H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_le_up â¦ HU2 â¦ HLK â¦ HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_tpss.etc new file mode 100644 index 000000000..d124bb32c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_tpss.etc @@ -0,0 +1,96 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,T2,d. L â¢ T1 â¶* [d, 1] T2 â L â¢ T1 â¶ [d, 1] T2. +#L #T1 #T2 #d #H @(tpss_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (tps_trans_ge â¦ IHT1 â¦ HT2 ?) // +qed-. + +(* Advanced properties ******************************************************) + +lemma tpss_strip_eq: âL,T0,T1,d1,e1. L â¢ T0 â¶* [d1, e1] T1 â + âT2,d2,e2. L â¢ T0 â¶ [d2, e2] T2 â + ââT. L â¢ T1 â¶ [d2, e2] T & L â¢ T2 â¶* [d1, e1] T. +/3 width=3/ qed. + +lemma tpss_strip_neq: âL1,T0,T1,d1,e1. L1 â¢ T0 â¶* [d1, e1] T1 â + âL2,T2,d2,e2. L2 â¢ T0 â¶ [d2, e2] T2 â + (d1 + e1 â¤ d2 â¨ d2 + e2 â¤ d1) â + ââT. L2 â¢ T1 â¶ [d2, e2] T & L1 â¢ T2 â¶* [d1, e1] T. +/3 width=3/ qed. + +lemma tpss_strap1_down: âL,T1,T0,d1,e1. L â¢ T1 â¶* [d1, e1] T0 â + âT2,d2,e2. L â¢ T0 â¶ [d2, e2] T2 â d2 + e2 â¤ d1 â + ââT. L â¢ T1 â¶ [d2, e2] T & L â¢ T â¶* [d1, e1] T2. +/3 width=3/ qed. + +lemma tpss_strap2_down: âL,T1,T0,d1,e1. L â¢ T1 â¶ [d1, e1] T0 â + âT2,d2,e2. L â¢ T0 â¶* [d2, e2] T2 â d2 + e2 â¤ d1 â + ââT. L â¢ T1 â¶* [d2, e2] T & L â¢ T â¶ [d1, e1] T2. +/3 width=3/ qed. + +lemma tpss_split_up: âL,T1,T2,d,e. L â¢ T1 â¶* [d, e] T2 â + âi. d â¤ i â i â¤ d + e â + ââT. L â¢ T1 â¶* [d, i - d] T & L â¢ T â¶* [i, d + e - i] T2. +#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind â¦ H) -T2 +[ /2 width=3/ +| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 + elim (tps_split_up â¦ HT12 â¦ Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02 + elim (tpss_strap1_down â¦ HT3 â¦ 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: âL,U1,U2,dt,et. L â¢ U1 â¶* [dt, et] U2 â + âK,d,e. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â + d â¤ dt â dt â¤ d + e â d + e â¤ dt + et â + ââT2. K â¢ T1 â¶* [d, dt + et - (d + e)] T2 & + â§[d, e] T2 â¡ U2. +#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet +elim (tpss_split_up â¦ HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 +lapply (tpss_weak â¦ HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 +lapply (tpss_inv_lift1_eq â¦ HU1 â¦ HTU1) -HU1 #HU1 destruct +elim (tpss_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1 ?) -HU2 -HLK -HTU1 // (lift_mono â¦ H1 â¦ 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 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HVW2 â¦ HWU2) -W2 // plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct + elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpys_flat/ +] +qed. + +lemma cpys_inv_lift1: âG. 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 â¦ H) -H #H destruct /2 width=3 by cpys_atom, lift_sort, ex2_intro/ + | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hid #H destruct + [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV + elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 + elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // (length_inv_zero_dx â¦ H) -L1 // +| #I1 #I2 #L1 #L2 #V #_ #IHL12 #H lapply (injective_plus_l â¦ H) -H + /3 width=1 by lsuby_pair/ +] +qed-. + +(* Basic inversion lemmas ***************************************************) + +fact lsuby_inv_atom1_aux: âL1,L2. L1 â L2 â L1 = â â L2 = â. +#L1 #L2 * -L1 -L2 // +#I1 #I2 #L1 #L2 #V #_ #H destruct +qed-. + +lemma lsuby_inv_atom1: âL2. â â L2 â L2 = â. +/2 width=3 by lsuby_inv_atom1_aux/ qed-. + +fact lsuby_inv_pair1_aux: âL1,L2. L1 â L2 â âJ1,K1,W. L1 = K1.â{J1}W â + L2 = â â¨ ââI2,K2. K1 â K2 & L2 = K2.â{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: âI1,K1,L2,W. K1.â{I1}W â L2 â + L2 = â â¨ ââI2,K2. K1 â K2 & L2 = K2.â{I2}W. +/2 width=4 by lsuby_inv_pair1_aux/ qed-. + +fact lsuby_inv_pair2_aux: âL1,L2. L1 â L2 â âJ2,K2,W. L2 = K2.â{J2}W â + ââI1,K1. K1 â K2 & L1 = K1.â{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: âI2,L1,K2,W. L1 â K2.â{I2}W â + ââI1,K1. K1 â K2 & L1 = K1.â{I1}W. +/2 width=4 by lsuby_inv_pair2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lsuby_fwd_length: âL1,L2. L1 â L2 â |L2| â¤ |L1|. +#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/ +qed-. + +lemma lsuby_ldrop_trans: âL1,L2. L1 â L2 â + âI2,K2,W,s,i. â©[s, 0, i] L2 â¡ K2.â{I2}W â + ââI1,K1. K1 â K2 & â©[s, 0, i] L1 â¡ K1.â{I1}W. +#L1 #L2 #H elim H -L1 -L2 +[ #L #J2 #K2 #W #s #i #H + elim (ldrop_inv_atom1 â¦ H) -H #H destruct +| #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #s #i #H + elim (ldrop_inv_O1_pair1 â¦ H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ] + [ /3 width=4 by ldrop_pair, ex2_2_intro/ + | elim (IHL12 â¦ HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_drop_lt, ex2_2_intro/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/lsuby_lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/lsuby_lsuby.etc new file mode 100644 index 000000000..254d62928 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/lsuby_lsuby.etc @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ lsuby. +#L1 #L #H elim H -L1 -L +[ #L1 #X #H lapply (lsuby_inv_atom1 â¦ H) -H // +| #I1 #I #L1 #L #V #_ #IHL1 #X #H elim (lsuby_inv_pair1 â¦ H) -H // * + #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/psubstsnstar_3.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/psubstsnstar_3.etc new file mode 100644 index 000000000..0d6dffee6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/psubstsnstar_3.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¶ * break term 46 L2 )" + non associative with precedence 45 + for @{ 'PSubstSnStar $G $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/psubststar_4.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/psubststar_4.etc new file mode 100644 index 000000000..0e76c6f71 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/psubststar_4.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 break â¶ * break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $G $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpx_cpys.etc new file mode 100644 index 000000000..3b38ee6ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpx_cpys.etc @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G. lsub_trans â¦ (cpx h g G) (lsuby 0 (â)). +#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 â¦ HL12 â¦ 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: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. +#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt â¦ H) -G -L -T1 -T2 -d -e +/2 width=7 by cpx_delta, cpx_bind, cpx_flat/ +qed. + +lemma cpy_cpx: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. +/3 width=3 by cpy_cpys, cpys_cpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpx_cpys2.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpx_cpys2.etc new file mode 100644 index 000000000..46abd5464 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpx_cpys2.etc @@ -0,0 +1,70 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. +#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt â¦ H) -G -L -T1 -T2 -d -e +/2 width=7 by cpx_delta, cpx_bind, cpx_flat/ +qed. + +lemma cpy_cpx: âh,g,G,L,T1,T2,d,e. â¦G, Lâ¦ â¢ T1 â¶[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. +/3 width=3 by cpy_cpys, cpys_cpx/ qed. + +lemma cpx_cpy_trans: âh,g,G,L,T1,T. â¦G, Lâ¦ â¢ T1 â¡[h, g] T â + âT2,d,e. â¦G, Lâ¦ â¢ T â¶[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[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 â¦ H) // + * /2 width=3 by cpy_cpx/ +| #G #L #k #l #Hkl #X #d #e #H >(cpy_inv_sort1 â¦ 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 â¦ HLK) #H0 + lapply (cpy_weak â¦ H 0 (â) ? ?) -H // #H + elim (cpy_inv_lift1_be â¦ H â¦ H0 â¦ 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 â¦ 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 â¦ 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 â¦ HT2 0 (â) ? ?) -HT2 // #HT2 + elim (lift_total T2 0 1) #U2 #HTU2 + lapply (cpy_lift_be â¦ HT2 (L.âV1) â¦ (â») â¦ HTU â¦ 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 â¦ HX) -HX #Y #T2 #HY #HT2 #H destruct + elim (cpy_inv_flat1 â¦ 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 â¦ HX) -HX #W2 #Y #HW2 #HY #H destruct + elim (cpy_inv_flat1 â¦ HY) -HY #U2 #T2 #HU2 #HT2 #H destruct + lapply (cpy_weak â¦ HU2 0 (â) ? ?) -HU2 // #HU2 + elim (cpy_inv_lift1_be â¦ HU2 L â¦ HVU) -U + /5 width=7 by cpx_theta, lsuby_cpy_trans, lsuby_succ, ldrop_drop/ +] +qed-. + +lemma cpx_cpys_trans: âh,g,G,L,T1,T. â¦G, Lâ¦ â¢ T1 â¡[h, g] T â + âT2,d,e. â¦G, Lâ¦ â¢ T â¶*[d, e] T2 â â¦G, Lâ¦ â¢ T1 â¡[h, g] T2. +#h #g #G #L #T1 #T #HT1 #T2 #d #e #H @(cpys_ind â¦ H) -T2 +/2 width=5 by cpx_cpy_trans/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpxs_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpxs_cpys.etc new file mode 100644 index 000000000..881030572 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/cpxs_cpys.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G. lsub_trans â¦ (cpxs h g G) (lsuby 0 (â)). +/3 width=5 by lsuby_cpx_trans, LTC_lsub_trans/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/lpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/lpx_cpys.etc new file mode 100644 index 000000000..d3c490a89 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/lpx_cpys.etc @@ -0,0 +1,69 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T1,T. â¦G, L1â¦ â¢ T1 â¡[h, g] T â + âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â + âT2,d,e. â¦G, L2â¦ â¢ T â¶[d, e] T2 â â¦G, L1â¦ â¢ T1 â¡[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 â¦ H) -H // + * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct + elim (lpx_ldrop_trans_O1 â¦ HL12 â¦ HLK2) -L2 #X #HLK1 #H + elim (lpx_inv_pair2 â¦ 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 â¦ 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 â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 + elim (lpx_inv_pair1 â¦ H) -H #K2 #V0 #HK12 #_ #H destruct + lapply (ldrop_fwd_drop2 â¦ HLK2) -V0 #HLK2 + lapply (cpy_weak â¦ HT2 0 (â) ? ?) -HT2 // #HT2 + elim (cpy_inv_lift1_be â¦ HT2 â¦ HLK2 â¦ 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 â¦ 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 â¦ 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 â¦ HT2 0 (â) ? ?) -HT2 // #HT2 + elim (lift_total T2 0 1) #U2 #HTU2 + lapply (cpy_lift_be â¦ HT2 (L2.âV1) â¦ (â») â¦ HTU â¦ 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 â¦ HX) -HX #Y #T2 #HY #HT2 #H destruct + elim (cpy_inv_flat1 â¦ 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 â¦ HX) -HX #W2 #Y #HW2 #HY #H destruct + elim (cpy_inv_flat1 â¦ HY) -HY #U2 #T2 #HU2 #HT2 #H destruct + lapply (cpy_weak â¦ HU2 0 (â) ? ?) -HU2 // #HU2 + elim (cpy_inv_lift1_be â¦ HU2 L2 â¦ HVU) -U + /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/ +] +qed-. + +lemma cpx_cpys_trans_lpx: âh,g,G,L1,T1,T. â¦G, L1â¦ â¢ T1 â¡[h, g] T â + âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â + âT2,d,e. â¦G, L2â¦ â¢ T â¶*[d, e] T2 â â¦G, L1â¦ â¢ T1 â¡[h, g] T2. +#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind â¦ H) -T2 +/2 width=7 by cpx_cpy_trans_lpx/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/lsuby.etc new file mode 100644 index 000000000..657ad6b6e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpys2/lsuby.etc @@ -0,0 +1,15 @@ +(* +lemma lsuby_weak: âL1,L2,d1,e1. L1 âÃ[d1, e1] L2 â + âd2,e2. d1 â¤ d2 â e2 â¤ e1 â L1 âÃ[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 â¦ 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 â¦ He21) -He21 #He21 + elim (ynat_cases d2) /3 width=1 by lsuby_pair/ + * #d0 #H destruct @lsuby_succ @IHL12 // + [ destruct + +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csup.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csup.etc new file mode 100644 index 000000000..0a980f6a7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csup.etc @@ -0,0 +1,88 @@ +(* Basic forward lemmas *****************************************************) + +lemma csup_fwd_ldrop: âL1,L2,T1,T2. â¦L1, T1â¦ > â¦L2, T2â¦ â + âi. â©[0, i] L1 â¡ L2 â¨ â©[0, i] L2 â¡ 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 â¦ HLK1) -HLK1 /3 width=2/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lift_csup_trans_eq: âT1,U1,d,e. â§[d, e] T1 â¡ U1 â + âL,U2. â¦L, U1â¦ > â¦L, U2â¦ â + âT2. â§[d, e] T2 â¡ 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 â¦ H) -H * + [ #_ #H destruct /2 width=2/ + | #H elim (discr_lpair_x_xy â¦ H) + ] +|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H + elim (csup_inv_flat1 â¦ H) -H #_ * #H destruct /2 width=2/ +] +#i #d #e [2,3: #_ ] #L #X #H +elim (csup_inv_atom1 â¦ H) -H #I #j #HL #H destruct +lapply (ldrop_pair2_fwd_cw â¦ HL X) -HL #H +elim (lt_refl_false â¦ H) +qed-. +(* +lemma lift_csup_trans_gt: âL1,L2,U1,U2. â¦L1, U1â¦ > â¦L2, U2â¦ â + â©[0, 1] L2 â¡ L1 â âT1,d,e. â§[d, e] T1 â¡ U1 â + âT2. â§[d + 1, e] T2 â¡ U2. +#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2 +[ #I #L1 #K1 #V #i #HLK1 #HKL1 + lapply (ldrop_fwd_lw â¦ HLK1) -HLK1 #HLK1 + lapply (ldrop_fwd_lw â¦ HKL1) -HKL1 #HKL1 + lapply (transitive_le â¦ HLK1 HKL1) -L1 normalize #H + + +| #a +| #a +] +#I #L1 #W1 #U1 #HL1 + + + + #X #d #e #H + lapply (ldrop_inv_refl â¦ HL1) -HL1 +| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H + lapply (ldrop_inv_ldrop1 â¦ HL1) + + elim (lift_inv_bind2 â¦ H) -H #W2 #U2 #HW21 #HU21 #H destruct + + + /3 width=2/ /4 width=2/ + +*) + + + +(* Advanced inversion lemmas ************************************************) + +lemma csup_inv_lref2_be: âL,U,i. â¦L, Uâ¦ > â¦L, #iâ¦ â + âT,d,e. â§[d, e] T â¡ U â d â¤ i â i < d + e â â¥. +#L #U #i #H #T #d #e #HTU #Hdi #Hide +elim (lift_csup_trans_eq â¦ HTU â¦ H) -H -T #T #H +elim (lift_inv_lref2_be â¦ H ? ?) // +qed-. + + +fact csup_inv_all4_refl_aux: âL1,L2,T1,T2. â¦L1, T1â¦ > â¦L2, T2â¦ â L1 = L2 â + â¨â¨ ââa,I,U. T1 = â{a,I}T2.U + | ââI,W. T1 = â{I}W.T2 + | ââI,U. T1 = â{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 â¦ HLK V) -HLK #H + elim (lt_refl_false â¦ H) +| #a #I #L #V #T #H + elim (discr_lpair_x_xy â¦ H) +] +qed-. + +lemma csup_inv_all4_refl: âL,T1,T2. â¦L, T1â¦ > â¦L, T2â¦ â + â¨â¨ ââa,I,U. T1 = â{a,I}T2.U + | ââI,W. T1 = â{I}W.T2 + | ââI,U. T1 = â{I}T2.U. +/2 width=4 by csup_inv_all4_refl_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csup_csup.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csup_csup.etc new file mode 100644 index 000000000..661d89697 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csup_csup.etc @@ -0,0 +1,33 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,L0,T1,T2. â¦L, T1â¦ > â¦L0, T2â¦ â + âT3. â¦L0, T2â¦ > â¦L, T3â¦ â + L = L0 â¨ â¦L, T1â¦ > â¦L, T3â¦. +#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ +[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H + lapply (ldrop_pair2_fwd_cw â¦ HLK0 T3) -HLK0 #H1 + lapply (csup_fwd_cw â¦ H) -H #H2 + lapply (transitive_lt â¦ H1 H2) -H1 -H2 #H + elim (lt_refl_false â¦ H) +| #a #I #L0 #V2 #T2 #T3 #HT23 + elim (csup_inv_ldrop â¦ HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ + qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csups_csups.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csups_csups.etc new file mode 100644 index 000000000..9978429d7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/csup/csups_csups.etc @@ -0,0 +1,57 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,L0,T1,T2. â¦L, T1â¦ > â¦L0, T2â¦ â + âT3. â¦L0, T2â¦ >+ â¦L, T3â¦ â + L = L0 â¨ â¦L, T1â¦ >+ â¦L, T3â¦. +#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ +[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H + lapply (ldrop_pair2_fwd_cw â¦ HLK0 T3) -HLK0 #H1 + lapply (csupp_fwd_cw â¦ H) -H #H2 + lapply (transitive_lt â¦ H1 H2) -H1 -H2 #H + elim (lt_refl_false â¦ H) +| #a #I #L0 #V2 #T2 #T3 #HT23 + /3 width=5/ + + elim (csup_inv_ldrop â¦ HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ + qed-. + + + + + + + + +lemma csups_strap1_fwd_refl: âL,L0,T1,T2. â¦L, T1â¦ >* â¦L0, T2â¦ â + âT3. â¦L0, T2â¦ > â¦L, T3â¦ â L = L0. +#L #L0 #T1 #T2 #H @(csups_ind_dx â¦ H) -L -T1 // +#L1 #L #T1 #T #HL1 #_ #IHL0 #T3 #HL0 +lapply (csup_trans_fwd_refl â¦ HL10) â¦ HL0) -T2 +*) +lemma lift_csups_trans_aux: âT1,U1,d,e. â§[d, e] T1 â¡ U1 â + âL1,L2,U2. â¦L1, U1â¦ >* â¦L2, U2â¦ â L1 = L2 â + âT2. â§[d, e] T2 â¡ U2. +#T1 #U1 #d #e #HTU1 #L1 #L2 #U2 #H @(csups_ind â¦ H) -L2 -U2 /2 width=2/ -T1 +#L #L2 #U #U2 #HL1 #HL2 #IHL1 #H destruct + +* -T1 -U1 -d -e diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cpcs_delift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cpcs_delift.etc new file mode 100644 index 000000000..7012ec11e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cpcs_delift.etc @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,V,T1,T2. L.âV â¢ â¼*[O, 1] T1 â¡ T2 â + L â¢ T2 â¬* +âV.T1. +/3 width=1/ qed. + +(* Basic_1: was only: pc3_gen_cabbr *) +lemma thin_cpcs_delift_mono: âL,U1,U2. L â¢ U1 â¬* U2 â + âK,d,e. â¼*[d, e] L â¡ K â âT1. L â¢ â¼*[d, e] U1 â¡ T1 â + âT2. L â¢ â¼*[d, e] U2 â¡ T2 â K â¢ T1 â¬* T2. +#L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2 +elim (cpcs_inv_cprs â¦ H) -H #U #HU1 #HU2 +elim (thin_cprs_delift_conf â¦ HU1 â¦ HLK â¦ HTU1) -U1 #T #HT1 #HUT +elim (thin_cprs_delift_conf â¦ HU2 â¦ HLK â¦ HTU2) -U2 -HLK #X #HT2 #H +lapply (delift_mono â¦ H â¦ HUT) -L #H destruct /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cpr_delift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cpr_delift.etc new file mode 100644 index 000000000..04eb4ae23 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cpr_delift.etc @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,U1,U2. L â¢ U1 â¡ U2 â + âK,d,e. â¼*[d, e] L â¡ K â âT1. L â¢ â¼*[d, e] U1 â¡ T1 â + ââT2. K â¢ T1 â¡ T2 & L â¢ â¼*[d, e] U2 â¡ T2. +#L #U1 #U2 * #U #HU1 #HU2 #K #d #e #HLK #T1 #HTU1 +elim (tpr_delift_conf â¦ HU1 â¦ HTU1) -U1 #T #HT1 #HUT +elim (le_or_ge (|L|) d) #Hd +[ elim (thin_delift_tpss_conf_le â¦ HU2 â¦ HUT â¦ HLK ?) +| elim (le_or_ge (|L|) (d+e)) #Hde + [ elim (thin_delift_tpss_conf_le_up â¦ HU2 â¦ HUT â¦ HLK ? ? ?) + | elim (thin_delift_tpss_conf_be â¦ HU2 â¦ HUT â¦ HLK ? ?) + ] +] -U -HLK // -Hd [2,3: -Hde] #T2 #HT2 +lapply (cpr_intro â¦ HT1 HT2) -T /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cprs_delift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cprs_delift.etc new file mode 100644 index 000000000..6b7892611 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/cprs_delift.etc @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,V,T1,T2. L.âV â¢ â¼*[O, 1] T1 â¡ T2 â L â¢ +âV.T1 â¡* T2. +#L #V #T1 #T2 * #T #HT1 #HT2 +@(cprs_strap2 â¦ (+âV.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: âL,U1,U2. L â¢ U1 â¡* U2 â + âK,d,e. â¼*[d, e] L â¡ K â âT1. L â¢ â¼*[d, e] U1 â¡ T1 â + ââT2. K â¢ T1 â¡* T2 & L â¢ â¼*[d, e] U2 â¡ T2. +#L #U1 #U2 #H @(cprs_ind â¦ H) -U2 /2 width=3/ +#U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1 +elim (IHU1 â¦ HLK â¦ HTU1) -U1 #T #HT1 #HUT +elim (thin_cpr_delift_conf â¦ HU2 â¦ HLK â¦ HUT) -U -HLK /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift.etc new file mode 100644 index 000000000..eda839607 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift.etc @@ -0,0 +1,112 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break â¼ * [ term 46 d , break term 46 e ] break term 46 T1 â¡ 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 â nat â lenv â relation term â + Î»d,e,L,T1,T2. ââT. L â¢ T1 â¶* [d, e] T & â§[d, e] T2 â¡ T. + +interpretation "inverse basic relocation (term)" + 'TSubst L T1 d e T2 = (delift d e L T1 T2). + +(* Basic properties *********************************************************) + +lemma lift_delift: âT1,T2,d,e. â§[d, e] T1 â¡ T2 â + âL. L â¢ â¼*[d, e] T2 â¡ T1. +/2 width=3/ qed. + +lemma delift_refl_O2: âL,T,d. L â¢ â¼*[d, 0] T â¡ T. +/2 width=3/ qed. + +lemma delift_lsubr_trans: âL1,T1,T2,d,e. L1 â¢ â¼*[d, e] T1 â¡ T2 â + âL2. L2 â [d, e] L1 â L2 â¢ â¼*[d, e] T1 â¡ T2. +#L1 #T1 #T2 #d #e * /3 width=3/ +qed. + +lemma delift_sort: âL,d,e,k. L â¢ â¼*[d, e] âk â¡ âk. +/2 width=3/ qed. + +lemma delift_lref_lt: âL,d,e,i. i < d â L â¢ â¼*[d, e] #i â¡ #i. +/3 width=3/ qed. + +lemma delift_lref_ge: âL,d,e,i. d + e â¤ i â L â¢ â¼*[d, e] #i â¡ #(i - e). +/3 width=3/ qed. + +lemma delift_gref: âL,d,e,p. L â¢ â¼*[d, e] Â§p â¡ Â§p. +/2 width=3/ qed. + +lemma delift_bind: âa,I,L,V1,V2,T1,T2,d,e. + L â¢ â¼*[d, e] V1 â¡ V2 â L. â{I} V2 â¢ â¼*[d+1, e] T1 â¡ T2 â + L â¢ â¼*[d, e] â{a,I} V1. T1 â¡ â{a,I} V2. T2. +#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2 +lapply (tpss_lsubr_trans â¦ HT1 (L. â{I} V) ?) -HT1 /2 width=1/ /3 width=5/ +qed. + +lemma delift_flat: âI,L,V1,V2,T1,T2,d,e. + L â¢ â¼*[d, e] V1 â¡ V2 â L â¢ â¼*[d, e] T1 â¡ T2 â + L â¢ â¼*[d, e] â{I} V1. T1 â¡ â{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: âL,U2,d,e,k. L â¢ â¼*[d, e] âk â¡ U2 â U2 = âk. +#L #U2 #d #e #k * #U #HU +>(tpss_inv_sort1 â¦ HU) -HU #HU2 +>(lift_inv_sort2 â¦ HU2) -HU2 // +qed-. + +lemma delift_inv_gref1: âL,U2,d,e,p. L â¢ â¼*[d, e] Â§p â¡ U2 â U2 = Â§p. +#L #U #d #e #p * #U #HU +>(tpss_inv_gref1 â¦ HU) -HU #HU2 +>(lift_inv_gref2 â¦ HU2) -HU2 // +qed-. + +lemma delift_inv_bind1: âa,I,L,V1,T1,U2,d,e. L â¢ â¼*[d, e] â{a,I} V1. T1 â¡ U2 â + ââV2,T2. L â¢ â¼*[d, e] V1 â¡ V2 & + L. â{I} V2 â¢ â¼*[d+1, e] T1 â¡ T2 & + U2 = â{a,I} V2. T2. +#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 +elim (tpss_inv_bind1 â¦ HU) -HU #V #T #HV1 #HT1 #X destruct +elim (lift_inv_bind2 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 +lapply (tpss_lsubr_trans â¦ HT1 (L. â{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ +qed-. + +lemma delift_inv_flat1: âI,L,V1,T1,U2,d,e. L â¢ â¼*[d, e] â{I} V1. T1 â¡ U2 â + ââV2,T2. L â¢ â¼*[d, e] V1 â¡ V2 & + L â¢ â¼*[d, e] T1 â¡ T2 & + U2 = â{I} V2. T2. +#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 +elim (tpss_inv_flat1 â¦ HU) -HU #V #T #HV1 #HT1 #X destruct +elim (lift_inv_flat2 â¦ HU2) -HU2 /3 width=5/ +qed-. + +lemma delift_inv_refl_O2: âL,T1,T2,d. L â¢ â¼*[d, 0] T1 â¡ T2 â T1 = T2. +#L #T1 #T2 #d * #T #HT1 +>(tpss_inv_refl_O2 â¦ HT1) -HT1 #HT2 +>(lift_inv_refl_O2 â¦ HT2) -HT2 // +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma delift_fwd_tw: âL,T1,T2,d,e. L â¢ â¼*[d, e] T1 â¡ T2 â â¯{T1} â¤ â¯{T2}. +#L #T1 #T2 #d #e * #T #HT1 #HT2 +>(lift_fwd_tw â¦ HT2) -T2 /2 width=4 by tpss_fwd_tw/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_alt.etc new file mode 100644 index 000000000..e22911032 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_alt.etc @@ -0,0 +1,104 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break â¼ â¼ * [ term 46 d , break term 46 e ] break term 46 T1 â¡ 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 â nat â lenv â relation term â +| delifta_sort : âL,d,e,k. delifta d e L (âk) (âk) +| delifta_lref_lt: âL,d,e,i. i < d â delifta d e L (#i) (#i) +| delifta_lref_be: âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â + â©[0, i] L â¡ K. âV1 â delifta 0 (d + e - i - 1) K V1 V2 â + â§[0, d] V2 â¡ W2 â delifta d e L (#i) W2 +| delifta_lref_ge: âL,d,e,i. d + e â¤ i â delifta d e L (#i) (#(i - e)) +| delifta_gref : âL,d,e,p. delifta d e L (Â§p) (Â§p) +| delifta_bind : âL,a,I,V1,V2,T1,T2,d,e. + delifta d e L V1 V2 â delifta (d + 1) e (L. â{I} V2) T1 T2 â + delifta d e L (â{a,I} V1. T1) (â{a,I} V2. T2) +| delifta_flat : âL,I,V1,V2,T1,T2,d,e. + delifta d e L V1 V2 â delifta d e L T1 T2 â + delifta d e L (â{I} V1. T1) (â{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: âL1,T1,T2,d,e. L1 â¢ â¼â¼*[d, e] T1 â¡ T2 â + âL2. L2 â [d, e] L1 â L2 â¢ â¼â¼*[d, e] T1 â¡ 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 â¦ HL12 â¦ HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ +| /4 width=1/ +| /3 width=1/ +] +qed. + +lemma delift_delifta: âL,T1,T2,d,e. L â¢ â¼*[d, e] T1 â¡ T2 â L â¢ â¼â¼*[d, e] T1 â¡ T2. +#L #T1 @(f2_ind â¦ fw â¦ L T1) -L -T1 #n #IH #L * +[ * #i #Hn #T2 #d #e #H destruct + [ >(delift_inv_sort1 â¦ H) -H // + | elim (delift_inv_lref1 â¦ H) -H * /2 width=1/ + #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2 + lapply (ldrop_pair2_fwd_fw â¦ HLK) #H + lapply (IH â¦ HV12) // -H /2 width=6/ + | >(delift_inv_gref1 â¦ H) -H // + ] +| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H + [ elim (delift_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (delift_lsubr_trans â¦ HT12 (L.â{I}V1) ?) -HT12 /2 width=1/ #HT12 + lapply (IH â¦ HV12) -HV12 // #HV12 + lapply (IH â¦ HT12) -IH -HT12 /2 width=1/ #HT12 + lapply (delifta_lsubr_trans â¦ HT12 (L.â{I}V2) ?) -HT12 /2 width=1/ + | elim (delift_inv_flat1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH â¦ HV12) -HV12 // + lapply (IH â¦ HT12) -IH -HT12 // /2 width=1/ + ] +] +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma delifta_delift: âL,T1,T2,d,e. L â¢ â¼â¼*[d, e] T1 â¡ T2 â L â¢ â¼*[d, e] T1 â¡ 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: âR:ââââlenvârelation term. + (âL,d,e,k. R d e L (âk) (âk)) â + (âL,d,e,i. i < d â R d e L (#i) (#i)) â + (âL,K,V1,V2,W2,i,d,e. d â¤ i â i < d + e â + â©[O, i] L â¡ K.âV1 â K â¢ â¼*[O, d + e - i - 1] V1 â¡ V2 â + â§[O, d] V2 â¡ W2 â R O (d+e-i-1) K V1 V2 â R d e L (#i) W2 + ) â + (âL,d,e,i. d + e â¤ i â R d e L (#i) (#(i - e))) â + (âL,d,e,p. R d e L (Â§p) (Â§p)) â + (âL,a,I,V1,V2,T1,T2,d,e. L â¢ â¼*[d, e] V1 â¡ V2 â + L.â{I}V2 â¢ â¼*[d + 1, e] T1 â¡ T2 â R d e L V1 V2 â + R (d+1) e (L.â{I}V2) T1 T2 â R d e L (â{a,I}V1.T1) (â{a,I}V2.T2) + ) â + (âL,I,V1,V2,T1,T2,d,e. L â¢ â¼*[d, e] V1 â¡ V2 â + Lâ¢ â¼*[d, e] T1 â¡ T2 â R d e L V1 V2 â + R d e L T1 T2 â R d e L (â{I}V1.T1) (â{I}V2.T2) + ) â + âd,e,L,T1,T2. L â¢ â¼*[d, e] T1 â¡ T2 â R d e L T1 T2. +#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_delift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_delift.etc new file mode 100644 index 000000000..a5c563565 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_delift.etc @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T,T1,T2,d,e. + L â¢ â¼*[d, e] T â¡ T1 â L â¢ â¼*[d, e] T â¡ T2 â T1 = T2. +#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2 +elim (tpss_conf_eq â¦ H1TU1 â¦ H1TU2) -T #U #HU1 #HU2 +lapply (tpss_inv_lift1_eq â¦ HU1 â¦ H2TU1) -HU1 #H destruct +lapply (tpss_inv_lift1_eq â¦ HU2 â¦ H2TU2) -HU2 #H destruct +lapply (lift_inj â¦ H2TU1 â¦ H2TU2) // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_lift.etc new file mode 100644 index 000000000..8abaeb9f9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/delift/delift_lift.etc @@ -0,0 +1,164 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,K,V1,V2,U2,i,d,e. d â¤ i â i < d + e â + â©[0, i] L â¡ K. âV1 â K â¢ â¼*[0, d + e - i - 1] V1 â¡ V2 â + â§[0, d] V2 â¡ U2 â L â¢ â¼*[d, e] #i â¡ 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 â¦ HV2 â¦ HVU ? ?) -HV2 // >minus_plus commutative_plus in â¢ (??%??â?); 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 â¦ V1 â¦ Hde HL) // #V2 #HV12 + elim (IH (L.â{I}V1) T1 â¦ (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12 + lapply (delift_lsubr_trans â¦ HT12 (L.â{I}V2) ?) -HT12 /2 width=1/ /3 width=4/ +| #I #V1 #T1 #H #d #e #Hde #HL destruct + elim (IH â¦ V1 â¦ Hde HL) // #V2 #HV12 + elim (IH â¦ T1 â¦ Hde HL) -IH -Hde -HL // /3 width=2/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma delift_inv_lref1_lt: âL,U2,i,d,e. L â¢ â¼*[d, e] #i â¡ U2 â i < d â U2 = #i. +#L #U2 #i #d #e * #U #HU #HU2 #Hid +elim (tpss_inv_lref1 â¦ HU) -HU +[ #H destruct >(lift_inv_lref2_lt â¦ HU2) // +| * #K #V1 #V2 #Hdi + lapply (lt_to_le_to_lt â¦ Hid Hdi) -Hid -Hdi #Hi + elim (lt_refl_false â¦ Hi) +] +qed-. + +lemma delift_inv_lref1_be: âL,U2,d,e,i. L â¢ â¼*[d, e] #i â¡ U2 â + d â¤ i â i < d + e â + ââK,V1,V2. â©[0, i] L â¡ K. âV1 & + K â¢ â¼*[0, d + e - i - 1] V1 â¡ V2 & + â§[0, d] V2 â¡ U2. +#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide +elim (tpss_inv_lref1 â¦ HU) -HU +[ #H destruct elim (lift_inv_lref2_be â¦ HU2 ? ?) // +| * #K #V1 #V #_ #_ #HLK #HV1 #HVU + elim (lift_div_be â¦ HVU â¦ HU2 ? ?) -U // /2 width=1/ /3 width=6/ +] +qed-. + +lemma delift_inv_lref1_ge: âL,U2,i,d,e. L â¢ â¼*[d, e] #i â¡ U2 â + d + e â¤ i â U2 = #(i - e). +#L #U2 #i #d #e * #U #HU #HU2 #Hdei +elim (tpss_inv_lref1 â¦ HU) -HU +[ #H destruct >(lift_inv_lref2_ge â¦ HU2) // +| * #K #V1 #V2 #_ #Hide + lapply (lt_to_le_to_lt â¦ Hide Hdei) -Hide -Hdei #Hi + elim (lt_refl_false â¦ Hi) +] +qed-. + +lemma delift_inv_lref1: âL,U2,i,d,e. L â¢ â¼*[d, e] #i â¡ U2 â + â¨â¨ (i < d â§ U2 = #i) + | (ââK,V1,V2. d â¤ i & i < d + e & + â©[0, i] L â¡ K. âV1 & + K â¢ â¼*[0, d + e - i - 1] V1 â¡ V2 & + â§[0, d] V2 â¡ U2 + ) + | (d + e â¤ i â§ U2 = #(i - e)). +#L #U2 #i #d #e #H +elim (lt_or_ge i d) #Hdi +[ elim (delift_inv_lref1_lt â¦ H Hdi) -H /3 width=1/ +| elim (lt_or_ge i (d+e)) #Hide + [ elim (delift_inv_lref1_be â¦ H Hdi Hide) -H /3 width=6/ + | elim (delift_inv_lref1_ge â¦ H Hide) -H /3 width=1/ + ] +] +qed-. + +(* Properties on basic term relocation **************************************) + +lemma delift_lift_le: âK,T1,T2,dt,et. K â¢ â¼*[dt, et] T1 â¡ T2 â + âL,U1,d,e. dt + et â¤ d â â©[d, e] L â¡ K â + â§[d, e] T1 â¡ U1 â âU2. â§[d - et, e] T2 â¡ U2 â + L â¢ â¼*[dt, et] U1 â¡ 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 â¦ HT1 â¦ HLK HTU1 â¦ HTU) -T1 -HLK // #HU1 +elim (lift_trans_ge â¦ HT2 â¦ HTU ?) -T // -Hdetd #T #HT2 #HTU +>(lift_mono â¦ HTU2 â¦ HT2) -T2 /2 width=3/ +qed. + +lemma delift_lift_be: âK,T1,T2,dt,et. K â¢ â¼*[dt, et] T1 â¡ T2 â + âL,U1,d,e. dt â¤ d â d â¤ dt + et â + â©[d, e] L â¡ K â â§[d, e] T1 â¡ U1 â + L â¢ â¼*[dt, et + e] U1 â¡ 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 â¦ HT1 â¦ HLK HTU1 â¦ HTU) -T1 -HLK // #HU1 +lapply (lift_trans_be â¦ HT2 â¦ HTU ? ?) -T // -Hdtd -Hddet /2 width=3/ +qed. + +lemma delift_lift_ge: âK,T1,T2,dt,et. K â¢ â¼*[dt, et] T1 â¡ T2 â + âL,U1,d,e. d â¤ dt â â©[d, e] L â¡ K â + â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â + L â¢ â¼*[dt + e, et] U1 â¡ 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 â¦ HT1 â¦ HLK HTU1 â¦ HTU) -T1 -HLK // #HU1 +elim (lift_trans_le â¦ HT2 â¦ HTU ?) -T // -Hddt #T #HT2 #HTU +>(lift_mono â¦ HTU2 â¦ HT2) -T2 /2 width=3/ +qed. + +lemma delift_inv_lift1_eq: âL,U1,T2,d,e. L â¢ â¼*[d, e] U1 â¡ T2 â + âK. â©[d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â T1 = T2. +#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1 +lapply (tpss_inv_lift1_eq â¦ HU12 â¦ HTU1) -L -K #H destruct +lapply (lift_inj â¦ HTU1 â¦ HTU2) -U2 // +qed-. + +lemma delift_lift_div_be: âL,T1,T,d,e,i. L â¢ â¼*[i, d + e - i] T1 â¡ T â + âT2. â§[d, i - d] T2 â¡ T â d â¤ i â i â¤ d + e â + L â¢ â¼*[d, e] T1 â¡ T2. +#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide +lapply (tpss_weak â¦ HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10 +lapply (lift_trans_be â¦ HT2 â¦ HT0 ? ?) -T // +>commutative_plus >commutative_plus in â¢ (? ? (? % ?) ? ? â ?); +commutative_plus in H; >plus_plus_comm_23 #H + elim (le_plus_xySz_x_false â¦ H) + ] + ] +| #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2 + elim (cpr_inv_bind1 â¦ H1) -H1 * + [ #V #T #HV1 #HT1 #H destruct + elim (cpr_inv_bind1 â¦ H2) -H2 * + [ #V2 #T2 #HV2 #HT2 #H destruct + elim (IH â¦ HV1 â¦ HV2) // #HV12 destruct + [ @or_introl #H destruct /2 width=1/ + | elim (IH â¦ HT1 â¦ HT2) // /2 width=1/ -L1 -L2 #HT12 destruct + @or_introl #H destruct /2 width=1/ + ] + | #T2 #HT2 #HXT2 #H1 #H2 destruct + elim (IH â¦ HT1 â¦ 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 (+âV1.T1) 0 1) #Y2 #HXY2 + lapply (cpr_lift â¦ H2 (L2.âV1) â¦ HXY1 â¦ HXY2) /2 width=1/ -X1 /4 width=5/ + ] +| #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2 + elim (cpr_inv_flat1 â¦ H1) -H1 * + [ #V #T #HV1 #HT1 #H destruct + elim (cpr_inv_flat1 â¦ H2) -H2 * + [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ + | #HX2 #H destruct /3 width=5/ + ] + | #HX1 #H destruct /3 width=5/ +] +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/etc/ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/etc/ldrop.etc new file mode 100644 index 000000000..e4caef0c8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/etc/ldrop.etc @@ -0,0 +1,11 @@ +lemma ldrop_bor1: âL1,L2,s1,s2,d,e. â©[s1, d, e] L1 â¡ L2 â â©[s1 â¨ s2, d, e] L1 â¡ L2. +#L1 #L2 * /2 width=1 by ldrop_gen/ +qed. + +lemma ldrop_bor2: âL1,L2,s1,s2,d,e. â©[s2, d, e] L1 â¡ L2 â â©[s1 â¨ s2, d, e] L1 â¡ L2. +#L1 #L2 #s1 #s2 >commutative_orb /2 width=1 by ldrop_bor1/ +qed. + +(* Basic_1: was: drop_conf_rev *) +axiom ldrop_div: âe1,L1,L. â©[0, e1] L1 â¡ L â âe2,L2. â©[0, e2] L2 â¡ L â + ââL0. â©[0, e1] L0 â¡ L2 & â©[e1, e2] L0 â¡ L1. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fleq/fpbc_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fleq/fpbc_lift.etc new file mode 100644 index 000000000..286290135 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fleq/fpbc_lift.etc @@ -0,0 +1,9 @@ +include "basic_2/reduction/lpx_ldrop.ma". + +lemma lpx_fpbc_trans: âh,g,G1,G2,K1,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â + â¦G1, K1â¦ â¢ â¡[h, g] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡[h, g] T & â¦G1, K1, Tâ¦ â»[h, g] â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g] L2. +#h #g #G1 #G2 #K1 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 +[ #G2 #L2 #T2 #H12 #HKL1 elim (lpx_fqu_trans â¦ H12 â¦ HKL1) -L1 + /3 width=5 by fpbc_fqu, ex3_2_intro/ +| #T2 #HT12 #H #HKL1 diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fleq/fpbs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fleq/fpbs.etc new file mode 100644 index 000000000..017421feb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fleq/fpbs.etc @@ -0,0 +1,49 @@ +lemma fpbs_fpbu_sn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â¡[0] â¦G2, L2, T2â¦ â¨ + ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦. +(* ALTERNATIVE PROOF +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx â¦ 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 â¦ H1) -H1 #H1 + [ /3 width=1 by +*) +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt â¦ H) -H +#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct +[ -HT1 elim (fqus_inv_gen â¦ HT2) -HT2 + [ #H elim (fqup_inv_step_sn â¦ 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 â¦ 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 â¦ 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: âh,g,G1,G2,L1,L,L2,T1,T2. â¦G1, L1â¦ â¢â¡ [h, g] L â + â¦G1, L, T1â¦ â[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â[h, g] â¦G2, L2, T2â¦. + +lemma fpbs_fwd_fpb_sn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â[h, g] â¦G2, L2, T2â¦ â¨ + ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx â¦ 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 â¦ 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 + diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpa.etc new file mode 100644 index 000000000..8f72937c4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpa.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| fpa_fwd: âL,K,T,U,d,e. â©[s, d, e] L â¡ K â â§[d, e] T â¡ U â fpa s K T L U +| fpa_bwd: âL,K,T,U,d,e. â©[s, d, e] L â¡ K â â§[d, e] T â¡ U â 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: âs. bi_reflexive â¦ (fpa s). +/2 width=4 by fpa_fwd/ qed. + +lemma fpa_sym: âs. bi_symmetric â¦ (fpa s). +#s #L1 #L2 #T1 #T2 * /2 width=4 by fpa_fwd, fpa_bwd/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpa_fpa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpa_fpa.etc new file mode 100644 index 000000000..c62b8bfb9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpa_fpa.etc @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âs. bi_confluent â¦ (fpa s). +/3 width=4 by fpa_sym, ex2_2_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas.etc new file mode 100644 index 000000000..6817d8c90 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas.etc @@ -0,0 +1,59 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â bi_relation lenv term â Î»s. bi_TC â¦ (fpa s). + +interpretation + "multiple ajustment (restricted closure)" + 'RAjustStar L1 T1 s L2 T2 = (fpas s L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma fpas_ind: âL1,T1,s. âR:relation2 â¦. R L1 T1 â + (âL,L2,T,T2. â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³[s] â¦L2, T2â¦ â R L T â R L2 T2) â + âL2,T2. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â R L2 T2. +#L1 #T1 #s #R #IH1 #IH2 #L2 #T2 #H +@(bi_TC_star_ind â¦ IH1 IH2 L2 T2 H) // +qed-. + +lemma fpas_ind_dx: âL2,T2,s. âR:relation2 â¦. R L2 T2 â + (âL1,L,T1,T. â¦L1, T1â¦ â³[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³*[s] â¦L2, T2â¦ â R L T â R L1 T1) â + âL1,T1. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â R L1 T1. +#L2 #T2 #s #R #IH1 #IH2 #L1 #T1 #H +@(bi_TC_star_ind_dx â¦ IH1 IH2 L1 T1 H) // +qed-. + +(* Basic properties *********************************************************) + +lemma fpas_refl: âs. bi_reflexive â¦ (fpas s). +/2 width=1 by bi_inj/ qed. + +lemma fpas_sym: âs. bi_symmetric â¦ (fpas s). +/3 width=1 by fpa_sym, bi_TC_symmetric/ qed-. + +lemma fpa_fpas: âL1,L2,T1,T2,s. â¦L1, T1â¦ â³[s] â¦L2, T2â¦ â â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. +/2 width=1 by bi_inj/ qed. + +lemma fpas_strap1: âL1,L,L2,T1,T,T2,s. â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³[s] â¦L2, T2â¦ â + â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. +/2 width=4 by bi_step/ qed-. + +lemma fpas_strap2: âL1,L,L2,T1,T,T2,s. â¦L1, T1â¦ â³[s] â¦L, Tâ¦ â â¦L, Tâ¦ â³*[s] â¦L2, T2â¦ â + â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. +/2 width=4 by bi_TC_strap/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas_fpas.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas_fpas.etc new file mode 100644 index 000000000..28d648e7b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas_fpas.etc @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,L1,L2,T0,T1,T2,s. â¦L0, T0â¦ â³[s] â¦L1, T1â¦ â â¦L0, T0â¦ â³*[s] â¦L2, T2â¦ â + ââL,T. â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ & â¦L2, T2â¦ â³[s] â¦L, Tâ¦. +/3 width=4 by fpa_conf, bi_TC_strip/ qed-. + +(* Main properties **********************************************************) + +theorem fpas_conf: âs. bi_confluent â¦ (fpas s). +/3 width=4 by fpa_conf, bi_TC_confluent/ qed-. + +theorem fpas_trans: âs. bi_transitive â¦ (fpas s). +/2 width=4 by bi_TC_transitive/ qed-. + +theorem fpas_canc_sn: âL,L1,L2,T,T1,T2,s. + â¦L, Tâ¦ â³*[s] â¦L1, T1â¦â â¦L, Tâ¦ â³*[s] â¦L2, T2â¦ â â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. +/3 width=4 by fpas_trans, fpas_sym/ qed-. + +theorem fpas_canc_dx: âL1,L2,L,T1,T2,T,s. + â¦L1, T1â¦ â³*[s] â¦L, Tâ¦ â â¦L2, T2â¦ â³*[s] â¦L, Tâ¦ â â¦L1, T1â¦ â³*[s] â¦L2, T2â¦. +/3 width=4 by fpas_trans, fpas_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas_vector.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas_vector.etc new file mode 100644 index 000000000..65e8190e4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/fpas_vector.etc @@ -0,0 +1,56 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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) â +| fpasv_nil : âL1,L2,T1,T2. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â fpasv s L1 (â) L2 (â) +| fpasv_cons: âL1,L2,T1s,T2s,T1,T2. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â + fpasv s L1 T1s L2 T2s â + 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: âL1,L2,V1s,T1,X,s. â¦L1, â¶ V1s.T1â¦ â³*[s] â¦L2, Xâ¦ â + ââV2s,T2. â¦L1, V1sâ¦ â³*[s] â¦L2, V2sâ¦ & â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ & + X = â¶ V2s.T2. +#L1 #L2 #V1s elim V1s -V1s +[ #T1 #X #s #H + @(ex3_2_intro â¦ (â) X) /2 width=3 by fpasv_nil/ (**) (* explicit constructor *) +| #V1 #V1s #IHV1s #T1 #X #s #H + elim (lifts_inv_flat1 â¦ H) -H #V2 #Y #HV12 #HY #H destruct + elim (IHV1s â¦ 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: âV1s,V2s,des. â§*[des] V1s â¡ V2s â + âT1,T2. â§*[des] T1 â¡ T2 â + â§*[des] â¶ V1s. T1 â¡ â¶ V2s. T2. +#V1s #V2s #des #H elim H -V1s -V2s /3 width=1 by lifts_flat/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/gcp.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/gcp.etc new file mode 100644 index 000000000..b9b2db820 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/gcp.etc @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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] â relation3 genv lenv term. + +definition CP0 â Î»RR:relation4 genv lenv term term. Î»RS:relation term. + âG,L1,L2,T1,T2,s. â¦L1, T1â¦ â³[s] â¦L2, T2â¦ â + NF â¦ (RR G L1) RS T1 â NF â¦ (RR G L2) RS T2. + +definition CP0s â Î»RR:relation4 genv lenv term term. Î»RS:relation term. + âG,L1,L2,T1,T2,s. â¦L1, T1â¦ â³*[s] â¦L2, T2â¦ â + NF â¦ (RR G L1) RS T1 â NF â¦ (RR G L2) RS T2. + +definition CP1 â Î»RR:relation4 genv lenv term term. Î»RS:relation term. + âG,L. âk. NF â¦ (RR G L) RS (âk). + +definition CP2 â Î»RP:candidate. + âG,L,T,k. RP G L (ââk.T) â RP G L T. + +(* requirements for generic computation properties *) +record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop â +{ cp0: CP0 RR RS; + cp1: CP1 RR RS; + cp2: CP2 RP +}. + +(* Basic properties *********************************************************) + +(* Basic_1: was just: nf2_lift1 *) +lemma gcp_fpas: âRR,RS. CP0 RR RS â CP0s RR RS. +#RR #RS #HRR #G #L1 #L2 #T1 #T2 #s #H @(fpas_ind â¦ H) -L2 -T2 /3 width=5 by/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/gcp_cr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/gcp_cr.etc new file mode 100644 index 000000000..76490987f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/gcp_cr.etc @@ -0,0 +1,185 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â Î»C:candidate. âG,L1,L2,T1,T2. â¦L1, T1â¦ â³[â»] â¦L2, T2â¦ â + C G L1 T1 â C G L2 T2. + +definition S0s â Î»C:candidate. âG,L1,L2,T1,T2. â¦L1, T1â¦ â³*[â»] â¦L2, T2â¦ â + C G L1 T1 â C G L2 T2. + +(* Note: this is Girard's CR1 *) +definition S1 â Î»RP,C:candidate. + âG,L,T. C G L T â RP G L T. + +(* Note: this is Tait's iii, or Girard's CR4 *) +definition S2 â Î»RR:relation4 genv lenv term term. Î»RS:relation term. Î»RP,C:candidate. + âG,L,Vs. all â¦ (RP G L) Vs â + âT. ðâ¦Tâ¦ â NF â¦ (RR G L) RS T â C G L (â¶Vs.T). + +(* Note: this generalizes Tait's ii *) +definition S3 â Î»C:candidate. + âa,G,L,Vs,V,T,W. + C G L (â¶Vs.â{a}âW.V.T) â C G L (â¶Vs.âV.â{a}W.T). + +definition S4 â Î»RP,C:candidate. + âG,L,Vs. all â¦ (RP G L) Vs â âk. C G L (â¶Vs.âk). + +definition S5 â Î»C:candidate. âI,G,L,K,Vs,V1,V2,i. + C G L (â¶Vs.V2) â â§[0, i+1] V1 â¡ V2 â + â©[i] L â¡ K.â{I}V1 â C G L (â¶Vs.#i). + +definition S6 â Î»RP,C:candidate. + âG,L,V1s,V2s. â§[0, 1] V1s â¡ V2s â + âa,V,T. C G (L.âV) (â¶V2s.T) â RP G L V â C G L (â¶V1s.â{a}V.T). + +definition S7 â Î»C:candidate. + âG,L,Vs,T,W. C G L (â¶Vs.T) â C G L (â¶Vs.W) â C G L (â¶Vs.âW.T). + +(* requirements for the generic reducibility candidate *) +record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop â +{ 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 â candidate â candidate â + Î»C1,C2,G,K,T. âL,V,U. â¦K, Tâ¦ â³*[â»] â¦L, Uâ¦ â + C1 G L V â C2 G L (âV.U). + +(* the reducibility candidate associated to an atomic arity *) +let rec acr (RP:candidate) (A:aarity) on A: candidate â +match A with +[ AAtom â RP +| APair B A â 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: âC. S0 C â S0s C. +#C #HC #G #L1 #L2 #T1 #T2 #H @(fpas_ind â¦ H) -L2 -T2 /3 width=5 by/ +qed. +(* +lemma rp_lifts: âRR,RS,RP. gcr RR RS RP RP â + âdes,G,L0,L,V,V0. â©*[â», des] L0 â¡ L â â§*[des] V â¡ V0 â + RP G L V â RP G L0 V0. +#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV +@gcr_lifts /width=7 by/ +@(s0 â¦ HRP) +qed. + +(* Basic_1: was only: sns3_lifts1 *) +lemma rp_liftsv_all: âRR,RS,RP. gcr RR RS RP RP â + âdes,G,L0,L,Vs,V0s. â©*[â», des] L0 â¡ L â â§*[des] Vs â¡ V0s â + all â¦ (RP G L) Vs â all â¦ (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: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â + âA. 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 â¦ H1RP G L) #k #HK + lapply (H L (âk) T ? ?) -H // + [ @(s2 â¦ IHB â¦ (â)) // + | #H @(cp2 â¦ H1RP â¦ k) @(s1 â¦ IHA) // + ] +| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #H #HB + elim (lifts_inv_applv1 â¦ H) -H #V0s #T0 #HV0s #HT0 #H destruct + lapply (s1 â¦ IHB â¦ HB) #HV0 + @(s2 â¦ IHA â¦ (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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct + elim (lifts_inv_flat1 â¦ HY) -HY #U0 #X #HU0 #HX #H destruct + elim (lifts_inv_bind1 â¦ HX) -HX #W0 #T0 #HW0 #HT0 #H destruct + @(s3 â¦ IHA â¦ (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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct + >(lifts_inv_sort1 â¦ HY) -Y + lapply (s1 â¦ IHB â¦ HB) #HV0 + @(s4 â¦ IHA â¦ (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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct + elim (lifts_inv_lref1 â¦ HY) -HY #i0 #Hi0 #H destruct + elim (drops_drop_trans â¦ HL0 â¦ HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0 + >(at_mono â¦ Hi1 â¦ Hi0) in HL02; -i1 #HL02 + elim (drops_inv_skip2 â¦ Hdes0 â¦ H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct + elim (lift_total W1 0 (i0 + 1)) #W2 #HW12 + elim (lifts_lift_trans â¦ Hdes0 â¦ HVW1 â¦ HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2 + >(lift_mono â¦ HV13 â¦ HV12) in HVW2; -V3 #HVW2 + @(s5 â¦ IHA â¦ (V0 @ V0s) â¦ 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 â¦ H) -H #V10s #Y #HV10s #HY #H destruct + elim (lifts_inv_bind1 â¦ 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 â¦ IHA â¦ (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/ + @(HA â¦ (des + 1)) /2 width=2 by drops_skip/ + [ @lifts_applv // + elim (liftsv_liftv_trans_le â¦ HV10s â¦ HV120s) -V10s #V10s #HV10s #HV120s + >(liftv_mono â¦ HV12s â¦ HV10s) -V1s // + | @(s0 â¦ IHB â¦ HB â¦ 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 â¦ H) -H #V0s #Y #HV0s #HY #H destruct + elim (lifts_inv_flat1 â¦ HY) -HY #W0 #T0 #HW0 #HT0 #H destruct + @(s7 â¦ IHA â¦ (V0 @ V0s)) /3 width=5 by lifts_applv/ +] +qed. + +lemma acr_abst: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â + âa,G,L,W,T,A,B. â¦G, L, Wâ¦ Ïµ[RP] ãBã â ( + âL0,V0,W0,T0,des. â©*[â», des] L0 â¡ L â â§*[des] W â¡ W0 â â§*[des + 1] T â¡ T0 â + â¦G, L0, V0â¦ Ïµ[RP] ãBã â â¦G, L0, W0â¦ Ïµ[RP] ãBã â â¦G, L0.ââW0.V0, T0â¦ Ïµ[RP] ãAã + ) â + â¦G, L, â{a}W.Tâ¦ Ïµ[RP] ãâ¡B.Aã. +#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HL0 #H #HB +lapply (acr_gcr â¦ H1RP H2RP A) #HCA +lapply (acr_gcr â¦ H1RP H2RP B) #HCB +elim (lifts_inv_bind1 â¦ H) -H #W0 #T0 #HW0 #HT0 #H destruct +lapply (gcr_lifts â¦ HL0 â¦ HW0 HW) -HW [ @(s0 â¦ HCB) ] #HW0 +@(s3 â¦ HCA â¦ (â)) +@(s6 â¦ HCA â¦ (â) (â)) // +[ @(HA â¦ HL0) // +| lapply (s1 â¦ HCB) -HCB #HCB + @(s7 â¦ H2RP â¦ (â)) /2 width=1 by/ +] +qed. + +(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *) +(* Basic_1: removed local theorems 1: sc3_sn3_abst *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/rajust_5.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/rajust_5.etc new file mode 100644 index 000000000..ef0fbc49b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/rajust_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 L1, break term 46 T1 â¦ â³ [ break term 46 s ] break â¦ term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'RAjust $L1 $T1 $s $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/rajuststar_5.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/rajuststar_5.etc new file mode 100644 index 000000000..7cf9191d6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpa/rajuststar_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 L1, break term 46 T1 â¦ â³ * [ break term 46 s ] break â¦ term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'RAjustStar $L1 $T1 $s $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbc.etc new file mode 100644 index 000000000..9c803f041 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbc.etc @@ -0,0 +1,33 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â tri_relation genv lenv term â + Î»h,g,G1,L1,T1,G2,L2,T2. + ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¡[0] â¦G2, L2, T2â¦. + +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: âh,g,G1,G2,L1,L2,T1,T2. + â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦. +/2 width=5 by ex2_3_intro/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbc_fleq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbc_fleq.etc new file mode 100644 index 000000000..aa5cd7f2b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbc_fleq.etc @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G, L, Tâ¦ â + â¦G, L, Tâ¦ â¡[0] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦. +#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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¡[0] â¦G, L, Tâ¦ â + â¦G, L, Tâ¦ â»â¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 * +#G0 #L0 #T0 #H0 #H02 elim (fleq_fpbu_trans â¦ H1 â¦ H0) -G -L -T +/3 width=9 by fleq_trans, ex2_3_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg.etc new file mode 100644 index 000000000..574c35e10 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg.etc @@ -0,0 +1,61 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â tri_relation genv lenv term â + Î»h,g. tri_TC â¦ (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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +/2 width=1 by tri_inj/ qed. + +lemma fpbg_strap1: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. + â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â»â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +/2 width=5 by tri_step/ qed. + +lemma fpbg_strap2: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. + â¦G1, L1, T1â¦ â»â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +/2 width=5 by tri_TC_strap/ qed. + +lemma fpbu_fpbg: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»[h,g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +/3 width=1 by fpbu_fpbc, fpbc_fpbg/ qed. + +(* Basic eliminators ********************************************************) + +lemma fpbg_ind: âh,g,G1,L1,T1. âR:relation3 â¦. + (âG2,L2,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â R G2 L2 T2) â + (âG,G2,L,L2,T,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â»â¡[h, g] â¦G2, L2, T2â¦ â R G L T â R G2 L2 T2) â + âG2,L2,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â R G2 L2 T2. +#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H +@(tri_TC_ind â¦ IH1 IH2 G2 L2 T2 H) +qed-. + +lemma fpbg_ind_dx: âh,g,G2,L2,T2. âR:relation3 â¦. + (âG1,L1,T1. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â R G1 L1 T1) â + (âG1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â R G L T â R G1 L1 T1) â + âG1,L1,T1. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â R G1 L1 T1. +#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H +@(tri_TC_ind_dx â¦ IH1 IH2 G1 L1 T1 H) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fleq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fleq.etc new file mode 100644 index 000000000..754811ec2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fleq.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G1,G,L1,L,T1,T. â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â + âG2,L2,T2. â¦G, L, Tâ¦ â¡[0] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind â¦ 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: âh,g,G,G2,L,L2,T,T2. â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â + âG1,L1,T1. â¦G1, L1, T1â¦ â¡[0] â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx â¦ H) -G -L -T +[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/ +| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fpbg.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fpbg.etc new file mode 100644 index 000000000..57dd5ebb4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fpbg.etc @@ -0,0 +1,53 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â + ââG,L,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ & â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx â¦ 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind â¦ H) -G2 -L2 -T2 +[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans â¦ IH1) -IH1 ] (**) (* full auto fails *) +/2 width=1 by fpbc_fpbs/ +qed-. + +(* Advanced properties ******************************************************) + +lemma fpbs_fpbu_trans: âh,g,F1,F2,K1,K2,T1,T2. â¦F1, K1, T1â¦ â¥[h, g] â¦F2, K2, T2â¦ â + âG2,L2,U2. â¦F2, K2, T2â¦ â»[h, g] â¦G2, L2, U2â¦ â + ââG1,L1,U1. â¦F1, K1, T1â¦ â»[h, g] â¦G1, L1, U1â¦ & â¦G1, L1, U1â¦ â¥[h, g] â¦G2, L2, U2â¦. +/5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-. + +(* Man properties ***********************************************************) + +theorem fpbg_trans: âh,g. tri_transitive â¦ (fpbg h g). +/2 width=5 by tri_TC_transitive/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fpbs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fpbs.etc new file mode 100644 index 000000000..4b8662295 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_fpbs.etc @@ -0,0 +1,88 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. + â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â½[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu â¦ H2) -H2 +/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/ +qed-. + +lemma fpb_fpbg_trans: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. + â¦G1, L1, T1â¦ â½[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu â¦ 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: âh,g,G1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â¥[h, g] â¦G, L, Tâ¦ â + âG2,L2,T2. â¦G, L, Tâ¦ >â¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind â¦ H) -G -L -T /3 width=5 by fpb_fpbg_trans/ +qed-. + +(* Note: this is used in the closure proof *) +lemma fpbg_fpbs_trans: âh,g,G,G2,L,L2,T,T2. â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦ â + âG1,L1,T1. â¦G1, L1, T1â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx â¦ H) -G -L -T /3 width=5 by fpbg_fpb_trans/ +qed-. + +lemma fpbu_fpbs_fpbg: âh,g,G1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â»[h, g] â¦G, L, Tâ¦ â + âG2,L2,T2. â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed. + +(* Note: this is used in the closure proof *) +lemma fqup_fpbg: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn â¦ H) -H +/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/ +qed. + +lemma cpxs_fpbg: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡*[h, g] T2 â + (T1 = T2 â â¥) â â¦G, L, T1â¦ >â¡[h, g] â¦G, L, T2â¦. +#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn â¦ H â¦ H0) -H -H0 +/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/ +qed. + +lemma lstas_fpbg: âh,g,G,L,T1,T2,l2. â¦G, Lâ¦ â¢ T1 â¢*[h, l2] T2 â (T1 = T2 â â¥) â + âl1. l2 â¤ l1 â â¦G, Lâ¦ â¢ T1 âª[h, g] l1 â â¦G, L, T1â¦ >â¡[h, g] â¦G, L, T2â¦. +/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed. + +lemma lpxs_fpbg: âh,g,G,L1,L2,T. â¦G, L1â¦ â¢ â¡*[h, g] L2 â + (L1 â¡[T, 0] L2 â â¥) â â¦G, L1, Tâ¦ >â¡[h, g] â¦G, L2, Tâ¦. +#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn â¦ H â¦ H0) -H -H0 +/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/ +qed. + +lemma fpbs_fpbg: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â¡[0] â¦G2, L2, T2â¦ â¨ + â¦G1, L1, T1â¦ >â¡[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind â¦ H) -G2 -L2 -T2 +[ /2 width=1 by or_introl/ +| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_lift.etc new file mode 100644 index 000000000..58ecd1200 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbg_lift.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 âª[h, g] l+1 â + â¦G, Lâ¦ â¢ T1 â¢*[h, 1] T2 â â¦G, L, T1â¦ >â¡[h, g] â¦G, L, T2â¦. +/4 width=2 by fpbu_fpbg, sta_fpbu/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbs_fpbc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbs_fpbc.etc new file mode 100644 index 000000000..a88d51925 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/fpbs_fpbc.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 * +/3 width=5 by fpbu_fwd_fpb, fpbs_strap2, fleq_fpbs/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/lazybtpredproper_8.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/lazybtpredproper_8.etc new file mode 100644 index 000000000..6abdbeb59 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpbc/lazybtpredproper_8.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â»â¡ break [ term 46 h, break term 46 g ] break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'LazyBTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq.etc new file mode 100644 index 000000000..c909e2ed6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq.etc @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â + Î»G1,L1,T1,G2,L2,T2. + â§â§ 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 â¦ bteq. +/2 width=1 by and3_intro/ qed. + +lemma bteq_sym: tri_symmetric â¦ bteq. +#G1 #G2 #L1 #L2 #T1 #T2 * // +qed-. + +lemma bteq_dec: âG1,G2,L1,L2,T1,T2. Decidable (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦). +#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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq_6.etc new file mode 100644 index 000000000..5ee962fe5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'BTEq $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq_bteq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq_bteq.etc new file mode 100644 index 000000000..7870b2644 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/bteq_bteq.etc @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ bteq. +#G1 #G #L1 #L #T1 #T * // +qed-. + +theorem bteq_canc_sn: âG,G1,G2,L,L1,L2,T,T1,T2. â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ â + â¦G, L, Tâ¦ â â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦. +/3 width=5 by bteq_trans, bteq_sym/ qed-. + +theorem bteq_canc_dx: âG1,G2,G,L1,L2,L,T1,T2,T. â¦G1, L1, T1â¦ â â¦G, L, Tâ¦ â + â¦G2, L2, T2â¦ â â¦G, L, Tâ¦ â â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦. +/3 width=5 by bteq_trans, bteq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/btpredsn_8.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/btpredsn_8.etc new file mode 100644 index 000000000..0350d1495 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/btpredsn_8.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â¢ â â¡ break [ term 46 h , break term 46 g ] break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'BTPRedSn $h $g $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpb.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpb.etc new file mode 100644 index 000000000..e6698349d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpb.etc @@ -0,0 +1,14 @@ +include "basic_2/reduction/fpn.ma". + +(* Basic forward lemmas *****************************************************) + +lemma fpb_bteq_fwd_fpn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G2, L2, T2â¦. +#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 â¦ H) -H + [ #H1 #H2 elim (fqu_fwd_bteq â¦ H1 H2) + | * #HG #HL #HT #_ destruct // + ] +| #T2 #HT12 * // +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpbc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpbc.etc new file mode 100644 index 000000000..1ed366cdd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpbc.etc @@ -0,0 +1,24 @@ +lemma fpb_fpbc_or_fpn: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â¨ + â¦G1, L1, T1â¦ â¢ â¡[h,g] â¦G2, L2, T2â¦. +#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 â¦ 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â + (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â + â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 elim (fpb_fpbc_or_fpn â¦ H) -H // +#H elim H0 -H0 /2 width=3 by fpn_fwd_bteq/ +qed. + +lemma fpbc_fwd_bteq: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥. +#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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpbs_conj.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpbs_conj.etc new file mode 100644 index 000000000..8d0f137df --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpbs_conj.etc @@ -0,0 +1,34 @@ +(* +inclade "basic_2/computation/fpns.ma". +inclade "basic_2/computation/fpbs.ma". +inclade "basic_2/reduction/fpbc.ma". + +lemma fpn_dec: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â + Decidable (â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G2, L2, T2â¦). +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fpb_fpbc_or_fpn â¦ H) -H /2 width=1 by or_introl/ +#H12 @or_intror +#H @(fpbc_fwd_bteq â¦ H12) -H12 @(fpn_fwd_bteq â¦ H) +qed-. +*) +(* +lemma fpns_dec: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + Decidable (â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦). +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind â¦ H) -G2 -L2 -T2 /2 width=1 by or_introl/ +#G #G2 #L #L2 #T #T2 #H #H2 * +#H1 elim (fpn_dec â¦ 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind â¦ 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpn.etc new file mode 100644 index 000000000..406e6eb8e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpn.etc @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| fpn_intro: âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â L1 â[T] L2 â 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: âh,g. tri_reflexive â¦ (fpn h g). +/2 width=1 by fpn_intro/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma fpn_inv_gen: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â + â§â§ G1 = G2 & â¦G1, L1â¦ â¢ â¡[h, g] L2 & L1 â[T1] L2 & T1 = T2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and4_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpn_fpn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpn_fpn.etc new file mode 100644 index 000000000..adda4b538 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpn_fpn.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â + âG2,L2,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â + ââF2,K2,V2. â¦F1, K1, V1â¦ â â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. +#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 +#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqu_trans â¦ H12 â¦ HKL1 HV1) -L1 +/3 width=5 by fpn_intro, ex2_3_intro/ +qed-. + +lemma fpn_fquq_trans: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â + âG2,L2,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â + ââF2,K2,V2. â¦F1, K1, V1â¦ ââ¸® â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. +#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 +#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fquq_trans â¦ H12 â¦ HKL1 HV1) -L1 +/3 width=5 by fpn_intro, ex2_3_intro/ +qed-. + +lemma fpn_fqup_trans: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â + âG2,L2,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â + ââF2,K2,V2. â¦F1, K1, V1â¦ â+ â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. +#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 +#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqup_trans â¦ H12 â¦ HKL1 HV1) -L1 +/3 width=5 by fpn_intro, ex2_3_intro/ +qed-. + +lemma fpn_fqus_trans: âh,g,F1,G1,K1,L1,V1,T1. â¦F1, K1, V1â¦ â¢ ââ¡[h, g] â¦G1, L1, T1â¦ â + âG2,L2,T2. â¦G1, L1, T1â¦ â* â¦G2, L2, T2â¦ â + ââF2,K2,V2. â¦F1, K1, V1â¦ â* â¦F2, K2, V2â¦ & â¦F2, K2, V2â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦. +#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1 +#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqus_trans â¦ H12 â¦ HKL1 HV1) -L1 +/3 width=5 by fpn_intro, ex2_3_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns.etc new file mode 100644 index 000000000..0a208fedf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns.etc @@ -0,0 +1,58 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â tri_relation genv lenv term â + Î»h,g. tri_TC â¦ (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: âh,g,G1,L1,T1. âR:relation3 â¦. R G1 L1 T1 â + (âG,G2,L,L2,T,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â R G L T â R G2 L2 T2) â + âG2,L2,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â R G2 L2 T2. +#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H +lapply (tri_TC_star_ind â¦ IH1 IH2 G2 L2 T2 H) // +qed-. + +lemma fpns_ind_dx: âh,g,G2,L2,T2. âR:relation3 â¦. R G2 L2 T2 â + (âG1,G,L1,L,T1,T. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G, L, Tâ¦ â â¦G, L, Tâ¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â R G L T â R G1 L1 T1) â + âG1,L1,T1. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â R G1 L1 T1. +#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H +@(tri_TC_star_ind_dx â¦ IH1 IH2 G1 L1 T1 H) // +qed-. + +(* Basic_properties *********************************************************) + +lemma fpns_refl: âh,g. tri_reflexive â¦ (fpns h g). +/2 width=1 by tri_inj/ qed. + +lemma fpn_fpns: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦. +/2 width=1 by tri_inj/ qed. + +lemma fpns_strap1: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G, L, Tâ¦ â + â¦G, L, Tâ¦ â¢ ââ¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦. +/2 width=5 by tri_step/ qed-. + +lemma fpns_strap2: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ ââ¡[h, g] â¦G, L, Tâ¦ â + â¦G, L, Tâ¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦. +/2 width=5 by tri_TC_strap/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns_fpns.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns_fpns.etc new file mode 100644 index 000000000..5aaea6a9e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns_fpns.etc @@ -0,0 +1,74 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ ââ¡*[h, g] â¦G2, L2, T2â¦ â + â§â§ G1 = G2 & â¦G1, L1â¦ â¢ â¡*[h, g] L2 & L1 â[T1] L2 & T1 = T2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpns_ind â¦ 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 â¦ H2) -H2 #HG #HL2 #HT #HT2 destruct +/3 width=3 by lpxs_strap1, lleq_trans, and4_intro/ +qed-. + +(* Advanced properties ******************************************************) + +lemma lpxs_lleq_fpns: âh,g,G,L1,L2,T. â¦G, L1â¦ â¢ â¡*[h, g] L2 â L1 â[T] L2 â + â¦G, L1, Tâ¦ â¢ ââ¡*[h, g] â¦G, L2, Tâ¦. +#h #g #G #L1 #L2 #T #H @(lpxs_ind â¦ 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 â¦ fqus. +/2 width=5 by tri_TC_transitive/ qed-. + + +(* +lemma fpns_refl: âh,g. tri_reflexive â¦ (fpns h g). +/2 width=1 by and3_intro/ qed. + +lemma fpn_fpns: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=1 by lpx_lpxs, and3_intro/ +qed. + +lemma fpns_strap1: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G, L, Tâ¦ â + â¦G, L, Tâ¦ â¢ â¡[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. +#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: âh,g,G1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1â¦ â¢ â¡[h, g] â¦G, L, Tâ¦ â + â¦G, L, Tâ¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦ â â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦. +#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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â¢ â¡*[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=4 by lpxs_fwd_length, and3_intro/ +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fqu.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fqu.etc new file mode 100644 index 000000000..1e4defb3c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fqu.etc @@ -0,0 +1,14 @@ +include "basic_2/grammar/bteq.ma". + +lemma fqu_fwd_bteq: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥. +#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 â¦ H) +| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_x â¦ H) +| #a #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y â¦ H) +| #I #G #L #V #T * #_ #_ #H elim (discr_tpair_xy_y â¦ H) +| #G #L #K #T #U #e #HLK #_ * #_ #H + lapply (ldrop_fwd_length_lt4 â¦ HLK ?) // >H -L #H + elim (lt_refl_false â¦ H) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fsb.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fsb.etc new file mode 100644 index 000000000..3cd6861ad --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fsb.etc @@ -0,0 +1,12 @@ +(* Basic eliminators ********************************************************) + +theorem fsb_ind_alt: âh,g. âR: relation3 â¦. ( + âG1,L1,T1. â¦G1, L1â¦ â¢ â¦¥[h,g] T1 â ( + âG2,L2,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â + (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â R G2 L2 T2 + ) â R G1 L1 T1 + ) â + âG,L,T. â¦G, Lâ¦ â¢ â¦¥[h, g] T â 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fsb_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fsb_alt.etc new file mode 100644 index 000000000..8c69a001d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fsb_alt.etc @@ -0,0 +1,105 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| fsba_intro: âG1,L1,T1. ( + âG2,L2,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â fsba h g G2 L2 T2 + ) â 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: âh,g. âR: relation3 â¦. ( + âG1,L1,T1. â¦G1, L1â¦ â¢ â¦¥â¦¥[h,g] T1 â ( + âG2,L2,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â + (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â R G2 L2 T2 + ) â R G1 L1 T1 + ) â + âG,L,T. â¦G, Lâ¦ â¢ â¦¥â¦¥[h, g] T â 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: âh,g,G1,L1,T1. ( + âG,G2,L,L2,T,T2. â¦G, L, Tâ¦ â¥[h, g] â¦G2, L2, T2â¦ â + â¦G1, L1, T1â¦ â â¦G, L, Tâ¦ â + (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â fsba h g G2 L2 T2 + ) â fsba h g G1 L1 T1. +/4 width=5 by fsba_intro/ qed-. + +lemma fsba_fpbs_trans: âh,g,G1,L1,T1. â¦G1, L1â¦ â¢ â¦¥â¦¥[h, g] T1 â + âG2,L2,T2. â¦G1, L1, T1â¦ â¥[h, g] â¦G2, L2, T2â¦ â â¦G2, L2â¦ â¢ â¦¥â¦¥[h, g] T2. +#h #g #G1 #L1 #T1 #H @(fsba_ind_alt â¦ H) -G1 -L1 -T1 +#G1 #L1 #T1 #H0 #IH0 #G #L #T #H1 @fsba_intro +#G2 #L2 #T2 #H2 #_ lapply (fpbs_trans â¦ H1 â¦ 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 â¦ H) -IH0 -H // @(fpbs_trans â¦ H1 â¦ H2) + +lemma fsba_intro_fpb: âh,g,G1,L1,T1. ( + âG2,L2,T2. â¦G1, L1, T1â¦ â½[h, g] â¦G2, L2, T2â¦ â + (â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â â¥) â â¦G2, L2â¦ â¢ â¦¥â¦¥[h, g] T2 + ) â â¦G1, L1â¦ â¢ â¦¥â¦¥[h, g] T1. +#h #g #G1 #L1 #T1 #IH1 @fsba_intro_aux +#G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx â¦ 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 â¦ H21) -IH1 -H21 + [ + | -H10 -H210 #H +(* +(* Main inversion lemmas ****************************************************) + +theorem fsba_inv_fsb: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¦¥â¦¥[h, g] T â â¦G, Lâ¦ â¢ â¦¥[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: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¦¥[h, g] T â â¦G, Lâ¦ â¢ â¦¥â¦¥[h, g] T. +#h #g #G #L #T #H @(fsb_ind_alt â¦ H) -G -L -T +/4 width=1 by fsba_intro_fpb/ +qed. +(* +| fsba_intro: âG1,L1,T1. ( + âG2,L2,T2. â¦G1, L1, T1â¦ â»[h, g] â¦G2, L2, T2â¦ â fsb h g G2 L2 T2 + ) â fsb h g G1 L1 T1 +. + + + +(****************************************************************************) + +include "basic_2/substitution/fqup.ma". + +lemma fsb_csx: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¬*[h, g] T â â¦G, Lâ¦ â¢ â¦¥[h, g] T. +#h #g #G #L #T #H @(csx_ind â¦ H) -T +*)*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/lazyeq_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/lazyeq_6.etc new file mode 100644 index 000000000..105a47363 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/lazyeq_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G1, break term 46 L1, break term 46 T1 â¦ â break â¦ term 46 G2, break term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'LazyEq $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr.etc new file mode 100644 index 000000000..f58a52c37 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr.etc @@ -0,0 +1,59 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break â¦ term 46 L1, break term 46 T1 â¦ â¡ break â¦ term 46 L2 , break term 46 T2 â¦ )" + 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 â bi_relation lenv term â + Î»L,L1,T1,L2,T2. |L1| = |L2| â§ L â¢ L1 @@ T1 â¡ 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: âL. bi_reflexive â¦ (cfpr L). +/2 width=1/ qed. + +lemma fpr_cfpr: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â â¢ â¦L1, T1â¦ â¡ â¦L2, T2â¦. +#L1 #L2 #T1 #T2 * /3 width=1/ +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma cfpr_inv_atom1: âL,L2,T1,T2. L â¢ â¦â, T1â¦ â¡ â¦L2, T2â¦ â L â¢ T1 â¡ T2 â§ L2 = â. +#L #L2 #T1 #T2 * #H >(length_inv_zero_sn â¦ H) /2 width=1/ +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma fpr_inv_pair1_sn: âI,K1,L2,V1,T1,T2. â¦â.â{I}V1@@K1, T1â¦ â¡ â¦L2, T2â¦ â + ââK2,V2. V1 â¡ V2 & + â.â{I}V2 â¢ â¦K1, T1â¦ â¡ â¦K2, T2â¦ & + L2 = â.â{I}V2@@K2. +#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H +elim (length_inv_pos_sn_append â¦ H) -H #I2 #K2 #V2 #HK12 #H destruct +>shift_append_assoc >shift_append_assoc normalize in â¢ (%â?); #H +elim (tpr_inv_bind1 â¦ H) -H * +[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/ +| #T0 #_ #_ #H destruct +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_aaa.etc new file mode 100644 index 000000000..72802e499 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_aaa.etc @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T1,A. L1 â¢ T1 â A â + âL2,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â L2 â¢ T2 â A. +#L1 #T1 #A #HT1 #L2 #T2 #H +elim (fpr_inv_all â¦ H) -H +/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_cfpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_cfpr.etc new file mode 100644 index 000000000..f442be28b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_cfpr.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL. bi_confluent â¦ (cfpr L). +#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02 +elim (cpr_conf â¦ HT01 HT02) -L0 -T0 #X #H1 #H2 +elim (cpr_fwd_shift1 â¦ H1) #L0 #T0 #HL10 #H destruct /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_cpr.etc new file mode 100644 index 000000000..a1e20a025 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_cpr.etc @@ -0,0 +1,64 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L. L1 â¡ L â âL2,T1,T2. L â¢ T1 â¡ T2 â + L â¢ â¶* [0, |L|] L2 â â¦L1, T1â¦ â¡ â¦L2, T2â¦. +#L1 #L #H elim H -L1 -L +[ #L2 #T1 #T2 #HT12 #HL2 + lapply (ltpss_sn_inv_atom1 â¦ HL2) -HL2 #H destruct + lapply (cpr_inv_atom â¦ HT12) -HT12 /2 width=1/ +| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H + elim (ltpss_sn_inv_tpss21 â¦ H ?) -H // append_length >append_length #H + lapply (injective_plus_r â¦ H) -H #H + @(ex3_intro â¦ (â.â{I}V@@Y)) append_length H -H >commutative_plus /3 width=1/ +] +qed-. + +lemma fpr_inv_all: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â + ââL. L1 â¡ L & L â¢ T1 â¡ T2 & L â¢ â¶* [0, |L|] L2. +#L1 #L2 #T1 #T2 #H +lapply (fpr_cfpr â¦ H) -H #H +elim (cfpr_inv_all â¦ H) -H /2 width=4/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_ltpss.etc new file mode 100644 index 000000000..66e102e63 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/cfpr_ltpss.etc @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âI,L,K1,L2,V1,T1,T2. L â¢ â¦â.â{I}V1@@K1, T1â¦ â¡ â¦L2, T2â¦ â + ââK2,V,V2. V1 â¡ V & L â¢ V â¶* [0, |L|] V2 & + L.â{I}V â¢ â¦K1, T1â¦ â¡ â¦K2, T2â¦ & + L2 = â.â{I}V2@@K2. +* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H +elim (length_inv_pos_sn_append â¦ H) -H #I2 #K2 #V2 #HK12 #H destruct +>shift_append_assoc >shift_append_assoc normalize in â¢ (??%%â?); #H +[ elim (cpr_inv_abbr1 â¦ H) -H * + [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/ + | #T0 #_ #_ #H destruct + ] +| elim (cpr_inv_abst1 â¦ H Abst V2) -H + #V #T * #V0 #HV10 #HV0 #HT1 #H destruct + lapply (ltpss_sn_cpr_trans (L.âV0) â¦ 0 (|L|+1) â¦ HT1) -HT1 /2 width=1/ #HT12 + /3 width=7/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpc.etc new file mode 100644 index 000000000..ee362471b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpc.etc @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1 , break term 46 T1 â¦ â¬ break â¦ term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }. + +notation "hvbox( â¦ term 46 L1 , break term 46 T1 â¦ â¬ â¬ break â¦ term 46 L2 , break term 46 T2 â¦ )" + 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 â + Î»L1,T1,L2,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â¨ â¦L2, T2â¦ â¡ â¦L1, T1â¦. + +interpretation + "context-free parallel conversion (closure)" + 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2). + +(* Basic properties *********************************************************) + +lemma fpc_refl: bi_reflexive â¦ fpc. +/2 width=1/ qed. + +lemma fpc_sym: bi_symmetric â¦ fpc. +#L1 #L2 #T1 #T2 * /2 width=1/ +qed. + +(* Basic forward lemmas *****************************************************) + +lemma fpc_fwd_fpr: âL1,L2,T1,T2. â¦L1, T1â¦ â¬ â¦L2, T2â¦ â + ââL,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ & â¦L2, T2â¦ â¡ â¦L, Tâ¦. +#L1 #L2 #T1 #T2 * /2 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpc_fpc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpc_fpc.etc new file mode 100644 index 000000000..22fc16f37 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpc_fpc.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,L1,T0,T1. â¦L0, T0â¦ â¬ â¦L1, T1â¦ â + âL2,T2. â¦L0, T0â¦ â¬ â¦L2, T2â¦ â + ââL,T. â¦L1, T1â¦ â¬ â¦L, Tâ¦ & â¦L2, T2â¦ â¬ â¦L, Tâ¦. +/3 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs.etc new file mode 100644 index 000000000..68c946ce8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs.etc @@ -0,0 +1,84 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1 , break term 46 T1 â¦ â¬ * break â¦ term 46 L2 , break term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }. + +notation "hvbox( â¦ term 46 L1 , break term 46 T1 â¦ â¬ â¬ * break â¦ term 46 L2 , break term 46 T2 â¦ )" + 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 â bi_TC â¦ fpc. + +interpretation "context-free parallel equivalence (closure)" + 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma fpcs_ind: âL1,T1. âR:relation2 lenv term. R L1 T1 â + (âL,L2,T,T2. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â â¦L, Tâ¦ â¬ â¦L2, T2â¦ â R L T â R L2 T2) â + âL2,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â R L2 T2. +/3 width=7 by bi_TC_star_ind/ qed-. + +lemma fpcs_ind_dx: âL2,T2. âR:relation2 lenv term. R L2 T2 â + (âL1,L,T1,T. â¦L1, T1â¦ â¬ â¦L, Tâ¦ â â¦L, Tâ¦ â¬* â¦L2, T2â¦ â R L T â R L1 T1) â + âL1,T1. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â R L1 T1. +/3 width=7 by bi_TC_star_ind_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma fpcs_refl: bi_reflexive â¦ fpcs. +/2 width=1/ qed. + +lemma fpcs_sym: bi_symmetric â¦ fpcs. +/3 width=1/ qed. + +lemma fpc_fpcs: âL1,L2,T1,T2. â¦L1, T1â¦ â¬ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/2 width=1/ qed. + +lemma fpcs_strap1: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â â¦L, Tâ¦ â¬ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/2 width=4/ qed. + +lemma fpcs_strap2: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¬ â¦L, Tâ¦ â â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/2 width=4/ qed. + +lemma fpcs_fpr_dx: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=1/ qed. + +lemma fpcs_fpr_sn: âL1,L2,T1,T2. â¦L2, T2â¦ â¡ â¦L1, T1â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=1/ qed. + +lemma fpcs_fpr_strap1: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â + âL2,T2. â¦L, Tâ¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4/ qed. + +lemma fpcs_fpr_strap2: âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â + âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4/ qed. + +lemma fpcs_fpr_div: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â + âL2,T2. â¦L2, T2â¦ â¡ â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4/ qed. + +lemma fpr_div: âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡ â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4/ qed-. + +lemma fpcs_fpr_conf: âL1,L,T1,T. â¦L, Tâ¦ â¡ â¦L1, T1â¦ â + âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_aaa.etc new file mode 100644 index 000000000..9f4327bff --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_aaa.etc @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â + âA1. L1 â¢ T1 â A1 â âA2. L2 â¢ T2 â A2 â + A1 = A2. +#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2 +elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 +lapply (aaa_fprs_conf â¦ HT1 â¦ H1) -L1 -T1 #HT1 +lapply (aaa_fprs_conf â¦ HT2 â¦ H2) -L2 -T2 #HT2 +lapply (aaa_mono â¦ HT1 â¦ HT2) -L -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_cpcs.etc new file mode 100644 index 000000000..b1288f751 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_cpcs.etc @@ -0,0 +1,86 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â âV1,V2. V1 â¡ V2 â + âI. â¦L1, â{I}V1.T1â¦ â¬* â¦L2, â{I}V2.T2â¦. +#L1 #L2 #T1 #T2 #HT12 +elim (fpcs_inv_fprs â¦ HT12) -HT12 +/3 width=6 by fprs_flat_dx_tpr, fprs_div/ (**) (* auto too slow without trace *) +qed. + +lemma fpcs_shift: âI,L1,L2,V1,V2,T1,T2. â¦L1, -â{I}V1.T1â¦ â¬* â¦L2, -â{I}V2.T2â¦ â + â¦L1.â{I}V1, T1â¦ â¬* â¦L2.â{I}V2, T2â¦. +#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 +elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 +elim (fprs_bind2_minus â¦ H1) -H1 #W1 #U1 #HTU1 #H destruct +elim (fprs_bind2_minus â¦ H2) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/ +qed. + +(* Advanced inversion lemmas ************************************************) + +lemma fpcs_inv_shift: âI,L1,L2,V1,V2,T1,T2. â¦L1.â{I}V1, T1â¦ â¬* â¦L2.â{I}V2, T2â¦ â + â¦L1, -â{I}V1.T1â¦ â¬* â¦L2, -â{I}V2.T2â¦. +#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 +elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 +elim (fprs_inv_pair1 â¦ H1) -H1 #K1 #U1 #_ #HTU1 #H destruct +elim (fprs_inv_pair1 â¦ H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma fpcs_fwd_bind_minus: âI,L1,L2,V1,V2,T1,T2. â¦L1, -â{I}V1.T1â¦ â¬* â¦L2, -â{I}V2.T2â¦ â + âb. â¦L1, â{b,I}V1.T1â¦ â¬* â¦L2, â{b,I}V2.T2â¦. +#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b +elim (fpcs_inv_fprs â¦ H12) -H12 #L #T #H1 #H2 +elim (fprs_fwd_bind2_minus â¦ H1 b) -H1 #W1 #U1 #HTU1 #H destruct +elim (fprs_fwd_bind2_minus â¦ H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/ +qed-. + +lemma fpcs_fwd_shift: âI,L1,L2,V1,V2,T1,T2. â¦L1.â{I}V1, T1â¦ â¬* â¦L2.â{I}V2, T2â¦ â + âb. â¦L1, â{b,I}V1.T1â¦ â¬* â¦L2, â{b,I}V2.T2â¦. +/3 width=1 by fpcs_inv_shift, fpcs_fwd_bind_minus/ qed-. + +lemma fpcs_fwd_abst24: âa,L1,L2,V1,V2,T1,T2. â¦L1, â{a}V1.T1â¦ â¬* â¦L2, â{a}V2.T2â¦ â + âb,I,W. â¦L1, â{b,I}W.T1â¦ â¬* â¦L2, â{b,I}W.T2â¦. +#a #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b #I #W +elim (fpcs_inv_fprs â¦ H12) -H12 #L #U #H1 #H2 +elim (fprs_fwd_abst2 â¦ H1 b I W) -H1 #W1 #U1 #HTU1 #H destruct +elim (fprs_fwd_abst2 â¦ H2 b I W) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/ +qed-. + +lemma fpcs_fwd_abst13: âL1,L2,V1,V2,T1,T2. â¦L1.âV1, T1â¦ â¬* â¦L2.âV2, T2â¦ â + âI,W. â¦L1.â{I}W, T1â¦ â¬* â¦L2.â{I}W, T2â¦. +/4 width=4 by fpcs_fwd_shift, fpcs_fwd_abst24, fpcs_shift/ qed-. + +(* Properties on context-sensitive parallel equivalence for terms ***********) + +lemma cpcs_fpcs: âL,T1,T2. L â¢ T1 â¬* T2 â â¦L, T1â¦ â¬* â¦L, T2â¦. +#L #T1 #T2 #H +elim (cpcs_inv_cprs â¦ 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: âL,T1,T2. â¦L, T1â¦ â¬* â¦L, T2â¦ â L â¢ T1 â¬* T2. +#L #T1 #T2 #H +elim (fpcs_inv_fprs â¦ H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fpcs.etc new file mode 100644 index 000000000..270e8dc40 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fpcs.etc @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â + ââL,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ & â¦L2, T2â¦ â¡* â¦L, Tâ¦. +#L1 #L2 #T1 #T2 #H @(fpcs_ind â¦ H) -L2 -T2 +[ /3 width=4/ +| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 + [ elim (fprs_strip â¦ HT2 â¦ HT0) -L -T #L #T #HT2 #HT0 + lapply (fprs_strap1 â¦ HT10 â¦ HT0) -L0 -T0 /2 width=4/ + | lapply (fprs_strap2 â¦ HT2 â¦ HT0) -L -T /2 width=4/ + ] +] +qed-. + +(* Advanced properties ******************************************************) + +lemma fpr_fprs_conf: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 +elim (fprs_strip â¦ HT2 â¦ HT1) /2 width=4 by fpr_fprs_div/ +qed-. + +lemma fprs_fpr_conf: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â â¦L2, T2â¦ â¬* â¦L1, T1â¦. +#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 +elim (fprs_strip â¦ HT2 â¦ HT1) /2 width=4 by fprs_fpr_div/ +qed-. + +lemma fprs_conf: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â â¦L, Tâ¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 +elim (fprs_conf â¦ HT1 â¦ HT2) /2 width=4/ +qed-. + +lemma fpcs_strip: âL0,L1,T0,T1. â¦L0, T0â¦ â¬ â¦L1, T1â¦ â + âL2,T2. â¦L0, T0â¦ â¬* â¦L2, T2â¦ â + ââL,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ & â¦L2, T2â¦ â¬ â¦L, Tâ¦. +/3 width=4/ qed. + +(* Main properties **********************************************************) + +theorem fpcs_trans: bi_transitive â¦ fpcs. +/2 width=4/ qed. + +theorem fpcs_canc_sn: âL,L1,L2,T,T1,T2. â¦L, Tâ¦ â¬* â¦L1, T1â¦ â â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) + +theorem fpcs_canc_dx: âL1,L2,L,T1,T2,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â â¦L2, T2â¦ â¬* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fprs.etc new file mode 100644 index 000000000..43239eadd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fprs.etc @@ -0,0 +1,55 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L2 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 /width=1/ /3 width=4/ +qed. + +lemma fpcs_fprs_sn: âL1,L2,T1,T2. â¦L2, T2â¦ â¡* â¦L1, T1â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L2 #T1 #T2 #H @(fprs_ind_dx â¦ H) -L2 -T2 /width=1/ /3 width=4/ +qed. + +lemma fpcs_fprs_strap1: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â âL2,T2. â¦L, Tâ¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind â¦ H) -L2 -T2 /width=1/ /2 width=4/ +qed. + +lemma fpcs_fprs_strap2: âL1,L,T1,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx â¦ H) -L1 -T1 /width=1/ /2 width=4/ +qed. + +lemma fpcs_fprs_div: âL1,L,T1,T. â¦L1, T1â¦ â¬* â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx â¦ H) -L2 -T2 /width=1/ /2 width=4/ +qed. + +lemma fpcs_fprs_conf: âL1,L,T1,T. â¦L, Tâ¦ â¡* â¦L1, T1â¦ â âL2,T2. â¦L, Tâ¦ â¬* â¦L2, T2â¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind â¦ H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *) +qed. + +lemma fprs_div: âL1,L,T1,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx â¦ H) -L2 -T2 /2 width=1/ /2 width=4/ +qed. + +lemma fprs_fpr_div: âL1,L,T1,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡ â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=7 by bi_step, fprs_div/ qed-. + +lemma fpr_fprs_div: âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â âL2,T2. â¦L2, T2â¦ â¡* â¦L, Tâ¦ â â¦L1, T1â¦ â¬* â¦L2, T2â¦. +/3 width=4 by bi_step, fprs_div/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr.etc new file mode 100644 index 000000000..3bb168f9e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr.etc @@ -0,0 +1,68 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1, break term 46 T1 â¦ â¡ break â¦ term 46 L2 , break term 46 T2 â¦ )" + 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 â + Î»L1,T1,L2,T2. |L1| = |L2| â§ L1 @@ T1 â¡ 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 â¦ fpr. +/2 width=1/ qed. + +lemma fpr_shift: âI1,I2,L1,L2,V1,V2,T1,T2. + â¦L1, -â{I1}V1.T1â¦ â¡ â¦L2, -â{I2}V2.T2â¦ â + â¦L1.â{I1}V1, T1â¦ â¡ â¦L2.â{I2}V2, T2â¦. +#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12 +@conj // normalize // (**) (* explicit constructor *) +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma fpr_inv_atom1: âL2,T1,T2. â¦â, T1â¦ â¡ â¦L2, T2â¦ â T1 â¡ T2 â§ L2 = â. +#L2 #T1 #T2 * #H +lapply (length_inv_zero_sn â¦ H) -H #H destruct /2 width=1/ +qed-. + +lemma fpr_inv_atom3: âL1,T1,T2. â¦L1,T1â¦ â¡ â¦â,T2â¦ â T1 â¡ T2 â§ L1 = â. +#L1 #T1 #T2 * #H +lapply (length_inv_zero_dx â¦ H) -H #H destruct /2 width=1/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma fpr_fwd_pair1: âI1,K1,L2,V1,T1,T2. â¦K1.â{I1}V1, T1â¦ â¡ â¦L2, T2â¦ â + ââI2,K2,V2. â¦K1, -â{I1}V1.T1â¦ â¡ â¦K2, -â{I2}V2.T2â¦ & + L2 = K2.â{I2}V2. +#I1 #K1 #L2 #V1 #T1 #T2 * #H +elim (length_inv_pos_sn â¦ H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/ +qed-. + +lemma fpr_fwd_pair3: âI2,L1,K2,V2,T1,T2. â¦L1, T1â¦ â¡ â¦K2.â{I2}V2, T2â¦ â + ââI1,K1,V1. â¦K1, -â{I1}V1.T1â¦ â¡ â¦K2, -â{I2}V2.T2â¦ & + L1 = K1.â{I1}V1. +#I2 #L1 #K2 #V2 #T1 #T2 * #H +elim (length_inv_pos_dx â¦ H) -H #I1 #K1 #V1 #HK12 #H destruct /3 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr_cpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr_cpr.etc new file mode 100644 index 000000000..baa630f5b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr_cpr.etc @@ -0,0 +1,125 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. L1 â¡ L2 â âT1,T2. T1 â¡ T2 â â¦L1, T1â¦ â¡ â¦L2, T2â¦. +/3 width=4/ qed. + +lemma cpr_fpr: âL,T1,T2. L â¢ T1 â¡ T2 â â¦L, T1â¦ â¡ â¦L, T2â¦. +/2 width=4/ qed. + +lemma fpr_lift: âK1,K2,T1,T2. â¦K1, T1â¦ â¡ â¦K2, T2â¦ â + âd,e,L1. â©[d, e] L1 â¡ K1 â + âU1,U2. â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â + ââL2. â¦L1, U1â¦ â¡ â¦L2, U2â¦ & â©[d, e] L2 â¡ K2. +#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2 +elim (fpr_inv_all â¦ HT12) -HT12 #K #HK1 #HT12 #HK2 +elim (ldrop_ltpr_trans â¦ HLK1 â¦ HK1) -K1 #L #HL1 #HLK +lapply (cpr_lift â¦ HLK â¦ HTU1 â¦ HTU2 HT12) -T1 -T2 #HU12 +elim (le_or_ge (|K|) d) #Hd +[ elim (ldrop_ltpss_sn_trans_ge â¦ HLK â¦ HK2 â¦) +| elim (ldrop_ltpss_sn_trans_be â¦ HLK â¦ HK2 â¦) +] // -Hd #L2 #HL2 #HLK2 +lapply (ltpss_sn_weak_full â¦ HL2) -K /3 width=4/ +qed-. + +(* Advanced properties ******************************************************) + +lemma fpr_flat_dx: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â âV1,V2. V1 â¡ V2 â + âI. â¦L1, â{I}V1.T1â¦ â¡ â¦L2, â{I}V2.T2â¦. +#L1 #L2 #T1 #T2 #HT12 +elim (fpr_inv_all â¦ HT12) -HT12 /4 width=4/ +qed. + +lemma fpr_bind_sn: âL1,L2,V1,V2. â¦L1, V1â¦ â¡ â¦L2, V2â¦ â âT1,T2. T1 â¡ T2 â + âa,I. â¦L1, â{a,I}V1.T1â¦ â¡ â¦L2, â{a,I}V2.T2â¦. +#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I +elim (fpr_inv_all â¦ H) /3 width=4/ +qed. + +lemma fpr_bind2_minus: âI,L1,L2,V1,T1,U2. â¦L1, -â{I}V1.T1â¦ â¡ â¦L2, U2â¦ â + ââV2,T2. â¦L1.â{I}V1, T1â¦ â¡ â¦L2.â{I}V2, T2â¦ & + U2 = -â{I}V2.T2. +#I1 #L1 #L2 #V1 #T1 #U2 #H +elim (fpr_inv_all â¦ H) -H #L #HL1 #H #HL2 +elim (cpr_fwd_bind1_minus â¦ H false) -H /4 width=4/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma fpr_fwd_bind2_minus: âI,L1,L,V1,T1,T. â¦L1, -â{I}V1.T1â¦ â¡ â¦L, Tâ¦ â âb. + ââV2,T2. â¦L1, â{b,I}V1.T1â¦ â¡ â¦L, â{b,I}V2.T2â¦ & + T = -â{I}V2.T2. +#I #L1 #L #V1 #T1 #T #H1 #b +elim (fpr_inv_all â¦ H1) -H1 #L0 #HL10 #HT1 #HL0 +elim (cpr_fwd_bind1_minus â¦ HT1 b) -HT1 /3 width=4/ +qed-. + +lemma fpr_fwd_shift_bind_minus: âI1,I2,L1,L2,V1,V2,T1,T2. + â¦L1, -â{I1}V1.T1â¦ â¡ â¦L2, -â{I2}V2.T2â¦ â + â¦L1, V1â¦ â¡ â¦L2, V2â¦ â§ I1 = I2. +* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H +elim (fpr_inv_all â¦ H) -H #L #HL1 #H #HL2 +[ elim (cpr_inv_abbr1 â¦ H) -H * + [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/ + | #T #_ #_ #H destruct + ] +| elim (cpr_inv_abst1 â¦ H Abst V2) -H + #V #T #HV1 #_ #H destruct /3 width=4/ +] +qed-. + +lemma fpr_fwd_abst2: âa,L1,L2,V1,T1,U2. â¦L1, â{a}V1.T1â¦ â¡ â¦L2, U2â¦ â âb,I,W. + ââV2,T2. â¦L1, â{b,I}W.T1â¦ â¡ â¦L2, â{b,I}W.T2â¦ & + U2 = â{a}V2.T2. +#a #L1 #L2 #V1 #T1 #U2 #H +elim (fpr_inv_all â¦ H) #L #HL1 #H #HL2 #b #I #W +elim (cpr_fwd_abst1 â¦ H b I W) -H /3 width=4/ +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma fpr_inv_pair1: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡ â¦L2, T2â¦ â + ââK2,V2. â¦K1, V1â¦ â¡ â¦K2, V2â¦ & + â¦K1, -â{I}V1.T1â¦ â¡ â¦K2, -â{I}V2.T2â¦ & + L2 = K2.â{I}V2. +#I1 #K1 #X #V1 #T1 #T2 #H +elim (fpr_fwd_pair1 â¦ H) -H #I2 #K2 #V2 #HT12 #H destruct +elim (fpr_fwd_shift_bind_minus â¦ HT12) #HV12 #H destruct /2 width=5/ +qed-. + +lemma fpr_inv_pair3: âI,L1,K2,V2,T1,T2. â¦L1, T1â¦ â¡ â¦K2.â{I}V2, T2â¦ â + ââK1,V1. â¦K1, V1â¦ â¡ â¦K2, V2â¦ & + â¦K1, -â{I}V1.T1â¦ â¡ â¦K2, -â{I}V2.T2â¦ & + L1 = K1.â{I}V1. +#I2 #X #K2 #V2 #T1 #T2 #H +elim (fpr_fwd_pair3 â¦ H) -H #I1 #K1 #V1 #HT12 #H destruct +elim (fpr_fwd_shift_bind_minus â¦ HT12) #HV12 #H destruct /2 width=5/ +qed-. + +(* More advanced forward lemmas *********************************************) + +lemma fpr_fwd_pair1_full: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡ â¦L2, T2â¦ â + âb. ââK2,V2. â¦K1, V1â¦ â¡ â¦K2, V2â¦ & + â¦K1, â{b,I}V1.T1â¦ â¡ â¦K2, â{b,I}V2.T2â¦ & + L2 = K2.â{I}V2. +#I #K1 #L2 #V1 #T1 #T2 #H #b +elim (fpr_inv_pair1 â¦ H) -H #K2 #V2 #HV12 #HT12 #H destruct +elim (fpr_fwd_bind2_minus â¦ HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr_fpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr_fpr.etc new file mode 100644 index 000000000..3f7ac2ceb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr_fpr.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ fpr. +#L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02 +elim (tpr_conf â¦ HT01 HT02) -L0 -T0 #X #H1 #H2 +elim (tpr_fwd_shift1 â¦ H1) #L #T #HL1 #H destruct /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs.etc new file mode 100644 index 000000000..f42bdf7e2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs.etc @@ -0,0 +1,58 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1 , term 46 T1 â¦ â¡ * break â¦ term 46 L2 , term 46 T2 â¦ )" + non associative with precedence 45 + for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }. + +notation "hvbox( â¦ term 46 L1 , term 46 T1 â¦ â¡ â¡ * break â¦ term 46 L2 , term 46 T2 â¦ )" + 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 â bi_TC â¦ fpr. + +interpretation "context-free parallel computation (closure)" + 'FocalizedPRedStar L1 T1 L2 T2 = (fprs L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma fprs_ind: âL1,T1. âR:relation2 lenv term. R L1 T1 â + (âL,L2,T,T2. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â R L T â R L2 T2) â + âL2,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â R L2 T2. +/3 width=7 by bi_TC_star_ind/ qed-. + +lemma fprs_ind_dx: âL2,T2. âR:relation2 lenv term. R L2 T2 â + (âL1,L,T1,T. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â â¦L, Tâ¦ â¡* â¦L2, T2â¦ â R L T â R L1 T1) â + âL1,T1. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â R L1 T1. +/3 width=7 by bi_TC_star_ind_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma fpr_fprs: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1, T1â¦ â¡* â¦L2, T2â¦. +/2 width=1/ qed. + +lemma fprs_refl: bi_reflexive â¦ fprs. +/2 width=1/ qed. + +lemma fprs_strap1: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¡* â¦L, Tâ¦ â â¦L, Tâ¦ â¡ â¦L2, T2â¦ â + â¦L1, T1â¦ â¡* â¦L2, T2â¦. +/2 width=4/ qed. + +lemma fprs_strap2: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â¡ â¦L, Tâ¦ â â¦L, Tâ¦ â¡* â¦L2, T2â¦ â + â¦L1, T1â¦ â¡* â¦L2, T2â¦. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_aaa.etc new file mode 100644 index 000000000..b76637ff7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_aaa.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T1,A. L1 â¢ T1 â A â + âL2,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â L2 â¢ T2 â A. +#L1 #T1 #A #HT1 #L2 #T2 #HLT12 +@(bi_TC_Conf3 â¦ HT1 ?? HLT12) /2 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_cprs.etc new file mode 100644 index 000000000..9d4d954b8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_cprs.etc @@ -0,0 +1,138 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â âV1,V2. V1 â¡ V2 â + âI. â¦L1, â{I}V1.T1â¦ â¡* â¦L2, â{I}V2.T2â¦. +#L1 #L2 #T1 #T2 #HT12 @(fprs_ind â¦ HT12) -L2 -T2 /3 width=1/ +#L #L2 #T #T2 #_ #HT2 #IHT2 #V1 #V2 #HV12 #I +lapply (IHT2 â¦ HV12 I) -IHT2 -HV12 /3 width=6/ +qed. + +lemma fprs_bind2_minus: âI,L1,L2,V1,T1,U2. â¦L1, -â{I}V1.T1â¦ â¡* â¦L2, U2â¦ â + ââV2,T2. â¦L1.â{I}V1, T1â¦ â¡* â¦L2.â{I}V2, T2â¦ & + U2 = -â{I}V2.T2. +#I #L1 #L2 #V1 #T1 #U2 #H @(fprs_ind â¦ H) -L2 -U2 /2 width=4/ +#L #L2 #U #U2 #_ #HU2 * #V #T #HT1 #H destruct +elim (fpr_bind2_minus â¦ HU2) -HU2 /3 width=4/ +qed-. + +lemma fprs_lift: âK1,K2,T1,T2. â¦K1, T1â¦ â¡* â¦K2, T2â¦ â + âd,e,L1. â©[d, e] L1 â¡ K1 â + âU1,U2. â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â + ââL2. â¦L1, U1â¦ â¡* â¦L2, U2â¦ & â©[d, e] L2 â¡ K2. +#K1 #K2 #T1 #T2 #HT12 @(fprs_ind â¦ HT12) -K2 -T2 +[ #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2 + >(lift_mono â¦ HTU2 â¦ 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 â¦ HLK1 â¦ HTU1 HTU) -K1 -T1 #L #HU1 #HKL + elim (fpr_lift â¦ HT2 â¦ HKL â¦ HTU HTU2) -K -T -T2 /3 width=4/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma fprs_inv_pair1: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡* â¦L2, T2â¦ â + ââK2,V2. â¦K1, V1â¦ â¡* â¦K2, V2â¦ & + â¦K1, -â{I}V1.T1â¦ â¡* â¦K2, -â{I}V2.T2â¦ & + L2 = K2.â{I}V2. +#I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 /2 width=5/ +#L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct +elim (fpr_inv_pair1 â¦ HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/ +qed-. + +lemma fprs_inv_pair3: âI,L1,K2,V2,T1,T2. â¦L1, T1â¦ â¡* â¦K2.â{I}V2, T2â¦ â + ââK1,V1. â¦K1, V1â¦ â¡* â¦K2, V2â¦ & + â¦K1, -â{I}V1.T1â¦ â¡* â¦K2, -â{I}V2.T2â¦ & + L1 = K1.â{I}V1. +#I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx â¦ H) -L1 -T1 /2 width=5/ +#L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct +elim (fpr_inv_pair3 â¦ HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma fprs_fwd_bind2_minus: âI,L1,L,V1,T1,T. â¦L1, -â{I}V1.T1â¦ â¡* â¦L, Tâ¦ â âb. + ââV2,T2. â¦L1, â{b,I}V1.T1â¦ â¡* â¦L, â{b,I}V2.T2â¦ & + T = -â{I}V2.T2. +#I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind â¦ H1) -L -T /2 width=4/ +#L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct +elim (fpr_fwd_bind2_minus â¦ H0 b) -H0 /3 width=4/ +qed-. + +lemma fprs_fwd_pair1_full: âI,K1,L2,V1,T1,T2. â¦K1.â{I}V1, T1â¦ â¡* â¦L2, T2â¦ â + âb. ââK2,V2. â¦K1, V1â¦ â¡* â¦K2, V2â¦ & + â¦K1, â{b,I}V1.T1â¦ â¡* â¦K2, â{b,I}V2.T2â¦ & + L2 = K2.â{I}V2. +#I #K1 #L2 #V1 #T1 #T2 #H #b +elim (fprs_inv_pair1 â¦ H) -H #K2 #V2 #HV12 #HT12 #H destruct +elim (fprs_fwd_bind2_minus â¦ HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/ +qed-. + +lemma fprs_fwd_abst2: âa,L1,L2,V1,T1,U2. â¦L1, â{a}V1.T1â¦ â¡* â¦L2, U2â¦ â âb,I,W. + ââV2,T2. â¦L1, â{b,I}W.T1â¦ â¡* â¦L2, â{b,I}W.T2â¦ & + U2 = â{a}V2.T2. +#a #L1 #L2 #V1 #T1 #U2 #H #b #I #W @(fprs_ind â¦ H) -L2 -U2 /2 width=4/ +#L #L2 #U #U2 #_ #H2 * #V #T #HT1 #H destruct +elim (fpr_fwd_abst2 â¦ H2 b I W) -H2 /3 width=4/ +qed-. + +(* Properties on context-sensitive parallel computation for terms ***********) + +lemma cprs_fprs: âL,T1,T2. L â¢ T1 â¡* T2 â â¦L, T1â¦ â¡* â¦L, T2â¦. +#L #T1 #T2 #H @(cprs_ind â¦ H) -T2 // /3 width=4/ +qed. + +(* Forward lemmas on context-sensitive parallel computation for terms *******) + +lemma fprs_fwd_cprs: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â L1 â¢ T1 â¡* T2. +#L1 #L2 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 // +#L #L2 #T #T2 #H1 #H2 #IH1 +elim (fpr_inv_all â¦ H2) -H2 #L0 #HL0 #HT2 #_ -L2 +lapply (lfprs_cpr_trans L1 â¦ HT2) -HT2 /3 width=3/ +qed-. +(* +(* Advanced properties ******************************************************) + +lamma fpr_bind_sn: âL1,L2,V1,V2. â¦L1, V1â¦ â¡ â¦L2, V2â¦ â âT1,T2. T1 â¡ T2 â + âa,I. â¦L1, â{a,I}V1.T1â¦ â¡ â¦L2, â{a,I}V2.T2â¦. +#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I +elim (fpr_inv_all â¦ H) /3 width=4/ +qed. + +(* Advanced forward lemmas **************************************************) + +lamma fpr_fwd_shift_bind_minus: âI1,I2,L1,L2,V1,V2,T1,T2. + â¦L1, -â{I1}V1.T1â¦ â¡ â¦L2, -â{I2}V2.T2â¦ â + â¦L1, V1â¦ â¡ â¦L2, V2â¦ â§ I1 = I2. +* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H +elim (fpr_inv_all â¦ H) -H #L #HL1 #H #HL2 +[ elim (cpr_inv_abbr1 â¦ H) -H * + [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/ + | #T #_ #_ #H destruct + ] +| elim (cpr_inv_abst1 â¦ H Abst V2) -H + #V #T #HV1 #_ #H destruct /3 width=4/ +] +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_fprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_fprs.etc new file mode 100644 index 000000000..e0c1b3058 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fprs_fprs.etc @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,L1,T0,T1. â¦L0, T0â¦ â¡ â¦L1, T1â¦ â + âL2,T2. â¦L0, T0â¦ â¡* â¦L2, T2â¦ â + ââL,T. â¦L1, T1â¦ â¡* â¦L, Tâ¦ & â¦L2, T2â¦ â¡ â¦L, Tâ¦. +#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 +/2 width=4/ qed. + +(* Main propertis ***********************************************************) + +theorem fprs_conf: bi_confluent â¦ fprs. +/2 width=4/ qed. + +theorem fprs_trans: bi_transitive â¦ fprs. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lenv_px_bi.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lenv_px_bi.etc new file mode 100644 index 000000000..931d075ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lenv_px_bi.etc @@ -0,0 +1,88 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| lpx_bi_stom: lpx_bi R (â) (â) +| lpx_bi_pair: âI,K1,K2,V1,V2. + lpx_bi R K1 K2 â R K1 V1 K2 V2 â + lpx_bi R (K1. â{I} V1) (K2. â{I} V2) +. + +(* Basic inversion lemmas ***************************************************) + +fact lpx_bi_inv_atom1_aux: âR,L1,L2. lpx_bi R L1 L2 â L1 = â â L2 = â. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_bi_inv_atom1: âR,L2. lpx_bi R (â) L2 â L2 = â. +/2 width=4 by lpx_bi_inv_atom1_aux/ qed-. + +fact lpx_bi_inv_pair1_aux: âR,L1,L2. lpx_bi R L1 L2 â + âI,K1,V1. L1 = K1. â{I} V1 â + ââK2,V2. lpx_bi R K1 K2 & + R K1 V1 K2 V2 & L2 = K2. â{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: âR,I,K1,V1,L2. lpx_bi R (K1. â{I} V1) L2 â + ââK2,V2. lpx_bi R K1 K2 & R K1 V1 K2 V2 & + L2 = K2. â{I} V2. +/2 width=3 by lpx_bi_inv_pair1_aux/ qed-. + +fact lpx_bi_inv_atom2_aux: âR,L1,L2. lpx_bi R L1 L2 â L2 = â â L1 = â. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_bi_inv_atom2: âR,L1. lpx_bi R L1 (â) â L1 = â. +/2 width=4 by lpx_bi_inv_atom2_aux/ qed-. + +fact lpx_bi_inv_pair2_aux: âR,L1,L2. lpx_bi R L1 L2 â + âI,K2,V2. L2 = K2. â{I} V2 â + ââK1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & + L1 = K1. â{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: âR,I,L1,K2,V2. lpx_bi R L1 (K2. â{I} V2) â + ââK1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & + L1 = K1. â{I} V1. +/2 width=3 by lpx_bi_inv_pair2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lpx_bi_fwd_length: âR,L1,L2. lpx_bi R L1 L2 â |L1| = |L2|. +#R #L1 #L2 #H elim H -L1 -L2 normalize // +qed-. + +(* Basic properties *********************************************************) + +lemma lpx_bi_refl: âR. bi_reflexive ? ? R â reflexive â¦ (lpx_bi R). +#R #HR #L elim L -L // /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfpr_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfpr_alt.etc new file mode 100644 index 000000000..95ec60ba2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfpr_alt.etc @@ -0,0 +1,87 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ â¡ â¡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'PRedSnAlt $L1 $L2 }. + +notation "hvbox( â¦ term 46 L1 â¦ â¡ â¡ break â¦ term 46 L2 â¦ )" + 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 â lpx_bi fpr. + +interpretation + "focalized parallel reduction (environment) alternative" + 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2). + +(* Basic properties *********************************************************) + +lemma lfpra_refl: reflexive â¦ lfpra. +/2 width=1/ qed. + +lemma fpr_lfpra: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1â¦ â¡â¡ â¦L2â¦. +#L1 elim L1 -L1 +[ #L2 #T1 #T2 #H + elim (fpr_inv_atom1 â¦ H) -H #_ #H destruct // +| #L1 #I #V1 #IH #L2 #T1 #T2 #H + elim (fpr_inv_pair1 â¦ H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma lfpra_inv_atom1: âL2. â¦ââ¦ â¡â¡ â¦L2â¦ â L2 = â. +/2 width=2 by lpx_bi_inv_atom1/ qed-. + +lemma lfpra_inv_pair1: âK1,I,V1,L2. â¦K1. â{I} V1â¦ â¡â¡ â¦L2â¦ â + ââK2,V2. â¦K1â¦ â¡â¡ â¦K2â¦ & â¦K1, V1â¦ â¡ â¦K2, V2â¦ & + L2 = K2. â{I} V2. +/2 width=1 by lpx_bi_inv_pair1/ qed-. + +lemma lfpra_inv_atom2: âL1. â¦L1â¦ â¡â¡ â¦ââ¦ â L1 = â. +/2 width=2 by lpx_bi_inv_atom2/ qed-. + +lemma lfpra_inv_pair2: âL1,K2,I,V2. â¦L1â¦ â¡â¡ â¦K2. â{I} V2â¦ â + ââK1,V1. â¦K1â¦ â¡â¡ â¦K2â¦ & â¦K1, V1â¦ â¡ â¦K2, V2â¦ & + L1 = K1. â{I} V1. +/2 width=1 by lpx_bi_inv_pair2/ qed-. + +lemma lfpra_inv_fpr: âL1,L2. â¦L1â¦ â¡â¡ â¦L2â¦ â âT.â¦L1, Tâ¦ â¡ â¦L2, Tâ¦. +#L1 #L2 * -L1 -L2 // /3 width=1/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lfpra_fwd_length: âL1,L2. â¦L1â¦ â¡â¡ â¦L2â¦ â |L1| = |L2|. +/2 width=2 by lpx_bi_fwd_length/ qed-. + +(* Main properties **********************************************************) + +theorem lfpr_lfpra: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â â¦L1â¦ â¡â¡ â¦L2â¦. +#L1 #L2 #H +lapply (lfpr_inv_fpr â¦ H (â0)) -H /2 width=3/ +qed. + +theorem lfpra_lfpr: âL1,L2. â¦L1â¦ â¡â¡ â¦L2â¦ â â¦L1â¦ â¡ â¦L2â¦. +#L1 #L2 #H +lapply (lfpra_inv_fpr â¦ H (â0)) -H /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfpr_fpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfpr_fpr.etc new file mode 100644 index 000000000..f798d566b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfpr_fpr.etc @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T1,T2. â¦L1, T1â¦ â¡ â¦L2, T2â¦ â â¦L1â¦ â¡ â¦L2â¦. +#L1 #L2 #T1 #T2 #H +elim (fpr_inv_all â¦ H) -H /2 width=3/ +qed. + +(* Inversion lemmas on context-free parallel reduction for closures *********) + +lemma lfpr_inv_fpr: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â âT. â¦L1, Tâ¦ â¡ â¦L2, Tâ¦. +#L1 #L2 * /2 width=4/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfprs_fprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfprs_fprs.etc new file mode 100644 index 000000000..a5c42ba65 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/lfprs_fprs.etc @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. â¦L1â¦ â¡* â¦L2â¦ â âT. â¦L1, Tâ¦ â¡* â¦L2, Tâ¦. +#L1 #L2 #H @(lfprs_ind â¦ H) -L2 // +#L #L2 #_ #HL2 #IHL1 #T +lapply (lfpr_inv_fpr â¦ HL2 T) -HL2 /3 width=4/ +qed-. + +(* Properties on context-free parallel computation for closures *************) + +lemma fprs_lfprs: âL1,L2,T1,T2. â¦L1, T1â¦ â¡* â¦L2, T2â¦ â â¦L1â¦ â¡* â¦L2â¦. +#L1 #L2 #T1 #T2 #H @(fprs_ind â¦ H) -L2 -T2 // /3 width=5/ +qed. + +lemma lfprs_fprs_trans: âL1,L,L2,T1,T2. â¦L1â¦ â¡* â¦Lâ¦ â â¦L, T1â¦ â¡* â¦L2, T2â¦ â â¦L1, T1â¦ â¡* â¦L2, T2â¦. +#L1 #L #L2 #T1 #T2 #HL1 #HL2 +lapply (lfprs_inv_fprs â¦ HL1 T1) -HL1 /2 width=4/ +qed. +(* +lamma lfprs_cprs_conf: âL1,L,L2,T1,T2. â¦L1â¦ â¡* â¦L2â¦ â L1 â¢ T1 â¡* T2 â â¦L1, T1â¦ â¡* â¦L2, T2â¦. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsup.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsup.etc new file mode 100644 index 000000000..077fd7422 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsup.etc @@ -0,0 +1,123 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1, break term 46 T1 â¦ â§ break â¦ term 46 L2 , break term 46 T2 â¦ )" + 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 â +| frsup_bind_sn: âa,I,L,V,T. frsup L (â{a,I}V.T) L V +| frsup_bind_dx: âa,I,L,V,T. frsup L (â{a,I}V.T) (L.â{I}V) T +| frsup_flat_sn: âI,L,V,T. frsup L (â{I}V.T) L V +| frsup_flat_dx: âI,L,V,T. frsup L (â{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: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â + âJ. T1 = âª{J} â â¥. +#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: âJ,L1,L2,T2. â¦L1, âª{J}â¦ â§ â¦L2, T2â¦ â â¥. +/2 width=7 by frsup_inv_atom1_aux/ qed-. + +fact frsup_inv_bind1_aux: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â + âb,J,W,U. T1 = â{b,J}W.U â + (L2 = L1 â§ T2 = W) â¨ + (L2 = L1.â{J}W â§ 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: âb,J,L1,L2,W,U,T2. â¦L1, â{b,J}W.Uâ¦ â§ â¦L2, T2â¦ â + (L2 = L1 â§ T2 = W) â¨ + (L2 = L1.â{J}W â§ T2 = U). +/2 width=4 by frsup_inv_bind1_aux/ qed-. + +fact frsup_inv_flat1_aux: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â + âJ,W,U. T1 = â{J}W.U â + L2 = L1 â§ (T2 = W â¨ 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: âJ,L1,L2,W,U,T2. â¦L1, â{J}W.Uâ¦ â§ â¦L2, T2â¦ â + L2 = L1 â§ (T2 = W â¨ T2 = U). +/2 width=4 by frsup_inv_flat1_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma frsup_fwd_fw: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¯{L2, T2} < â¯{L1, T1}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ +qed-. + +lemma frsup_fwd_lw: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¯{L1} â¤ â¯{L2}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ +qed-. + +lemma frsup_fwd_tw: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¯{T2} < â¯{T1}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/ +qed-. + +lemma frsup_fwd_append: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â âL. L2 = L1 @@ L. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #a +| #a #I #L #V #_ @(ex_intro â¦ (â.â{I}V)) // +] +#I #L #V #T @(ex_intro â¦ (â)) // +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lift_frsup_trans: âT1,U1,d,e. â§[d, e] T1 â¡ U1 â + âL,K,U2. â¦L, U1â¦ â§ â¦L @@ K, U2â¦ â + âT2. â§[d + |K|, e] T2 â¡ 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 â¦ H) -H * + [ -HTU1 #H1 #H2 destruct + >(append_inv_refl_dx â¦ H1) -L -K normalize /2 width=2/ + | -HVW1 #H1 #H2 destruct + >(append_inv_pair_dx â¦ 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 â¦ H) -H #H1 * #H2 destruct + >(append_inv_refl_dx â¦ H1) -L -K normalize /2 width=2/ +] +#i #d #e [2,3: #_ ] #L #K #X #H +elim (frsup_inv_atom1 â¦ H) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsupp.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsupp.etc new file mode 100644 index 000000000..9f7a8dc9d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsupp.etc @@ -0,0 +1,110 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1, break term 46 T1 â¦ â§ + break â¦ term 46 L2 , break term 46 T2 â¦ )" + 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 â bi_TC â¦ frsup. + +interpretation "plus-iterated restricted structural predecessor (closure)" + 'RestSupTermPlus L1 T1 L2 T2 = (frsupp L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma frsupp_ind: âL1,T1. âR:relation2 lenv term. + (âL2,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â R L2 T2) â + (âL,T,L2,T2. â¦L1, T1â¦ â§+ â¦L, Tâ¦ â â¦L, Tâ¦ â§ â¦L2, T2â¦ â R L T â R L2 T2) â + âL2,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â R L2 T2. +#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H +@(bi_TC_ind â¦ IH1 IH2 ? ? H) +qed-. + +lemma frsupp_ind_dx: âL2,T2. âR:relation2 lenv term. + (âL1,T1. â¦L1, T1â¦ â§ â¦L2, T2â¦ â R L1 T1) â + (âL1,L,T1,T. â¦L1, T1â¦ â§ â¦L, Tâ¦ â â¦L, Tâ¦ â§+ â¦L2, T2â¦ â R L T â R L1 T1) â + âL1,T1. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â R L1 T1. +#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H +@(bi_TC_ind_dx â¦ IH1 IH2 ? ? H) +qed-. + +(* Baic inversion lemmas ****************************************************) + +lemma frsupp_inv_dx: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¦L1, T1â¦ â§ â¦L2, T2â¦ â¨ + ââL,T. â¦L1, T1â¦ â§+ â¦L, Tâ¦ & â¦L, Tâ¦ â§ â¦L2, T2â¦. +/2 width=1 by bi_TC_decomp_r/ qed-. + +lemma frsupp_inv_sn: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¦L1, T1â¦ â§ â¦L2, T2â¦ â¨ + ââL,T. â¦L1, T1â¦ â§ â¦L, Tâ¦ & â¦L, Tâ¦ â§+ â¦L2, T2â¦. +/2 width=1 by bi_TC_decomp_l/ qed-. + +(* Basic properties *********************************************************) + +lemma frsup_frsupp: âL1,L2,T1,T2. â¦L1, T1â¦ â§ â¦L2, T2â¦ â â¦L1, T1â¦ â§+ â¦L2, T2â¦. +/2 width=1/ qed. + +lemma frsupp_strap1: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â§+ â¦L, Tâ¦ â â¦L, Tâ¦ â§ â¦L2, T2â¦ â + â¦L1, T1â¦ â§+ â¦L2, T2â¦. +/2 width=4/ qed. + +lemma frsupp_strap2: âL1,L,L2,T1,T,T2. â¦L1, T1â¦ â§ â¦L, Tâ¦ â â¦L, Tâ¦ â§+ â¦L2, T2â¦ â + â¦L1, T1â¦ â§+ â¦L2, T2â¦. +/2 width=4/ qed. + +(* Basic forward lemmas *****************************************************) + +lemma frsupp_fwd_fw: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¯{L2, T2} < â¯{L1, T1}. +#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 +/3 width=3 by frsup_fwd_fw, transitive_lt/ +qed-. + +lemma frsupp_fwd_lw: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¯{L1} â¤ â¯{L2}. +#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ 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 â¦ HL2) -HL2 /2 width=3 by transitive_le/ +qed-. + +lemma frsupp_fwd_tw: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â â¯{T2} < â¯{T1}. +#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 +/3 width=3 by frsup_fwd_tw, transitive_lt/ +qed-. + +lemma frsupp_fwd_append: âL1,L2,T1,T2. â¦L1, T1â¦ â§+ â¦L2, T2â¦ â âL. L2 = L1 @@ L. +#L1 #L2 #T1 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 /2 width=3 by frsup_fwd_append/ +#L #T #L2 #T2 #_ #HL2 * #K1 #H destruct +elim (frsup_fwd_append â¦ HL2) -HL2 #K2 #H destruct /2 width=2/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lift_frsupp_trans: âL,U1,K,U2. â¦L, U1â¦ â§+ â¦L @@ K, U2â¦ â + âT1,d,e. â§[d, e] T1 â¡ U1 â + âT2. â§[d + |K|, e] T2 â¡ U2. +#L #U1 @(f2_ind â¦ fw â¦ L U1) -L -U1 #n #IH +#L #U1 #Hn #K #U2 #H #T1 #d #e #HTU1 destruct +elim (frsupp_inv_sn â¦ H) -H /2 width=5 by lift_frsup_trans/ * +#L0 #U0 #HL0 #HL +elim (frsup_fwd_append â¦ HL0) #K0 #H destruct +elim (frsupp_fwd_append â¦ HL) #L0 >append_assoc #H +elim (append_inj_dx â¦ H ?) -H // #_ #H destruct +(append_inv_refl_dx â¦ (sym_eq â¦ H1)) -H1 normalize /2 width=2/ +| /2 width=5 by lift_frsupp_trans/ +] +qed-. + +(* Advanced inversion lemmas for frsupp **************************************) + +lemma frsupp_inv_atom1_frsups: âJ,L1,L2,T2. â¦L1, âª{J}â¦ â§+ â¦L2, T2â¦ â â¥. +#J #L1 #L2 #T2 #H @(frsupp_ind â¦ H) -L2 -T2 // +#L2 #T2 #H elim (frsup_inv_atom1 â¦ H) +qed-. + +lemma frsupp_inv_bind1_frsups: âb,J,L1,L2,W,U,T2. â¦L1, â{b,J}W.Uâ¦ â§+ â¦L2, T2â¦ â + â¦L1, Wâ¦ â§* â¦L2, T2â¦ â¨ â¦L1.â{J}W, Uâ¦ â§* â¦L2, T2â¦. +#b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind â¦ H) -L2 -T2 +[ #L2 #T2 #H + elim (frsup_inv_bind1 â¦ H) -H * #H1 #H2 destruct /2 width=1/ +| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ +] +qed-. + +lemma frsupp_inv_flat1_frsups: âJ,L1,L2,W,U,T2. â¦L1, â{J}W.Uâ¦ â§+ â¦L2, T2â¦ â + â¦L1, Wâ¦ â§* â¦L2, T2â¦ â¨ â¦L1, Uâ¦ â§* â¦L2, T2â¦. +#J #L1 #L2 #W #U #T2 #H @(frsupp_ind â¦ H) -L2 -T2 +[ #L2 #T2 #H + elim (frsup_inv_flat1 â¦ H) -H #H1 * #H2 destruct /2 width=1/ +| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsups_frsups.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsups_frsups.etc new file mode 100644 index 000000000..e7b7de26e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsups_frsups.etc @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ frsups. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/ssta_frsups.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/ssta_frsups.etc new file mode 100644 index 000000000..9eb1fbd8b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/ssta_frsups.etc @@ -0,0 +1,70 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L,T,U,l. â¦h, Lâ¦ â¢ T â¢[g] â¦l, Uâ¦ â â¦L, Uâ¦ â§+ â¦L, Tâ¦ â â¥. +#h #g #L #T #U #l #H elim H -L -T -U -l +[ #L #k #l #_ #H + elim (frsupp_inv_atom1_frsups â¦ H) +| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H + elim (lift_frsupp_trans â¦ (â) â¦ H â¦ HWU) -U #X #H + elim (lift_inv_lref2_be â¦ H ? ?) -H // +| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H + elim (lift_frsupp_trans â¦ (â) â¦ H â¦ HWU) -U #X #H + elim (lift_inv_lref2_be â¦ H ? ?) -H // +| #a #I #L #V #T #U #l #_ #IHTU #H + elim (frsupp_inv_bind1_frsups â¦ H) -H #H [2: /4 width=4/ ] -IHTU + lapply (frsups_fwd_fw â¦ H) -H normalize + (lcoeq_inv_O2 â¦ HL1) -HL1 // +| #I1 #I #L1 #L #V1 #V #e #_ #IHL1 #X #H elim (lcoeq_inv_pair1 â¦ 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 â¦ H) -H // + #L2 #HL2 #H destruct /3 width=1 by lcoeq_succ/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lcoeq/ldrop_lcoeq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lcoeq/ldrop_lcoeq.etc new file mode 100644 index 000000000..4891949c2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lcoeq/ldrop_lcoeq.etc @@ -0,0 +1,51 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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: âL1,L2,d,e. L1 â[d, e] L2 â + âI,K2,W,s,i. â©[s, 0, i] L2 â¡ K2.â{I}W â i < d â + ââK1. K1 â[â«°(d-i), e] K2 & â©[s, 0, i] L1 â¡ K1.â{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 â¦ H) -H #H destruct +| #I #L1 #L2 #V #_ #_ #J #K2 #W #s #i #_ #H + elim (ylt_yle_false â¦ H) // +| #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #J #K2 #W #s #i #_ #H + elim (ylt_yle_false â¦ H) // +| #I #L1 #L2 #V #d #e #HL12 #IHL12 #J #K2 #W #s #i #H + elim (ldrop_inv_O1_pair1 â¦ 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 yminus_succ (cpys_inv_sort1 â¦ H) -H // +qed. + +lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â[Â§p, d] L2. +#L1 #L2 #d #k #HL12 @conj // -HL12 +#U @conj #H >(cpys_inv_gref1 â¦ H) -H // +qed. + +lemma lleq_bind: âa,I,L1,L2,V,T,d. + L1 â[V, d] L2 â L1.â{I}V â[T, â«¯d] L2.â{I}V â + L1 â[â{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 â¦ 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: âI,L1,L2,V,T,d. + L1 â[V, d] L2 â L1 â[T, d] L2 â L1 â[â{I}V.T, d] L2. +#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12 +#X @conj #H elim (cpys_inv_flat1 â¦ 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: âL1,L2,U,dt. L1 â[U, dt] L2 â âT,d,e. â§[d, e] T â¡ U â + d â¤ dt â dt â¤ d + e â L1 â[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 â¦ HU0 â¦ HTU) // -HU0 /4 width=5 by cpys_weak/ +qed-. + +lemma lsuby_lleq_trans: âL2,L,T,d. L2 â[T, d] L â + âL1. L1 âÃ[d, â] L2 â |L1| = |L2| â L1 â[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 â¦ 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: âL,L1,T,d. L â[T, d] L1 â + âL2. L1 âÃ[d, â] L2 â |L1| = |L2| â L â[T, d] L2. +/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-. + +lemma lleq_lsuby_repl: âL1,L2,T,d. L1 â[T, d] L2 â + âK1. K1 âÃ[d, â] L1 â |K1| = |L1| â + âK2. L2 âÃ[d, â] K2 â |L2| = |K2| â + K1 â[T, d] K2. +/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lleq_fwd_length: âL1,L2,T,d. L1 â[T, d] L2 â |L1| = |L2|. +#L1 #L2 #T #d * // +qed-. + +lemma lleq_fwd_ldrop_sn: âL1,L2,T,d. L1 â[d, T] L2 â âK1,i. â©[i] L1 â¡ K1 â + âK2. â©[i] L2 â¡ K2. +#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length â¦ H) -H +#HL12 lapply (ldrop_fwd_length_le2 â¦ HLK1) -HLK1 /2 width=1 by ldrop_O1_le/ +qed-. + +lemma lleq_fwd_ldrop_dx: âL1,L2,T,d. L1 â[d, T] L2 â âK2,i. â©[i] L2 â¡ K2 â + âK1. â©[i] L1 â¡ K1. +/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-. + +lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,d. + L1 â[â{a,I}V.T, d] L2 â L1 â[V, d] L2. +#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12 +#U elim (H (â{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 â¦ H) -H +#X #Y #H1 #H2 #H destruct // +qed-. + +lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,d. + L1 â[â{a,I}V.T, d] L2 â L1.â{I}V â[T, â«¯d] L2.â{I}V. +#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12 +#U elim (H (â{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 â¦ H) -H +#X #Y #H1 #H2 #H destruct // +qed-. + +lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,d. + L1 â[â{I}V.T, d] L2 â L1 â[V, d] L2. +#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12 +#U elim (H (â{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 â¦ H) -H +#X #Y #H1 #H2 #H destruct // +qed-. + +lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,d. + L1 â[â{I}V.T, d] L2 â L1 â[T, d] L2. +#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12 +#U elim (H (â{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 â¦ H) -H +#X #Y #H1 #H2 #H destruct // +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma lleq_inv_bind: âa,I,L1,L2,V,T,d. L1 â[â{a,I}V.T, d] L2 â + L1 â[V, d] L2 â§ L1.â{I}V â[T, â«¯d] L2.â{I}V. +/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-. + +lemma lleq_inv_flat: âI,L1,L2,V,T,d. L1 â[â{I}V.T, d] L2 â + L1 â[V, d] L2 â§ L1 â[T, d] L2. +/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_alt.etc new file mode 100644 index 000000000..a8fb526e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_alt.etc @@ -0,0 +1,85 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| lleqa_sort: âL1,L2,d,k. |L1| = |L2| â lleqa d (âk) L1 L2 +| lleqa_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â lleqa d (#i) L1 L2 +| lleqa_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â + lleqa (yinj 0) V K1 K2 â lleqa d (#i) L1 L2 +| lleqa_free: âL1,L2,d,i. |L1| = |L2| â |L1| â¤ i â |L2| â¤ i â lleqa d (#i) L1 L2 +| lleqa_gref: âL1,L2,d,p. |L1| = |L2| â lleqa d (Â§p) L1 L2 +| lleqa_bind: âa,I,L1,L2,V,T,d. + lleqa d V L1 L2 â lleqa (â«¯d) T (L1.â{I}V) (L2.â{I}V) â + lleqa d (â{a,I}V.T) L1 L2 +| lleqa_flat: âI,L1,L2,V,T,d. + lleqa d V L1 L2 â lleqa d T L1 L2 â lleqa d (â{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: âL1,L2,T,d. L1 ââ[T, d] L2 â L1 â[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: âL1,T,L2,d. L1 â[T, d] L2 â L1 ââ[T, d] L2. +#L1 #T @(f2_ind â¦ rfw â¦ 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 â¦ 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 â¦ H) -H /3 width=1 by lleqa_bind/ +| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat â¦ H) -H /3 width=1 by lleqa_flat/ +] +qed. + +(* Advanced eliminators *****************************************************) + +lemma lleq_ind_alt: âR:relation4 ynat term lenv lenv. ( + âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2 + ) â ( + âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2 + ) â ( + âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â + K1 â[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2 + ) â ( + âL1,L2,d,i. |L1| = |L2| â |L1| â¤ i â |L2| â¤ i â R d (#i) L1 L2 + ) â ( + âL1,L2,d,p. |L1| = |L2| â R d (Â§p) L1 L2 + ) â ( + âa,I,L1,L2,V,T,d. + L1 â[V, d]L2 â L1.â{I}V â[T, â«¯d] L2.â{I}V â + R d V L1 L2 â R (â«¯d) T (L1.â{I}V) (L2.â{I}V) â R d (â{a,I}V.T) L1 L2 + ) â ( + âI,L1,L2,V,T,d. + L1 â[V, d]L2 â L1 â[T, d] L2 â + R d V L1 L2 â R d T L1 L2 â R d (â{I}V.T) L1 L2 + ) â + âd,T,L1,L2. L1 â[T, d] L2 â R d T L1 L2. +#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa â¦ H) -H +/3 width=9 by lleqa_inv_lleq/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_ext.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_ext.etc new file mode 100644 index 000000000..d6bb03fa7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_ext.etc @@ -0,0 +1,91 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T,d0. L1 â[T, d0] L2 â âd. d0 = d + 1 â + âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â + K1 â[V, 0] K2 â L1 â[T, d] L2. +#L1 #L2 #T #d0 #H @(lleq_ind_alt â¦ 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: âT,L1,L2,d. L1 â[T, d+1] L2 â + âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â + K1 â[V, 0] K2 â L1 â[T, d] L2. +/2 width=7 by lleq_inv_S_aux/ qed-. + +lemma lleq_inv_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â + L1 â[V, 0] L2 â§ L1.â{I}V â[T, 0] L2.â{I}V. +#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind â¦ H) -H +/3 width=7 by ldrop_pair, conj, lleq_inv_S/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lleq_fwd_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â + L1.â{I}V â[T, 0] L2.â{I}V. +#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O â¦ H) -H // +qed-. + +(* Advanced properties ******************************************************) + +lemma lleq_ge: âL1,L2,T,d1. L1 â[T, d1] L2 â âd2. d1 â¤ d2 â L1 â[T, d2] L2. +#L1 #L2 #T #d1 #H @(lleq_ind_alt â¦ 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 â¦ HV) #HK12 #Hid2 + lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) + normalize in â¢ (%â%â?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/ + | /3 width=8 by lleq_lref, yle_trans/ + ] +] +qed-. + +lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â[V, 0] L2 â L1.â{I}V â[T, 0] L2.â{I}V â + L1 â[â{a,I}V.T, 0] L2. +/3 width=3 by lleq_ge, lleq_bind/ qed. + +lemma lleq_bind_repl_SO: âI1,I2,L1,L2,V1,V2,T. L1.â{I1}V1 â[T, 0] L2.â{I2}V2 â + âJ1,J2,W1,W2. L1.â{J1}W1 â[T, 1] L2.â{J2}W2. +#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge â¦ HT 1 ?) // -HT +#HT @(lleq_lsuby_repl â¦ HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *) +qed-. + +lemma lleq_bind_repl_O: âI,L1,L2,V,T. L1.â{I}V â[T, 0] L2.â{I}V â + âJ,W. L1 â[W, 0] L2 â L1.â{J}W â[T, 0] L2.â{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: âa,I,L1,L2,V,T. (L1 â[â{a,I}V.T, 0] L2 â â¥) â + (L1 â[V, 0] L2 â â¥) â¨ (L1.â{I}V â[T, 0] L2.â{I}V â â¥). +#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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_fqus.etc new file mode 100644 index 000000000..18744a539 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_fqus.etc @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ â â¦G2, K2, Uâ¦ â + âL1. L1 â[T, 0] L2 â + ââK1. â¦G1, L1, Tâ¦ â â¦G2, K1, Uâ¦ & K1 â[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 â¦ H â¦ I L2 V) -H // + #I1 #K1 #H1 #H2 lapply (ldrop_inv_O2 â¦ 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 â¦ H) + | elim (lleq_inv_flat â¦ H) + ] -H + /2 width=3 by fqu_pair_sn, ex2_intro/ +| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O â¦ H) -H + #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/ +| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat â¦ 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 â¦ HLK2) -K2 + lapply (lleq_fwd_length â¦ HL12) -T -U // + ] +] +qed-. + +lemma lleq_fquq_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ ââ¸® â¦G2, K2, Uâ¦ â + âL1. L1 â[T, 0] L2 â + ââK1. â¦G1, L1, Tâ¦ ââ¸® â¦G2, K1, Uâ¦ & K1 â[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen â¦ H) -H +[ #H elim (lleq_fqu_trans â¦ H â¦ 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: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ â+ â¦G2, K2, Uâ¦ â + âL1. L1 â[T, 0] L2 â + ââK1. â¦G1, L1, Tâ¦ â+ â¦G2, K1, Uâ¦ & K1 â[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind â¦ H) -G2 -K2 -U +[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans â¦ HTU â¦ HL12) -L2 + /3 width=3 by fqu_fqup, ex2_intro/ +| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU â¦ HL12) -L2 + #K1 #HTU #HK1 elim (lleq_fqu_trans â¦ HU2 â¦ HK1) -K + /3 width=5 by fqup_strap1, ex2_intro/ +] +qed-. + +lemma lleq_fqus_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, Tâ¦ â* â¦G2, K2, Uâ¦ â + âL1. L1 â[T, 0] L2 â + ââK1. â¦G1, L1, Tâ¦ â* â¦G2, K1, Uâ¦ & K1 â[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen â¦ H) -H +[ #H elim (lleq_fqup_trans â¦ H â¦ HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/ +| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_ldrop.etc new file mode 100644 index 000000000..9cb597c6f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_ldrop.etc @@ -0,0 +1,123 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â[#i, d] L2. +#L1 #L2 #d #i #Hid #HL12 @conj // -HL12 +#U @conj #H elim (cpys_inv_lref1 â¦ H) -H // * +#I #Z #Y #X #H elim (ylt_yle_false â¦ Hid â¦ H) +qed. + +lemma lleq_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â + K1 â[V, 0] K2 â L1 â[#i, d] L2. +#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ] +[ lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H1 + lapply (ldrop_fwd_length â¦ HLK2) -HLK2 #H2 + >H1 >H2 -H1 -H2 normalize // +| #U @conj #H elim (cpys_inv_lref1 â¦ H) -H // * + >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU + [ letin HLK â HLK1 | letin HLK â HLK2 ] + lapply (ldrop_mono â¦ H â¦ HLK) -H #H destruct elim (IH W) + /3 width=7 by cpys_subst_Y2/ +] +qed. + +lemma lleq_free: âL1,L2,d,i. |L1| â¤ i â |L2| â¤ i â |L1| = |L2| â L1 â[#i, d] L2. +#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12 +#U @conj #H elim (cpys_inv_lref1 â¦ H) -H // * +#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 â¦ 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: âK1,K2,T,dt. K1 â[T, dt] K2 â + âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âU. â§[d, e] T â¡ U â dt â¤ d â L1 â[U, dt] L2. +#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd +lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) +#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0 +[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ] +elim (cpys_inv_lift1_be â¦ HU0 â¦ HLKA â¦ 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: âK1,K2,T,dt. K1 â[T, dt] K2 â + âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âU. â§[d, e] T â¡ U â d â¤ dt â L1 â[U, dt+e] L2. +#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt +lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) +#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0 +[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ] +elim (cpys_inv_lift1_ge â¦ HU0 â¦ HLKA â¦ 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: âL1,L2,U,dt. L1 â[U, dt] L2 â + âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âT. â§[d, e] T â¡ U â dt â¤ d â K1 â[T, dt] K2. +#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd +lapply (ldrop_fwd_length_minus2 â¦ HLK1) lapply (ldrop_fwd_length_minus2 â¦ 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 â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ] +lapply (cpys_lift_be â¦ HT0 â¦ HLKA â¦ HTU â¦ HTU0) // -HT0 +>yplus_Y1 #HU0 elim (cpys_inv_lift1_be â¦ (H0 HU0) â¦ HLKB â¦ HTU) // -L1 -L2 -U -Hdtd +#X #HT0 #HX lapply (lift_inj â¦ HX â¦ HTU0) -U0 // +qed-. + +lemma lleq_inv_lift_ge: âL1,L2,U,dt. L1 â[U, dt] L2 â + âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âT. â§[d, e] T â¡ U â yinj d + e â¤ dt â K1 â[T, dt-e] K2. +#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt +lapply (ldrop_fwd_length_minus2 â¦ HLK1) lapply (ldrop_fwd_length_minus2 â¦ HLK2) +#H2 #H1 @conj // -HL12 -H1 -H2 +elim (yle_inv_plus_inj2 â¦ Hdedt) #Hddt #Hedt +#T0 elim (lift_total T0 d e) +#U0 #HTU0 elim (IH U0) -IH +#H12 #H21 @conj #HT0 +[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ] +lapply (cpys_lift_ge â¦ HT0 â¦ HLKA â¦ HTU â¦ HTU0) // -HT0 -Hddt +>ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge â¦ (H0 HU0) â¦ HLKB â¦ HTU) // -L1 -L2 -U -Hdedt -Hedt +#X #HT0 #HX lapply (lift_inj â¦ HX â¦ HTU0) -U0 // +qed-. + +lemma lleq_inv_lift_be: âL1,L2,U,dt. L1 â[U, dt] L2 â + âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âT. â§[d, e] T â¡ U â d â¤ dt â dt â¤ yinj d + e â K1 â[T, d] K2. +#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde +lapply (ldrop_fwd_length_minus2 â¦ HLK1) lapply (ldrop_fwd_length_minus2 â¦ 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 â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ] +lapply (cpys_lift_ge â¦ HT0 â¦ HLKA â¦ HTU â¦ HTU0) // -HT0 +#HU0 lapply (cpys_weak â¦ HU0 dt (â) ? ?) // -HU0 +#HU0 lapply (H0 HU0) +#HU0 lapply (cpys_weak â¦ HU0 d (â) ? ?) // -HU0 +#HU0 elim (cpys_inv_lift1_ge_up â¦ HU0 â¦ HLKB â¦ HTU) // -L1 -L2 -U -Hddt -Hdtde +#X #HT0 #HX lapply (lift_inj â¦ HX â¦ HTU0) -U0 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_lleq.etc new file mode 100644 index 000000000..2a8873d9b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lleq_alt/lleq_lleq.etc @@ -0,0 +1,175 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. âd:ynat. âi:nat. L1 â[#i, d] L2 â + â¨â¨ |L1| â¤ i â§ |L2| â¤ i + | yinj i < d + | ââI1,I2,K1,K2,V. â©[i] L1 â¡ K1.â{I1}V & + â©[i] L2 â¡ K2.â{I2}V & + K1 â[V, yinj 0] K2 & d â¤ 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 â¦ Hi) #I1 #K1 #V1 #HLK1 +elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2 +lapply (ldrop_fwd_length_minus2 â¦ HLK2) #H +lapply (ldrop_fwd_length_minus2 â¦ HLK1) >HL12 yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/ +qed-. + +lemma lleq_fwd_lref_dx: âL1,L2,d,i. L1 â[#i, d] L2 â + âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â + i < d â¨ + ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2 & d â¤ i. +#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref â¦ H) -H [ * || * ] +[ #_ #H elim (lt_refl_false i) + lapply (ldrop_fwd_length_lt2 â¦ 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 â¦ HLK22 â¦ HLK2) -L2 + #H destruct /3 width=5 by ex3_2_intro, or_intror/ +] +qed-. + +lemma lleq_fwd_lref_sn: âL1,L2,d,i. L1 â[#i, d] L2 â + âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â + i < d â¨ + ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2 & d â¤ i. +#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 â¦ d â¦ 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: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â + âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â + ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2. +#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx â¦ H â¦ HLK2) -L2 +[ #H elim (ylt_yle_false â¦ H Hdi) +| * /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lleq_inv_lref_ge_sn: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â + âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â + ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2. +#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 â¦ Hdi â¦ HLK1) -Hdi -HLK1 +/3 width=4 by lleq_sym, ex2_2_intro/ +qed-. + +lemma lleq_inv_lref_ge_gen: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â + âI1,I2,K1,K2,V1,V2. + â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â + V1 = V2 â§ K1 â[V2, 0] K2. +#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 +elim (lleq_inv_lref_ge_sn â¦ HL12 â¦ HLK1) // -L1 -d +#J #Y #HY lapply (ldrop_mono â¦ HY â¦ HLK2) -L2 -i #H destruct /2 width=1 by conj/ +qed-. + +lemma lleq_inv_lref_ge: âL1,L2,d,i. L1 â[#i, d] L2 â d â¤ i â + âI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â + K1 â[V, 0] K2. +#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2 +elim (lleq_inv_lref_ge_gen â¦ HL12 â¦ HLK1 HLK2) // +qed-. + +(* Advanced properties ******************************************************) + +lemma lleq_dec: âT,L1,L2,d. Decidable (L1 â[T, d] L2). +#T #L1 @(f2_ind â¦ rfw â¦ 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 â¦ HiL2) #I2 #K2 #V2 #HLK2 + elim (ldrop_O1_lt â¦ HiL1) #I1 #K1 #V1 #HLK1 + elim (eq_term_dec V2 V1) + [ #H3 elim (IH K1 V1 â¦ K2 0) destruct + /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/ + ] + -IH #H3 @or_intror + #H elim (lleq_fwd_lref â¦ 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 â¦ HLY1 â¦ HLK1) -HLY1 -HLK1 + lapply (ldrop_mono â¦ HLY2 â¦ 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 â¦ L2 d) /2 width=1 by/ + elim (IH (L1.â{I}V) T â¦ (L2.â{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/ + #H1 #H2 @or_intror + #H elim (lleq_inv_bind â¦ H) -H /2 width=1 by/ +| #I #V #T #Hn #L2 #d destruct + elim (IH L1 V â¦ L2 d) /2 width=1 by/ + elim (IH L1 T â¦ L2 d) -IH /3 width=1 by or_introl, lleq_flat/ + #H1 #H2 @or_intror + #H elim (lleq_inv_flat â¦ H) -H /2 width=1 by/ +] +-n /4 width=3 by lleq_fwd_length, or_intror/ +qed-. + +(* Main properties **********************************************************) + +theorem lleq_trans: âd,T. Transitive â¦ (lleq d T). +#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/ +qed-. + +theorem lleq_canc_sn: âL,L1,L2,T,d. L â[d, T] L1â L â[d, T] L2 â L1 â[d, T] L2. +/3 width=3 by lleq_trans, lleq_sym/ qed-. + +theorem lleq_canc_dx: âL1,L2,L,T,d. L1 â[d, T] L â L2 â[d, T] L â L1 â[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: âa,I,L1,L2,V,T,d. (L1 â[â{a,I}V.T, d] L2 â â¥) â + (L1 â[V, d] L2 â â¥) â¨ (L1.â{I}V â[T, â«¯d] L2.â{I}V â â¥). +#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: âI,L1,L2,V,T,d. (L1 â[â{I}V.T, d] L2 â â¥) â + (L1 â[V, d] L2 â â¥) â¨ (L1 â[T, d] L2 â â¥). +#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: âd,T,L1,L. L1â[T, d] L â + âL2. (L â[T, d] L2 â â¥) â (L1 â[T, d] L2 â â¥). +/3 width=3 by lleq_canc_sn/ qed-. +works with /4 width=8/ so lleq_canc_sn is more convenient +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/lazynegatedeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/lazynegatedeq_4.etc new file mode 100644 index 000000000..6b0cb85e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/lazynegatedeq_4.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( L1 â§£ break [ term 46 T , break term 46 d ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LazyNegatedEq $T $d $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/lazynegatedeqalt_4.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/lazynegatedeqalt_4.etc new file mode 100644 index 000000000..b150f63ff --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/lazynegatedeqalt_4.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( L1 â§£ â§£ break [ term 46 T , break term 46 d ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LazyNegatedEqAlt $T $d $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq.etc new file mode 100644 index 000000000..d41a4e83c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â + Î»d,T,L1,L2. |L1| = |L2| â§ (L1 â[T, d] L2 â â¥). + +interpretation + "negated lazy equivalence (local environment)" + 'LazyNegatedEq T d L1 L2 = (llneq d T L1 L2). diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq_alt.etc new file mode 100644 index 000000000..00eb2c34e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq_alt.etc @@ -0,0 +1,68 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| llneqa_neq: âI1,I2,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â + â©[i]L1 â¡ K1.â{I1}V1 â â©[i]L2 â¡ K2.â{I2}V2 â + |K1| = |K2| â (V1 = V2 â â¥) â llneqa d (#i) L1 L2 +| llneqa_eq : âI1,I2,L1,L2,K1,K2,V,d,i. d â¤ yinj i â + â©[i]L1 â¡ K1.â{I1}V â â©[i]L2 â¡ K2.â{I2}V â + llneqa 0 (V) K1 K2 â llneqa d (#i) L1 L2 +| llneqa_bind_sn: âa,I,L1,L2,V,T,d. + llneqa d V L1 L2 â llneqa d (â{a,I}V.T) L1 L2 +| llneqa_bind_dx: âa,I,L1,L2,V,T,d. + llneqa (â«¯d) T (L1.â{I}V) (L2.â{I}V) â llneqa d (â{a,I}V.T) L1 L2 +| llneqa_flat_sn: âI,L1,L2,V,T,d. + llneqa d V L1 L2 â llneqa d (â{I}V.T) L1 L2 +| llneqa_flat_dx: âI,L1,L2,V,T,d. + llneqa d T L1 L2 â llneqa d (â{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: âT,L1,L2,d. L1 â§£[T, d] L2 â L1 â§£â§£[T, d] L2. +#T #L1 @(f2_ind â¦ rfw â¦ 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 â¦ HiL1) #I1 #K1 #V1 #HLK1 + elim (ldrop_O1_lt L2 i) /2 width=1 by/ #I2 #K2 #V2 #HLK2 + lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) normalize + elim (eq_term_dec V1 V2) #HnV12 destruct + [2: #H @(llneqa_neq â¦ HLK1 â¦ 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 â¦ HLK1 â¦ 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 â¦ 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 â¦ H) -H + /5 width=1 by llneqa_flat_dx, llneqa_flat_sn, conj/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq_ext.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq_ext.etc new file mode 100644 index 000000000..235d869c2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llneq/llneq_ext.etc @@ -0,0 +1,33 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2,T,d. L1 â§£â§£[T, d] L2 â |L1| â¤ d â â¥. +#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 â¦ HLK1) -HLK1 normalize + #H lapply (yle_trans â¦ H â¦ 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 â¦ HLK1) -HLK1 normalize + #H lapply (yle_trans â¦ H â¦ Hdi) -d + /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/ +| #a #I #L1 (cprs_inv_sort1 â¦ H1) -T #H2 +lapply (cprs_inv_sort1 â¦ H2) -L #H destruct // +qed-. + +lemma cpcs_inv_abst1: âa,G,L,W1,T1,T. â¦G, Lâ¦ â¢ â{a}W1.T1 â¬* T â + ââW2,T2. â¦G, Lâ¦ â¢ T â¡* â{a}W2.T2 & â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* â{a}W2.T2. +#a #G #L #W1 #T1 #T #H +elim (cpcs_inv_cprs â¦ H) -H #X #H1 #H2 +elim (cprs_inv_abst1 â¦ H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct +/3 width=6 by cprs_bind, ex2_2_intro/ +qed-. + +lemma cpcs_inv_abst2: âa,G,L,W1,T1,T. â¦G, Lâ¦ â¢ T â¬* â{a}W1.T1 â + ââW2,T2. â¦G, Lâ¦ â¢ T â¡* â{a}W2.T2 & â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* â{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: âa,G,L,W,T,k. â¦G, Lâ¦ â¢ âk â¬* â{a}W.T â â¥. +#a #G #L #W #T #k #H +elim (cpcs_inv_cprs â¦ H) -H #X #H1 +>(cprs_inv_sort1 â¦ H1) -X #H2 +elim (cprs_inv_abst1 â¦ H2) -H2 #W0 #T0 #_ #_ #H destruct +qed-. + +(* Basic_1: was: pc3_gen_lift *) +lemma cpcs_inv_lift: âG,L,K,s,d,e. â©[s, d, e] L â¡ K â + âT1,U1. â§[d, e] T1 â¡ U1 â âT2,U2. â§[d, e] T2 â¡ U2 â + â¦G, Lâ¦ â¢ U1 â¬* U2 â â¦G, Kâ¦ â¢ T1 â¬* T2. +#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12 +elim (cpcs_inv_cprs â¦ HU12) -HU12 #U #HU1 #HU2 +elim (cprs_inv_lift1 â¦ HU1 â¦ HLK â¦ HTU1) -U1 #T #HTU #HT1 +elim (cprs_inv_lift1 â¦ HU2 â¦ HLK â¦ HTU2) -L -U2 #X #HXU +>(lift_inj â¦ HXU â¦ HTU) -X -U -d -e /2 width=3 by cprs_div/ +qed-. + +(* Advanced properties ******************************************************) + +lemma llpr_cpcs_trans: âG,L1,L2,T1,T2. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â â¦G, L1â¦ â¢ â¡[T2, 0] L2 â + â¦G, L2â¦ â¢ T1 â¬* T2 â â¦G, L1â¦ â¢ T1 â¬* T2. +#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs â¦ H) -H +/4 width=5 by cprs_div, cprs_llpr_trans/ +qed-. + +lemma llprs_cpcs_trans: âG,L1,L2,T1,T2. â¦G, L1â¦ â¢ â¡*[T1, 0] L2 â â¦G, L1â¦ â¢ â¡*[T2, 0] L2 â + â¦G, L2â¦ â¢ T1 â¬* T2 â â¦G, L1â¦ â¢ T1 â¬* T2. +#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs â¦ H) -H +/4 width=5 by cprs_div, llprs_cprs_trans/ +qed-. + +lemma cpr_cprs_conf_cpcs: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¡* T1 â â¦G, Lâ¦ â¢ T â¡ T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. +#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip â¦ HT1 â¦ HT2) -HT1 -HT2 +/2 width=3 by cpr_cprs_div/ +qed-. + +lemma cprs_cpr_conf_cpcs: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¡* T1 â â¦G, Lâ¦ â¢ T â¡ T2 â â¦G, Lâ¦ â¢ T2 â¬* T1. +#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip â¦ HT1 â¦ HT2) -HT1 -HT2 +/2 width=3 by cprs_cpr_div/ +qed-. + +lemma cprs_conf_cpcs: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¡* T1 â â¦G, Lâ¦ â¢ T â¡* T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. +#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf â¦ HT1 â¦ 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: âG,L1,L2,T1. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â + âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â â¦G, L2â¦ â¢ T1 â¬* T2. +#G #L1 #L2 #T1 #HL12 #T2 #HT12 elim (cprs_llpr_conf_dx â¦ HT12 â¦ 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: âG,L1,L2,T1. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â + âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â â¦G, L2â¦ â¢ T1 â¬* T2. +/3 width=5 by llpr_cprs_conf, cpr_cprs/ qed-. + +(* Basic_1: was only: pc3_thin_dx *) +lemma cpcs_flat: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¬* V2 â âT1,T2. â¦G, Lâ¦ â¢ T1 â¬* T2 â + âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¬* â{I}V2.T2. +#G #L #V1 #V2 #HV12 #T1 #T2 #HT12 +elim (cpcs_inv_cprs â¦ HV12) -HV12 +elim (cpcs_inv_cprs â¦ HT12) -HT12 +/3 width=5 by cprs_flat, cprs_div/ +qed. + +lemma cpcs_flat_dx_cpr_rev: âG,L,V1,V2. â¦G, Lâ¦ â¢ V2 â¡ V1 â âT1,T2. â¦G, Lâ¦ â¢ T1 â¬* T2 â + âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¬* â{I}V2.T2. +/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed. + +lemma cpcs_bind_dx: âa,I,G,L,V,T1,T2. â¦G, L.â{I}Vâ¦ â¢ T1 â¬* T2 â + â¦G, Lâ¦ â¢ â{a,I}V.T1 â¬* â{a,I}V.T2. +#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs â¦ HT12) -HT12 +/3 width=5 by cprs_div, cprs_bind/ +qed. + +lemma cpcs_bind_sn: âa,I,G,L,V1,V2,T. â¦G, Lâ¦ â¢ V1 â¬* V2 â â¦G, Lâ¦ â¢ â{a,I}V1. T â¬* â{a,I}V2. T. +#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs â¦ HV12) -HV12 +/3 width=5 by cprs_div, cprs_bind/ +qed. + +lemma lsubr_cpcs_trans: âG,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¬* T2 â + âL2. L2 â L1 â â¦G, L2â¦ â¢ T1 â¬* T2. +#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs â¦ HT12) -HT12 +/3 width=5 by cprs_div, lsubr_cprs_trans/ +qed-. + +(* Basic_1: was: pc3_lift *) +lemma cpcs_lift: âG,L,K,s,d,e. â©[s, d, e] L â¡ K â + âT1,U1. â§[d, e] T1 â¡ U1 â âT2,U2. â§[d, e] T2 â¡ U2 â + â¦G, Kâ¦ â¢ T1 â¬* T2 â â¦G, Lâ¦ â¢ U1 â¬* U2. +#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12 +elim (cpcs_inv_cprs â¦ HT12) -HT12 #T #HT1 #HT2 +elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/ +qed. + +lemma cpcs_strip: âG,L,T1,T. â¦G, Lâ¦ â¢ T â¬* T1 â âT2. â¦G, Lâ¦ â¢ T â¬ T2 â + ââT0. â¦G, Lâ¦ â¢ T1 â¬ T0 & â¦G, Lâ¦ â¢ T2 â¬* T0. +#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-. + +(* More inversion lemmas ****************************************************) + +axiom cpcs_inv_abst_sn: âa1,a2,G,L,W1,W2,T1,T2. â¦G, Lâ¦ â¢ â{a1}W1.T1 â¬* â{a2}W2.T2 â + â§â§ â¦G, Lâ¦ â¢ W1 â¬* W2 & â¦G, L.âW1â¦ â¢ T1 â¬* T2 & a1 = a2. +(* +#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H +elim (cpcs_inv_cprs â¦ H) -H #T #H1 #H2 +elim (cprs_inv_abst1 â¦ H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct +elim (cprs_inv_abst1 â¦ H2) -H2 #W #T #HW2 #HT2 #H destruct +lapply (llprs_cprs_conf â¦ (L.âW) â¦ HT2) /2 width=1 by llprs_pair/ -HT2 #HT2 +lapply (llprs_cpcs_trans â¦ (L.âW1) â¦ 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: âa1,a2,G,L,W1,W2,T1,T2. â¦G, Lâ¦ â¢ â{a1}W1.T1 â¬* â{a2}W2.T2 â + â§â§ â¦G, Lâ¦ â¢ W1 â¬* W2 & â¦G, L.âW2â¦ â¢ T1 â¬* T2 & a1 = a2. +#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym â¦ HT12) -HT12 +#HT12 elim (cpcs_inv_abst_sn â¦ HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/ +qed-. + +(* Main properties **********************************************************) + +(* Basic_1: was pc3_t *) +theorem cpcs_trans: âG,L,T1,T. â¦G, Lâ¦ â¢ T1 â¬* T â âT2. â¦G, Lâ¦ â¢ T â¬* T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. +#G #L #T1 #T #HT1 #T2 @(trans_TC â¦ HT1) qed-. + +theorem cpcs_canc_sn: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T â¬* T1 â â¦G, Lâ¦ â¢ T â¬* T2 â â¦G, Lâ¦ â¢ T1 â¬* T2. +/3 width=3 by cpcs_trans, cpcs_sym/ qed-. + +theorem cpcs_canc_dx: âG,L,T,T1,T2. â¦G, Lâ¦ â¢ T1 â¬* T â â¦G, Lâ¦ â¢ T2 â¬* T â â¦G, Lâ¦ â¢ T1 â¬* T2. +/3 width=3 by cpcs_trans, cpcs_sym/ qed-. + +lemma cpcs_bind1: âa,I,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¬* V2 â + âT1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¬* T2 â + â¦G, Lâ¦ â¢ â{a,I}V1. T1 â¬* â{a,I}V2. T2. +/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed. + +lemma cpcs_bind2: âa,I,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¬* V2 â + âT1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¬* T2 â + â¦G, Lâ¦ â¢ â{a,I}V1. T1 â¬* â{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: âG,L1,L2,T1,T2. â¦G, L1â¦ â¢ â¡[T1, 0] L2 â â¦G, L1â¦ â¢ â¡[T2, 0] L2 â + â¦G, L1â¦ â¢ T1 â¬* T2 â â¦G, L2â¦ â¢ T1 â¬* T2. +#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs â¦ H) -H +/3 width=5 by cpcs_canc_dx, llpr_cprs_conf/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/cprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/cprs_cprs.etc new file mode 100644 index 000000000..a85488cdf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/cprs_cprs.etc @@ -0,0 +1,166 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L. Transitive â¦ (cprs G L). +normalize /2 width=3 by trans_TC/ qed-. + +(* Basic_1: was: pr3_confluence *) +(* Basic_1: includes: pr1_confluence *) +theorem cprs_conf: âG,L. confluent2 â¦ (cprs G L) (cprs G L). +normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-. + +theorem cprs_bind: âa,I,G,L,V1,V2,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ V1 â¡* V2 â + â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{a,I}V2.T2. +#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind â¦ H) -V2 +/3 width=5 by cprs_trans, cprs_bind_dx/ +qed. + +(* Basic_1: was: pr3_flat *) +theorem cprs_flat: âI,G,L,V1,V2,T1,T2. â¦G, Lâ¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ V1 â¡* V2 â + â¦G, Lâ¦ â¢ â{I}V1.T1 â¡* â{I}V2.T2. +#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind â¦ H) -V2 +/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/ +qed. + +theorem cprs_beta_rc: âa,G,L,V1,V2,W1,W2,T1,T2. + â¦G, Lâ¦ â¢ V1 â¡ V2 â â¦G, L.âW1â¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ W1 â¡* W2 â + â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}âW2.V2.T2. +#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind â¦ H) -W2 /2 width=1 by cprs_beta_dx/ +#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *) +@(cprs_trans â¦ IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/ +qed. + +theorem cprs_beta: âa,G,L,V1,V2,W1,W2,T1,T2. + â¦G, L.âW1â¦ â¢ T1 â¡* T2 â â¦G, Lâ¦ â¢ W1 â¡* W2 â â¦G, Lâ¦ â¢ V1 â¡* V2 â + â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}âW2.V2.T2. +#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind â¦ H) -V2 /2 width=1 by cprs_beta_rc/ +#V #V2 #_ #HV2 #IHV1 +@(cprs_trans â¦ IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/ +qed. + +theorem cprs_theta_rc: âa,G,L,V1,V,V2,W1,W2,T1,T2. + â¦G, Lâ¦ â¢ V1 â¡ V â â§[0, 1] V â¡ V2 â â¦G, L.âW1â¦ â¢ T1 â¡* T2 â + â¦G, Lâ¦ â¢ W1 â¡* W2 â â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}W2.âV2.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: âa,G,L,V1,V,V2,W1,W2,T1,T2. + â§[0, 1] V â¡ V2 â â¦G, Lâ¦ â¢ W1 â¡* W2 â â¦G, L.âW1â¦ â¢ T1 â¡* T2 â + â¦G, Lâ¦ â¢ V1 â¡* V â â¦G, Lâ¦ â¢ âV1.â{a}W1.T1 â¡* â{a}W2.âV2.T2. +#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx â¦ 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: âG,L,V1,T1,U2. â¦G, Lâ¦ â¢ âV1.T1 â¡* U2 â + â¨â¨ ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡* V2 & â¦G, Lâ¦ â¢ T1 â¡* T2 & + U2 = âV2. T2 + | ââa,W,T. â¦G, Lâ¦ â¢ T1 â¡* â{a}W.T & + â¦G, Lâ¦ â¢ â{a}âW.V1.T â¡* U2 + | ââa,V0,V2,V,T. â¦G, Lâ¦ â¢ V1 â¡* V0 & â§[0,1] V0 â¡ V2 & + â¦G, Lâ¦ â¢ T1 â¡* â{a}V.T & + â¦G, Lâ¦ â¢ â{a}V.âV2.T â¡* U2. +#G #L #V1 #T1 #U2 #H @(cprs_ind â¦ H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/ +#U #U2 #_ #HU2 * * +[ #V0 #T0 #HV10 #HT10 #H destruct + elim (cpr_inv_appl1 â¦ 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 â¦ HV10 â¦ HV02) -V0 #HV12 + lapply (lsubr_cpr_trans â¦ HT2 (L.ââW.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 â¦ 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: âG. s_r_transitive â¦ (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 â¦ HL12 â¦ 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 â¦ HL12) -HL12 /4 width=1 by cprs_bind/ +| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12 + elim (llpr_inv_flat â¦ HL12) -HL12 /3 width=1 by cprs_flat/ +| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12 + elim (llpr_inv_bind_O â¦ HL12) /3 width=3 by cprs_zeta/ +| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12 + elim (llpr_inv_flat â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 + elim (llpr_inv_bind_O â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 + elim (llpr_inv_bind_O â¦ HL12) /3 width=3 by cprs_theta/ +] +qed-. + +lemma cpr_bind2: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡ V2 â âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡ T2 â + âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{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: âG. s_rs_transitive â¦ (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: âG,L. confluent2 â¦ (cprs G L) (cpr G L). +normalize /4 width=3 by cpr_conf, TC_strip1/ qed-. + +lemma cprs_llpr_conf_dx: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡* T1 â âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â + ââT. â¦G, L1â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. +#G #L0 #T0 #T1 #H @(cprs_ind_dx â¦ T0 H) -T0 /2 width=3 by ex2_intro/ +#T0 #T #HT0 #_ #IHT1 #L1 #HL01 +elim (IHT1 â¦ L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2 +elim (llpr_cpr_conf_dx â¦ HT0 â¦ HL01) -L0 #T3 #HT03 #HT3 +elim (cprs_strip â¦ HT2 â¦ HT3) -T +/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/ +qed-. + +lemma cprs_llpr_conf_sn: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡* T1 â + âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â + ââT. â¦G, L0â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. +#G #L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cprs_llpr_conf_dx â¦ HT01 â¦ HL01) +/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/ +qed-. + +lemma cprs_bind2_dx: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡ V2 â + âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡* T2 â + âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{a,I}V2.T2. +/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/cprs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/cprs_lift.etc new file mode 100644 index 000000000..27f0bb6e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/cprs_lift.etc @@ -0,0 +1,63 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,K,V,V2,i. + â©[i] L â¡ K.âV â â¦G, Kâ¦ â¢ V â¡* V2 â + âW2. â§[0, i + 1] V2 â¡ W2 â â¦G, Lâ¦ â¢ #i â¡* 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 â¦ HLK) -HLK #HLK +elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/ +qed. + +lemma cprs_llpr_conf: âG. s_r_confluent1 â¦ (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: âG,L,T2,i. â¦G, Lâ¦ â¢ #i â¡* T2 â + T2 = #i â¨ + ââK,V1,T1. â©[i] L â¡ K.âV1 & â¦G, Kâ¦ â¢ V1 â¡* T1 & + â§[0, i + 1] T1 â¡ T2. +#G #L #T2 #i #H @(cprs_ind â¦ H) -T2 /2 width=1 by or_introl/ +#T #T2 #_ #HT2 * +[ #H destruct + elim (cpr_inv_lref1 â¦ 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 â¦ HLK) #H0LK + elim (cpr_inv_lift1 â¦ HT2 â¦ H0LK â¦ 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: âG. l_liftable (cprs G). +/3 width=10 by l_liftable_LTC, cpr_lift/ qed. + +(* Basic_1: was: pr3_gen_lift *) +lemma cprs_inv_lift1: âG. l_deliftable_sn (cprs G). +/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/lazypredsn_5.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/lazypredsn_5.etc new file mode 100644 index 000000000..01ce586f8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/lazypredsn_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡ break [ term 46 T , break term 46 d ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LazyPRedSn $G $L1 $L2 $T $d }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/lazypredsnstar_5.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/lazypredsnstar_5.etc new file mode 100644 index 000000000..824933fba --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/lazypredsnstar_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡* break [ term 46 T , break term 46 d ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LazyPRedSnStar $G $L1 $L2 $T $d }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr.etc new file mode 100644 index 000000000..e4c1a01be --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr.etc @@ -0,0 +1,51 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â relation4 ynat term lenv lenv â Î»G. 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: âI,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[â{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡[V, d] L2 â§ â¦G, L1â¦ â¢ â¡[T, d] L2. +/2 width=2 by llpx_sn_inv_flat/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma llpr_fwd_length: âG,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[T, d] L2 â |L1| = |L2|. +/2 width=4 by llpx_sn_fwd_length/ qed-. + +(* Basic properties *********************************************************) + +lemma llpr_lref: âI,G,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I}V1 â â©[i] L2 â¡ K2.â{I}V2 â + â¦G, K1â¦ â¢ â¡[V1, 0] K2 â â¦G, K1â¦ â¢ V1 â¡ V2 â â¦G, L1â¦ â¢ â¡[#i, d] L2. +/2 width=9 by llpx_sn_lref/ qed. + +(* Note: lemma 250 *) +lemma llpr_refl: âG,T,d. reflexive â¦ (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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr_ldrop.etc new file mode 100644 index 000000000..74f7cc6b3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr_ldrop.etc @@ -0,0 +1,97 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[#i, d] L2 â d â¤ i â + âI,K2,V2. â©[i] L2 â¡ K2.â{I}V2 â + ââK1,V1. â©[i] L1 â¡ K1.â{I}V1 & + â¦G, K1â¦ â¢ â¡[V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡ V2. +/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-. + +lemma llpr_inv_lref_ge_sn: âG,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[#i, d] L2 â d â¤ i â + âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â + ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & + â¦G, K1â¦ â¢ â¡[V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡ V2. +/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-. + +lemma llpr_inv_lref_ge_bi: âG,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[#i, d] L2 â d â¤ i â + âI1,I2,K1,K2,V1,V2. + â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â + â§â§ I1 = I2 & â¦G, K1â¦ â¢ â¡[V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡ V2. +/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. + +lemma llpr_inv_bind_O: âa,I,G,L1,L2,V,T. â¦G, L1â¦ â¢ â¡ [â{a,I}V.T, 0] L2 â + â¦G, L1â¦ â¢ â¡[V, 0] L2 â§ â¦G, L1.â{I}Vâ¦ â¢ â¡[T, 0] L2.â{I}V. +/2 width=2 by llpx_sn_inv_bind_O/ qed-. + +lemma llpr_bind_repl_O: âI,G,L1,L2,V1,V2,T. â¦G, L1.â{I}V1â¦ â¢ â¡[T, 0] L2.â{I}V2 â + âJ,W1,W2. â¦G, L1â¦ â¢ â¡[W1, 0] L2 â â¦G, L1â¦ â¢ W1 â¡ W2 â â¦G, L1.â{J}W1â¦ â¢ â¡[T, 0] L2.â{J}W2. +/2 width=4 by llpx_sn_bind_repl_O/ qed-. + +(* Advanced properties ******************************************************) + +lemma llpr_ssta_conf: âh,g,G. s_r_confluent1 â¦ (ssta h g G) (llpr G 0). +/3 width=10 by ssta_llpx_sn_conf, cpr_lift/ qed-. + +lemma llpr_cpr_conf: âG. s_r_confluent1 â¦ (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: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â + âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â + ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, Lâ¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ â â¦G2, L2, U2â¦. +#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 â¦ HU2 â¦ HLK â¦ HUT â¦ HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/ +qed-. + +lemma fquq_cpr_trans_dx: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â + âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â + ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, Lâ¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ ââ¸® â¦G2, L2, U2â¦. +#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen â¦ H) -H +[ #HT12 elim (fqu_cpr_trans_dx â¦ HT12 â¦ 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: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â + âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â + ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, L1â¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ â â¦G2, L2, U2â¦. +#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 â¦ HU2 â¦ HLK â¦ HUT â¦ HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/ +qed-. + +lemma fquq_cpr_trans_sn: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â + âU2. â¦G2, L2â¦ â¢ T2 â¡ U2 â + ââL,U1. â¦G1, L1â¦ â¢ â¡[T1, 0] L & â¦G1, L1â¦ â¢ T1 â¡ U1 & â¦G1, L, U1â¦ ââ¸® â¦G2, L2, U2â¦. +#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen â¦ H) -H +[ #HT12 elim (fqu_cpr_trans_sn â¦ HT12 â¦ HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/ +| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr_llpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr_llpr.etc new file mode 100644 index 000000000..4e5b9e2be --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpr_llpr.etc @@ -0,0 +1,375 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: + âI,G,L1,L2. ââT. â¦G, L1â¦ â¢ âª{I} â¡ T & â¦G, L2â¦ â¢ âª{I} â¡ T. +/2 width=3 by cpr_atom, ex2_intro/ qed-. + +fact cpr_conf_llpr_atom_delta: + âG,L0,i. ( + âL,T. â¦G, L0, #iâ¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âK0,V0. â©[i] L0 â¡ K0.âV0 â + âV2. â¦G, K0â¦ â¢ V0 â¡ V2 â âT2. â§[O, i + 1] V2 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[#i, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[#i, 0] L2 â + ââT. â¦G, L1â¦ â¢ #i â¡ T & â¦G, L2â¦ â¢ T2 â¡ T. +#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +elim (llpr_inv_lref_ge_sn â¦ HL01 â¦ HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01 +elim (llpr_inv_lref_ge_sn â¦ HL02 â¦ HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_ +lapply (ldrop_fwd_drop2 â¦ HLK2) -W2 #HLK2 +lapply (fqup_lref â¦ G â¦ HLK0) -HLK0 #HLK0 +elim (IH â¦ HLK0 â¦ HV01 â¦ HV02 â¦ HK01 â¦ 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: + âG,L0,i. ( + âL,T. â¦G, L0, #iâ¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âK0,V0. â©[i] L0 â¡ K0.âV0 â + âV1. â¦G, K0â¦ â¢ V0 â¡ V1 â âT1. â§[O, i + 1] V1 â¡ T1 â + âKX,VX. â©[i] L0 â¡ KX.âVX â + âV2. â¦G, KXâ¦ â¢ VX â¡ V2 â âT2. â§[O, i + 1] V2 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[#i, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[#i, 0] L2 â + ââT. â¦G, L1â¦ â¢ T1 â¡ T & â¦G, L2â¦ â¢ T2 â¡ 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 â¦ H â¦ HLK0) -H #H destruct +elim (llpr_inv_lref_ge_sn â¦ HL01 â¦ HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_ +lapply (ldrop_fwd_drop2 â¦ HLK1) -W1 #HLK1 +elim (llpr_inv_lref_ge_sn â¦ HL02 â¦ HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_ +lapply (ldrop_fwd_drop2 â¦ HLK2) -W2 #HLK2 +lapply (fqup_lref â¦ G â¦ HLK0) -HLK0 #HLK0 +elim (IH â¦ HLK0 â¦ HV01 â¦ HV02 â¦ HK01 â¦ 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: + âa,I,G,L0,V0,T0. ( + âL,T. â¦G, L0, â{a,I}V0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0.â{I}V0â¦ â¢ T0 â¡ T1 â + âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âT2. â¦G, L0.â{I}V0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[â{a,I}V0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[â{a,I}V0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ â{a,I}V1.T1 â¡ T & â¦G, L2â¦ â¢ â{a,I}V2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #H1T0 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2V0 #H2T0 +elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) // +elim (IH â¦ HT01 â¦ HT02 (L1.â{I}V1) â¦ (L2.â{I}V2)) -IH +/3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/ +qed-. + +fact cpr_conf_llpr_bind_zeta: + âG,L0,V0,T0. ( + âL,T. â¦G, L0, +âV0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0.âV0â¦ â¢ T0 â¡ T1 â + âT2. â¦G, L0.âV0â¦ â¢ T0 â¡ T2 â âX2. â§[O, 1] X2 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ +âV1.T1 â¡ T & â¦G, L2â¦ â¢ X2 â¡ T. +#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1V0 #H1T0 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2V0 #H2T0 +elim (IH â¦ HT01 â¦ HT02 (L1.âV1) â¦ (L2.âV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2 +elim (cpr_inv_lift1 â¦ HT2 L2 â¦ HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/ +qed-. + +fact cpr_conf_llpr_zeta_zeta: + âG,L0,V0,T0. ( + âL,T. â¦G, L0, +âV0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âT1. â¦G, L0.âV0â¦ â¢ T0 â¡ T1 â âX1. â§[O, 1] X1 â¡ T1 â + âT2. â¦G, L0.âV0â¦ â¢ T0 â¡ T2 â âX2. â§[O, 1] X2 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[+âV0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ X1 â¡ T & â¦G, L2â¦ â¢ X2 â¡ T. +#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1V0 #H1T0 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2V0 #H2T0 +elim (IH â¦ HT01 â¦ HT02 (L1.âV0) â¦ (L2.âV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2 +elim (cpr_inv_lift1 â¦ HT1 L1 â¦ HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1 +elim (cpr_inv_lift1 â¦ HT2 L2 â¦ HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2 +lapply (lift_inj â¦ HT2 â¦ HT1) -T #H destruct /2 width=3 by ex2_intro/ +qed-. + +fact cpr_conf_llpr_flat_flat: + âI,G,L0,V0,T0. ( + âL,T. â¦G, L0, â{I}V0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0â¦ â¢ T0 â¡ T1 â + âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âT2. â¦G, L0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[â{I}V0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[â{I}V0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ â{I}V1.T1 â¡ T & â¦G, L2â¦ â¢ â{I}V2.T2 â¡ T. +#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (llpr_inv_flat â¦ HL01) -HL01 #H1V0 #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #H2T0 +elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) // +elim (IH â¦ HT01 â¦ HT02 â¦ H1T0 â¦ H2T0) /3 width=5 by cpr_flat, ex2_intro/ +qed-. + +fact cpr_conf_llpr_flat_tau: + âG,L0,V0,T0. ( + âL,T. â¦G, L0, âV0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â âT2. â¦G, L0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ âV1.T1 â¡ T & â¦G, L2â¦ â¢ T2 â¡ T. +#G #L0 #V0 #T0 #IH #V1 #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (llpr_inv_flat â¦ HL01) -HL01 #_ #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #_ #H2T0 +elim (IH â¦ HT01 â¦ HT02 â¦ H1T0 â¦ H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/ +qed-. + +fact cpr_conf_llpr_tau_tau: + âG,L0,V0,T0. ( + âL,T. â¦G, L0, âV0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âT1. â¦G, L0â¦ â¢ T0 â¡ T1 â âT2. â¦G, L0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ T1 â¡ T & â¦G, L2â¦ â¢ T2 â¡ T. +#G #L0 #V0 #T0 #IH #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (llpr_inv_flat â¦ HL01) -HL01 #_ #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #_ #H2T0 +elim (IH â¦ HT01 â¦ HT02 â¦ H1T0 â¦ H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/ +qed-. + +fact cpr_conf_llpr_flat_beta: + âa,G,L0,V0,W0,T0. ( + âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0â¦ â¢ â{a}W0.T0 â¡ T1 â + âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ âV1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}âW2.V2.T2 â¡ 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 â¦ H) -H #W1 #T1 #HW01 #HT01 #H destruct +elim (llpr_inv_flat â¦ HL01) -HL01 #H1V0 #HL01 +elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 +elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1 by/ #W #HW1 #HW2 +elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (lsubr_cpr_trans â¦ HT2 (L2.ââW2.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: + âa,G,L0,V0,W0,T0. ( + âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âT1. â¦G, L0â¦ â¢ â{a}W0.T0 â¡ T1 â + âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âU2. â§[O, 1] V2 â¡ U2 â + âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ âV1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}W2.âU2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #HL01 +elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 +elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (lift_total V 0 1) #U #HVU +lapply (cpr_lift â¦ HV2 (L2.âW2) â¦ HVU2 â¦ HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2 +elim (cpr_inv_abbr1 â¦ H) -H * +[ #W1 #T1 #HW01 #HT01 #H destruct + elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1 by/ + elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /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 â¦ HT01 â¦ HT02 (L1.âW2) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 + elim (cpr_inv_lift1 â¦ HT1 L1 â¦ HXT1) -HXT1 + /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/ +] +qed-. + +fact cpr_conf_llpr_beta_beta: + âa,G,L0,V0,W0,T0. ( + âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âW1. â¦G, L0â¦ â¢ W0 â¡ W1 â âT1. â¦G, L0.âW0â¦ â¢ T0 â¡ T1 â + âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ â{a}âW1.V1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}âW2.V2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #HL01 +elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 +elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1/ #W #HW1 #HW2 +elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (lsubr_cpr_trans â¦ HT1 (L1.ââW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/ +lapply (lsubr_cpr_trans â¦ HT2 (L2.ââW2.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: + âa,G,L0,V0,W0,T0. ( + âL,T. â¦G, L0, âV0.â{a}W0.T0â¦ â+ â¦G, L, Tâ¦ â + âT1. â¦G, Lâ¦ â¢ T â¡ T1 â âT2. â¦G, Lâ¦ â¢ T â¡ T2 â + âL1. â¦G, Lâ¦ â¢ â¡[T, 0] L1 â âL2. â¦G, Lâ¦ â¢ â¡[T, 0] L2 â + ââT0. â¦G, L1â¦ â¢ T1 â¡ T0 & â¦G, L2â¦ â¢ T2 â¡ T0 + ) â + âV1. â¦G, L0â¦ â¢ V0 â¡ V1 â âU1. â§[O, 1] V1 â¡ U1 â + âW1. â¦G, L0â¦ â¢ W0 â¡ W1 â âT1. â¦G, L0.âW0â¦ â¢ T0 â¡ T1 â + âV2. â¦G, L0â¦ â¢ V0 â¡ V2 â âU2. â§[O, 1] V2 â¡ U2 â + âW2. â¦G, L0â¦ â¢ W0 â¡ W2 â âT2. â¦G, L0.âW0â¦ â¢ T0 â¡ T2 â + âL1. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L1 â âL2. â¦G, L0â¦ â¢ â¡[âV0.â{a}W0.T0, 0] L2 â + ââT. â¦G, L1â¦ â¢ â{a}W1.âU1.T1 â¡ T & â¦G, L2â¦ â¢ â{a}W2.âU2.T2 â¡ 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 â¦ HL01) -HL01 #H1V0 #HL01 +elim (llpr_inv_bind_O â¦ HL01) -HL01 #H1W0 #H1T0 +elim (llpr_inv_flat â¦ HL02) -HL02 #H2V0 #HL02 +elim (llpr_inv_bind_O â¦ HL02) -HL02 #H2W0 #H2T0 +elim (IH â¦ HV01 â¦ HV02 â¦ H1V0 â¦ H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (IH â¦ HW01 â¦ HW02 â¦ H1W0 â¦ H2W0) /2 width=1 by/ +elim (IH â¦ HT01 â¦ HT02 (L1.âW1) â¦ (L2.âW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 +elim (lift_total V 0 1) #U #HVU +lapply (cpr_lift â¦ HV1 (L1.âW1) â¦ HVU1 â¦ HVU) -HVU1 /2 width=2 by ldrop_drop/ +lapply (cpr_lift â¦ HV2 (L2.âW2) â¦ HVU2 â¦ 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: âG. llpx_sn_confluent2 (cpr G) (cpr G). +#G #L0 #T0 @(fqup_wf_ind_eq â¦ 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 â¦ H1) -H1 + elim (cpr_inv_atom1 â¦ 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 â¦ H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #T1 #HT01 #HXT1 #H11 #H12 + ] + elim (cpr_inv_bind1 â¦ 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 â¦ 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 â¦ 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: âG,L. confluent â¦ (cpr G L). +/2 width=6 by cpr_conf_llpr/ qed-. + +(* Properties on context-sensitive parallel reduction for terms *************) + +lemma llpr_cpr_conf_dx: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â + ââT. â¦G, L1â¦ â¢ T0 â¡ T & â¦G, L1â¦ â¢ T1 â¡ T. +#G #L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_conf_llpr â¦ HT01 T0 â¦ HL01 â¦ HL01) /2 width=3 by ex2_intro/ +qed-. + +lemma llpr_cpr_conf_sn: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â âL1. â¦G, L0â¦ â¢ â¡[T0, 0] L1 â + ââT. â¦G, L1â¦ â¢ T0 â¡ T & â¦G, L0â¦ â¢ T1 â¡ T. +#G #L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_conf_llpr â¦ HT01 T0 â¦ L0 â¦ HL01) /2 width=3 by ex2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs.etc new file mode 100644 index 000000000..d3c2a6f27 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs.etc @@ -0,0 +1,61 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â relation4 ynat term lenv lenv â + Î»G,d. LTC â¦ (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: âG,L1,T,d. âR:predicate lenv. R L1 â + (âL,L2. â¦G, L1â¦ â¢ â¡*[T, d] L â â¦G, Lâ¦ â¢ â¡[T, d] L2 â R L â R L2) â + âL2. â¦G, L1â¦ â¢ â¡*[T, d] L2 â R L2. +#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12 +@(TC_star_ind â¦ HL1 IHL1 â¦ HL12) // +qed-. + +lemma llprs_ind_dx: âG,L2,T,d. âR:predicate lenv. R L2 â + (âL1,L. â¦G, L1â¦ â¢ â¡[T, d] L â â¦G, Lâ¦ â¢ â¡*[T, d] L2 â R L â R L1) â + âL1. â¦G, L1â¦ â¢ â¡*[T, d] L2 â R L1. +#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12 +@(TC_star_ind_dx â¦ HL2 IHL2 â¦ HL12) // +qed-. + +(* Basic properties *********************************************************) + +lemma lpr_llprs: âG,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[T, d] L2 â â¦G, L1â¦ â¢ â¡*[T, d] L2. +/2 width=1 by inj/ qed. + +lemma llprs_refl: âG,L,T,d. â¦G, Lâ¦ â¢ â¡*[T, d] L. +/2 width=1 by lpr_llprs/ qed. + +lemma llprs_strap1: âG,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡*[T, d] L â â¦G, Lâ¦ â¢ â¡[T, d] L2 â â¦G, L1â¦ â¢ â¡*[T, d] L2. +normalize /2 width=3 by step/ qed-. + +lemma llprs_strap2: âG,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡[T, d] L â â¦G, Lâ¦ â¢ â¡*[T, d] L2 â â¦G, L1â¦ â¢ â¡*[T, d] L2. +normalize /2 width=3 by TC_strap/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma llprs_fwd_length: âG,L1,L2,T,d. â¦G, L1â¦ â¢ â¡*[T, d] L2 â |L1| = |L2|. +#G #L1 #L2 #T #d #H @(llprs_ind â¦ H) -L2 +/3 width=6 by llpr_fwd_length, trans_eq/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs_cprs.etc new file mode 100644 index 000000000..b9bf0f963 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs_cprs.etc @@ -0,0 +1,122 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡* V2 â + âI,T. â¦G, L.â{I}V1â¦ â¢ â¡*[T, 0] L.â{I}V2. +/2 width=1 by llpx_sn_TC_pair_dx/ qed. + +(* Properties on context-sensitive parallel computation for terms ***********) + +lemma llprs_cpr_trans: âG. s_r_transitive â¦ (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: âG. s_rs_transitive â¦ (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: âG2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡* U2 â + âL0. â¦G2, L0â¦ â¢ â¡*[T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[U2, O] L2. +#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx â¦ H) -L0 // +#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 â¦ IHL2) -IHL2 +lapply (llprs_cprs_trans â¦ HTU2 â¦ 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: âG2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡ U2 â + âL0. â¦G2, L0â¦ â¢ â¡*[T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[U2, O] L2. +#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx â¦ H) -L0 // +#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 â¦ IHL2) -IHL2 +lapply (llprs_cpr_trans â¦ HTU2 â¦ HL2) -L2 #HTU2 +/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/ +qed-. + +lemma llprs_cprs_conf_sn: âG,L0,L1,T0. â¦G, L0â¦ â¢ â¡*[T0, 0] L1 â + âT1. â¦G, L0â¦ â¢ T0 â¡* T1 â + ââT. â¦G, L0â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. +#G #L0 #L1 #T0 #H @(llprs_ind_dx â¦ H) -L0 /2 width=3 by ex2_intro/ +#L0 #L #HL0 #HL1 #IHL1 #T1 #HT01 elim (cprs_llpr_conf_sn â¦ HT01 â¦ HL0) +#T2 #HT12 #HT02 elim (IHL1 â¦ HT02) -IHL1 -HT02 +lapply (cprs_trans â¦ HT01 â¦ HT12) #HT02 +lapply (cprs_llpr_conf â¦ HT02 â¦ HL0) -HT02 -HL0 +/4 width=5 by cprs_llpr_trans, cprs_trans, ex2_intro/ +qed-. + +lemma llprs_cpr_conf_sn: âG,L0,T0,T1. â¦G, L0â¦ â¢ T0 â¡ T1 â + âL1. â¦G, L0â¦ â¢ â¡*[T0, 0] L1 â + ââT. â¦G, L0â¦ â¢ T1 â¡* T & â¦G, L1â¦ â¢ T0 â¡* T. +/3 width=3 by llprs_cprs_conf_sn, cpr_cprs/ qed-. + +lemma cprs_bind2: âG,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡* V2 â + âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡* T2 â + âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡* â{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: âa,G,L,W1,T1,U2. â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* U2 â + ââW2,T2. â¦G, Lâ¦ â¢ W1 â¡* W2 & â¦G, L.âW1â¦ â¢ T1 â¡* T2 & + U2 = â{a}W2.T2. +#a #G #L #V1 #T1 #U2 #H @(cprs_ind â¦ H) -U2 /2 width=5 by ex3_2_intro/ +#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct +elim (cpr_inv_abst1 â¦ HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct +lapply (llprs_cpr_trans â¦ HT02 (L.âV1) ?) +/3 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro/ +qed-. + +lemma cprs_inv_abst: âa,G,L,W1,W2,T1,T2. â¦G, Lâ¦ â¢ â{a}W1.T1 â¡* â{a}W2.T2 â + â¦G, Lâ¦ â¢ W1 â¡* W2 â§ â¦G, L.âW1â¦ â¢ T1 â¡* T2. +#a #G #L #W1 #W2 #T1 #T2 #H +elim (cprs_inv_abst1 â¦ H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/ +qed-. + +(* Basic_1: was pr3_gen_abbr *) +lemma cprs_inv_abbr1: âa,G,L,V1,T1,U2. â¦G, Lâ¦ â¢ â{a}V1.T1 â¡* U2 â ( + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡* V2 & â¦G, L.âV1â¦ â¢ T1 â¡* T2 & + U2 = â{a}V2.T2 + ) â¨ + ââT2. â¦G, L.âV1â¦ â¢ T1 â¡* T2 & â§[0, 1] U2 â¡ T2 & a = true. +#a #G #L #V1 #T1 #U2 #H @(cprs_ind â¦ H) -U2 /3 width=5 by ex3_2_intro, or_introl/ +#U0 #U2 #_ #HU02 * * +[ #V0 #T0 #HV10 #HT10 #H destruct + elim (cpr_inv_abbr1 â¦ HU02) -HU02 * + [ #V2 #T2 #HV02 #HT02 #H destruct + lapply (llprs_cpr_trans â¦ HT02 (L.âV1) ?) + /4 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/ + | #T2 #HT02 #HUT2 + lapply (llprs_cpr_trans â¦ HT02 (L.âV1) ?) -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 â¦ HU02 (L.âV1) â¦ HU01 â¦ 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs_llprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs_llprs.etc new file mode 100644 index 000000000..4f97672f3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs_llprs.etc @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âG,T,d. Transitive â¦ (llprs G d T). +normalize /2 width=3 by trans_TC/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpx_sn_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpx_sn_llpx_sn.etc new file mode 100644 index 000000000..0df9690ee --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llpx_sn_llpx_sn.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 (lenvârelation term) â Î»R1,R2. + âL0,T0,T1. R1 L0 T0 T1 â âT2. R2 L0 T0 T2 â + âL1. llpx_sn R1 0 T0 L0 L1 â âL2. llpx_sn R2 0 T0 L0 L2 â + ââT. R2 L1 T1 T & R1 L2 T2 T. + +(* Note: we miss llpx_sn_conf and derivatives: lpr_conf lprs_conf *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/cpxs_llpx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/cpxs_llpx.etc new file mode 100644 index 000000000..bec908ef7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/cpxs_llpx.etc @@ -0,0 +1,53 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G. s_r_confluent1 â¦ (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: âh,g,G. s_r_transitive â¦ (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 â¦ HL12 â¦ 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 â¦ HL12) -HL12 /4 width=1 by cpxs_bind/ +| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12 + elim (llpx_inv_flat â¦ HL12) -HL12 /3 width=1 by cpxs_flat/ +| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12 + elim (llpx_inv_bind_O â¦ HL12) /3 width=3 by cpxs_zeta/ +| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12 + elim (llpx_inv_flat â¦ HL12) /3 width=1 by cpxs_tau/ +| #G #L2 #V1 #V2 #T2 #HV12 #IHV12 #L1 #HL12 + elim (llpx_inv_flat â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 + elim (llpx_inv_bind_O â¦ 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 â¦ HL12) -HL12 #HV1 #HL12 + elim (llpx_inv_bind_O â¦ HL12) /3 width=3 by cpxs_theta/ +] +qed-. + +lemma llpx_cpxs_trans: âh,g,G. s_rs_transitive â¦ (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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/csx_llpx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/csx_llpx.etc new file mode 100644 index 000000000..c14492f21 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/csx_llpx.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T. â¦G, L1â¦ â¢ â¬*[h, g] T â + âL2. â¦G, L1â¦ â¢ â¡[h, g, T, 0] L2 â â¦G, L2â¦ â¢ â¬*[h, g] T. +#h #g #G #L1 #T #H @(csx_ind_alt â¦ H) -T +/5 width=3 by csx_intro_cpxs, llpx_cpxs_trans, cpxs_llpx_conf/ (* 2 cpxs_llpx_trans *) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/csx_llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/csx_llpxs.etc new file mode 100644 index 000000000..aad3e8bf0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/csx_llpxs.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T. â¦G, L1â¦ â¢ â¬*[h, g] T â + âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, 0] L2 â â¦G, L2â¦ â¢ â¬*[h, g] T. +#h #g #G #L1 #T #HT #L2 #H @(llpxs_ind â¦ H) -L2 /3 by llpxs_strap1, csx_llpx_conf/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/fpbs_lpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/fpbs_lpr.etc new file mode 100644 index 000000000..34b60ad1d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/fpbs_lpr.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2,T1,T2. â¦G, L1â¦ â¢ T1 â¡ T2 â â¦G, L1â¦ â¢ â¡ L2 â + â¦G, L1, T1â¦ â¥[h, g] â¦G, L2, T2â¦. +/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: âh,g,G,L1,L2,T1,T2,U2,l2. + â¦G, L1â¦ â¢ T1 â¡ T2 â â¦G, L1â¦ â¢ â¡ L2 â + â¦G, L2â¦ â¢ T2 âª[h, g] l2+1 â â¦G, L2â¦ â¢ T2 â¢[h, g] U2 â + â¦G, L1, T1â¦ â¥[h, g] â¦G, L2, U2â¦. +/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, ssta_cpx, fpb_cpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazypredsn_7.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazypredsn_7.etc new file mode 100644 index 000000000..2350f5efc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazypredsn_7.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡ 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazypredsnstar_7.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazypredsnstar_7.etc new file mode 100644 index 000000000..f8209e2f8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazypredsnstar_7.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L1 â¦ â¢ â¡* 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazysn_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazysn_6.etc new file mode 100644 index 000000000..0c52a6ea5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazysn_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( G â¢ â â¬ * 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazysnalt_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazysnalt_6.etc new file mode 100644 index 000000000..0976a3136 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/lazysnalt_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( G â¢ â â¬ â¬ * 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx.etc new file mode 100644 index 000000000..3276008b4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx.etc @@ -0,0 +1,69 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â genv â relation4 ynat term lenv lenv â + Î»h,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: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡[h, g, V, d] L2 â§ â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. +/2 width=2 by llpx_sn_inv_flat/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma llpx_fwd_length: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â |L1| = |L2|. +/2 width=4 by llpx_sn_fwd_length/ qed-. + +lemma llpx_fwd_flat_dx: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. +/2 width=3 by llpx_sn_fwd_flat_dx/ qed-. + +lemma llpx_fwd_pair_sn: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â¡{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡[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: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{a,I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡[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: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{a,I}V.T, d] L2 â + â¦G, L1.â{I}Vâ¦ â¢ â¡[h, g, T, â«¯d] L2.â{I}V. +/2 width=2 by llpx_sn_fwd_bind_dx/ qed-. + +(* Note: this might be removed *) +lemma llpx_fwd_flat_sn: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡[h, g, â{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡[h, g, V, d] L2. +/2 width=3 by llpx_sn_fwd_flat_sn/ qed-. + +(* Basic properties *********************************************************) + +lemma llpx_lref: âh,g,I,G,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I}V1 â â©[i] L2 â¡ K2.â{I}V2 â + â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 â â¦G, K1â¦ â¢ V1 â¡[h, g] V2 â â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2. +/2 width=9 by llpx_sn_lref/ qed. + +lemma llpx_refl: âh,g,G,T,d. reflexive â¦ (llpx h g G d T). +/2 width=1 by llpx_sn_refl/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_aaa.etc new file mode 100644 index 000000000..99eef8d55 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_aaa.etc @@ -0,0 +1,82 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T1,A. â¦G, L1â¦ â¢ T1 â A â + âT2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â + âL2. â¦G, L1â¦ â¢ â¡[h, g, T1, 0] L2 â â¦G, L2â¦ â¢ T2 â A. +#h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A +[ #g #L1 #k #X #H elim (cpx_inv_sort1 â¦ H) -H // * // +| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12 + elim (cpx_inv_lref1 â¦ H) -H + [ #H destruct + elim (llpx_inv_lref_ge_sn â¦ HL12 â¦ HLK1) -L1 /3 width=6 by aaa_lref/ + | * #J #Y #Z #V2 #H #HV12 #HV2 + lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct + elim (llpx_inv_lref_ge_sn â¦ HL12 â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 + elim (cpx_inv_abbr1 â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 + elim (cpx_inv_abst1 â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 + elim (cpx_inv_appl1 â¦ 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 â¦ HV12 â¦ HV1) -IHV1 -HV12 #HV2 + lapply (IHT1 (â{b}W2.U2) â¦ HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H + elim (aaa_inv_abst â¦ 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 â¦ B â¦ (L2.âW2) â¦ HV2) -HV2 /2 width=2 by ldrop_drop/ #HV2 + lapply (IHT1 (â{b}W2.U2) â¦ HT1) -IHT1 /2 width=1 by cpx_bind/ -L1 #H + elim (aaa_inv_abbr â¦ 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 â¦ HL12) -HL12 #HV1 #HT1 + elim (cpx_inv_cast1 â¦ 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: âh,g,G,L,T1,A. â¦G, Lâ¦ â¢ T1 â A â âT2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â â¦G, Lâ¦ â¢ T2 â A. +/2 width=7 by aaa_cpx_llpx_conf/ qed-. + +lemma aaa_llpx_conf: âh,g,G,L1,T,A. â¦G, L1â¦ â¢ T â A â âL2. â¦G, L1â¦ â¢ â¡[h, g, T, 0] L2 â â¦G, L2â¦ â¢ T â A. +/2 width=7 by aaa_cpx_llpx_conf/ qed-. + +lemma aaa_cpr_conf: âG,L,T1,A. â¦G, Lâ¦ â¢ T1 â A â âT2. â¦G, Lâ¦ â¢ T1 â¡ T2 â â¦G, Lâ¦ â¢ T2 â A. +/3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-. +(* +lamma aaa_llpr_conf: âG,L1,T,A. â¦G, L1â¦ â¢ T â A â âL2. â¦G, L1â¦ â¢ â¡[T, 0] L2 â â¦G, L2â¦ â¢ T â A. +/3 width=5 by aaa_llpx_conf, llpr_llpx/ qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_ldrop.etc new file mode 100644 index 000000000..3e4bcab8f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_ldrop.etc @@ -0,0 +1,90 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2 â d â¤ i â + âI,K2,V2. â©[i] L2 â¡ K2.â{I}V2 â + ââK1,V1. â©[i] L1 â¡ K1.â{I}V1 & + â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡[h, g] V2. +/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-. + +lemma llpx_inv_lref_ge_sn: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2 â d â¤ i â + âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â + ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & + â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡[h, g] V2. +/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-. + +lemma llpx_inv_lref_ge_bi: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡[h, g, #i, d] L2 â d â¤ i â + âI1,I2,K1,K2,V1,V2. + â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â + â§â§ I1 = I2 & â¦G, K1â¦ â¢ â¡[h, g, V1, 0] K2 & â¦G, K1â¦ â¢ V1 â¡[h, g] V2. +/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. + +lemma llpx_inv_bind_O: âh,g,a,I,G,L1,L2,V,T. â¦G, L1â¦ â¢ â¡ [h, g, â{a,I}V.T, 0] L2 â + â¦G, L1â¦ â¢ â¡[h, g, V, 0] L2 â§ â¦G, L1.â{I}Vâ¦ â¢ â¡[h, g, T, 0] L2.â{I}V. +/2 width=2 by llpx_sn_inv_bind_O/ qed-. + +lemma llpx_bind_repl_O: âh,g,I,G,L1,L2,V1,V2,T. â¦G, L1.â{I}V1â¦ â¢ â¡[h, g, T, 0] L2.â{I}V2 â + âJ,W1,W2. â¦G, L1â¦ â¢ â¡[h, g, W1, 0] L2 â â¦G, L1â¦ â¢ W1 â¡[h, g] W2 â â¦G, L1.â{J}W1â¦ â¢ â¡[h, g, T, 0] L2.â{J}W2. +/2 width=4 by llpx_sn_bind_repl_O/ qed-. + +(* Advanced forward lemmas **************************************************) + +lemma llpx_fwd_bind_O_dx: âh,g,a,I,G,L1,L2,V,T. â¦G, L1â¦ â¢ â¡[h, g, â{a,I}V.T, 0] L2 â + â¦G, L1.â{I}Vâ¦ â¢ â¡[h, g, T, 0] L2.â{I}V. +/2 width=3 by llpx_sn_fwd_bind_O_dx/ qed-. + +(* Advanced properties ******************************************************) + +lemma llpx_cpx_conf: âh,g,G. s_r_confluent1 â¦ (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: âh,g,G,L1,L2,U. â¦G, L1â¦ â¢ â¡[h, g, U, 0] L2 â + âK2,e. â©[e] L2 â¡ K2 â âT. â§[0, e] T â¡ U â + ââK1. â©[e] L1 â¡ K1 & â¦G, K1â¦ â¢ â¡[h, g, T, 0] K2. +/2 width=5 by llpx_sn_ldrop_trans_O/ qed-. + +(* Properties on supclosure *************************************************) + +lemma llpx_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡[h, g] T & â¦G1, K1, Tâ¦ â â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[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 â¦ 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 â¦ HU1 â¦ HLK1) -L1 + /3 width=5 by fqu_drop, ex3_2_intro/ +] +qed-. + +lemma llpx_fquq_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡[h, g] T & â¦G1, K1, Tâ¦ ââ¸® â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g, T2, 0] L2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen â¦ H) -H +[ #HT12 elim (llpx_fqu_trans â¦ HT12 â¦ HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/ +| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_lleq.etc new file mode 100644 index 000000000..e936cc531 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_lleq.etc @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. +/2 width=1 by llpx_sn_lrefl/ qed-. + +lemma lleq_llpx_trans: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â + âL.â¦G, L2â¦ â¢ â¡[h, g, T, d] L â â¦G, L1â¦ â¢ â¡[h, g, T, d] L. +/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_trans/ qed-. + +lemma lleq_llpx_conf: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â + âL.â¦G, L1â¦ â¢ â¡[h, g, T, d] L â â¦G, L2â¦ â¢ â¡[h, g, T, d] L. +/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_lpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_lpr.etc new file mode 100644 index 000000000..6ceb64b51 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_lpr.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L1â¦ â¢ â¡[h, g, T, d] L2. +/3 width=4 by cpr_cpx, lpx_sn_llpx_sn, llpx_sn_co/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs.etc new file mode 100644 index 000000000..750d0b5b1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs.etc @@ -0,0 +1,89 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â genv â relation4 ynat term lenv lenv â + Î»h,g,G,d. LTC â¦ (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: âh,g,G,L1,T,d. âR:predicate lenv. R L1 â + (âL,L2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡[h, g, T, d] L2 â R L â R L2) â + âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â R L2. +#h #g #G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12 +@(TC_star_ind â¦ HL1 IHL1 â¦ HL12) // +qed-. + +lemma llpxs_ind_dx: âh,g,G,L2,T,d. âR:predicate lenv. R L2 â + (âL1,L. â¦G, L1â¦ â¢ â¡[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L2 â R L â R L1) â + âL1. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â R L1. +#h #g #G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12 +@(TC_star_ind_dx â¦ HL2 IHL2 â¦ HL12) // +qed-. + +(* Basic properties *********************************************************) + +lemma llpx_llpxs: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. +normalize /2 width=1 by inj/ qed. + +lemma llpxs_refl: âh,g,G,L,T,d. â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L. +/2 width=1 by llpx_llpxs/ qed. + +lemma llpxs_strap1: âh,g,G,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. +normalize /2 width=3 by step/ qed. + +lemma llpxs_strap2: âh,g,G,L1,L,L2,T,d. â¦G, L1â¦ â¢ â¡[h, g, T, d] L â â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. +normalize /2 width=3 by TC_strap/ qed. + +(* Basic forward lemmas *****************************************************) + +lemma llpxs_fwd_length: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â |L1| = |L2|. +#h #g #G #L1 #L2 #T #d #H @(llpxs_ind â¦ H) -L2 +/3 width=8 by llpx_fwd_length, trans_eq/ +qed-. + +(* Note: this might be moved *) +lemma llpxs_fwd_bind_sn: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{a,I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡*[h, g, V, d] L2. +#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 +/3 width=6 by llpx_fwd_bind_sn, llpxs_strap1/ +qed-. + +(* Note: this might be moved *) +lemma llpxs_fwd_bind_dx: âh,g,a,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{a,I}V.T, d] L2 â + â¦G, L1.â{I}Vâ¦ â¢ â¡*[h, g, T, â«¯d] L2.â{I}V. +#h #g #a #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 +/3 width=6 by llpx_fwd_bind_dx, llpxs_strap1/ +qed-. + +(* Note: this might be moved *) +lemma llpxs_fwd_flat_sn: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡*[h, g, V, d] L2. +#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 +/3 width=6 by llpx_fwd_flat_sn, llpxs_strap1/ +qed-. + +(* Note: this might be moved *) +lemma llpxs_fwd_flat_dx: âh,g,I,G,L1,L2,V,T,d. â¦G, L1â¦ â¢ â¡*[h, g, â{I}V.T, d] L2 â + â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. +#h #g #I #G #L1 #L2 #V #T #d #H @(llpxs_ind â¦ H) -L2 +/3 width=6 by llpx_fwd_flat_dx, llpxs_strap1/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_aaa.etc new file mode 100644 index 000000000..ededb03a2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_aaa.etc @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T,A. â¦G, L1â¦ â¢ T â A â + âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, 0] L2 â â¦G, L2â¦ â¢ T â A. +#h #g #G #L1 #T #A #HT #L2 #HL12 +@(TC_Conf3 â¦ (Î»L,A. â¦G, Lâ¦ â¢ T â A) â¦ HT ? HL12) /2 width=5 by aaa_llpx_conf/ +qed-. +(* +lamma aaa_llprs_conf: âG,L1,T,A. â¦G, L1â¦ â¢ T â A â + âL2. â¦G, L1â¦ â¢ â¡*[T, 0] L2 â â¦G, L2â¦ â¢ T â A. +/3 width=5 by aaa_llpxs_conf, llprs_llpxs/ qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_cpxs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_cpxs.etc new file mode 100644 index 000000000..35472d7e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_cpxs.etc @@ -0,0 +1,172 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 â + âI,T. â¦G, L.â{I}V1â¦ â¢ â¡*[h, g, T, 0] L.â{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: âh,g,G. s_r_transitive â¦ (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: âh,g,G. s_rs_transitive â¦ (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: âh,g,G2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡*[h, g] U2 â + âL0. â¦G2, L0â¦ â¢ â¡*[h, g, T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[h, g, U2, O] L2. +#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx â¦ H) -L0 // +#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 â¦ IHL2) -IHL2 +lapply (llpxs_cpxs_trans â¦ HTU2 â¦ 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: âh,g,G2,L2,T2,U2. â¦G2, L2â¦ â¢ T2 â¡[h, g] U2 â + âL0. â¦G2, L0â¦ â¢ â¡*[h, g, T2, O] L2 â â¦G2, L0â¦ â¢ â¡*[h, g, U2, O] L2. +#h #g #G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llpxs_ind_dx â¦ H) -L0 // +#L0 #L #HL0 #HL2 #IHL2 @(llpxs_strap2 â¦ IHL2) -IHL2 +lapply (llpxs_cpx_trans â¦ HTU2 â¦ HL2) -L2 #HTU2 +/3 width=3 by llpx_cpxs_trans, cpxs_llpx_conf/ +qed-. + +lemma cpxs_bind2: âh,g,G,L,V1,V2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 â + âI,T1,T2. â¦G, L.â{I}V2â¦ â¢ T1 â¡*[h, g] T2 â + âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¡*[h, g] â{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: âh,g,G,L1,L2,d,i. â¦G, L1â¦ â¢ â¡*[h, g, #i, d] L2 â d â¤ i â + âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â + ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & + â¦G, K1â¦ â¢ â¡*[h, g, V2, 0] K2 & â¦G, K1â¦ â¢ V1 â¡*[h, g] V2. +#h #g #G #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 @(llpxs_ind â¦ H) -L2 /2 width=5 by ex3_2_intro/ -HLK1 +#L #L2 #_ #HL2 * #K #V #HLK #HK1 #HV1 elim (llpx_inv_lref_ge_sn â¦ HL2 â¦ HLK) // -HL2 -HLK -Hdi +#K2 #V2 #HLK2 #HK2 #HV2 +@(ex3_2_intro â¦ 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: âh,g,a,G,L,V1,T1,U2. â¦G, Lâ¦ â¢ â{a}V1.T1 â¡*[h, g] U2 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 & â¦G, L.âV1â¦ â¢ T1 â¡*[h, g] T2 & + U2 = â{a}V2.T2. +#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind â¦ H) -U2 /2 width=5 by ex3_2_intro/ +#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct +elim (cpx_inv_abst1 â¦ HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct +lapply (llpxs_cpx_trans â¦ HT02 (L.âV1) ?) +/3 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro/ +qed-. + +lemma cpxs_inv_abbr1: âh,g,a,G,L,V1,T1,U2. â¦G, Lâ¦ â¢ â{a}V1.T1 â¡*[h, g] U2 â ( + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¡*[h, g] V2 & â¦G, L.âV1â¦ â¢ T1 â¡*[h, g] T2 & + U2 = â{a}V2.T2 + ) â¨ + ââT2. â¦G, L.âV1â¦ â¢ T1 â¡*[h, g] T2 & â§[0, 1] U2 â¡ T2 & a = true. +#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind â¦ H) -U2 /3 width=5 by ex3_2_intro, or_introl/ +#U0 #U2 #_ #HU02 * * +[ #V0 #T0 #HV10 #HT10 #H destruct + elim (cpx_inv_abbr1 â¦ HU02) -HU02 * + [ #V2 #T2 #HV02 #HT02 #H destruct + lapply (llpxs_cpx_trans â¦ HT02 (L.âV1) ?) + /4 width=5 by llpxs_pair_dx, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/ + | #T2 #HT02 #HUT2 + lapply (llpxs_cpx_trans â¦ HT02 (L.âV1) ?) -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: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â+ â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g, T2, 0] L2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind â¦ H) -G2 -L2 -T2 +[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (llpx_fqu_trans â¦ H12 â¦ 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 â¦ HLK1) -L1 + #L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fqu_trans â¦ H2 â¦ HL0) -L + #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans â¦ HT0 â¦ HT3) -T + /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/ +] +qed-. + +lemma llpx_fqus_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â* â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â* â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡[h, g, T2, 0] L2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind â¦ H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/ +#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 â¦ HLK1) -L1 +#L0 #T0 #HT10 #HT0 #HL0 elim (llpx_fquq_trans â¦ H2 â¦ HL0) -L +#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans â¦ HT0 â¦ HT3) -T +/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/ +qed-. + +lemma llpxs_fquq_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ ââ¸® â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡*[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ ââ¸® â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡*[h, g, T2, 0] L2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx â¦ H) -K1 +[ /2 width=5 by ex3_2_intro/ +| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12 + lapply (llpx_cpxs_trans â¦ HT1 â¦ HK1) -HT1 #HT1 + lapply (cpxs_llpx_conf â¦ HT1 â¦ HK1) -HK1 #HK1 + elim (llpx_fquq_trans â¦ HT2 â¦ HK1) -K + /3 width=7 by llpxs_strap2, cpxs_strap1, ex3_2_intro/ +] +qed-. + +lemma llpxs_fqup_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â+ â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡*[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â+ â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡*[h, g, T2, 0] L2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx â¦ H) -K1 +[ /2 width=5 by ex3_2_intro/ +| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12 + lapply (llpx_cpxs_trans â¦ HT1 â¦ HK1) -HT1 #HT1 + lapply (cpxs_llpx_conf â¦ HT1 â¦ HK1) -HK1 #HK1 + elim (llpx_fqup_trans â¦ HT2 â¦ HK1) -K + /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/ +] +qed-. + +lemma llpxs_fqus_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1â¦ â* â¦G2, L2, T2â¦ â + âK1. â¦G1, K1â¦ â¢ â¡*[h, g, T1, 0] L1 â + ââK2,T. â¦G1, K1â¦ â¢ T1 â¡*[h, g] T & â¦G1, K1, Tâ¦ â* â¦G2, K2, T2â¦ & â¦G2, K2â¦ â¢ â¡*[h, g, T2, 0] L2. +#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(llpxs_ind_dx â¦ H) -K1 +[ /2 width=5 by ex3_2_intro/ +| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12 + lapply (llpx_cpxs_trans â¦ HT1 â¦ HK1) -HT1 #HT1 + lapply (cpxs_llpx_conf â¦ HT1 â¦ HK1) -HK1 #HK1 + elim (llpx_fqus_trans â¦ HT2 â¦ HK1) -K + /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_lleq.etc new file mode 100644 index 000000000..a38ba395a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_lleq.etc @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2,T,d. L1 â[T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. +/3 width=1 by llpx_lrefl, llpx_llpxs/ qed-. + +lemma lleq_llpxs_trans: âh,g,G,L2,L,T,d. â¦G, L2â¦ â¢ â¡*[h, g, T, d] L â + âL1. L1 â[T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L. +#h #g #G #L2 #L #T #d #H @(llpxs_ind â¦ H) -L +/3 width=3 by llpxs_strap1, llpxs_lrefl/ +qed-. + +lemma lleq_llpxs_conf: âh,g,G,L1,L,T,d. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L â + âL2. L1 â[T, d] L2 â â¦G, L2â¦ â¢ â¡*[h, g, T, d] L. +/3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-. +(* +foct leq_lpxs_trans_lleq_aux: âh,g,G,L1,L0,d,e. L1 â[d, e] L0 â e = â â + âL2. â¦G, L0â¦ â¢ â¡*[h, g] L2 â + ââL. L â[d, e] L2 & â¦G, L1â¦ â¢ â¡*[h, g] L & + (âT. L0 â[T, d] L2 â L1 â[T, d] L). +#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e +[ #d #e #_ #L2 #H >(lpxs_inv_atom1 â¦ 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 â¦ H) -H #L2 #V2 #HL02 #HV02 #H destruct + lapply (ysucc_inv_Y_dx â¦ He) -He #He + elim (IHL10 â¦ HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH + @(ex3_intro â¦ (L.â{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/ + #T elim (IH T) #HL0dx #HL0sn + @conj #H @(lleq_leq_repl â¦ 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 â¦ H) -H #L2 #V2 #HL02 #HV02 #H destruct + elim (IHL10 â¦ HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH + @(ex3_intro â¦ (L.â{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/ + #T elim (IH T) #HL0dx #HL0sn + @conj #H @(lleq_leq_repl â¦ H) -H /3 width=1 by leq_sym, leq_succ/ +] +qed-. + +lamma leq_lpxs_trans_lleq: âh,g,G,L1,L0,d. L1 â[d, â] L0 â + âL2. â¦G, L0â¦ â¢ â¡*[h, g] L2 â + ââL. L â[d, â] L2 & â¦G, L1â¦ â¢ â¡*[h, g] L & + (âT. L0 â[T, d] L2 â L1 â[T, d] L). +/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_llpxs.etc new file mode 100644 index 000000000..d316c5207 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_llpxs.etc @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,T,d. Transitive â¦ (llpxs h g G d T). +normalize /2 width=3 by trans_TC/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_lprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_lprs.etc new file mode 100644 index 000000000..32d7bbc38 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpxs_lprs.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,L2,T,d. â¦G, L1â¦ â¢ â¡* L2 â â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2. +normalize /3 width=3 by lpr_llpx, monotonic_TC/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx.etc new file mode 100644 index 000000000..286d0645f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx.etc @@ -0,0 +1,57 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â relation4 ynat term genv lenv â + Î»h,g,d,T,G. SN â¦ (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: âh,g,G,T,d. âR:predicate lenv. + (âL1. G â¢ ââ¬*[h, g, T, d] L1 â + (âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â R L2) â + R L1 + ) â + âL. G â¢ ââ¬*[h, g, T, d] L â 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: âh,g,G,L1,T,d. + (âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬*[h, g, T, d] L2) â + G â¢ ââ¬*[h, g, T, d] L1. +/5 width=1 by lleq_sym, SN_intro/ qed. + +lemma llsx_sort: âh,g,G,L,d,k. G â¢ ââ¬*[h, g, âk, 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: âh,g,G,L,d,p. G â¢ ââ¬*[h, g, Â§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. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_alt.etc new file mode 100644 index 000000000..0d23439b9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_alt.etc @@ -0,0 +1,107 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh. sd h â relation4 ynat term genv lenv â + Î»h,g,d,T,G. SN â¦ (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: âh,g,G,T,d. âR:predicate lenv. + (âL1. G â¢ ââ¬â¬*[h, g, T, d] L1 â + (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â R L2) â + R L1 + ) â + âL. G â¢ ââ¬â¬*[h, g, T, d] L â 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: âh,g,G,L1,T,d. + (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬â¬*[h, g, T, d] L2) â + G â¢ ââ¬â¬*[h, g, T, d] L1. +/5 width=1 by lleq_sym, SN_intro/ qed. + +fact llsxa_intro_aux: âh,g,G,L1,T,d. + (âL,L2. â¦G, Lâ¦ â¢ â¡*[h, g, T, d] L2 â L1 â[T, d] L â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬â¬*[h, g, T, d] L2) â + G â¢ ââ¬â¬*[h, g, T, d] L1. +/4 width=3 by llsxa_intro/ qed-. + +lemma llsxa_llpxs_trans: âh,g,G,L1,T,d. G â¢ ââ¬â¬*[h, g, T, d] L1 â + âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â G â¢ ââ¬â¬*[h, g, T, d] L2. +#h #g #G #L1 #T #d #H @(llsxa_ind â¦ 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: âh,g,G,L1,T,d. + (âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬â¬*[h, g, T, d] L2) â + G â¢ ââ¬â¬*[h, g, T, d] L1. +#h #g #G #L1 #T #d #IH @llsxa_intro_aux +#L #L2 #H @(llpxs_ind_dx â¦ 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: âh,g,G,L,T,d. G â¢ ââ¬*[h, g, T, d] L â G â¢ ââ¬â¬*[h, g, T, d] L. +#h #g #G #L #T #d #H @(llsx_ind â¦ H) -L +/4 width=1 by llsxa_intro_llpx/ +qed. + +(* Main inversion lemmas ****************************************************) + +theorem llsxa_inv_llsx: âh,g,G,L,T,d. G â¢ ââ¬â¬*[h, g, T, d] L â G â¢ ââ¬*[h, g, T, d] L. +#h #g #G #L #T #d #H @(llsxa_ind â¦ H) -L +/4 width=1 by llsx_intro, llpx_llpxs/ +qed-. + +(* Advanced properties ******************************************************) + +lemma llsx_intro_alt: âh,g,G,L1,T,d. + (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â G â¢ ââ¬*[h, g, T, d] L2) â + G â¢ ââ¬*[h, g, T, d] L1. +/6 width=1 by llsxa_inv_llsx, llsx_llsxa, llsxa_intro/ qed. + +lemma llsx_llpxs_trans: âh,g,G,L1,T,d. G â¢ ââ¬*[h, g, T, d] L1 â + âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â G â¢ ââ¬*[h, g, T, d] L2. +/4 width=3 by llsxa_inv_llsx, llsx_llsxa, llsxa_llpxs_trans/ +qed-. + +(* Advanced eliminators *****************************************************) + +lemma llsx_ind_alt: âh,g,G,T,d. âR:predicate lenv. + (âL1. G â¢ ââ¬*[h, g, T, d] L1 â + (âL2. â¦G, L1â¦ â¢ â¡*[h, g, T, d] L2 â (L1 â[T, d] L2 â â¥) â R L2) â + R L1 + ) â + âL. G â¢ ââ¬*[h, g, T, d] L â R L. +#h #g #G #T #d #R #IH #L #H @(llsxa_ind h g G T d â¦ L) +/4 width=1 by llsxa_inv_llsx, llsx_llsxa/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_csx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_csx.etc new file mode 100644 index 000000000..9c284732c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_csx.etc @@ -0,0 +1,96 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T1,T2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â + âL2. â¦G, L1â¦â¢ â¡[h, g, T2, O] L2 â + ââL. â¦G, L1â¦â¢ â¡[h, g, T1, O] L & L â[T2, 0] L2. +(* +fact llsx_cpx_trans_aux: âh,g,G,L0,T1,T2. â¦G, L0â¦ â¢ T1 â¡[h, g] T2 â + âL1,d. G â¢ ââ¬*[h, g, T1, d] L1 â d = 0 â + L0 â[T1, d] L1 â âL2. L1 â[T2, d] L2 â + G â¢ ââ¬*[h, g, T2, d] L2. +#h #g #G #L0 #T1 #T2 #HT12 #L1 #d #H @(llsx_ind â¦ H) -L1 +#L1 #_ #IHL1 #Hd #He011 #L2 #He122 @llsx_intro +#L3 #Hx223 #Hn223 destruct +lapply (lleq_cpx_conf_sn â¦ HT12 â¦ He011) #He021 +lapply (lleq_cpx_conf â¦ HT12 â¦ He011) -HT12 #HT12 +lapply (lleq_llpx_trans â¦ He122 â¦ Hx223) -Hx223 #Hx123 +elim (cpx_llpx_trans â¦ HT12 â¦ Hx123) -Hx123 #L4 #Hx114 #He423 +(* lapply (lleq_cpx_conf â¦ Hx114 â¦ He011) #He120 *) +@(IHL1 â¦ Hx114) // -IHL1 +[ #HL13 @HnL2 -HnL2 +*) + +fact llsx_cpx_trans_aux: âh,g,G,L1,T1,d. G â¢ ââ¬*[h, g, T1, d] L1 â d = 0 â + âT2. â¦G, L1â¦ â¢ T1 â¡[h, g] T2 â + âL2. L1 â[T1, d] L2 â G â¢ ââ¬*[h, g, T2, 0] L2. +#h #g #G #L1 #T1 #d #H @(llsx_ind â¦ H) -L1 +#L1 #_ #IHL1 #Hd #T2 #HT12 #L2 #He112 @llsx_intro +#L3 #Hx223 #Hn223 destruct +lapply (lleq_cpx_conf_sn â¦ HT12 â¦ He112) #He122 +lapply (lleq_cpx_conf â¦ HT12 â¦ He112) -HT12 #HT12 +elim (cpx_llpx_trans â¦ HT12 â¦ Hx223) #L4 #Hx214 #He423 +@(IHL1 â¦ L4) // +*) +axiom llsx_cpx_trans_O: âh,g,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¡[h, g] T2 â + G â¢ ââ¬*[h, g, T1, 0] L â G â¢ ââ¬*[h, g, T2, 0] L. + +(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************) + +(* Advanced properties ******************************************************) + +lemma llsx_lref_be_lpxs: âh,g,I,G,K1,V,i,d. d â¤ yinj i â â¦G, K1â¦ â¢ â¬*[h, g] V â + âK2. G â¢ ââ¬*[h, g, V, 0] K2 â â¦G, K1â¦ â¢ â¡*[h, g, V, 0] K2 â + âL2. â©[i] L2 â¡ K2.â{I}V â G â¢ ââ¬*[h, g, #i, d] L2. +#h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt â¦ H) -V +#V0 #_ #IHV0 #K2 #H @(llsx_ind â¦ H) -K2 +#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @llsx_intro +#L2 #HL02 #HnL02 elim (llpx_inv_lref_ge_sn â¦ HL02 â¦ 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 â¦ HV02 â¦ HK02) -HK02 #HK02 + @(IHV0 â¦ HnV02 â¦ 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: âh,g,I,G,K,V,i,d. d â¤ yinj i â â¦G, Kâ¦ â¢ â¬*[h, g] V â + G â¢ ââ¬*[h, g, V, 0] K â + âL. â©[i] L â¡ K.â{I}V â G â¢ ââ¬*[h, g, #i, d] L. +/2 width=8 by llsx_lref_be_lpxs/ qed. + +(* Main properties **********************************************************) + +theorem csx_llsx: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¬*[h, g] T â âd. G â¢ ââ¬*[h, g, T, d] L. +#h #g #G #L #T @(fqup_wf_ind_eq â¦ 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 â¦ Hi) -Hi + #I #K #V #HLK lapply (csx_inv_lref_bind â¦ HLK â¦ 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 â¦ H) -H /3 width=1 by llsx_bind/ +| #I #V #T #HG #HL #HT #H #d destruct + elim (csx_fwd_flat â¦ H) -H /3 width=1 by llsx_flat/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_ldrop.etc new file mode 100644 index 000000000..2d16e3240 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_ldrop.etc @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L,d,i. |L| â¤ i â G â¢ ââ¬*[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: âh,g,G,L,d,i. yinj i < d â G â¢ ââ¬*[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. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_llpx.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_llpx.etc new file mode 100644 index 000000000..9392afba2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_llpx.etc @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L1,T,d. G â¢ ââ¬*[h, g, T, d] L1 â + âL2. â¦G, L1â¦ â¢ â¡[h, g, T, d] L2 â G â¢ ââ¬*[h, g, T, d] L2. +#h #g #G #L1 #T #d #H @(llsx_ind â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_llpxs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_llpxs.etc new file mode 100644 index 000000000..af37a372f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llsx_llpxs.etc @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,a,I,G,L1,V,d. G â¢ ââ¬*[h, g, V, d] L1 â + âY,T. G â¢ ââ¬*[h, g, T, â«¯d] Y â + âL2. Y = L2.â{I}V â â¦G, L1â¦ â¢ â¡*[h, g, â{a,I}V.T, d] L2 â + G â¢ ââ¬*[h, g, â{a,I}V.T, d] L2. +#h #g #a #I #G #L1 #V #d #H @(llsx_ind_alt â¦ H) -L1 +#L1 #HL1 #IHL1 #Y #T #H @(llsx_ind_alt â¦ H) -Y +#Y #HY #IHY #L2 #H #HL12 destruct @llsx_intro_alt +#L0 #HL20 lapply (llpxs_fwd_bind_dx â¦ HL20) +lapply (llpxs_trans â¦ HL12 â¦ HL20) +#HL10 #HT #H elim (nlleq_inv_bind â¦ H) -H [ -HL1 -IHY | -HY -IHL1 ] +[ #HnV elim (lleq_dec V L1 L2 d) + [ -HL20 #HV @(IHL1 â¦ 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: âh,g,a,I,G,L,V,d. G â¢ ââ¬*[h, g, V, d] L â + âT. G â¢ ââ¬*[h, g, T, â«¯d] L.â{I}V â + G â¢ ââ¬*[h, g, â{a,I}V.T, d] L. +/2 width=3 by llsx_bind_llpxs_aux/ qed. + +lemma llsx_flat_llpxs: âh,g,I,G,L1,V,d. G â¢ ââ¬*[h, g, V, d] L1 â + âL2,T. G â¢ ââ¬*[h, g, T, d] L2 â â¦G, L1â¦ â¢ â¡*[h, g, â{I}V.T, d] L2 â + G â¢ ââ¬*[h, g, â{I}V.T, d] L2. +#h #g #I #G #L1 #V #d #H @(llsx_ind_alt â¦ H) -L1 +#L1 #HL1 #IHL1 #L2 #T #H @(llsx_ind_alt â¦ H) -L2 +#L2 #HL2 #IHL2 #HL12 @llsx_intro_alt +#L0 #HL20 lapply (llpxs_fwd_flat_dx â¦ HL20) +lapply (llpxs_trans â¦ HL12 â¦ HL20) +#HL10 #HT #H elim (nlleq_inv_flat â¦ H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ] +[ #HnV elim (lleq_dec V L1 L2 d) + [ #HV @(IHL1 â¦ 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: âh,g,I,G,L,V,d. G â¢ ââ¬*[h, g, V, d] L â + âT. G â¢ ââ¬*[h, g, T, d] L â G â¢ ââ¬*[h, g, â{I}V.T, d] L. +/2 width=3 by llsx_flat_llpxs/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx_sn/llpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx_sn/llpx_sn_tc.etc new file mode 100644 index 000000000..682b48c68 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx_sn/llpx_sn_tc.etc @@ -0,0 +1,160 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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) â Î»S,R. + âLs,T1,T2. S Ls T1 T2 â + âLd. TC â¦ (llpx_sn R 0 T1) Ls Ld â TC â¦ (llpx_sn R 0 T2) Ls Ld. + +lemma TC_llpx_sn_s_confluent: âS,R. (llpx_sn_confluent1 S R) â TC_llpx_sn_confluent1 S R. +#S #R #HSR #Ls #T1 #T2 #HT12 #Ld #H +generalize in match HT12; -HT12 +@(TC_ind_dx â¦ Ls H) -Ls +[ /3 width=3 by inj/ +| #Ls #L #HLs #_ #IHLd #HT12 + @(TC_strap â¦ L) /2 width=3 by/ @IHLd -IHLd + +lemma TC_llpx_sn_lref_refl: âR. (âL.reflexive â¦ (R L)) â + âI,L1,K1,K2,V,d,i. d â¤ yinj i â â©[i] L1 â¡ K1.â{I}V â + TC lenv (llpx_sn R 0 V) K1 K2 â + âL2. â©[i] L2 â¡ K2.â{I}V â TC â¦ (llpx_sn R d (#i)) L1 L2. +#R #HR #I #L1 #K1 #K2 #V #d #i #Hdi #HLK1 #H @(TC_star_ind â¦ 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 â¦ HLK2) + #H elim (ldrop_O1_ex (K.â{I}V) i L2) [2: normalize in H â¢ %; >(llpx_sn_fwd_length â¦ HV) ] + /4 width=11 by llpx_sn_lref, step/ +] +qed-. + +lemma TC_llpx_sn_lref: âR. (âL.reflexive â¦ (R L)) â (llpx_sn_confluent1 R R) â + âI,K1,V1,V2,d,i. d â¤ yinj i â LTC â¦ R K1 V1 V2 â + âK2. TC lenv (llpx_sn R 0 V1) K1 K2 â âL1. â©[i] L1 â¡ K1.â{I}V1 â + âL2. â©[i] L2 â¡ K2.â{I}V2 â TC â¦ (llpx_sn R d (#i)) L1 L2. +#R #H1R #H2R #I #K1 #V1 #V2 #d #i #Hdi #H @(TC_star_ind_dx â¦ 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 â¦ HLK1) + #H elim (ldrop_O1_ex (K1.â{I}V) i L1) [2: normalize in H â¢ %; // ] -H + #L #_ #HLK @(TC_strap â¦ L) + [ @(llpx_sn_lref â¦ HLK1 â¦ HLK) /2 width=1 by llpx_sn_refl/ + | @(IHV2 â¦ HLK â¦ HLK2) + -HLK1 -HLK2 -HLK -IHV2 -Hdi @(TC_llpx_sn_s_confluent R R â¦ HK12) // + ] +] + + +lemma llpx_sn_LTC_TC_llpx_sn: âR. (âL. reflexive â¦ (R L)) â + âL1,L2,T,d. llpx_sn (LTC â¦ R) d T L1 L2 â + TC â¦ (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: âR. (âL. reflexive â¦ (R L)) â + âI,L1,L2. TC â¦ (lpx_sn R) L1 L2 â + âV1,V2. LTC â¦ R L1 V1 V2 â + TC â¦ (lpx_sn R) (L1. â{I} V1) (L2. â{I} V2). +#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx â¦ 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: âR. (âL. reflexive â¦ (R L)) â + âL1,L2. lpx_sn (LTC â¦ R) L1 L2 â + TC â¦ (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: âR,L1. TC â¦ (lpx_sn R) L1 (â) â L1 = â. +#R #L1 #H @(TC_ind_dx â¦ 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: âR. s_rs_trans â¦ R (lpx_sn R) â + âI,L1,K2,V2. TC â¦ (lpx_sn R) L1 (K2.â{I}V2) â + ââK1,V1. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ R K1 V1 V2 & L1 = K1. â{I} V1. +#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx â¦ L1 H) -L1 +[ #L1 #H elim (lpx_sn_inv_pair2 â¦ H) -H /3 width=5 by inj, ex3_2_intro/ +| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct + elim (lpx_sn_inv_pair2 â¦ HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct + lapply (HR â¦ HV2 â¦ HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/ +] +qed-. + +lemma TC_lpx_sn_ind: âR. s_rs_trans â¦ R (lpx_sn R) â + âS:relation lenv. + S (â) (â) â ( + âI,K1,K2,V1,V2. + TC â¦ (lpx_sn R) K1 K2 â LTC â¦ R K1 V1 V2 â + S K1 K2 â S (K1.â{I}V1) (K2.â{I}V2) + ) â + âL2,L1. TC â¦ (lpx_sn R) L1 L2 â S L1 L2. +#R #HR #S #IH1 #IH2 #L2 elim L2 -L2 +[ #X #H >(TC_lpx_sn_inv_atom2 â¦ H) -X // +| #L2 #I #V2 #IHL2 #X #H + elim (TC_lpx_sn_inv_pair2 â¦ H) // -H -HR + #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/ +] +qed-. + +lemma TC_lpx_sn_inv_atom1: âR,L2. TC â¦ (lpx_sn R) (â) L2 â L2 = â. +#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: âR. s_rs_trans â¦ R (lpx_sn R) â + âL1,L2. TC â¦ (lpx_sn R) L1 L2 â + âI,K1,V1. L1 = K1.â{I}V1 â + ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ R K1 V1 V2 & L2 = K2. â{I} V2. +#R #HR #L1 #L2 #H @(TC_lpx_sn_ind â¦ 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: âR. s_rs_trans â¦ R (lpx_sn R) â + âI,K1,L2,V1. TC â¦ (lpx_sn R) (K1.â{I}V1) L2 â + ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ R K1 V1 V2 & L2 = K2. â{I} V2. +/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-. + +lemma TC_lpx_sn_inv_lpx_sn_LTC: âR. s_rs_trans â¦ R (lpx_sn R) â + âL1,L2. TC â¦ (lpx_sn R) L1 L2 â + lpx_sn (LTC â¦ 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: âR,L1,L2. TC â¦ (lpx_sn R) L1 L2 â |L1| = |L2|. +#R #L1 #L2 #H elim H -L2 +[ #L2 #HL12 >(lpx_sn_fwd_length â¦ HL12) -HL12 // +| #L #L2 #_ #HL2 #IHL1 + >IHL1 -L1 >(lpx_sn_fwd_length â¦ HL2) -HL2 // +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx_sn/lpx_conj.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx_sn/lpx_conj.etc new file mode 100644 index 000000000..5950f1b92 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/llpx_sn/lpx_conj.etc @@ -0,0 +1,120 @@ +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: âA:Type[0]. âR:predicate A. + ((âa. R a)ââ¥) â (âa. R a â â¥). +/3 width=2 by ex_intro/ qed-. + +lemma lt_repl_sn_trans_tw: âL1,L2,T1,T2. â¯{L1, T1} < â¯{L2, T2} â + âU1. â¯{U1} = â¯{T1} â â¯{L1, U1} < â¯{L2, T2}. +normalize in â¢ (?â?â?â?â?%%â?â?â?%%); // +qed-. + +axiom cpx_fwd_lift1: âh,g,G,L,U1,U2. â¦G, Lâ¦ â¢ U1 â¡[h, g] U2 â + âT1,d,e. â§[d, e] T1 â¡ U1 â âT2. â§[d, e] T2 â¡ 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 â¦ H) -H #H destruct /2 width=2 by ex_intro/ + | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=2 by lift_lref_ge_minus, lift_lref_lt, ex_intro/ + | lapply (lift_inv_gref2 â¦ H) -H #H destruct /2 width=2 by ex_intro/ + ] +| #G #L #k #l #Hkl #T1 #d #e #H + lapply (lift_inv_sort2 â¦ 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 â¦ H) -H * #Hid #H destruct + [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV + elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 + elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // (length_inv_zero_dx â¦ 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 â¦ H) -H #Is #L1s #W1s #_ #H destruct + elim (is_lift_dec U1 0 1) [ -IHU1 -HW12d | -HU1 ] + [ * #T1 #HTU1 lapply (lift_fwd_tw â¦ HTU1) #H + lapply (lleq_inv_lift_le â¦ HU1 L1s L1d â¦ HTU1 ?) -HU1 /2 width=1 by ldrop_drop/ + #HT1 elim (cpx_inv_lift1 â¦ HU12 L1d â¦ HTU1) -HU12 -HTU1 /2 width=4 by ldrop_drop/ + #T2 #HTU2 #HT12 elim (IH â¦ HT1 â¦ HL12d â¦ HT12) /2 width=3 by lt_repl_sn_trans_tw/ -IH -HT1 -HT12 -H + #L2s #HL12s #HT2 @(ex2_intro â¦ (L2s.â{Is}W1s)) + /3 width=10 by lleq_lift_le, lpx_pair, ldrop_drop/ (**) (* full auto too slow *) + | #HnU1 lapply (not_ex_to_all_not â¦ HnU1) -HnU1 #HnU1 + elim (IHU1 â¦ HnU1) [2,3,4: // |5,6,7,8,9,10: skip ] -HnU1 #H1 #H2 #HW1s destruct + elim (IH â¦ HW1s â¦ HL12d â¦ HW12d) // #L2s #HL12s #HW2d + @(ex2_intro â¦ (L2s.â{Id}W2d)) /3 width=3 by lleq_cpx_trans, lpx_pair/ + lapply (lleq_fwd_length â¦ HW2d) #HL2sd -HW12d -HW1s + @lleq_intro_alt [ normalize // ] -HL2sd + #I2s #I2d #K2s #K2d #V2s #V2d #i @(nat_ind_plus â¦ i) -i + [ #_ #_ #HLK2s #HLK2d -IH -IHU1 -HU12 -HL12s -HL12d + lapply (ldrop_inv_O2 â¦ HLK2s) -HLK2s #H destruct + lapply (ldrop_inv_O2 â¦ HLK2d) -HLK2d #H destruct /2 width=1 by and3_intro/ + | #i #_ #_ #HnU2 #HLK2s #HLK2d + lapply (cpx_fwd_nlift2 â¦ HU12 â¦ HnU2) -HU12 -HnU2 #HnU1 + lapply (ldrop_inv_drop1 â¦ HLK2s) -HLK2s #HLK2s + lapply (ldrop_inv_drop1 â¦ HLK2d) -HLK2d #HLK2d + elim (lpx_ldrop_trans_O1 â¦ HL12s â¦ HLK2s) -L2s #Y #HLK1s #H + elim (lpx_inv_pair2 â¦ H) -H #K1s #V1s #HK12s #HV12s #H destruct + elim (lpx_ldrop_trans_O1 â¦ HL12d â¦ HLK2d) -L2d #Y #HLK1d #H + elim (lpx_inv_pair2 â¦ H) -H #K1d #V1d #HK12d #HV12d #H destruct + elim (IHU1 â¦ 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 â¦ HLK1s) -HLK1s #H + elim (IH â¦ HV1d â¦ HK12d â¦ HV12d) // -IH -HV1d -HK12d -HV12d + [ #Y #H1Y #H2Y + + + diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpc.etc new file mode 100644 index 000000000..b80b290b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpc.etc @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1 â¦ â¬ â¬ break â¦ term 46 L2 â¦ )" + 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 â + Î»L1,L2. â¦L1â¦ â¡ â¦L2â¦ â¨ â¦L2â¦ â¡ â¦L1â¦. + +interpretation + "focalized parallel conversion (local environment)" + 'FocalizedPConv L1 L2 = (lfpc L1 L2). + +(* Basic properties *********************************************************) + +lemma lfpc_refl: âL. â¦Lâ¦ â¬ â¦Lâ¦. +/2 width=1/ qed. + +lemma lfpc_sym: âL1,L2. â¦L1â¦ â¬ â¦L2â¦ â â¦L2â¦ â¬ â¦L1â¦. +#L1 #L2 * /2 width=1/ +qed. + +lemma lfpc_lfpr: âL1,L2. â¦L1â¦ â¬ â¦L2â¦ â ââL. â¦L1â¦ â¡ â¦Lâ¦ & â¦L2â¦ â¡ â¦Lâ¦. +#L1 #L2 * /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpc_lfpc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpc_lfpc.etc new file mode 100644 index 000000000..69e444adb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpc_lfpc.etc @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL0,L1,L2. â¦L0â¦ â¬ â¦L1â¦ â â¦L0â¦ â¬ â¦L2â¦ â + ââL. â¦L1â¦ â¬ â¦Lâ¦ & â¦L2â¦ â¬ â¦Lâ¦. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs.etc new file mode 100644 index 000000000..3674637dd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs.etc @@ -0,0 +1,76 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 L1 â¦ â¬ â¬ * break â¦ term 46 L2 â¦ )" + 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 â TC â¦ lfpc. + +interpretation "focalized parallel equivalence (local environment)" + 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2). + +(* Basic eliminators ********************************************************) + +lemma lfpcs_ind: âL1. âR:predicate lenv. R L1 â + (âL,L2. â¦L1â¦ â¬* â¦Lâ¦ â â¦Lâ¦ â¬ â¦L2â¦ â R L â R L2) â + âL2. â¦L1â¦ â¬* â¦L2â¦ â R L2. +#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind â¦ HL1 IHL1 â¦ HL12) // +qed-. + +lemma lfpcs_ind_dx: âL2. âR:predicate lenv. R L2 â + (âL1,L. â¦L1â¦ â¬ â¦Lâ¦ â â¦Lâ¦ â¬* â¦L2â¦ â R L â R L1) â + âL1. â¦L1â¦ â¬* â¦L2â¦ â R L1. +#L2 #R #HL2 #IHL2 #L1 #HL12 +@(TC_star_ind_dx â¦ HL2 IHL2 â¦ HL12) // +qed-. + +(* Basic properties *********************************************************) + +lemma lfpcs_refl: reflexive â¦ lfpcs. +/2 width=1/ qed. + +lemma lfpcs_sym: symmetric â¦ lfpcs. +/3 width=1/ qed. + +lemma lfpc_lfpcs: âL1,L2. â¦L1â¦ â¬ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/2 width=1/ qed. + +lemma lfpcs_strap1: âL1,L,L2. â¦L1â¦ â¬* â¦Lâ¦ â â¦Lâ¦ â¬ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/2 width=3/ qed. + +lemma lfpcs_strap2: âL1,L,L2. â¦L1â¦ â¬ â¦Lâ¦ â â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/2 width=3/ qed. + +lemma lfpcs_lfpr_dx: âL1,L2. â¦L1â¦ â¡ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=1/ qed. + +lemma lfpcs_lfpr_sn: âL1,L2. â¦L2â¦ â¡ â¦L1â¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=1/ qed. + +lemma lfpcs_lfpr_strap1: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦Lâ¦ â¡ â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=3/ qed. + +lemma lfpcs_lfpr_strap2: âL1,L. â¦L1â¦ â¡ â¦Lâ¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=3/ qed. + +lemma lfpcs_lfpr_div: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦L2â¦ â¡ â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=3/ qed. + +lemma lfpcs_lfpr_conf: âL1,L. â¦Lâ¦ â¡ â¦L1â¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_aaa.etc new file mode 100644 index 000000000..b7cea0b7a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_aaa.etc @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â + âT,A1. L1 â¢ T â A1 â âA2. L2 â¢ T â A2 â + A1 = A2. +#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2 +elim (lfpcs_inv_lfprs â¦ HL12) -HL12 #L #HL1 #HL2 +lapply (aaa_lfprs_conf â¦ HT1 â¦ HL1) -L1 #HT1 +lapply (aaa_lfprs_conf â¦ HT2 â¦ HL2) -L2 #HT2 +lapply (aaa_mono â¦ HT1 â¦ HT2) -L -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_fpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_fpcs.etc new file mode 100644 index 000000000..8f3688890 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_fpcs.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â âT. â¦L1, Tâ¦ â¬* â¦L2, Tâ¦. +#L1 #L2 #HL12 #T +elim (lfpcs_inv_lfprs â¦ HL12) -HL12 #L #HL1 #HL2 +lapply (lfprs_inv_fprs â¦ HL1 T) -HL1 +lapply (lfprs_inv_fprs â¦ HL2 T) -HL2 /2 width=4/ +qed-. + +(* Properties on context-free parallel equivalence for closures *************) + +lemma fpcs_lfpcs: âL1,L2,T1,T2. â¦L1, T1â¦ â¬* â¦L2, T2â¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L2 #T1 #T2 #HT12 +elim (fpcs_inv_fprs â¦ HT12) -HT12 /3 width=5 by fprs_lfprs, lfprs_div/ (**) (* auto too slow without trace *) +qed. + +lemma fpcs_lift: âK1,K2,T1,T2. â¦K1, T1â¦ â¬* â¦K2, T2â¦ â + âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â + âd,e. â©[d, e] L1 â¡ K1 â â©[d, e] L2 â¡ K2 â + âU1,U2. â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â + â¦L1, U1â¦ â¬* â¦L2, U2â¦. +#K1 #K2 #T1 #T2 #HT12 #L1 #L2 #HL12 #d #e #HLK1 #HLK2 #U1 #U2 #HTU1 #HTU2 +elim (fpcs_inv_fprs â¦ HT12) -HT12 #K #T #HT1 #HT2 +elim (lift_total T d e) #U #HTU +elim (fprs_lift â¦ HT1 â¦ HLK1 â¦ HTU1 HTU) -K1 -T1 #K1 #HU1 #_ +elim (fprs_lift â¦ HT2 â¦ HLK2 â¦ HTU2 HTU) -K2 -T2 -T #K2 #HU2 #_ -K -d -e +lapply (lfpcs_lfprs_conf K1 â¦ HL12) -HL12 /2 width=3/ #H +lapply (lfpcs_lfprs_strap1 â¦ H K2 ?) -H /2 width=3/ #HK12 +lapply (lfpcs_inv_fpcs â¦ HK12 U) -HK12 #HU +/3 width=4 by fpcs_fprs_strap2, fpcs_fprs_div/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_lfpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_lfpcs.etc new file mode 100644 index 000000000..1e9521736 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_lfpcs.etc @@ -0,0 +1,50 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. â¦L1â¦ â¬* â¦L2â¦ â + ââL. â¦L1â¦ â¡* â¦Lâ¦ & â¦L2â¦ â¡* â¦Lâ¦. +#L1 #L2 #H @(lfpcs_ind â¦ H) -L2 +[ /3 width=3/ +| #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0 + [ elim (lfprs_strip â¦ HL0 â¦ HL2) -L #L #HL0 #HL2 + lapply (lfprs_strap1 â¦ HL10 â¦ HL0) -L0 /2 width=3/ + | lapply (lfprs_strap2 â¦ HL2 â¦ HL0) -L /2 width=3/ + ] +] +qed-. + +(* Advanced properties ******************************************************) + +lemma lfpcs_strip: âL,L1. â¦Lâ¦ â¬* â¦L1â¦ â âL2. â¦Lâ¦ â¬ â¦L2â¦ â + ââL0. â¦L1â¦ â¬ â¦L0â¦ & â¦L2â¦ â¬* â¦L0â¦. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem lfpcs_trans: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/2 width=3/ qed. + +theorem lfpcs_canc_sn: âL,L1,L2. â¦Lâ¦ â¬* â¦L1â¦ â â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=3 by lfpcs_trans, lfpcs_sym/ qed. + +theorem lfpcs_canc_dx: âL,L1,L2. â¦L1â¦ â¬* â¦Lâ¦ â â¦L2â¦ â¬* â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. +/3 width=3 by lfpcs_trans, lfpcs_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_lfprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_lfprs.etc new file mode 100644 index 000000000..baf2caf27 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpc/lfpcs_lfprs.etc @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,L2. â¦L1â¦ â¡* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L2 #H @(lfprs_ind â¦ H) -L2 /width=1/ /3 width=3/ +qed. + +lemma lfpcs_lfprs_sn: âL1,L2. â¦L2â¦ â¡* â¦L1â¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L2 #H @(lfprs_ind_dx â¦ H) -L2 /width=1/ /3 width=3/ +qed. + +lemma lfpcs_lfprs_strap1: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦Lâ¦ â¡* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L #HL1 #L2 #H @(lfprs_ind â¦ H) -L2 /width=1/ /2 width=3/ +qed. + +lemma lfpcs_lfprs_strap2: âL1,L. â¦L1â¦ â¡* â¦Lâ¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L #H #L2 #HL2 @(lfprs_ind_dx â¦ H) -L1 /width=1/ /2 width=3/ +qed. + +lemma lfpcs_lfprs_div: âL1,L. â¦L1â¦ â¬* â¦Lâ¦ â âL2. â¦L2â¦ â¡* â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L #HL1 #L2 #H @(lfprs_ind_dx â¦ H) -L2 /width=1/ /2 width=3/ +qed. + +lemma lfpcs_lfprs_conf: âL1,L. â¦Lâ¦ â¡* â¦L1â¦ â âL2. â¦Lâ¦ â¬* â¦L2â¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L #H #L2 #HL2 @(lfprs_ind â¦ H) -L1 /width=1/ /2 width=3/ +qed. + +lemma lfprs_div: âL1,L. â¦L1â¦ â¡* â¦Lâ¦ â âL2. â¦L2â¦ â¡* â¦Lâ¦ â â¦L1â¦ â¬* â¦L2â¦. +#L1 #L #HL1 #L2 #H @(lfprs_ind_dx â¦ H) -L2 /2 width=1/ /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/cpr_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/cpr_llpx_sn.etc new file mode 100644 index 000000000..4db858798 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/cpr_llpx_sn.etc @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âR. (âI,L.reflexive â¦ (R I L)) â + (âI.l_liftable (R I)) â + (âI.l_deliftable_sn (R I)) â + âG. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H + #Kd #V1d #HLKd #HV1s #HV1sd + lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs + lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd + @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H + #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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 â¦ H) -H + #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/cpx_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/cpx_llpx_sn.etc new file mode 100644 index 000000000..418106494 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/cpx_llpx_sn.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âR. (âI,L.reflexive â¦ (R I L)) â + (âI.l_liftable (R I)) â + (âI.l_deliftable_sn (R I)) â + âh,g,G. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H + #Kd #V1d #HLKd #HV1s #HV1sd + lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs + lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd + @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H /2 width=1 by/ +| #G #Ls #V1 #V2 #T #_ #IHV12 #Ld #H elim (llpx_sn_inv_flat â¦ 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 â¦ H) -H + #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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 â¦ H) -H + #HV1 #H elim (llpx_sn_inv_bind_O â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/lleq.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/lleq.etc new file mode 100644 index 000000000..dc138a492 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/lleq.etc @@ -0,0 +1,160 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â Î»I,L,T1,T2. T1 = T2. + +definition lleq: relation4 ynat term lenv lenv â 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) â + Î»R. âI,L2,T1,T2. R I L2 T1 T2 â âL1. L1 â¡[T1, 0] L2 â R I L1 T1 T2. + +(* Basic inversion lemmas ***************************************************) + +lemma lleq_ind: âR:relation4 ynat term lenv lenv. ( + âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2 + ) â ( + âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2 + ) â ( + âI,L1,L2,K1,K2,V,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â + K1 â¡[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2 + ) â ( + âL1,L2,d,i. |L1| = |L2| â |L1| â¤ i â |L2| â¤ i â R d (#i) L1 L2 + ) â ( + âL1,L2,d,p. |L1| = |L2| â R d (Â§p) L1 L2 + ) â ( + âa,I,L1,L2,V,T,d. + L1 â¡[V, d]L2 â L1.â{I}V â¡[T, â«¯d] L2.â{I}V â + R d V L1 L2 â R (â«¯d) T (L1.â{I}V) (L2.â{I}V) â R d (â{a,I}V.T) L1 L2 + ) â ( + âI,L1,L2,V,T,d. + L1 â¡[V, d]L2 â L1 â¡[T, d] L2 â + R d V L1 L2 â R d T L1 L2 â R d (â{I}V.T) L1 L2 + ) â + âd,T,L1,L2. L1 â¡[T, d] L2 â 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: âa,I,L1,L2,V,T,d. L1 â¡[â{a,I}V.T, d] L2 â + L1 â¡[V, d] L2 â§ L1.â{I}V â¡[T, â«¯d] L2.â{I}V. +/2 width=2 by llpx_sn_inv_bind/ qed-. + +lemma lleq_inv_flat: âI,L1,L2,V,T,d. L1 â¡[â{I}V.T, d] L2 â + L1 â¡[V, d] L2 â§ L1 â¡[T, d] L2. +/2 width=2 by llpx_sn_inv_flat/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lleq_fwd_length: âL1,L2,T,d. L1 â¡[T, d] L2 â |L1| = |L2|. +/2 width=4 by llpx_sn_fwd_length/ qed-. + +lemma lleq_fwd_lref: âL1,L2,d,i. L1 â¡[#i, d] L2 â + â¨â¨ |L1| â¤ i â§ |L2| â¤ i + | yinj i < d + | ââI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V & + â©[i] L2 â¡ K2.â{I}V & + K1 â¡[V, yinj 0] K2 & d â¤ yinj i. +#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref â¦ H) /2 width=1/ +* /3 width=7 by or3_intro2, ex4_4_intro/ +qed-. + +lemma lleq_fwd_ldrop_sn: âL1,L2,T,d. L1 â¡[d, T] L2 â âK1,i. â©[i] L1 â¡ K1 â + âK2. â©[i] L2 â¡ K2. +/2 width=7 by llpx_sn_fwd_ldrop_sn/ qed-. + +lemma lleq_fwd_ldrop_dx: âL1,L2,T,d. L1 â¡[d, T] L2 â âK2,i. â©[i] L2 â¡ K2 â + âK1. â©[i] L1 â¡ K1. +/2 width=7 by llpx_sn_fwd_ldrop_dx/ qed-. + +lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,d. + L1 â¡[â{a,I}V.T, d] L2 â L1 â¡[V, d] L2. +/2 width=4 by llpx_sn_fwd_bind_sn/ qed-. + +lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,d. + L1 â¡[â{a,I}V.T, d] L2 â L1.â{I}V â¡[T, â«¯d] L2.â{I}V. +/2 width=2 by llpx_sn_fwd_bind_dx/ qed-. + +lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,d. + L1 â¡[â{I}V.T, d] L2 â L1 â¡[V, d] L2. +/2 width=3 by llpx_sn_fwd_flat_sn/ qed-. + +lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,d. + L1 â¡[â{I}V.T, d] L2 â L1 â¡[T, d] L2. +/2 width=3 by llpx_sn_fwd_flat_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma lleq_sort: âL1,L2,d,k. |L1| = |L2| â L1 â¡[âk, d] L2. +/2 width=1 by llpx_sn_sort/ qed. + +lemma lleq_skip: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â¡[#i, d] L2. +/2 width=1 by llpx_sn_skip/ qed. + +lemma lleq_lref: âI,L1,L2,K1,K2,V,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â + K1 â¡[V, 0] K2 â L1 â¡[#i, d] L2. +/2 width=9 by llpx_sn_lref/ qed. + +lemma lleq_free: âL1,L2,d,i. |L1| â¤ i â |L2| â¤ i â |L1| = |L2| â L1 â¡[#i, d] L2. +/2 width=1 by llpx_sn_free/ qed. + +lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â¡[Â§p, d] L2. +/2 width=1 by llpx_sn_gref/ qed. + +lemma lleq_bind: âa,I,L1,L2,V,T,d. + L1 â¡[V, d] L2 â L1.â{I}V â¡[T, â«¯d] L2.â{I}V â + L1 â¡[â{a,I}V.T, d] L2. +/2 width=1 by llpx_sn_bind/ qed. + +lemma lleq_flat: âI,L1,L2,V,T,d. + L1 â¡[V, d] L2 â L1 â¡[T, d] L2 â L1 â¡[â{I}V.T, d] L2. +/2 width=1 by llpx_sn_flat/ qed. + +lemma lleq_refl: âd,T. reflexive â¦ (lleq d T). +/2 width=1 by llpx_sn_refl/ qed. + +lemma lleq_Y: âL1,L2,T. |L1| = |L2| â L1 â¡[T, â] L2. +/2 width=1 by llpx_sn_Y/ qed. + +lemma lleq_sym: âd,T. symmetric â¦ (lleq d T). +#d #T #L1 #L2 #H @(lleq_ind â¦ 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: âL1,L2,U,dt. L1 â¡[U, dt] L2 â + âT,d,e. â§[d, e] T â¡ U â + dt â¤ d + e â L1 â¡[U, d] L2. +/2 width=6 by llpx_sn_ge_up/ qed-. + +lemma lleq_ge: âL1,L2,T,d1. L1 â¡[T, d1] L2 â âd2. d1 â¤ d2 â L1 â¡[T, d2] L2. +/2 width=3 by llpx_sn_ge/ qed-. + +lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â¡[V, 0] L2 â L1.â{I}V â¡[T, 0] L2.â{I}V â + L1 â¡[â{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: âR. (âI,L. reflexive â¦ (R I L)) â + âL1,L2,T,d. L1 â¡[T, d] L2 â llpx_sn R d T L1 L2. +/2 width=3 by llpx_sn_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/llpx_sn.etc new file mode 100644 index 000000000..918b42605 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/llpx_sn.etc @@ -0,0 +1,209 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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 â +| llpx_sn_sort: âL1,L2,d,k. |L1| = |L2| â llpx_sn R d (âk) L1 L2 +| llpx_sn_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â llpx_sn R d (#i) L1 L2 +| llpx_sn_lref: âI,L1,L2,K1,K2,V1,V2,d,i. d â¤ yinj i â + â©[i] L1 â¡ K1.â{I}V1 â â©[i] L2 â¡ K2.â{I}V2 â + llpx_sn R (yinj 0) V1 K1 K2 â R I K1 V1 V2 â llpx_sn R d (#i) L1 L2 +| llpx_sn_free: âL1,L2,d,i. |L1| â¤ i â |L2| â¤ i â |L1| = |L2| â llpx_sn R d (#i) L1 L2 +| llpx_sn_gref: âL1,L2,d,p. |L1| = |L2| â llpx_sn R d (Â§p) L1 L2 +| llpx_sn_bind: âa,I,L1,L2,V,T,d. + llpx_sn R d V L1 L2 â llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{I}V) â + llpx_sn R d (â{a,I}V.T) L1 L2 +| llpx_sn_flat: âI,L1,L2,V,T,d. + llpx_sn R d V L1 L2 â llpx_sn R d T L1 L2 â llpx_sn R d (â{I}V.T) L1 L2 +. + +(* Basic inversion lemmas ***************************************************) + +fact llpx_sn_inv_bind_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â + âa,I,V,T. X = â{a,I}V.T â + llpx_sn R d V L1 L2 â§ llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{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: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â + llpx_sn R d V L1 L2 â§ llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{I}V). +/2 width=4 by llpx_sn_inv_bind_aux/ qed-. + +fact llpx_sn_inv_flat_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â + âI,V,T. X = â{I}V.T â + llpx_sn R d V L1 L2 â§ 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: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â + llpx_sn R d V L1 L2 â§ 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: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â |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 â¦ HLK1) -HLK1 +lapply (ldrop_fwd_length â¦ HLK2) -HLK2 +normalize // +qed-. + +lemma llpx_sn_fwd_ldrop_sn: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â + âK1,i. â©[i] L1 â¡ K1 â âK2. â©[i] L2 â¡ K2. +#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length â¦ H) -H +#HL12 lapply (ldrop_fwd_length_le2 â¦ HLK1) -HLK1 /2 width=1 by ldrop_O1_le/ +qed-. + +lemma llpx_sn_fwd_ldrop_dx: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â + âK2,i. â©[i] L2 â¡ K2 â âK1. â©[i] L1 â¡ K1. +#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length â¦ H) -H +#HL12 lapply (ldrop_fwd_length_le2 â¦ HLK2) -HLK2 /2 width=1 by ldrop_O1_le/ +qed-. + +fact llpx_sn_fwd_lref_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â âi. X = #i â + â¨â¨ |L1| â¤ i â§ |L2| â¤ i + | yinj i < d + | ââI,K1,K2,V1,V2. â©[i] L1 â¡ K1.â{I}V1 & + â©[i] L2 â¡ K2.â{I}V2 & + llpx_sn R (yinj 0) V1 K1 K2 & + R I K1 V1 V2 & d â¤ 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: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â + â¨â¨ |L1| â¤ i â§ |L2| â¤ i + | yinj i < d + | ââI,K1,K2,V1,V2. â©[i] L1 â¡ K1.â{I}V1 & + â©[i] L2 â¡ K2.â{I}V2 & + llpx_sn R (yinj 0) V1 K1 K2 & + R I K1 V1 V2 & d â¤ yinj i. +/2 width=3 by llpx_sn_fwd_lref_aux/ qed-. + +lemma llpx_sn_fwd_bind_sn: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â + llpx_sn R d V L1 L2. +#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind â¦ H) -H // +qed-. + +lemma llpx_sn_fwd_bind_dx: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â + llpx_sn R (â«¯d) T (L1.â{I}V) (L2.â{I}V). +#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind â¦ H) -H // +qed-. + +lemma llpx_sn_fwd_flat_sn: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â + llpx_sn R d V L1 L2. +#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat â¦ H) -H // +qed-. + +lemma llpx_sn_fwd_flat_dx: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â + llpx_sn R d T L1 L2. +#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat â¦ H) -H // +qed-. + +lemma llpx_sn_fwd_pair_sn: âR,I,L1,L2,V,T,d. llpx_sn R d (â¡{I}V.T) L1 L2 â + 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: âR. (âI,L. reflexive â¦ (R I L)) â âT,L,d. llpx_sn R d T L L. +#R #HR #T #L @(f2_ind â¦ rfw â¦ 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 â¦ HiL) -HiL destruct /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/ +qed-. + +lemma llpx_sn_Y: âR,T,L1,L2. |L1| = |L2| â llpx_sn R (â) T L1 L2. +#R #T #L1 @(f2_ind â¦ rfw â¦ 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: âR,L1,L2,U,dt. llpx_sn R dt U L1 L2 â âT,d,e. â§[d, e] T â¡ U â + dt â¤ d + e â 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 â¦ H) -H /2 width=1 by llpx_sn_sort/ +| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde + elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12 + elim (ylt_yle_false â¦ Hidt) -Hidt + @(yle_trans â¦ 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 â¦ H) -H * #Hid #H destruct + [ lapply (llpx_sn_fwd_length â¦ HW1) -HW1 #HK12 + lapply (ldrop_fwd_length â¦ HLK1) lapply (ldrop_fwd_length â¦ HLK2) + normalize in â¢ (%â%â?); -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 â¦ 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 â¦ H) -H #V #T #HVW >commutative_plus #HTU #H destruct + @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *) + @(IHT â¦ 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 â¦ H) -H #HVW #HTU #H destruct + /3 width=4 by llpx_sn_flat/ +] +qed-. + +(**) (* the minor premise comes first *) +lemma llpx_sn_ge: âR,L1,L2,T,d1,d2. d1 â¤ d2 â + llpx_sn R d1 T L1 L2 â 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: âR,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 â + llpx_sn R 0 T (L1.â{I}V) (L2.â{I}V) â + llpx_sn R 0 (â{a,I}V.T) L1 L2. +/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-. + +lemma llpx_sn_co: âR1,R2. (âI,L,T1,T2. R1 I L T1 T2 â R2 I L T1 T2) â + âL1,L2,T,d. llpx_sn R1 d T L1 L2 â 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/llpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/llpx_sn_alt.etc new file mode 100644 index 000000000..0916edb67 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/llpx_sn_alt.etc @@ -0,0 +1,62 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â relation4 ynat term lenv lenv â + Î»R,d,T,L1,L2. |L1| = |L2| â§ + (âI1,I2,K1,K2,V1,V2,i. d â¤ yinj i â (L1 â¢ i ~Ïµ ð*[d]â¦Tâ¦ â â¥) â + â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â + I1 = I2 â§ R I1 K1 V1 V2 + ). + +(* Main properties **********************************************************) + +theorem llpx_sn_llpx_sn_alt: âR,T,L1,L2,d. llpx_sn R d T L1 L2 â llpx_sn_alt R d T L1 L2. +#R #U #L1 @(f2_ind â¦ rfw â¦ L1 U) -L1 -U +#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r â¦ H) -H +#HL12 #IHU @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv_ge â¦ H) -H // +[ -n #HnU elim (IHU â¦ 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 â¦ HLK10) #H + lapply (ldrop_conf_ge â¦ H â¦ HLK1 ?) -H /2 width=1 by lt_to_le/ (minus_plus_m_m j (i+1)) in â¢ (%â?); >commutative_plus (lift_inv_sort1 â¦ H) -X + lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ H) -H + * #Hdi #H destruct + [ lapply (ldrop_fwd_length_eq2 â¦ HLK1 HLK2 HK12) -K1 -K2 -d + /2 width=1 by llpx_sn_skip/ + | elim (ylt_yle_false â¦ 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 â¦ H) -H + * #Hdi #H destruct [ -HK12 | -IHK12 ] + [ elim (ldrop_trans_lt â¦ HLK1 â¦ HK11) // -K1 + elim (ldrop_trans_lt â¦ HLK2 â¦ HK22) // -Hdi -K2 + /3 width=18 by llpx_sn_lref/ + | lapply (ldrop_trans_ge_comm â¦ HLK1 â¦ HK11 ?) // -K1 + lapply (ldrop_trans_ge_comm â¦ HLK2 â¦ 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 â¦ H) -H + * #Hid #H destruct + lapply (ldrop_fwd_length_eq2 â¦ HLK1 HLK2 HK12) -HK12 + [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/ + | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #HLK1 + lapply (ldrop_fwd_length â¦ 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 â¦ H) -X + lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ 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 â¦ H) -H + #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ +] +qed-. + +lemma llpx_sn_lift_ge: âR,K1,K2,T,d0. llpx_sn R d0 T K1 K2 â + âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âU. â§[d, e] T â¡ U â d â¤ d0 â 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 â¦ H) -X + lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ H) -H + * #_ #H destruct + lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ H) -H + * #Hid #H destruct + [ elim (ylt_yle_false â¦ 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 â¦ HLK1 â¦ HK11 ?) // -K1 + lapply (ldrop_trans_ge_comm â¦ HLK2 â¦ 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 â¦ H) -H + * #Hid #H destruct + lapply (ldrop_fwd_length_eq2 â¦ HLK1 HLK2 HK12) -HK12 + [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/ + | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #HLK1 + lapply (ldrop_fwd_length â¦ 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 â¦ H) -X + lapply (ldrop_fwd_length_eq2 â¦ 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 â¦ 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 â¦ 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: âR. (âI. l_deliftable_sn (R I)) â + âL1,L2,U,d0. llpx_sn R d0 U L1 L2 â + âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âT. â§[d, e] T â¡ U â d0 â¤ d â 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 â¦ H) -X + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H + * #_ #H destruct + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H + * #Hid #H destruct [ -HK12 | -IHK12 ] + [ elim (ldrop_conf_lt â¦ HLK1 â¦ HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1 + elim (ldrop_conf_lt â¦ HLK2 â¦ HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2 + elim (HR â¦ HW12 â¦ HKL11 â¦ HVW1) -HR #V0 #HV0 #HV12 + lapply (lift_inj â¦ HV0 â¦ HVW2) -HV0 -HVW2 #H destruct + /3 width=10 by llpx_sn_lref/ + | lapply (ldrop_conf_ge â¦ HLK1 â¦ HLK11 ?) // -L1 + lapply (ldrop_conf_ge â¦ HLK2 â¦ HLK22 ?) // -L2 -Hid0 + elim (le_inv_plus_l â¦ 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 â¦ H) -H + * #_ #H destruct + lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) + [ lapply (ldrop_fwd_length_le4 â¦ HLK1) -HLK1 + lapply (ldrop_fwd_length_le4 â¦ HLK2) -HLK2 + #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) + | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H >H in HL1; -H + lapply (ldrop_fwd_length â¦ 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 â¦ H) -X + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ 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 â¦ H) -H + #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ +] +qed-. + +lemma llpx_sn_inv_lift_be: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â + âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âT. â§[d, e] T â¡ U â d â¤ d0 â d0 â¤ yinj d + e â 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 â¦ H) -X + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H + * #Hid #H destruct + [ lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) -L1 -L2 + -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/ + | elim (ylt_yle_false â¦ 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 â¦ H) -H + * #Hid #H destruct + [ elim (ylt_yle_false â¦ Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0 + /3 width=3 by ylt_yle_trans, ylt_inj/ + | lapply (ldrop_conf_ge â¦ HLK1 â¦ HLK11 ?) // -L1 + lapply (ldrop_conf_ge â¦ HLK2 â¦ HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e + elim (le_inv_plus_l â¦ 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 â¦ H) -H + * #_ #H destruct + lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) + [ lapply (ldrop_fwd_length_le4 â¦ HLK1) -HLK1 + lapply (ldrop_fwd_length_le4 â¦ HLK2) -HLK2 + #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) + | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H >H in HL1; -H + lapply (ldrop_fwd_length â¦ 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 â¦ H) -X + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H + >commutative_plus #V #T #HVW #HTU #H destruct + @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) + @(IHU â¦ 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 â¦ H) -H + #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ +] +qed-. + +lemma llpx_sn_inv_lift_ge: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â + âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â + âT. â§[d, e] T â¡ U â yinj d + e â¤ d0 â 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 â¦ H) -X + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H + * #Hid #H destruct [ -Hid0 | -Hded0 ] + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ 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 â¦ H) -H + * #Hid #H destruct + [ elim (ylt_yle_false â¦ 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 â¦ HLK1 â¦ HLK11 ?) // -L1 + lapply (ldrop_conf_ge â¦ HLK2 â¦ 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 â¦ H) -H + * #_ #H destruct + lapply (ldrop_fwd_length_eq1 â¦ HLK1 HLK2 HL12) + [ lapply (ldrop_fwd_length_le4 â¦ HLK1) -HLK1 + lapply (ldrop_fwd_length_le4 â¦ HLK2) -HLK2 + #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) + | lapply (ldrop_fwd_length â¦ HLK1) -HLK1 #H >H in HL1; -H + lapply (ldrop_fwd_length â¦ 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 â¦ H) -X + lapply (ldrop_fwd_length_eq1 â¦ 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 â¦ H) -H + #V #T #HVW #HTU #H destruct + @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) + (lpx_sn_inv_atom1 â¦ H) -H + /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/ +| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 â¦ 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 â¦ H) -H + #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 â¦ 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 â¦ H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (HR â¦ HV12 â¦ HLK1 â¦ HWV1) -V1 + elim (IHLK1 â¦ HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/ +] +qed-. + +lemma lpx_sn_liftable_dedropable: âR. (âI,L. reflexive ? (R I L)) â + (âI. l_liftable (R I)) â 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 â¦ H) -H + /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/ +| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 â¦ H) -H + #K2 #V2 #HK12 #HV12 #H destruct + lapply (lpx_sn_fwd_length â¦ HK12) + #H @(ex3_intro â¦ (K2.â{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 â¦ 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 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (lift_total W2 d e) #V2 #HWV2 + lapply (H2R â¦ HW12 â¦ HLK1 â¦ HWV1 â¦ HWV2) -W1 + elim (IHLK1 â¦ HK12) -K1 + /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/ +] +qed-. + +fact lpx_sn_dropable_aux: âR,L2,K2,s,d,e. â©[s, d, e] L2 â¡ K2 â âL1. lpx_sn R L1 L2 â + d = 0 â ââK1. â©[s, 0, e] L1 â¡ 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 â¦ H) -H + /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/ +| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 â¦ 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 â¦ H) -H + #L1 #V1 #HL12 #HV12 #H destruct + elim (IHLK2 â¦ HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/ +| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_ + (lpx_sn_inv_atom1 â¦ H1) -X1 + >(lpx_sn_inv_atom1 â¦ 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 â¦ H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct + elim (lpx_sn_inv_pair1 â¦ H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct + elim (IH â¦ HL01 â¦ HL02) -IH normalize // #L #HL1 #HL2 + elim (HR12 â¦ HV01 â¦ HV02 â¦ HL01 â¦ HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/lpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/lpx_sn_tc.etc new file mode 100644 index 000000000..2a3a87cf4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/lpx_sn_tc.etc @@ -0,0 +1,119 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âR. (âI,L. reflexive â¦ (R I L)) â + âL1,L2. TC â¦ (lpx_sn R) L1 L2 â + âI,V. TC â¦ (lpx_sn R) (L1.â{I}V) (L2. â{I}V). +#R #HR #L1 #L2 #H @(TC_star_ind â¦ 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: âR. (âI,L. reflexive â¦ (R I L)) â + âI,L1,L2. TC â¦ (lpx_sn R) L1 L2 â + âV1,V2. LTC â¦ (R I) L1 V1 V2 â + TC â¦ (lpx_sn R) (L1.â{I}V1) (L2. â{I}V2). +#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx â¦ 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: âR. (âI,L. reflexive â¦ (R I L)) â + âL1,L2. lpx_sn (Î»I.LTC â¦ (R I)) L1 L2 â + TC â¦ (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: âR,L1. TC â¦ (lpx_sn R) L1 (â) â L1 = â. +#R #L1 #H @(TC_ind_dx â¦ 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: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â + âI,L1,K2,V2. TC â¦ (lpx_sn R) L1 (K2.â{I}V2) â + ââK1,V1. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ (R I) K1 V1 V2 & L1 = K1.â{I}V1. +#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx â¦ L1 H) -L1 +[ #L1 #H elim (lpx_sn_inv_pair2 â¦ H) -H /3 width=5 by inj, ex3_2_intro/ +| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct + elim (lpx_sn_inv_pair2 â¦ HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct + lapply (HR â¦ HV2 â¦ HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/ +] +qed-. + +lemma TC_lpx_sn_ind: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â + âS:relation lenv. + S (â) (â) â ( + âI,K1,K2,V1,V2. + TC â¦ (lpx_sn R) K1 K2 â LTC â¦ (R I) K1 V1 V2 â + S K1 K2 â S (K1.â{I}V1) (K2.â{I}V2) + ) â + âL2,L1. TC â¦ (lpx_sn R) L1 L2 â S L1 L2. +#R #HR #S #IH1 #IH2 #L2 elim L2 -L2 +[ #X #H >(TC_lpx_sn_inv_atom2 â¦ H) -X // +| #L2 #I #V2 #IHL2 #X #H + elim (TC_lpx_sn_inv_pair2 â¦ H) // -H -HR + #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/ +] +qed-. + +lemma TC_lpx_sn_inv_atom1: âR,L2. TC â¦ (lpx_sn R) (â) L2 â L2 = â. +#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: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â + âL1,L2. TC â¦ (lpx_sn R) L1 L2 â + âI,K1,V1. L1 = K1.â{I}V1 â + ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ (R I) K1 V1 V2 & L2 = K2. â{I} V2. +#R #HR #L1 #L2 #H @(TC_lpx_sn_ind â¦ 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: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â + âI,K1,L2,V1. TC â¦ (lpx_sn R) (K1.â{I}V1) L2 â + ââK2,V2. TC â¦ (lpx_sn R) K1 K2 & LTC â¦ (R I) K1 V1 V2 & L2 = K2. â{I} V2. +/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-. + +lemma TC_lpx_sn_inv_lpx_sn_LTC: âR. (âI.s_rs_transitive â¦ (R I) (Î»_.lpx_sn R)) â + âL1,L2. TC â¦ (lpx_sn R) L1 L2 â + lpx_sn (Î»I.LTC â¦ (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: âR,L1,L2. TC â¦ (lpx_sn R) L1 L2 â |L1| = |L2|. +#R #L1 #L2 #H elim H -L2 +[ #L2 #HL12 >(lpx_sn_fwd_length â¦ HL12) -HL12 // +| #L #L2 #_ #HL2 #IHL1 + >IHL1 -L1 >(lpx_sn_fwd_length â¦ HL2) -HL2 // +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/ssta_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/ssta_llpx_sn.etc new file mode 100644 index 000000000..3b1db6a9a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lpx_sn/ssta_llpx_sn.etc @@ -0,0 +1,44 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âR. (âI,L.reflexive â¦ (R I L)) â + (âI.l_liftable (R I)) â + âh,g,G. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H + #Kd #V1d #HLKd #HV1s #HV1sd + lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs + lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd + @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ H â¦ HLKs) // -H + #Kd #V1d #HLKd #HV1s #HV1sd + lapply (ldrop_fwd_drop2 â¦ HLKs) -HLKs #HLKs + lapply (ldrop_fwd_drop2 â¦ HLKd) -HLKd #HLKd + @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ H) -H + /3 width=1 by llpx_sn_flat/ +| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat â¦ H) -H + /3 width=1 by llpx_sn_flat/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn.etc new file mode 100644 index 000000000..c4359c35f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn.etc @@ -0,0 +1,118 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 L1 : â break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqN $h $L1 $L2 }. + +notation "hvbox( h â¢ break term 46 L1 : : â 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 â +| lsubn_atom: lsubn h (â) (â) +| lsubn_pair: âI,L1,L2,W. lsubn h L1 L2 â lsubn h (L1. â{I} W) (L2. â{I} W) +| lsubn_abbr: âL1,L2,V,W. â¦h, L1â¦ â¢ V : W â â¦h, L2â¦ â¢ V : W â + lsubn h L1 L2 â lsubn h (L1. âV) (L2. âW) +. + +interpretation + "local environment refinement (native type assigment)" + 'CrSubEqN h L1 L2 = (lsubn h L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubn_inv_atom1_aux: âh,L1,L2. h â¢ L1 :â L2 â L1 = â â L2 = â. +#h #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #_ #_ #_ #H destruct +] +qed. + +lemma lsubn_inv_atom1: âh,L2. h â¢ â :â L2 â L2 = â. +/2 width=4/ qed-. + +fact lsubn_inv_pair1_aux: âh,L1,L2. h â¢ L1 :â L2 â âI,K1,V. L1 = K1. â{I} V â + (ââK2. h â¢ K1 :â K2 & L2 = K2. â{I} V) â¨ + ââK2,W. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & + h â¢ K1 :â K2 & L2 = K2. âW & 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: âh,I,K1,L2,V. h â¢ K1. â{I} V :â L2 â + (ââK2. h â¢ K1 :â K2 & L2 = K2. â{I} V) â¨ + ââK2,W. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & + h â¢ K1 :â K2 & L2 = K2. âW & I = Abbr. +/2 width=3/ qed-. + +fact lsubn_inv_atom2_aux: âh,L1,L2. h â¢ L1 :â L2 â L2 = â â L1 = â. +#h #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #_ #_ #_ #H destruct +] +qed. + +lemma lsubc_inv_atom2: âh,L1. h â¢ L1 :â â â L1 = â. +/2 width=4/ qed-. + +fact lsubn_inv_pair2_aux: âh,L1,L2. h â¢ L1 :â L2 â âI,K2,W. L2 = K2. â{I} W â + (ââK1. h â¢ K1 :â K2 & L1 = K1. â{I} W) â¨ + ââK1,V. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & + h â¢ K1 :â K2 & L1 = K1. âV & 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: âh,I,L1,K2,W. h â¢ L1 :â K2. â{I} W â + (ââK1. h â¢ K1 :â K2 & L1 = K1. â{I} W) â¨ + ââK1,V. â¦h, K1â¦ â¢ V : W & â¦h, K2â¦ â¢ V : W & + h â¢ K1 :â K2 & L1 = K1. âV & I = Abst. +/2 width=3/ qed-. + +(* Basic_forward lemmas *****************************************************) + +lemma lsubn_fwd_lsubs1: âh,L1,L2. h â¢ L1 :â L2 â L1 â¼[0, |L1|] L2. +#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +lemma lsubn_fwd_lsubs2: âh,L1,L2. h â¢ L1 :â L2 â L1 â¼[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: âh,L. h â¢ L :â 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_cpcs.etc new file mode 100644 index 000000000..5f610bc96 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_cpcs.etc @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,L2. h â¢ L1 :â L2 â + âT1,T2. L2 â¢ T1 â¡ T2 â L1 â¢ T1 â¡ T2. +/3 width=4 by lsubn_fwd_lsubs2, cpr_lsubs_trans/ qed. + +lemma cprs_lsubn_trans: âh,L1,L2. h â¢ L1 :â L2 â + âT1,T2. L2 â¢ T1 â¡* T2 â L1 â¢ T1 â¡* T2. +/3 width=4 by lsubn_fwd_lsubs2, cprs_lsubs_trans/ qed. + +(* Basic_1: was: csubt_pc3 *) +lemma cpcs_lsubn_trans: âh,L1,L2. h â¢ L1 :â L2 â + âT1,T2. L2 â¢ T1 â¬* T2 â L1 â¢ T1 â¬* T2. +/3 width=4 by lsubn_fwd_lsubs2, cpcs_lsubs_trans/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_ldrop.etc new file mode 100644 index 000000000..a16fff618 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_ldrop.etc @@ -0,0 +1,64 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,L2. h â¢ L1 :â L2 â âK1,e. â©[0, e] L1 â¡ K1 â + ââK2. h â¢ K1 :â K2 & â©[0, e] L2 â¡ K2. +#h #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ 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: âh,L1,L2. h â¢ L1 :â L2 â âK2,e. â©[0, e] L2 â¡ K2 â + ââK1. h â¢ K1 :â K2 & â©[0, e] L1 â¡ K1. +#h #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_lsubn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_lsubn.etc new file mode 100644 index 000000000..9ef3dda88 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_lsubn.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,L. h â¢ L1 :â L â âL2. h â¢ L :â L2 â h â¢ L1 :â L2. +#h #L1 #L #H elim H -L1 -L +[ #X #H >(lsubn_inv_atom1 â¦ H) -H // +| #I #L1 #L #V #HL1 #H1W #IHL1 #X #H + elim (lsubn_inv_pair1 â¦ 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 â¦ H) -H * #L2 + [ #HL2 #H #HW destruct /3 width=1/ + | #V #_ #_ #_ #_ #H destruct + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_nta.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_nta.etc new file mode 100644 index 000000000..5832b00b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubn/lsubn_nta.etc @@ -0,0 +1,47 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L2,T,U. â¦h, L2â¦ â¢ T : U â âL1. h â¢ L1 :â L2 â + â¦h, L1â¦ â¢ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 + elim (lsubn_inv_pair2 â¦ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 + elim (lsubn_inv_pair2 â¦ 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/aaa_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/aaa_lpss.etc new file mode 100644 index 000000000..e930823ad --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/aaa_lpss.etc @@ -0,0 +1,54 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T1,A. L1 â¢ T1 â A â âT2. L1 â¢ T1 â¶* T2 â + âL2. L1 â¢ â¶* L2 â L2 â¢ T2 â A. +#L1 #T1 #A #H elim H -L1 -T1 -A +[ #L1 #k #X #H + >(cpss_inv_sort1 â¦ H) -H // +| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12 + elim (cpss_inv_lref1 â¦ H) -H + [ #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/ + | * #Y #Z #V2 #H #HV12 #HV2 + lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #Z #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #V0 #HK12 #_ #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) -V0 /3 width=7/ + ] +| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12 + elim (cpss_inv_bind1 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/ +] +qed-. + +lemma aaa_cpss_conf: âL,T1,A. L â¢ T1 â A â âT2. L â¢ T1 â¶* T2 â L â¢ T2 â A. +/2 width=5 by aaa_cpss_lpss_conf/ qed-. + +lemma aaa_lpss_conf: âL1,T,A. L1 â¢ T â A â âL2. L1 â¢ â¶* L2 â L2 â¢ T â A. +/2 width=5 by aaa_cpss_lpss_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpcs_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpcs_lpss.etc new file mode 100644 index 000000000..09d29754f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpcs_lpss.etc @@ -0,0 +1,50 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL1,T1,T2. L1 â¢ T1 â¬* T2 â + âL2. L1 â¢ â¶* L2 â L2 â¢ T1 â¬* T2. +#L1 #T1 #T2 #H #L2 #HL12 +elim (cpcs_inv_cprs â¦ H) -H #T #HT1 #HT2 +elim (cprs_lpss_conf_dx â¦ HT1 â¦ HL12) -HT1 #U1 #H1 #HTU1 +elim (cprs_lpss_conf_dx â¦ HT2 â¦ HL12) -L1 #U2 #H2 #HTU2 +elim (cpss_conf â¦ H1 â¦ H2) -T #U #HU1 #HU2 +lapply (cprs_cpss_trans â¦ HTU1 â¦ HU1) -U1 +lapply (cprs_cpss_trans â¦ HTU2 â¦ HU2) -U2 /2 width=3/ +qed-. + +lemma cpcs_cpss_lpss_conf: âL1,T,T2. L1 â¢ T â¬* T2 â âT1. L1 â¢ T â¶* T1 â + âL2. L1 â¢ â¶* L2 â L2 â¢ T1 â¬* T2. +#L1 #T #T2 #HT2 #T1 #HT1 #L2 #HL12 +lapply (cpcs_lpss_conf â¦ HT2 â¦ HL12) -HT2 #HT2 +elim (lpss_cpss_conf_dx â¦ HT1 â¦ HL12) -L1 #T0 #HT0 #HT10 +lapply (cpcs_cpss_conf â¦ HT0 â¦ HT2) -T #HT02 +lapply (cpcs_cpss_strap2 â¦ HT10 â¦ HT02) -T0 // +qed-. + +lemma cpcs_cpss2_lpss_conf: âL1,T1,T2. L1 â¢ T1 â¬* T2 â + âT3. L1 â¢ T1 â¶* T3 â âT4. L1 â¢ T2 â¶* T4 â + âL2. L1 â¢ â¶* L2 â L2 â¢ T3 â¬* T4. +#L1 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 #L2 #HL12 +lapply (cpcs_cpss_lpss_conf â¦ HT12 â¦ HT13 â¦ HL12) -T1 #HT32 +elim (lpss_cpss_conf_dx â¦ HT24 â¦ HL12) -L1 #T #HT2 #HT4 +lapply (cpcs_cpss_strap1 â¦ HT32 â¦ HT2) -T2 #HT3 +lapply (cpcs_cpss_div â¦ HT3 â¦ HT4) -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpqs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpqs.etc new file mode 100644 index 000000000..41120361a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpqs.etc @@ -0,0 +1,227 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 T1 â¤ * 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 â relation term â +| cpqs_atom : âI,L. cpqs L (âª{I}) (âª{I}) +| cpqs_delta: âL,K,V,V2,W2,i. + â©[0, i] L â¡ K. âV â cpqs K V V2 â + â§[0, i + 1] V2 â¡ W2 â cpqs L (#i) W2 +| cpqs_bind : âa,I,L,V1,V2,T1,T2. + cpqs L V1 V2 â cpqs (L. â{I} V1) T1 T2 â + cpqs L (â{a,I} V1. T1) (â{a,I} V2. T2) +| cpqs_flat : âI,L,V1,V2,T1,T2. + cpqs L V1 V2 â cpqs L T1 T2 â + cpqs L (â{I} V1. T1) (â{I} V2. T2) +| cpqs_zeta : âL,V,T1,T,T2. cpqs (L.âV) T1 T â + â§[0, 1] T2 â¡ T â cpqs L (+âV. T1) T2 +| cpqs_tau : âL,V,T1,T2. cpqs L T1 T2 â cpqs L (âV. 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 â¦ 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 â¦ HL12 â¦ HLK1) -HL12 -HLK1 /3 width=6/ +| /4 width=1/ +|4,6: /3 width=1/ +| /4 width=3/ +] +qed-. + +lemma cpss_cpqs: âL,T1,T2. L â¢ T1 â¶* T2 â L â¢ T1 â¤* T2. +#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/ +qed. + +lemma cpqs_refl: âT,L. L â¢ T â¤* T. +/2 width=1/ qed. + +lemma cpqs_delift: âL,K,V,T1,d. â©[0, d] L â¡ (K. âV) â + ââT2,T. L â¢ T1 â¤* T2 & â§[d, 1] T â¡ T2. +#L #K #V #T1 #d #HLK +elim (cpss_delift â¦ T1 â¦ HLK) -HLK /3 width=4/ +qed-. + +lemma cpqs_append: l_appendable_sn â¦ 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 â¦ HK0) #H +@(cpqs_delta â¦ (L@@K0) V1 â¦ HVW2) // +@(ldrop_O1_append_sn_le â¦ HK0) /2 width=2/ (**) (* /3/ does not work *) +qed. + +(* Basic inversion lemmas ***************************************************) + +fact cpqs_inv_atom1_aux: âL,T1,T2. L â¢ T1 â¤* T2 â âI. T1 = âª{I} â + T2 = âª{I} â¨ + ââK,V,V2,i. â©[O, i] L â¡ K. âV & + K â¢ V â¤* V2 & + â§[O, i + 1] V2 â¡ 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: âI,L,T2. L â¢ âª{I} â¤* T2 â + T2 = âª{I} â¨ + ââK,V,V2,i. â©[O, i] L â¡ K. âV & + K â¢ V â¤* V2 & + â§[O, i + 1] V2 â¡ T2 & + I = LRef i. +/2 width=3 by cpqs_inv_atom1_aux/ qed-. + +lemma cpqs_inv_sort1: âL,T2,k. L â¢ âk â¤* T2 â T2 = âk. +#L #T2 #k #H +elim (cpqs_inv_atom1 â¦ H) -H // +* #K #V #V2 #i #_ #_ #_ #H destruct +qed-. + +lemma cpqs_inv_lref1: âL,T2,i. L â¢ #i â¤* T2 â + T2 = #i â¨ + ââK,V,V2. â©[O, i] L â¡ K. âV & + K â¢ V â¤* V2 & + â§[O, i + 1] V2 â¡ T2. +#L #T2 #i #H +elim (cpqs_inv_atom1 â¦ H) -H /2 width=1/ +* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/ +qed-. + +lemma cpqs_inv_gref1: âL,T2,p. L â¢ Â§p â¤* T2 â T2 = Â§p. +#L #T2 #p #H +elim (cpqs_inv_atom1 â¦ H) -H // +* #K #V #V2 #i #_ #_ #_ #H destruct +qed-. + +fact cpqs_inv_bind1_aux: âL,U1,U2. L â¢ U1 â¤* U2 â + âa,I,V1,T1. U1 = â{a,I} V1. T1 â ( + ââV2,T2. L â¢ V1 â¤* V2 & + L. â{I} V1 â¢ T1 â¤* T2 & + U2 = â{a,I} V2. T2 + ) â¨ + ââT. L.âV1 â¢ T1 â¤* T & â§[0, 1] U2 â¡ 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: âa,I,L,V1,T1,U2. L â¢ â{a,I} V1. T1 â¤* U2 â ( + ââV2,T2. L â¢ V1 â¤* V2 & + L. â{I} V1 â¢ T1 â¤* T2 & + U2 = â{a,I} V2. T2 + ) â¨ + ââT. L.âV1 â¢ T1 â¤* T & â§[0, 1] U2 â¡ T & a = true & I = Abbr. +/2 width=3 by cpqs_inv_bind1_aux/ qed-. + +lemma cpqs_inv_abbr1: âa,L,V1,T1,U2. L â¢ â{a} V1. T1 â¤* U2 â ( + ââV2,T2. L â¢ V1 â¤* V2 & + L. â V1 â¢ T1 â¤* T2 & + U2 = â{a} V2. T2 + ) â¨ + ââT. L.âV1 â¢ T1 â¤* T & â§[0, 1] U2 â¡ T & a = true. +#a #L #V1 #T1 #U2 #H +elim (cpqs_inv_bind1 â¦ H) -H * /3 width=3/ /3 width=5/ +qed-. + +lemma cpqs_inv_abst1: âa,L,V1,T1,U2. L â¢ â{a} V1. T1 â¤* U2 â + ââV2,T2. L â¢ V1 â¤* V2 & + L. â V1 â¢ T1 â¤* T2 & + U2 = â{a} V2. T2. +#a #L #V1 #T1 #U2 #H +elim (cpqs_inv_bind1 â¦ H) -H * +[ /3 width=5/ +| #T #_ #_ #_ #H destruct +] +qed-. + +fact cpqs_inv_flat1_aux: âL,U1,U2. L â¢ U1 â¤* U2 â + âI,V1,T1. U1 = â{I} V1. T1 â ( + ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & + U2 = â{I} V2. T2 + ) â¨ + (L â¢ T1 â¤* U2 â§ 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: âI,L,V1,T1,U2. L â¢ â{I} V1. T1 â¤* U2 â ( + ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & + U2 = â{I} V2. T2 + ) â¨ + (L â¢ T1 â¤* U2 â§ I = Cast). +/2 width=3 by cpqs_inv_flat1_aux/ qed-. + +lemma cpqs_inv_appl1: âL,V1,T1,U2. L â¢ â V1. T1 â¤* U2 â + ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & + U2 = â V2. T2. +#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 â¦ H) -H * +[ /3 width=5/ +| #_ #H destruct +] +qed-. + +lemma cpqs_inv_cast1: âL,V1,T1,U2. L â¢ â V1. T1 â¤* U2 â ( + ââV2,T2. L â¢ V1 â¤* V2 & L â¢ T1 â¤* T2 & + U2 = â V2. T2 + ) â¨ + L â¢ T1 â¤* U2. +#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 â¦ H) -H * /2 width=1/ /3 width=5/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cpqs_fwd_shift1: âL1,L,T1,T. L â¢ L1 @@ T1 â¤* T â + ââL2,T2. |L1| = |L2| & T = L2 @@ T2. +#L1 @(lenv_ind_dx â¦ L1) -L1 normalize +[ #L #T1 #T #HT1 + @(ex2_2_intro â¦ (â)) // (**) (* explicit constructor *) +| #I #L1 #V1 #IH #L #T1 #X + >shift_append_assoc normalize #H + elim (cpqs_inv_bind1 â¦ H) -H * + [ #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) + | #T #_ #_ #H destruct + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpqs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpqs_lift.etc new file mode 100644 index 000000000..78529b6b1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpqs_lift.etc @@ -0,0 +1,81 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ H1 â¦ 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 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HVW2 â¦ HWU2) -W2 // plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct + elim (lift_inv_flat1 â¦ 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 â¦ H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct + elim (lift_conf_O1 â¦ HTU2 â¦ HT2) -T2 /4 width=5/ +| #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2 + elim (lift_inv_flat1 â¦ 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 â¦ H) -H #H destruct /2 width=3/ + | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hid #H destruct + [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV + elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 + elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // shift_append_assoc normalize #H + elim (cpss_inv_bind1 â¦ H) -H + #V0 #T0 #_ #HT10 #H destruct + elim (IH â¦ HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro â¦ (â.â{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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpss_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpss_lift.etc new file mode 100644 index 000000000..9745b6cde --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/cpss_lift.etc @@ -0,0 +1,71 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â¦ H1 â¦ 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 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HVW2 â¦ HWU2) -W2 // plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct + elim (lift_inv_flat1 â¦ 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 â¦ H) -H #H destruct /2 width=3/ + | elim (lift_inv_lref2 â¦ H) -H * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hid #H destruct + [ elim (ldrop_conf_lt â¦ HLK â¦ HLV) -L // #L #U #HKL #HLV #HUV + elim (IHV2 â¦ HLV â¦ HUV) -V #U2 #HUV2 #HU2 + elim (lift_trans_le â¦ HUV2 â¦ HVW2) -V2 // >minus_plus plus_minus // shift_append_assoc #H shift_append_assoc #H (cpss_inv_sort1 â¦ H) -X // +| #I #L1 #K1 #V1 #i #HLK1 #_ #IHV1 #W2 #H #L2 #HL12 + elim (cpss_inv_lref1 â¦ H) -H + [ #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct + lapply (IHV1 â¦ HV12 â¦ HK12) -IHV1 -HV12 -HK12 /2 width=5/ + | * #K0 #V0 #V2 #HLK0 #HV12 #HVW2 + lapply (ldrop_mono â¦ HLK0 â¦ HLK1) -HLK0 #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -L1 #X #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #V #HK12 #_ #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) -V #HLK2 + lapply (IHV1 â¦ HV12 â¦ HK12) -IHV1 -HV12 -HK12 /2 width=7/ + ] +| #a #I #L1 #V1 #T1 #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12 + elim (cpss_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IHV1 â¦ HV12 â¦ HL12) -IHV1 #HV2 + lapply (IHT1 â¦ HT12 (L2.â{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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IHV1 â¦ HV12 â¦ HL12) -IHV1 #HV2 + lapply (IHT1 â¦ HT12 â¦ HL12) -IHT1 #HT2 + elim (ssta_cpss_lpss_conf â¦ HVW1 â¦ HV12 â¦ HL12) -V1 #W2 #HVW2 #HW12 + elim (dxprs_cpss_lpss_conf â¦ HTU1 â¦ HT12 â¦ HL12) -T1 #X #HTU2 #H + elim (cpss_inv_bind1 â¦ H) -H #W #U2 #HW0 #_ #H -U1 destruct + elim (cprs_cpss2_lpss_conf_dx â¦ HW10 â¦ HW12 â¦ HW0 â¦ HL12) -L1 -W1 -W0 #W0 #HW20 #HW0 + lapply (dxprs_strap1 â¦ (â{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 â¦ H) -H #W2 #T2 #HW12 #HT12 #H destruct + lapply (IHW1 â¦ HW12 â¦ HL12) -IHW1 #HW2 + lapply (IHT1 â¦ HT12 â¦ HL12) -IHT1 #HT2 + elim (ssta_cpss_lpss_conf â¦ HTU1 â¦ HT12 â¦ HL12) -T1 #U2 #HTU2 #HU12 + lapply (cpcs_cpss2_lpss_conf â¦ HUW1 â¦ HU12 â¦ HW12 â¦ HL12) -L1 -W1 -U1 /2 width=4/ +] +qed-. + +lemma snv_lpss_conf: âh,g,L1,T. â¦h, L1â¦ â¢ T Â¡[g] â + âL2. L1 â¢ â¶* L2 â â¦h, L2â¦ â¢ T Â¡[g]. +#h #g #L1 #T #HT #L2 #HL12 +@(snv_cpss_lpss_conf â¦ HT â¦ HL12) // +qed-. + +lemma snv_cpss_conf: âh,g,L,T1. â¦h, Lâ¦ â¢ T1 Â¡[g] â + âT2. L â¢ T1 â¶* T2 â â¦h, Lâ¦ â¢ T2 Â¡[g]. +#h #g #L #T1 #HT1 #T2 #HT12 +@(snv_cpss_lpss_conf â¦ HT1 â¦ HT12) // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/ssta_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/ssta_lpss.etc new file mode 100644 index 000000000..7471b2304 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/ssta_lpss.etc @@ -0,0 +1,81 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 â¢[g] â¦l, U1â¦ â + âT2. L1 â¢ T1 â¶* T2 â âL2. L1 â¢ â¶* L2 â + ââU2. â¦h, L2â¦ â¢ T2 â¢[g] â¦l, U2â¦ & L1 â¢ U1 â¶* U2. +#h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l +[ #L1 #k1 #l1 #Hkl1 #X #H + >(cpss_inv_sort1 â¦ H) -H /3 width=3/ +| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #_ #HWU1 #IHVW1 #X #H #L2 #HL12 + elim (cpss_inv_lref1 â¦ H) -H + [ #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK1) -HLK1 #HLK1 + elim (IHVW1 â¦ HV12 â¦ HK12) -IHVW1 -HV12 -HK12 #W2 #HVW2 #HW12 + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (cpss_lift â¦ HW12 â¦ HLK1 â¦ HWU1 â¦ HWU2) -HW12 -HLK1 -HWU1 /3 width=6/ + | * #Y #Z #V2 #H #HV12 #HV2 + lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #Z #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #V0 #HK12 #_ #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) -V0 #HLK2 + lapply (ldrop_fwd_ldrop2 â¦ HLK1) -HLK1 #HLK1 + elim (IHVW1 â¦ HV12 â¦ HK12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (ssta_lift â¦ HVW2 â¦ HLK2 â¦ HV2 â¦ HWU2) -HVW2 -HLK2 -HV2 + lapply (cpss_lift â¦ HW12 â¦ HLK1 â¦ HWU1 â¦ 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 â¦ H) -H [ | -IHWV1 -HWU1 -HL12 ] + [ #H destruct + elim (lpss_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 + elim (lpss_inv_pair1 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK1) -HLK1 #HLK1 + elim (IHWV1 â¦ HW12 â¦ HK12) -IHWV1 -HK12 #V2 #HWV2 #_ + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (cpss_lift â¦ HW12 â¦ HLK1 â¦ HWU1 â¦ HWU2) -HW12 -HLK1 -HWU1 /3 width=6/ + | * #K2 #V2 #W2 #HLK2 #_ #_ + lapply (ldrop_mono â¦ HLK2 â¦ HLK1) -HLK1 -HLK2 #H destruct + ] +| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12 + elim (cpss_inv_bind1 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + elim (IHTU1 â¦ HT12 (L2.â{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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + elim (IHTU1 â¦ HT12 â¦ HL12) -IHTU1 -HT12 -HL12 /3 width=5/ +| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12 + elim (cpss_inv_flat1 â¦ H) -H #W2 #T2 #HW12 #HT12 #H destruct + elim (IHTU1 â¦ HT12 â¦ HL12) -IHTU1 -HT12 -HL12 /3 width=3/ +] +qed-. + +lemma ssta_cpss_conf: âh,g,L,T1,U1,l. â¦h, Lâ¦ â¢ T1 â¢[g] â¦l, U1â¦ â + âT2. L â¢ T1 â¶* T2 â + ââU2. â¦h, Lâ¦ â¢ T2 â¢[g] â¦l, U2â¦ & L â¢ U1 â¶* U2. +/2 width=3 by ssta_cpss_lpss_conf/ qed-. + +lemma ssta_lpss_conf: âh,g,L1,T,U1,l. â¦h, L1â¦ â¢ T â¢[g] â¦l, U1â¦ â + âL2. L1 â¢ â¶* L2 â + ââU2. â¦h, L2â¦ â¢ T â¢[g] â¦l, U2â¦ & L1 â¢ U1 â¶* U2. +/2 width=3 by ssta_cpss_lpss_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/sstas_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/sstas_lpss.etc new file mode 100644 index 000000000..3ecbc2256 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/sstas_lpss.etc @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,T1,U1. â¦h, L1â¦ â¢ T1 â¢*[g] U1 â + âT2. L1 â¢ T1 â¶* T2 â âL2. L1 â¢ â¶* L2 â + ââU2. â¦h, L2â¦ â¢ T2 â¢*[g] U2 & L1 â¢ U1 â¶* U2. +#h #g #L1 #T1 #U1 #H @(sstas_ind_dx â¦ H) -T1 /2 width=3/ +#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12 +elim (ssta_cpss_lpss_conf â¦ HTU0 â¦ HT0 â¦ HL12) -HTU0 -HT0 #U #HTU #HU0 +elim (IHU01 â¦ HU0 â¦ HL12) -IHU01 -U0 -HL12 /3 width=4/ +qed-. + +lemma sstas_cpss_conf: âh,g,L,T1,U1. â¦h, Lâ¦ â¢ T1 â¢*[g] U1 â + âT2. L â¢ T1 â¶* T2 â + ââU2. â¦h, Lâ¦ â¢ T2 â¢*[g] U2 & L â¢ U1 â¶* U2. +/2 width=3 by sstas_cpss_lpss_conf/ qed-. + +lemma sstas_lpss_conf: âh,g,L1,T,U1. â¦h, L1â¦ â¢ T â¢*[g] U1 â + âL2. L1 â¢ â¶* L2 â + ââU2. â¦h, L2â¦ â¢ T â¢*[g] U2 & L1 â¢ U1 â¶* U2. +/2 width=3 by sstas_cpss_lpss_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn.etc new file mode 100644 index 000000000..f6b54500e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn.etc @@ -0,0 +1,110 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 L1 : â [ ] 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 â +| lsubsn_atom: lsubsn h (â) (â) +| lsubsn_pair: âI,L1,L2,W. lsubsn h L1 L2 â + lsubsn h (L1. â{I} W) (L2. â{I} W) +| lsubsn_abbr: âL1,L2,V,W,l. â¦h, L1â¦ â¢ V :[l+1] W â â¦h, L2â¦ â¢ V :[l+1] W â + lsubsn h L1 L2 â lsubsn h (L1. âV) (L2. âW) +. + +interpretation + "local environment refinement (stratified native type assigment)" + 'StratifiedCrSubEqN h L1 L2 = (lsubsn h L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubsn_inv_atom1_aux: âh,L1,L2. h â¢ L1 :â[] L2 â L1 = â â L2 = â. +#h #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #l #_ #_ #_ #H destruct +] +qed. + +lemma lsubsn_inv_atom1: âh,L2. h â¢ â :â[] L2 â L2 = â. +/2 width=5/ qed-. + +fact lsubsn_inv_pair1_aux: âh,L1,L2. h â¢ L1 :â[] L2 â + âI,K1,V. L1 = K1. â{I} V â + (ââK2. h â¢ K1 :â[] K2 & L2 = K2. â{I} V) â¨ + ââK2,W,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & + h â¢ K1 :â[] K2 & L2 = K2. âW & 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: âh,I,K1,L2,V. h â¢ K1. â{I} V :â[] L2 â + (ââK2. h â¢ K1 :â[] K2 & L2 = K2. â{I} V) â¨ + ââK2,W,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & + h â¢ K1 :â[] K2 & L2 = K2. âW & I = Abbr. +/2 width=3/ qed-. + +fact lsubsn_inv_atom2_aux: âh,L1,L2. h â¢ L1 :â[] L2 â L2 = â â L1 = â. +#h #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #l #_ #_ #_ #H destruct +] +qed. + +lemma lsubsn_inv_atom2: âh,L1. h â¢ L1 :â[] â â L1 = â. +/2 width=5/ qed-. + +fact lsubsn_inv_pair2_aux: âh,L1,L2. h â¢ L1 :â[] L2 â + âI,K2,W. L2 = K2. â{I} W â + (ââK1. h â¢ K1 :â[] K2 & L1 = K1. â{I} W) â¨ + ââK1,V,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & + h â¢ K1 :â[] K2 & L1 = K1. âV & 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: âh,I,L1,K2,W. h â¢ L1 :â[] K2. â{I} W â + (ââK1. h â¢ K1 :â[] K2 & L1 = K1. â{I} W) â¨ + ââK1,V,l. â¦h, K1â¦ â¢ V :[l+1] W & â¦h, K2â¦ â¢ V :[l+1] W & + h â¢ K1 :â[] K2 & L1 = K1. âV & I = Abst. +/2 width=3/ qed-. + +(* Basic_forward lemmas *****************************************************) + +lemma lsubsn_fwd_lsubs1: âh,L1,L2. h â¢ L1 :â[] L2 â L1 â¼[0, |L1|] L2. +#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +lemma lsubsn_fwd_lsubs2: âh,L1,L2. h â¢ L1 :â[] L2 â L1 â¼[0, |L2|] L2. +#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +lemma lsubsn_refl: âh,L. h â¢ L :â[] L. +#h #L elim L -L // /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_cpcs.etc new file mode 100644 index 000000000..38f7e17bd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_cpcs.etc @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,L2. h â¢ L1 :â[] L2 â + âT1,T2. L2 â¢ T1 â¡ T2 â L1 â¢ T1 â¡ T2. +/3 width=5 by lsubsn_fwd_lsubs2, cpr_lsubs_trans/ qed-. + +lemma cprs_lsubsn_trans: âh,L1,L2. h â¢ L1 :â[] L2 â + âT1,T2. L2 â¢ T1 â¡* T2 â L1 â¢ T1 â¡* T2. +/3 width=5 by lsubsn_fwd_lsubs2, cprs_lsubs_trans/ qed-. + +lemma cpcs_lsubsn_trans: âh,L1,L2. h â¢ L1 :â[] L2 â + âT1,T2. L2 â¢ T1 â¬* T2 â L1 â¢ T1 â¬* T2. +/3 width=5 by lsubsn_fwd_lsubs2, cpcs_lsubs_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_ldrop.etc new file mode 100644 index 000000000..092883783 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_ldrop.etc @@ -0,0 +1,65 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,L2. h â¢ L1 :â[] L2 â + âK1,e. â©[0, e] L1 â¡ K1 â + ââK2. h â¢ K1 :â[] K2 & â©[0, e] L2 â¡ K2. +#h #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK1) -L1 /3 width=3/ + ] +] +qed. + +(* Note: the constant 0 cannot be generalized *) +lemma lsubsn_ldrop_O1_trans: âh,L1,L2. h â¢ L1 :â[] L2 â + âK2,e. â©[0, e] L2 â¡ K2 â + ââK1. h â¢ K1 :â[] K2 & â©[0, e] L1 â¡ K1. +#h #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 â¦ H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl â¦ H) in HL12; -H /3 width=3/ + | elim (IHL12 â¦ HLK2) -L2 /3 width=3/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_snta.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_snta.etc new file mode 100644 index 000000000..c6215302c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/lsubsn_snta.etc @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L2,T,U,l. â¦h, L2â¦ â¢ T :[l] U â + âL1. h â¢ L1 :â[] L2 â â¦h, L1â¦ â¢ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 + elim (lsubsn_inv_pair2 â¦ 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 â¦ HL12 â¦ HLK2) -L2 #X #H #HLK1 + elim (lsubsn_inv_pair2 â¦ H) -H * #K1 [ -HWV2 | -IHWV2 ] + [ #HK12 #H destruct /3 width=6/ + | #V1 #l0 #H1 #H2 #_ #H #_ destruct + elim (snta_fwd_correct â¦ H2) -H2 #V #H + elim (snta_mono â¦ HWV2 â¦ 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 â¦ HL12 â¦ HU12) -HU12 /3 width=3/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta.etc new file mode 100644 index 000000000..71ed42179 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta.etc @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ h , break L â¦ â¢ break term 46 T1 : * break [ l ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeStar $h $l $L $T1 $T2 }. + +notation "hvbox( â¦ h , break L â¦ â¢ 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 â lenv â relation term â +| snta_sort: âL,k. snta h 0 L (âk) (â(next h k)) +| snta_ldef: âL,K,V,W,U,i,l. â©[0, i] L â¡ K. âV â snta h l K V W â + â§[0, i + 1] W â¡ U â snta h l L (#i) U +| snta_ldec: âL,K,W,V,U,i,l. â©[0, i] L â¡ K. âW â snta h l K W V â + â§[0, i + 1] W â¡ U â snta h (l+1) L (#i) U +| snta_bind: âI,L,V,W,T,U,l1,l2. snta h l1 L V W â snta h l2 (L. â{I} V) T U â + snta h l2 L (â{I}V.T) (â{I}V.U) +| snta_appl: âL,V,W1,W2,T,U,l1,l2. snta h (l1+1) L V W2 â + snta h l2 L (âW1.T) (âW2.U) â + snta h l2 L (âV.âW1.T) (âV.âW2.U) +| snta_pure: âL,V,T,U,W,l. snta h (l+1) L T U â snta h l L (âV.U) W â + snta h (l+1) L (âV.T) (âV.U) +| snta_cast: âL,T,U,W,l1,l2. snta h l2 L T U â snta h l1 L U W â + snta h l2 L (âU.T) U +| snta_conv: âL,T,U1,U2,V2,l. snta h l L T U1 â L â¢ U1 â¬* U2 â + snta h (l-1) L U2 V2 â snta h l L T U2 +. + +interpretation "stratified native type assignment (term)" + 'StratifiedNativeType h l L T U = (snta h l L T U). diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_lift.etc new file mode 100644 index 000000000..424bfe89f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_lift.etc @@ -0,0 +1,238 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âk0. T = âk0 â + l = 0 â§ L â¢ â(next h k0) â¬* 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 â¦ Hk0 â¦ HU12) -U1 /2 width=1/ +] +qed. + +lemma snta_inv_sort1: âh,L,U,k,l. â¦h, Lâ¦ â¢ âk :[l] U â + l = 0 â§ L â¢ â(next h k) â¬* U. +/2 width=4/ qed-. + +fact snta_inv_lref1_aux: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âj. T = #j â + (ââK,V,W,U0. â©[0, j] L â¡ K. âV & â¦h, Kâ¦ â¢ V :[l] W & + â§[0, j + 1] W â¡ U0 & L â¢ U0 â¬* U + ) â¨ + (ââK,W,V,U0. â©[0, j] L â¡ K. âW & â¦h, Kâ¦ â¢ W :[l-1] V & + â§[0, j + 1] W â¡ U0 & l > 0 & L â¢ U0 â¬* 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 â¦ HU01 â¦ HU12) -U1 /3 width=8/ +] +qed. + +lemma snta_inv_lref1: âh,L,U,i,l. â¦h, Lâ¦ â¢ #i :[l] U â + (ââK,V,W,U0. â©[0, i] L â¡ K. âV & â¦h, Kâ¦ â¢ V :[l] W & + â§[0, i + 1] W â¡ U0 & L â¢ U0 â¬* U + ) â¨ + (ââK,W,V,U0. â©[0, i] L â¡ K. âW & â¦h, Kâ¦ â¢ W :[l-1] V & + â§[0, i + 1] W â¡ U0 & l > 0 & L â¢ U0 â¬* U + ). +/2 width=3/ qed-. + +fact snta_inv_bind1_aux: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âJ,X,Y. T = â{J}Y.X â + ââZ1,Z2,l0. â¦h, Lâ¦ â¢ Y :[l0] Z1 & + â¦h, L.â{J}Yâ¦ â¢ X :[l] Z2 & + L â¢ â{J}Y.Z2 â¬* 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 â¦ HU1 â¦ HU12) -U1 /2 width=3/ +] +qed. + +lemma snta_inv_bind1: âh,J,L,Y,X,U,l. â¦h, Lâ¦ â¢ â{J}Y.X :[l] U â + ââZ1,Z2,l0. â¦h, Lâ¦ â¢ Y :[l0] Z1 & â¦h, L.â{J}Yâ¦ â¢ X :[l] Z2 & + L â¢ â{J}Y.Z2 â¬* U. +/2 width=3/ qed-. + +fact snta_inv_cast1_aux: âh,L,T,U,l. â¦h, Lâ¦ â¢ T :[l] U â âX,Y. T = âY.X â + â¦h, Lâ¦ â¢ X :[l] Y â§ L â¢ Y â¬* 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 â¦ HU1 â¦ HU12) -U1 /2 width=1/ +] +qed. + +lemma snta_inv_cast1: âh,L,X,Y,U,l. â¦h, Lâ¦ â¢ âY.X :[l] U â + â¦h, Lâ¦ â¢ X :[l] Y â§ L â¢ Y â¬* U. +/2 width=3/ qed-. + +(* Properties on relocation *************************************************) + +lemma snta_lift: âh,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 :[l] U1 â + âL2,d,e. â©[d, e] L2 â¡ L1 â + âT2. â§[d, e] T1 â¡ T2 â âU2. â§[d, e] U1 â¡ U2 â + â¦h, L2â¦ â¢ 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 â¦ H1) -X1 + >(lift_inv_sort1 â¦ H2) -X2 // +| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HW1 â¦ HWU2 ?) -W // #W2 #HW12 #HWU2 + elim (ldrop_trans_le â¦ HL21 â¦ HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H + elim (ldrop_inv_skip2 â¦ H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct + /3 width=8/ + | lapply (lift_trans_be â¦ HW1 â¦ HWU2 ? ?) -W // /2 width=1/ #HW1U2 + lapply (ldrop_trans_ge â¦ HL21 â¦ 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 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HW1 â¦ HWU2 ?) -W // (tpr_inv_atom1 â¦ H) -H // +| #L1 #K1 #V1 #W #U #i1 #l #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H #Hl -IH + >(tpr_inv_atom1 â¦ H) -T2 + elim (ltpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 â¦ 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 â¦ H) -T2 + elim (ltpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 â¦ H) -H #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) #HLK + elim (lift_total V1 0 (i+1)) #W #HW + lapply (snta_lift h â¦ HLK â¦ HWU1 â¦ HW) /2 width=1/ -HLK -HW + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpr_lift â¦ HW12 â¦ HWU1 â¦ HWU2) -HWU1 #HU12 + @(snta_conv â¦ 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 â¦ H) -H * + [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct + lapply (IHVW1 â¦ HL12 â¦ HV12) #HV2W1 + lapply (IHVW1 L2 â¦ V1 ?) // -IHVW1 #HWV1 + lapply (IHTU1 (L2.â{I}V2) â¦ HT1) -HT1 /2 width=1/ #HTU1 + lapply (IHTU1 (L2.â{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H + lapply (tps_lsubs_trans â¦ HT2 (L2.â{I}V2) ?) -HT2 /2 width=1/ #HT2 + lapply (snta_tps_conf â¦ HTU1 â¦ HT2) -T #HT2U1 + elim (snta_fwd_correct â¦ H) -H #U2 #HU12 + @(snta_conv â¦ (â{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | #T #HT1 #HTX #H destruct + lapply (IHVW1 â¦ HL12 V1 ?) -IHVW1 // #HVW1 + lapply (IHTU1 (L2.âV1) â¦ HT1) -T1 /2 width=1/ -L1 #H + elim (snta_fwd_correct â¦ H) #T1 #HUT1 + elim (snta_ldrop_conf â¦ H L2 0 1 ? ?) -H // /2 width=1/ #T0 #U0 #HTU0 #H #HU10 + lapply (delift_inv_lift1_eq â¦ H L2 â¦ HTX) -H -HTX /2 width=1/ #H destruct + @(snta_conv â¦ HTU0) /2 width=2/ + ] +*) +| #L1 #V1 #W11 #W2 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH +(* + elim (tpr_inv_appl1 â¦ H) -H * + [ #V2 #Y #HV12 #HY #H destruct + elim (tpr_inv_abst1 â¦ HY) -HY #W2 #T2 #HW12 #HT12 #H destruct + lapply (IHTU1 L2 ? (âW1.T1) ?) // #H + elim (snta_fwd_correct â¦ H) -H #X #H + elim (snta_inv_bind1 â¦ H) -H #W #U #HW #HU #_ + @(snta_conv â¦ (âV2.âW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + lapply (IHVW1 â¦ HL12 â¦ HV12) #HVW2 + lapply (IHVW1 â¦ HL12 V1 ?) -IHVW1 // #HV1W2 + lapply (IHTU1 â¦ HL12 (âW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1 + elim (snta_fwd_correct â¦ H1) #T #H2 + elim (snta_inv_bind1 â¦ H1) -H1 #W #U2 #HW2 #HTU2 #H + elim (cpcs_inv_abst â¦ H Abst W2) -H #_ #HU21 + elim (snta_inv_bind1 â¦ H2) -H2 #W0 #U0 #_ #H #_ -T -W0 + lapply (lsubsn_snta_trans â¦ HTU2 (L2.âV2) ?) -HTU2 /2 width=1/ #HTU2 + @(snta_conv â¦ (âV2.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 â¦ H) -H * + [ #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpr â¦ HV12 L2) #HV + elim (snta_fwd_correct h L2 (âV1.T1) (âV1.U1) (l+1) ?) [2: /3 width=6/ ] #U + @(snta_conv â¦ (âV2.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 â¦ HL12 (âW0.T2) ? ?) -IHTU1 // /2 width=1/ -T0 #H1 + lapply (IH â¦ (âV2.U1) â¦ HUW1 HL12 ?) // /3 width=1/ #H2 + lapply (snta_pure â¦ H1 H2) -H2 #H + elim (snta_inv_bind1 â¦ H1) -H1 #V0 #U2 #l1 #HWV0 #HTU2 #HU21 + @(snta_conv â¦ (âV2.U2)) (**) (* explicit constructor *) + [2: +(* + @snta_bind /3 width=2/ /3 width=6/ (**) (* /4 width=6/ is a bit slow *) +*) + |3: @(cpcs_cpr_conf â¦ (âV1.âW0.U2)) /2 width=1/ + |4: /2 width=5/ + | skip + ] +(* + elim (snta_fwd_pure1 â¦ H) -H #T1 #W2 #HVW2 #HUT1 #HTW1 + + elim (cpcs_inv_abst1 â¦ HU21) #W3 #U3 #HU13 #H + elim (cprs_inv_abst â¦ H Abst W0) -H #HW03 #_ + elim (pippo â¦ IH â¦ 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 â¦ HV1 â¦ HL12 â¦ HV12) -HV1 -HV12 /width=5/ #HB + lapply (IH â¦ HB0 â¦ HL12 W2 ?) -HB0 /width=5/ #HB0 + lapply (IH â¦ HA0 â¦ (L2.âW2) â¦ HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ + +axiom pippo: â¦h, Lâ¦ â¢ âV.X : Y â + ââW,T. L â¢ X â¡* âW.T & â¦h, Lâ¦ â¢ âV : W. + +*) +(* SEGMENT 2 +| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 â¦ H) -H #U2 #T2 #HU12 #HT12 #H destruct + lapply (cpr_tpss â¦ HU12) /4 width=4/ +| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 + @(snta_conv â¦ U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) +] +qed. +*) + +(* SEGMENT 3 +fact snta_ltpr_tpr_conf_aux: âh,L,T,L1,T1,U. â¦h, L1â¦ â¢ T1 : U â L = L1 â T = T1 â + âL2. L1 â¡ L2 â âT2. T1 â¡ T2 â â¦h, L2â¦ â¢ T2 : U. + + + | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct + elim (snta_inv_abbr â¦ HT1) -HT1 #B0 #HW0 #HT0 + lapply (IH â¦ HW0 â¦ HL12 â¦ HW02) -HW0 /width=5/ #HW2 + lapply (IH â¦ HV1 â¦ HL12 â¦ HV10) -HV1 -HV10 /width=5/ #HV0 + lapply (IH â¦ HT0 â¦ (L2.âW2) â¦ HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2 + @(snta_abbr â¦ HW2) -HW2 + @(snta_appl â¦ 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 â¦ H) -H + [ * #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH â¦ HV1 â¦ HL12 â¦ HV12) -HV1 -HV12 /width=5/ #HV2 + lapply (IH â¦ HT1 â¦ HL12 â¦ HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ + | -HV1 #HT1X + lapply (IH â¦ HT1 â¦ HL12 â¦ HT1X) -IH -HT1 -HL12 -HT1X /width=5/ + ] +] +qed. + +lemma snta_ltpr_tpr_conf: âh,L1,T1,U. â¦h, L1â¦ â¢ T1 : U â âL2. L1 â¡ L2 â + âT2. T1 â¡ T2 â â¦h, L2â¦ â¢ T2 : U. + +/2 width=9/ qed. + +axiom snta_ltpr_conf: âL1,T,A. L1 â¢ T : A â âL2. L1 â¡ L2 â L2 â¢ T : A. +/2 width=5/ qed. + +axiom snta_tpr_conf: âL,T1,A. L â¢ T1 : A â âT2. T1 â¡ T2 â L â¢ T2 : A. +/2 width=5/ qed. +*) +*)*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_ltpss.etc new file mode 100644 index 000000000..0e5f3930e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_ltpss.etc @@ -0,0 +1,123 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,T1,U,l. â¦h, L1â¦ â¢ T1 :[l] U â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶* [d, e] T2 â â¦h, L2â¦ â¢ 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 â¦ H) -H // +| #L1 #K1 #V1 #W #U #i #l #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 â¦ H) -H + [ #H destruct + elim (lt_or_ge i d) #Hdi + [ elim (ltpss_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct + /3 width=7/ + | lapply (ltpss_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -L1 // -Hide /3 width=7/ + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 + elim (ltpss_ldrop_conf_be â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct + lapply (ldrop_mono â¦ HL2K0 â¦ HLK2) -HL2K0 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) -HLK2 #HLK2 + lapply (tpss_trans_eq â¦ 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 â¦ 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 â¦ HL12 â¦ HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 â¦ H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) #HLK + lapply (snta_lift h â¦ HLK â¦ HWU1 â¦ HW) [ /2 width=4/ | skip ] -HW #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge â¦ HW12 â¦ HLK â¦ HWU1 â¦ HWU2) -HLK -HWU1 // #HU12 + lapply (cpr_tpss â¦ HU12) -HU12 #HU12 + @(snta_conv â¦ 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 â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK2) #HLK + lapply (snta_lift h â¦ HLK â¦ HWU1 â¦ HW) [ /2 width=4/ | skip ] -HW #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge â¦ HW12 â¦ HLK â¦ HWU1 â¦ HWU2) -HLK -HWU1 // #HU12 + lapply (cpr_tpss â¦ HU12) -HU12 #HU12 + @(snta_conv â¦ U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | lapply (ltpss_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -L1 // -Hide /2 width=6/ + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_ + elim (ltpss_ldrop_conf_be â¦ HL12 â¦ HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_inv_tpss21 â¦ H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct + lapply (ldrop_mono â¦ HL2K0 â¦ 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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpss â¦ HV12) #HV + lapply (IHTU1 (L2.â{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H + elim (snta_fwd_correct â¦ H) -H #U2 #HU12 + @(snta_conv â¦ (â{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 â¦ H) -H #V2 #Y #HV12 #HY #H destruct + elim (tpss_inv_bind1 â¦ HY) -HY #W21 #T2 #HW121 #HT12 #H destruct + lapply (cpr_tpss â¦ HV12) #HVV12 + lapply (IHTU1 L2 d e ? (âW21.T2) ?) -IHTU1 // /2 width=1/ -HW121 -HT12 #H0 + elim (snta_fwd_correct â¦ H0) #X #H + elim (snta_inv_bind1 â¦ H) -H #W #U #l0 #HW #HU #_ + @(snta_conv â¦ (âV2.âW12.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 â¦ H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpss â¦ HV12) #HV + elim (snta_fwd_correct h L2 (âV1.T1) (âV1.U1) (l+1) ?) [2: /3 width=4/ ] #U + @(snta_conv â¦ (âV2.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 â¦ HTU1) -HTU1 #U #H + elim (snta_mono â¦ HUW1 â¦ H) -HUW1 -H #H #_ -U destruct + elim (tpss_inv_flat1 â¦ H) -H #U2 #T2 #HU12 #HT12 #H destruct + lapply (cpr_tpss â¦ HU12) #HU /4 width=4/ +| #L1 #T1 #U11 #U12 #U #l #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 + @(snta_conv â¦ U11) /2 width=5/ (**) (* explicit constructor, /3 width=7/ is too slow *) +] +qed. + +lemma snta_ltpss_tps_conf: âh,L1,T1,U,l. â¦h, L1â¦ â¢ T1 :[l] U â + âL2,d,e. L1 â¶* [d, e] L2 â + âT2. L2 â¢ T1 â¶ [d, e] T2 â â¦h, L2â¦ â¢ T2 :[l] U. +/3 width=7/ qed. + +lemma snta_ltpss_conf: âh,L1,T,U,l. â¦h, L1â¦ â¢ T :[l] U â + âL2,d,e. L1 â¶* [d, e] L2 â â¦h, L2â¦ â¢ T :[l] U. +/2 width=7/ qed. + +lemma snta_tpss_conf: âh,L,T1,U,l. â¦h, Lâ¦ â¢ T1 :[l] U â + âT2,d,e. L â¢ T1 â¶* [d, e] T2 â â¦h, Lâ¦ â¢ T2 :[l] U. +/2 width=7/ qed. + +lemma snta_tps_conf: âh,L,T1,U,l. â¦h, Lâ¦ â¢ T1 :[l] U â + âT2,d,e. L â¢ T1 â¶ [d, e] T2 â â¦h, Lâ¦ â¢ T2 :[l] U. +/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_snta.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_snta.etc new file mode 100644 index 000000000..db71e1192 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_snta.etc @@ -0,0 +1,65 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L,T,U1,l1. â¦h, Lâ¦ â¢ T :[l1] U1 â + âU2,l2. â¦h, Lâ¦ â¢ T :[l2] U2 â l1 = l2 â§ L â¢ U1 â¬* U2. +#h #L #T #U1 #l1 #H elim H -L -T -U1 -l1 +[ #L #k #X #l2 #H + lapply (snta_inv_sort1 â¦ H) -H * /2 width=1/ +| #L #K #V #W11 #W12 #i #l1 #HLK #_ #HW112 #IHVW11 #X #l2 #H + elim (snta_inv_lref1 â¦ H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX + lapply (ldrop_mono â¦ HLK0 â¦ HLK) -HLK0 #H destruct + lapply (ldrop_fwd_ldrop2 â¦ HLK) -HLK #HLK + elim (IHVW11 â¦ HVW21) -IHVW11 -HVW21 #Hl12 #HW121 + lapply (cpcs_lift â¦ HLK â¦ HW112 â¦ HW212 ?) // -K -W11 -W21 /3 width=3/ +| #L #K #W #V1 #V #i #l1 #HLK #_ #HWV #IHWV1 #X #l2 #H + elim (snta_inv_lref1 â¦ H) -H * #K0 #W0 #V2 #V0 #HLK0 #HW0V2 #HWV0 [2: #HL2 ] #HX + lapply (ldrop_mono â¦ HLK0 â¦ HLK) -HLK0 -HLK #H destruct + lapply (lift_mono â¦ HWV0 â¦ HWV) -HWV0 -HWV #H destruct + elim (IHWV1 â¦ HW0V2) -IHWV1 -HW0V2 /3 width=1/ +| #I #L #V #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H + elim (snta_inv_bind1 â¦ H) -H #W2 #U2 #l20 #_ #HTU2 #H + elim (IHTU1 â¦ HTU2) -IHTU1 -HTU2 #Hl12 #HU12 + lapply (cpcs_trans â¦ (â{I}V.U1) â¦ H) -H /2 width=1/ +| #L #V #W #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H + elim (snta_fwd_pure1 â¦ H) -H #U2 #W2 #l20 #_ #HTU2 #H + elim (IHTU1 â¦ HTU2) -IHTU1 -HTU2 #Hl12 #HU12 + lapply (cpcs_trans â¦ (âV.âW1.U1) â¦ H) -H /2 width=1/ +| #L #V #T #U1 #W1 #l1 #_ #_ #IHTU1 #_ #X #l2 #H + elim (snta_fwd_pure1 â¦ H) -H #U2 #W2 #l20 #_ #HTU2 #H + elim (IHTU1 â¦ HTU2) -IHTU1 -HTU2 #Hl12 #HU12 + lapply (cpcs_trans â¦ (âV.U1) â¦ H) -H /2 width=1/ +| #L #T #U1 #W1 #l10 #l1 #_ #_ #IHTU1 #_ #X #l2 #H + elim (snta_inv_cast1 â¦ H) -H #HTU1 + elim (IHTU1 â¦ HTU1) -IHTU1 -HTU1 /2 width=1/ +| #L #T #U11 #U12 #V12 #l1 #_ #HU112 #_ #IHTU11 #_ #U2 #l2 #HTU2 + elim (IHTU11 â¦ HTU2) -IHTU11 -HTU2 #Hl12 #H + lapply (cpcs_canc_sn â¦ HU112 â¦ H) -U11 /2 width=1/ +] +qed-. + +(* Advanced properties ******************************************************) + +lemma snta_cast_alt: âh,L,T,W,U,l. â¦h, Lâ¦ â¢ T :[l] W â â¦h, Lâ¦ â¢ T :[l] U â + â¦h, Lâ¦ â¢ âW.T :[l] U. +#h #L #T #W #U #l #HTW #HTU +elim (snta_mono â¦ HTW â¦ HTU) #_ #HWU +elim (snta_fwd_correct â¦ HTU) -HTU /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_thin.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_thin.etc new file mode 100644 index 000000000..ceb5375bf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snta/snta_thin.etc @@ -0,0 +1,116 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,L1,T1,U1,l. â¦h, L1â¦ â¢ T1 :[l] U1 â + âL2,d,e. â½ [d, e] L1 â L1 â¼*[d, e] â¡ L2 â + ââT2,U2. â¦h, L2â¦ â¢ T2 :[l] U2 & + L1 â¢ T1 â¼*[d, e] â¡ T2 & L1 â¢ U1 â¼*[d, e] â¡ 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 â¦ HLK1 â¦ HL1 ?) -HL1 /2 width=2/ #H + lapply (sfr_inv_skip â¦ H ?) -H /2 width=1/ #HK1 + elim (thin_ldrop_conf_le â¦ HL12 â¦ HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2 + elim (thin_inv_delift1 â¦ H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 â¦ HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12 + lapply (delift_mono â¦ H â¦ HV12) -H -HV12 #H destruct + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (ldrop_fwd_ldrop2 â¦ HLK1) -V1 #HLK1 + lapply (delift_lift_ge â¦ HW12 â¦ HLK1 HWU1 â¦ HWU2) -HW12 -HLK1 -HWU1 // + >minus_plus minus_plus #HU1 + lapply (lift_conf_be â¦ HWU2 â¦ HW2U ?) -W2 /2 width=1/ #HU2 + lapply (delift_lift_div_be â¦ HU1 â¦ HU2 ? ?) -U // /2 width=1/ /3 width=8/ + | lapply (transitive_le â¦ (i+1) Hide ?) /2 width=1/ #Hdei + lapply (thin_ldrop_conf_ge â¦ HL12 â¦ HLK1 ?) -HL12 -HLK1 // #HL2K1 + elim (lift_split â¦ HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W + commutative_plus minus_plus commutative_plus â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ T2 Â¡[h, g]. +/3 width=6 by lpr_lpx, cpr_cpx/ qed-. + +fact snv_sta_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + âG,L,T. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, Tâ¦ â â¦G, Lâ¦ â¢ T Â¡[h, g] â + âl. â¦G, Lâ¦ â¢ T âª[h, g] l+1 â + âU. â¦G, Lâ¦ â¢ T â¢[h] U â â¦G, Lâ¦ â¢ U Â¡[h, g]. +/3 width=6 by sta_cpx/ qed-. + +fact snv_cpxs_lpx_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âT2. â¦G, L1â¦ â¢ T1 â¡*[h, g] T2 â âL2. â¦G, L1â¦ â¢ â¡[h, g] L2 â â¦G, L2â¦ â¢ T2 Â¡[h, g]. +#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H +@(cpxs_ind â¦ H) -T2 /4 width=6 by fpbg_fpbs_trans, cpxs_fpbs/ +qed-. + +fact snv_cprs_lpr_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ T2 Â¡[h, g]. +/3 width=10 by snv_cpxs_lpx_aux, cprs_cpxs, lpr_lpx/ qed-. + +fact snv_lstas_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âl1,l2. l2 â¤ l1 â â¦G, L1â¦ â¢ T1 âª[h, g] l1 â + âU1. â¦G, L1â¦ â¢ T1 â¢*[h, l2] U1 â â¦G, L1â¦ â¢ U1 Â¡[h, g]. +/3 width=12 by snv_cpxs_lpx_aux, lstas_cpxs/ qed-. + +fact da_cprs_lpr_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âl. â¦G, L1â¦ â¢ T1 âª[h, g] l â + âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ T2 âª[h, g] l. +#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H +@(cprs_ind â¦ H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/ +qed-. + +fact da_cpcs_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + âG,L,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, T1â¦ â â¦G, Lâ¦ â¢ T1 Â¡[h, g] â + âT2. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, T2â¦ â â¦G, Lâ¦ â¢ T2 Â¡[h, g] â + âl1. â¦G, Lâ¦ â¢ T1 âª[h, g] l1 â âl2. â¦G, Lâ¦ â¢ T2 âª[h, g] l2 â + â¦G, Lâ¦ â¢ T1 â¬* T2 â 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 â¦ H) -H /4 width=18 by da_cprs_lpr_aux, da_mono/ +qed-. + +fact sta_cpr_lpr_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âl. â¦G, L1â¦ â¢ T1 âª[h, g] l+1 â + âU1. â¦G, L1â¦ â¢ T1 â¢[h] U1 â + âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â + ââU2. â¦G, L2â¦ â¢ T2 â¢[h] U2 & â¦G, L2â¦ â¢ U1 â¬* U2. +#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #H01 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12 +elim (IH â¦ H01 â¦ 1 â¦ Hl U1 â¦ HT12 â¦ HL12) -H01 -Hl -HT12 -HL12 +/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/ +qed-. + +fact lstas_cprs_lpr_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âl1,l2. l2 â¤ l1 â â¦G, L1â¦ â¢ T1 âª[h, g] l1 â + âU1. â¦G, L1â¦ â¢ T1 â¢*[h, l2] U1 â + âT2. â¦G, L1â¦ â¢ T1 â¡* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â + ââU2. â¦G, L2â¦ â¢ T2 â¢*[h, l2] U2 & â¦G, L2â¦ â¢ U1 â¬* U2. +#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H +@(cprs_ind â¦ H) -T2 [ /2 width=10 by/ ] +#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12 +elim (IHT1 L1) // -IHT1 #U #HTU #HU1 +elim (IH1 â¦ Hl21 â¦ HTU â¦ HTT2 â¦ 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: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âl,l1. l â¤ l1 â â¦G, L1â¦ â¢ T1 âª[h, g] l1 â âU1. â¦G, L1â¦ â¢ T1 â¢*[h, l] U1 â + âT2. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T2â¦ â â¦G, L1â¦ â¢ T2 Â¡[h, g] â + âl2. l â¤ l2 â â¦G, L1â¦ â¢ T2 âª[h, g] l2 â âU2. â¦G, L1â¦ â¢ T2 â¢*[h, l] U2 â + â¦G, L1â¦ â¢ T1 â¬* T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â â¦G, L2â¦ â¢ U1 â¬* 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 â¦ H) -H #T #H1 #H2 +elim (lstas_cprs_lpr_aux â¦ H01 HT1 â¦ Hl1 HTl1 â¦ HTU1 â¦ H1 â¦ HL12) -T1 /2 width=1 by/ #W1 #H1 #HUW1 +elim (lstas_cprs_lpr_aux â¦ H02 HT2 â¦ Hl2 HTl2 â¦ HTU2 â¦ H2 â¦ HL12) -T2 /2 width=1 by/ #W2 #H2 #HUW2 -L0 -T0 +lapply (lstas_mono â¦ H1 â¦ H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/ +qed-. + +fact lstas_cpds_aux: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + âG,L,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L, T1â¦ â â¦G, Lâ¦ â¢ T1 Â¡[h, g] â + âl1,l2. l2 â¤ l1 â â¦G, Lâ¦ â¢ T1 âª[h, g] l1 â + âU1. â¦G, Lâ¦ â¢ T1 â¢*[h, l2] U1 â âT2. â¦G, Lâ¦ â¢ T1 â¢*â¡*[h, g] T2 â + ââU2,l. l â¤ l2 & â¦G, Lâ¦ â¢ T2 â¢*[h, l] U2 & â¦G, Lâ¦ â¢ U1 â¢*â¬*[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 â¦ H â¦ Hl1) -H #H destruct +lapply (lstas_da_conf â¦ HTU1 â¦ Hl1) #Hl12 +elim (le_or_ge l2 l) #Hl2 +[ lapply (lstas_conf_le â¦ HTU1 â¦ HT1T) -HT1T + /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/ +| lapply (lstas_da_conf â¦ HT1T â¦ Hl1) #Hl1l + lapply (lstas_conf_le â¦ HT1T â¦ HTU1) -HTU1 // #HTU1 + elim (lstas_cprs_lpr_aux â¦ IH3 IH2 IH1 â¦ Hl1l â¦ HTU1 â¦ 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: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + âG,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G, L1, T1â¦ â â¦G, L1â¦ â¢ T1 Â¡[h, g] â + âU1. â¦G, L1â¦ â¢ T1 â¢*â¡*[h, g] U1 â + âT2. â¦G, L1â¦ â¢ T1 â¡ T2 â âL2. â¦G, L1â¦ â¢ â¡ L2 â + ââU2. â¦G, L2â¦ â¢ T2 â¢*â¡*[h, g] U2 & â¦G, L2â¦ â¢ U1 â¡* 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 â¦ H01 â¦ HTW1 â¦ HT12 â¦ HL12) -IH1 // #W2 #HTW2 #HW12 +lapply (IH2 â¦ H01 â¦ Hl1 â¦ HT12 â¦ HL12) -L0 -T0 // -T1 +lapply (lpr_cprs_conf â¦ HL12 â¦ HWU1) -L1 #HWU1 +lapply (cpcs_canc_sn â¦ HW12 HWU1) -W1 #H +elim (cpcs_inv_cprs â¦ H) -H /3 width=7 by ex4_3_intro, ex2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/snv/snv_da_lpr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/snv/snv_da_lpr.etc new file mode 100644 index 000000000..119a40243 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/snv/snv_da_lpr.etc @@ -0,0 +1,93 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G0,L0,T0. + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â 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 â¦ H2) -H2 + lapply (cpr_inv_sort1 â¦ 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 â¦ H1) -H1 #I0 #K0 #X0 #H #HX0 + elim (da_inv_lref â¦ H2) -H2 * #K1 [ #V1 | #W1 #l1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct + lapply (ldrop_mono â¦ H â¦ HLK1) -H #H destruct + elim (cpr_inv_lref1 â¦ H3) -H3 + [1,3: #H destruct + lapply (fqup_lref â¦ G1 â¦ HLK1) + elim (lpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #H #HLK2 + elim (lpr_inv_pair1 â¦ 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 â¦ H â¦ HLK1) -H #H destruct + lapply (fqup_lref â¦ G1 â¦ HLK1) + elim (lpr_ldrop_conf â¦ HLK1 â¦ HL12) -HLK1 -HL12 #X #H #HLK2 + elim (lpr_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #_ #H destruct + lapply (ldrop_fwd_drop2 â¦ HLK2) -V2 + /4 width=8 by da_lift, fqup_fpbg/ + ] +| #p #_ #_ #HT0 #H1 destruct -IH2 -IH1 + elim (snv_inv_gref â¦ H1) +| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2 + elim (snv_inv_bind â¦ H1) -H1 #_ #HT1 + lapply (da_inv_bind â¦ H2) -H2 + elim (cpr_inv_bind1 â¦ 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 â¦ H1) -H1 #b0 #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10 + lapply (da_inv_flat â¦ H2) -H2 #Hl + elim (cpr_inv_appl1 â¦ 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 â¦ HT1) -HT1 #HW #HU1 + lapply (da_inv_bind â¦ Hl) -Hl #Hl + elim (cpds_inv_abst1 â¦ HT10) -HT10 #W3 #U3 #HW3 #_ #H destruct -U3 + lapply (cprs_div â¦ HW3 â¦ HW10) -W3 #HWW1 + lapply (da_sta_conf â¦ HVW1 â¦ Hl0) â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â 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 â¦ H2) -H2 // * // +| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2 + elim (snv_inv_lref â¦ H1) -H1 #I #K1 #V1 #HLK1 #HV1 + elim (lpx_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 + elim (lpx_inv_pair1 â¦ H) -H #K2 #V2 #HK12 #HV12 #H destruct + lapply (fqup_lref â¦ G1 â¦ HLK1) #HKL + elim (cpx_inv_lref1 â¦ H2) -H2 + [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/ + | * #I0 #K0 #V0 #W0 #H #HVW0 #W0 -HV12 + lapply (ldrop_mono â¦ H â¦ HLK1) -HLK1 -H #H destruct + lapply (ldrop_fwd_drop2 â¦ 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 â¦ H1) +| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH3 -IH2 + elim (snv_inv_bind â¦ H1) -H1 #HV1 #HT1 + elim (cpx_inv_bind1 â¦ 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 â¦ H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1 + elim (cpx_inv_appl1 â¦ H2) -H2 * + [ #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH1 â¦ HV12 â¦ HL12) /2 width=1 by fqup_fpbg/ #HV2 + lapply (IH1 â¦ HT12 â¦ HL12) /2 width=1 by fqup_fpbg/ #HT2 +(* lapply (IH2 â¦ Hl0 â¦ HV12 â¦ HL12) /2 width=1 by fqup_fpbg/ #H2l0 + elim (sta_cpr_lpr_aux â¦ IH3 â¦ Hl0 â¦ HVW1 â¦ HV12 â¦ HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fqup_fpbg/ -HV1 #W2 #HVW2 #HW12 + elim (cpds_cpr_lpr_aux â¦ IH2 IH3 â¦ HTU1 â¦ HT12 â¦ HL12) /2 width=1 by fqup_fpbg/ -HT12 -HTU1 #X #HTU2 #H + elim (cprs_inv_abst1 â¦ H) -H #W20 #U2 #HW120 #_ #H destruct + lapply (lpr_cprs_conf â¦ HL12 â¦ HW10) -L1 #HW10 + lapply (cpcs_cprs_strap1 â¦ HW10 â¦ HW120) -W1 #HW120 + lapply (cpcs_canc_sn â¦ HW12 HW120) -W10 #HW20 + elim (cpcs_inv_cprs â¦ HW20) -HW20 #W0 #HW20 #HW200 + lapply (cpds_cprs_trans â¦ (â{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 â¦ HT1) -HT1 #HW20 #HT20 + elim (cpds_inv_abst1 â¦ HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30 + lapply (cprs_div â¦ HW10 â¦ HW230) -W30 #HW120 + lapply (snv_sta_aux â¦ IH1 â¦ Hl0 â¦ HVW1) /2 width=1 by fqup_fpbg/ #HW10 + lapply (da_sta_conf â¦ HVW1 â¦ Hl0) â¡[h, g] â¦G1, L1, T1â¦ â IH_snv_cpx_lpx h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_da_cpr_lpr h g G1 L1 T1) â + (âG1,L1,T1. â¦G0, L0, T0â¦ >â¡[h, g] â¦G1, L1, T1â¦ â IH_lstas_cpr_lpr h g G1 L1 T1) â + âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â 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 â¦ H2) -X2 + >(cpr_inv_sort1 â¦ H3) -X3 /2 width=3 by ex2_intro/ +| #i #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus â¦ l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH3 + [ lapply (lstas_inv_O â¦ H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ] + elim (snv_inv_lref â¦ H1) -H1 #I0 #K0 #X0 #HK0 #HX0 + elim (da_inv_lref â¦ Hl1) -Hl1 * #K1 [ #V1 | #W1 #l0 ] #HLK1 [ #HVl1 | #HWl1 #H destruct ] + lapply (ldrop_mono â¦ HK0 â¦ HLK1) -HK0 #H destruct + elim (lstas_inv_lref1 â¦ H2) -H2 * #K0 #V0 #W0 [2,4: #X0 ] #HK0 [1,2: #_ -X0 ] #HVW0 #HX2 + lapply (ldrop_mono â¦ HK0 â¦ HLK1) -HK0 #H destruct + [ lapply (le_plus_to_le_r â¦ Hl21) -Hl21 #Hl21 ] + lapply (fqup_lref â¦ G1 â¦ HLK1) #HKV1 + elim (lpr_ldrop_conf â¦ HLK1 â¦ HL12) -HL12 #X #H #HLK2 + elim (lpr_inv_pair1 â¦ H) -H #K2 [ #W2 | #V2 ] #HK12 [ #HW12 | #HV12 ] #H destruct + lapply (ldrop_fwd_drop2 â¦ HLK2) #H2 + elim (cpr_inv_lref1 â¦ H3) -H3 + [1,3: #H destruct -HLK1 + |2,4: * #K0 #V0 #X0 #H #HVX0 #HX0 + lapply (ldrop_mono â¦ H â¦ HLK1) -H -HLK1 #H destruct + ] + [ lapply (IH2 â¦ HWl1 â¦ HW12 â¦ HK12) /2 width=1 by fqup_fpbg/ -IH2 #H + elim (da_inv_sta â¦ H) -H + elim (IH1 â¦ HWl1 â¦ HVW0 â¦ HW12 â¦ 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 â¦ HVl1 â¦ HVW0 â¦ HV12 â¦ 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 â¦ HVl1 â¦ HVW0 â¦ HVX0 â¦ 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 â¦ 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 â¦ H1) -H1 #_ #HT1 + lapply (da_inv_bind â¦ Hl1) -Hl1 #Hl1 + elim (lstas_inv_bind1 â¦ H2) -H2 #U1 #HTU1 #H destruct + elim (cpr_inv_bind1 â¦ H3) -H3 * + [ #V2 #T2 #HV12 #HT12 #H destruct + elim (IH1 â¦ Hl1 â¦ HTU1 â¦ HT12 (L2.â{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 â¦ Hl1 â¦ HTU1 â¦ HT13 (L2.âV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13 + elim (lstas_inv_lift1 â¦ HTU3 L2 â¦ 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 â¦ H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU10 + lapply (da_inv_flat â¦ Hl1) -Hl1 #Hl1 + elim (lstas_inv_appl1 â¦ H2) -H2 #U1 #HTU1 #H destruct + elim (cpr_inv_appl1 â¦ H3) -H3 * + [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2 + elim (IH1 â¦ Hl1 â¦ HTU1 â¦ HT12 â¦ 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 â¦ HT1) -HT1 #HW2 #HT2 + lapply (da_inv_bind â¦ Hl1) -Hl1 #Hl1 + elim (lstas_inv_bind1 â¦ HTU1) -HTU1 #U2 #HTU2 #H destruct + elim (cpds_inv_abst1 â¦ HTU10) -HTU10 #W0 #U0 #HW20 #_ #H destruct + lapply (cprs_div â¦ HW10 â¦ HW20) -W0 #HW12 + lapply (da_sta_conf â¦ HVW1 â¦ Hl0) iter_SO // +qed-. + +lemma lstas_inv_gref1: âh,G,L,X,p,l. â¦G, Lâ¦ â¢ Â§p â¢*[h, l+1] X â â¥. +#h #G #L #X #p #l #H elim (lstas_inv_step_sn â¦ H) -H +#U #H #HUX elim (sta_inv_gref1 â¦ H) +qed-. + +lemma lstas_inv_bind1: âh,a,I,G,L,V,T,X,l. â¦G, Lâ¦ â¢ â{a,I}V.T â¢*[h, l] X â + ââU. â¦G, L.â{I}Vâ¦ â¢ T â¢*[h, l] U & X = â{a,I}V.U. +#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx â¦ H) -X -l /2 width=3 by ex2_intro/ +#l #X #X0 #_ #HX0 * #U #HTU #H destruct +elim (sta_inv_bind1 â¦ HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/ +qed-. + +lemma lstas_inv_appl1: âh,G,L,V,T,X,l. â¦G, Lâ¦ â¢ âV.T â¢*[h, l] X â + ââU. â¦G, Lâ¦ â¢ T â¢*[h, l] U & X = âV.U. +#h #G #L #V #T #X #l #H @(lstas_ind_dx â¦ H) -X -l /2 width=3 by ex2_intro/ +#l #X #X0 #_ #HX0 * #U #HTU #H destruct +elim (sta_inv_appl1 â¦ HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/ +qed-. + +lemma lstas_inv_cast1: âh,G,L,W,T,U,l. â¦G, Lâ¦ â¢ âW.T â¢*[h, l+1] U â â¦G, Lâ¦ â¢ T â¢*[h, l+1] U. +#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn â¦ H) -H +#U #H #HUX lapply (sta_inv_cast1 â¦ H) -H /2 width=3 by lstar_S/ +qed-. + +(* Basic properties *********************************************************) + +lemma lstas_refl: âh,G,L. reflexive â¦ (lstas h G L 0). +// qed. + +lemma sta_lstas: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â â¦G, Lâ¦ â¢ T â¢*[h, 1] U. +/2 width=1 by lstar_step/ qed. + +lemma lstas_step_sn: âh,G,L,T1,U1,U2,l. â¦G, Lâ¦ â¢ T1 â¢[h] U1 â â¦G, Lâ¦ â¢ U1 â¢*[h, l] U2 â + â¦G, Lâ¦ â¢ T1 â¢*[h, l+1] U2. +/2 width=3 by lstar_S/ qed. + +lemma lstas_step_dx: âh,G,L,T1,T2,U2,l. â¦G, Lâ¦ â¢ T1 â¢*[h, l] T2 â â¦G, Lâ¦ â¢ T2 â¢[h] U2 â + â¦G, Lâ¦ â¢ T1 â¢*[h, l+1] U2. +/2 width=3 by lstar_dx/ qed. + +lemma lstas_split: âh,G,L. inv_ltransitive â¦ (lstas h G L). +/2 width=1 by lstar_inv_ltransitive/ qed-. + +lemma lstas_sort: âh,G,L,l,k. â¦G, Lâ¦ â¢ âk â¢*[h, l] â((next h)^l k). +#h #G #L #l @(nat_ind_plus â¦ l) -l // +#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/ +qed. + +lemma lstas_bind: âh,I,G,L,V,T,U,l. â¦G, L.â{I}Vâ¦ â¢ T â¢*[h, l] U â + âa. â¦G, Lâ¦ â¢ â{a,I}V.T â¢*[h, l] â{a,I}V.U. +#h #I #G #L #V #T #U #l #H @(lstas_ind_dx â¦ H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/ +qed. + +lemma lstas_appl: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l] U â + âV.â¦G, Lâ¦ â¢ âV.T â¢*[h, l] âV.U. +#h #G #L #T #U #l #H @(lstas_ind_dx â¦ H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/ +qed. + +lemma lstas_cast: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l+1] U â + âW. â¦G, Lâ¦ â¢ âW.T â¢*[h, l+1] U. +#h #G #L #T #U #l #H elim (lstas_inv_step_sn â¦ 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_aaa.etc new file mode 100644 index 000000000..7497f44b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_aaa.etc @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,G,L,l. Conf3 â¦ (aaa G L) (lstas h G L l). +/3 width=6 by sta_aaa_conf, lstar_Conf3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_alt.etc new file mode 100644 index 000000000..0ad7b9bbc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_alt.etc @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â relation4 lenv nat term term â +| lstasa_O : âG,L,T. lstasa h G L 0 T T +| lstasa_sort: âG,L,l,k. lstasa h G L l (âk) (â((next h)^l k)) +| lstasa_ldef: âG,L,K,V,W,U,i,l. â©[i] L â¡ K.âV â lstasa h G K (l+1) V W â + â§[0, i+1] W â¡ U â lstasa h G L (l+1) (#i) U +| lstasa_ldec: âG,L,K,W,V,V0,U,i,l. â©[i] L â¡ K.âW â â¦G, Kâ¦ â¢ W â¢[h] V0 â + lstasa h G K l W V â â§[0, i+1] V â¡ U â lstasa h G L (l+1) (#i) U +| lstasa_bind: âa,I,G,L,V,T,U,l. lstasa h G (L.â{I}V) l T U â + lstasa h G L l (â{a,I}V.T) (â{a,I}V.U) +| lstasa_appl: âG,L,V,T,U,l. lstasa h G L l T U â lstasa h G L l (âV.T) (âV.U) +| lstasa_cast: âG,L,W,T,U,l. lstasa h G L (l+1) T U â lstasa h G L (l+1) (âW.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: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â â¦G, Lâ¦ â¢ T â¢â¢*[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: âh,G,L,T1,T,l. â¦G, Lâ¦ â¢ T1 â¢â¢*[h, l] T â + âT2. â¦G, Lâ¦ â¢ T â¢[h] T2 â â¦G, Lâ¦ â¢ T1 â¢â¢*[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 â¦ H) -X >commutative_plus // +| #G #L #K #V #W #U #i #l #HLK #_ #HWU #IHVW #U2 #HU2 + lapply (drop_fwd_drop2 â¦ HLK) #H + elim (sta_inv_lift1 â¦ HU2 â¦ H â¦ 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 â¦ HLK) #H + elim (sta_inv_lift1 â¦ HU2 â¦ H â¦ HVU) -H -U /3 width=8 by lstasa_ldec/ +| #a #I #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H + elim (sta_inv_bind1 â¦ 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 â¦ H) -H #U #HU1 #H destruct /3 width=1 by lstasa_appl/ +| /3 width=1 by lstasa_cast/ +] +qed. + +(* Main properties **********************************************************) + +theorem lstas_lstasa: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l] U â â¦G, Lâ¦ â¢ T â¢â¢*[h, l] U. +#h #G #L #T #U #l #H @(lstas_ind_dx â¦ H) -U -l /2 width=3 by lstasa_step_dx, lstasa_O/ +qed. + +(* Main inversion lemmas ****************************************************) + +theorem lstasa_inv_lstas: âh,G,L,T,U,l. â¦G, Lâ¦ â¢ T â¢â¢*[h, l] U â â¦G, Lâ¦ â¢ T â¢*[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: âh. âR:genvârelation4 lenv nat term term. + (âG,L,T. R G L O T T) â + (âG,L,l,k. R G L l (âk) (â((next h)^l k))) â ( + âG,L,K,V,W,U,i,l. + â©[i] L â¡ K.âV â â¦G, Kâ¦ â¢ V â¢*[h, l+1] W â â§[O, i+1] W â¡ U â + R G K (l+1) V W â R G L (l+1) (#i) U + ) â ( + âG,L,K,W,V,V0,U,i,l. + â©[i] L â¡ K.âW â â¦G, Kâ¦ â¢ W â¢[h] V0 â + â¦G, Kâ¦ â¢ W â¢*[h, l]V â â§[O, i+1] V â¡ U â + R G K l W V â R G L (l+1) (#i) U + ) â ( + âa,I,G,L,V,T,U,l. â¦G, L.â{I}Vâ¦ â¢ T â¢*[h, l] U â + R G (L.â{I}V) l T U â R G L l (â{a,I}V.T) (â{a,I}V.U) + ) â ( + âG,L,V,T,U,l. â¦G, Lâ¦ â¢ T â¢*[h, l] U â + R G L l T U â R G L l (âV.T) (âV.U) + ) â ( + âG,L,W,T,U,l. â¦G, Lâ¦â¢ T â¢*[h, l+1] U â + R G L (l+1) T U â R G L (l+1) (âW.T) U + ) â + âG,L,l,T,U. â¦G, Lâ¦ â¢ T â¢*[h, l] U â R G L l T U. +#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #G #L #l #T #U #H +elim (lstas_lstasa â¦ H) /3 width=10 by lstasa_inv_lstas/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_da.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_da.etc new file mode 100644 index 000000000..8b11d1dc0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/lstas_da.etc @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,G,L,T,U,l1. â¦G, Lâ¦ â¢ T â¢*[h, l1] U â + âl2. â¦G, Lâ¦ â¢ T âª[h, g] l2 â â¦G, Lâ¦ â¢ U âª[h, g] l2-l1. +#h #g #G #L #T #U #l1 #H @(lstas_ind_dx â¦ H) -U -l1 // +#l1 #U #U0 #_ #HU0 #IHTU #l2 #HT +(plus_minus_m_m â¦ Hl12) in â¢ (%â?); -Hl12 >commutative_plus #H +elim (lstas_split â¦ H) -H #U #HTU +>(lstas_mono â¦ HTU â¦ HTU1) -T // +qed-. + +(* Advanced properties ******************************************************) + +lemma lstas_sta_conf_pos: âh,G,L,T,U1. â¦G, Lâ¦ â¢ T â¢[h] U1 â + âU2,l. â¦G, Lâ¦ â¢ T â¢*[h, l+1] U2 â â¦G, Lâ¦ â¢ U1 â¢*[h, l] U2. +#h #G #L #T #U1 #HTU1 #U2 #l #HTU2 +lapply (lstas_conf_le â¦ T U1 1 â¦ HTU2) -HTU2 /2 width=1 by sta_lstas/ +qed-. + +lemma lstas_strip_pos: âh,G,L,T1,U1. â¦G, Lâ¦ â¢ T1 â¢[h] U1 â + âT2,l. â¦G, Lâ¦ â¢ T1 â¢*[h, l+1] T2 â + ââU2. â¦G, Lâ¦ â¢ T2 â¢[h] U2 & â¦G, Lâ¦ â¢ U1 â¢*[h, l+1] U2. +#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12 +elim (lstas_fwd_correct â¦ HTU1 â¦ HT12) +lapply (lstas_sta_conf_pos â¦ HTU1 â¦ HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta.etc new file mode 100644 index 000000000..fc6bf6a2a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta.etc @@ -0,0 +1,143 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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 â +| sta_sort: âG,L,k. sta h G L (âk) (â(next h k)) +| sta_ldef: âG,L,K,V,W,U,i. â©[i] L â¡ K.âV â sta h G K V W â + â§[0, i + 1] W â¡ U â sta h G L (#i) U +| sta_ldec: âG,L,K,W,V,U,i. â©[i] L â¡ K.âW â sta h G K W V â + â§[0, i + 1] W â¡ U â sta h G L (#i) U +| sta_bind: âa,I,G,L,V,T,U. sta h G (L.â{I}V) T U â + sta h G L (â{a,I}V.T) (â{a,I}V.U) +| sta_appl: âG,L,V,T,U. sta h G L T U â sta h G L (âV.T) (âV.U) +| sta_cast: âG,L,W,T,U. sta h G L T U â sta h G L (âW.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: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âk0. T = âk0 â + U = â(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: âh,G,L,U,k. â¦G, Lâ¦ â¢ âk â¢[h] U â U = â(next h k). +/2 width=5 by sta_inv_sort1_aux/ qed-. + +fact sta_inv_lref1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âj. T = #j â + (ââK,V,W. â©[j] L â¡ K.âV & â¦G, Kâ¦ â¢ V â¢[h] W & + â§[0, j+1] W â¡ U + ) â¨ + (ââK,W,V. â©[j] L â¡ K.âW & â¦G, Kâ¦ â¢ W â¢[h] V & + â§[0, j+1] W â¡ 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: âh,G,L,U,i. â¦G, Lâ¦ â¢ #i â¢[h] U â + (ââK,V,W. â©[i] L â¡ K.âV & â¦G, Kâ¦ â¢ V â¢[h] W & + â§[0, i+1] W â¡ U + ) â¨ + (ââK,W,V. â©[i] L â¡ K.âW & â¦G, Kâ¦ â¢ W â¢[h] V & + â§[0, i+1] W â¡ U + ). +/2 width=3 by sta_inv_lref1_aux/ qed-. + +fact sta_inv_gref1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âp0. T = Â§p0 â â¥. +#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: âh,G,L,U,p. â¦G, Lâ¦ â¢ Â§p â¢[h] U â â¥. +/2 width=8 by sta_inv_gref1_aux/ qed-. + +fact sta_inv_bind1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âb,J,X,Y. T = â{b,J}Y.X â + ââZ. â¦G, L.â{J}Yâ¦ â¢ X â¢[h] Z & U = â{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: âh,b,J,G,L,Y,X,U. â¦G, Lâ¦ â¢ â{b,J}Y.X â¢[h] U â + ââZ. â¦G, L.â{J}Yâ¦ â¢ X â¢[h] Z & U = â{b,J}Y.Z. +/2 width=3 by sta_inv_bind1_aux/ qed-. + +fact sta_inv_appl1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âX,Y. T = âY.X â + ââZ. â¦G, Lâ¦ â¢ X â¢[h] Z & U = âY.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: âh,G,L,Y,X,U. â¦G, Lâ¦ â¢ âY.X â¢[h] U â + ââZ. â¦G, Lâ¦ â¢ X â¢[h] Z & U = âY.Z. +/2 width=3 by sta_inv_appl1_aux/ qed-. + +fact sta_inv_cast1_aux: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âX,Y. T = âY.X â + â¦G, Lâ¦ â¢ X â¢[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: âh,G,L,X,Y,U. â¦G, Lâ¦ â¢ âY.X â¢[h] U â â¦G, Lâ¦ â¢ X â¢[h] U. +/2 width=4 by sta_inv_cast1_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_aaa.etc new file mode 100644 index 000000000..affd2eded --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_aaa.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,G,L,T,A. â¦G, Lâ¦ â¢ T â A â âU. â¦G, Lâ¦ â¢ T â¢[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: âh,G,L. Conf3 â¦ (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 â¦ H) -H #H destruct // +| #I #G #L #K #V #B #i #HLK #HV #IHV #U #H + elim (sta_inv_lref1 â¦ H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HU + lapply (drop_mono â¦ HLK0 â¦ HLK) -HLK0 #H0 destruct + lapply (drop_fwd_drop2 â¦ HLK) -HLK #HLK + @(aaa_lift â¦ HLK â¦ HU) -HU -L /2 width=2 by/ +| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H + elim (sta_inv_bind1 â¦ 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 â¦ 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 â¦ 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 â¦ H) -H /2 width=2 by/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_lift.etc new file mode 100644 index 000000000..4d342878f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_lift.etc @@ -0,0 +1,113 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,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 â¦ H1) -X1 + >(lift_inv_sort1 â¦ 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 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HW1 â¦ HWU2) -W // #W2 #HW12 #HWU2 + elim (drop_trans_le â¦ HL21 â¦ HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H + elim (drop_inv_skip2 â¦ 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 â¦ HW1 â¦ HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2 + lapply (drop_trans_ge â¦ HL21 â¦ 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 â¦ H) * #Hid #H destruct + [ elim (lift_trans_ge â¦ HW1 â¦ HWU2) -W // (lift_inv_sort2 â¦ 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 â¦ H) * #Hid #H destruct [ -HVW2 | -IHVW2 ] + [ elim (drop_conf_lt â¦ HL21 â¦ HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 + elim (IHVW2 â¦ HK21 â¦ HV12) -K2 -V2 #W1 #HW12 #HVW1 + elim (lift_trans_le â¦ HW12 â¦ HW2) -W2 // >minus_plus 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 â¦ H) * #Hid #H destruct [ -HWV2 | -IHWV2 ] + [ elim (drop_conf_lt â¦ HL21 â¦ HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12 + elim (IHWV2 â¦ HK21 â¦ HW12) -K2 #V1 #_ #HWV1 + elim (lift_trans_le â¦ HW12 â¦ HW2) -W2 // >minus_plus 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 â¦ H) -H #V1 #T1 #HV12 #HT12 #H destruct + elim (IHTU2 (L1.â{I}V1) â¦ 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 â¦ H) -H #V1 #T1 #HV12 #HT12 #H destruct + elim (IHTU2 â¦ HL21 â¦ 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 â¦ H) -H #W1 #T1 #_ #HT12 #H destruct + elim (IHTU2 â¦ HL21 â¦ HT12) -L2 -HT12 /3 width=3 by sta_cast, ex2_intro/ +] +qed-. + +(* Advanced forward lemmas **************************************************) + +(* Basic_1: was: sty0_correct *) +lemma sta_fwd_correct: âh,G,L,T,U. â¦G, Lâ¦ â¢ T â¢[h] U â âT0. â¦G, Lâ¦ â¢ U â¢[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 â¦ 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 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_llpx_sn.etc new file mode 100644 index 000000000..3adf28fc4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_llpx_sn.etc @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âR. (âL. reflexive â¦ (R L)) â l_liftable R â + âh,G. s_r_confluent1 â¦ (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 â¦ H â¦ HLKs) // -H + #Kd #V1d #HLKd #HV1s #HV1sd + lapply (drop_fwd_drop2 â¦ HLKs) -HLKs #HLKs + lapply (drop_fwd_drop2 â¦ HLKd) -HLKd #HLKd + @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ H â¦ HLKs) // -H + #Kd #V1d #HLKd #HV1s #HV1sd + lapply (drop_fwd_drop2 â¦ HLKs) -HLKs #HLKs + lapply (drop_fwd_drop2 â¦ HLKd) -HLKd #HLKd + @(llpx_sn_lift_le â¦ HLKs HLKd â¦ 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 â¦ 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 â¦ H) -H + /3 width=1 by llpx_sn_flat/ +| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat â¦ H) -H + /3 width=1 by llpx_sn_flat/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_sta.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_sta.etc new file mode 100644 index 000000000..6d54d56d7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/sta_sta.etc @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,G,L. singlevalued â¦ (sta h G L). +#h #G #L #T #U1 #H elim H -G -L -T -U1 +[ #G #L #k #X #H >(sta_inv_sort1 â¦ H) -X // +| #G #L #K #V #W #U1 #i #HLK #_ #HWU1 #IHVW #U2 #H + elim (sta_inv_lref1 â¦ H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HW0U2 + lapply (drop_mono â¦ HLK0 â¦ HLK) -HLK -HLK0 #H destruct + lapply (IHVW â¦ HVW0) -IHVW -HVW0 #H destruct + >(lift_mono â¦ HWU1 â¦ HW0U2) -W0 -U1 // +| #G #L #K #W #V #U1 #i #HLK #_ #HWU1 #IHWV #U2 #H + elim (sta_inv_lref1 â¦ H) -H * #K0 #W0 #V0 #HLK0 #HWV0 #HV0U2 + lapply (drop_mono â¦ HLK0 â¦ HLK) -HLK -HLK0 #H destruct + lapply (IHWV â¦ HWV0) -IHWV -HWV0 #H destruct + >(lift_mono â¦ HWU1 â¦ HV0U2) -W -U1 // +| #a #I #G #L #V #T #U1 #_ #IHTU1 #X #H + elim (sta_inv_bind1 â¦ H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ +| #G #L #V #T #U1 #_ #IHTU1 #X #H + elim (sta_inv_appl1 â¦ H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ +| #G #L #W #T #U1 #_ #IHTU1 #U2 #H + lapply (sta_inv_cast1 â¦ H) -H /2 width=1 by/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/statictype_5.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/statictype_5.etc new file mode 100644 index 000000000..9249be4e0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/statictype_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 â¢ break [ term 46 h ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'StaticType $h $G $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/statictypestaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/statictypestaralt_6.etc new file mode 100644 index 000000000..a6e46be62 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/sta/statictypestaralt_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 â¢ â¢ * 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 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/chnf.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/chnf.etc new file mode 100644 index 000000000..f6106d38d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/chnf.etc @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break ðð â¦ term 46 T â¦ )" + 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 â predicate term â Î»L. NF â¦ (cpr L) tshf. + +interpretation + "context-sensitive head normality (term)" + 'HdNormal L T = (chnf L T). + +(* Basic inversion lemmas ***************************************************) + +lemma chnf_inv_tshf: âL,T. L â¢ ððâ¦Tâ¦ â T â T. +normalize /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +lemma tshf_thnf: âT. T â T â â â¢ ððâ¦Tâ¦. +#T #HT #T2 #H elim (cpr_fwd_tshf1 â¦ H) -H // +#H elim H // +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/cpr_tshf.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/cpr_tshf.etc new file mode 100644 index 000000000..0fc5e1d0c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/cpr_tshf.etc @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âL,T1,T2. L â¢ T1 â¡ T2 â T1 â T1 â + T2 â T1 â¨ (L = â â â¥). +#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 â¦ HLK) -HLK #H destruct +| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H + elim (tshf_inv_bind1 â¦ H) -H #W2 #U2 #H1 * #H2 destruct /2 width=1/ +| #I #L #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H + elim (tshf_inv_flat1 â¦ H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct + lapply (IHT12 HT1U2) -IHT12 -HT1U2 * #HUT2 /3 width=1/ + lapply (simple_tshf_repl_sn â¦ HUT2 HT1) /3 width=1/ +| #L #V #T #T1 #T2 #_ #_ #_ #H + elim (tshf_inv_bind1 â¦ H) -H #W2 #U2 #H1 * #H2 destruct +| #L #V #T1 #T2 #_ #_ #H + elim (tshf_inv_flat1 â¦ H) -H #W2 #U2 #_ #_ #_ #H destruct +| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H + elim (tshf_inv_flat1 â¦ H) -H #W2 #U2 #_ #H + elim (simple_inv_bind â¦ H) +| #a #L #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H + elim (tshf_inv_flat1 â¦ H) -H #U1 #U2 #_ #H + elim (simple_inv_bind â¦ H) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/tshf.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/tshf.etc new file mode 100644 index 000000000..583c3f46f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/tshf/tshf.etc @@ -0,0 +1,98 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 â¢ break term 46 T1 â break term 46 T2 )" + non associative with precedence 45 + for @{ 'Hom $L $T1 $T2 }. + +notation "hvbox( L â¢ break ðð â¦ term 46 T â¦ )" + non associative with precedence 45 + for @{ 'HdReducible $L $T }. + +notation "hvbox( L â¢ break ðð â¦ term 46 T â¦ )" + non associative with precedence 45 + for @{ 'NotHdReducible $L $T }. + +include "basic_2/grammar/term_simple.ma". + +(* SAME HEAD TERM FORMS *****************************************************) + +inductive tshf: relation term â + | tshf_atom: âI. tshf (âª{I}) (âª{I}) + | tshf_abbr: âV1,V2,T1,T2. tshf (-âV1. T1) (-âV2. T2) + | tshf_abst: âa,V1,V2,T1,T2. tshf (â{a}V1. T1) (â{a}V2. T2) + | tshf_appl: âV1,V2,T1,T2. tshf T1 T2 â ðâ¦T1â¦ â ðâ¦T2â¦ â + tshf (âV1. T1) (âV2. T2) +. + +interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2). + +(* Basic properties *********************************************************) + +lemma tshf_sym: âT1,T2. T1 â T2 â T2 â T1. +#T1 #T2 #H elim H -T1 -T2 /2 width=1/ +qed. + +lemma tshf_refl2: âT1,T2. T1 â T2 â T2 â T2. +#T1 #T2 #H elim H -T1 -T2 // /2 width=1/ +qed. + +lemma tshf_refl1: âT1,T2. T1 â T2 â T1 â T1. +/3 width=2/ qed. + +lemma simple_tshf_repl_dx: âT1,T2. T1 â T2 â ðâ¦T1â¦ â ðâ¦T2â¦. +#T1 #T2 #H elim H -T1 -T2 // +[ #V1 #V2 #T1 #T2 #H + elim (simple_inv_bind â¦ H) +| #a #V1 #V2 #T1 #T2 #H + elim (simple_inv_bind â¦ H) +] +qed. (**) (* remove from index *) + +lemma simple_tshf_repl_sn: âT1,T2. T1 â T2 â ðâ¦T2â¦ â ðâ¦T1â¦. +/3 width=3/ qed-. + +(* Basic inversion lemmas ***************************************************) + +fact tshf_inv_bind1_aux: âT1,T2. T1 â T2 â âa,I,W1,U1. T1 = â{a,I}W1.U1 â + ââW2,U2. T2 = â{a,I}W2. U2 & + (Bind2 a I = Bind2 false Abbr â¨ 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: âa,I,W1,U1,T2. â{a,I}W1.U1 â T2 â + ââW2,U2. T2 = â{a,I}W2. U2 & + (Bind2 a I = Bind2 false Abbr â¨ I = Abst). +/2 width=5/ qed-. + +fact tshf_inv_flat1_aux: âT1,T2. T1 â T2 â âI,W1,U1. T1 = â{I}W1.U1 â + ââW2,U2. U1 â U2 & ðâ¦U1â¦ & ðâ¦U2â¦ & + I = Appl & T2 = âW2. 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: âI,W1,U1,T2. â{I}W1.U1 â T2 â + ââW2,U2. U1 â U2 & ðâ¦U1â¦ & ðâ¦U2â¦ & + I = Appl & T2 = âW2. U2. +/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_aaa.etc new file mode 100644 index 000000000..98fe01347 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_aaa.etc @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L,T,A. L â¢ T â A â âU. â¦h, Lâ¦ â¢ T â¢â¡[g] U â L â¢ U â A. +#h #g #L #T #A #HT #U * /2 width=3/ /2 width=6/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_lift.etc new file mode 100644 index 000000000..8f5deca51 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_lift.etc @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,a,L,V1,T1,U2. â¦h, Lâ¦ â¢ â{a}V1.T1 â¢â¡[g] U2 â + ââV2,T2. L â¢ V1 â¡ V2 & â¦h, L. âV1â¦ â¢ T1 â¢â¡[g] T2 & + U2 = â{a}V2. T2. +#h #g #a #L #V1 #T1 #U2 * +[ #H elim (cpr_inv_abst1 â¦ H Abst V1) /3 width=5/ +| #l #H elim (ssta_inv_bind1 â¦ H) /3 width=5/ +] +qed-. + +(* Relocation properties ****************************************************) + +lemma xpr_lift: âL,K,d,e. â©[d, e] L â¡ K â + âT1,U1. â§[d, e] T1 â¡ U1 â âT2,U2. â§[d, e] T2 â¡ U2 â + âh,g. â¦h, Kâ¦ â¢ T1 â¢â¡[g] T2 â â¦h, Lâ¦ â¢ U1 â¢â¡[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: âL,K,d,e. â©[d, e] L â¡ K â + âT1,U1. â§[d, e] T1 â¡ U1 â âh,g,U2. â¦h, Lâ¦ â¢ U1 â¢â¡[g] U2 â + ââT2. â§[d, e] T2 â¡ U2 & â¦h, Kâ¦ â¢ T1 â¢â¡[g] T2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 * [ #HU12 | #l #HU12 ] +[ elim (cpr_inv_lift1 â¦ HLK â¦ HTU1 â¦ HU12) -L -U1 /3 width=3/ +| elim (ssta_inv_lift1 â¦ HU12 â¦ HLK â¦ HTU1) -L -U1 /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_lsubss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_lsubss.etc new file mode 100644 index 000000000..bc66cd2cd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xpr_lsubss.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,L2. h â¢ L1 â¢â[g] L2 â + âT1,T2. â¦h, L2â¦ â¢ T1 â¢â¡[g] T2 â â¦h, L1â¦ â¢ T1 â¢â¡[g] T2. +#h #g #L1 #L2 #HL12 #T1 #T2 * [ | #l ] #HT12 +[ lapply (lsubss_fwd_lsubs2 â¦ HL12) -HL12 /3 width=3/ +| /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs.etc new file mode 100644 index 000000000..bffc4e735 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs.etc @@ -0,0 +1,72 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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( â¦ term 46 h , break term 46 L â¦ â¢ break term 46 T1 â¢ â¡ * break [ term 46 g ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'XPRedStar $h $g $L $T1 $T2 }. + +notation "hvbox( â¦ term 46 h , break term 46 L â¦ â¢ â¢ â¬ * 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: âh. sd h â lenv â relation term â + Î»h,g,L. TC â¦ (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: âh,g,L,T1. âR:predicate term. R T1 â + (âT,T2. â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T â â¦h, Lâ¦ â¢ T â¢â¡[g] T2 â R T â R T2) â + âT2. â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2 â R T2. +#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // +qed-. + +lemma xprs_ind_dx: âh,g,L,T2. âR:predicate term. R T2 â + (âT1,T. â¦h, Lâ¦ â¢ T1 â¢â¡[g] T â â¦h, Lâ¦ â¢ T â¢â¡*[g] T2 â R T â R T1) â + âT1. â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2 â R T1. +#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma xprs_refl: âh,g,L. reflexive â¦ (xprs h g L). +/2 width=1/ qed. + +lemma xprs_strap1: âh,g,L,T1,T,T2. + â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T â â¦h, Lâ¦ â¢ T â¢â¡[g] T2 â â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2. +/2 width=3/ qed. + +lemma xprs_strap2: âh,g,L,T1,T,T2. + â¦h, Lâ¦ â¢ T1 â¢â¡[g] T â â¦h, Lâ¦ â¢ T â¢â¡*[g] T2 â â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2. +/2 width=3/ qed. + +(* Basic inversion lemmas ***************************************************) +(* +axiom xprs_inv_cnf1: âL,T,U. L â¢ T â¡* U â L â¢ ðâ¦Tâ¦ â T = U. +#L #T #U #H @(xprs_ind_dx â¦ H) -T // +#T0 #T #H1T0 #_ #IHT #H2T0 +lapply (H2T0 â¦ H1T0) -H1T0 #H destruct /2 width=1/ +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_aaa.etc new file mode 100644 index 000000000..5beb8fe19 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_aaa.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L,T,A. L â¢ T â A â âU. â¦h, Lâ¦ â¢ T â¢â¡*[g] U â L â¢ U â A. +#h #g #L #T #A #HT #U #H @(xprs_ind â¦ H) -U // /2 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_cprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_cprs.etc new file mode 100644 index 000000000..13a4f8889 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_cprs.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L,T1,T2. L â¢ T1 â¡* T2 â â¦h, Lâ¦ â¢ T1 â¢â¡*[g] T2. +#h #g #L #T1 #T2 #H @(cprs_ind â¦ H) -T2 // /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_lift.etc new file mode 100644 index 000000000..cb151a194 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_lift.etc @@ -0,0 +1,50 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,a,L,V1,T1,U2. â¦h, Lâ¦ â¢ â{a}V1. T1 â¢â¡*[g] U2 â + ââV2,T2. L â¢ V1 â¡* V2 & U2 = â{a}V2. T2. +#h #g #a #L #V1 #T1 #U2 #H @(xprs_ind â¦ H) -U2 /2 width=4/ +#U #U2 #_ #HU2 * #V #T #HV1 #H destruct +elim (xpr_inv_abst1 â¦ HU2) -HU2 #V2 #T2 #HV2 #_ #H destruct /3 width=4/ +qed-. + +(* Relocation properties ****************************************************) + +lemma xprs_lift: âL,K,d,e. â©[d, e] L â¡ K â âT1,U1. â§[d, e] T1 â¡ U1 â + âh,g,T2. â¦h, Kâ¦ â¢ T1 â¢â¡*[g] T2 â âU2. â§[d, e] T2 â¡ U2 â + â¦h, Lâ¦ â¢ U1 â¢â¡*[g] U2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 #HT12 @(xprs_ind â¦ HT12) -T2 +[ -HLK #T2 #HT12 + <(lift_mono â¦ HTU1 â¦ HT12) -T1 // +| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (xpr_lift â¦ HLK â¦ HTU â¦ HTU2 â¦ HT2) -T2 -HLK /3 width=3/ +] +qed. + +lemma xprs_inv_lift1: âL,K,d,e. â©[d, e] L â¡ K â + âT1,U1. â§[d, e] T1 â¡ U1 â âh,g,U2. â¦h, Lâ¦ â¢ U1 â¢â¡*[g] U2 â + ââT2. â§[d, e] T2 â¡ U2 & â¦h, Kâ¦ â¢ T1 â¢â¡*[g] T2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 #HU12 @(xprs_ind â¦ HU12) -U2 /2 width=3/ +-HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 +elim (xpr_inv_lift1 â¦ HLK â¦ HTU â¦ HU2) -U -HLK /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_lsubss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_lsubss.etc new file mode 100644 index 000000000..c883c14f3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_lsubss.etc @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L1,L2. h â¢ L1 â¢â[g] L2 â + âT1,T2. â¦h, L2â¦ â¢ T1 â¢â¡*[g] T2 â â¦h, L1â¦ â¢ T1 â¢â¡*[g] T2. +#h #g #L1 #L2 #HL12 #T1 #T2 #H @(xprs_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (lsubss_xpr_trans â¦ HL12 â¦ HT2) -L2 /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_xprs.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_xprs.etc new file mode 100644 index 000000000..9593f0550 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/xpr/xprs_xprs.etc @@ -0,0 +1,20 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: âh,g,L. transitive â¦ (xprs h g L). +/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy.etc new file mode 100644 index 000000000..7f36b140a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy.etc @@ -0,0 +1,303 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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 â ynat â relation4 genv lenv term term â +| cpy_atom : âI,G,L,l,m. cpy l m G L (âª{I}) (âª{I}) +| cpy_subst: âI,G,L,K,V,W,i,l,m. l â¤ yinj i â i < l+m â + â¬[i] L â¡ K.â{I}V â â¬[0, i+1] V â¡ W â cpy l m G L (#i) W +| cpy_bind : âa,I,G,L,V1,V2,T1,T2,l,m. + cpy l m G L V1 V2 â cpy (âl) m G (L.â{I}V1) T1 T2 â + cpy l m G L (â{a,I}V1.T1) (â{a,I}V2.T2) +| cpy_flat : âI,G,L,V1,V2,T1,T2,l,m. + cpy l m G L V1 V2 â cpy l m G L T1 T2 â + cpy l m G L (â{I}V1.T1) (â{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: âG,l,m. lsub_trans â¦ (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 â¦ HL12 â¦ 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: âG,T,L,l,m. â¦G, Lâ¦ â¢ T â¶[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: âI,G,K,V,T1,L,l. â¬[l] L â¡ K.â{I}V â + ââT2,T. â¦G, Lâ¦ â¢ T1 â¶[l, 1] T2 & â¬[l, 1] T â¡ 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 â¦ 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 â¦ HLK) -IHW1 #W2 #W #HW12 #HW2 + [ elim (IHU1 (L.â{J}W1) (l+1)) -IHU1 + /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/ + | elim (IHU1 â¦ HLK) -IHU1 -HLK + /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/ + ] +] +qed-. + +lemma cpy_weak: âG,L,T1,T2,l1,m1. â¦G, Lâ¦ â¢ T1 â¶[l1, m1] T2 â + âl2,m2. l2 â¤ l1 â l1 + m1 â¤ l2 + m2 â + â¦G, Lâ¦ â¢ T1 â¶[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: âG,L,T1,T2,l,m. + â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶[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 â¦ HLK) #Hil + lapply (ylt_inj â¦ Hil) -Hil #Hil + lapply (yle_ylt_trans â¦ Hil Hli) #Hl + @(cpy_subst â¦ Hli â¦ HLK HVW) + yplus_SO2 >yplus_succ1 #T #HT1 #HT2 + lapply (lsuby_cpy_trans â¦ HT2 (L.â{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: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â âi. i â¤ l + m â + ââT. â¦G, Lâ¦ â¢ T1 â¶[i, l+m-i] T & â¦G, Lâ¦ â¢ T â¶[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 â¦ HT2 (L.â{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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â + âT1,l,m. â¬[l, m] T1 â¡ U1 â + l â¤ lt â l + m â¤ lt + mt â + ââT2. â¦G, Lâ¦ â¢ U1 â¶[l+m, lt+mt-(l+m)] U2 & â¬[l, m] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 â¦ H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ] + [ elim (ylt_yle_false â¦ Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/ + | elim (le_inv_plus_l â¦ Hil) #Hlim #Hmi + elim (lift_split â¦ HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim + #T2 #_ >plus_minus // 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 â¦ HVW1) -V1 -IHW12 // + elim (IHU12 â¦ HTU1) -T1 -IHU12 /2 width=1 by yle_succ/ + 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 â¦ HVW1) -V1 -IHW12 // elim (IHU12 â¦ HTU1) -T1 -IHU12 + /3 width=2 by cpy_flat, lift_flat, ex2_intro/ +] +qed-. + +lemma cpy_fwd_tw: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â â¯{T1} â¤ â¯{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: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â âJ. T1 = âª{J} â + T2 = âª{J} â¨ + ââI,K,V,i. l â¤ yinj i & i < l + m & + â¬[i] L â¡ K.â{I}V & + â¬[O, i+1] V â¡ 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: âI,G,L,T2,l,m. â¦G, Lâ¦ â¢ âª{I} â¶[l, m] T2 â + T2 = âª{I} â¨ + ââJ,K,V,i. l â¤ yinj i & i < l + m & + â¬[i] L â¡ K.â{J}V & + â¬[O, i+1] V â¡ T2 & + I = LRef i. +/2 width=4 by cpy_inv_atom1_aux/ qed-. + +(* Basic_1: was: subst1_gen_sort *) +lemma cpy_inv_sort1: âG,L,T2,k,l,m. â¦G, Lâ¦ â¢ âk â¶[l, m] T2 â T2 = âk. +#G #L #T2 #k #l #m #H +elim (cpy_inv_atom1 â¦ H) -H // +* #I #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +(* Basic_1: was: subst1_gen_lref *) +lemma cpy_inv_lref1: âG,L,T2,i,l,m. â¦G, Lâ¦ â¢ #i â¶[l, m] T2 â + T2 = #i â¨ + ââI,K,V. l â¤ i & i < l + m & + â¬[i] L â¡ K.â{I}V & + â¬[O, i+1] V â¡ T2. +#G #L #T2 #i #l #m #H +elim (cpy_inv_atom1 â¦ 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: âG,L,T2,p,l,m. â¦G, Lâ¦ â¢ Â§p â¶[l, m] T2 â T2 = Â§p. +#G #L #T2 #p #l #m #H +elim (cpy_inv_atom1 â¦ H) -H // +* #I #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +fact cpy_inv_bind1_aux: âG,L,U1,U2,l,m. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â + âa,I,V1,T1. U1 = â{a,I}V1.T1 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & + â¦G, L. â{I}V1â¦ â¢ T1 â¶[âl, m] T2 & + U2 = â{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: âa,I,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{a,I} V1. T1 â¶[l, m] U2 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & + â¦G, L.â{I}V1â¦ â¢ T1 â¶[âl, m] T2 & + U2 = â{a,I}V2.T2. +/2 width=3 by cpy_inv_bind1_aux/ qed-. + +fact cpy_inv_flat1_aux: âG,L,U1,U2,l,m. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â + âI,V1,T1. U1 = â{I}V1.T1 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & + â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 & + U2 = â{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: âI,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{I} V1. T1 â¶[l, m] U2 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & + â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 & + U2 = â{I}V2.T2. +/2 width=3 by cpy_inv_flat1_aux/ qed-. + + +fact cpy_inv_refl_O2_aux: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â m = 0 â 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 â¦ Hli) -Hli // +| /3 width=1 by eq_f2/ +| /3 width=1 by eq_f2/ +] +qed-. + +lemma cpy_inv_refl_O2: âG,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 â¶[l, 0] T2 â T1 = T2. +/2 width=6 by cpy_inv_refl_O2_aux/ qed-. + +(* Basic_1: was: subst1_gen_lift_eq *) +lemma cpy_inv_lift1_eq: âG,T1,U1,l,m. â¬[l, m] T1 â¡ U1 â + âL,U2. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â U1 = U2. +#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up â¦ HU12 â¦ 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_cpy.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_cpy.etc new file mode 100644 index 000000000..3a8857155 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_cpy.etc @@ -0,0 +1,122 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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: âG,L,T0,T1,l1,m1. â¦G, Lâ¦ â¢ T0 â¶[l1, m1] T1 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â + ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T2 â¶[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 â¦ H) -H + [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/ + | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2 + lapply (drop_mono â¦ HLK1 â¦ HLK2) -HLK1 -HLK2 #H destruct + >(lift_mono â¦ HVT1 â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02) -IHV01 -HV02 #V #HV1 #HV2 + elim (IHT01 â¦ HT02) -T0 #T #HT1 #HT2 + lapply (lsuby_cpy_trans â¦ HT1 (L.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans â¦ HT2 (L.â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02) -V0 + elim (IHT01 â¦ HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/ +] +qed-. + +(* Basic_1: was: subst1_confluence_neq *) +theorem cpy_conf_neq: âG,L1,T0,T1,l1,m1. â¦G, L1â¦ â¢ T0 â¶[l1, m1] T1 â + âL2,T2,l2,m2. â¦G, L2â¦ â¢ T0 â¶[l2, m2] T2 â + (l1 + m1 â¤ l2 â¨ l2 + m2 â¤ l1) â + ââT. â¦G, L2â¦ â¢ T1 â¶[l2, m2] T & â¦G, L1â¦ â¢ T2 â¶[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 â¦ 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 â¦ Hlm1) -Hlm1 /2 width=3 by yle_trans/ + | elim (ylt_yle_false â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02 H) -IHV01 -HV02 #V #HV1 #HV2 + elim (IHT01 â¦ HT02) -T0 + [ -H #T #HT1 #HT2 + lapply (lsuby_cpy_trans â¦ HT1 (L2.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans â¦ HT2 (L1.â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 â¦ HV02 H) -V0 + elim (IHT01 â¦ 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: âG,L,T1,T0,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T0 â + âT2. â¦G, Lâ¦ â¢ T0 â¶[l, 1] T2 â 1 â¤ m â â¦G, Lâ¦ â¢ T1 â¶[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 â¦ H) -H + [ #H destruct // + | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct + lapply (ylt_yle_trans â¦ (l+m) â¦ 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 â¦ HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/ + >yplus_inj #HVT2 <(cpy_inv_lift1_eq â¦ HVW â¦ 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 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct + lapply (lsuby_cpy_trans â¦ HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 + lapply (IHT10 â¦ 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 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/ +] +qed-. + +theorem cpy_trans_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶[l1, m1] T0 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â l2 + m2 â¤ l1 â + ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T â¶[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 â¦ Hlm2l1 â¦ Hli1) -Hlm2l1 #Hlm2i1 + lapply (cpy_weak â¦ HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1 + >yplus_inj #HWT2 <(cpy_inv_lift1_eq â¦ HVW â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (lsuby_cpy_trans â¦ HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 + elim (IHV10 â¦ HV02) -IHV10 -HV02 // #V + elim (IHT10 â¦ HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2 + lapply (lsuby_cpy_trans â¦ HT2 (L.â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV10 â¦ HV02) -V0 // + elim (IHT10 â¦ HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_lift.etc new file mode 100644 index 000000000..a188129b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_lift.etc @@ -0,0 +1,249 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 â + âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â + â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â + lt + mt â¤ l â â¦G, Lâ¦ â¢ U1 â¶[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 â¦ H1 â¦ 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 â¦ Hlmtl â¦ Hilmt) -Hlmtl #Hil + lapply (ylt_inv_inj â¦ Hil) -Hil #Hil + lapply (lift_inv_lref1_lt â¦ H â¦ Hil) -H #H destruct + elim (lift_trans_ge â¦ HVW â¦ HWU2) -W // (lift_mono â¦ HVY â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /3 width=7 by cpy_flat/ +] +qed-. + +lemma cpy_lift_be: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 â + âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â + â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â + lt â¤ l â l â¤ lt + mt â â¦G, Lâ¦ â¢ U1 â¶[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 â¦ H1 â¦ 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 â¦ H) -H * #Hil #H destruct + [ -Hltl + lapply (ylt_yle_trans â¦ (lt+mt+m) â¦ Hilmt) // -Hilmt #Hilmtm + elim (lift_trans_ge â¦ HVW â¦ HWU2) -W // (lift_mono â¦ HVY â¦ HVW) -V #H destruct /2 width=5 by cpy_subst/ + | -Hlti + elim (yle_inv_inj2 â¦ Hltl) -Hltl #ltt #Hltl #H destruct + lapply (transitive_le â¦ Hltl Hil) -Hltl #Hlti + lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2 + lapply (drop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 â¦ 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: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 â + âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â + â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â + l â¤ lt â â¦G, Lâ¦ â¢ U1 â¶[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 â¦ H1 â¦ 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 â¦ Hllt â¦ Hlti) -Hllt #Hil + elim (yle_inv_inj2 â¦ Hil) -Hil #ll #Hlli #H0 destruct + lapply (lift_inv_lref1_ge â¦ H â¦ Hlli) -H #H destruct + lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2 + lapply (drop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 â¦ 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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt + mt â¤ l â + ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 & â¬[l, m] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 â¦ H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 â¦ 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 â¦ Hlmtl â¦ Hilmt) -Hlmtl #Hil + lapply (ylt_inv_inj â¦ Hil) -Hil #Hil + lapply (lift_inv_lref2_lt â¦ H â¦ Hil) -H #H destruct + elim (drop_conf_lt â¦ HLK â¦ HLKV) -L // #L #U #HKL #_ #HUV + elim (lift_trans_le â¦ HUV â¦ HVW) -V // >minus_plus yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm + elim (drop_conf_lt â¦ HLK â¦ HLKV) -L // #L #U #HKL #_ #HUV + elim (lift_trans_le â¦ HUV â¦ HVW) -V // >minus_plus plus_minus // 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 â¦ HLK â¦ HVW1) -IHW12 // #V2 #HV12 #HVW2 + elim (IHU12 â¦ 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 â¦ HLK â¦ HVW1) -W1 // + elim (IHU12 â¦ HLK â¦ 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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + yinj l + m â¤ lt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt-m, mt] T2 & â¬[l, m] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 â¦ H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 â¦ 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 â¦ Hlmlt â¦ Hlti) #Hlmi + elim (yle_inv_plus_inj2 â¦ Hlmlt) -Hlmlt #_ #Hmlt + elim (yle_inv_plus_inj2 â¦ Hlmi) #Hlim #Hmi + lapply (lift_inv_lref2_ge â¦ H ?) -H /2 width=1 by yle_inv_inj/ #H destruct + lapply (drop_conf_ge â¦ HLK â¦ HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV + elim (lift_split â¦ 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/ 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 â¦ HLK â¦ HVW1) -W1 // + elim (IHU12 â¦ HLK â¦ 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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + l â¤ lt â lt â¤ yinj l + m â yinj l + m â¤ lt + mt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶[l, lt + mt - (yinj l + m)] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt +elim (cpy_split_up â¦ HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 +lapply (cpy_weak â¦ HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 +lapply (cpy_inv_lift1_eq â¦ HTU1 â¦ HU1) -HU1 #HU1 destruct +elim (cpy_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1) -U -L /2 width=3 by ex2_intro/ +qed-. + +lemma cpy_inv_lift1_be_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt â¤ l â lt + mt â¤ yinj l + m â + ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt, l-lt] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm +lapply (cpy_weak â¦ 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 â¦ HU12 â¦ HLK â¦ HTU1) -U1 -L /2 width=3 by ex2_intro/ +qed-. + +lemma cpy_inv_lift1_le_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt â¤ l â l â¤ lt + mt â lt + mt â¤ yinj l + m â + ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm +elim (cpy_split_up â¦ HU12 l) -HU12 // #U #HU1 #HU2 +elim (cpy_inv_lift1_le â¦ HU1 â¦ HLK â¦ HTU1) -U1 +[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU +lapply (cpy_weak â¦ HU2 l m ? ?) -HU2 // +[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2 +lapply (cpy_inv_lift1_eq â¦ HTU â¦ HU2) -L #H destruct /2 width=3 by ex2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_nlift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_nlift.etc new file mode 100644 index 000000000..ded7af4f5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpy/cpy_nlift.etc @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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: âG,L,U1,U2,l,m. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â + âi. l â¤ yinj i â (âT2. â¬[i, 1] T2 â¡ U2 â â¥) â + (âT1. â¬[i, 1] T1 â¡ U1 â â¥) â¨ + ââI,K,W,j. l â¤ yinj j & j < i & â¬[j]L â¡ K.â{I}W & + (âV. â¬[i-j-1, 1] V â¡ W â â¥) & (âT1. â¬[j, 1] T1 â¡ U1 â â¥). +#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 â¦ HLK) // -HLK + [ #X #HXV elim (lift_trans_le â¦ HXV â¦ HVW ?) -V // + #Y #HXY >minus_plus (plus_minus_m_m j 1) in â¢ (%â?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ] + #HnU1 (cpy_inv_sort1 â¦ HT2) -HT2 // +qed-. + +(* Note: this can be derived from cpys_inv_atom1 *) +lemma cpys_inv_gref1: âG,L,T2,p,l,m. â¦G, Lâ¦ â¢ Â§p â¶*[l, m] T2 â T2 = Â§p. +#G #L #T2 #p #l #m #H @(cpys_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 destruct +>(cpy_inv_gref1 â¦ HT2) -HT2 // +qed-. + +lemma cpys_inv_bind1: âa,I,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[l, m] U2 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 & + â¦G, L.â{I}V1â¦ â¢ T1 â¶*[âl, m] T2 & + U2 = â{a,I}V2.T2. +#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind â¦ H) -U2 +[ /2 width=5 by ex3_2_intro/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (cpy_inv_bind1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (lsuby_cpy_trans â¦ HT2 (L.â{I}V1) ?) -HT2 + /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/ +] +qed-. + +lemma cpys_inv_flat1: âI,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[l, m] U2 â + ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 & â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 & + U2 = â{I}V2.T2. +#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind â¦ H) -U2 +[ /2 width=5 by ex3_2_intro/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (cpy_inv_flat1 â¦ HU2) -HU2 + /3 width=5 by cpys_strap1, ex3_2_intro/ +] +qed-. + +lemma cpys_inv_refl_O2: âG,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 â¶*[l, 0] T2 â T1 = T2. +#G #L #T1 #T2 #l #H @(cpys_ind â¦ H) -T2 // +#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 â¦ HT2) -HT2 // +qed-. + +lemma cpys_inv_lift1_eq: âG,L,U1,U2. âl,m:nat. + â¦G, Lâ¦ â¢ U1 â¶*[l, m] U2 â âT1. â¬[l, m] T1 â¡ U1 â U1 = U2. +#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind â¦ H) -U2 +/2 width=7 by cpy_inv_lift1_eq/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_alt.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_alt.etc new file mode 100644 index 000000000..8c4266107 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_alt.etc @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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 â ynat â relation4 genv lenv term term â +| cpysa_atom : âI,G,L,l,m. cpysa l m G L (âª{I}) (âª{I}) +| cpysa_subst: âI,G,L,K,V1,V2,W2,i,l,m. l â¤ yinj i â i < l+m â + â¬[i] L â¡ K.â{I}V1 â cpysa 0 (â(l+m-i)) G K V1 V2 â + â¬[0, i+1] V2 â¡ W2 â cpysa l m G L (#i) W2 +| cpysa_bind : âa,I,G,L,V1,V2,T1,T2,l,m. + cpysa l m G L V1 V2 â cpysa (âl) m G (L.â{I}V1) T1 T2 â + cpysa l m G L (â{a,I}V1.T1) (â{a,I}V2.T2) +| cpysa_flat : âI,G,L,V1,V2,T1,T2,l,m. + cpysa l m G L V1 V2 â cpysa l m G L T1 T2 â + cpysa l m G L (â{I}V1.T1) (â{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: âG,l,m. lsub_trans â¦ (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 â¦ HL12 â¦ 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: âG,T,L,l,m. â¦G, Lâ¦ â¢ T â¶â¶*[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: âG,L,T1,T,l,m. â¦G, Lâ¦ â¢ T1 â¶â¶*[l, m] T â + âT2. â¦G, Lâ¦ â¢ T â¶[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶â¶*[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 â¦ 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 â¦ HLK) #H0LK + lapply (cpy_weak â¦ H 0 (l+m) ? ?) -H // #H + elim (cpy_inv_lift1_be â¦ H â¦ H0LK â¦ 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 â¦ 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 â¦ H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/ +] +qed-. + +lemma cpys_cpysa: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶â¶*[l, m] T2. +/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma cpysa_inv_cpys: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[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: âR:ynatâynatârelation4 genv lenv term term. + (âI,G,L,l,m. R l m G L (âª{I}) (âª{I})) â + (âI,G,L,K,V1,V2,W2,i,l,m. l â¤ yinj i â i < l + m â + â¬[i] L â¡ K.â{I}V1 â â¦G, Kâ¦ â¢ V1 â¶*[O, â(l+m-i)] V2 â + â¬[O, i+1] V2 â¡ W2 â R O (â(l+m-i)) G K V1 V2 â R l m G L (#i) W2 + ) â + (âa,I,G,L,V1,V2,T1,T2,l,m. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 â + â¦G, L.â{I}V1â¦ â¢ T1 â¶*[âl, m] T2 â R l m G L V1 V2 â + R (âl) m G (L.â{I}V1) T1 T2 â R l m G L (â{a,I}V1.T1) (â{a,I}V2.T2) + ) â + (âI,G,L,V1,V2,T1,T2,l,m. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 â + â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â R l m G L V1 V2 â + R l m G L T1 T2 â R l m G L (â{I}V1.T1) (â{I}V2.T2) + ) â + âl,m,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â R l m G L T1 T2. +#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa â¦ H) -G -L -T1 -T2 -l -m +/3 width=8 by cpysa_inv_cpys/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_cpys.etc new file mode 100644 index 000000000..a01999d3f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_cpys.etc @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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: âG,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 â¶*[l, 1] T2 â â¦G, Lâ¦ â¢ T1 â¶[l, 1] T2. +#G #L #T1 #T2 #l #H @(cpys_ind â¦ H) -T2 /2 width=3 by cpy_trans_ge/ +qed-. + +(* Advanced properties ******************************************************) + +lemma cpys_strip_eq: âG,L,T0,T1,l1,m1. â¦G, Lâ¦ â¢ T0 â¶*[l1, m1] T1 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â + ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T2 â¶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-. + +lemma cpys_strip_neq: âG,L1,T0,T1,l1,m1. â¦G, L1â¦ â¢ T0 â¶*[l1, m1] T1 â + âL2,T2,l2,m2. â¦G, L2â¦ â¢ T0 â¶[l2, m2] T2 â + (l1 + m1 â¤ l2 â¨ l2 + m2 â¤ l1) â + ââT. â¦G, L2â¦ â¢ T1 â¶[l2, m2] T & â¦G, L1â¦ â¢ T2 â¶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-. + +lemma cpys_strap1_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶*[l1, m1] T0 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â l2 + m2 â¤ l1 â + ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T â¶*[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed. + +lemma cpys_strap2_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶[l1, m1] T0 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶*[l2, m2] T2 â l2 + m2 â¤ l1 â + ââT. â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T & â¦G, Lâ¦ â¢ T â¶[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-. + +lemma cpys_split_up: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â + âi. l â¤ i â i â¤ l + m â + ââT. â¦G, Lâ¦ â¢ T1 â¶*[l, i - l] T & â¦G, Lâ¦ â¢ T â¶*[i, l + m - i] T2. +#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind â¦ H) -T2 +[ /2 width=3 by ex2_intro/ +| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 + elim (cpy_split_up â¦ HT12 â¦ Hilm) -HT12 -Hilm #T0 #HT0 #HT02 + elim (cpys_strap1_down â¦ HT3 â¦ HT0) -T /3 width=5 by cpys_strap1, ex2_intro/ + >ymax_pre_sn_comm // +] +qed-. + +lemma cpys_inv_lift1_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + l â¤ lt â lt â¤ yinj l + m â yinj l + m â¤ lt + mt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[l, lt + mt - (yinj l + m)] T2 & + â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt +elim (cpys_split_up â¦ HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 +lapply (cpys_weak â¦ HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 +lapply (cpys_inv_lift1_eq â¦ HU1 â¦ HTU1) -HU1 #HU1 destruct +elim (cpys_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1) -HU2 -HLK -HTU1 // +>yplus_minus_inj /2 width=3 by ex2_intro/ +qed-. + +(* Main properties **********************************************************) + +theorem cpys_conf_eq: âG,L,T0,T1,l1,m1. â¦G, Lâ¦ â¢ T0 â¶*[l1, m1] T1 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶*[l2, m2] T2 â + ââT. â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T & â¦G, Lâ¦ â¢ T2 â¶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-. + +theorem cpys_conf_neq: âG,L1,T0,T1,l1,m1. â¦G, L1â¦ â¢ T0 â¶*[l1, m1] T1 â + âL2,T2,l2,m2. â¦G, L2â¦ â¢ T0 â¶*[l2, m2] T2 â + (l1 + m1 â¤ l2 â¨ l2 + m2 â¤ l1) â + ââT. â¦G, L2â¦ â¢ T1 â¶*[l2, m2] T & â¦G, L1â¦ â¢ T2 â¶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-. + +theorem cpys_trans_eq: âG,L,T1,T,T2,l,m. + â¦G, Lâ¦ â¢ T1 â¶*[l, m] T â â¦G, Lâ¦ â¢ T â¶*[l, m] T2 â + â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2. +normalize /2 width=3 by trans_TC/ qed-. + +theorem cpys_trans_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶*[l1, m1] T0 â + âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶*[l2, m2] T2 â l2 + m2 â¤ l1 â + ââT. â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T & â¦G, Lâ¦ â¢ T â¶*[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-. + +theorem cpys_antisym_eq: âG,L1,T1,T2,l,m. â¦G, L1â¦ â¢ T1 â¶*[l, m] T2 â + âL2. â¦G, L2â¦ â¢ T2 â¶*[l, m] T1 â T1 = T2. +#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt â¦ 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 â¦ HW2) -HW2 #HW2 + lapply (cpys_weak â¦ HW2 0 (i+1) ? ?) -HW2 // + [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi + #HW2 >(cpys_inv_lift1_eq â¦ HW2) -HW2 // + | elim (drop_O1_le (â») â¦ Hi) -Hi #K2 #HLK2 + elim (cpys_inv_lift1_ge_up â¦ HW2 â¦ HLK2 â¦ HVW2 ? ? ?) -HW2 -HLK2 -HVW2 + /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hli -Hilm + #X #_ #H elim (lift_inv_lref2_be â¦ H) -H // + ] +| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 â¦ H) -H + #V #T #HV2 #HT2 #H destruct + lapply (IHV12 â¦ 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 â¦ H) -H + #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_lift.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_lift.etc new file mode 100644 index 000000000..48c45f565 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/cpys_lift.etc @@ -0,0 +1,228 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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: âI,G,L,K,V,U1,i,l,m. + l â¤ yinj i â i < l + m â + â¬[i] L â¡ K.â{I}V â â¦G, Kâ¦ â¢ V â¶*[0, â(l+m-i)] U1 â + âU2. â¬[0, i+1] U1 â¡ U2 â â¦G, Lâ¦ â¢ #i â¶*[l, m] U2. +#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind â¦ 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 â¦ HU0) -IHU #H + lapply (drop_fwd_drop2 â¦ HLK) -HLK #HLK + lapply (cpy_lift_ge â¦ HU1 â¦ HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 + lapply (cpy_weak â¦ HU02 l m ? ?) -HU02 + [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] + >yplus_O1 ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/ +] +qed. + +lemma cpys_subst_Y2: âI,G,L,K,V,U1,i,l. + l â¤ yinj i â + â¬[i] L â¡ K.â{I}V â â¦G, Kâ¦ â¢ V â¶*[0, â] U1 â + âU2. â¬[0, i+1] U1 â¡ U2 â â¦G, Lâ¦ â¢ #i â¶*[l, â] U2. +#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12 +@(cpys_subst â¦ HLK â¦ HU12) >yminus_Y_inj // +qed. + +(* Advanced inversion lemmas *************************************************) + +lemma cpys_inv_atom1: âI,G,L,T2,l,m. â¦G, Lâ¦ â¢ âª{I} â¶*[l, m] T2 â + T2 = âª{I} â¨ + ââJ,K,V1,V2,i. l â¤ yinj i & i < l + m & + â¬[i] L â¡ K.â{J}V1 & + â¦G, Kâ¦ â¢ V1 â¶*[0, â(l+m-i)] V2 & + â¬[O, i+1] V2 â¡ T2 & + I = LRef i. +#I #G #L #T2 #l #m #H @(cpys_ind â¦ H) -T2 +[ /2 width=1 by or_introl/ +| #T #T2 #_ #HT2 * + [ #H destruct + elim (cpy_inv_atom1 â¦ 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 â¦ HLK) #H + elim (cpy_inv_lift1_ge_up â¦ HT2 â¦ H â¦ 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: âG,L,T2,i,l,m. â¦G, Lâ¦ â¢ #i â¶*[l, m] T2 â + T2 = #i â¨ + ââI,K,V1,V2. l â¤ i & i < l + m & + â¬[i] L â¡ K.â{I}V1 & + â¦G, Kâ¦ â¢ V1 â¶*[0, â(l+m-i)] V2 & + â¬[O, i+1] V2 â¡ T2. +#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 â¦ 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: âG,L,T2,i,l. â¦G, Lâ¦ â¢ #i â¶*[l, â] T2 â + T2 = #i â¨ + ââI,K,V1,V2. l â¤ i & â¬[i] L â¡ K.â{I}V1 & + â¦G, Kâ¦ â¢ V1 â¶*[0, â] V2 & â¬[O, i+1] V2 â¡ T2. +#G #L #T2 #i #l #H elim (cpys_inv_lref1 â¦ 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: âG,L,T2,i,l,m. â¦G, Lâ¦ â¢ #i â¶*[l, m] T2 â + âI,K,V1. â¬[i] L â¡ K.â{I}V1 â + âV2. â¬[O, i+1] V2 â¡ T2 â + â§â§ â¦G, Kâ¦ â¢ V1 â¶*[0, â(l+m-i)] V2 + & l â¤ i + & i < l + m. +#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 â¦ H) -H +[ #H destruct elim (lift_inv_lref2_be â¦ HVT2) -HVT2 -HLK // +| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2 + lapply (lift_inj â¦ HXT2 â¦ HVT2) -T2 #H destruct + lapply (drop_mono â¦ HLY â¦ HLK) -L #H destruct + /2 width=1 by and3_intro/ +] +qed-. + +(* Properties on relocation *************************************************) + +lemma cpys_lift_le: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 â + âL,U1,s,l,m. lt + mt â¤ yinj l â â¬[s, l, m] L â¡ K â + â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â + â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind â¦ H) -T2 +[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT â¦ HTU) -IHT #HU1 + lapply (cpy_lift_le â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +lemma cpys_lift_be: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 â + âL,U1,s,l,m. lt â¤ yinj l â l â¤ lt + mt â + â¬[s, l, m] L â¡ K â â¬[l, m] T1 â¡ U1 â + âU2. â¬[l, m] T2 â¡ U2 â â¦G, Lâ¦ â¢ U1 â¶*[lt, mt + m] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind â¦ H) -T2 +[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT â¦ HTU) -IHT #HU1 + lapply (cpy_lift_be â¦ HT2 â¦ HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +lemma cpys_lift_ge: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 â + âL,U1,s,l,m. yinj l â¤ lt â â¬[s, l, m] L â¡ K â + â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â + â¦G, Lâ¦ â¢ U1 â¶*[lt+m, mt] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind â¦ H) -T2 +[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT â¦ HTU) -IHT #HU1 + lapply (cpy_lift_ge â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +(* Inversion lemmas for relocation ******************************************) + +lemma cpys_inv_lift1_le: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt + mt â¤ l â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind â¦ H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_le â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_be: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt â¤ l â yinj l + m â¤ lt + mt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt - m] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind â¦ H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_be â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_ge: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + yinj l + m â¤ lt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt - m, mt] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind â¦ H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +(* Advanced inversion lemmas on relocation **********************************) + +lemma cpys_inv_lift1_ge_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + l â¤ lt â lt â¤ yinj l + m â yinj l + m â¤ lt + mt â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[l, lt + mt - (yinj l + m)] T2 & + â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind â¦ H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_ge_up â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_be_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt â¤ l â lt + mt â¤ yinj l + m â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind â¦ H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_be_up â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_le_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â + âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â + lt â¤ l â l â¤ lt + mt â lt + mt â¤ yinj l + m â + ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind â¦ H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_le_up â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_subst: âG,L,W1,W2,l,m. â¦G, Lâ¦ â¢ W1 â¶*[l, m] W2 â + âK,V1,i. â¬[i+1] L â¡ K â â¬[O, i+1] V1 â¡ W1 â + l â¤ yinj i â i < l + m â + ââV2. â¦G, Kâ¦ â¢ V1 â¶*[O, â(l+m-i)] V2 & â¬[O, i+1] V2 â¡ W2. +#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm +elim (cpys_inv_lift1_ge_up â¦ HW12 â¦ HLK â¦ HVW1 ? ? ?) // +>yplus_O1 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/psubststar_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/psubststar_6.etc new file mode 100644 index 000000000..2dd1f2d6e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/psubststar_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 break â¶ * [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $G $L $T1 $l $m $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/psubststaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/psubststaralt_6.etc new file mode 100644 index 000000000..97227dccf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/cpys/psubststaralt_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( â¦ term 46 G , break term 46 L â¦ â¢ break term 46 T1 break â¶ â¶ * [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStarAlt $G $L $T1 $l $m $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lrsubeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lrsubeq_4.etc new file mode 100644 index 000000000..994e6b22b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lrsubeq_4.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 Î»Î´ ****************************************) + +notation "hvbox( L1 break â [ term 46 l , break term 46 m ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LRSubEq $L1 $l $m $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lsuby.etc new file mode 100644 index 000000000..bd5c2a2c1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lsuby.etc @@ -0,0 +1,239 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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 â +| lsuby_atom: âL,l,m. lsuby l m L (â) +| lsuby_zero: âI1,I2,L1,L2,V1,V2. + lsuby 0 0 L1 L2 â lsuby 0 0 (L1.â{I1}V1) (L2.â{I2}V2) +| lsuby_pair: âI1,I2,L1,L2,V,m. lsuby 0 m L1 L2 â + lsuby 0 (âm) (L1.â{I1}V) (L2.â{I2}V) +| lsuby_succ: âI1,I2,L1,L2,V1,V2,l,m. + lsuby l m L1 L2 â lsuby (âl) m (L1.â{I1}V1) (L2.â{I2}V2) +. + +interpretation + "local environment refinement (extended substitution)" + 'LRSubEq L1 l m L2 = (lsuby l m L1 L2). + +(* Basic properties *********************************************************) + +lemma lsuby_pair_lt: âI1,I2,L1,L2,V,m. L1 â[0, âm] L2 â 0 < m â + L1.â{I1}V â[0, m] L2.â{I2}V. +#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 â¦ Hm) /2 width=1 by lsuby_pair/ +qed. + +lemma lsuby_succ_lt: âI1,I2,L1,L2,V1,V2,l,m. L1 â[âl, m] L2 â 0 < l â + L1.â{I1}V1 â[l, m] L2. â{I2}V2. +#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 â¦ Hl) /2 width=1 by lsuby_succ/ +qed. + +lemma lsuby_pair_O_Y: âL1,L2. L1 â[0, â] L2 â + âI1,I2,V. L1.â{I1}V â[0,â] L2.â{I2}V. +#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 â¦ V â¦ HL12) -HL12 // +qed. + +lemma lsuby_refl: âL,l,m. L â[l, m] L. +#L elim L -L // +#L #I #V #IHL #l elim (ynat_cases â¦ l) [| * #x ] +#Hl destruct /2 width=1 by lsuby_succ/ +#m elim (ynat_cases â¦ m) [| * #x ] +#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/ +qed. + +lemma lsuby_O2: âL2,L1,l. |L2| â¤ |L1| â L1 â[l, yinj 0] L2. +#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize +[ #l #H elim (le_plus_xSy_O_false â¦ H) +| #L1 #I1 #V1 #l #H lapply (le_plus_to_le_r â¦ H) -H #HL12 + elim (ynat_cases l) /3 width=1 by lsuby_zero/ + * /3 width=1 by lsuby_succ/ +] +qed. + +lemma lsuby_sym: âl,m,L1,L2. L1 â[l, m] L2 â |L1| = |L2| â L2 â[l, m] L1. +#l #m #L1 #L2 #H elim H -l -m -L1 -L2 +[ #L1 #l #m #H >(length_inv_zero_dx â¦ H) -L1 // +| /2 width=1 by lsuby_O2/ +| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l â¦ H) + /3 width=1 by lsuby_pair/ +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l â¦ H) + /3 width=1 by lsuby_succ/ +] +qed-. + +(* Basic inversion lemmas ***************************************************) + +fact lsuby_inv_atom1_aux: âL1,L2,l,m. L1 â[l, m] L2 â L1 = â â L2 = â. +#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: âL2,l,m. â â[l, m] L2 â L2 = â. +/2 width=5 by lsuby_inv_atom1_aux/ qed-. + +fact lsuby_inv_zero1_aux: âL1,L2,l,m. L1 â[l, m] L2 â + âJ1,K1,W1. L1 = K1.â{J1}W1 â l = 0 â m = 0 â + L2 = â â¨ + ââJ2,K2,W2. K1 â[0, 0] K2 & L2 = K2.â{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 â¦ H) +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H + elim (ysucc_inv_O_dx â¦ H) +] +qed-. + +lemma lsuby_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 â[0, 0] L2 â + L2 = â â¨ + ââI2,K2,V2. K1 â[0, 0] K2 & L2 = K2.â{I2}V2. +/2 width=9 by lsuby_inv_zero1_aux/ qed-. + +fact lsuby_inv_pair1_aux: âL1,L2,l,m. L1 â[l, m] L2 â + âJ1,K1,W. L1 = K1.â{J1}W â l = 0 â 0 < m â + L2 = â â¨ + ââJ2,K2. K1 â[0, âm] K2 & L2 = K2.â{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 â¦ 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 â¦ H) +] +qed-. + +lemma lsuby_inv_pair1: âI1,K1,L2,V,m. K1.â{I1}V â[0, m] L2 â 0 < m â + L2 = â â¨ + ââI2,K2. K1 â[0, âm] K2 & L2 = K2.â{I2}V. +/2 width=6 by lsuby_inv_pair1_aux/ qed-. + +fact lsuby_inv_succ1_aux: âL1,L2,l,m. L1 â[l, m] L2 â + âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < l â + L2 = â â¨ + ââJ2,K2,W2. K1 â[âl, m] K2 & L2 = K2.â{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 â¦ H) // +| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H + elim (ylt_yle_false â¦ 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: âI1,K1,L2,V1,l,m. K1.â{I1}V1 â[l, m] L2 â 0 < l â + L2 = â â¨ + ââI2,K2,V2. K1 â[âl, m] K2 & L2 = K2.â{I2}V2. +/2 width=5 by lsuby_inv_succ1_aux/ qed-. + +fact lsuby_inv_zero2_aux: âL1,L2,l,m. L1 â[l, m] L2 â + âJ2,K2,W2. L2 = K2.â{J2}W2 â l = 0 â m = 0 â + ââJ1,K1,W1. K1 â[0, 0] K2 & L1 = K1.â{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 â¦ H) +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H + elim (ysucc_inv_O_dx â¦ H) +] +qed-. + +lemma lsuby_inv_zero2: âI2,K2,L1,V2. L1 â[0, 0] K2.â{I2}V2 â + ââI1,K1,V1. K1 â[0, 0] K2 & L1 = K1.â{I1}V1. +/2 width=9 by lsuby_inv_zero2_aux/ qed-. + +fact lsuby_inv_pair2_aux: âL1,L2,l,m. L1 â[l, m] L2 â + âJ2,K2,W. L2 = K2.â{J2}W â l = 0 â 0 < m â + ââJ1,K1. K1 â[0, âm] K2 & L1 = K1.â{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 â¦ 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 â¦ H) +] +qed-. + +lemma lsuby_inv_pair2: âI2,K2,L1,V,m. L1 â[0, m] K2.â{I2}V â 0 < m â + ââI1,K1. K1 â[0, âm] K2 & L1 = K1.â{I1}V. +/2 width=6 by lsuby_inv_pair2_aux/ qed-. + +fact lsuby_inv_succ2_aux: âL1,L2,l,m. L1 â[l, m] L2 â + âJ2,K2,W2. L2 = K2.â{J2}W2 â 0 < l â + ââJ1,K1,W1. K1 â[âl, m] K2 & L1 = K1.â{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 â¦ H) // +| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H + elim (ylt_yle_false â¦ 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: âI2,K2,L1,V2,l,m. L1 â[l, m] K2.â{I2}V2 â 0 < l â + ââI1,K1,V1. K1 â[âl, m] K2 & L1 = K1.â{I1}V1. +/2 width=5 by lsuby_inv_succ2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lsuby_fwd_length: âL1,L2,l,m. L1 â[l, m] L2 â |L2| â¤ |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: âL1,L2,l,m. L1 â[l, m] L2 â + âI2,K2,W,s,i. â¬[s, 0, i] L2 â¡ K2.â{I2}W â + l â¤ i â i < l + m â + ââI1,K1. K1 â[0, â(l+m-i)] K2 & â¬[s, 0, i] L1 â¡ K1.â{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 â¦ H) -H #H destruct +| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H + elim (ylt_yle_false â¦ H) // +| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1 + elim (drop_inv_O1_pair1 â¦ H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ] + [ #_ destruct -I2 >ypred_succ + /2 width=4 by drop_pair, ex2_2_intro/ + | <(S_pred â¦ Hi) 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 â¦ Hli) -Hli #Hli #Hi + lapply (ylt_inv_inj â¦ Hi) -Hi #Hi + <(S_pred â¦ Hi) >yplus_succ1 minus_SO_dx + /3 width=4 by drop_drop_lt, ex2_2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lsuby_lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lsuby_lsuby.etc new file mode 100644 index 000000000..8e45e98c9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/etc_yminus/lsuby/lsuby_lsuby.etc @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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: âl,m. Transitive â¦ (lsuby l m). +#l #m #L1 #L2 #H elim H -L1 -L2 -l -m +[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 â¦ H) -H + #H destruct // +| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 â¦ 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 â¦ 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 â¦ H) -H // + * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma index 835d8f837..9888366fe 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma @@ -16,7 +16,7 @@ * 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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma index 235b741bf..e96c34523 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma @@ -12,7 +12,6 @@ (* *) (**************************************************************************) -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". @@ -22,19 +21,22 @@ include "basic_2A/grammar/term.ma". (* GLOBAL ENVIRONMENTS ******************************************************) (* global environments *) -definition genv â list2 bind2 term. +inductive genv: Type[0] â +| GAtom: genv (* empty *) +| GPair: genv â bind2 â term â 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 *********************************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv_length.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv_length.ma new file mode 100644 index 000000000..15cf81749 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv_length.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||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 â match L with +[ GAtom â 0 +| GPair G _ _ â glength G + 1 +]. + +interpretation "length (global environment)" 'card G = (glength G). diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma index 88d5cf57a..97207c56c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma @@ -12,8 +12,8 @@ (* *) (**************************************************************************) -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 ********************************************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma index 25273f9f2..40f5e530f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma index cefc9b049..5346ca04b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "ground_2/xoa/ex_2_3.ma". include "basic_2A/grammar/lenv.ma". (* LENGTH OF A LOCAL ENVIRONMENT ********************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma index 820eaa46b..081396d0a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -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". @@ -23,9 +23,9 @@ inductive lreq: relation4 ynat ynat lenv lenv â | lreq_zero: âI1,I2,L1,L2,V1,V2. lreq 0 0 L1 L2 â lreq 0 0 (L1.â{I1}V1) (L2.â{I2}V2) | lreq_pair: âI,L1,L2,V,m. lreq 0 m L1 L2 â - lreq 0 (â«¯m) (L1.â{I}V) (L2.â{I}V) + lreq 0 (âm) (L1.â{I}V) (L2.â{I}V) | lreq_succ: âI1,I2,L1,L2,V1,V2,l,m. - lreq l m L1 L2 â lreq (â«¯l) m (L1.â{I1}V1) (L2.â{I2}V2) + lreq l m L1 L2 â lreq (âl) m (L1.â{I1}V1) (L2.â{I2}V2) . interpretation @@ -34,12 +34,12 @@ interpretation (* Basic properties *********************************************************) -lemma lreq_pair_lt: âI,L1,L2,V,m. L1 â©¬[0, â«°m] L2 â 0 < m â +lemma lreq_pair_lt: âI,L1,L2,V,m. L1 â©¬[0, âm] L2 â 0 < m â L1.â{I}V â©¬[0, m] L2.â{I}V. #I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 â¦ Hm) /2 width=1 by lreq_pair/ qed. -lemma lreq_succ_lt: âI1,I2,L1,L2,V1,V2,l,m. L1 â©¬[â«°l, m] L2 â 0 < l â +lemma lreq_succ_lt: âI1,I2,L1,L2,V1,V2,l,m. L1 â©¬[âl, m] L2 â 0 < l â L1.â{I1}V1 â©¬[l, m] L2. â{I2}V2. #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 â¦ Hl) /2 width=1 by lreq_succ/ qed. @@ -104,7 +104,7 @@ lemma lreq_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 â©¬[0, 0] L2 â fact lreq_inv_pair1_aux: âL1,L2,l,m. L1 â©¬[l, m] L2 â âJ,K1,W. L1 = K1.â{J}W â l = 0 â 0 < m â - ââK2. K1 â©¬[0, â«°m] K2 & L2 = K2.â{J}W. + ââK2. K1 â©¬[0, âm] K2 & L2 = K2.â{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 @@ -117,18 +117,18 @@ fact lreq_inv_pair1_aux: âL1,L2,l,m. L1 â©¬[l, m] L2 â qed-. lemma lreq_inv_pair1: âI,K1,L2,V,m. K1.â{I}V â©¬[0, m] L2 â 0 < m â - ââK2. K1 â©¬[0, â«°m] K2 & L2 = K2.â{I}V. + ââK2. K1 â©¬[0, âm] K2 & L2 = K2.â{I}V. /2 width=6 by lreq_inv_pair1_aux/ qed-. lemma lreq_inv_pair: âI1,I2,L1,L2,V1,V2,m. L1.â{I1}V1 â©¬[0, m] L2.â{I2}V2 â 0 < m â - â§â§ L1 â©¬[0, â«°m] L2 & I1 = I2 & V1 = V2. + â§â§ L1 â©¬[0, âm] L2 & I1 = I2 & V1 = V2. #I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 â¦ H) -H // #Y #HL12 #H destruct /2 width=1 by and3_intro/ qed-. fact lreq_inv_succ1_aux: âL1,L2,l,m. L1 â©¬[l, m] L2 â âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < l â - ââJ2,K2,W2. K1 â©¬[â«°l, m] K2 & L2 = K2.â{J2}W2. + ââJ2,K2,W2. K1 â©¬[âl, m] K2 & L2 = K2.â{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 @@ -141,7 +141,7 @@ fact lreq_inv_succ1_aux: âL1,L2,l,m. L1 â©¬[l, m] L2 â qed-. lemma lreq_inv_succ1: âI1,K1,L2,V1,l,m. K1.â{I1}V1 â©¬[l, m] L2 â 0 < l â - ââI2,K2,V2. K1 â©¬[â«°l, m] K2 & L2 = K2.â{I2}V2. + ââI2,K2,V2. K1 â©¬[âl, m] K2 & L2 = K2.â{I2}V2. /2 width=5 by lreq_inv_succ1_aux/ qed-. lemma lreq_inv_atom2: âL1,l,m. L1 â©¬[l, m] â â L1 = â. @@ -149,7 +149,7 @@ lemma lreq_inv_atom2: âL1,l,m. L1 â©¬[l, m] â â L1 = â. qed-. lemma lreq_inv_succ: âI1,I2,L1,L2,V1,V2,l,m. L1.â{I1}V1 â©¬[l, m] L2.â{I2}V2 â 0 < l â - L1 â©¬[â«°l, m] L2. + L1 â©¬[âl, m] L2. #I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 â¦ H) -H // #Z #Y #X #HL12 #H destruct // qed-. @@ -161,13 +161,13 @@ lemma lreq_inv_zero2: âI2,K2,L1,V2. L1 â©¬[0, 0] K2.â{I2}V2 â qed-. lemma lreq_inv_pair2: âI,K2,L1,V,m. L1 â©¬[0, m] K2.â{I}V â 0 < m â - ââK1. K1 â©¬[0, â«°m] K2 & L1 = K1.â{I}V. + ââK1. K1 â©¬[0, âm] K2 & L1 = K1.â{I}V. #I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 â¦ (lreq_sym â¦ H)) -H /3 width=3 by lreq_sym, ex2_intro/ qed-. lemma lreq_inv_succ2: âI2,K2,L1,V2,l,m. L1 â©¬[l, m] K2.â{I2}V2 â 0 < l â - ââI1,K1,V1. K1 â©¬[â«°l, m] K2 & L1 = K1.â{I1}V1. + ââI1,K1,V1. K1 â©¬[âl, m] K2 & L1 = K1.â{I1}V1. #I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 â¦ (lreq_sym â¦ H)) -H /3 width=5 by lreq_sym, ex2_3_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma index 700b87d04..0076c81a8 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -include "ground_2A/ynat/ynat_plus.ma". +include "ground_2/ynat/ynat_plus.ma". include "basic_2A/grammar/lreq.ma". (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma index f6b163371..417414595 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma index 80311fe76..3e9315de3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "ground_2/xoa/ex_1_2.ma". include "basic_2A/notation/relations/topiso_2.ma". include "basic_2A/grammar/term_simple.ma". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma deleted file mode 100644 index 6f3f953f9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma +++ /dev/null @@ -1,166 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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 â ynat â relation4 genv lenv term term â - Î»l,m,G. LTC â¦ (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: âG,L,T1,l,m. âR:predicate term. R T1 â - (âT,T2. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T â â¦G, Lâ¦ â¢ T â¶[l, m] T2 â R T â R T2) â - âT2. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â R T2. -#G #L #T1 #l #m #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind â¦ HT1 IHT1 â¦ HT12) // -qed-. - -lemma cpys_ind_dx: âG,L,T2,l,m. âR:predicate term. R T2 â - (âT1,T. â¦G, Lâ¦ â¢ T1 â¶[l, m] T â â¦G, Lâ¦ â¢ T â¶*[l, m] T2 â R T â R T1) â - âT1. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â R T1. -#G #L #T2 #l #m #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx â¦ HT2 IHT2 â¦ HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma cpy_cpys: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2. -/2 width=1 by inj/ qed. - -lemma cpys_strap1: âG,L,T1,T,T2,l,m. - â¦G, Lâ¦ â¢ T1 â¶*[l, m] T â â¦G, Lâ¦ â¢ T â¶[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2. -normalize /2 width=3 by step/ qed-. - -lemma cpys_strap2: âG,L,T1,T,T2,l,m. - â¦G, Lâ¦ â¢ T1 â¶[l, m] T â â¦G, Lâ¦ â¢ T â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2. -normalize /2 width=3 by TC_strap/ qed-. - -lemma lsuby_cpys_trans: âG,l,m. lsub_trans â¦ (cpys l m G) (lsuby l m). -/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/ -qed-. - -lemma cpys_refl: âG,L,l,m. reflexive â¦ (cpys l m G L). -/2 width=1 by cpy_cpys/ qed. - -lemma cpys_bind: âG,L,V1,V2,l,m. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 â - âI,T1,T2. â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯l, m] T2 â - âa. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[l, m] â{a,I}V2.T2. -#G #L #V1 #V2 #l #m #HV12 @(cpys_ind â¦ HV12) -V2 -[ #I #T1 #T2 #HT12 @(cpys_ind â¦ HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/ -| /3 width=5 by cpys_strap1, cpy_bind/ -] -qed. - -lemma cpys_flat: âG,L,V1,V2,l,m. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 â - âT1,T2. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â - âI. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[l, m] â{I}V2.T2. -#G #L #V1 #V2 #l #m #HV12 @(cpys_ind â¦ HV12) -V2 -[ #T1 #T2 #HT12 @(cpys_ind â¦ HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/ -| /3 width=5 by cpys_strap1, cpy_flat/ -qed. - -lemma cpys_weak: âG,L,T1,T2,l1,m1. â¦G, Lâ¦ â¢ T1 â¶*[l1, m1] T2 â - âl2,m2. l2 â¤ l1 â l1 + m1 â¤ l2 + m2 â - â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T2. -#G #L #T1 #T2 #l1 #m1 #H #l1 #l2 #Hl21 #Hlm12 @(cpys_ind â¦ H) -T2 -/3 width=7 by cpys_strap1, cpy_weak/ -qed-. - -lemma cpys_weak_top: âG,L,T1,T2,l,m. - â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[l, |L| - l] T2. -#G #L #T1 #T2 #l #m #H @(cpys_ind â¦ H) -T2 -/3 width=4 by cpys_strap1, cpy_weak_top/ -qed-. - -lemma cpys_weak_full: âG,L,T1,T2,l,m. - â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[0, |L|] T2. -#G #L #T1 #T2 #l #m #H @(cpys_ind â¦ H) -T2 -/3 width=5 by cpys_strap1, cpy_weak_full/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cpys_fwd_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âT1,l,m. â¬[l, m] T1 â¡ U1 â - l â¤ lt â l + m â¤ lt + mt â - ââT2. â¦G, Lâ¦ â¢ U1 â¶*[l+m, lt+mt-(l+m)] U2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #T1 #l #m #HTU1 #Hllt #Hlmlmt @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU - elim (cpy_fwd_up â¦ HU2 â¦ HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_fwd_tw: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â â¯{T1} â¤ â¯{T2}. -#G #L #T1 #T2 #l #m #H @(cpys_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw â¦ 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: âG,L,T2,k,l,m. â¦G, Lâ¦ â¢ âk â¶*[l, m] T2 â T2 = âk. -#G #L #T2 #k #l #m #H @(cpys_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 destruct ->(cpy_inv_sort1 â¦ HT2) -HT2 // -qed-. - -(* Note: this can be derived from cpys_inv_atom1 *) -lemma cpys_inv_gref1: âG,L,T2,p,l,m. â¦G, Lâ¦ â¢ Â§p â¶*[l, m] T2 â T2 = Â§p. -#G #L #T2 #p #l #m #H @(cpys_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 destruct ->(cpy_inv_gref1 â¦ HT2) -HT2 // -qed-. - -lemma cpys_inv_bind1: âa,I,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{a,I}V1.T1 â¶*[l, m] U2 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 & - â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯l, m] T2 & - U2 = â{a,I}V2.T2. -#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind â¦ H) -U2 -[ /2 width=5 by ex3_2_intro/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (cpy_inv_bind1 â¦ HU2) -HU2 #V2 #T2 #HV2 #HT2 #H - lapply (lsuby_cpy_trans â¦ HT2 (L.â{I}V1) ?) -HT2 - /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/ -] -qed-. - -lemma cpys_inv_flat1: âI,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{I}V1.T1 â¶*[l, m] U2 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 & â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 & - U2 = â{I}V2.T2. -#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind â¦ H) -U2 -[ /2 width=5 by ex3_2_intro/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (cpy_inv_flat1 â¦ HU2) -HU2 - /3 width=5 by cpys_strap1, ex3_2_intro/ -] -qed-. - -lemma cpys_inv_refl_O2: âG,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 â¶*[l, 0] T2 â T1 = T2. -#G #L #T1 #T2 #l #H @(cpys_ind â¦ H) -T2 // -#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 â¦ HT2) -HT2 // -qed-. - -lemma cpys_inv_lift1_eq: âG,L,U1,U2. âl,m:nat. - â¦G, Lâ¦ â¢ U1 â¶*[l, m] U2 â âT1. â¬[l, m] T1 â¡ U1 â U1 = U2. -#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind â¦ H) -U2 -/2 width=7 by cpy_inv_lift1_eq/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_alt.ma deleted file mode 100644 index 7d1274a91..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_alt.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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 â ynat â relation4 genv lenv term term â -| cpysa_atom : âI,G,L,l,m. cpysa l m G L (âª{I}) (âª{I}) -| cpysa_subst: âI,G,L,K,V1,V2,W2,i,l,m. l â¤ yinj i â i < l+m â - â¬[i] L â¡ K.â{I}V1 â cpysa 0 (â«°(l+m-i)) G K V1 V2 â - â¬[0, i+1] V2 â¡ W2 â cpysa l m G L (#i) W2 -| cpysa_bind : âa,I,G,L,V1,V2,T1,T2,l,m. - cpysa l m G L V1 V2 â cpysa (â«¯l) m G (L.â{I}V1) T1 T2 â - cpysa l m G L (â{a,I}V1.T1) (â{a,I}V2.T2) -| cpysa_flat : âI,G,L,V1,V2,T1,T2,l,m. - cpysa l m G L V1 V2 â cpysa l m G L T1 T2 â - cpysa l m G L (â{I}V1.T1) (â{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: âG,l,m. lsub_trans â¦ (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 â¦ HL12 â¦ 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: âG,T,L,l,m. â¦G, Lâ¦ â¢ T â¶â¶*[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: âG,L,T1,T,l,m. â¦G, Lâ¦ â¢ T1 â¶â¶*[l, m] T â - âT2. â¦G, Lâ¦ â¢ T â¶[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶â¶*[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 â¦ 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 â¦ HLK) #H0LK - lapply (cpy_weak â¦ H 0 (l+m) ? ?) -H // #H - elim (cpy_inv_lift1_be â¦ H â¦ H0LK â¦ 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 â¦ 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 â¦ H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/ -] -qed-. - -lemma cpys_cpysa: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶â¶*[l, m] T2. -/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma cpysa_inv_cpys: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶â¶*[l, m] T2 â â¦G, Lâ¦ â¢ T1 â¶*[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: âR:ynatâynatârelation4 genv lenv term term. - (âI,G,L,l,m. R l m G L (âª{I}) (âª{I})) â - (âI,G,L,K,V1,V2,W2,i,l,m. l â¤ yinj i â i < l + m â - â¬[i] L â¡ K.â{I}V1 â â¦G, Kâ¦ â¢ V1 â¶*[O, â«°(l+m-i)] V2 â - â¬[O, i+1] V2 â¡ W2 â R O (â«°(l+m-i)) G K V1 V2 â R l m G L (#i) W2 - ) â - (âa,I,G,L,V1,V2,T1,T2,l,m. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 â - â¦G, L.â{I}V1â¦ â¢ T1 â¶*[â«¯l, m] T2 â R l m G L V1 V2 â - R (â«¯l) m G (L.â{I}V1) T1 T2 â R l m G L (â{a,I}V1.T1) (â{a,I}V2.T2) - ) â - (âI,G,L,V1,V2,T1,T2,l,m. â¦G, Lâ¦ â¢ V1 â¶*[l, m] V2 â - â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â R l m G L V1 V2 â - R l m G L T1 T2 â R l m G L (â{I}V1.T1) (â{I}V2.T2) - ) â - âl,m,G,L,T1,T2. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â R l m G L T1 T2. -#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa â¦ H) -G -L -T1 -T2 -l -m -/3 width=8 by cpysa_inv_cpys/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_cpys.ma deleted file mode 100644 index a01999d3f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_cpys.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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: âG,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 â¶*[l, 1] T2 â â¦G, Lâ¦ â¢ T1 â¶[l, 1] T2. -#G #L #T1 #T2 #l #H @(cpys_ind â¦ H) -T2 /2 width=3 by cpy_trans_ge/ -qed-. - -(* Advanced properties ******************************************************) - -lemma cpys_strip_eq: âG,L,T0,T1,l1,m1. â¦G, Lâ¦ â¢ T0 â¶*[l1, m1] T1 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â - ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T2 â¶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-. - -lemma cpys_strip_neq: âG,L1,T0,T1,l1,m1. â¦G, L1â¦ â¢ T0 â¶*[l1, m1] T1 â - âL2,T2,l2,m2. â¦G, L2â¦ â¢ T0 â¶[l2, m2] T2 â - (l1 + m1 â¤ l2 â¨ l2 + m2 â¤ l1) â - ââT. â¦G, L2â¦ â¢ T1 â¶[l2, m2] T & â¦G, L1â¦ â¢ T2 â¶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-. - -lemma cpys_strap1_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶*[l1, m1] T0 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â l2 + m2 â¤ l1 â - ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T â¶*[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed. - -lemma cpys_strap2_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶[l1, m1] T0 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶*[l2, m2] T2 â l2 + m2 â¤ l1 â - ââT. â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T & â¦G, Lâ¦ â¢ T â¶[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-. - -lemma cpys_split_up: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2 â - âi. l â¤ i â i â¤ l + m â - ââT. â¦G, Lâ¦ â¢ T1 â¶*[l, i - l] T & â¦G, Lâ¦ â¢ T â¶*[i, l + m - i] T2. -#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind â¦ H) -T2 -[ /2 width=3 by ex2_intro/ -| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 - elim (cpy_split_up â¦ HT12 â¦ Hilm) -HT12 -Hilm #T0 #HT0 #HT02 - elim (cpys_strap1_down â¦ HT3 â¦ HT0) -T /3 width=5 by cpys_strap1, ex2_intro/ - >ymax_pre_sn_comm // -] -qed-. - -lemma cpys_inv_lift1_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - l â¤ lt â lt â¤ yinj l + m â yinj l + m â¤ lt + mt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[l, lt + mt - (yinj l + m)] T2 & - â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt -elim (cpys_split_up â¦ HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 -lapply (cpys_weak â¦ HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 -lapply (cpys_inv_lift1_eq â¦ HU1 â¦ HTU1) -HU1 #HU1 destruct -elim (cpys_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1) -HU2 -HLK -HTU1 // ->yplus_minus_inj /2 width=3 by ex2_intro/ -qed-. - -(* Main properties **********************************************************) - -theorem cpys_conf_eq: âG,L,T0,T1,l1,m1. â¦G, Lâ¦ â¢ T0 â¶*[l1, m1] T1 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶*[l2, m2] T2 â - ââT. â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T & â¦G, Lâ¦ â¢ T2 â¶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-. - -theorem cpys_conf_neq: âG,L1,T0,T1,l1,m1. â¦G, L1â¦ â¢ T0 â¶*[l1, m1] T1 â - âL2,T2,l2,m2. â¦G, L2â¦ â¢ T0 â¶*[l2, m2] T2 â - (l1 + m1 â¤ l2 â¨ l2 + m2 â¤ l1) â - ââT. â¦G, L2â¦ â¢ T1 â¶*[l2, m2] T & â¦G, L1â¦ â¢ T2 â¶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-. - -theorem cpys_trans_eq: âG,L,T1,T,T2,l,m. - â¦G, Lâ¦ â¢ T1 â¶*[l, m] T â â¦G, Lâ¦ â¢ T â¶*[l, m] T2 â - â¦G, Lâ¦ â¢ T1 â¶*[l, m] T2. -normalize /2 width=3 by trans_TC/ qed-. - -theorem cpys_trans_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶*[l1, m1] T0 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶*[l2, m2] T2 â l2 + m2 â¤ l1 â - ââT. â¦G, Lâ¦ â¢ T1 â¶*[l2, m2] T & â¦G, Lâ¦ â¢ T â¶*[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-. - -theorem cpys_antisym_eq: âG,L1,T1,T2,l,m. â¦G, L1â¦ â¢ T1 â¶*[l, m] T2 â - âL2. â¦G, L2â¦ â¢ T2 â¶*[l, m] T1 â T1 = T2. -#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt â¦ 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 â¦ HW2) -HW2 #HW2 - lapply (cpys_weak â¦ HW2 0 (i+1) ? ?) -HW2 // - [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi - #HW2 >(cpys_inv_lift1_eq â¦ HW2) -HW2 // - | elim (drop_O1_le (â») â¦ Hi) -Hi #K2 #HLK2 - elim (cpys_inv_lift1_ge_up â¦ HW2 â¦ HLK2 â¦ HVW2 ? ? ?) -HW2 -HLK2 -HVW2 - /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hli -Hilm - #X #_ #H elim (lift_inv_lref2_be â¦ H) -H // - ] -| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 â¦ H) -H - #V #T #HV2 #HT2 #H destruct - lapply (IHV12 â¦ 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 â¦ H) -H - #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma deleted file mode 100644 index d3f292a7f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma +++ /dev/null @@ -1,226 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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: âI,G,L,K,V,U1,i,l,m. - l â¤ yinj i â i < l + m â - â¬[i] L â¡ K.â{I}V â â¦G, Kâ¦ â¢ V â¶*[0, â«°(l+m-i)] U1 â - âU2. â¬[0, i+1] U1 â¡ U2 â â¦G, Lâ¦ â¢ #i â¶*[l, m] U2. -#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind â¦ 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 â¦ HU0) -IHU #H - lapply (drop_fwd_drop2 â¦ HLK) -HLK #HLK - lapply (cpy_lift_ge â¦ HU1 â¦ HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 - lapply (cpy_weak â¦ HU02 l m ? ?) -HU02 - [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] - >yplus_O1 ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/ -] -qed. - -lemma cpys_subst_Y2: âI,G,L,K,V,U1,i,l. - l â¤ yinj i â - â¬[i] L â¡ K.â{I}V â â¦G, Kâ¦ â¢ V â¶*[0, â] U1 â - âU2. â¬[0, i+1] U1 â¡ U2 â â¦G, Lâ¦ â¢ #i â¶*[l, â] U2. -#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12 -@(cpys_subst â¦ HLK â¦ HU12) >yminus_Y_inj // -qed. - -(* Advanced inversion lemmas *************************************************) - -lemma cpys_inv_atom1: âI,G,L,T2,l,m. â¦G, Lâ¦ â¢ âª{I} â¶*[l, m] T2 â - T2 = âª{I} â¨ - ââJ,K,V1,V2,i. l â¤ yinj i & i < l + m & - â¬[i] L â¡ K.â{J}V1 & - â¦G, Kâ¦ â¢ V1 â¶*[0, â«°(l+m-i)] V2 & - â¬[O, i+1] V2 â¡ T2 & - I = LRef i. -#I #G #L #T2 #l #m #H @(cpys_ind â¦ H) -T2 -[ /2 width=1 by or_introl/ -| #T #T2 #_ #HT2 * - [ #H destruct - elim (cpy_inv_atom1 â¦ 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 â¦ HLK) #H - elim (cpy_inv_lift1_ge_up â¦ HT2 â¦ H â¦ 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: âG,L,T2,i,l,m. â¦G, Lâ¦ â¢ #i â¶*[l, m] T2 â - T2 = #i â¨ - ââI,K,V1,V2. l â¤ i & i < l + m & - â¬[i] L â¡ K.â{I}V1 & - â¦G, Kâ¦ â¢ V1 â¶*[0, â«°(l+m-i)] V2 & - â¬[O, i+1] V2 â¡ T2. -#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 â¦ 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: âG,L,T2,i,l. â¦G, Lâ¦ â¢ #i â¶*[l, â] T2 â - T2 = #i â¨ - ââI,K,V1,V2. l â¤ i & â¬[i] L â¡ K.â{I}V1 & - â¦G, Kâ¦ â¢ V1 â¶*[0, â] V2 & â¬[O, i+1] V2 â¡ T2. -#G #L #T2 #i #l #H elim (cpys_inv_lref1 â¦ 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: âG,L,T2,i,l,m. â¦G, Lâ¦ â¢ #i â¶*[l, m] T2 â - âI,K,V1. â¬[i] L â¡ K.â{I}V1 â - âV2. â¬[O, i+1] V2 â¡ T2 â - â§â§ â¦G, Kâ¦ â¢ V1 â¶*[0, â«°(l+m-i)] V2 - & l â¤ i - & i < l + m. -#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 â¦ H) -H -[ #H destruct elim (lift_inv_lref2_be â¦ HVT2) -HVT2 -HLK // -| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2 - lapply (lift_inj â¦ HXT2 â¦ HVT2) -T2 #H destruct - lapply (drop_mono â¦ HLY â¦ HLK) -L #H destruct - /2 width=1 by and3_intro/ -] -qed-. - -(* Properties on relocation *************************************************) - -lemma cpys_lift_le: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 â - âL,U1,s,l,m. lt + mt â¤ yinj l â â¬[s, l, m] L â¡ K â - â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â - â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind â¦ H) -T2 -[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT â¦ HTU) -IHT #HU1 - lapply (cpy_lift_le â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -lemma cpys_lift_be: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 â - âL,U1,s,l,m. lt â¤ yinj l â l â¤ lt + mt â - â¬[s, l, m] L â¡ K â â¬[l, m] T1 â¡ U1 â - âU2. â¬[l, m] T2 â¡ U2 â â¦G, Lâ¦ â¢ U1 â¶*[lt, mt + m] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind â¦ H) -T2 -[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT â¦ HTU) -IHT #HU1 - lapply (cpy_lift_be â¦ HT2 â¦ HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -lemma cpys_lift_ge: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 â - âL,U1,s,l,m. yinj l â¤ lt â â¬[s, l, m] L â¡ K â - â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â - â¦G, Lâ¦ â¢ U1 â¶*[lt+m, mt] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind â¦ H) -T2 -[ #U2 #H >(lift_mono â¦ HTU1 â¦ H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT â¦ HTU) -IHT #HU1 - lapply (cpy_lift_ge â¦ HT2 â¦ HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -(* Inversion lemmas for relocation ******************************************) - -lemma cpys_inv_lift1_le: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt + mt â¤ l â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_le â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_be: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt â¤ l â yinj l + m â¤ lt + mt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, mt - m] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_be â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_ge: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - yinj l + m â¤ lt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt - m, mt] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -(* Advanced inversion lemmas on relocation **********************************) - -lemma cpys_inv_lift1_ge_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - l â¤ lt â lt â¤ yinj l + m â yinj l + m â¤ lt + mt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[l, lt + mt - (yinj l + m)] T2 & - â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_ge_up â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_be_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt â¤ l â lt + mt â¤ yinj l + m â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_be_up â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_le_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶*[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt â¤ l â l â¤ lt + mt â lt + mt â¤ yinj l + m â - ââT2. â¦G, Kâ¦ â¢ T1 â¶*[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind â¦ H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_le_up â¦ HU2 â¦ HLK â¦ HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_subst: âG,L,W1,W2,l,m. â¦G, Lâ¦ â¢ W1 â¶*[l, m] W2 â - âK,V1,i. â¬[i+1] L â¡ K â â¬[O, i+1] V1 â¡ W1 â - l â¤ yinj i â i < l + m â - ââV2. â¦G, Kâ¦ â¢ V1 â¶*[O, â«°(l+m-i)] V2 & â¬[O, i+1] V2 â¡ W2. -#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm -elim (cpys_inv_lift1_ge_up â¦ HW12 â¦ HLK â¦ HVW1 ? ? ?) // ->yplus_O1 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma index a86afce84..2e096edaf 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma @@ -12,18 +12,19 @@ (* *) (**************************************************************************) +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 â relation lenv â +inductive drops (s:bool): mr2 â relation lenv â | drops_nil : âL. drops s (â) L L | drops_cons: âL1,L,L2,cs,l,m. - drops s cs L1 L â â¬[s, l, m] L â¡ L2 â drops s ({l, m} @ cs) L1 L2 + drops s cs L1 L â â¬[s, l, m] L â¡ L2 â drops s (â¨l, mâ©; cs) L1 L2 . interpretation "iterated slicing (local environment) abstract" @@ -58,7 +59,7 @@ lemma drops_inv_nil: âL1,L2,s. â¬*[s, â] L1 â¡ L2 â L1 = L2. /2 width=4 by drops_inv_nil_aux/ qed-. fact drops_inv_cons_aux: âL1,L2,s,cs. â¬*[s, cs] L1 â¡ L2 â - âl,m,tl. cs = {l, m} @ tl â + âl,m,tl. cs = â¨l, mâ©; tl â ââL. â¬*[s, tl] L1 â¡ L & â¬[s, l, m] L â¡ L2. #L1 #L2 #s #cs * -L1 -L2 -cs [ #L #l #m #tl #H destruct @@ -68,13 +69,13 @@ fact drops_inv_cons_aux: âL1,L2,s,cs. â¬*[s, cs] L1 â¡ L2 â qed-. (* Basic_1: was: drop1_gen_pcons *) -lemma drops_inv_cons: âL1,L2,s,l,m,cs. â¬*[s, {l, m} @ cs] L1 â¡ L2 â +lemma drops_inv_cons: âL1,L2,s,l,m,cs. â¬*[s, â¨l, mâ©; cs] L1 â¡ L2 â ââL. â¬*[s, cs] L1 â¡ L & â¬[s, l, m] L â¡ L2. /2 width=3 by drops_inv_cons_aux/ qed-. -lemma drops_inv_skip2: âI,s,cs,cs2,i. cs â i â¡ cs2 â +lemma drops_inv_skip2: âI,s,cs,cs2,i. cs â i â cs2 â âL1,K2,V2. â¬*[s, cs2] L1 â¡ K2. â{I} V2 â - ââK1,V1,cs1. cs + 1 â i + 1 â¡ cs1 + 1 & + ââK1,V1,cs1. cs + 1 â i + 1 â cs1 + 1 & â¬*[s, cs1] K1 â¡ K2 & â¬*[cs1] V2 â¡ V1 & L1 = K1. â{I} V1. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma index 2e938ac75..2502ebc9e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma @@ -21,7 +21,7 @@ include "basic_2A/multiple/drops.ma". lemma drops_drop_trans: âL1,L,cs. â¬*[â», cs] L1 â¡ L â âL2,i. â¬[i] L â¡ L2 â ââL0,cs0,i0. â¬[i0] L1 â¡ L0 & â¬*[â», cs0] L0 â¡ L2 & - @â¦i, csâ¦ â¡ i0 & cs â i â¡ cs0. + @âªi, csâ« â i0 & cs â i â 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 diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma index 03473dc91..f984e37e9 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "ground_2/relocation/mr2_append.ma". include "basic_2A/multiple/drops_drop.ma". (* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma index 7a26437f6..5af757542 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma @@ -12,7 +12,9 @@ (* *) (**************************************************************************) -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". @@ -84,7 +86,7 @@ lemma frees_inv_lref_lt: âL,l,j,i.L â¢ i Ïµ ð*[l]â¦#jâ¦ â j < i â qed-. lemma frees_inv_bind: âa,I,L,W,U,l,i. L â¢ i Ïµ ð*[l]â¦â{a,I}W.Uâ¦ â - L â¢ i Ïµ ð*[l]â¦Wâ¦ â¨ L.â{I}W â¢ i+1 Ïµ ð*[â«¯l]â¦Uâ¦ . + L â¢ i Ïµ ð*[l]â¦Wâ¦ â¨ L.â{I}W â¢ i+1 Ïµ ð*[âl]â¦Uâ¦ . #a #J #L #V #U #l #i #H elim (frees_inv â¦ H) -H [ #HnX elim (nlift_inv_bind â¦ HnX) -HnX /4 width=2 by frees_eq, or_intror, or_introl/ @@ -124,15 +126,18 @@ lemma frees_bind_sn: âa,I,L,W,U,l,i. L â¢ i Ïµ ð*[l]â¦Wâ¦ â /4 width=9 by frees_be, frees_eq, nlift_bind_sn/ qed. -lemma frees_bind_dx: âa,I,L,W,U,l,i. L.â{I}W â¢ i+1 Ïµ ð*[â«¯l]â¦Uâ¦ â +lemma pippo (j) (i): O < j â j < i+1 â âj< i. +/2 width=1 by lt_plus_to_minus/ qed-. + +lemma frees_bind_dx: âa,I,L,W,U,l,i. L.â{I}W â¢ i+1 Ïµ ð*[âl]â¦Uâ¦ â L â¢ i Ïµ ð*[l]â¦â{a,I}W.Uâ¦. #a #J #L #V #U #l #i #H elim (frees_inv â¦ H) -H [ /4 width=9 by frees_eq, nlift_bind_dx/ | * #I #K #W #j #Hlj #Hji #HnU #HLK #HW - elim (yle_inv_succ1 â¦ Hlj) -Hlj (plus_minus_m_m j 1) in HnU; // (HW â¦ HW0) -W0 >(HT â¦ HT0) -T0 // +<(HW â¦ HW0) -W0 <(HT â¦ HT0) -T0 // qed. (* Basic_1: was only: nf2_appl_lref *) lemma cnr_appl_simple: âG,L,V,T. â¦G, Lâ¦ â¢ â¡ ðâ¦Vâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦âV.Tâ¦. #G #L #V #T #HV #HT #HS #X #H elim (cpr_inv_appl1_simple â¦ H) -H // #V0 #T0 #HV0 #HT0 #H destruct ->(HV â¦ HV0) -V0 >(HT â¦ HT0) -T0 // +<(HV â¦ HV0) -V0 <(HT â¦ HT0) -T0 // qed. (* Basic_1: was: nf2_dec *) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma index 786611f54..a6a90214d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma @@ -20,7 +20,7 @@ include "basic_2A/reduction/cnr_crr.ma". (* Main properties on irreducibility ****************************************) theorem cir_cnr: âG,L,T. â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦. -/2 width=4 by cpr_fwd_cir/ qed. +/3 width=4 by cpr_fwd_cir, sym_eq/ qed. (* Main inversion lemmas on irreducibility **********************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma index 4ff641f4f..aedc330d1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma @@ -34,7 +34,7 @@ lemma cnr_lift: âG,L0,L,T,T0,s,l,m. â¦G, Lâ¦ â¢ â¡ ðâ¦Tâ¦ â â¬[s, l, m] L0 â¡ L â â¬[l, m] T â¡ T0 â â¦G, L0â¦ â¢ â¡ ðâ¦T0â¦. #G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H elim (cpr_inv_lift1 â¦ H â¦ HL0 â¦ HT0) -L0 #T1 #HT10 #HT1 -<(HLT â¦ HT1) in HT0; -L #HT0 +>(HLT â¦ HT1) in HT0; -L #HT0 >(lift_mono â¦ HT10 â¦ HT0) -T1 -X // qed. @@ -44,6 +44,6 @@ lemma cnr_inv_lift: âG,L0,L,T,T0,s,l,m. â¦G, L0â¦ â¢ â¡ ðâ¦T0â¦ â #G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H elim (lift_total X l m) #X0 #HX0 lapply (cpr_lift â¦ H â¦ HL0 â¦ HT0 â¦ HX0) -L #HTX0 ->(HLT0 â¦ HTX0) in HX0; -L0 -X0 #H +<(HLT0 â¦ HTX0) in HX0; -L0 -X0 #H >(lift_inj â¦ H â¦ HT0) -T0 -X -s -l -m // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma index e754c0a1f..4a03a7130 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma @@ -33,7 +33,7 @@ lemma cnx_inv_sort: âh,g,G,L,k. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦âkâ¦ â de lapply (H (â(next h k)) ?) -H /2 width=2 by cpx_st/ -L -d #H lapply (destruct_tatom_tatom_aux â¦ H) -H #H (**) (* destruct lemma needed *) lapply (destruct_sort_sort_aux â¦ H) -H #H (**) (* destruct lemma needed *) -lapply (next_lt h k) >H -H #H elim (lt_refl_false â¦ H) +lapply (next_lt h k) (HW â¦ HW0) -W0 >(HT â¦ HT0) -T0 // +<(HW â¦ HW0) -W0 <(HT â¦ HT0) -T0 // qed. lemma cnx_appl_simple: âh,g,G,L,V,T. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Vâ¦ â â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦âV.Tâ¦. #h #g #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple â¦ H) -H // #V0 #T0 #HV0 #HT0 #H destruct ->(HV â¦ HV0) -V0 >(HT â¦ HT0) -T0 // +<(HV â¦ HV0) -V0 <(HT â¦ HT0) -T0 // qed. axiom cnx_dec: âh,g,G,L,T1. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦T1â¦ â¨ diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma index 40f69e006..f8a1aea91 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma @@ -20,7 +20,7 @@ include "basic_2A/reduction/cnx_crx.ma". (* Main properties on irreducibility ****************************************) theorem cix_cnx: âh,g,G,L,T. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦. -/2 width=6 by cpx_fwd_cix/ qed. +/3 width=6 by cpx_fwd_cix, sym_eq/ qed. (* Main inversion lemmas on irreducibility **********************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma index 08395110a..b4f94974a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma @@ -23,7 +23,7 @@ lemma cnx_lift: âh,g,G,L0,L,T,T0,s,l,m. â¦G, Lâ¦ â¢ â¡[h, g] ðâ¦Tâ¦ â¬[l, m] T â¡ T0 â â¦G, L0â¦ â¢ â¡[h, g] ðâ¦T0â¦. #h #g #G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H elim (cpx_inv_lift1 â¦ H â¦ HL0 â¦ HT0) -L0 #T1 #HT10 #HT1 -<(HLT â¦ HT1) in HT0; -L #HT0 +>(HLT â¦ HT1) in HT0; -L #HT0 >(lift_mono â¦ HT10 â¦ HT0) -T1 -X // qed. @@ -32,6 +32,6 @@ lemma cnx_inv_lift: âh,g,G,L0,L,T,T0,s,l,m. â¦G, L0â¦ â¢ â¡[h, g] ðâ¦T #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 â¦ H â¦ HL0 â¦ HT0 â¦ HX0) -L #HTX0 ->(HLT0 â¦ HTX0) in HX0; -L0 -X0 #H +<(HLT0 â¦ HTX0) in HX0; -L0 -X0 #H >(lift_inj â¦ H â¦ HT0) -T0 -X -l -m // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma index d79fba9fd..9b9186531 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma @@ -12,6 +12,12 @@ (* *) (**************************************************************************) +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". @@ -97,8 +103,7 @@ fact lstas_cpr_aux: âh,G,L,T1,T2,d. â¦G, Lâ¦ â¢ T1 â¢*[h, d] T2 â d = 0 â â¦G, Lâ¦ â¢ T1 â¡ 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 #_ #_ #_ #_ yplus_SO2 >yplus_succ1 #T #HT1 #HT2 - lapply (lsuby_cpy_trans â¦ HT2 (L.â{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: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â âi. i â¤ l + m â - ââT. â¦G, Lâ¦ â¢ T1 â¶[i, l+m-i] T & â¦G, Lâ¦ â¢ T â¶[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 â¦ HT2 (L.â{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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â - âT1,l,m. â¬[l, m] T1 â¡ U1 â - l â¤ lt â l + m â¤ lt + mt â - ââT2. â¦G, Lâ¦ â¢ U1 â¶[l+m, lt+mt-(l+m)] U2 & â¬[l, m] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 â¦ H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 â¦ 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 â¦ H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ] - [ elim (ylt_yle_false â¦ Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/ - | elim (le_inv_plus_l â¦ Hil) #Hlim #Hmi - elim (lift_split â¦ HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim - #T2 #_ >plus_minus // 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 â¦ HVW1) -V1 -IHW12 // - elim (IHU12 â¦ HTU1) -T1 -IHU12 /2 width=1 by yle_succ/ - 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 â¦ HVW1) -V1 -IHW12 // elim (IHU12 â¦ HTU1) -T1 -IHU12 - /3 width=2 by cpy_flat, lift_flat, ex2_intro/ -] -qed-. - -lemma cpy_fwd_tw: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â â¯{T1} â¤ â¯{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: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â âJ. T1 = âª{J} â - T2 = âª{J} â¨ - ââI,K,V,i. l â¤ yinj i & i < l + m & - â¬[i] L â¡ K.â{I}V & - â¬[O, i+1] V â¡ 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: âI,G,L,T2,l,m. â¦G, Lâ¦ â¢ âª{I} â¶[l, m] T2 â - T2 = âª{I} â¨ - ââJ,K,V,i. l â¤ yinj i & i < l + m & - â¬[i] L â¡ K.â{J}V & - â¬[O, i+1] V â¡ T2 & - I = LRef i. -/2 width=4 by cpy_inv_atom1_aux/ qed-. - -(* Basic_1: was: subst1_gen_sort *) -lemma cpy_inv_sort1: âG,L,T2,k,l,m. â¦G, Lâ¦ â¢ âk â¶[l, m] T2 â T2 = âk. -#G #L #T2 #k #l #m #H -elim (cpy_inv_atom1 â¦ H) -H // -* #I #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -(* Basic_1: was: subst1_gen_lref *) -lemma cpy_inv_lref1: âG,L,T2,i,l,m. â¦G, Lâ¦ â¢ #i â¶[l, m] T2 â - T2 = #i â¨ - ââI,K,V. l â¤ i & i < l + m & - â¬[i] L â¡ K.â{I}V & - â¬[O, i+1] V â¡ T2. -#G #L #T2 #i #l #m #H -elim (cpy_inv_atom1 â¦ 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: âG,L,T2,p,l,m. â¦G, Lâ¦ â¢ Â§p â¶[l, m] T2 â T2 = Â§p. -#G #L #T2 #p #l #m #H -elim (cpy_inv_atom1 â¦ H) -H // -* #I #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -fact cpy_inv_bind1_aux: âG,L,U1,U2,l,m. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â - âa,I,V1,T1. U1 = â{a,I}V1.T1 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & - â¦G, L. â{I}V1â¦ â¢ T1 â¶[â«¯l, m] T2 & - U2 = â{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: âa,I,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{a,I} V1. T1 â¶[l, m] U2 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & - â¦G, L.â{I}V1â¦ â¢ T1 â¶[â«¯l, m] T2 & - U2 = â{a,I}V2.T2. -/2 width=3 by cpy_inv_bind1_aux/ qed-. - -fact cpy_inv_flat1_aux: âG,L,U1,U2,l,m. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â - âI,V1,T1. U1 = â{I}V1.T1 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & - â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 & - U2 = â{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: âI,G,L,V1,T1,U2,l,m. â¦G, Lâ¦ â¢ â{I} V1. T1 â¶[l, m] U2 â - ââV2,T2. â¦G, Lâ¦ â¢ V1 â¶[l, m] V2 & - â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 & - U2 = â{I}V2.T2. -/2 width=3 by cpy_inv_flat1_aux/ qed-. - - -fact cpy_inv_refl_O2_aux: âG,L,T1,T2,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T2 â m = 0 â 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 â¦ Hli) -Hli // -| /3 width=1 by eq_f2/ -| /3 width=1 by eq_f2/ -] -qed-. - -lemma cpy_inv_refl_O2: âG,L,T1,T2,l. â¦G, Lâ¦ â¢ T1 â¶[l, 0] T2 â T1 = T2. -/2 width=6 by cpy_inv_refl_O2_aux/ qed-. - -(* Basic_1: was: subst1_gen_lift_eq *) -lemma cpy_inv_lift1_eq: âG,T1,U1,l,m. â¬[l, m] T1 â¡ U1 â - âL,U2. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â U1 = U2. -#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up â¦ HU12 â¦ 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 -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_cpy.ma deleted file mode 100644 index 3a8857155..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_cpy.ma +++ /dev/null @@ -1,122 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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: âG,L,T0,T1,l1,m1. â¦G, Lâ¦ â¢ T0 â¶[l1, m1] T1 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â - ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T2 â¶[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 â¦ H) -H - [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/ - | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2 - lapply (drop_mono â¦ HLK1 â¦ HLK2) -HLK1 -HLK2 #H destruct - >(lift_mono â¦ HVT1 â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02) -IHV01 -HV02 #V #HV1 #HV2 - elim (IHT01 â¦ HT02) -T0 #T #HT1 #HT2 - lapply (lsuby_cpy_trans â¦ HT1 (L.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ - lapply (lsuby_cpy_trans â¦ HT2 (L.â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02) -V0 - elim (IHT01 â¦ HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/ -] -qed-. - -(* Basic_1: was: subst1_confluence_neq *) -theorem cpy_conf_neq: âG,L1,T0,T1,l1,m1. â¦G, L1â¦ â¢ T0 â¶[l1, m1] T1 â - âL2,T2,l2,m2. â¦G, L2â¦ â¢ T0 â¶[l2, m2] T2 â - (l1 + m1 â¤ l2 â¨ l2 + m2 â¤ l1) â - ââT. â¦G, L2â¦ â¢ T1 â¶[l2, m2] T & â¦G, L1â¦ â¢ T2 â¶[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 â¦ 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 â¦ Hlm1) -Hlm1 /2 width=3 by yle_trans/ - | elim (ylt_yle_false â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02 H) -IHV01 -HV02 #V #HV1 #HV2 - elim (IHT01 â¦ HT02) -T0 - [ -H #T #HT1 #HT2 - lapply (lsuby_cpy_trans â¦ HT1 (L2.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ - lapply (lsuby_cpy_trans â¦ HT2 (L1.â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 â¦ HV02 H) -V0 - elim (IHT01 â¦ 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: âG,L,T1,T0,l,m. â¦G, Lâ¦ â¢ T1 â¶[l, m] T0 â - âT2. â¦G, Lâ¦ â¢ T0 â¶[l, 1] T2 â 1 â¤ m â â¦G, Lâ¦ â¢ T1 â¶[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 â¦ H) -H - [ #H destruct // - | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct - lapply (ylt_yle_trans â¦ (l+m) â¦ 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 â¦ HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/ - >yplus_inj #HVT2 <(cpy_inv_lift1_eq â¦ HVW â¦ 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 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct - lapply (lsuby_cpy_trans â¦ HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 - lapply (IHT10 â¦ 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 â¦ H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/ -] -qed-. - -theorem cpy_trans_down: âG,L,T1,T0,l1,m1. â¦G, Lâ¦ â¢ T1 â¶[l1, m1] T0 â - âT2,l2,m2. â¦G, Lâ¦ â¢ T0 â¶[l2, m2] T2 â l2 + m2 â¤ l1 â - ââT. â¦G, Lâ¦ â¢ T1 â¶[l2, m2] T & â¦G, Lâ¦ â¢ T â¶[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 â¦ Hlm2l1 â¦ Hli1) -Hlm2l1 #Hlm2i1 - lapply (cpy_weak â¦ HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1 - >yplus_inj #HWT2 <(cpy_inv_lift1_eq â¦ HVW â¦ 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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - lapply (lsuby_cpy_trans â¦ HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 - elim (IHV10 â¦ HV02) -IHV10 -HV02 // #V - elim (IHT10 â¦ HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2 - lapply (lsuby_cpy_trans â¦ HT2 (L.â{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 â¦ HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV10 â¦ HV02) -V0 // - elim (IHT10 â¦ HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_lift.ma deleted file mode 100644 index a188129b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_lift.ma +++ /dev/null @@ -1,249 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 â - âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â - â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â - lt + mt â¤ l â â¦G, Lâ¦ â¢ U1 â¶[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 â¦ H1 â¦ 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 â¦ Hlmtl â¦ Hilmt) -Hlmtl #Hil - lapply (ylt_inv_inj â¦ Hil) -Hil #Hil - lapply (lift_inv_lref1_lt â¦ H â¦ Hil) -H #H destruct - elim (lift_trans_ge â¦ HVW â¦ HWU2) -W // (lift_mono â¦ HVY â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 â¦ H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /3 width=7 by cpy_flat/ -] -qed-. - -lemma cpy_lift_be: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 â - âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â - â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â - lt â¤ l â l â¤ lt + mt â â¦G, Lâ¦ â¢ U1 â¶[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 â¦ H1 â¦ 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 â¦ H) -H * #Hil #H destruct - [ -Hltl - lapply (ylt_yle_trans â¦ (lt+mt+m) â¦ Hilmt) // -Hilmt #Hilmtm - elim (lift_trans_ge â¦ HVW â¦ HWU2) -W // (lift_mono â¦ HVY â¦ HVW) -V #H destruct /2 width=5 by cpy_subst/ - | -Hlti - elim (yle_inv_inj2 â¦ Hltl) -Hltl #ltt #Hltl #H destruct - lapply (transitive_le â¦ Hltl Hil) -Hltl #Hlti - lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2 - lapply (drop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 â¦ 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: âG,K,T1,T2,lt,mt. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 â - âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â - â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â - l â¤ lt â â¦G, Lâ¦ â¢ U1 â¶[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 â¦ H1 â¦ 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 â¦ Hllt â¦ Hlti) -Hllt #Hil - elim (yle_inv_inj2 â¦ Hil) -Hil #ll #Hlli #H0 destruct - lapply (lift_inv_lref1_ge â¦ H â¦ Hlli) -H #H destruct - lapply (lift_trans_be â¦ HVW â¦ HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2 - lapply (drop_trans_ge_comm â¦ HLK â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 â¦ 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 â¦ H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 â¦ 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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt + mt â¤ l â - ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt, mt] T2 & â¬[l, m] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 â¦ H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 â¦ 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 â¦ Hlmtl â¦ Hilmt) -Hlmtl #Hil - lapply (ylt_inv_inj â¦ Hil) -Hil #Hil - lapply (lift_inv_lref2_lt â¦ H â¦ Hil) -H #H destruct - elim (drop_conf_lt â¦ HLK â¦ HLKV) -L // #L #U #HKL #_ #HUV - elim (lift_trans_le â¦ HUV â¦ HVW) -V // >minus_plus yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm - elim (drop_conf_lt â¦ HLK â¦ HLKV) -L // #L #U #HKL #_ #HUV - elim (lift_trans_le â¦ HUV â¦ HVW) -V // >minus_plus plus_minus // 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 â¦ HLK â¦ HVW1) -IHW12 // #V2 #HV12 #HVW2 - elim (IHU12 â¦ 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 â¦ HLK â¦ HVW1) -W1 // - elim (IHU12 â¦ HLK â¦ 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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - yinj l + m â¤ lt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt-m, mt] T2 & â¬[l, m] T2 â¡ 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 â¦ H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 â¦ H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 â¦ 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 â¦ Hlmlt â¦ Hlti) #Hlmi - elim (yle_inv_plus_inj2 â¦ Hlmlt) -Hlmlt #_ #Hmlt - elim (yle_inv_plus_inj2 â¦ Hlmi) #Hlim #Hmi - lapply (lift_inv_lref2_ge â¦ H ?) -H /2 width=1 by yle_inv_inj/ #H destruct - lapply (drop_conf_ge â¦ HLK â¦ HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV - elim (lift_split â¦ 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/ 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 â¦ H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 â¦ HLK â¦ HVW1) -W1 // - elim (IHU12 â¦ HLK â¦ 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: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - l â¤ lt â lt â¤ yinj l + m â yinj l + m â¤ lt + mt â - ââT2. â¦G, Kâ¦ â¢ T1 â¶[l, lt + mt - (yinj l + m)] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt -elim (cpy_split_up â¦ HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 -lapply (cpy_weak â¦ HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 -lapply (cpy_inv_lift1_eq â¦ HTU1 â¦ HU1) -HU1 #HU1 destruct -elim (cpy_inv_lift1_ge â¦ HU2 â¦ HLK â¦ HTU1) -U -L /2 width=3 by ex2_intro/ -qed-. - -lemma cpy_inv_lift1_be_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt â¤ l â lt + mt â¤ yinj l + m â - ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt, l-lt] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm -lapply (cpy_weak â¦ 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 â¦ HU12 â¦ HLK â¦ HTU1) -U1 -L /2 width=3 by ex2_intro/ -qed-. - -lemma cpy_inv_lift1_le_up: âG,L,U1,U2,lt,mt. â¦G, Lâ¦ â¢ U1 â¶[lt, mt] U2 â - âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â - lt â¤ l â l â¤ lt + mt â lt + mt â¤ yinj l + m â - ââT2. â¦G, Kâ¦ â¢ T1 â¶[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm -elim (cpy_split_up â¦ HU12 l) -HU12 // #U #HU1 #HU2 -elim (cpy_inv_lift1_le â¦ HU1 â¦ HLK â¦ HTU1) -U1 -[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU -lapply (cpy_weak â¦ HU2 l m ? ?) -HU2 // -[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2 -lapply (cpy_inv_lift1_eq â¦ HTU â¦ HU2) -L #H destruct /2 width=3 by ex2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_nlift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_nlift.ma deleted file mode 100644 index ded7af4f5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_nlift.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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: âG,L,U1,U2,l,m. â¦G, Lâ¦ â¢ U1 â¶[l, m] U2 â - âi. l â¤ yinj i â (âT2. â¬[i, 1] T2 â¡ U2 â â¥) â - (âT1. â¬[i, 1] T1 â¡ U1 â â¥) â¨ - ââI,K,W,j. l â¤ yinj j & j < i & â¬[j]L â¡ K.â{I}W & - (âV. â¬[i-j-1, 1] V â¡ W â â¥) & (âT1. â¬[j, 1] T1 â¡ U1 â â¥). -#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 â¦ HLK) // -HLK - [ #X #HXV elim (lift_trans_le â¦ HXV â¦ HVW ?) -V // - #Y #HXY >minus_plus (plus_minus_m_m j 1) in â¢ (%â?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ] - #HnU1 ypred_succ /2 width=3 by drop_pair, ex2_intro/ - | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/ - #H yminus_succ 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 â¦ Hli) -Hli - #Hli #Hi yplus_succ1 #H lapply (ylt_inv_succ â¦ H) -H - #Hilm lapply (drop_inv_drop1_lt â¦ HLK2 ?) -HLK2 /2 width=1 by ylt_O/ - #HLK1 elim (IHL12 â¦ HLK1) -IHL12 -HLK1 yminus_SO2 - /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/ + elim (yle_inv_succ_sn_lt â¦ Hli) -Hli #Hli #Hi + lapply (ylt_inv_inj â¦ Hi) -Hi #Hi + <(S_pred â¦ Hi) >yplus_succ1 minus_SO_dx + /3 width=3 by drop_drop_lt, ex2_intro/ ] qed-. lemma lreq_drop_conf_be: âL1,L2,l,m. L1 â©¬[l, m] L2 â âI,K1,W,s,i. â¬[s, 0, i] L1 â¡ K1.â{I}W â l â¤ i â i < l + m â - ââK2. K1 â©¬[0, â«°(l+m-i)] K2 & â¬[s, 0, i] L2 â¡ K2.â{I}W. + ââK2. K1 â©¬[0, â(l+m-i)] K2 & â¬[s, 0, i] L2 â¡ K2.â{I}W. #L1 #L2 #l #m #HL12 #I #K1 #W #s #i #HLK1 #Hli #Hilm elim (lreq_drop_trans_be â¦ (lreq_sym â¦ HL12) â¦ HLK1) // -L1 -Hli -Hilm /3 width=3 by lreq_sym, ex2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma index b11b0af4c..993bbb963 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma @@ -13,7 +13,7 @@ (**************************************************************************) include "basic_2A/notation/relations/rdrop_3.ma". -include "basic_2A/grammar/genv.ma". +include "basic_2A/grammar/genv_length.ma". (* GLOBAL ENVIRONMENT READING ***********************************************) @@ -65,14 +65,14 @@ fact gget_inv_lt_aux: âI,G,G1,G2,V,m. â¬[m] G â¡ G2 â G = G1. â{I} V â qed-. lemma gget_inv_lt: âI,G1,G2,V,m. - â¬[m] G1. â{I} V â¡ G2 â m < |G1| â â¬[m] G1 â¡ G2. + â¬[m] G1. â{I} V â¡ G2 â m < |G1| â â¬[m] G1 â¡ G2. /2 width=5 by gget_inv_lt_aux/ qed-. (* Basic properties *********************************************************) lemma gget_total: âm,G1. âG2. â¬[m] G1 â¡ 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/ diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma index 63e82758a..a93de4161 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +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". @@ -345,7 +346,7 @@ lemma lift_split: âl1,m2,T1,T2. â¬[l1, m2] T1 â¡ T2 â | /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 â¦ Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b - elim (IHT (l2+1) â¦ ? ? Hm12) /3 width=5 by lift_bind, le_S_S, ex2_intro/ + elim (IHT (l2+1) â¦ ? ? 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 â¦ Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b elim (IHT l2 â¦ ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma index bbc747aa7..d40925d25 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma @@ -136,7 +136,7 @@ theorem lift_trans_be: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21 elim (lift_inv_bind1 â¦ HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct lapply (IHV12 â¦ HV20 ? ?) // -IHV12 -HV20 #HV10 - lapply (IHT12 â¦ HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *) + lapply (IHT12 â¦ 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 â¦ HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct lapply (IHV12 â¦ HV20 ? ?) // -IHV12 -HV20 #HV10 @@ -190,7 +190,7 @@ theorem lift_trans_ge: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml elim (lift_inv_bind1 â¦ HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct elim (IHV12 â¦ HV20) -IHV12 -HV20 // - elim (IHT12 â¦ HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T + elim (IHT12 â¦ HT20) -IHT12 -HT20 [2: (length_inv_zero_dx â¦ H) -L1 // -| /2 width=1 by lsuby_O2/ -| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l â¦ H) - /3 width=1 by lsuby_pair/ -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l â¦ H) - /3 width=1 by lsuby_succ/ -] -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact lsuby_inv_atom1_aux: âL1,L2,l,m. L1 â[l, m] L2 â L1 = â â L2 = â. -#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: âL2,l,m. â â[l, m] L2 â L2 = â. -/2 width=5 by lsuby_inv_atom1_aux/ qed-. - -fact lsuby_inv_zero1_aux: âL1,L2,l,m. L1 â[l, m] L2 â - âJ1,K1,W1. L1 = K1.â{J1}W1 â l = 0 â m = 0 â - L2 = â â¨ - ââJ2,K2,W2. K1 â[0, 0] K2 & L2 = K2.â{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 â¦ H) -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H - elim (ysucc_inv_O_dx â¦ H) -] -qed-. - -lemma lsuby_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 â[0, 0] L2 â - L2 = â â¨ - ââI2,K2,V2. K1 â[0, 0] K2 & L2 = K2.â{I2}V2. -/2 width=9 by lsuby_inv_zero1_aux/ qed-. - -fact lsuby_inv_pair1_aux: âL1,L2,l,m. L1 â[l, m] L2 â - âJ1,K1,W. L1 = K1.â{J1}W â l = 0 â 0 < m â - L2 = â â¨ - ââJ2,K2. K1 â[0, â«°m] K2 & L2 = K2.â{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 â¦ 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 â¦ H) -] -qed-. - -lemma lsuby_inv_pair1: âI1,K1,L2,V,m. K1.â{I1}V â[0, m] L2 â 0 < m â - L2 = â â¨ - ââI2,K2. K1 â[0, â«°m] K2 & L2 = K2.â{I2}V. -/2 width=6 by lsuby_inv_pair1_aux/ qed-. - -fact lsuby_inv_succ1_aux: âL1,L2,l,m. L1 â[l, m] L2 â - âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < l â - L2 = â â¨ - ââJ2,K2,W2. K1 â[â«°l, m] K2 & L2 = K2.â{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 â¦ H) // -| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H - elim (ylt_yle_false â¦ 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: âI1,K1,L2,V1,l,m. K1.â{I1}V1 â[l, m] L2 â 0 < l â - L2 = â â¨ - ââI2,K2,V2. K1 â[â«°l, m] K2 & L2 = K2.â{I2}V2. -/2 width=5 by lsuby_inv_succ1_aux/ qed-. - -fact lsuby_inv_zero2_aux: âL1,L2,l,m. L1 â[l, m] L2 â - âJ2,K2,W2. L2 = K2.â{J2}W2 â l = 0 â m = 0 â - ââJ1,K1,W1. K1 â[0, 0] K2 & L1 = K1.â{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 â¦ H) -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H - elim (ysucc_inv_O_dx â¦ H) -] -qed-. - -lemma lsuby_inv_zero2: âI2,K2,L1,V2. L1 â[0, 0] K2.â{I2}V2 â - ââI1,K1,V1. K1 â[0, 0] K2 & L1 = K1.â{I1}V1. -/2 width=9 by lsuby_inv_zero2_aux/ qed-. - -fact lsuby_inv_pair2_aux: âL1,L2,l,m. L1 â[l, m] L2 â - âJ2,K2,W. L2 = K2.â{J2}W â l = 0 â 0 < m â - ââJ1,K1. K1 â[0, â«°m] K2 & L1 = K1.â{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 â¦ 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 â¦ H) -] -qed-. - -lemma lsuby_inv_pair2: âI2,K2,L1,V,m. L1 â[0, m] K2.â{I2}V â 0 < m â - ââI1,K1. K1 â[0, â«°m] K2 & L1 = K1.â{I1}V. -/2 width=6 by lsuby_inv_pair2_aux/ qed-. - -fact lsuby_inv_succ2_aux: âL1,L2,l,m. L1 â[l, m] L2 â - âJ2,K2,W2. L2 = K2.â{J2}W2 â 0 < l â - ââJ1,K1,W1. K1 â[â«°l, m] K2 & L1 = K1.â{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 â¦ H) // -| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H - elim (ylt_yle_false â¦ 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: âI2,K2,L1,V2,l,m. L1 â[l, m] K2.â{I2}V2 â 0 < l â - ââI1,K1,V1. K1 â[â«°l, m] K2 & L1 = K1.â{I1}V1. -/2 width=5 by lsuby_inv_succ2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lsuby_fwd_length: âL1,L2,l,m. L1 â[l, m] L2 â |L2| â¤ |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: âL1,L2,l,m. L1 â[l, m] L2 â - âI2,K2,W,s,i. â¬[s, 0, i] L2 â¡ K2.â{I2}W â - l â¤ i â i < l + m â - ââI1,K1. K1 â[0, â«°(l+m-i)] K2 & â¬[s, 0, i] L1 â¡ K1.â{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 â¦ H) -H #H destruct -| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H - elim (ylt_yle_false â¦ H) // -| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1 - elim (drop_inv_O1_pair1 â¦ 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 yminus_succ yplus_succ1 #H lapply (ylt_inv_succ â¦ H) -H - #Hilm lapply (drop_inv_drop1_lt â¦ HLK2 ?) -HLK2 /2 width=1 by ylt_O/ - #HLK1 elim (IHL12 â¦ HLK1) -IHL12 -HLK1 yminus_SO2 - /4 width=4 by ylt_O, drop_drop_lt, ex2_2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby_lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby_lsuby.ma deleted file mode 100644 index 8e45e98c9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby_lsuby.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||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: âl,m. Transitive â¦ (lsuby l m). -#l #m #L1 #L2 #H elim H -L1 -L2 -l -m -[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 â¦ H) -H - #H destruct // -| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 â¦ 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 â¦ 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 â¦ H) -H // - * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma index 20cce2f5c..6bb2eb6ba 100644 --- a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma +++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma @@ -12,6 +12,8 @@ (* *) (**************************************************************************) +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". diff --git a/matita/matita/contribs/lambdadelta/basic_2A/web/basic_2A.ldw.xml b/matita/matita/contribs/lambdadelta/basic_2A/web/basic_2A.ldw.xml new file mode 100644 index 000000000..819c9260b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/web/basic_2A.ldw.xml @@ -0,0 +1,104 @@ + + + + + + Summary of the Specification + Here is a numerical account of the specification's contents + and its timeline. + +

+ + + Î»Î´-2A is repackaged on the basis of Î»Î´-ground-2. + + + Î»Î´-2A is repackaged (was Î»Î´-2A1). + + + Î»Î´-2A appears too complex and is dismissed. + + + Î»Î´-2A is released. + + + Iterated static type assignment defined (more elegantly) + as a primitive notion. + + + Preservation of stratified native validity + for context-sensitive computation on terms. + + + Strong qrst-normalization + for simply typed terms. + + + Lazy equivalence on local environments + added as q-step to rst-computation on closures + (anniversary milestone). + + + Parametrized slicing on local environments + comprises both versions of this operation + (one from basic_1, the other used in basic_2 till now). + + + Passive support for global environments. + + + Reaxiomatized Î²-reductum as in rt-reduction. + + + Context-sensitive strong rt-normalization + for simply typed terms. + + + Reaxiomatized substitution and reduction + commute with respect to subclosure + (anniversary milestone). + + + Mutual recursive preservation of stratified native validity + for rst-computation on closures. + + + Confluence for context-free parallel reduction on closures. + + + Term binders polarized to control Î¶-reduction (not released). + + + Context-sensitive subject equivalence + for atomic arity assignment + (anniversary milestone). + + + Context-sensitive strong normalization + for simply typed terms. + + + Generic candidates of reducibility. + + + Confluence for context-sensitive parallel reduction on terms. + + + Confluence for context-free parallel reduction on terms. + + + Î»Î´-2A is started. + + +

+ + Specification becomes an independent package. + Centralized xoa infrastructure removed. diff --git a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl index 629e8999d..0dfbaaecf 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl +++ b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl @@ -40,7 +40,7 @@ table { [ "trace ( â¥?â¥ )" "trace_at ( @âª?,?â« â ? )" "trace_after ( ? â ? â ? )" "trace_isid ( ðâª?â« )" "trace_isun ( ðâª?â« )" "trace_sle ( ? â ? )" "trace_sor ( ? â ? â ? )" "trace_snot ( â ? )" * ] *) - [ "mr2 ( â ) ( â¨?,?â©;? )" "mr2_at ( @âª?,?â« â ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? â ? â ? )" * ] + [ "mr2 ( â ) ( â¨?,?â©;? )" "mr2_append ( ? @@? )" "mr2_at ( @âª?,?â« â ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? â ? â ? )" * ] } ] } @@ -50,7 +50,7 @@ table { [ { "" * } { [ "ynat ( â )" "ynat_pred ( â? )" "ynat_succ ( â? )" "ynat_le ( ? â¤ ? )" "ynat_lt ( ? < ? )" - "ynat_plus ( ? + ? )" * + "ynat_plus ( ? + ? )" "ynat_minus_sn ( ? - ? )" * ] } ] @@ -63,7 +63,7 @@ table { [ "list ( âº{?} ) ( ? â¨®{?} ? )" "list_length ( |?| )" * ] [ "bool ( â» ) ( â )" "arith ( ?^? ) ( â? ) ( â? ) ( ? â¨ ? ) ( ? â§ ? )" "arith_2a" "arith_2b" * ] [ "ltc" "ltc_ctc" * ] - [ "logic ( â¥ ) ( â¤ )" "relations ( ? â ? )" "functions" "exteq ( ? â{?,?} ? )" "star" * ] + [ "logic ( â¥ ) ( â¤ )" "relations ( ? â ? )" "functions" "exteq ( ? â{?,?} ? )" "star" "lstar_2a" * ] } ] } diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml b/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml index ba6470477..e7546ade3 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml @@ -29,16 +29,20 @@ 5 55 65 7 + 6 36 46 56 66 76 86 9 + 7 3 + 7 47 57 67 77 10 + 8 48 59 33 diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_6_3.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_6_3.ma new file mode 100644 index 000000000..7e18738b8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_6_3.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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:A0âA1âA2âProp) : Prop â + | ex6_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â P3 x0 x1 x2 â P4 x0 x1 x2 â P5 x0 x1 x2 â ex6_3 ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 3)" 'Ex3 P0 P1 P2 P3 P4 P5 = (ex6_3 ? ? ? P0 P1 P2 P3 P4 P5). + diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_7_3.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_7_3.ma new file mode 100644 index 000000000..582b06cab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_7_3.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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:A0âA1âA2âProp) : Prop â + | ex7_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â P3 x0 x1 x2 â P4 x0 x1 x2 â P5 x0 x1 x2 â P6 x0 x1 x2 â 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). + diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_7_4.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_7_4.ma new file mode 100644 index 000000000..f1bbb932d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_7_4.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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:A0âA1âA2âA3âProp) : Prop â + | ex7_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â P4 x0 x1 x2 x3 â P5 x0 x1 x2 x3 â P6 x0 x1 x2 x3 â 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). + diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_8_4.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_8_4.ma new file mode 100644 index 000000000..489c1f9e7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_8_4.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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:A0âA1âA2âA3âProp) : Prop â + | ex8_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â P4 x0 x1 x2 x3 â P5 x0 x1 x2 x3 â P6 x0 x1 x2 x3 â P7 x0 x1 x2 x3 â 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). + diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma index 846522994..5a1bb14dd 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma @@ -135,6 +135,11 @@ lemma ylt_O1: âx:ynat. ââx = x â 0 < x. #H destruct qed. +lemma yle_inv_succ_sn_lt (x:ynat) (y:ynat): + âx â¤ y â â§â§ x â¤ ây & 0 < y. +#x #y #H elim (yle_inv_succ1 â¦ H) -H /3 width=2 by ylt_O1, conj/ +qed-. + (* Properties on predecessor ************************************************) lemma ylt_pred: âm,n:ynat. m < n â 0 < m â âm < ân. @@ -144,7 +149,7 @@ qed. (* Properties on successor **************************************************) -lemma ylt_O_succ: ân. 0 < ân. +lemma ylt_O_succ: âx:ynat. 0 < âx. * /2 width=1 by ylt_inj/ qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus_sn.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus_sn.ma new file mode 100644 index 000000000..8b6f79602 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus_sn.ma @@ -0,0 +1,221 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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 â ypred^y x. + +interpretation "ynat left minus" 'minus x y = (yminus_sn x y). + +lemma yminus_O2: âm:ynat. m - 0 = m. +// qed. + +lemma yminus_S2: âm:ynat. ân:nat. m - S n = â(m - n). +// qed. + +(* Basic properties *********************************************************) + +lemma yminus_inj: âm,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: ân. â - n = â. +#n elim n -n // +qed. + +lemma yminus_O1: âx:nat. yinj 0 - x = 0. +// qed. + +lemma yminus_refl: âx:nat. yinj x - x = 0. +// qed. + +lemma yminus_minus_comm: âx:ynat. ây,z. x - y - z = x - z - y. +* // qed. + +(* Properties on predecessor ************************************************) + +lemma yminus_SO2: âm:ynat. m - 1 = âm. +// qed. + +lemma yminus_pred1: âx,y. âx - y = â(x-y). +#x * // #y elim y -y // +qed. + +lemma yminus_pred: âm:ynat. ân. 0 < m â 0 < n â âm - ân = 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: âm:ynat. ân. âm - ân = m - n. +* // qed. + +lemma yminus_succ1_inj: ân:nat. âm:ynat. n â¤ m â âm - n = â(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: âx:ynat. ây. x - ây = â(x-y). +* // +qed. + +(* Properties on order ******************************************************) + +lemma yle_minus_sn: âm:ynat. ân. m - n â¤ m. +* // #n /2 width=1 by yle_inj/ +qed. + +lemma yle_to_minus: âm:ynat. ân:nat. m â¤ n â 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 â¦ H) -H #H destruct +] +qed-. + +lemma yminus_to_le: âm:ynat. ân. m - n = 0 â m â¤ n. +* [2: #n >yminus_Y_inj #H destruct ] +#m #n >yminus_inj #H +lapply (yinj_inj â¦ H) -H (**) (* destruct lemma needed *) +/2 width=1 by yle_inj/ +qed. + +lemma monotonic_yle_minus_dx: âx,y. x â¤ y â âz. x - z â¤ y - z. +#x #y * /3 width=1 by yle_inj, monotonic_le_minus_l2/ +qed. + +(* Properties on strict order ***********************************************) + +lemma ylt_to_minus: ây:ynat. âx. yinj x < y â 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: ây:ynat. âx. 0 < y - x â 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: âx,y:ynat. x < y â âz:nat. z â¤ x â 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: âm:ynat. ân:nat. m + n - n = m. +#m #n elim n -n // +#n #IHn >(yplus_succ2 m n) >(yminus_succ â¦ n) // +qed. + +lemma yminus_plus2: âx:ynat. ây,z. x - (y + z) = x - y - z. +* // qed. + +(* Forward lemmas on minus **************************************************) + +lemma yle_plus1_to_minus_inj2: âx,z:ynat. ây:nat. x + y â¤ z â x â¤ z - y. +#x #z #y #H lapply (monotonic_yle_minus_dx â¦ H y) -H // +qed-. + +lemma yle_plus1_to_minus_inj1: âx,z:ynat. ây:nat. y + x â¤ z â x â¤ z - y. +/2 width=1 by yle_plus1_to_minus_inj2/ qed-. + +lemma yle_plus2_to_minus_inj2: âx,y:ynat. âz:nat. x â¤ y + z â x - z â¤ y. +/2 width=1 by monotonic_yle_minus_dx/ qed-. + +lemma yle_plus2_to_minus_inj1: âx,y:ynat. âz:nat. x â¤ z + y â x - z â¤ y. +/2 width=1 by yle_plus2_to_minus_inj2/ qed-. + +lemma yminus_plus (x:ynat) (y:nat): y â¤ x â 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: âx:nat. ây,z:ynat. x â¤ y â 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: âz:ynat. âx,y:nat. x â¤ y â 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: ây:nat. âx,z:ynat. y â¤ x â x + z - y = x - y + z. +#y * // #x * // +#z #Hxy >yplus_inj >yminus_inj yplus_inj #H destruct +qed-. + +lemma yplus_inv_minus: + âx1,y2:ynat.ây1,x2:nat. + y1 â¤ x1 â x1 + x2 = y2 + y1 â â§â§ x1 - y1 = y2 - x2 & x2 â¤ 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 â¦ H1) -H1 #Hyx1 + lapply (yinj_inj â¦ 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 â¦ H) -H #H destruct /2 width=1 by conj/ +] +qed-. + +(* Inversion lemmas on minus ************************************************) + +lemma yle_inv_plus_inj2: âx,z:ynat. ây:nat. x + y â¤ z â x â¤ z - y â§ y â¤ z. +/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-. + +lemma yle_inv_plus_inj1: âx,z:ynat. ây:nat. y + x â¤ z â x â¤ z - y â§ y â¤ z. +/2 width=1 by yle_inv_plus_inj2/ qed-. + +lemma yle_inv_plus_inj_dx: âz,x:ynat. ây:nat. x + y â¤ z â + â§â§ x â¤ z - y & y â¤ z. +* [| /2 width=1 by conj/ ] +#z * [| #y >yplus_Y1 #H >(yle_inv_Y1 â¦ 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-. diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma index 2e1d05077..9e0db4161 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma @@ -47,7 +47,7 @@ qed. lemma yplus_succ_swap: âm,n. m + ân = âm + n. // qed. -lemma yplus_SO2: âm. m + 1 = âm. +lemma yplus_SO2: âm:ynat. m + 1 = âm. * // qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/arith.ma deleted file mode 100644 index 0c1a561d5..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/lib/arith.ma +++ /dev/null @@ -1,208 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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: ân,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: âx,y,z. y â¤ z â z â¤ x â (x - z) + (z - y) = x - y. -#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus // -qed-. - -fact plus_minus_minus_be_aux: âi,x,y,z. y â¤ z â z â¤ x â i = z - y â x - z + i = x - y. -/2 width=1 by plus_minus_minus_be/ qed-. - -lemma plus_n_2: ân. n + 2 = n + 1 + 1. -// qed. - -lemma le_plus_minus: âm,n,p. p â¤ n â m + n - p = m + (n - p). -/2 by plus_minus/ qed. - -lemma le_plus_minus_comm: ân,m,p. p â¤ m â m + n - p = m - p + n. -/2 by plus_minus/ qed. - -lemma minus_minus_comm3: ân,x,y,z. n-x-y-z = n-y-z-x. -// qed. - -lemma arith_b1: âa,b,c1. c1 â¤ b â a - c1 - (b - c1) = a - b. -#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm // -qed. - -lemma arith_b2: âa,b,c1,c2. c1 + c2 â¤ b â 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: âx,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: âa1,a2,b,c1. c1 â¤ a1 â c1 â¤ b â - 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: âx,y,z. y < x â x+z-y-1 = x-y-1+z. -/2 width=1 by plus_minus/ qed-. - -(* Properties ***************************************************************) - -lemma eq_nat_dec: ân1,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: âm,n. â¨â¨ 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: âx,y,z:nat. x â¤ z â x = y â y â¤ z. -// qed-. - -fact le_repl_sn_trans_aux: âx,y,z:nat. x â¤ z â y = x â y â¤ z. -// qed-. - -lemma monotonic_le_minus_l2: âx1,x2,y,z. x1 â¤ x2 â x1 - y - z â¤ x2 - y - z. -/3 width=1 by monotonic_le_minus_l/ qed. - -(* Note: this might interfere with nat.ma *) -lemma monotonic_lt_pred: âm,n. m < n â O < m â pred m < pred n. -#m #n #Hmn #Hm whd >(S_pred â¦ Hm) -@le_S_S_to_le >S_pred /2 width=3 by transitive_lt/ -qed. - -lemma arith_j: âx,y,z. x-y-1 â¤ x-(y-z)-1. -/3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed. - -lemma arith_k_sn: âz,x,y,n. z < x â x+n â¤ y â x-z-1+n â¤ 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: âz,x,y,n. z < x â y â¤ x+n â y-z-1 â¤ 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: âx,y. x + y = y â x = 0. -// qed-. - -lemma lt_plus_SO_to_le: âx,y. x < y + 1 â x â¤ y. -/2 width=1 by monotonic_pred/ qed-. - -lemma lt_refl_false: ân. n < n â â¥. -#n #H elim (lt_to_not_eq â¦ H) -H /2 width=1 by/ -qed-. - -lemma lt_zero_false: ân. n < 0 â â¥. -#n #H elim (lt_to_not_le â¦ H) -H /2 width=1 by/ -qed-. - -lemma pred_inv_refl: âm. pred m = m â m = 0. -* // normalize #m #H elim (lt_refl_false m) // -qed-. - -lemma le_plus_xSy_O_false: âx,y. x + S y â¤ 0 â â¥. -#x #y #H lapply (le_n_O_to_eq â¦ H) -H minus_plus_plus_l -#H lapply (discr_plus_xy_minus_xz â¦ H) -H -#H destruct -qed-. - -lemma zero_eq_plus: âx,y. 0 = x + y â 0 = x â§ 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 â B) (nil: B) â - match n with - [ O â nil - | S k â op (iter k B op nil) - ]. - -interpretation "iterated function" 'exp op n = (iter n ? op). - -lemma iter_SO: âB:Type[0]. âf:BâB. âb,l. f^(l+1) b = f (f^l b). -#B #f #b #l >commutative_plus // -qed. - -lemma iter_n_Sm: âB:Type[0]. âf:BâB. âb,l. f^l (f b) = f (f^l b). -#B #f #b #l elim l -l normalize // -qed. - -lemma iter_plus: âB:Type[0]. âf:BâB. âb,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 â - match n1 with - [ O â match n2 with [ O â a2 | S n2 â a1 ] - | S n1 â match n2 with [ O â a3 | S n2 â tri A n1 n2 a1 a2 a3 ] - ]. - -lemma tri_lt: âA,a1,a2,a3,n2,n1. n1 < n2 â tri A n1 n2 a1 a2 a3 = a1. -#A #a1 #a2 #a3 #n2 elim n2 -n2 -[ #n1 #H elim (lt_zero_false â¦ H) -| #n2 #IH #n1 elim n1 -n1 /3 width=1 by monotonic_lt_pred/ -] -qed. - -lemma tri_eq: âA,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: âA,a1,a2,a3,n1,n2. n2 < n1 â tri A n1 n2 a1 a2 a3 = a3. -#A #a1 #a2 #a3 #n1 elim n1 -n1 -[ #n2 #H elim (lt_zero_false â¦ H) -| #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/bool.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/bool.ma deleted file mode 100644 index 293940b30..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/lib/bool.ma +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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: âb1,b2:bool. (b1 â¨ b2) = false â b1 = false â§ b2 = false. -* normalize /2 width=1 by conj/ #b2 #H destruct -qed-. - -lemma commutative_orb: commutative â¦ orb. -* * // qed. - -lemma eq_bool_dec: âb1,b2:bool. Decidable (b1 = b2). -* * /2 width=1 by or_introl/ -@or_intror #H destruct -qed-. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/list.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/list.ma deleted file mode 100644 index f572a3f58..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/lib/list.ma +++ /dev/null @@ -1,59 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â list A â 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 â - match l with - [ nil â â¤ - | cons hd tl â R hd â§ all A R tl - ]. - -inductive list2 (A1,A2:Type[0]) : Type[0] := - | nil2 : list2 A1 A2 - | cons2: A1 â A2 â list2 A1 A2 â 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 â match l1 with -[ nil2 â l2 -| cons2 a1 a2 tl â {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 â match l with -[ nil2 â 0 -| cons2 _ _ l â length2 A1 A2 l + 1 -]. - -interpretation "length (list of pairs)" - 'card l = (length2 ? ? l). diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/lstar.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/lstar.ma deleted file mode 100644 index ae707f2b9..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/lib/lstar.ma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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: âA:Type[0]. âB. (Aârelation B) â nat â (Aârelation B) â - Î»A,B,R,l,a. lstar â¦ (R a) l. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma deleted file mode 100644 index 0f193f0b1..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma +++ /dev/null @@ -1,329 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â Prop â Î»R. R â¨ (R â â¥). - -definition Transitive: âA. âR: relation A. Prop â Î»A,R. - âa1,a0. R a1 a0 â âa2. R a0 a2 â R a1 a2. - -definition confluent2: âA. âR1,R2: relation A. Prop â Î»A,R1,R2. - âa0,a1. R1 a0 a1 â âa2. R2 a0 a2 â - ââa. R2 a1 a & R1 a2 a. - -definition transitive2: âA. âR1,R2: relation A. Prop â Î»A,R1,R2. - âa1,a0. R1 a1 a0 â âa2. R2 a0 a2 â - ââa. R2 a1 a & R1 a a2. - -definition bi_confluent: âA,B. âR: bi_relation A B. Prop â Î»A,B,R. - âa0,a1,b0,b1. R a0 b0 a1 b1 â âa2,b2. R a0 b0 a2 b2 â - ââa,b. R a1 b1 a b & R a2 b2 a b. - -definition LTC: âA:Type[0]. âB. (Aârelation B) â (Aârelation B) â - Î»A,B,R,a. TC â¦ (R a). - -definition lsub_trans: âA,B. relation2 (Aârelation B) (relation A) â Î»A,B,R1,R2. - âL2,T1,T2. R1 L2 T1 T2 â âL1. R2 L1 L2 â R1 L1 T1 T2. - -definition s_r_transitive: âA,B. relation2 (Aârelation B) (Bârelation A) â Î»A,B,R1,R2. - âL2,T1,T2. R1 L2 T1 T2 â âL1. R2 T1 L1 L2 â LTC â¦ R1 L1 T1 T2. - -definition s_rs_transitive: âA,B. relation2 (Aârelation B) (Bârelation A) â Î»A,B,R1,R2. - âL2,T1,T2. LTC â¦ R1 L2 T1 T2 â âL1. R2 T1 L1 L2 â LTC â¦ R1 L1 T1 T2. - -definition s_r_confluent1: âA,B. relation2 (Aârelation B) (Bârelation A) â Î»A,B,R1,R2. - âL1,T1,T2. R1 L1 T1 T2 â âL2. R2 T1 L1 L2 â R2 T2 L1 L2. - -lemma TC_strip1: âA,R1,R2. confluent2 A R1 R2 â - âa0,a1. TC â¦ R1 a0 a1 â âa2. R2 a0 a2 â - ââa. R2 a1 a & TC â¦ R1 a2 a. -#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 -[ #a1 #Ha01 #a2 #Ha02 - elim (HR12 â¦ Ha01 â¦ Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 - elim (IHa0 â¦ Ha02) -a0 #a0 #Ha0 #Ha20 - elim (HR12 â¦ Ha1 â¦ Ha0) -HR12 -a /4 width=5 by step, ex2_intro/ -] -qed. - -lemma TC_strip2: âA,R1,R2. confluent2 A R1 R2 â - âa0,a2. TC â¦ R2 a0 a2 â âa1. R1 a0 a1 â - ââa. TC â¦ R2 a1 a & R1 a2 a. -#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 -[ #a2 #Ha02 #a1 #Ha01 - elim (HR12 â¦ Ha01 â¦ Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01 - elim (IHa0 â¦ Ha01) -a0 #a0 #Ha10 #Ha0 - elim (HR12 â¦ Ha0 â¦ Ha2) -HR12 -a /4 width=3 by step, ex2_intro/ -] -qed. - -lemma TC_confluent2: âA,R1,R2. - confluent2 A R1 R2 â confluent2 A (TC â¦ R1) (TC â¦ R2). -#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 -[ #a1 #Ha01 #a2 #Ha02 - elim (TC_strip2 â¦ HR12 â¦ Ha02 â¦ Ha01) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 - elim (IHa0 â¦ Ha02) -a0 #a0 #Ha0 #Ha20 - elim (TC_strip2 â¦ HR12 â¦ Ha0 â¦ Ha1) -HR12 -a /4 width=5 by step, ex2_intro/ -] -qed. - -lemma TC_strap1: âA,R1,R2. transitive2 A R1 R2 â - âa1,a0. TC â¦ R1 a1 a0 â âa2. R2 a0 a2 â - ââa. R2 a1 a & TC â¦ R1 a a2. -#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 -[ #a0 #Ha10 #a2 #Ha02 - elim (HR12 â¦ Ha10 â¦ Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 - elim (HR12 â¦ Ha0 â¦ Ha02) -HR12 -a0 #a0 #Ha0 #Ha02 - elim (IHa â¦ Ha0) -a /4 width=5 by step, ex2_intro/ -] -qed. - -lemma TC_strap2: âA,R1,R2. transitive2 A R1 R2 â - âa0,a2. TC â¦ R2 a0 a2 â âa1. R1 a1 a0 â - ââa. TC â¦ R2 a1 a & R1 a a2. -#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 -[ #a2 #Ha02 #a1 #Ha10 - elim (HR12 â¦ Ha10 â¦ Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a2 #_ #Ha02 #IHa #a1 #Ha10 - elim (IHa â¦ Ha10) -a0 #a0 #Ha10 #Ha0 - elim (HR12 â¦ Ha0 â¦ Ha02) -HR12 -a /4 width=3 by step, ex2_intro/ -] -qed. - -lemma TC_transitive2: âA,R1,R2. - transitive2 A R1 R2 â transitive2 A (TC â¦ R1) (TC â¦ R2). -#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 -[ #a0 #Ha10 #a2 #Ha02 - elim (TC_strap2 â¦ HR12 â¦ Ha02 â¦ Ha10) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 - elim (TC_strap2 â¦ HR12 â¦ Ha02 â¦ Ha0) -HR12 -a0 #a0 #Ha0 #Ha02 - elim (IHa â¦ Ha0) -a /4 width=5 by step, ex2_intro/ -] -qed. - -definition NF: âA. relation A â relation A â predicate A â - Î»A,R,S,a1. âa2. R a1 a2 â S a2 a1. - -definition NF_dec: âA. relation A â relation A â Prop â - Î»A,R,S. âa1. NF A R S a1 â¨ - ââa2. R â¦ a1 a2 & (S a2 a1 â â¥). - -inductive SN (A) (R,S:relation A): predicate A â -| SN_intro: âa1. (âa2. R a1 a2 â (S a2 a1 â â¥) â SN A R S a2) â SN A R S a1 -. - -lemma NF_to_SN: âA,R,S,a. NF A R S a â 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: âA,R,S. NF_dec A R S â - âa1. SN A R S a1 â - ââa2. star â¦ 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 â¦ Ha10 Ha01) -IHa1 -Ha01 /3 width=3 by star_compl, ex2_intro/ -qed-. - -definition NF_sn: âA. relation A â relation A â predicate A â - Î»A,R,S,a2. âa1. R a1 a2 â S a2 a1. - -inductive SN_sn (A) (R,S:relation A): predicate A â -| SN_sn_intro: âa2. (âa1. R a1 a2 â (S a2 a1 â â¥) â SN_sn A R S a1) â SN_sn A R S a2 -. - -lemma NF_to_SN_sn: âA,R,S,a. NF_sn A R S a â 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: âA,B,R,S. lsub_trans A B R S â lsub_trans A B (LTC â¦ 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 â¦ HT2 â¦ HL12) -HRS -HT2 /3 width=3 by step/ -qed-. - -lemma s_r_conf1_LTC1: âA,B,S,R. s_r_confluent1 A B S R â s_r_confluent1 A B (LTC â¦ S) R. -#A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx â¦ T1 H) -T1 /3 width=3 by/ -qed-. - -lemma s_r_trans_LTC1: âA,B,S,R. s_r_confluent1 A B S R â - s_r_transitive A B S R â s_rs_transitive A B S R. -#A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx â¦ T1 H) -T1 /2 width=3 by/ -#T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR â¦ HT1 â¦ HL12) -H2SR -HT1 -/4 width=5 by s_r_conf1_LTC1, trans_TC/ -qed-. - -lemma s_r_trans_LTC2: âA,B,S,R. s_rs_transitive A B S R â s_r_transitive A B S (LTC â¦ R). -#A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx â¦ L1 H) -L1 /3 width=3 by inj/ -qed-. - -lemma s_r_to_s_rs_trans: âA,B,S,R. s_r_transitive A B (LTC â¦ S) R â - s_rs_transitive A B S R. -#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1 -elim (TC_idem â¦ (S L1) â¦ T1 T2) -#_ #H @H @HSR // -qed-. - -lemma s_rs_to_s_r_trans: âA,B,S,R. s_rs_transitive A B S R â - s_r_transitive A B (LTC â¦ S) R. -#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1 -elim (TC_idem â¦ (S L1) â¦ T1 T2) -#H #_ @H @HSR // -qed-. - -lemma s_rs_trans_TC1: âA,B,S,R. s_rs_transitive A B S R â - s_rs_transitive A B (LTC â¦ S) R. -#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1 -elim (TC_idem â¦ (S L1) â¦ T1 T2) -elim (TC_idem â¦ (S L2) â¦ T1 T2) -#_ #H1 #H2 #_ @H2 @HSR /3 width=3 by/ -qed-. - -(* relations on unboxed pairs ***********************************************) - -lemma bi_TC_strip: âA,B,R. bi_confluent A B R â - âa0,a1,b0,b1. R a0 b0 a1 b1 â âa2,b2. bi_TC â¦ R a0 b0 a2 b2 â - ââa,b. bi_TC â¦ 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 â¦ H01 â¦ H02) -HR -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/ -| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2 - elim (HR â¦ H23 â¦ H2) -HR -a0 -b0 -a2 -b2 /3 width=4 by ex2_2_intro, bi_step/ -] -qed. - -lemma bi_TC_confluent: âA,B,R. bi_confluent A B R â - bi_confluent A B (bi_TC â¦ R). -#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1 -[ #a1 #b1 #H01 #a2 #b2 #H02 - elim (bi_TC_strip â¦ HR â¦ H01 â¦ H02) -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/ -| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02 - elim (IH â¦ H02) -a0 -b0 #a0 #b0 #H10 #H20 - elim (bi_TC_strip â¦ HR â¦ H13 â¦ H10) -a1 -b1 /3 width=7 by ex2_2_intro, bi_step/ -] -qed. - -lemma bi_TC_decomp_r: âA,B. âR:bi_relation A B. - âa1,a2,b1,b2. bi_TC â¦ R a1 b1 a2 b2 â - R a1 b1 a2 b2 â¨ - ââa,b. bi_TC â¦ 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: âA,B. âR:bi_relation A B. - âa1,a2,b1,b2. bi_TC â¦ R a1 b1 a2 b2 â - R a1 b1 a2 b2 â¨ - ââa,b. R a1 b1 a b & bi_TC â¦ R a b a2 b2. -#A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx â¦ 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: âA,B,C. tri_relation A B C â tri_relation A B C â - Î»A,B,C,R,a1,b1,c1,a2,b2,c2. R â¦ a1 b1 c1 a2 b2 c2 â¨ - â§â§ a1 = a2 & b1 = b2 & c1 = c2. - -lemma tri_RC_reflexive: âA,B,C,R. tri_reflexive A B C (tri_RC â¦ R). -/3 width=1 by and3_intro, or_intror/ qed. - -definition tri_star: âA,B,C,R. tri_relation A B C â - Î»A,B,C,R. tri_RC A B C (tri_TC â¦ R). - -lemma tri_star_tri_reflexive: âA,B,C,R. tri_reflexive A B C (tri_star â¦ R). -/2 width=1 by/ qed. - -lemma tri_TC_to_tri_star: âA,B,C,R,a1,b1,c1,a2,b2,c2. - tri_TC A B C R a1 b1 c1 a2 b2 c2 â - 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: âA,B,C,R,a1,b1,c1,a2,b2,c2. - R a1 b1 c1 a2 b2 c2 â 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: âA,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. - tri_star A B C R a1 b1 c1 a b c â - R a b c a2 b2 c2 â 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: âA,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c â - tri_star A B C R a b c a2 b2 c2 â - 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: âA,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. - tri_star A B C R a1 b1 c1 a b c â - tri_TC A B C R a b c a2 b2 c2 â - 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: âA,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. - tri_TC A B C R a1 b1 c1 a b c â - tri_star A B C R a b c a2 b2 c2 â - 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: âA,B,C,R. tri_transitive A B C (tri_star â¦ 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: âA,B,C,R,a1,b1,c1. âP:relation3 A B C. P a1 b1 c1 â - (âa,a2,b,b2,c,c2. tri_star â¦ R a1 b1 c1 a b c â R a b c a2 b2 c2 â P a b c â P a2 b2 c2) â - âa2,b2,c2. tri_star â¦ R a1 b1 c1 a2 b2 c2 â 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: âA,B,C,R,a2,b2,c2. âP:relation3 A B C. P a2 b2 c2 â - (âa1,a,b1,b,c1,c. R a1 b1 c1 a b c â tri_star â¦ R a b c a2 b2 c2 â P a b c â P a1 b1 c1) â - âa1,b1,c1. tri_star â¦ R a1 b1 c1 a2 b2 c2 â P a1 b1 c1. -#A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 * -[ #H12 @(tri_TC_ind_dx â¦ a1 b1 c1 H12) -a1 -b1 -c1 /3 width=6 by tri_TC_to_tri_star/ -| * #H1 #H2 #H3 destruct // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_2.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_2.ma deleted file mode 100644 index c95c57d8e..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_2.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "hvbox( hd @ break tl )" - right associative with precedence 47 - for @{ 'Cons $hd $tl }. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_3.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_3.ma deleted file mode 100644 index cfa556e68..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_3.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "hvbox( { term 46 hd1 , break term 46 hd2 } @ break term 46 tl )" - non associative with precedence 47 - for @{ 'Cons $hd1 $hd2 $tl }. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/infinity_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/infinity_0.ma deleted file mode 100644 index f5c849158..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/infinity_0.ma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "â" - non associative with precedence 55 - for @{ 'Infinity }. - diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/nil_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/nil_0.ma deleted file mode 100644 index 6ea5151a9..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/nil_0.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "â" - non associative with precedence 46 - for @{ 'Nil }. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/no_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/no_0.ma deleted file mode 100644 index af692211e..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/no_0.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "â»" - non associative with precedence 55 - for @{'no}. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/yes_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/yes_0.ma deleted file mode 100644 index c321749ae..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/yes_0.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "â" - non associative with precedence 55 - for @{'yes}. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/append_2.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/append_2.ma deleted file mode 100644 index f6d95184b..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/append_2.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "hvbox( l1 @@ break l2 )" - right associative with precedence 47 - for @{ 'Append $l1 $l2 }. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/predecessor_1.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/predecessor_1.ma deleted file mode 100644 index cf94d0497..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/predecessor_1.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "hvbox( â«° term 70 T )" - non associative with precedence 70 - for @{ 'Predecessor $T }. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/successor_1.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/successor_1.ma deleted file mode 100644 index 05e2c3146..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/successor_1.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "hvbox( â«¯ term 70 T )" - non associative with precedence 70 - for @{ 'Successor $T }. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/false_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/false_0.ma deleted file mode 100644 index b96432510..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/false_0.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "â¥" - non associative with precedence 19 - for @{'false}. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/true_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/true_0.ma deleted file mode 100644 index 7a9ad4366..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/true_0.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 Î»Î´ ****************************) - -notation "â¤" - non associative with precedence 19 - for @{'true}. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa2_notation.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa2_notation.ma deleted file mode 100644 index a7651b7d3..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa2_notation.ma +++ /dev/null @@ -1,16 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 *********************) - diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa_notation.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa_notation.ma deleted file mode 100644 index 6054aa308..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa_notation.ma +++ /dev/null @@ -1,326 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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(ââ ident x0 , ident x1 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.$P0) }. - -notation < "hvbox(ââ ident x0 , ident x1 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P0) }. - -(* multiple existental quantifier (1, 3) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) }. - -(* multiple existental quantifier (2, 2) *) - -notation > "hvbox(ââ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.$P0) (Î»${ident x0}.Î»${ident x1}.$P1) }. - -notation < "hvbox(ââ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P1) }. - -(* multiple existental quantifier (2, 3) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P1) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P1) }. - -(* multiple existental quantifier (3, 1) *) - -notation > "hvbox(ââ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.$P0) (Î»${ident x0}.$P1) (Î»${ident x0}.$P2) }. - -notation < "hvbox(ââ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.$P0) (Î»${ident x0}:$T0.$P1) (Î»${ident x0}:$T0.$P2) }. - -(* multiple existental quantifier (3, 2) *) - -notation > "hvbox(ââ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.$P0) (Î»${ident x0}.Î»${ident x1}.$P1) (Î»${ident x0}.Î»${ident x1}.$P2) }. - -notation < "hvbox(ââ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P2) }. - -(* multiple existental quantifier (3, 3) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P2) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P2) }. - -(* multiple existental quantifier (3, 4) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P2) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P2) }. - -(* multiple existental quantifier (4, 1) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.$P0) (Î»${ident x0}.$P1) (Î»${ident x0}.$P2) (Î»${ident x0}.$P3) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.$P0) (Î»${ident x0}:$T0.$P1) (Î»${ident x0}:$T0.$P2) (Î»${ident x0}:$T0.$P3) }. - -(* multiple existental quantifier (4, 2) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.$P0) (Î»${ident x0}.Î»${ident x1}.$P1) (Î»${ident x0}.Î»${ident x1}.$P2) (Î»${ident x0}.Î»${ident x1}.$P3) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P3) }. - -(* multiple existental quantifier (4, 3) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P3) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P3) }. - -(* multiple existental quantifier (4, 4) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P3) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P3) }. - -(* multiple existental quantifier (4, 5) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P3) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P3) }. - -(* multiple existental quantifier (5, 2) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.$P0) (Î»${ident x0}.Î»${ident x1}.$P1) (Î»${ident x0}.Î»${ident x1}.$P2) (Î»${ident x0}.Î»${ident x1}.$P3) (Î»${ident x0}.Î»${ident x1}.$P4) }. - -notation < "hvbox(ââ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.$P4) }. - -(* multiple existental quantifier (5, 3) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P4) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P4) }. - -(* multiple existental quantifier (5, 4) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P4) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P4) }. - -(* multiple existental quantifier (5, 5) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P4) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P4) }. - -(* multiple existental quantifier (5, 6) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P4) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P4) }. - -(* multiple existental quantifier (6, 3) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P5) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P5) }. - -(* multiple existental quantifier (6, 4) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P5) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P5) }. - -(* multiple existental quantifier (6, 5) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P5) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P5) }. - -(* multiple existental quantifier (6, 6) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.$P5) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.$P5) }. - -(* multiple existental quantifier (6, 7) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P5) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P5) }. - -(* multiple existental quantifier (7, 3) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P5) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.$P6) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P5) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.$P6) }. - -(* multiple existental quantifier (7, 4) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P5) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P6) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P5) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P6) }. - -(* multiple existental quantifier (7, 7) *) - -notation > "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P5) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.Î»${ident x5}.Î»${ident x6}.$P6) }. - -notation < "hvbox(ââ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" - non associative with precedence 20 - for @{ 'Ex (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P5) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.Î»${ident x5}:$T5.Î»${ident x6}:$T6.$P6) }. - -(* multiple existental quantifier (8, 4) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P5) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P6) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.$P7) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P5) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P6) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.$P7) }. - -(* multiple existental quantifier (8, 5) *) - -notation > "hvbox(ââ 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 (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P0) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P1) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P2) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P3) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P4) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P5) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P6) (Î»${ident x0}.Î»${ident x1}.Î»${ident x2}.Î»${ident x3}.Î»${ident x4}.$P7) }. - -notation < "hvbox(ââ 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 (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P0) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P1) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P2) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P3) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P4) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P5) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P6) (Î»${ident x0}:$T0.Î»${ident x1}:$T1.Î»${ident x2}:$T2.Î»${ident x3}:$T3.Î»${ident x4}:$T4.$P7) }. - -(* multiple disjunction connective (3) *) - -notation "hvbox(â¨â¨ 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(â¨â¨ 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(â¨â¨ 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(â§â§ 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(â§â§ 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 }. - diff --git a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa.ma b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa.ma deleted file mode 100644 index 27da042cd..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa.ma +++ /dev/null @@ -1,293 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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:A0âA1âProp) : Prop â - | ex1_2_intro: âx0,x1. P0 x0 x1 â 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:A0âA1âA2âProp) : Prop â - | ex1_3_intro: âx0,x1,x2. P0 x0 x1 x2 â 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:A0âA1âProp) : Prop â - | ex2_2_intro: âx0,x1. P0 x0 x1 â P1 x0 x1 â 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:A0âA1âA2âProp) : Prop â - | ex2_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â 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:A0âProp) : Prop â - | ex3_intro: âx0. P0 x0 â P1 x0 â P2 x0 â 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:A0âA1âProp) : Prop â - | ex3_2_intro: âx0,x1. P0 x0 x1 â P1 x0 x1 â P2 x0 x1 â 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:A0âA1âA2âProp) : Prop â - | ex3_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â 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:A0âA1âA2âA3âProp) : Prop â - | ex3_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â 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:A0âProp) : Prop â - | ex4_intro: âx0. P0 x0 â P1 x0 â P2 x0 â P3 x0 â 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:A0âA1âProp) : Prop â - | ex4_2_intro: âx0,x1. P0 x0 x1 â P1 x0 x1 â P2 x0 x1 â P3 x0 x1 â 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:A0âA1âA2âProp) : Prop â - | ex4_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â P3 x0 x1 x2 â 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:A0âA1âA2âA3âProp) : Prop â - | ex4_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â 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:A0âA1âA2âA3âA4âProp) : Prop â - | ex4_5_intro: âx0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 â P1 x0 x1 x2 x3 x4 â P2 x0 x1 x2 x3 x4 â P3 x0 x1 x2 x3 x4 â 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:A0âA1âProp) : Prop â - | ex5_2_intro: âx0,x1. P0 x0 x1 â P1 x0 x1 â P2 x0 x1 â P3 x0 x1 â P4 x0 x1 â 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:A0âA1âA2âProp) : Prop â - | ex5_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â P3 x0 x1 x2 â P4 x0 x1 x2 â 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:A0âA1âA2âA3âProp) : Prop â - | ex5_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â P4 x0 x1 x2 x3 â 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:A0âA1âA2âA3âA4âProp) : Prop â - | ex5_5_intro: âx0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 â P1 x0 x1 x2 x3 x4 â P2 x0 x1 x2 x3 x4 â P3 x0 x1 x2 x3 x4 â P4 x0 x1 x2 x3 x4 â 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:A0âA1âA2âA3âA4âA5âProp) : Prop â - | ex5_6_intro: âx0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 â P1 x0 x1 x2 x3 x4 x5 â P2 x0 x1 x2 x3 x4 x5 â P3 x0 x1 x2 x3 x4 x5 â P4 x0 x1 x2 x3 x4 x5 â 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:A0âA1âA2âProp) : Prop â - | ex6_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â P3 x0 x1 x2 â P4 x0 x1 x2 â P5 x0 x1 x2 â 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:A0âA1âA2âA3âProp) : Prop â - | ex6_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â P4 x0 x1 x2 x3 â P5 x0 x1 x2 x3 â 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:A0âA1âA2âA3âA4âProp) : Prop â - | ex6_5_intro: âx0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 â P1 x0 x1 x2 x3 x4 â P2 x0 x1 x2 x3 x4 â P3 x0 x1 x2 x3 x4 â P4 x0 x1 x2 x3 x4 â P5 x0 x1 x2 x3 x4 â 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:A0âA1âA2âA3âA4âA5âProp) : Prop â - | ex6_6_intro: âx0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 â P1 x0 x1 x2 x3 x4 x5 â P2 x0 x1 x2 x3 x4 x5 â P3 x0 x1 x2 x3 x4 x5 â P4 x0 x1 x2 x3 x4 x5 â P5 x0 x1 x2 x3 x4 x5 â 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:A0âA1âA2âA3âA4âA5âA6âProp) : Prop â - | ex6_7_intro: âx0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 â P1 x0 x1 x2 x3 x4 x5 x6 â P2 x0 x1 x2 x3 x4 x5 x6 â P3 x0 x1 x2 x3 x4 x5 x6 â P4 x0 x1 x2 x3 x4 x5 x6 â P5 x0 x1 x2 x3 x4 x5 x6 â 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:A0âA1âA2âProp) : Prop â - | ex7_3_intro: âx0,x1,x2. P0 x0 x1 x2 â P1 x0 x1 x2 â P2 x0 x1 x2 â P3 x0 x1 x2 â P4 x0 x1 x2 â P5 x0 x1 x2 â P6 x0 x1 x2 â 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:A0âA1âA2âA3âProp) : Prop â - | ex7_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â P4 x0 x1 x2 x3 â P5 x0 x1 x2 x3 â P6 x0 x1 x2 x3 â 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:A0âA1âA2âA3âA4âA5âA6âProp) : Prop â - | ex7_7_intro: âx0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 â P1 x0 x1 x2 x3 x4 x5 x6 â P2 x0 x1 x2 x3 x4 x5 x6 â P3 x0 x1 x2 x3 x4 x5 x6 â P4 x0 x1 x2 x3 x4 x5 x6 â P5 x0 x1 x2 x3 x4 x5 x6 â P6 x0 x1 x2 x3 x4 x5 x6 â 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:A0âA1âA2âA3âProp) : Prop â - | ex8_4_intro: âx0,x1,x2,x3. P0 x0 x1 x2 x3 â P1 x0 x1 x2 x3 â P2 x0 x1 x2 x3 â P3 x0 x1 x2 x3 â P4 x0 x1 x2 x3 â P5 x0 x1 x2 x3 â P6 x0 x1 x2 x3 â P7 x0 x1 x2 x3 â 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:A0âA1âA2âA3âA4âProp) : Prop â - | ex8_5_intro: âx0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 â P1 x0 x1 x2 x3 x4 â P2 x0 x1 x2 x3 x4 â P3 x0 x1 x2 x3 x4 â P4 x0 x1 x2 x3 x4 â P5 x0 x1 x2 x3 x4 â P6 x0 x1 x2 x3 x4 â P7 x0 x1 x2 x3 x4 â 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 â - | or3_intro0: P0 â or3 ? ? ? - | or3_intro1: P1 â or3 ? ? ? - | or3_intro2: P2 â 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 â - | or4_intro0: P0 â or4 ? ? ? ? - | or4_intro1: P1 â or4 ? ? ? ? - | or4_intro2: P2 â or4 ? ? ? ? - | or4_intro3: P3 â 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 â - | or5_intro0: P0 â or5 ? ? ? ? ? - | or5_intro1: P1 â or5 ? ? ? ? ? - | or5_intro2: P2 â or5 ? ? ? ? ? - | or5_intro3: P3 â or5 ? ? ? ? ? - | or5_intro4: P4 â 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 â - | and3_intro: P0 â P1 â P2 â 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 â - | and4_intro: P0 â P1 â P2 â P3 â and4 ? ? ? ? -. - -interpretation "multiple conjunction connective (4)" 'And P0 P1 P2 P3 = (and4 P0 P1 P2 P3). - diff --git a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa2.ma b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa2.ma deleted file mode 100644 index 3975bd4a2..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa2.ma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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". - diff --git a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa_props.ma b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa_props.ma deleted file mode 100644 index 8ab935a9a..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa_props.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat.ma deleted file mode 100644 index 1d802e296..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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] â -| yinj: nat â ynat -| Y : ynat -. - -coercion yinj. - -interpretation "ynat infinity" 'Infinity = Y. - -(* Inversion lemmas *********************************************************) - -lemma yinj_inj: âm,n. yinj m = yinj n â m = n. -#m #n #H destruct // -qed-. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma deleted file mode 100644 index 4a8e89e3d..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma +++ /dev/null @@ -1,136 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â -| yle_inj: âm,n. m â¤ n â yle m n -| yle_Y : âm. yle m (â) -. - -interpretation "ynat 'less or equal to'" 'leq x y = (yle x y). - -(* Basic inversion lemmas ***************************************************) - -fact yle_inv_inj2_aux: âx,y. x â¤ y â ân. y = yinj n â - ââm. m â¤ 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: âx,n. x â¤ yinj n â ââm. m â¤ n & x = yinj m. -/2 width=3 by yle_inv_inj2_aux/ qed-. - -lemma yle_inv_inj: âm,n. yinj m â¤ yinj n â m â¤ n. -#m #n #H elim (yle_inv_inj2 â¦ H) -H -#x #Hxn #H destruct // -qed-. - -fact yle_inv_O2_aux: âm:ynat. âx:ynat. m â¤ x â x = 0 â 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: âm:ynat. m â¤ 0 â m = 0. -/2 width =3 by yle_inv_O2_aux/ qed-. - -fact yle_inv_Y1_aux: âx,n. x â¤ n â x = â â n = â. -#x #n * -x -n // -#x #n #_ #H destruct -qed-. - -lemma yle_inv_Y1: ân. â â¤ n â n = â. -/2 width=3 by yle_inv_Y1_aux/ qed-. - -(* Inversion lemmas on successor ********************************************) - -fact yle_inv_succ1_aux: âx,y. x â¤ y â âm. x = â«¯m â m â¤ â«°y â§ â«¯â«°y = y. -#x #y * -x -y -[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn â¦ H) -H - #n #H1 #H2 destruct elim (le_inv_S1 â¦ 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: âm,y. â«¯m â¤ y â m â¤ â«°y â§ â«¯â«°y = y. -/2 width=3 by yle_inv_succ1_aux/ qed-. - -lemma yle_inv_succ: âm,n. â«¯m â¤ â«¯n â m â¤ n. -#m #n #H elim (yle_inv_succ1 â¦ H) -H // -qed-. - -(* Basic properties *********************************************************) - -lemma le_O1: ân:ynat. 0 â¤ n. -* /2 width=1 by yle_inj/ -qed. - -lemma yle_refl: reflexive â¦ yle. -* /2 width=1 by le_n, yle_inj/ -qed. - -lemma yle_split: âx,y:ynat. x â¤ y â¨ y â¤ 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: âm,n. m â¤ n â â«°m â¤ n. -#m #n * -m -n /3 width=3 by transitive_le, yle_inj/ -qed. - -lemma yle_refl_pred_sn: âx. â«°x â¤ x. -/2 width=1 by yle_refl, yle_pred_sn/ qed. - -lemma yle_pred: âm,n. m â¤ n â â«°m â¤ â«°n. -#m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/ -qed. - -(* Properties on successor **************************************************) - -lemma yle_succ: âm,n. m â¤ n â â«¯m â¤ â«¯n. -#m #n * -m -n /3 width=1 by yle_inj, le_S_S/ -qed. - -lemma yle_succ_dx: âm,n. m â¤ n â m â¤ â«¯n. -#m #n * -m -n /3 width=1 by le_S, yle_inj/ -qed. - -lemma yle_refl_S_dx: âx. x â¤ â«¯x. -/2 width=1 by yle_succ_dx/ qed. - -lemma yle_refl_SP_dx: âx. x â¤ â«¯â«°x. -* // * // -qed. - -(* Main properties **********************************************************) - -theorem yle_trans: Transitive â¦ yle. -#x #y * -x -y -[ #x #y #Hxy * // - #z #H lapply (yle_inv_inj â¦ H) -H - /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *) -| #x #z #H lapply (yle_inv_Y1 â¦ H) // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma deleted file mode 100644 index 850fae157..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â -| ylt_inj: âm,n. m < n â ylt m n -| ylt_Y : âm:nat. ylt m (â) -. - -interpretation "ynat 'less than'" 'lt x y = (ylt x y). - -(* Basic forward lemmas *****************************************************) - -lemma ylt_fwd_gen: âx,y. x < y â âm. x = yinj m. -#x #y * -x -y /2 width=2 by ex_intro/ -qed-. - -lemma ylt_fwd_le_succ: âx,y. x < y â â«¯x â¤ y. -#x #y * -x -y /2 width=1 by yle_inj/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact ylt_inv_inj2_aux: âx,y. x < y â ân. y = yinj n â - ââm. m < n & x = yinj m. -#x #y * -x -y -[ #x #y #Hxy #n #Hy elim (le_inv_S1 â¦ Hxy) -Hxy - #m #Hm #H destruct /3 width=3 by le_S_S, ex2_intro/ -| #x #n #Hy destruct -] -qed-. - -lemma ylt_inv_inj2: âx,n. x < yinj n â - ââm. m < n & x = yinj m. -/2 width=3 by ylt_inv_inj2_aux/ qed-. - -lemma ylt_inv_inj: âm,n. yinj m < yinj n â m < n. -#m #n #H elim (ylt_inv_inj2 â¦ H) -H -#x #Hx #H destruct // -qed-. - -lemma ylt_inv_Y1: ân. â < n â â¥. -#n #H elim (ylt_fwd_gen â¦ H) -H -#y #H destruct -qed-. - -lemma ylt_inv_O1: ân. 0 < n â â«¯â«°n = n. -* // #n #H lapply (ylt_inv_inj â¦ H) -H normalize -/3 width=1 by S_pred, eq_f/ -qed-. - -(* Inversion lemmas on successor ********************************************) - -fact ylt_inv_succ1_aux: âx,y. x < y â âm. x = â«¯m â m < â«°y â§ â«¯â«°y = y. -#x #y * -x -y -[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn â¦ H) -H - #n #H1 #H2 destruct elim (le_inv_S1 â¦ Hxy) -Hxy - #m #Hnm #H destruct /3 width=1 by ylt_inj, conj/ -| #x #y #H elim (ysucc_inv_inj_sn â¦ H) -H - #m #H #_ destruct /2 width=1 by ylt_Y, conj/ -] -qed-. - -lemma ylt_inv_succ1: âm,y. â«¯m < y â m < â«°y â§ â«¯â«°y = y. -/2 width=3 by ylt_inv_succ1_aux/ qed-. - -lemma ylt_inv_succ: âm,n. â«¯m < â«¯n â m < n. -#m #n #H elim (ylt_inv_succ1 â¦ H) -H // -qed-. - -(* Forward lemmas on successor **********************************************) - -fact ylt_fwd_succ2_aux: âx,y. x < y â ân. y = â«¯n â x â¤ n. -#x #y * -x -y -[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn â¦ 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 â¦ H) -H // -] -qed-. - -lemma ylt_fwd_succ2: âm,n. m < â«¯n â m â¤ n. -/2 width=3 by ylt_fwd_succ2_aux/ qed-. - -(* inversion and forward lemmas on yle **************************************) - -lemma ylt_fwd_le_succ1: âm,n. m < n â â«¯m â¤ n. -#m #n * -m -n /2 width=1 by yle_inj/ -qed-. - -lemma ylt_fwd_le: âm:ynat. ân:ynat. m < n â m â¤ n. -#m #n * -m -n /3 width=1 by lt_to_le, yle_inj/ -qed-. - -lemma ylt_yle_false: âm:ynat. ân:ynat. m < n â n â¤ m â â¥. -#m #n * -m -n -[ #m #n #Hmn #H lapply (yle_inv_inj â¦ H) -H - #H elim (lt_refl_false n) /2 width=3 by le_to_lt_to_lt/ -| #m #H lapply (yle_inv_Y1 â¦ H) -H - #H destruct -] -qed-. - -(* Basic properties *********************************************************) - -lemma ylt_O: âx. â«¯â«°(yinj x) = yinj x â 0 < x. -* /2 width=1 by/ normalize -#H destruct -qed. - -(* Properties on predecessor ************************************************) - -lemma ylt_pred: âm,n. m < n â 0 < m â â«°m < â«°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: ân. 0 < â«¯n. -* /2 width=1 by ylt_inj/ -qed. - -lemma ylt_succ: âm,n. m < n â â«¯m < â«¯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: âm:ynat. ân:ynat. m â¤ n â m < n â¨ m = n. -#m #n * -m -n -[ #m #n #Hmn elim (le_to_or_lt_eq â¦ Hmn) -Hmn - /3 width=1 by or_introl, ylt_inj/ -| * /2 width=1 by or_introl, ylt_Y/ -] -qed-. - -lemma ylt_split: âm,n:ynat. m < n â¨ n â¤ m.. -#m #n elim (yle_split m n) /2 width=1 by or_intror/ -#H elim (yle_split_eq â¦ H) -H /2 width=1 by or_introl, or_intror/ -qed-. - -lemma ylt_yle_trans: âx:ynat. ây:ynat. âz:ynat. y â¤ z â x < y â x < z. -#x #y #z * -y -z -[ #y #z #Hyz #H elim (ylt_inv_inj2 â¦ H) -H - #m #Hm #H destruct /3 width=3 by ylt_inj, lt_to_le_to_lt/ -| #y * // -] -qed-. - -lemma yle_ylt_trans: âx:ynat. ây:ynat. âz:ynat. y < z â x â¤ y â x < z. -#x #y #z * -y -z -[ #y #z #Hyz #H elim (yle_inv_inj2 â¦ H) -H - #m #Hm #H destruct /3 width=3 by ylt_inj, le_to_lt_to_lt/ -| #y #H elim (yle_inv_inj2 â¦ H) -H // -] -qed-. - -(* Main properties **********************************************************) - -theorem ylt_trans: Transitive â¦ ylt. -#x #y * -x -y -[ #x #y #Hxy * // - #z #H lapply (ylt_inv_inj â¦ H) -H - /3 width=3 by transitive_lt, ylt_inj/ (**) (* full auto too slow *) -| #x #z #H elim (ylt_yle_false â¦ H) // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_max.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_max.ma deleted file mode 100644 index 05ce327cd..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_max.ma +++ /dev/null @@ -1,58 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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: âx,y. x â¤ y â x - y + y = y. -#x #y * -x -y // -#x #y #Hxy >yminus_inj >(eq_minus_O â¦ Hxy) -Hxy // -qed-. - -lemma ymax_pre_sn: âx,y. y â¤ x â 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: ây,x. y â¤ x - y + y. -// qed. - -lemma ymax_pre_i_sn: ây,x. x â¤ x - y + y. -* // #y * /2 width=1 by yle_inj/ -qed. - -lemma ymax_pre_e: âx,z. x â¤ z â ây. y â¤ z â x - y + y â¤ z. -#x #z #Hxz #y #Hyz elim (yle_split x y) -[ #Hxy >(ymax_pre_dx â¦ Hxy) -x // -| #Hyx >(ymax_pre_sn â¦ Hyx) -y // -] -qed. - -lemma ymax_pre_dx_comm: âx,y. x â¤ y â y + (x - y) = y. -/2 width=1 by ymax_pre_dx/ qed-. - -lemma ymax_pre_sn_comm: âx,y. y â¤ x â y + (x - y) = x. -/2 width=1 by ymax_pre_sn/ qed-. - -lemma ymax_pre_i_dx_comm: ây,x. y â¤ y + (x - y). -// qed. - -lemma ymax_pre_i_sn_comm: ây,x. x â¤ y + (x - y). -/2 width=1 by ymax_pre_i_sn/ qed. - -lemma ymax_pre_e_comm: âx,z. x â¤ z â ây. y â¤ z â y + (x - y) â¤ z. -/2 width=1 by ymax_pre_e/ qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma deleted file mode 100644 index 4ec861b95..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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: âx,y. y â¤ x â x - (x - y) â¤ 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: âx,y. x â¤ y â x - (x - y) = x. -#x #y * -x -y // -#x #y #Hxy >yminus_inj >(eq_minus_O â¦ Hxy) -Hxy // -qed-. - -lemma ymin_pre_i_dx: âx,y. x - (x - y) â¤ y. -#x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/ -#Hxy >(ymin_pre_sn â¦ Hxy) // -qed. - -lemma ymin_pre_i_sn: âx,y. x - (x - y) â¤ x. -// qed. - -lemma ymin_pre_dx: âx,y. y â¤ yinj x â yinj x - (yinj x - y) = y. -#x #y #H elim (yle_inv_inj2 â¦ H) -H -#z #Hzx #H destruct >yminus_inj -/3 width=4 by minus_le_minus_minus_comm, eq_f/ -qed-. - -lemma ymin_pre_e: âz,x. z â¤ yinj x â ây. z â¤ y â - z â¤ yinj x - (yinj x - y). -#z #x #Hzx #y #Hzy elim (yle_split x y) -[ #H >(ymin_pre_sn â¦ H) -y // -| #H >(ymin_pre_dx â¦ H) -x // -] -qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma deleted file mode 100644 index e4763f40e..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma +++ /dev/null @@ -1,109 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â ynat â ynat â Î»x,y. match y with -[ yinj n â ypred^n x -| Y â yinj 0 -]. - -interpretation "ynat minus" 'minus x y = (yminus x y). - -(* Basic properties *********************************************************) - -lemma yminus_inj: ân,m. yinj m - yinj n = yinj (m - n). -#n elim n -n /2 width=3 by trans_eq/ -qed. - -lemma yminus_Y_inj: ân. â - yinj n = â. -#n elim n -n // normalize -#n #IHn >IHn // -qed. - -lemma yminus_O1: âx:ynat. 0 - x = 0. -* // qed. - -lemma yminus_refl: âx:ynat. x - x = 0. -* // qed. - -lemma yminus_minus_comm: ây,z,x. x - y - z = x - z - y. -* #y [ * #z [ * // ] ] >yminus_O1 // -qed. - -(* Properties on predecessor ************************************************) - -lemma yminus_SO2: âm. m - 1 = â«°m. -* // -qed. - -lemma yminus_pred: ân,m. 0 < m â 0 < n â â«°m - â«°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: ân,m. â«¯m - â«¯n = m - n. -* // #n * [2: >yminus_Y_inj // ] -#m >yminus_inj // -qed. - -lemma yminus_succ1_inj: ân:nat. âm:ynat. n â¤ m â â«¯m - n = â«¯(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: ây,x. x - â«¯y = â«°(x-y). -* // -qed. - -(* Properties on order ******************************************************) - -lemma yle_minus_sn: ân,m. m - n â¤ m. -* // #n * /2 width=1 by yle_inj/ -qed. - -lemma yle_to_minus: âm:ynat. ân:ynat. m â¤ n â m - n = 0. -#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/ -qed-. - -lemma yminus_to_le: ân:ynat. âm:ynat. m - n = 0 â m â¤ n. -* // #n * -[ #m >yminus_inj #H lapply (yinj_inj â¦ H) -H (**) (* destruct lemma needed *) - /2 width=1 by yle_inj/ -| >yminus_Y_inj #H destruct -] -qed. - -lemma monotonic_yle_minus_dx: âx,y. x â¤ y â âz. x - z â¤ 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: âx,y:ynat. x < y â âz:nat. z â¤ x â x - z < y - z. -#x #y * -x -y -/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/ -qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_plus.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_plus.ma deleted file mode 100644 index 761fc9965..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_plus.ma +++ /dev/null @@ -1,203 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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 â ynat â ynat â Î»x,y. match y with -[ yinj n â ysucc^n x -| Y â Y -]. - -interpretation "ynat plus" 'plus x y = (yplus x y). - -(* Properties on successor **************************************************) - -lemma yplus_succ2: âm,n. m + â«¯n = â«¯(m + n). -#m * // -qed. - -lemma yplus_succ1: âm,n. â«¯m + n = â«¯(m + n). -#m * normalize // -qed. - -lemma yplus_succ_swap: âm,n. m + â«¯n = â«¯m + n. -// qed. - -lemma yplus_SO2: âm. m + 1 = â«¯m. -* // -qed. - -(* Basic properties *********************************************************) - -lemma yplus_inj: ân,m. yinj m + yinj n = yinj (m + n). -#n elim n -n [ normalize // ] -#n #IHn #m >(yplus_succ2 ? n) >IHn -IHn -ysucc_iter_Y // -qed. - -lemma yplus_assoc: associative â¦ yplus. -#x #y * // #z cases y -y -[ #y >yplus_inj whd in â¢ (??%%); yplus_Y1 // -] -qed. - -lemma yplus_O1: ân:ynat. 0 + n = n. -#n >yplus_comm // qed. - -(* Basic inversion lemmas ***************************************************) - -lemma yplus_inv_inj: âz,y,x. x + y = yinj z â - ââm,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: ân,m. n â¤ m + n. -* // -#n elim n -n // -#n #IHn #m >(yplus_succ2 ? n) @(yle_succ n) // (**) (* full auto fails *) -qed. - -lemma yle_plus_dx1: ân,m. m â¤ m + n. -// qed. - -lemma yle_plus_dx1_trans: âx,z. z â¤ x â ây. z â¤ x + y. -/2 width=3 by yle_trans/ qed. - -lemma yle_plus_dx2_trans: ây,z. z â¤ y â âx. z â¤ x + y. -/2 width=3 by yle_trans/ qed. - -lemma monotonic_yle_plus_dx: âx,y. x â¤ y â âz. x + z â¤ y + z. -#x #y #Hxy * // -#z elim z -z /3 width=1 by yle_succ/ -qed. - -lemma monotonic_yle_plus_sn: âx,y. x â¤ y â âz. z + x â¤ z + y. -/2 width=1 by monotonic_yle_plus_dx/ qed. - -lemma monotonic_yle_plus: âx1,y1. x1 â¤ y1 â âx2,y2. x2 â¤ y2 â - x1 + x2 â¤ y1 + y2. -/3 width=3 by monotonic_yle_plus_dx, yle_trans/ qed. - -(* Forward lemmas on order **************************************************) - -lemma yle_fwd_plus_sn2: âx,y,z. x + y â¤ z â y â¤ z. -/2 width=3 by yle_trans/ qed-. - -lemma yle_fwd_plus_sn1: âx,y,z. x + y â¤ z â x â¤ z. -/2 width=3 by yle_trans/ qed-. - -lemma yle_inv_monotonic_plus_dx: âx,y:ynat.âz:nat. x + z â¤ y + z â x â¤ y. -#x #y #z elim z -z /3 width=1 by yle_inv_succ/ -qed-. - -lemma yle_inv_monotonic_plus_sn: âx,y:ynat.âz:nat. z + x â¤ z + y â x â¤ y. -/2 width=2 by yle_inv_monotonic_plus_dx/ qed-. - -lemma yle_fwd_plus_ge: âm1,m2:nat. m2 â¤ m1 â ân1,n2:ynat. m1 + n1 â¤ m2 + n2 â n1 â¤ n2. -#m1 #m2 #Hm12 #n1 #n2 #H -lapply (monotonic_yle_plus â¦ Hm12 â¦ H) -Hm12 -H -/2 width=2 by yle_inv_monotonic_plus_sn/ -qed-. - -lemma yle_fwd_plus_ge_inj: âm1:nat. âm2,n1,n2:ynat. m2 â¤ m1 â m1 + n1 â¤ m2 + n2 â n1 â¤ n2. -#m2 #m1 #n1 #n2 #H elim (yle_inv_inj2 â¦ 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: âx,y,z. x + z < y + z â x < y. -* [2: #y #z >yplus_comm #H elim (ylt_inv_Y1 â¦ H) ] -#x * // #y * [2: #H elim (ylt_inv_Y1 â¦ 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: âx,z. z < x â ây. z < x + yinj y. -/2 width=3 by ylt_yle_trans/ qed. - -lemma ylt_plus_dx2_trans: ây,z. z < y â âx. z < yinj x + y. -/2 width=3 by ylt_yle_trans/ qed. - -lemma monotonic_ylt_plus_dx: âx,y. x < y â âz: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: âx,y. x < y â âz:nat. yinj z + x < yinj z + y. -/2 width=1 by monotonic_ylt_plus_dx/ qed. - -(* Properties on minus ******************************************************) - -lemma yplus_minus_inj: âm:ynat. ân:nat. m + n - n = m. -#m #n elim n -n // -#n #IHn >(yplus_succ2 m n) >(yminus_succ â¦ n) // -qed. - -lemma yplus_minus: âm,n. m + n - n â¤ m. -#m * // -qed. - -(* Forward lemmas on minus **************************************************) - -lemma yle_plus1_to_minus_inj2: âx,z:ynat. ây:nat. x + y â¤ z â x â¤ z - y. -/2 width=1 by monotonic_yle_minus_dx/ qed-. - -lemma yle_plus1_to_minus_inj1: âx,z:ynat. ây:nat. y + x â¤ z â x â¤ z - y. -/2 width=1 by yle_plus1_to_minus_inj2/ qed-. - -lemma yle_plus2_to_minus_inj2: âx,y:ynat. âz:nat. x â¤ y + z â x - z â¤ y. -/2 width=1 by monotonic_yle_minus_dx/ qed-. - -lemma yle_plus2_to_minus_inj1: âx,y:ynat. âz:nat. x â¤ z + y â x - z â¤ y. -/2 width=1 by yle_plus2_to_minus_inj2/ qed-. - -lemma yplus_minus_assoc_inj: âx:nat. ây,z:ynat. x â¤ y â 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: ây:nat. âx,z:ynat. y â¤ x â x + z - y = x - y + z. -#y * // #x * // -#z #Hxy >yplus_inj >yminus_inj IHm // -qed. - -(* Inversion lemmas *********************************************************) - -lemma ysucc_inj: âm,n. â«¯m = â«¯n â m = n. -#m #n #H <(ypred_succ m) <(ypred_succ n) // -qed-. - -lemma ysucc_inv_refl: âm. â«¯m = m â m = â. -* // normalize -#m #H lapply (yinj_inj â¦ H) -H (**) (* destruct lemma needed *) -#H elim (lt_refl_false m) // -qed-. - -lemma ysucc_inv_inj_sn: âm2,n1. yinj m2 = â«¯n1 â - ââm1. 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: âm2,n1. â«¯n1 = yinj m2 â - ââm1. n1 = yinj m1 & m2 = S m1. -/2 width=1 by ysucc_inv_inj_sn/ qed-. - -lemma ysucc_inv_Y_sn: âm. â = â«¯m â m = â. -* // normalize -#m #H destruct -qed-. - -lemma ysucc_inv_Y_dx: âm. â«¯m = â â m = â. -/2 width=1 by ysucc_inv_Y_sn/ qed-. - -lemma ysucc_inv_O_sn: âm. yinj 0 = â«¯m â â¥. (**) (* explicit coercion *) -#m #H elim (ysucc_inv_inj_sn â¦ H) -H -#n #_ #H destruct -qed-. - -lemma ysucc_inv_O_dx: âm. â«¯m = 0 â â¥. -/2 width=2 by ysucc_inv_O_sn/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma index a88ef7706..b97d967fa 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma @@ -95,7 +95,7 @@ lemma lveq_inv_void_dx_length: âL1,L2,n1,n2. L1 ââ§*[n1,n2] L2.â§ â |L1| ââm2. L1 â â§*[n1,m2] L2 & 0 = n1 & âm2 = n2. #L1 #L2 #n1 #n2 #H #HL12 lapply (lveq_fwd_length_plus â¦ H) normalize >plus_n_Sm #H0 -lapply (plus2_inv_le_sn â¦ H0 HL12) -H0 -HL12 #H0 +lapply (plus2_le_sn_sn â¦ H0 HL12) -H0 -HL12 #H0 elim (le_inv_S1 â¦ H0) -H0 #m2 #_ #H0 destruct elim (lveq_inv_void_succ_dx â¦ H) -H /2 width=3 by ex3_intro/ qed-.