From: Ferruccio Guidi Date: Tue, 25 Dec 2012 21:48:14 +0000 (+0000) Subject: - lambda_delta: programmed renaming to lambdadelta X-Git-Tag: make_still_working~1372 X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=commitdiff_plain;h=e8998d29ab83e7b6aa495a079193705b2f6743d3 - lambda_delta: programmed renaming to lambdadelta - nat: general weight-based eliminator added for lambdadelta --- diff --git a/matita/matita/contribs/lambda_delta/Makefile b/matita/matita/contribs/lambda_delta/Makefile deleted file mode 100644 index 7e267a2f3..000000000 --- a/matita/matita/contribs/lambda_delta/Makefile +++ /dev/null @@ -1,130 +0,0 @@ -H = @ -XOA_DIR = ../../../components/binaries/xoa -XOA = xoa.native -DEP_DIR = ../../../components/binaries/matitadep -DEP = matitadep.native -MAC_DIR = ../../../components/binaries/mac -MAC = mac.native - -XOA_CONF = ground_2/xoa.conf.xml -XOA_TARGETS = ground_2/xoa_notation.ma ground_2/xoa.ma - -ORIG = . ./orig.sh - -ORIGS = basic_2/basic_1.orig - -PACKAGES = ground_2 basic_2 apps_2 - -all: - -# xoa ######################################################################## - -xoa: $(XOA_TARGETS) - -$(XOA_TARGETS): $(XOA_CONF) - @echo " EXEC $(XOA) $(XOA_CONF)" - $(H)MATITA_RT_BASE_DIR=../.. $(XOA_DIR)/$(XOA) $(XOA_CONF) - -# orig ####################################################################### - -orig: $(ORIGS) - @echo " ORIG basic_2" - $(H)$(ORIG) basic_2 < $(ORIGS) - -# dep ######################################################################## - -deps: MAS = $(shell find $* -name "*.ma") - -deps: $(DEP_DIR)/$(DEP) - @echo " MATITADEP" - $(H)grep "include \"" $(MAS) | $< - -# stats ###################################################################### - -stats: $(PACKAGES:%=%.stats) - -%.stats: MAS = $(shell find $* -name "*.ma") - -%.stats: CHARS = $(shell $(MAC_DIR)/$(MAC) $(MAS)) - -%.stats: - @printf '\x1B[1;40;37m' - @printf '%-15s %-40s' 'Statistics for:' $* - @printf '\x1B[0m\n' - @printf '\x1B[1;40;35m' - @printf '%-8s %6i' Chars $(CHARS) - @printf ' %-8s %3i' Pages `echo $$(($(CHARS) / 5120))` - @printf ' %-23s' '' - @printf '\x1B[0m\n' - @printf '\x1B[1;40;36m' - @printf '%-8s %6i' Sources `ls $(MAS) | wc -l` - @printf ' %-38s' '' -# @printf ' %-8s %5i' Objs `ls *.vo | wc -l` -# @printf ' %-6s %3i' Files `ls *.v | wc -l` - @printf '\x1B[0m\n' - @printf '\x1B[1;40;32m' - @printf '%-8s %6i' Theorems `grep "theorem " $(MAS) | wc -l` - @printf ' %-8s %3i' Lemmas `grep "lemma " $(MAS) | wc -l` - @printf ' %-5s %3i' Facts `grep "fact " $(MAS) | wc -l` - @printf ' %-6s %4i' Proofs `grep qed $(MAS) | wc -l` - @printf '\x1B[0m\n' - @printf '\x1B[1;40;33m' - @printf '%-8s %6i' Declared `grep "inductive \|record " $(MAS) | wc -l` - @printf ' %-8s %3i' Defined `grep "definition \|let rec " $(MAS) | wc -l` - @printf ' %-23s' '' -# @printf ' %-8s %5i' Local `grep "Local" *.v | wc -l` - @printf '\x1B[0m\n' - @printf '\x1B[1;40;31m' - @printf '%-8s %6i' Axioms `grep axiom $(MAS) | wc -l` - @printf ' %-8s %3i' Comments `grep "(\*[^*:]*$$" $(MAS) | wc -l` - @printf ' %-5s %3i' Marks `grep "(\*\*)" $(MAS) | wc -l` - @printf ' %-11s' '' - @printf '\x1B[0m\n' - -# summary #################################################################### - -define SUMMARY_TEMPLATE - TBL_$(1) := $(1)/$(1)_sum.tbl - MAS_$(1) := $$(shell find $(1) -name "*.ma") - TBLS += $$(TBL_$(1)) - - $$(TBL_$(1)): V1 := $$(shell ls $$(MAS_$(1)) | wc -l) - $$(TBL_$(1)): V2 := $$(shell $$(MAC_DIR)/$$(MAC) $$(MAS_$(1))) - $$(TBL_$(1)): C1 := $$(shell grep "inductive \|record " $$(MAS_$(1)) | wc -l) - $$(TBL_$(1)): C2 := $$(shell grep "definition \|let rec " $$(MAS_$(1)) | wc -l) - $$(TBL_$(1)): C3 := $$(shell grep "inductive \|record \|definition \|let rec " $$(MAS_$(1)) | wc -l) - $$(TBL_$(1)): P1 := $$(shell grep "theorem " $$(MAS_$(1)) | wc -l) - $$(TBL_$(1)): P2 := $$(shell grep "lemma " $$(MAS_$(1)) | wc -l) - $$(TBL_$(1)): P3 := $$(shell grep "lemma \|theorem " $$(MAS_$(1)) | wc -l) - - $$(TBL_$(1)): $$(MAS_$(1)) - @printf ' SUMMARY $(1)\n' - @printf 'name "$$(basename $$(@F))"\n\n' > $$@ - @printf 'table {\n' >> $$@ - @printf ' class "grey" [ "category"\n' >> $$@ - @printf ' [ "objects" * ]\n' >> $$@ - @printf ' ]\n' >> $$@ - @printf ' class "cyan" [ "sizes"\n' >> $$@ - @printf ' [ "files" "$$(V1)" ]\n' >> $$@ - @printf ' [ "characters" "$$(V2)" ]\n' >> $$@ - @printf ' [ * ]\n' >> $$@ - @printf ' ]\n' >> $$@ - @printf ' class "green" [ "propositions"\n' >> $$@ - @printf ' [ "theorems" "$$(P1)" ]\n' >> $$@ - @printf ' [ "lemmas" "$$(P2)" ]\n' >> $$@ - @printf ' [ "total" "$$(P3)" ]\n' >> $$@ - @printf ' ]\n' >> $$@ - @printf ' class "yellow" [ "concepts"\n' >> $$@ - @printf ' [ "declared" "$$(C1)" ]\n' >> $$@ - @printf ' [ "defined" "$$(C2)" ]\n' >> $$@ - @printf ' [ "total" "$$(C3)" ]\n' >> $$@ - @printf ' ]\n' >> $$@ - @printf '}\n\n' >> $$@ - @printf 'class "component" { 0 }\n\n' >> $$@ - @printf 'class "plane" { 1 } { 3 } { 5 }\n\n' >> $$@ - @printf 'class "number" { 2 } { 4 } { 6 }\n\n' >> $$@ -endef - -$(foreach PKG, $(PACKAGES), $(eval $(call SUMMARY_TEMPLATE,$(PKG)))) - -tbls: $(TBLS) diff --git a/matita/matita/contribs/lambda_delta/apps_2/functional/dsubst.ma b/matita/matita/contribs/lambda_delta/apps_2/functional/dsubst.ma deleted file mode 100644 index f5847371c..000000000 --- a/matita/matita/contribs/lambda_delta/apps_2/functional/dsubst.ma +++ /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/unfold/delift_lift.ma". -include "apps_2/functional/lift.ma". - -(* FUNCTIONAL DELIFTING SUBSTITUTION ****************************************) - -let rec fdsubst W d U on U ≝ match U with -[ TAtom I ⇒ match I with - [ Sort _ ⇒ U - | LRef i ⇒ tri … i d (#i) (↑[0, i] W) (#(i-1)) - | GRef _ ⇒ U - ] -| TPair I V T ⇒ match I with - [ Bind2 a I ⇒ ⓑ{a,I} (fdsubst W d V). (fdsubst W (d+1) T) - | Flat2 I ⇒ ⓕ{I} (fdsubst W d V). (fdsubst W d T) - ] -]. - -interpretation - "functional delifting substitution" - 'DSubst V d T = (fdsubst V d T). - -(* Main properties **********************************************************) - -theorem fdsubst_delift: ∀K,V,T,L,d. - ⇩[0, d] L ≡ K. ⓓV → L ⊢ ▼*[d, 1] T ≡ [d ⬐ V] T. -#K #V #T elim T -T -[ * #i #L #d #HLK normalize in ⊢ (? ? ? ? ? %); /2 width=3/ - elim (lt_or_eq_or_gt i d) #Hid - [ -HLK >(tri_lt ?????? Hid) /3 width=3/ - | destruct >tri_eq /4 width=4 by tpss_strap2, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *) - | -HLK >(tri_gt ?????? Hid) /3 width=3/ - ] -| * /3 width=1/ /4 width=1/ -] -qed. - -(* Main inversion properties ************************************************) - -theorem fdsubst_inv_delift: ∀K,V,T1,L,T2,d. ⇩[0, d] L ≡ K. ⓓV → - L ⊢ ▼*[d, 1] T1 ≡ T2 → [d ⬐ V] T1 = T2. -#K #V #T1 elim T1 -T1 -[ * #i #L #T2 #d #HLK #H - [ -HLK >(delift_inv_sort1 … H) -H // - | elim (lt_or_eq_or_gt i d) #Hid normalize - [ -HLK >(delift_inv_lref1_lt … H) -H // /2 width=1/ - | destruct - elim (delift_inv_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0 - lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H >minus_plus (delift_inv_refl_O2 … HV2) -V >(flift_inv_lift … HVT2) -V2 // - | -HLK >(delift_inv_lref1_ge … H) -H // /2 width=1/ - ] - | -HLK >(delift_inv_gref1 … H) -H // - ] -| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H - [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // /2 width=1/ - | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/apps_2/functional/lift.ma b/matita/matita/contribs/lambda_delta/apps_2/functional/lift.ma deleted file mode 100644 index bf05ea36a..000000000 --- a/matita/matita/contribs/lambda_delta/apps_2/functional/lift.ma +++ /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/substitution/lift.ma". -include "apps_2/functional/notation.ma". - -(* FUNCTIONAL RELOCATION ****************************************************) - -let rec flift d e U on U ≝ match U with -[ TAtom I ⇒ match I with - [ Sort _ ⇒ U - | LRef i ⇒ #(tri … i d i (i + e) (i + e)) - | GRef _ ⇒ U - ] -| TPair I V T ⇒ match I with - [ Bind2 a I ⇒ ⓑ{a,I} (flift d e V). (flift (d+1) e T) - | Flat2 I ⇒ ⓕ{I} (flift d e V). (flift d e T) - ] -]. - -interpretation "functional relocation" 'Lift d e T = (flift d e T). - -(* Main properties **********************************************************) - -theorem flift_lift: ∀T,d,e. ⇧[d, e] T ≡ ↑[d, e] T. -#T elim T -T -[ * #i #d #e // - elim (lt_or_eq_or_gt i d) #Hid normalize - [ >(tri_lt ?????? Hid) /2 width=1/ - | /2 width=1/ - | >(tri_gt ?????? Hid) /3 width=2/ - ] -| * /2/ -] -qed. - -(* Main inversion properties ************************************************) - -theorem flift_inv_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ↑[d, e] T1 = T2. -#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize // -[ #i #d #e #Hid >(tri_lt ?????? Hid) // -| #i #d #e #Hid - elim (le_to_or_lt_eq … Hid) -Hid #Hid - [ >(tri_gt ?????? Hid) // - | destruct // - ] -] -qed-. - -(* Derived properties *******************************************************) - -lemma flift_join: ∀e1,e2,T. ⇧[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T. -#e1 #e2 #T -lapply (flift_lift T 0 (e1+e2)) #H -elim (lift_split … H e1 e1 ? ? ?) -H // #U #H ->(flift_inv_lift … H) -H // -qed. diff --git a/matita/matita/contribs/lambda_delta/apps_2/functional/notation.ma b/matita/matita/contribs/lambda_delta/apps_2/functional/notation.ma deleted file mode 100644 index 1c60d6c18..000000000 --- a/matita/matita/contribs/lambda_delta/apps_2/functional/notation.ma +++ /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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE "functional" COMPONENT ********************************) - -notation "hvbox( ↑ [ term 46 d , break term 46 e ] break term 46 T )" - non associative with precedence 46 - for @{ 'Lift $d $e $T }. - -notation "hvbox( [ term 46 d ⬐ break term 46 V ] break term 46 T )" - non associative with precedence 46 - for @{ 'DSubst $V $d $T }. - -notation "hvbox( T1 ⇨ break term 46 T2 )" - non associative with precedence 45 - for @{ 'SRed $T1 $T2 }. diff --git a/matita/matita/contribs/lambda_delta/apps_2/functional/rtm.ma b/matita/matita/contribs/lambda_delta/apps_2/functional/rtm.ma deleted file mode 100644 index c7acff72e..000000000 --- a/matita/matita/contribs/lambda_delta/apps_2/functional/rtm.ma +++ /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/grammar/term_vector.ma". -include "basic_2/grammar/genv.ma". - -(* REDUCTION AND TYPE MACHINE ***********************************************) - -(* machine local environment *) -inductive xenv: Type[0] ≝ -| XAtom: xenv (* empty *) -| XQuad: xenv → bind2 → nat → xenv → term → xenv (* entry *) -. - -interpretation "atom (ext. local environment)" - 'Star = XAtom. - -interpretation "environment construction (quad)" - 'DxItem4 L I u K V = (XQuad L I u K V). - -(* machine stack *) -definition stack: Type[0] ≝ list2 xenv term. - -(* machine status *) -record rtm: Type[0] ≝ -{ rg: genv; (* global environment *) - ru: nat; (* current de Bruijn's level *) - re: xenv; (* extended local environment *) - rs: stack; (* application stack *) - rt: term (* code *) -}. - -(* initial state *) -definition rtm_i: genv → term → rtm ≝ - λG,T. mk_rtm G 0 (⋆) (⟠) T. - -(* update code *) -definition rtm_t: rtm → term → rtm ≝ - λM,T. match M with - [ mk_rtm G u E _ _ ⇒ mk_rtm G u E (⟠) T - ]. - -(* update closure *) -definition rtm_u: rtm → xenv → term → rtm ≝ - λM,E,T. match M with - [ mk_rtm G u _ _ _ ⇒ mk_rtm G u E (⟠) T - ]. - -(* get global environment *) -definition rtm_g: rtm → genv ≝ - λM. match M with - [ mk_rtm G _ _ _ _ ⇒ G - ]. - -(* get local reference level *) -definition rtm_l: rtm → nat ≝ - λM. match M with - [ mk_rtm _ u E _ _ ⇒ match E with - [ XAtom ⇒ u - | XQuad _ _ u _ _ ⇒ u - ] - ]. - -(* get stack *) -definition rtm_s: rtm → stack ≝ - λM. match M with - [ mk_rtm _ _ _ S _ ⇒ S - ]. - -(* get code *) -definition rtm_c: rtm → term ≝ - λM. match M with - [ mk_rtm _ _ _ _ T ⇒ T - ]. diff --git a/matita/matita/contribs/lambda_delta/apps_2/functional/rtm_step.ma b/matita/matita/contribs/lambda_delta/apps_2/functional/rtm_step.ma deleted file mode 100644 index ed16d5091..000000000 --- a/matita/matita/contribs/lambda_delta/apps_2/functional/rtm_step.ma +++ /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 "apps_2/functional/rtm.ma". - -(* REDUCTION AND TYPE MACHINE ***********************************************) - -(* transitions *) -inductive rtm_step: relation rtm ≝ -| rtm_ldrop : ∀G,u,E,I,t,F,V,S,i. - rtm_step (mk_rtm G u (E. ④{I} {t, F, V}) S (#(i + 1))) - (mk_rtm G u E S (#i)) -| rtm_ldelta: ∀G,u,E,t,F,V,S. - rtm_step (mk_rtm G u (E. ④{Abbr} {t, F, V}) S (#0)) - (mk_rtm G u F S V) -| rtm_ltype : ∀G,u,E,t,F,V,S. - rtm_step (mk_rtm G u (E. ④{Abst} {t, F, V}) S (#0)) - (mk_rtm G u F S V) -| rtm_gdrop : ∀G,I,V,u,E,S,p. p < |G| → - rtm_step (mk_rtm (G. ⓑ{I} V) u E S (§p)) - (mk_rtm G u E S (§p)) -| rtm_gdelta: ∀G,V,u,E,S,p. p = |G| → - rtm_step (mk_rtm (G. ⓓV) u E S (§p)) - (mk_rtm G u E S V) -| rtm_gtype : ∀G,V,u,E,S,p. p = |G| → - rtm_step (mk_rtm (G. ⓛV) u E S (§p)) - (mk_rtm G u E S V) -| rtm_tau : ∀G,u,E,S,W,T. - rtm_step (mk_rtm G u E S (ⓝW. T)) - (mk_rtm G u E S T) -| rtm_appl : ∀G,u,E,S,V,T. - rtm_step (mk_rtm G u E S (ⓐV. T)) - (mk_rtm G u E ({E, V} @ S) T) -| rtm_beta : ∀G,u,E,F,V,S,W,T. - rtm_step (mk_rtm G u E ({F, V} @ S) (+ⓛW. T)) - (mk_rtm G u (E. ④{Abbr} {u, F, V}) S T) -| rtm_push : ∀G,u,E,W,T. - rtm_step (mk_rtm G u E ⟠ (+ⓛW. T)) - (mk_rtm G (u + 1) (E. ④{Abst} {u, E, W}) ⟠ T) -| rtm_theta : ∀G,u,E,S,V,T. - rtm_step (mk_rtm G u E S (+ⓓV. T)) - (mk_rtm G u (E. ④{Abbr} {u, E, V}) S T) -. - -interpretation "sequential reduction (RTM)" - 'SRed O1 O2 = (rtm_step O1 O2). diff --git a/matita/matita/contribs/lambda_delta/basic_2/basic_1.orig b/matita/matita/contribs/lambda_delta/basic_2/basic_1.orig deleted file mode 100644 index 0b48e942f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/basic_1.orig +++ /dev/null @@ -1,712 +0,0 @@ -aplus/props / aplus_ahead_simpl -aplus/props / aplus_asort_le_simpl -aplus/props / aplus_asort_O_simpl -aplus/props / aplus_asort_simpl -aplus/props / aplus_assoc -aplus/props / aplus_asucc -aplus/props / aplus_asucc_false -aplus/props / aplus_inj -aplus/props / aplus_reg_r -aplus/props / aplus_sort_O_S_simpl -aplus/props / aplus_sort_S_S_simpl -aprem/fwd / aprem_gen_head_O -aprem/fwd / aprem_gen_head_S -aprem/fwd / aprem_gen_sort -aprem/props / aprem_asucc -aprem/props / aprem_repl -arity/aprem / arity_aprem -arity/cimp / arity_cimp_conf -arity/fwd / arity_gen_abst -arity/fwd / arity_gen_appl -arity/fwd / arity_gen_appls -arity/fwd / arity_gen_bind -arity/fwd / arity_gen_cast -arity/fwd / arity_gen_lift -arity/fwd / arity_gen_lref -arity/fwd / arity_gen_sort -arity/lift1 / arity_lift1 -arity/pr3 / arity_sred_pr2 -arity/pr3 / arity_sred_pr3 -arity/pr3 / arity_sred_wcpr0_pr0 -arity/pr3 / arity_sred_wcpr0_pr1 -arity/props / arity_appls_abbr -arity/props / arity_appls_bind -arity/props / arity_appls_cast -arity/props / arity_lift -arity/props / arity_mono -arity/props / arity_repellent -arity/props / node_inh -arity/subst0 / arity_fsubst0 -arity/subst0 / arity_gen_cvoid -arity/subst0 / arity_gen_cvoid_subst0 -arity/subst0 / arity_subst0 -asucc/fwd / asucc_gen_head -asucc/fwd / asucc_gen_sort -cimp/props / cimp_bind -cimp/props / cimp_flat_dx -cimp/props / cimp_flat_sx -cimp/props / cimp_getl_conf -clear/drop / drop_clear -clear/drop / drop_clear_O -clear/drop / drop_clear_S -clear/fwd / clear_gen_all -clear/fwd / clear_gen_bind -clear/fwd / clear_gen_flat -clear/fwd / clear_gen_flat_r -clear/fwd / clear_gen_sort -clear/props / clear_cle -clear/props / clear_clear -clear/props / clear_ctail -clear/props / clear_mono -clear/props / clear_trans -clen/getl / getl_ctail_clen -clen/getl / getl_gen_tail -cnt/props / cnt_lift -C/props / chead_ctail -C/props / clt_cong -C/props / clt_head -C/props / clt_thead -C/props / clt_wf_ind -C/props clt_wf q_ind -C/props / c_tail_ind -csuba/arity / arity_appls_appl -csuba/arity / csuba_arity -csuba/arity / csuba_arity_rev -csuba/clear / csuba_clear_conf -csuba/clear / csuba_clear_trans -csuba/drop / csuba_drop_abbr -csuba/drop / csuba_drop_abbr_rev -csuba/drop / csuba_drop_abst -csuba/drop / csuba_drop_abst_rev -csuba/fwd / csuba_gen_abbr -csuba/fwd / csuba_gen_abbr_rev -csuba/fwd / csuba_gen_abst -csuba/fwd / csuba_gen_abst_rev -csuba/fwd / csuba_gen_bind -csuba/fwd / csuba_gen_bind_rev -csuba/fwd / csuba_gen_flat -csuba/fwd / csuba_gen_flat_rev -csuba/fwd / csuba_gen_void -csuba/fwd / csuba_gen_void_rev -csuba/getl / csuba_getl_abbr -csuba/getl / csuba_getl_abbr_rev -csuba/getl / csuba_getl_abst -csuba/getl / csuba_getl_abst_rev -csuba/props / csuba_refl -csubc/arity / csubc_arity_conf -csubc/arity / csubc_arity_trans -csubc/clear / csubc_clear_conf -csubc/csuba / csubc_csuba -csubc/drop1 / csubc_drop1_conf_rev -csubc/drop1 / drop1_csubc_trans -csubc/drop / csubc_drop_conf_O -csubc/drop / csubc_drop_conf_rev -csubc/drop / drop_csubc_trans -csubc/fwd / csubc_gen_head_l -csubc/fwd / csubc_gen_head_r -csubc/fwd / csubc_gen_sort_l -csubc/fwd / csubc_gen_sort_r -csubc/getl / csubc_getl_conf -csubc/props / csubc_refl -csubst0/clear / csubst0_clear_O -csubst0/clear / csubst0_clear_O_back -csubst0/clear / csubst0_clear_S -csubst0/clear / csubst0_clear_trans -csubst0/drop / csubst0_drop_eq -csubst0/drop / csubst0_drop_eq_back -csubst0/drop / csubst0_drop_gt -csubst0/drop / csubst0_drop_gt_back -csubst0/drop / csubst0_drop_lt -csubst0/drop / csubst0_drop_lt_back -csubst0/fwd / csubst0_gen_head -csubst0/fwd / csubst0_gen_S_bind_2 -csubst0/fwd / csubst0_gen_sort -csubst0/getl / csubst0_getl_ge -csubst0/getl / csubst0_getl_ge_back -csubst0/getl / csubst0_getl_lt -csubst0/getl / csubst0_getl_lt_back -csubst0/props / csubst0_both_bind -csubst0/props / csubst0_fst_bind -csubst0/props / csubst0_snd_bind -csubst1/fwd / csubst1_gen_head -csubst1/getl / csubst1_getl_ge -csubst1/getl / csubst1_getl_ge_back -csubst1/getl / csubst1_getl_lt -csubst1/getl / getl_csubst1 -csubst1/props / csubst1_bind -csubst1/props / csubst1_flat -csubst1/props / csubst1_head -csubt/clear / csubt_clear_conf -csubt/csuba / csubt_csuba -csubt/drop / csubt_drop_abbr -csubt/drop / csubt_drop_abst -csubt/drop / csubt_drop_flat -csubt/fwd / csubt_gen_abbr -csubt/fwd / csubt_gen_abst -csubt/fwd / csubt_gen_bind -csubt/fwd / csubt_gen_flat -csubt/getl / csubt_getl_abbr -csubt/getl / csubt_getl_abst -csubt/pc3 / csubt_pc3 -csubt/pc3 / csubt_pr2 -csubt/props / csubt_refl -csubt/ty3 / csubt_ty3 -csubt/ty3 / csubt_ty3_ld -csubv/clear / csubv_clear_conf -csubv/clear / csubv_clear_conf_void -csubv/drop / csubv_drop_conf -csubv/getl / csubv_getl_conf -csubv/getl / csubv_getl_conf_void -csubv/props / csubv_bind_same -csubv/props / csubv_refl -drop1/fwd / drop1_gen_pcons -drop1/fwd / drop1_gen_pnil -drop1/getl / drop1_getl_trans -drop1/props / drop1_cons_tail -drop1/props / drop1_skip_bind -drop1/props / drop1_trans -drop/fwd / drop_gen_drop -drop/fwd / drop_gen_refl -drop/fwd / drop_gen_skip_l -drop/fwd / drop_gen_skip_r -drop/fwd / drop_gen_sort -drop/props / drop_conf_ge -drop/props / drop_conf_lt -drop/props / drop_conf_rev -drop/props / drop_ctail -drop/props / drop_mono -drop/props / drop_S -drop/props / drop_skip_bind -drop/props / drop_skip_flat -drop/props / drop_trans_ge -drop/props / drop_trans_le -ex0/props / aplus_gz_ge -ex0/props / aplus_gz_le -ex0/props / leq_leqz -ex0/props / leqz_leq -ex0/props / next_plus_gz -ex1/props / ex1_arity -ex1/props ex1 leq_sort_SS -ex1/props / ex1_ty3 -ex2/props / ex2_arity -ex2/props / ex2_nf2 -flt/props / flt_arith0 -flt/props / flt_arith1 -flt/props / flt_arith2 -flt/props / flt_shift -flt/props / flt_thead_dx -flt/props / flt_thead_sx -flt/props / flt_trans -flt/props / flt_wf_ind -flt/props flt_wf q_ind -fsubst0/fwd / fsubst0_gen_base -getl/clear / clear_getl_trans -getl/clear / getl_clear_bind -getl/clear / getl_clear_conf -getl/clear / getl_clear_trans -getl/dec / getl_dec -getl/drop / drop_getl_trans_ge -getl/drop / drop_getl_trans_le -getl/drop / drop_getl_trans_lt -getl/drop / getl_conf_ge_drop -getl/drop / getl_drop -getl/drop / getl_drop_conf_ge -getl/drop / getl_drop_conf_lt -getl/drop / getl_drop_conf_rev -getl/drop / getl_drop_trans -getl/flt / getl_flt -getl/fwd / getl_gen_2 -getl/fwd / getl_gen_all -getl/fwd / getl_gen_bind -getl/fwd / getl_gen_flat -getl/fwd / getl_gen_O -getl/fwd / getl_gen_S -getl/fwd / getl_gen_sort -getl/getl / getl_conf_le -getl/getl / getl_trans -getl/props / getl_ctail -getl/props / getl_flat -getl/props / getl_head -getl/props / getl_mono -getl/props / getl_refl -iso/fwd / iso_flats_flat_bind_false -iso/fwd / iso_flats_lref_bind_false -iso/fwd / iso_gen_head -iso/fwd / iso_gen_lref -iso/fwd / iso_gen_sort -iso/props / iso_refl -iso/props / iso_trans -leq/asucc / asucc_inj -leq/asucc / asucc_repl -leq/asucc / leq_ahead_asucc_false -leq/asucc / leq_asucc -leq/asucc / leq_asucc_false -leq/fwd / leq_gen_head1 -leq/fwd / leq_gen_head2 -leq/fwd / leq_gen_sort1 -leq/fwd / leq_gen_sort2 -leq/props / ahead_inj_snd -leq/props / leq_ahead_false_1 -leq/props / leq_ahead_false_2 -leq/props / leq_eq -leq/props / leq_refl -leq/props / leq_sym -leq/props / leq_trans -lift1/fwd / lift1_bind -lift1/fwd / lift1_cons_tail -lift1/fwd / lift1_flat -lift1/fwd / lift1_lref -lift1/fwd / lift1_sort -lift1/fwd / lifts1_cons -lift1/fwd / lifts1_flat -lift1/fwd / lifts1_nil -lift1/props / lift1_free -lift1/props / lift1_lift1 -lift1/props / lift1_xhg -lift1/props / lifts1_xhg -lift/fwd / lift_bind -lift/fwd / lift_flat -lift/fwd / lift_gen_bind -lift/fwd / lift_gen_flat -lift/fwd / lift_gen_head -lift/fwd / lift_gen_lref -lift/fwd / lift_gen_lref_false -lift/fwd / lift_gen_lref_ge -lift/fwd / lift_gen_lref_lt -lift/fwd / lift_gen_sort -lift/fwd / lift_head -lift/fwd / lift_lref_ge -lift/fwd / lift_lref_lt -lift/fwd / lift_sort -lift/props / lift_d -lift/props / lift_free -lift/props / lift_gen_lift -lift/props / lift_inj -lift/props / lift_lref_gt -lift/props / lift_r -lift/props / lifts_inj -lift/props / lifts_tapp -lift/props / thead_x_lift_y_y -lift/tlt / lift_tlt_dx -lift/tlt / lift_weight -lift/tlt / lift_weight_add -lift/tlt / lift_weight_add_O -lift/tlt / lift_weight_map -llt/props / llt_head_dx -llt/props / llt_head_sx -llt/props / llt_repl -llt/props / llt_trans -llt/props / llt_wf_ind -llt/props llt_wf q_ind -llt/props / lweight_repl -next_plus/props / next_plus_assoc -next_plus/props / next_plus_lt -next_plus/props / next_plus_next -nf2/arity / arity_nf2_inv_all -nf2/dec / nf2_dec -nf2/fwd / nf2_gen_abbr -nf2/fwd / nf2_gen_abst -nf2/fwd / nf2_gen_beta -nf2/fwd / nf2_gen_cast -nf2/fwd / nf2_gen_flat -nf2/fwd / nf2_gen_lref -nf2/fwd nf2_gen nf2_gen_aux -nf2/fwd / nf2_gen_void -nf2/iso / nf2_iso_appls_lref -nf2/lift1 / nf2_lift1 -nf2/pr3 / nf2_pr3_confluence -nf2/pr3 / nf2_pr3_unfold -nf2/props / nf2_abst -nf2/props / nf2_abst_shift -nf2/props / nf2_appl_lref -nf2/props / nf2_appls_lref -nf2/props / nf2_csort_lref -nf2/props / nf2_lift -nf2/props / nf2_lref_abst -nf2/props / nf2_sort -nf2/props / nfs2_tapp -pc1/props / pc1_head -pc1/props / pc1_head_1 -pc1/props / pc1_head_2 -pc1/props / pc1_pr0_r -pc1/props / pc1_pr0_u -pc1/props / pc1_pr0_u2 -pc1/props / pc1_pr0_x -pc1/props / pc1_refl -pc1/props / pc1_s -pc1/props / pc1_t -pc3/dec / pc3_abst_dec -pc3/dec / pc3_dec -pc3/fsubst0 / pc3_fsubst0 -pc3/fsubst0 / pc3_pr2_fsubst0 -pc3/fsubst0 / pc3_pr2_fsubst0_back -pc3/fwd / pc3_gen_abst -pc3/fwd / pc3_gen_abst_shift -pc3/fwd / pc3_gen_lift -pc3/fwd / pc3_gen_lift_abst -pc3/fwd / pc3_gen_not_abst -pc3/fwd / pc3_gen_sort -pc3/fwd / pc3_gen_sort_abst -pc3/left / pc3_ind_left -pc3/left pc3_ind_left pc3_left_pc3 -pc3/left pc3_ind_left pc3_left_pr3 -pc3/left pc3_ind_left pc3_left_sym -pc3/left pc3_ind_left pc3_left_trans -pc3/left pc3_ind_left pc3_pc3_left -pc3/nf2 / pc3_nf2 -pc3/nf2 / pc3_nf2_unfold -pc3/pc1 / pc3_pc1 -pc3/props / clear_pc3_trans -pc3/props / pc3_eta -pc3/props / pc3_head_1 -pc3/props / pc3_head_12 -pc3/props / pc3_head_2 -pc3/props / pc3_head_21 -pc3/props / pc3_lift -pc3/props / pc3_pr0_pr2_t -pc3/props / pc3_pr2_pr2_t -pc3/props / pc3_pr2_pr3_t -pc3/props / pc3_pr2_r -pc3/props / pc3_pr2_u -pc3/props / pc3_pr2_u2 -pc3/props / pc3_pr2_x -pc3/props / pc3_pr3_conf -pc3/props / pc3_pr3_pc3_t -pc3/props / pc3_pr3_r -pc3/props / pc3_pr3_t -pc3/props / pc3_pr3_x -pc3/props / pc3_refl -pc3/props / pc3_s -pc3/props / pc3_t -pc3/props / pc3_thin_dx -pc3/subst1 / pc3_gen_cabbr -pc3/wcpr0 / pc3_wcpr0 -pc3/wcpr0 pc3_wcpr0 pc3_wcpr0_t_aux -pc3/wcpr0 / pc3_wcpr0_t -pr0/dec / nf0_dec -pr0/fwd / pr0_gen_abbr -pr0/fwd / pr0_gen_abst -pr0/fwd / pr0_gen_appl -pr0/fwd / pr0_gen_cast -pr0/fwd / pr0_gen_lift -pr0/fwd / pr0_gen_lref -pr0/fwd / pr0_gen_sort -pr0/fwd / pr0_gen_void -pr0/pr0 / pr0_confluence -pr0/pr0 pr0_confluence pr0_cong_delta -pr0/pr0 pr0_confluence pr0_cong_upsilon_cong -pr0/pr0 pr0_confluence pr0_cong_upsilon_delta -pr0/pr0 pr0_confluence pr0_cong_upsilon_refl -pr0/pr0 pr0_confluence pr0_cong_upsilon_zeta -pr0/pr0 pr0_confluence pr0_delta_delta -pr0/pr0 pr0_confluence pr0_delta_tau -pr0/pr0 pr0_confluence pr0_upsilon_upsilon -pr0/props / pr0_lift -pr0/props / pr0_subst0 -pr0/props / pr0_subst0_back -pr0/props / pr0_subst0_fwd -pr0/subst1 / pr0_delta1 -pr0/subst1 / pr0_subst1 -pr0/subst1 / pr0_subst1_back -pr0/subst1 / pr0_subst1_fwd -pr1/pr1 / pr1_confluence -pr1/pr1 / pr1_strip -pr1/props / pr1_comp -pr1/props / pr1_eta -pr1/props / pr1_head_1 -pr1/props / pr1_head_2 -pr1/props / pr1_pr0 -pr1/props / pr1_t -pr2/clen / pr2_gen_cbind -pr2/clen / pr2_gen_cflat -pr2/clen / pr2_gen_ctail -pr2/fwd / pr2_gen_abbr -pr2/fwd / pr2_gen_abst -pr2/fwd / pr2_gen_appl -pr2/fwd / pr2_gen_cast -pr2/fwd / pr2_gen_csort -pr2/fwd / pr2_gen_lift -pr2/fwd / pr2_gen_lref -pr2/fwd / pr2_gen_sort -pr2/fwd / pr2_gen_void -pr2/pr2 / pr2_confluence -pr2/pr2 pr2_confluence pr2_delta_delta -pr2/pr2 pr2_confluence pr2_free_delta -pr2/pr2 pr2_confluence pr2_free_free -pr2/props / clear_pr2_trans -pr2/props / pr2_cflat -pr2/props / pr2_change -pr2/props / pr2_ctail -pr2/props / pr2_head_1 -pr2/props / pr2_head_2 -pr2/props / pr2_lift -pr2/props / pr2_thin_dx -pr2/subst1 / pr2_delta1 -pr2/subst1 / pr2_gen_cabbr -pr2/subst1 / pr2_subst1 -pr3/fwd / pr3_gen_abbr -pr3/fwd / pr3_gen_abst -pr3/fwd / pr3_gen_appl -pr3/fwd / pr3_gen_bind -pr3/fwd / pr3_gen_cast -pr3/fwd / pr3_gen_lift -pr3/fwd / pr3_gen_lref -pr3/fwd / pr3_gen_sort -pr3/fwd / pr3_gen_void -pr3/iso / pr3_iso_appl_bind -pr3/iso / pr3_iso_appls_abbr -pr3/iso / pr3_iso_appls_appl_bind -pr3/iso / pr3_iso_appls_beta -pr3/iso / pr3_iso_appls_bind -pr3/iso / pr3_iso_appls_cast -pr3/iso / pr3_iso_beta -pr3/pr1 / pr3_pr1 -pr3/pr3 / pr3_confluence -pr3/pr3 / pr3_strip -pr3/props / clear_pr3_trans -pr3/props / pr3_cflat -pr3/props / pr3_eta -pr3/props / pr3_flat -pr3/props / pr3_head_1 -pr3/props / pr3_head_12 -pr3/props / pr3_head_2 -pr3/props / pr3_head_21 -pr3/props / pr3_lift -pr3/props / pr3_pr0_pr2_t -pr3/props / pr3_pr2 -pr3/props / pr3_pr2_pr2_t -pr3/props / pr3_pr2_pr3_t -pr3/props / pr3_pr3_pr3_t -pr3/props / pr3_t -pr3/props / pr3_thin_dx -pr3/subst1 / pr3_gen_cabbr -pr3/subst1 / pr3_subst1 -pr3/wcpr0 / pr3_wcpr0_t -r/props / r_arith0 -r/props / r_arith1 -r/props / r_dis -r/props / r_minus -r/props / r_plus -r/props / r_plus_sym -r/props / r_S -r/props / s_r -sc3/arity / sc3_arity -sc3/arity / sc3_arity_csubc -sc3/props / sc3_abbr -sc3/props / sc3_abst -sc3/props / sc3_appl -sc3/props / sc3_arity_gen -sc3/props / sc3_bind -sc3/props / sc3_cast -sc3/props / sc3_lift -sc3/props / sc3_lift1 -sc3/props sc3_props sc3_sn3_abst -sc3/props / sc3_repl -sc3/props / sc3_sn3 -sn3/fwd / sn3_gen_bind -sn3/fwd / sn3_gen_cflat -sn3/fwd / sn3_gen_flat -sn3/fwd / sn3_gen_head -sn3/fwd / sn3_gen_lift -sn3/lift1 / sns3_lifts1 -sn3/nf2 / nf2_sn3 -sn3/nf2 / sn3_nf2 -sn3/props / sn3_abbr -sn3/props / sn3_appl_abbr -sn3/props / sn3_appl_appl -sn3/props / sn3_appl_appls -sn3/props / sn3_appl_beta -sn3/props / sn3_appl_bind -sn3/props / sn3_appl_cast -sn3/props / sn3_appl_lref -sn3/props / sn3_appls_abbr -sn3/props / sn3_appls_beta -sn3/props / sn3_appls_bind -sn3/props / sn3_appls_cast -sn3/props / sn3_appls_lref -sn3/props / sn3_beta -sn3/props / sn3_bind -sn3/props / sn3_cast -sn3/props / sn3_cdelta -sn3/props / sn3_cflat -sn3/props / sn3_change -sn3/props / sn3_cpr3_trans -sn3/props / sn3_gen_def -sn3/props / sn3_lift -sn3/props / sn3_pr2_intro -sn3/props / sn3_pr3_trans -sn3/props / sn3_shift -sn3/props / sns3_lifts -s/props / minus_s_s -s/props / s_arith0 -s/props / s_arith1 -s/props / s_inc -s/props / s_inj -s/props / s_le -s/props / s_lt -s/props / s_minus -s/props / s_plus -s/props / s_plus_sym -s/props / s_S -sty0/fwd / sty0_gen_appl -sty0/fwd / sty0_gen_bind -sty0/fwd / sty0_gen_cast -sty0/fwd / sty0_gen_lref -sty0/fwd / sty0_gen_sort -sty0/props / sty0_correct -sty0/props / sty0_lift -sty1/cnt / sty1_cnt -sty1/props / sty1_abbr -sty1/props / sty1_appl -sty1/props / sty1_bind -sty1/props / sty1_cast2 -sty1/props / sty1_correct -sty1/props / sty1_lift -sty1/props / sty1_trans -subst0/dec / dnf_dec -subst0/dec / dnf_dec2 -subst0/fwd / subst0_gen_head -subst0/fwd / subst0_gen_lift_false -subst0/fwd / subst0_gen_lift_ge -subst0/fwd / subst0_gen_lift_lt -subst0/fwd / subst0_gen_lref -subst0/fwd / subst0_gen_sort -subst0/props / subst0_lift_ge -subst0/props / subst0_lift_ge_s -subst0/props / subst0_lift_ge_S -subst0/props / subst0_lift_lt -subst0/props / subst0_refl -subst0/subst0 / subst0_confluence_eq -subst0/subst0 / subst0_confluence_lift -subst0/subst0 / subst0_confluence_neq -subst0/subst0 / subst0_subst0 -subst0/subst0 / subst0_subst0_back -subst0/subst0 / subst0_trans -subst0/tlt / subst0_tlt -subst0/tlt / subst0_tlt_head -subst0/tlt / subst0_weight_le -subst0/tlt / subst0_weight_lt -subst1/fwd / subst1_gen_head -subst1/fwd / subst1_gen_lift_eq -subst1/fwd / subst1_gen_lift_ge -subst1/fwd / subst1_gen_lift_lt -subst1/fwd / subst1_gen_lref -subst1/fwd / subst1_gen_sort -subst1/props / subst1_ex -subst1/props / subst1_head -subst1/props / subst1_lift_ge -subst1/props / subst1_lift_lt -subst1/props / subst1_lift_S -subst1/subst1 / subst1_confluence_eq -subst1/subst1 / subst1_confluence_lift -subst1/subst1 / subst1_confluence_neq -subst1/subst1 / subst1_subst1 -subst1/subst1 / subst1_subst1_back -subst1/subst1 / subst1_trans -subst/fwd / subst_head -subst/fwd / subst_lref_eq -subst/fwd / subst_lref_gt -subst/fwd / subst_lref_lt -subst/fwd / subst_sort -subst/props / subst_lift_SO -subst/props / subst_subst0 -T/dec / abst_dec -T/dec / bind_dec_not -T/dec / binder_dec -T/dec / term_dec -T/dec terms_props bind_dec -T/dec terms_props flat_dec -T/dec terms_props kind_dec -tlist/props / tcons_tapp_ex -tlist/props / theads_tapp -tlist/props / tlist_ind_rev -tlist/props / tslt_wf_ind -tlist/props tslt_wf q_ind -tlt/props / tlt_head_dx -tlt/props / tlt_head_sx -tlt/props / tlt_trans -tlt/props / tlt_wf_ind -tlt/props tlt_wf q_ind -tlt/props / wadd_le -tlt/props / wadd_lt -tlt/props / wadd_O -tlt/props / weight_add_O -tlt/props / weight_add_S -tlt/props / weight_eq -tlt/props / weight_le -T/props / not_abbr_abst -T/props / not_abbr_void -T/props / not_abst_void -T/props / not_void_abst -T/props / thead_x_y_y -T/props / tweight_lt -ty3/arity / ty3_arity -ty3/arity_props / ty3_acyclic -ty3/arity_props / ty3_predicative -ty3/arity_props / ty3_repellent -ty3/arity_props / ty3_sn3 -ty3/dec / ty3_inference -ty3/fsubst0 / ty3_csubst0 -ty3/fsubst0 / ty3_fsubst0 -ty3/fsubst0 / ty3_subst0 -ty3/fwd / ty3_gen_appl -ty3/fwd / ty3_gen_bind -ty3/fwd / ty3_gen_cast -ty3/fwd / ty3_gen_lref -ty3/fwd / ty3_gen_sort -ty3/fwd / tys3_gen_cons -ty3/fwd / tys3_gen_nil -ty3/fwd_nf2 / ty3_gen_appl_nf2 -ty3/fwd_nf2 / ty3_inv_appls_lref_nf2 -ty3/fwd_nf2 / ty3_inv_lref_lref_nf2 -ty3/fwd_nf2 / ty3_inv_lref_nf2 -ty3/fwd_nf2 / ty3_inv_lref_nf2_pc3 -ty3/nf2 ty3_nf2_gen ty3_nf2_inv_abst_aux -ty3/nf2 / ty3_nf2_inv_abst -ty3/nf2 / ty3_nf2_inv_abst_premise_csort -ty3/nf2 / ty3_nf2_inv_all -ty3/nf2 / ty3_nf2_inv_sort -ty3/pr3 / ty3_sred_pr0 -ty3/pr3 / ty3_sred_pr1 -ty3/pr3 / ty3_sred_pr2 -ty3/pr3 / ty3_sred_pr3 -ty3/pr3 / ty3_sred_wcpr0_pr0 -ty3/pr3_props / ty3_cred_pr2 -ty3/pr3_props / ty3_cred_pr3 -ty3/pr3_props / ty3_gen_lift -ty3/pr3_props / ty3_sconv -ty3/pr3_props / ty3_sconv_pc3 -ty3/pr3_props / ty3_sred_back -ty3/pr3_props / ty3_tred -ty3/props / ty3_correct -ty3/props / ty3_gen_abst_abst -ty3/props / ty3_getl_subst0 -ty3/props / ty3_lift -ty3/props / ty3_typecheck -ty3/props / ty3_unique -ty3/sty0 / ty3_sty0 -ty3/subst1 / ty3_gen_cabbr -ty3/subst1 / ty3_gen_cvoid -wcpr0/fwd / wcpr0_gen_head -wcpr0/fwd / wcpr0_gen_sort -wcpr0/getl / wcpr0_drop -wcpr0/getl / wcpr0_drop_back -wcpr0/getl / wcpr0_getl -wcpr0/getl / wcpr0_getl_back -wf3/clear / clear_wf3_trans -wf3/clear / wf3_clear_conf -wf3/fwd / wf3_gen_bind1 -wf3/fwd / wf3_gen_flat1 -wf3/fwd / wf3_gen_head2 -wf3/fwd / wf3_gen_sort1 -wf3/getl / getl_wf3_trans -wf3/getl / wf3_getl_conf -wf3/props / ty3_shift1 -wf3/props / wf3_idem -wf3/props / wf3_mono -wf3/props / wf3_total -wf3/props / wf3_ty3 -wf3/ty3 / wf3_pc3_conf -wf3/ty3 / wf3_pr2_conf -wf3/ty3 / wf3_pr3_conf -wf3/ty3 / wf3_ty3_conf diff --git a/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt b/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt deleted file mode 100644 index d64855d0d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt +++ /dev/null @@ -1,244 +0,0 @@ -# waiting #################################################################### - -aplus/props aplus_reg_r -aplus/props aplus_assoc -aplus/props aplus_asucc -aplus/props aplus_sort_O_S_simpl -aplus/props aplus_sort_S_S_simpl -aplus/props aplus_asort_O_simpl -aplus/props aplus_asort_le_simpl -aplus/props aplus_asort_simpl -aplus/props aplus_ahead_simpl -aplus/props aplus_asucc_false -aplus/props aplus_inj -aprem/fwd aprem_gen_sort -aprem/fwd aprem_gen_head_O -aprem/fwd aprem_gen_head_S -aprem/props aprem_repl -aprem/props aprem_asucc -arity/aprem arity_aprem -arity/cimp arity_cimp_conf -arity/fwd arity_gen_sort -arity/fwd arity_gen_lref -arity/fwd arity_gen_bind -arity/fwd arity_gen_abst -arity/fwd arity_gen_appl -arity/fwd arity_gen_cast -arity/fwd arity_gen_appls -arity/fwd arity_gen_lift -arity/lift1 arity_lift1 -arity/pr3 arity_sred_wcpr0_pr0 -arity/pr3 arity_sred_wcpr0_pr1 -arity/pr3 arity_sred_pr2 -arity/pr3 arity_sred_pr3 -arity/props node_inh -arity/props arity_lift -arity/props arity_mono -arity/props arity_repellent -arity/props arity_appls_cast -arity/props arity_appls_abbr -arity/props arity_appls_bind -arity/subst0 arity_gen_cvoid_subst0 -arity/subst0 arity_gen_cvoid -arity/subst0 arity_fsubst0 -arity/subst0 arity_subst0 -asucc/fwd asucc_gen_sort -asucc/fwd asucc_gen_head -cnt/props cnt_lift -C/props clt_wf__q_ind -C/props clt_wf_ind - -csuba/arity csuba_arity -csuba/arity csuba_arity_rev -csuba/arity arity_appls_appl -csuba/clear csuba_clear_conf -csuba/clear csuba_clear_trans -csuba/drop csuba_drop_abbr -csuba/drop csuba_drop_abst -csuba/drop csuba_drop_abst_rev -csuba/drop csuba_drop_abbr_rev -csuba/fwd csuba_gen_abbr -csuba/fwd csuba_gen_void -csuba/fwd csuba_gen_abst -csuba/fwd csuba_gen_flat -csuba/fwd csuba_gen_bind -csuba/fwd csuba_gen_abst_rev -csuba/fwd csuba_gen_void_rev -csuba/fwd csuba_gen_abbr_rev -csuba/fwd csuba_gen_flat_rev -csuba/fwd csuba_gen_bind_rev -csuba/getl csuba_getl_abbr -csuba/getl csuba_getl_abst -csuba/getl csuba_getl_abst_rev -csuba/getl csuba_getl_abbr_rev -csuba/props csuba_refl - -csubc/arity csubc_arity_conf -csubc/arity csubc_arity_trans -csubc/drop1 drop1_csubc_trans -csubc/drop drop_csubc_trans - -csubt/csuba csubt_csuba -csubt/fwd csubt_gen_abbr -csubt/fwd csubt_gen_abst - -csubv/clear csubv_clear_conf -csubv/clear csubv_clear_conf_void -csubv/drop csubv_drop_conf -csubv/getl csubv_getl_conf -csubv/getl csubv_getl_conf_void -csubv/props csubv_bind_same -csubv/props csubv_refl -drop1/props drop1_cons_tail -ex0/props aplus_gz_le -ex0/props aplus_gz_ge -ex0/props next_plus_gz -ex0/props leqz_leq -ex0/props leq_leqz -ex1/props ex1__leq_sort_SS -ex1/props ex1_arity -ex1/props ex1_ty3 -ex2/props ex2_nf2 -ex2/props ex2_arity -leq/asucc asucc_repl -leq/asucc asucc_inj -leq/asucc leq_asucc -leq/asucc leq_ahead_asucc_false -leq/asucc leq_asucc_false -leq/fwd leq_gen_sort1 -leq/fwd leq_gen_head1 -leq/fwd leq_gen_sort2 -leq/fwd leq_gen_head2 -leq/props ahead_inj_snd -leq/props leq_refl -leq/props leq_eq -leq/props leq_sym -leq/props leq_trans -leq/props leq_ahead_false_1 -leq/props leq_ahead_false_2 -lift1/fwd lift1_cons_tail -lift1/fwd lifts1_nil -lift1/fwd lifts1_cons -lift/props thead_x_lift_y_y -lift/props lifts_tapp -lift/props lifts_inj -llt/props lweight_repl -llt/props llt_repl -llt/props llt_trans -llt/props llt_head_sx -llt/props llt_head_dx -llt/props llt_wf__q_ind -llt/props llt_wf_ind -next_plus/props next_plus_assoc -next_plus/props next_plus_next -next_plus/props next_plus_lt -nf2/arity arity_nf2_inv_all -nf2/fwd nf2_gen_lref -nf2/fwd nf2_gen_abst -nf2/fwd nf2_gen_cast -nf2/fwd nf2_gen_beta -nf2/fwd nf2_gen_flat -nf2/fwd nf2_gen__nf2_gen_aux -nf2/fwd nf2_gen_abbr -nf2/fwd nf2_gen_void -nf2/props nfs2_tapp -nf2/props nf2_appls_lref -pc1/props pc1_pr0_r -pc1/props pc1_pr0_x -pc1/props pc1_refl -pc1/props pc1_pr0_u -pc1/props pc1_s -pc1/props pc1_head_1 -pc1/props pc1_head_2 -pc1/props pc1_t -pc1/props pc1_pr0_u2 -pc1/props pc1_head - -pc3/dec pc3_dec -pc3/dec pc3_abst_dec -pc3/fwd pc3_gen_not_abst -pc3/fwd pc3_gen_lift_abst -pc3/nf2 pc3_nf2 -pc3/nf2 pc3_nf2_unfold -pc3/pc1 pc3_pc1 -pc3/props pc3_pr2_pr2_t -pc3/props pc3_pr2_pr3_t -pc3/props pc3_pr3_pc3_t -pc3/props pc3_eta - -pr0/fwd pr0_gen_void -pr0/dec nf0_dec - -pr1/props pr1_eta - -pr2/fwd pr2_gen_void -pr3/fwd pr3_gen_void -pr3/props pr3_eta -sn3/props sns3_lifts -sty1/cnt sty1_cnt -subst/fwd subst_sort -subst/fwd subst_lref_lt -subst/fwd subst_lref_eq -subst/fwd subst_lref_gt -subst/fwd subst_head -subst/props subst_lift_SO -subst/props subst_subst0 -T/dec binder_dec -T/dec abst_dec -tlist/props tslt_wf__q_ind -tlist/props tslt_wf_ind -tlist/props theads_tapp -tlist/props tcons_tapp_ex -tlist/props tlist_ind_rev -ty3/arity ty3_arity -ty3/arity_props ty3_predicative -ty3/arity_props ty3_repellent -ty3/arity_props ty3_acyclic -ty3/dec ty3_inference -ty3/fwd tys3_gen_nil -ty3/fwd tys3_gen_cons -ty3/fwd_nf2 ty3_gen_appl_nf2 -ty3/fwd_nf2 ty3_inv_lref_nf2_pc3 -ty3/fwd_nf2 ty3_inv_lref_nf2 -ty3/fwd_nf2 ty3_inv_appls_lref_nf2 -ty3/fwd_nf2 ty3_inv_lref_lref_nf2 -ty3/nf2 ty3_nf2_inv_abst_premise_csort -ty3/nf2 ty3_nf2_inv_all -ty3/nf2 ty3_nf2_inv_sort -ty3/nf2 ty3_nf2_gen__ty3_nf2_inv_abst_aux -ty3/nf2 ty3_nf2_inv_abst -ty3/pr3 ty3_sred_wcpr0_pr0 -ty3/pr3 ty3_sred_pr0 -ty3/pr3 ty3_sred_pr1 -ty3/pr3 ty3_sred_pr2 -ty3/pr3 ty3_sred_pr3 -ty3/pr3_props ty3_cred_pr2 -ty3/pr3_props ty3_cred_pr3 -ty3/pr3_props ty3_gen_lift -ty3/pr3_props ty3_tred -ty3/pr3_props ty3_sconv_pc3 -ty3/pr3_props ty3_sred_back -ty3/pr3_props ty3_sconv -ty3/props ty3_gen_abst_abst -ty3/sty0 ty3_sty0 -ty3/subst1 ty3_gen_cvoid - -wf3/clear wf3_clear_conf -wf3/clear clear_wf3_trans -wf3/fwd wf3_gen_sort1 -wf3/fwd wf3_gen_bind1 -wf3/fwd wf3_gen_flat1 -wf3/fwd wf3_gen_head2 -wf3/getl wf3_getl_conf -wf3/getl getl_wf3_trans -wf3/props wf3_mono -wf3/props wf3_total -wf3/props ty3_shift1 -wf3/props wf3_idem -wf3/props wf3_ty3 -wf3/ty3 wf3_pr2_conf -wf3/ty3 wf3_pr3_conf -wf3/ty3 wf3_pc3_conf -wf3/ty3 wf3_ty3_conf - -# check ###################################################################### diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/acp.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/acp.ma deleted file mode 100644 index dc046b094..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/acp.ma +++ /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/unfold/ldrops.ma". - -(* ABSTRACT COMPUTATION PROPERTIES ******************************************) - -definition CP1 ≝ λRR:lenv→relation term. λRS:relation term. - ∀L,k. NF … (RR L) RS (⋆k). - -definition CP2 ≝ λRR:lenv→relation term. λRS:relation term. - ∀L,K,W,i. ⇩[0,i] L ≡ K. ⓛW → NF … (RR L) RS (#i). - -definition CP3 ≝ λRR:lenv→relation term. λRP:lenv→predicate term. - ∀L,V,k. RP L (ⓐ⋆k.V) → RP L V. - -definition CP4 ≝ λRR:lenv→relation term. λRS:relation term. - ∀L0,L,T,T0,d,e. NF … (RR L) RS T → - ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → NF … (RR L0) RS T0. - -definition CP4s ≝ λRR:lenv→relation term. λRS:relation term. - ∀L0,L,des. ⇩*[des] L0 ≡ L → - ∀T,T0. ⇧*[des] T ≡ T0 → - NF … (RR L) RS T → NF … (RR L0) RS T0. - -(* requirements for abstract computation properties *) -record acp (RR:lenv->relation term) (RS:relation term) (RP:lenv→predicate term) : Prop ≝ -{ cp1: CP1 RR RS; - cp2: CP2 RR RS; - cp3: CP3 RR RP; - cp4: CP4 RR RS -}. - -(* Basic properties *********************************************************) - -(* Basic_1: was: nf2_lift1 *) -lemma acp_lifts: ∀RR,RS. CP4 RR RS → CP4s RR RS. -#RR #RS #HRR #L1 #L2 #des #H elim H -L1 -L2 -des -[ #L #T1 #T2 #H #HT1 - <(lifts_inv_nil … H) -H // -| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2 - elim (lifts_inv_cons … H) -H /3 width=9/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/acp_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/acp_aaa.ma deleted file mode 100644 index f4da11310..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/acp_aaa.ma +++ /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/lifts_lifts.ma". -include "basic_2/unfold/ldrops_ldrops.ma". -include "basic_2/static/aaa_lifts.ma". -include "basic_2/static/aaa_aaa.ma". -include "basic_2/computation/lsubc_ldrops.ma". - -(* ABSTRACT COMPUTATION PROPERTIES ******************************************) - -(* Main propertis ***********************************************************) - -(* Basic_1: was: sc3_arity_csubc *) -theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP. - acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L0,des. ⇩*[des] L0 ≡ L1 → - ∀T0. ⇧*[des] T ≡ T0 → ∀L2. L2 ⊑[RP] L0 → - ⦃L2, T0⦄ ϵ[RP] 〚A〛. -#RR #RS #RP #H1RP #H2RP #L1 #T #A #H elim H -L1 -T -A -[ #L #k #L0 #des #HL0 #X #H #L2 #HL20 - >(lifts_inv_sort1 … H) -H - lapply (aacr_acr … H1RP H2RP ⓪) #HAtom - @(s2 … HAtom … ◊) // /2 width=2/ -| #I #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20 - lapply (aacr_acr … H1RP H2RP B) #HB - elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK1) #HK1b - elim (ldrops_ldrop_trans … HL01 … HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1 - >(at_mono … Hi1 … Hi0) -i1 - elim (ldrops_inv_skip2 … Hdes1 … H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct - elim (lsubc_ldrop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H - elim (lsubc_inv_pair2 … H) -H * - [ #K2 #HK20 #H destruct - generalize in match HLK2; generalize in match I; -HLK2 -I * #HLK2 - [ elim (lift_total V0 0 (i0 +1)) #V #HV0 - elim (lifts_lift_trans … Hi0 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2 - @(s4 … HB … ◊ … HV0 HLK2) /3 width=7/ (* uses IHB HL20 V2 HV0 *) - | @(s2 … HB … ◊) // /2 width=3/ - ] - | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0 - #K2 #V2 #A2 #HKV2A #HKV0A #_ #H1 #H2 destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK2b - lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B - >(aaa_mono … HKV0A … HKV0B) in HKV2A; -HKV0A -HKV0B #HKV2B - elim (lift_total V2 0 (i0 +1)) #V #HV2 - @(s4 … HB … ◊ … HV2 HLK2) - @(s7 … HB … HKV2B) // - ] -| #a #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20 - elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct - lapply (aacr_acr … H1RP H2RP A) #HA - lapply (aacr_acr … H1RP H2RP B) #HB - lapply (s1 … HB) -HB #HB - @(s5 … HA … ◊ ◊) // /3 width=5/ -| #a #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02 - elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct - @(aacr_abst … H1RP H2RP) - [ lapply (aacr_acr … H1RP H2RP B) #HB - @(s1 … HB) /2 width=5/ - | -IHB - #L3 #V3 #T3 #des3 #HL32 #HT03 #HB - elim (lifts_total des3 W0) #W2 #HW02 - elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20 - lapply (aaa_lifts … L2 W2 … (des @@ des3) … HLWB) -HLWB /2 width=3/ #HLW2B - @(IHA (L2. ⓛW2) … (des + 1 @@ des3 + 1)) -IHA - /2 width=3/ /3 width=5/ - ] -| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20 - elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct - /3 width=10/ -| #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20 - elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct - lapply (aacr_acr … H1RP H2RP A) #HA - lapply (s1 … HA) #H - @(s6 … HA … ◊) /2 width=5/ /3 width=5/ -] -qed. - -(* Basic_1: was: sc3_arity *) -lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀L,T,A. L ⊢ T ⁝ A → ⦃L, T⦄ ϵ[RP] 〚A〛. -/2 width=8/ qed. - -lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀L,T,A. L ⊢ T ⁝ A → RP L T. -#RR #RS #RP #H1RP #H2RP #L #T #A #HT -lapply (aacr_acr … H1RP H2RP A) #HA -@(s1 … HA) /2 width=4/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/acp_cr.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/acp_cr.ma deleted file mode 100644 index b0b15e665..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/acp_cr.ma +++ /dev/null @@ -1,174 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/aarity.ma". -include "basic_2/unfold/gr2_gr2.ma". -include "basic_2/unfold/lifts_lift_vector.ma". -include "basic_2/unfold/ldrops_ldrop.ma". -include "basic_2/computation/acp.ma". - -(* ABSTRACT COMPUTATION PROPERTIES ******************************************) - -(* Note: this is Girard's CR1 *) -definition S1 ≝ λRP,C:lenv→predicate term. - ∀L,T. C L T → RP L T. - -(* Note: this is Tait's iii, or Girard's CR4 *) -definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term. - ∀L,Vs. all … (RP L) Vs → - ∀T. 𝐒⦃T⦄ → NF … (RR L) RS T → C L (ⒶVs.T). - -(* Note: this is Tait's ii *) -definition S3 ≝ λRP,C:lenv→predicate term. - ∀a,L,Vs,V,T,W. C L (ⒶVs. ⓓ{a}V. T) → RP L W → C L (ⒶVs. ⓐV. ⓛ{a}W. T). - -definition S4 ≝ λRP,C:lenv→predicate term. ∀L,K,Vs,V1,V2,i. - C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 → - ⇩[0, i] L ≡ K. ⓓV1 → C L (Ⓐ Vs. #i). - -definition S5 ≝ λRP,C:lenv→predicate term. - ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s → - ∀a,V,T. C (L. ⓓV) (ⒶV2s. T) → RP L V → C L (ⒶV1s. ⓓ{a}V. T). - -definition S6 ≝ λRP,C:lenv→predicate term. - ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓝW. T). - -definition S7 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e. - C L1 T1 → ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2. - -definition S7s ≝ λC:lenv→predicate term. - ∀L1,L2,des. ⇩*[des] L2 ≡ L1 → - ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2. - -(* properties of the abstract candidate of reducibility *) -record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝ -{ s1: S1 RP C; - s2: S2 RR RS RP C; - s3: S3 RP C; - s4: S4 RP C; - s5: S5 RP C; - s6: S6 RP C; - s7: S7 C -}. - -(* the abstract candidate of reducibility associated to an atomic arity *) -let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝ -λT. match A with -[ AAtom ⇒ RP L T -| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 → - aacr RP A L0 (ⓐV0. T0) -]. - -interpretation - "candidate of reducibility of an atomic arity (abstract)" - 'InEInt RP L T A = (aacr RP A L T). - -(* Basic properties *********************************************************) - -(* Basic_1: was: sc3_lift1 *) -lemma acr_lifts: ∀C. S7 C → S7s C. -#C #HC #L1 #L2 #des #H elim H -L1 -L2 -des -[ #L #T1 #T2 #H #HT1 - <(lifts_inv_nil … H) -H // -| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2 - elim (lifts_inv_cons … H) -H /3 width=9/ -] -qed. - -lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) → - ∀des,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[des] V ≡ V0 → - RP L V → RP L0 V0. -#RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV -@acr_lifts /width=6/ -@(s7 … HRP) -qed. - -(* Basic_1: was only: sns3_lifts1 *) -lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) → - ∀des,L0,L,Vs,V0s. ⇧*[des] Vs ≡ V0s → ⇩*[des] L0 ≡ L → - all … (RP L) Vs → all … (RP L0) V0s. -#RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize // -#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s -@conj /2 width=1/ /2 width=6 by rp_lifts/ -qed. - -(* Basic_1: was: - sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift -*) -lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀A. acr RR RS RP (aacr RP A). -#RR #RS #RP #H1RP #H2RP #A elim A -A normalize // -#B #A #IHB #IHA @mk_acr normalize -[ #L #T #H - lapply (H ? (⋆0) ? ⟠ ? ? ?) -H - [1,3: // |2,4: skip - | @(s2 … IHB … ◊) // /2 width=2/ - | #H @(cp3 … H1RP … 0) @(s1 … IHA) // - ] -| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H - elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct - lapply (s1 … IHB … HB) #HV0 - @(s2 … IHA … (V0 @ V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/ -| #a #L #Vs #U #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H - elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct - elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct - elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct - @(s3 … IHA … (V0 @ V0s)) /2 width=6 by rp_lifts/ /4 width=5/ -| #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H - elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct - elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct - elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0 - >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02 - elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct - elim (lift_total W1 0 (i0 + 1)) #W2 #HW12 - elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2 - >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2 - @(s4 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=4/ -| #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H - elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct - elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct - elim (lift_total V10 0 1) #V20 #HV120 - elim (liftv_total 0 1 V10s) #V20s #HV120s - @(s5 … IHA … (V10 @ V10s) (V20 @ V20s)) /2 width=1/ /2 width=6 by rp_lifts/ - @(HA … (des + 1)) /2 width=1/ - [ @(s7 … IHB … HB … HV120) /2 width=1/ - | @lifts_applv // - elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s - >(liftv_mono … HV12s … HV10s) -V1s // - ] -| #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H - elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct - elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct - @(s6 … IHA … (V0 @ V0s)) /2 width=6 by rp_lifts/ /3 width=4/ -| /3 width=7/ -] -qed. - -lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀a,L,W,T,A,B. RP L W → ( - ∀L0,V0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des + 1] T ≡ T0 → - ⦃L0, V0⦄ ϵ[RP] 〚B〛 → ⦃L0. ⓓV0, T0⦄ ϵ[RP] 〚A〛 - ) → - ⦃L, ⓛ{a}W. T⦄ ϵ[RP] 〚②B. A〛. -#RR #RS #RP #H1RP #H2RP #a #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H -lapply (aacr_acr … H1RP H2RP A) #HCA -lapply (aacr_acr … H1RP H2RP B) #HCB -elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct -lapply (s1 … HCB) -HCB #HCB -@(s3 … HCA … ◊) /2 width=6 by rp_lifts/ -@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/ -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/lambda_delta/basic_2/computation/cpe.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cpe.ma deleted file mode 100644 index 285e6e4fc..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cpe.ma +++ /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/computation/cprs.ma". -include "basic_2/computation/csn.ma". - -(* CONTEXT-SENSITIVE PARALLEL EVALUATION ON TERMS **************************) - -definition cpe: lenv → relation term ≝ - λL,T1,T2. L ⊢ T1 ➡* T2 ∧ L ⊢ 𝐍⦃T2⦄. - -interpretation "context-sensitive parallel evaluation (term)" - 'PEval L T1 T2 = (cpe L T1 T2). - -(* Basic_properties *********************************************************) - -(* Basic_1: was: nf2_sn3 *) -lemma cpe_csn: ∀L,T1. L ⊢ ⬊* T1 → ∃T2. L ⊢ T1 ➡* 𝐍⦃T2⦄. -#L #T1 #H @(csn_ind … H) -T1 -#T1 #_ #IHT1 -elim (cnf_dec L T1) /3 width=3/ -* #T #H1T1 #H2T1 -elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1 #T2 * /4 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cpe_cpe.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cpe_cpe.ma deleted file mode 100644 index ec770787b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cpe_cpe.ma +++ /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/computation/cprs_cprs.ma". -include "basic_2/computation/cpe.ma". - -(* CONTEXT-SENSITIVE PARALLEL EVALUATION ON TERMS **************************) - -(* Main properties *********************************************************) - -(* Basic_1: was: nf2_pr3_confluence *) -theorem cpe_mono: ∀L,T,T1. L ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. L ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2. -#L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2 -elim (cprs_conf … H1T1 … H1T2) -T #T #HT1 ->(cprs_inv_cnf1 … HT1 H2T1) -T1 #HT2 ->(cprs_inv_cnf1 … HT2 H2T2) -T2 // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs.ma deleted file mode 100644 index ae0c1ae62..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs.ma +++ /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 *) -(* *) -(**************************************************************************) - -include "basic_2/reducibility/cnf.ma". -include "basic_2/computation/tprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Basic_1: includes: pr3_pr2 *) -definition cprs: lenv → relation term ≝ - λL. TC … (cpr L). - -interpretation "context-sensitive parallel computation (term)" - 'PRedStar L T1 T2 = (cprs L T1 T2). - -(* Basic eliminators ********************************************************) - -lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 → - (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) → - ∀T2. L ⊢ T1 ➡* T2 → R T2. -#L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind … HT1 IHT1 … HT12) // -qed-. - -lemma cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 → - (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) → - ∀T1. L ⊢ T1 ➡* T2 → R T1. -#L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx … HT2 IHT2 … HT12) // -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: pr3_refl *) -lemma cprs_refl: ∀L,T. L ⊢ T ➡* T. -/2 width=1/ qed. - -lemma cprs_strap1: ∀L,T1,T,T2. - L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2. -/2 width=3/ qed. - -(* Basic_1: was: pr3_step *) -lemma cprs_strap2: ∀L,T1,T,T2. - L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2. -/2 width=3/ qed. - -(* Note: it does not hold replacing |L1| with |L2| *) -lemma cprs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 → - ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ➡* T2. -/3 width=3/ -qed. - -(* Basic_1: was only: pr3_thin_dx *) -lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 → - L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. -#I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/ -#T #T2 #_ #HT2 #IHT2 -@(cprs_strap1 … IHT2) -IHT2 /2 width=1/ -qed. - -(* Basic_1: was: pr3_pr1 *) -lemma tprs_cprs: ∀T1,T2. T1 ➡* T2 → ∀L. L ⊢ T1 ➡* T2. -#T1 #T2 #H @(tprs_ind … H) -T2 /2 width=1/ /3 width=3/ -qed. - -(* Basic inversion lemmas ***************************************************) - -(* Basic_1: was: pr3_gen_sort *) -lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k. -#L #U2 #k #H @(cprs_ind … H) -U2 // -#U2 #U #_ #HU2 #IHU2 destruct ->(cpr_inv_sort1 … HU2) -HU2 // -qed-. - -(* Basic_1: was: pr3_gen_cast *) -lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓝW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨ - ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓝW2.T2. -#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ -#U2 #U #_ #HU2 * /3 width=3/ * -#W #T #HW1 #HT1 #H destruct -elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ * -#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/ -qed-. - -(* Basic_1: was: nf2_pr3_unfold *) -lemma cprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U. -#L #T #U #H @(cprs_ind_dx … H) -T // -#T0 #T #H1T0 #_ #IHT #H2T0 -lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/ -qed-. - -lemma tprs_inv_cnf1: ∀T,U. T ➡* U → ⋆ ⊢ 𝐍⦃T⦄ → T = U. -/3 width=3 by tprs_cprs, cprs_inv_cnf1/ qed-. - -(* Basic_1: removed theorems 10: - clear_pr3_trans pr3_cflat pr3_gen_bind - pr3_head_1 pr3_head_2 pr3_head_21 pr3_head_12 - pr3_iso_appl_bind pr3_iso_appls_appl_bind pr3_iso_appls_bind -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_aaa.ma deleted file mode 100644 index e04e3c784..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_aaa.ma +++ /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/reducibility/cpr_aaa.ma". -include "basic_2/computation/cprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Properties about atomic arity assignment on terms ************************) - -lemma aaa_cprs_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ T2 ⁝ A. -#L #T1 #A #HT1 #T2 #HT12 -@(TC_Conf3 … HT1 ? HT12) /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma deleted file mode 100644 index 8f94b0fa7..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma +++ /dev/null @@ -1,150 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_cpr.ma". -include "basic_2/reducibility/lfpr_cpr.ma". -include "basic_2/computation/cprs_lfpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Advanced properties ******************************************************) - -lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. - L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2. -#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HT12) -T2 -[ /3 width=2/ -| /3 width=6 by cprs_strap1, cpr_abst/ (**) (* /3 width=6/ is too slow *) -] -qed. - -lemma cprs_abbr1_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 -[ /3 width=5/ -| #T1 #T #HT1 #_ #IHT1 - @(cprs_strap2 … IHT1) -IHT1 /2 width=1/ -] -qed. - -lemma cpr_abbr1: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -/3 width=1/ qed. - -lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 -lapply (lfpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/ -qed. - -(* Basic_1: was: pr3_strip *) -lemma cprs_strip: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡ T2 → - ∃∃T0. L ⊢ T1 ➡ T0 & L ⊢ T2 ➡* T0. -/3 width=3/ qed. - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was pr3_gen_appl *) -lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 → - ∨∨ ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 & - U2 = ⓐV2. T2 - | ∃∃a,V2,W,T. L ⊢ V1 ➡* V2 & - L ⊢ T1 ➡* ⓛ{a}W. T & L ⊢ ⓓ{a}V2. T ➡* U2 - | ∃∃a,V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 & - L ⊢ T1 ➡* ⓓ{a}V. T & L ⊢ ⓓ{a}V. ⓐV2. T ➡* U2. -#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ -#U #U2 #_ #HU2 * * -[ #V0 #T0 #HV10 #HT10 #H destruct - elim (cpr_inv_appl1 … HU2) -HU2 * - [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ - | #a #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/ - | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct - @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *) - ] -| /4 width=9/ -| /4 width=11/ -] -qed-. - -(* Main propertis ***********************************************************) - -(* Basic_1: was: pr3_confluence *) -theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 → - ∃∃T0. L ⊢ T1 ➡* T0 & L ⊢ T2 ➡* T0. -/3 width=3/ qed. - -(* Basic_1: was: pr3_t *) -theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2. -/2 width=3/ qed. - -(* Basic_1: was: pr3_flat *) -lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 → - L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. -#I #L #T1 #T2 #HT12 #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 -@(cprs_trans … IHV1) -IHV1 /2 width=1/ -qed. - -lemma cprs_abst: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2. - L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2. -#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HV12) -V2 -[ lapply (cprs_lsubs_trans … HT12 (L.ⓛV1) ?) -HT12 /2 width=2/ -| #V0 #V2 #_ #HV02 #IHV01 - @(cprs_trans … IHV01) -V1 /2 width=2/ -] -qed. - -lemma cprs_abbr1: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 → - ∀a.L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #T1 #T2 #HT12 #V2 #HV12 #a @(cprs_ind … HV12) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 -@(cprs_trans … IHV1) -IHV1 /2 width=1/ -qed. - -lemma cprs_abbr2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 -[ /2 width=1/ -| #T1 #T #HT1 #_ #IHT1 - lapply (lfpr_cpr_trans (L. ⓓV1) … HT1) -HT1 /2 width=1/ #HT1 - @(cprs_trans … IHT1) -IHT1 /2 width=1/ -] -qed. - -lemma cprs_abbr2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 #a -lapply (IHV1 T1 T1 ? a) -IHV1 // #HV1 -@(cprs_trans … HV1) -HV1 /2 width=1/ -qed. - -lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2. - L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 → - ∀a.L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2. -#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #a @(cprs_ind … HT12) -T2 -[ /3 width=1/ -| -HV12 #T #T2 #_ #HT2 #IHT1 - lapply (cpr_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 - @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/ -] -qed. - -(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *) -lemma lcpr_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT2 -@(cprs_trans … IHT2) /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_delift.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_delift.ma deleted file mode 100644 index 6b7892611..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_delift.ma +++ /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: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 → L ⊢ +ⓓV.T1 ➡* T2. -#L #V #T1 #T2 * #T #HT1 #HT2 -@(cprs_strap2 … (+ⓓV.T)) [ /3 width=3/ | @inj /3 width=3/ ] (**) (* explicit constructor, /5 width=3/ is too slow *) -qed. - -(* Basic_1: was only: pr3_gen_cabbr *) -lemma thin_cprs_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡* U2 → - ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 → - ∃∃T2. K ⊢ T1 ➡* T2 & L ⊢ ▼*[d, e] U2 ≡ T2. -#L #U1 #U2 #H @(cprs_ind … H) -U2 /2 width=3/ -#U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1 -elim (IHU1 … HLK … HTU1) -U1 #T #HT1 #HUT -elim (thin_cpr_delift_conf … HU2 … HLK … HUT) -U -HLK /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lfpr.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lfpr.ma deleted file mode 100644 index a06643577..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lfpr.ma +++ /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/reducibility/ltpr_tps.ma". -include "basic_2/reducibility/cpr_ltpss.ma". -include "basic_2/reducibility/lfpr.ma". -include "basic_2/computation/cprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Properties concerning focalized parallel reduction on local environments *) - -lemma ltpr_tpss_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2,d,e. L2 ⊢ T1 ▶* [d, e] T2 → - ∃∃T. L1 ⊢ T1 ▶* [d, e] T & L1 ⊢ T ➡* T2. -#L1 #L2 #HL12 #T1 #T2 #d #e #H @(tpss_ind … H) -T2 -[ /2 width=3/ -| #T #T2 #_ #HT2 * #T0 #HT10 #HT0 - elim (ltpr_tps_trans … HT2 … HL12) -L2 #T3 #HT3 #HT32 - @(ex2_1_intro … HT10) -T1 (**) (* explicit constructors *) - @(cprs_strap1 … T3 …) /2 width=1/ -HT32 - @(cprs_strap1 … HT0) -HT0 /3 width=3/ -] -qed. - -(* Basic_1: was just: pr3_pr0_pr2_t *) -lemma ltpr_cpr_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2. -#L1 #L2 #HL12 #T1 #T2 * #T #HT1 -<(ltpr_fwd_length … HL12) #HT2 -elim (ltpr_tpss_trans … HL12 … HT2) -L2 /3 width=3/ -qed. - -(* Basic_1: was just: pr3_pr2_pr2_t *) -lemma lfpr_cpr_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2. -#L1 #L2 * /3 width=7/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lfprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lfprs.ma deleted file mode 100644 index 33620d01c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lfprs.ma +++ /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/computation/cprs_cprs.ma". -include "basic_2/computation/lfprs_lfprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Properties on focalized computation for local environments ***************) - -(* Basic_1: was just: pr3_pr3_pr3_t *) -lemma lfprs_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -#L1 #L2 #HL12 @(lfprs_ind … HL12) -L2 // /3 width=3/ -qed. - -lemma lfprs_cpr_trans: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → - ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2. -/3 width=3 by lfprs_cprs_trans, inj/ qed. - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was pr3_gen_abbr *) -lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡* U2 → - (∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 & - U2 = ⓓ{a}V2. T2 - ) ∨ - ∃∃T2. L. ⓓV1 ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true. -#a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ -#U0 #U2 #_ #HU02 * * -[ #V0 #T0 #HV10 #HT10 #H destruct - elim (cpr_inv_abbr1 … HU02) -HU02 * - [ #V #V2 #T2 #HV0 #HV2 #HT02 #H destruct - lapply (cpr_intro … HV0 … HV2) -HV2 #HV02 - lapply (ltpr_cpr_trans (L.ⓓV0) … HT02) /2 width=1/ -V #HT02 - lapply (lfprs_cprs_trans (L. ⓓV1) … HT02) -HT02 /2 width=1/ /4 width=5/ - | #T2 #HT02 #HUT2 - lapply (lfprs_cpr_trans (L.ⓓV1) … HT02) -HT02 /2 width=1/ -V0 #HT02 - lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3/ - ] -| #U1 #HTU1 #HU01 - elim (lift_total U2 0 1) #U #HU2 - lapply (cpr_lift (L.ⓓV1) … HU01 … HU2 HU02) -U0 /2 width=1/ /4 width=3/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lift.ma deleted file mode 100644 index 36ce0ef83..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lift.ma +++ /dev/null @@ -1,78 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/computation/cprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was: pr3_gen_lref *) -lemma cprs_inv_lref1: ∀L,T2,i. L ⊢ #i ➡* T2 → - T2 = #i ∨ - ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 & - K ⊢ V1 ➡* T1 & - ⇧[0, i + 1] T1 ≡ T2 & - i < |L|. -#L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1/ -#T #T2 #_ #HT2 * -[ #H destruct - elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1/ - * #K #V1 #T1 #HLK #HVT1 #HT12 #Hi - @or_intror @(ex4_3_intro … HLK … HT12) // /3 width=3/ (**) (* explicit constructors *) -| * #K #V1 #T1 #HLK #HVT1 #HT1 #Hi - lapply (ldrop_fwd_ldrop2 … HLK) #H0LK - elim (cpr_inv_lift1 … H0LK … HT1 … HT2) -H0LK -T /4 width=6/ -] -qed-. - -(* Basic_1: was: pr3_gen_abst *) -lemma cprs_inv_abst1: ∀I,W,a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡* U2 → - ∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓑ{I} W ⊢ T1 ➡* T2 & - U2 = ⓛ{a}V2. T2. -#I #W #a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/ -#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -elim (cpr_inv_abst1 … HU2 I W) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ -qed-. - -lemma cprs_inv_abst: ∀a,L,V1,V2,T1,T2. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 → ∀I,W. - L ⊢ V1 ➡* V2 ∧ L. ⓑ{I} W ⊢ T1 ➡* T2. -#a #L #V1 #V2 #T1 #T2 #H #I #W -elim (cprs_inv_abst1 I W … H) -H #V #T #HV1 #HT1 #H destruct /2 width=1/ -qed-. - -(* Relocation properties ****************************************************) - -(* Basic_1: was: pr3_lift *) -lemma cprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 → - ∀T2. K ⊢ T1 ➡* T2 → ∀U2. ⇧[d, e] T2 ≡ U2 → - L ⊢ U1 ➡* U2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #HT12 @(cprs_ind … HT12) -T2 -[ -HLK #T2 #HT12 - <(lift_mono … HTU1 … HT12) -T1 // -| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (cpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/ -] -qed. - -(* Basic_1: was: pr3_gen_lift *) -lemma cprs_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K → - ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡* U2 → - ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡* T2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 #HU12 @(cprs_ind … HU12) -U2 /2 width=3/ --HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 -elim (cpr_inv_lift1 … HLK … HTU … HU2) -U -HLK /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_ltpr.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_ltpr.ma deleted file mode 100644 index 2682a7609..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_ltpr.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_2/reducibility/cpr_ltpr.ma". -include "basic_2/computation/cprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Properties concerning parallel unfold on terms ***************************) - -(* Basic_1: was only: pr3_subst1 *) -lemma cprs_tpss_ltpr: ∀L1,T1,U1,d,e. L1 ⊢ T1 ▶* [d, e] U1 → - ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → - ∃∃U2. L2 ⊢ U1 ➡* U2 & L2 ⊢ T2 ▶* [d, e] U2. -#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2 -[ #T2 #HT12 - elim (cpr_tpss_ltpr … HL12 … HT12 … HTU1) -L1 -T1 /3 width=3/ -| #T #T2 #_ #HT2 * #U #HU1 #HTU - elim (cpr_tpss_ltpr … HT2 … HTU) -L1 -T // /3 width=3/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_tstc.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_tstc.ma deleted file mode 100644 index f7afb8df7..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_tstc.ma +++ /dev/null @@ -1,92 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/tstc.ma". -include "basic_2/computation/cprs_lift.ma". -include "basic_2/computation/cprs_lfprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Forward lemmas involving same top term constructor ***********************) - -lemma cprs_fwd_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → ∀U. L ⊢ T ➡* U → T ≃ U. -#L #T #HT #U #H ->(cprs_inv_cnf1 … H HT) -L -T // -qed-. - -(* Basic_1: was: pr3_iso_beta *) -lemma cprs_fwd_beta: ∀a,L,V,W,T,U. L ⊢ ⓐV. ⓛ{a}W. T ➡* U → - ⓐV. ⓛ{a}W. T ≃ U ∨ L ⊢ ⓓ{a}V. T ➡* U. -#a #L #V #W #T #U #H -elim (cprs_inv_appl1 … H) -H * -[ #V0 #T0 #_ #_ #H destruct /2 width=1/ -| #b #V0 #W0 #T0 #HV0 #HT0 #HU - elim (cprs_inv_abst1 Abbr V … HT0) -HT0 #W1 #T1 #_ #HT1 #H destruct -W1 - @or_intror -W - @(cprs_trans … HU) -U /2 width=1/ (**) (* explicit constructor *) -| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_ - elim (cprs_inv_abst1 Abbr V … HT1) -HT1 #W2 #T2 #_ #_ #H destruct -] -qed-. - -(* Note: probably this is an inversion lemma *) -lemma cprs_fwd_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 → - ∀V2. ⇧[0, i + 1] V1 ≡ V2 → - ∀U. L ⊢ #i ➡* U → - #i ≃ U ∨ L ⊢ V2 ➡* U. -#L #K #V1 #i #HLK #V2 #HV12 #U #H -elim (cprs_inv_lref1 … H) -H /2 width=1/ -* #K0 #V0 #U0 #HLK0 #HVU0 #HU0 #_ -lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct -lapply (ldrop_fwd_ldrop2 … HLK) -HLK /3 width=9/ -qed-. - -lemma cprs_fwd_theta: ∀a,L,V1,V,T,U. L ⊢ ⓐV1. ⓓ{a}V. T ➡* U → - ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1. ⓓ{a}V. T ≃ U ∨ - L ⊢ ⓓ{a}V. ⓐV2. T ➡* U. -#a #L #V1 #V #T #U #H #V2 #HV12 -elim (cprs_inv_appl1 … H) -H * -[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1/ -| #b #V0 #W #T0 #HV10 #HT0 #HU - elim (cprs_inv_abbr1 … HT0) -HT0 * - [ #V3 #T3 #_ #_ #H destruct - | #X #HT2 #H #H0 destruct - elim (lift_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct - @or_intror @(cprs_trans … HU) -U (**) (* explicit constructor *) - @(cprs_trans … (+ⓓV.ⓐV2.ⓛ{b}W2.T2)) [ /3 width=1/ ] -T - @(cprs_strap2 … (ⓐV1.ⓛ{b}W.T0)) [ /5 width=7/ ] -V -V2 -W2 -T2 - @(cprs_strap2 … (ⓓ{b}V1.T0)) [ /3 width=1/ ] -W /2 width=1/ - ] -| #b #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU - @or_intror @(cprs_trans … HU) -U (**) (* explicit constructor *) - elim (cprs_inv_abbr1 … HT0) -HT0 * - [ #V5 #T5 #HV5 #HT5 #H destruct - lapply (cprs_lift (L.ⓓV) … HV12 … HV13 … HV34) -V1 -V3 /2 width=1/ - /3 width=1/ - | #X #HT1 #H #H0 destruct - elim (lift_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct - lapply (cprs_lift (L.ⓓV0) … HV12 … HV13 … HV34) -V3 /2 width=1/ #HV24 - @(cprs_trans … (+ⓓV.ⓐV2.ⓓ{b}V5.T5)) [ /3 width=1/ ] -T - @(cprs_strap2 … (ⓐV1.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V5 -T5 - @(cprs_strap2 … (ⓓ{b}V0.ⓐV2.T0)) [ /3 width=3/ ] -V1 /3 width=1/ - ] -] -qed-. - -lemma cprs_fwd_tau: ∀L,W,T,U. L ⊢ ⓝW. T ➡* U → - ⓝW. T ≃ U ∨ L ⊢ T ➡* U. -#L #W #T #U #H -elim (cprs_inv_cast1 … H) -H /2 width=1/ * -#W0 #T0 #_ #_ #H destruct /2 width=1/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_tstc_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_tstc_vector.ma deleted file mode 100644 index fd3eb585e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_tstc_vector.ma +++ /dev/null @@ -1,157 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/tstc_vector.ma". -include "basic_2/substitution/lift_vector.ma". -include "basic_2/computation/cprs_tstc.ma". - -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Vector form of forward lemmas involving same top term constructor ********) - -(* Basic_1: was just: nf2_iso_appls_lref *) -lemma cprs_fwd_cnf_vector: ∀L,T. 𝐒⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ∀Vs,U. L ⊢ ⒶVs.T ➡* U → ⒶVs.T ≃ U. -#L #T #H1T #H2T #Vs elim Vs -Vs [ @(cprs_fwd_cnf … H2T) ] (**) (* /2 width=3 by cprs_fwd_cnf/ does not work *) -#V #Vs #IHVs #U #H -elim (cprs_inv_appl1 … H) -H * -[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/ -| #a #V0 #W0 #T0 #HV0 #HT0 #HU - lapply (IHVs … HT0) -IHVs -HT0 #HT0 - elim (tstc_inv_bind_appls_simple … HT0 ?) // -| #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU - lapply (IHVs … HT0) -IHVs -HT0 #HT0 - elim (tstc_inv_bind_appls_simple … HT0 ?) // -] -qed-. - -(* Basic_1: was: pr3_iso_appls_beta *) -lemma cprs_fwd_beta_vector: ∀a,L,Vs,V,W,T,U. L ⊢ ⒶVs. ⓐV. ⓛ{a}W. T ➡* U → - ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ L ⊢ ⒶVs. ⓓ{a}V. T ➡* U. -#a #L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_beta/ -#V0 #Vs #IHVs #V #W #T #U #H -elim (cprs_inv_appl1 … H) -H * -[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/ -| #b #V1 #W1 #T1 #HV01 #HT1 #HU - elim (IHVs … HT1) -IHVs -HT1 #HT1 - [ elim (tstc_inv_bind_appls_simple … HT1 ?) // - | @or_intror -W (**) (* explicit constructor *) - @(cprs_trans … HU) -U - @(cprs_strap1 … (ⓐV1.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=1/ - ] -| #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU - elim (IHVs … HT1) -IHVs -HT1 #HT1 - [ elim (tstc_inv_bind_appls_simple … HT1 ?) // - | @or_intror -W (**) (* explicit constructor *) - @(cprs_trans … HU) -U - @(cprs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=3/ - ] -] -qed-. - -lemma cprs_fwd_delta_vector: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 → - ∀V2. ⇧[0, i + 1] V1 ≡ V2 → - ∀Vs,U. L ⊢ ⒶVs.#i ➡* U → - ⒶVs.#i ≃ U ∨ L ⊢ ⒶVs.V2 ➡* U. -#L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=4 by cprs_fwd_delta/ -#V #Vs #IHVs #U #H -K -V1 -elim (cprs_inv_appl1 … H) -H * -[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/ -| #b #V0 #W0 #T0 #HV0 #HT0 #HU - elim (IHVs … HT0) -IHVs -HT0 #HT0 - [ elim (tstc_inv_bind_appls_simple … HT0 ?) // - | @or_intror -i (**) (* explicit constructor *) - @(cprs_trans … HU) -U - @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=1/ - ] -| #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU - elim (IHVs … HT0) -IHVs -HT0 #HT0 - [ elim (tstc_inv_bind_appls_simple … HT0 ?) // - | @or_intror -i (**) (* explicit constructor *) - @(cprs_trans … HU) -U - @(cprs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=3/ - ] -] -qed-. - -(* Basic_1: was: pr3_iso_appls_abbr *) -lemma cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s → - ∀a,V,T,U. L ⊢ ⒶV1s. ⓓ{a}V. T ➡* U → - ⒶV1s. ⓓ{a}V. T ≃ U ∨ L ⊢ ⓓ{a}V. ⒶV2s. T ➡* U. -#L #V1s #V2s * -V1s -V2s /3 width=1/ -#V1s #V2s #V1a #V2a #HV12a #HV12s #a -generalize in match HV12a; -HV12a -generalize in match V2a; -V2a -generalize in match V1a; -V1a -elim HV12s -V1s -V2s /2 width=1 by cprs_fwd_theta/ -#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H -elim (cprs_inv_appl1 … H) -H * -[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1/ -| #b #V0a #W0 #T0 #HV10a #HT0 #HU - elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0 - [ -HV12a -HV12b -HV10a -HU - elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct - | @or_intror -V1s (**) (* explicit constructor *) - @(cprs_trans … HU) -U - elim (cprs_inv_abbr1 … HT0) -HT0 * - [ -HV12a -HV12b -HV10a #V1 #T1 #_ #_ #H destruct - | -V1b #X #HT1 #H #H0 destruct - elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct - @(cprs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1/ ] -T -V2b -V2s - @(cprs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [ /5 width=7/ ] -V -V2a -W1 -T1 - @(cprs_strap2 … (ⓓ{b}V1a.T0)) [ /3 width=1/ ] -W0 /2 width=1/ - ] - ] -| #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU - elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0 - [ -HV12a -HV10a -HV0a -HU - elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct - | @or_intror -V1s -V1b (**) (* explicit constructor *) - @(cprs_trans … HU) -U - elim (cprs_inv_abbr1 … HT0) -HT0 * - [ #V1 #T1 #HV1 #HT1 #H destruct - lapply (cprs_lift (L.ⓓV) … HV12a … HV10a … HV0a) -V1a -V0a [ /2 width=1/ ] #HV2a - @(cprs_trans … (ⓓ{a}V.ⓐV2a.T1)) [ /3 width=1/ ] -T -V2b -V2s /3 width=1/ - | #X #HT1 #H #H0 destruct - elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct - lapply (cprs_lift (L.ⓓV0) … HV12a … HV10a … HV0a) -V0a [ /2 width=1/ ] #HV2a - @(cprs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1/ ] -T -V2b -V2s - @(cprs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V1 -T1 - @(cprs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) [ /3 width=3/ ] -V1a /3 width=1/ - ] - ] -] -qed-. - -(* Basic_1: was: pr3_iso_appls_cast *) -lemma cprs_fwd_tau_vector: ∀L,Vs,W,T,U. L ⊢ ⒶVs. ⓝW. T ➡* U → - ⒶVs. ⓝW. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U. -#L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_tau/ -#V #Vs #IHVs #W #T #U #H -elim (cprs_inv_appl1 … H) -H * -[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/ -| #b #V0 #W0 #T0 #HV0 #HT0 #HU - elim (IHVs … HT0) -IHVs -HT0 #HT0 - [ elim (tstc_inv_bind_appls_simple … HT0 ?) // - | @or_intror -W (**) (* explicit constructor *) - @(cprs_trans … HU) -U - @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=1/ - ] -| #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU - elim (IHVs … HT0) -IHVs -HT0 #HT0 - [ elim (tstc_inv_bind_appls_simple … HT0 ?) // - | @or_intror -W (**) (* explicit constructor *) - @(cprs_trans … HU) -U - @(cprs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=3/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn.ma deleted file mode 100644 index 3ed310164..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn.ma +++ /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/reducibility/cnf.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) - -definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …). - -interpretation - "context-sensitive strong normalization (term)" - 'SN L T = (csn L T). - -(* Basic eliminators ********************************************************) - -lemma csn_ind: ∀L. ∀R:predicate term. - (∀T1. L ⊢ ⬊* T1 → - (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → R T2) → - R T1 - ) → - ∀T. L ⊢ ⬊* T → R T. -#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1 -@H0 -H0 /3 width=1/ -IHT1 /4 width=1/ -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: sn3_pr2_intro *) -lemma csn_intro: ∀L,T1. - (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊* T2) → L ⊢ ⬊* T1. -/4 width=1/ qed. - -(* Basic_1: was: sn3_nf2 *) -lemma csn_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ ⬊* T. -/2 width=1/ qed. - -lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬊* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬊* T2. -#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12 -@csn_intro #T #HLT2 #HT2 -elim (term_eq_dec T1 T2) #HT12 -[ -IHT1 -HLT12 destruct /3 width=1/ -| -HT1 -HT2 /3 width=4/ -qed. - -(* Basic_1: was: sn3_cast *) -lemma csn_cast: ∀L,W. L ⊢ ⬊* W → ∀T. L ⊢ ⬊* T → L ⊢ ⬊* ⓝW. T. -#L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT -@csn_intro #X #H1 #H2 -elim (cpr_inv_cast1 … H1) -H1 -[ * #W0 #T0 #HLW0 #HLT0 #H destruct - elim (eq_false_inv_tpair_sn … H2) -H2 - [ /3 width=3/ - | -HLW0 * #H destruct /3 width=1/ - ] -| /3 width=3/ -] -qed. - -(* Basic forward lemmas *****************************************************) - -fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬊* T. -#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct -@csn_intro #T2 #HLT2 #HT2 -@(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/ -qed. - -(* Basic_1: was: sn3_gen_flat *) -lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬊* ⓕ{I} V. T → L ⊢ ⬊* T. -/2 width=5/ qed-. - -(* Basic_1: removed theorems 14: - sn3_cdelta - sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change - sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr - sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_aaa.ma deleted file mode 100644 index 67744a098..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_aaa.ma +++ /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/acp_aaa.ma". -include "basic_2/computation/csn_tstc_vector.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) - -(* Properties concerning atomic arity assignment ****************************) - -lemma csn_aaa: ∀L,T,A. L ⊢ T ⁝ A → L ⊢ ⬊* T. -#L #T #A #H -@(acp_aaa … csn_acp csn_acr … H) -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_alt.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_alt.ma deleted file mode 100644 index eeba707dc..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_alt.ma +++ /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/computation/cprs.ma". -include "basic_2/computation/csn.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) - -(* alternative definition of csn *) -definition csna: lenv → predicate term ≝ λL. SN … (cprs L) (eq …). - -interpretation - "context-sensitive strong normalization (term) alternative" - 'SNAlt L T = (csna L T). - -(* Basic eliminators ********************************************************) - -lemma csna_ind: ∀L. ∀R:predicate term. - (∀T1. L ⊢ ⬊⬊* T1 → - (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → R T2) → R T1 - ) → - ∀T. L ⊢ ⬊⬊* T → R T. -#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1 -@H0 -H0 /3 width=1/ -IHT1 /4 width=1/ -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: sn3_intro *) -lemma csna_intro: ∀L,T1. - (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → L ⊢ ⬊⬊* T2) → L ⊢ ⬊⬊* T1. -/4 width=1/ qed. - -fact csna_intro_aux: ∀L,T1. - (∀T,T2. L ⊢ T ➡* T2 → T1 = T → (T1 = T2 → ⊥) → L ⊢ ⬊⬊* T2) → L ⊢ ⬊⬊* T1. -/4 width=3/ qed-. - -(* Basic_1: was: sn3_pr3_trans (old version) *) -lemma csna_cprs_trans: ∀L,T1. L ⊢ ⬊⬊* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬊⬊* T2. -#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12 -@csna_intro #T #HLT2 #HT2 -elim (term_eq_dec T1 T2) #HT12 -[ -IHT1 -HLT12 destruct /3 width=1/ -| -HT1 -HT2 /3 width=4/ -qed. - -(* Basic_1: was: sn3_pr2_intro (old version) *) -lemma csna_intro_cpr: ∀L,T1. - (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊⬊* T2) → - L ⊢ ⬊⬊* T1. -#L #T1 #H -@csna_intro_aux #T #T2 #H @(cprs_ind_dx … H) -T -[ -H #H destruct #H - elim (H ?) // -| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct - elim (term_eq_dec T0 T) #HT0 - [ -HLT1 -HLT2 -H /3 width=1/ - | -IHT -HT12 /4 width=3/ - ] -] -qed. - -(* Main properties **********************************************************) - -theorem csn_csna: ∀L,T. L ⊢ ⬊* T → L ⊢ ⬊⬊* T. -#L #T #H @(csn_ind … H) -T /4 width=1/ -qed. - -theorem csna_csn: ∀L,T. L ⊢ ⬊⬊* T → L ⊢ ⬊* T. -#L #T #H @(csna_ind … H) -T /4 width=1/ -qed. - -(* Basic_1: was: sn3_pr3_trans *) -lemma csn_cprs_trans: ∀L,T1. L ⊢ ⬊* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬊* T2. -/4 width=3/ qed. - -(* Main eliminators *********************************************************) - -lemma csn_ind_alt: ∀L. ∀R:predicate term. - (∀T1. L ⊢ ⬊* T1 → - (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → R T2) → R T1 - ) → - ∀T. L ⊢ ⬊* T → R T. -#L #R #H0 #T1 #H @(csna_ind … (csn_csna … H)) -T1 #T1 #HT1 #IHT1 -@H0 -H0 /2 width=1/ -HT1 /3 width=1/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_cpr.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_cpr.ma deleted file mode 100644 index ccfd6015b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_cpr.ma +++ /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/cpr_cpr.ma". -include "basic_2/computation/csn.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) - -(* Advanced forvard lemmas **************************************************) - -fact csn_fwd_pair_sn_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ②{I} V. T → L ⊢ ⬊* V. -#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct -@csn_intro #V2 #HLV2 #HV2 -@(IH (②{I} V2. T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/ -qed. - -(* Basic_1: was: sn3_gen_head *) -lemma csn_fwd_pair_sn: ∀I,L,V,T. L ⊢ ⬊* ②{I} V. T → L ⊢ ⬊* V. -/2 width=5/ qed. - -fact csn_fwd_bind_dx_aux: ∀L,U. L ⊢ ⬊* U → - ∀a,I,V,T. U = ⓑ{a,I} V. T → L. ⓑ{I} V ⊢ ⬊* T. -#L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct -@csn_intro #T2 #HLT2 #HT2 -@(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/ -qed. - -(* Basic_1: was: sn3_gen_bind *) -lemma csn_fwd_bind_dx: ∀a,I,L,V,T. L ⊢ ⬊* ⓑ{a,I} V. T → L. ⓑ{I} V ⊢ ⬊* T. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_cpr_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_cpr_vector.ma deleted file mode 100644 index 70c00eb11..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_cpr_vector.ma +++ /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/csn_cpr.ma". -include "basic_2/computation/csn_vector.ma". - -(* Advanced forward lemmas **************************************************) - -lemma csn_fwd_applv: ∀L,T,Vs. L ⊢ ⬊* Ⓐ Vs. T → L ⊢ ⬊* Vs ∧ L ⊢ ⬊* T. -#L #T #Vs elim Vs -Vs /2 width=1/ -#V #Vs #IHVs #HVs -lapply (csn_fwd_pair_sn … HVs) #HV -lapply (csn_fwd_flat_dx … HVs) -HVs #HVs -elim (IHVs HVs) -IHVs -HVs /3 width=1/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lfpr.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lfpr.ma deleted file mode 100644 index 444dcf8f3..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lfpr.ma +++ /dev/null @@ -1,147 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/tstc_tstc.ma". -include "basic_2/computation/cprs_cprs.ma". -include "basic_2/computation/csn_lift.ma". -include "basic_2/computation/csn_cpr.ma". -include "basic_2/computation/csn_alt.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) - -(* Advanced properties ******************************************************) - -lemma csn_lfpr_conf: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. L1 ⊢ ⬊* T → L2 ⊢ ⬊* T. -#L1 #L2 #HL12 #T #H @(csn_ind_alt … H) -T #T #_ #IHT -@csn_intro #T0 #HLT0 #HT0 -@IHT /2 width=2/ -IHT -HT0 /2 width=3/ -qed. - -lemma csn_abbr: ∀a,L,V. L ⊢ ⬊* V → ∀T. L. ⓓV ⊢ ⬊* T → L ⊢ ⬊* ⓓ{a}V. T. -#a #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_alt … HT) -T #T #HT #IHT -@csn_intro #X #H1 #H2 -elim (cpr_inv_abbr1 … H1) -H1 * -[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct - lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1 - lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1 - elim (eq_false_inv_tpair_sn … H2) -H2 - [ #HV1 @IHV // /2 width=1/ -HV1 - @(csn_lfpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/ - | -IHV -HLV1 * #H destruct /3 width=1/ - ] -| -IHV -IHT -H2 #T0 #HLT0 #HT0 - lapply (csn_cpr_trans … HT … HLT0) -T #HLT0 - lapply (csn_inv_lift … HLT0 … HT0) -T0 /2 width=3/ -] -qed. - -fact csn_appl_beta_aux: ∀a,L,W. L ⊢ ⬊* W → ∀U. L ⊢ ⬊* U → - ∀V,T. U = ⓓ{a}V. T → L ⊢ ⬊* ⓐV. ⓛ{a}W. T. -#a #L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V #T #H destruct -lapply (csn_fwd_pair_sn … HVT) #HV -lapply (csn_fwd_bind_dx … HVT) #HT -HVT -@csn_intro #X #H #H2 -elim (cpr_inv_appl1 … H) -H * -[ #V0 #Y #HLV0 #H #H0 destruct - elim (cpr_inv_abst1 … H Abbr V) -H #W0 #T0 #HLW0 #HLT0 #H destruct - elim (eq_false_inv_beta … H2) -H2 - [ -IHVT #HW0 @IHW -IHW [1,5: // |3: skip ] -HLW0 /2 width=1/ -HW0 - @csn_abbr /2 width=3/ -HV - @(csn_lfpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/ - | -IHW -HLW0 -HV -HT * #H #HVT0 destruct - @(IHVT … HVT0) -IHVT -HVT0 // /2 width=1/ - ] -| -IHW -IHVT -H2 #b #V0 #W0 #T0 #T1 #HLV0 #HLT01 #H1 #H2 destruct - lapply (lfpr_cpr_trans (L. ⓓV) … HLT01) -HLT01 /2 width=1/ #HLT01 - @csn_abbr /2 width=3/ -HV - @(csn_lfpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/ -| -IHW -IHVT -HV -HT -H2 #b #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct -] -qed. - -(* Basic_1: was: sn3_beta *) -lemma csn_appl_beta: ∀a,L,W. L ⊢ ⬊* W → ∀V,T. L ⊢ ⬊* ⓓ{a}V. T → - L ⊢ ⬊* ⓐV. ⓛ{a}W. T. -/2 width=3/ qed. - -fact csn_appl_theta_aux: ∀a,L,U. L ⊢ ⬊* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 → - ∀V,T. U = ⓓ{a}V. ⓐV2. T → L ⊢ ⬊* ⓐV1. ⓓ{a}V. T. -#a #L #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct -lapply (csn_fwd_pair_sn … HVT) #HV -lapply (csn_fwd_bind_dx … HVT) -HVT #HVT -@csn_intro #X #HL #H -elim (cpr_inv_appl1 … HL) -HL * -[ -HV #V0 #Y #HLV10 #HL #H0 destruct - elim (cpr_inv_abbr1 … HL) -HL * - [ #V3 #V4 #T3 #HV3 #HLV34 #HLT3 #H0 destruct - lapply (cpr_intro … HV3 HLV34) -HLV34 #HLV34 - elim (lift_total V0 0 1) #V5 #HV05 - elim (term_eq_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V4.ⓐV5.T3)) - [ -IHVT #H0 destruct - elim (eq_false_inv_tpair_sn … H) -H - [ -HLV10 -HLV34 -HV3 -HLT3 -HVT - >(lift_inj … HV12 … HV05) -V5 - #H elim (H ?) // - | * #_ #H elim (H ?) // - ] - | -H -HVT #H - lapply (cpr_lift (L. ⓓV) … HV12 … HV05 HLV10) -HLV10 -HV12 /2 width=1/ #HV25 - lapply (ltpr_cpr_trans (L. ⓓV) … HLT3) /2 width=1/ -HLT3 #HLT3 - @(IHVT … H … HV05) -IHVT // -H -HV05 /3 width=1/ - ] - | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct - lapply (csn_cpr_trans … HVT (ⓐV2.T0) ?) /2 width=1/ -T #HVT0 - lapply (csn_inv_lift L … 1 HVT0 ? ? ?) -HVT0 [ /2 width=4/ |2,3: skip | /2 width=1/ ] -V2 -T0 #HVY - @(csn_cpr_trans … HVY) /2 width=1/ - ] -| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #T0 #T1 #_ #_ #H destruct -| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HLW01 #HLT01 #HV03 #H1 #H2 destruct - lapply (cpr_lift (L. ⓓW0) … HV12 … HV03 HLV10) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23 - lapply (lfpr_cpr_trans (L. ⓓW0) … HLT01) -HLT01 /2 width=1/ #HLT01 - @csn_abbr /2 width=3/ -HV - @(csn_lfpr_conf (L. ⓓW0)) /2 width=1/ -W1 - @(csn_cprs_trans … HVT) -HVT /2 width=1/ -] -qed. - -lemma csn_appl_theta: ∀a,V1,V2. ⇧[0, 1] V1 ≡ V2 → - ∀L,V,T. L ⊢ ⬊* ⓓ{a}V. ⓐV2. T → L ⊢ ⬊* ⓐV1. ⓓ{a}V. T. -/2 width=5/ qed. - -(* Basic_1: was only: sn3_appl_appl *) -lemma csn_appl_simple_tstc: ∀L,V. L ⊢ ⬊* V → ∀T1. - L ⊢ ⬊* T1 → - (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → ⊥) → L ⊢ ⬊* ⓐV. T2) → - 𝐒⦃T1⦄ → L ⊢ ⬊* ⓐV. T1. -#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1 -@csn_intro #X #HL #H -elim (cpr_inv_appl1_simple … HL ?) -HL // -#V0 #T0 #HLV0 #HLT10 #H0 destruct -elim (eq_false_inv_tpair_sn … H) -H -[ -IHT1 #HV0 - @(csn_cpr_trans … (ⓐV0.T1)) /2 width=1/ -HLT10 - @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0 - #T2 #HLT12 #HT12 - @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0 - @H2T1 -H2T1 // -HLT12 /2 width=1/ -| -IHV -H1T1 -HLV0 * #H #H1T10 destruct - elim (tstc_dec T1 T0) #H2T10 - [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1 - #T2 #HLT02 #HT02 - @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/ - | -IHT1 -H3T1 -H1T10 - @H2T1 -H2T1 /2 width=1/ - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lift.ma deleted file mode 100644 index 4fd784d75..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lift.ma +++ /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/reducibility/cnf_lift.ma". -include "basic_2/computation/acp.ma". -include "basic_2/computation/csn.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) - -(* Relocation properties ****************************************************) - -(* Basic_1: was: sn3_lift *) -lemma csn_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬊* T1 → - ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → L2 ⊢ ⬊* T2. -#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12 -@csn_intro #T #HLT2 #HT2 -elim (cpr_inv_lift1 … HL21 … HT12 … HLT2) -HLT2 #T0 #HT0 #HLT10 -@(IHT1 … HLT10) // -L1 -L2 #H destruct ->(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1/ -qed. - -(* Basic_1: was: sn3_gen_lift *) -lemma csn_inv_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬊* T1 → - ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → L2 ⊢ ⬊* T2. -#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21 -@csn_intro #T #HLT2 #HT2 -elim (lift_total T d e) #T0 #HT0 -lapply (cpr_lift … HL12 … HT21 … HT0 HLT2) -HLT2 #HLT10 -@(IHT1 … HLT10) // -L1 -L2 #H destruct ->(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1/ -qed. - -(* Advanced properties ******************************************************) - -(* Basic_1: was: sn3_abbr *) -lemma csn_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → K ⊢ ⬊* V → L ⊢ ⬊* #i. -#L #K #V #i #HLK #HV -@csn_intro #X #H #Hi -elim (cpr_inv_lref1 … H) -H -[ #H destruct elim (Hi ?) // -| -Hi * #K0 #V0 #V1 #HLK0 #HV01 #HV1 #_ - lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct - lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK - @(csn_lift … HLK HV1) -HLK -HV1 - @(csn_cpr_trans … HV) -HV - @(cpr_intro … HV01) -HV01 // -] -qed. - -lemma csn_abst: ∀a,L,W. L ⊢ ⬊* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬊* T → L ⊢ ⬊* ⓛ{a}W. T. -#a #L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT -@csn_intro #X #H1 #H2 -elim (cpr_inv_abst1 … H1 I V) -H1 -#W0 #T0 #HLW0 #HLT0 #H destruct -elim (eq_false_inv_tpair_sn … H2) -H2 -[ /3 width=5/ -| -HLW0 * #H destruct /3 width=1/ -] -qed. - -lemma csn_appl_simple: ∀L,V. L ⊢ ⬊* V → ∀T1. - (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊* ⓐV. T2) → - 𝐒⦃T1⦄ → L ⊢ ⬊* ⓐV. T1. -#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1 -@csn_intro #X #H1 #H2 -elim (cpr_inv_appl1_simple … H1 ?) // -H1 -#V0 #T0 #HLV0 #HLT10 #H destruct -elim (eq_false_inv_tpair_dx … H2) -H2 -[ -IHV -HT1 #HT10 - @(csn_cpr_trans … (ⓐV.T0)) /2 width=1/ -HLV0 - @IHT1 -IHT1 // /2 width=1/ -| -HLT10 * #H #HV0 destruct - @IHV -IHV // -HT1 /2 width=1/ -HV0 - #T2 #HLT02 #HT02 - @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0 - @IHT1 -IHT1 // -HLT02 /2 width=1/ -] -qed. - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was: sn3_gen_def *) -lemma csn_inv_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → L ⊢ ⬊* #i → K ⊢ ⬊* V. -#L #K #V #i #HLK #Hi -elim (lift_total V 0 (i+1)) #V0 #HV0 -lapply (ldrop_fwd_ldrop2 … HLK) #H0LK -@(csn_inv_lift … H0LK … HV0) -H0LK -@(csn_cpr_trans … Hi) -Hi /2 width=6/ -qed-. - -(* Main properties **********************************************************) - -theorem csn_acp: acp cpr (eq …) (csn …). -@mk_acp -[ /2 width=1/ -| /2 width=3/ -| /2 width=5/ -| @cnf_lift -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_tstc_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_tstc_vector.ma deleted file mode 100644 index cfee668b9..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_tstc_vector.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_2/computation/acp_cr.ma". -include "basic_2/computation/cprs_tstc_vector.ma". -include "basic_2/computation/csn_lfpr.ma". -include "basic_2/computation/csn_vector.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************) - -(* Advanced properties ******************************************************) - -(* Basic_1: was only: sn3_appls_lref *) -lemma csn_applv_cnf: ∀L,T. 𝐒⦃T⦄ → L ⊢ 𝐍⦃T⦄ → - ∀Vs. L ⊢ ⬊* Vs → L ⊢ ⬊* ⒶVs.T. -#L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(csn_cnf … H2T) ] (**) (* /2 width=1/ does not work *) -#V #Vs #IHV #H -elim (csnv_inv_cons … H) -H #HV #HVs -@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -HVs -#X #H #H0 -lapply (cprs_fwd_cnf_vector … H) -H // -H1T -H2T #H -elim (H0 ?) -H0 // -qed. - -(* Basic_1: was: sn3_appls_beta *) -lemma csn_applv_beta: ∀a,L,W. L ⊢ ⬊* W → - ∀Vs,V,T. L ⊢ ⬊* ⒶVs.ⓓ{a}V.T → - L ⊢ ⬊* ⒶVs. ⓐV.ⓛ{a}W. T. -#a #L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW -#V0 #Vs #IHV #V #T #H1T -lapply (csn_fwd_pair_sn … H1T) #HV0 -lapply (csn_fwd_flat_dx … H1T) #H2T -@csn_appl_simple_tstc // -HV0 /2 width=1/ -IHV -H2T -#X #H #H0 -elim (cprs_fwd_beta_vector … H) -H #H -[ -H1T elim (H0 ?) -H0 // -| -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/ -] -qed. - -lemma csn_applv_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 → - ∀V2. ⇧[0, i + 1] V1 ≡ V2 → - ∀Vs.L ⊢ ⬊* (ⒶVs. V2) → L ⊢ ⬊* (ⒶVs. #i). -#L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs -[ #H - lapply (ldrop_fwd_ldrop2 … HLK) #HLK0 - lapply (csn_inv_lift … H … HLK0 HV12) -V2 -HLK0 /2 width=4/ -| #V #Vs #IHV #H1T - lapply (csn_fwd_pair_sn … H1T) #HV - lapply (csn_fwd_flat_dx … H1T) #H2T - @csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T - #X #H #H0 - elim (cprs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H - [ -H1T elim (H0 ?) -H0 // - | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/ - ] -] -qed. - -(* Basic_1: was: sn3_appls_abbr *) -lemma csn_applv_theta: ∀a,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s → - ∀V,T. L ⊢ ⬊* ⓓ{a}V. ⒶV2s. T → L ⊢ ⬊* V → - L ⊢ ⬊* ⒶV1s. ⓓ{a}V. T. -#a #L #V1s #V2s * -V1s -V2s /2 width=1/ -#V1s #V2s #V1 #V2 #HV12 #H -generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1 -elim H -V1s -V2s /2 width=3/ -#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H #HV -lapply (csn_appl_theta … HW12 … H) -H -HW12 #H -lapply (csn_fwd_pair_sn … H) #HW1 -lapply (csn_fwd_flat_dx … H) #H1 -@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -HV -H1 #X #H1 #H2 -elim (cprs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12 -[ -H #H elim (H2 ?) -H2 // -| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/ -] -qed. - -(* Basic_1: was: sn3_appls_cast *) -lemma csn_applv_tau: ∀L,W. L ⊢ ⬊* W → - ∀Vs,T. L ⊢ ⬊* ⒶVs. T → - L ⊢ ⬊* ⒶVs. ⓝW. T. -#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW -#V #Vs #IHV #T #H1T -lapply (csn_fwd_pair_sn … H1T) #HV -lapply (csn_fwd_flat_dx … H1T) #H2T -@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T -#X #H #H0 -elim (cprs_fwd_tau_vector … H) -H #H -[ -H1T elim (H0 ?) -H0 // -| -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/ -] -qed. - -theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬊* T). -@mk_acr // -[ /3 width=1/ -| /2 width=1/ -| /2 width=6/ -| #L #V1 #V2 #HV12 #a #V #T #H #HVT - @(csn_applv_theta … HV12) -HV12 // - @(csn_abbr) // -| /2 width=1/ -| @csn_lift -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_vector.ma deleted file mode 100644 index 7c26ef429..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_vector.ma +++ /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/grammar/term_vector.ma". -include "basic_2/computation/csn.ma". - -(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************) - -definition csnv: lenv → predicate (list term) ≝ - λL. all … (csn L). - -interpretation - "context-sensitive strong normalization (term vector)" - 'SN L Ts = (csnv L Ts). - -(* Basic inversion lemmas ***************************************************) - -lemma csnv_inv_cons: ∀L,T,Ts. L ⊢ ⬊* T @ Ts → L ⊢ ⬊* T ∧ L ⊢ ⬊* Ts. -normalize // qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/fprs.ma deleted file mode 100644 index 61c720754..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs.ma +++ /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/reducibility/fpr.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************) - -definition fprs: bi_relation lenv term ≝ bi_TC … fpr. - -interpretation "context-free parallel computation (closure)" - 'FocalizedPRedStar L1 T1 L2 T2 = (fprs L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma fprs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 → - (∀L,L2,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → R L T → R L2 T2) → - ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L2 T2. -/3 width=7 by bi_TC_star_ind/ qed-. - -lemma fprs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 → - (∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → R L T → R L1 T1) → - ∀L1,T1. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L1 T1. -/3 width=7 by bi_TC_star_ind_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma fprs_refl: bi_reflexive … fprs. -/2 width=1/ qed. - -lemma fprs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → - ⦃L1, T1⦄ ➡* ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma fprs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → - ⦃L1, T1⦄ ➡* ⦃L2, T2⦄. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_aaa.ma deleted file mode 100644 index b76637ff7..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_aaa.ma +++ /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: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → - ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A. -#L1 #T1 #A #HT1 #L2 #T2 #HLT12 -@(bi_TC_Conf3 … HT1 ?? HLT12) /2 width=4/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_cprs.ma deleted file mode 100644 index 4e7a633c3..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_cprs.ma +++ /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/reducibility/fpr_cpr.ma". -include "basic_2/computation/cprs.ma". -include "basic_2/computation/fprs.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************) - -(* Properties on context-sensitive parallel computation for terms ***********) - -lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄. -#L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/ -qed. -(* -(* Advanced propertis *******************************************************) - -lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 → - ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄. -#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I -elim (fpr_inv_all … H) /3 width=4/ -qed. - -(* Advanced forward lemmas **************************************************) - -lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2. - ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ → - ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2. -* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H -elim (fpr_inv_all … H) -H #L #HL1 #H #HL2 -[ elim (cpr_inv_abbr1 … H) -H * - [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/ - | #T #_ #_ #H destruct - ] -| elim (cpr_inv_abst1 … H Abst V2) -H - #V #T #HV1 #_ #H destruct /3 width=4/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -lamma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ → - ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & - ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ & - L2 = K2.ⓑ{I}V2. -#I1 #K1 #X #V1 #T1 #T2 #H -elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct -elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/ -qed-. - -lamma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ → - ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & - ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ & - L1 = K1.ⓑ{I}V1. -#I2 #X #K2 #V2 #T1 #T2 #H -elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct -elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/ -qed-. -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_fprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_fprs.ma deleted file mode 100644 index e0c1b3058..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/fprs_fprs.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 "basic_2/reducibility/fpr_fpr.ma". -include "basic_2/computation/fprs.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************) - -(* Advanced properties ******************************************************) - -lemma fprs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ➡ ⦃L1, T1⦄ → - ∀L2,T2. ⦃L0, T0⦄ ➡* ⦃L2, T2⦄ → - ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄. -#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 -/2 width=4/ qed. - -(* Main propertis ***********************************************************) - -theorem fprs_conf: bi_confluent … fprs. -/2 width=4/ qed. - -theorem fprs_trans: bi_transitive … fprs. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs.ma deleted file mode 100644 index a193f3c0a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs.ma +++ /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/lfpr.ma". - -(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) - -definition lfprs: relation lenv ≝ TC … lfpr. - -interpretation - "focalized parallel computation (environment)" - 'FocalizedPRedStar L1 L2 = (lfprs L1 L2). - -(* Basic eliminators ********************************************************) - -lemma lfprs_ind: ∀L1. ∀R:predicate lenv. R L1 → - (∀L,L2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L⦄ ➡ ⦃L2⦄ → R L → R L2) → - ∀L2. ⦃L1⦄ ➡* ⦃L2⦄ → R L2. -#L1 #R #HL1 #IHL1 #L2 #HL12 -@(TC_star_ind … HL1 IHL1 … HL12) // -qed-. - -lemma lfprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 → - (∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ⦃L⦄ ➡* ⦃L2⦄ → R L → R L1) → - ∀L1. ⦃L1⦄ ➡* ⦃L2⦄ → R L1. -#L2 #R #HL2 #IHL2 #L1 #HL12 -@(TC_star_ind_dx … HL2 IHL2 … HL12) // -qed-. - -(* Basic properties *********************************************************) - -lemma lfprs_refl: ∀L. ⦃L⦄ ➡* ⦃L⦄. -/2 width=1/ qed. - -lemma lfprs_strap1: ∀L1,L,L2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄. -/2 width=3/ qed. - -lemma lfprs_strap2: ∀L1,L,L2. ⦃L1⦄ ➡ ⦃L⦄ → ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄. -/2 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_aaa.ma deleted file mode 100644 index 5c6cd31cb..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_aaa.ma +++ /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/reducibility/lfpr_aaa.ma". -include "basic_2/computation/lfprs.ma". - -(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) - -(* Properties about atomic arity assignment on terms ************************) - -lemma aaa_lfprs_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡* ⦃L2⦄ → L2 ⊢ T ⁝ A. -#L1 #T #A #HT #L2 #HL12 -@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_cprs.ma deleted file mode 100644 index b0f7c4a94..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_cprs.ma +++ /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/lfpr_cpr.ma". -include "basic_2/computation/cprs.ma". -include "basic_2/computation/lfprs.ma". - -(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) - -(* Advanced properties ******************************************************) - -lemma lfprs_pair_dx: ∀I,L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡* V2 → - ⦃L1. ⓑ{I} V1⦄ ➡* ⦃L2. ⓑ{I} V2⦄. -#I #L1 #L2 #HL12 #V1 #V2 #H @(cprs_ind … H) -V2 -/3 width=1/ /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_lfprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_lfprs.ma deleted file mode 100644 index e3866fd2e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lfprs_lfprs.ma +++ /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/reducibility/lfpr_lfpr.ma". -include "basic_2/computation/lfprs_cprs.ma". - -(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) - -(* Advanced properties ******************************************************) - -lemma lfprs_strip: ∀L,L1. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → - ∃∃L0. ⦃L1⦄ ➡ ⦃L0⦄ & ⦃L2⦄ ➡* ⦃L0⦄. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem lfprs_conf: ∀L,L1. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → - ∃∃L0. ⦃L1⦄ ➡* ⦃L0⦄ & ⦃L2⦄ ➡* ⦃L0⦄. -/3 width=3/ qed. - -theorem lfprs_trans: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄. -/2 width=3/ qed. - -lemma lfprs_pair: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡* V2 → - ∀I. ⦃L1. ⓑ{I} V1⦄ ➡* ⦃L2. ⓑ{I} V2⦄. -#L1 #L2 #H @(lfprs_ind … H) -L2 /2 width=1/ -#L #L2 #_ #HL2 #IHL1 #V1 #V2 #HV12 #I -@(lfprs_trans … (L.ⓑ{I}V1)) /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc.ma deleted file mode 100644 index bcf6c7714..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc.ma +++ /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/static/aaa.ma". -include "basic_2/computation/acp_cr.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) - -inductive lsubc (RP:lenv→predicate term): relation lenv ≝ -| lsubc_atom: lsubc RP (⋆) (⋆) -| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V) -| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ ϵ[RP] 〚A〛 → L2 ⊢ W ⁝ A → - lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW) -. - -interpretation - "local environment refinement (abstract candidates of reducibility)" - 'CrSubEq L1 RP L2 = (lsubc RP L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L1 = ⋆ → L2 = ⋆. -#RP #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #A #_ #_ #_ #H destruct -] -qed. - -(* Basic_1: was: csubc_gen_sort_r *) -lemma lsubc_inv_atom1: ∀RP,L2. ⋆ ⊑[RP] L2 → L2 = ⋆. -/2 width=4/ qed-. - -fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V → - (∃∃K2. K1 ⊑[RP] K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & - K1 ⊑[RP] K2 & - L2 = K2. ⓛW & I = Abbr. -#RP #L1 #L2 * -L1 -L2 -[ #I #K1 #V #H destruct -| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/ -| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/ -] -qed. - -(* Basic_1: was: csubc_gen_head_r *) -lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V ⊑[RP] L2 → - (∃∃K2. K1 ⊑[RP] K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & - K1 ⊑[RP] K2 & - L2 = K2. ⓛW & I = Abbr. -/2 width=3/ qed-. - -fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L2 = ⋆ → L1 = ⋆. -#RP #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #A #_ #_ #_ #H destruct -] -qed. - -(* Basic_1: was: csubc_gen_sort_l *) -lemma lsubc_inv_atom2: ∀RP,L1. L1 ⊑[RP] ⋆ → L1 = ⋆. -/2 width=4/ qed-. - -fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W → - (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & - K1 ⊑[RP] K2 & - L1 = K1. ⓓV & I = Abst. -#RP #L1 #L2 * -L1 -L2 -[ #I #K2 #W #H destruct -| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/ -| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/ -] -qed. - -(* Basic_1: was: csubc_gen_head_l *) -lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 ⊑[RP] K2. ⓑ{I} W → - (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & - K1 ⊑[RP] K2 & - L1 = K1. ⓓV & I = Abst. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: csubc_refl *) -lemma lsubc_refl: ∀RP,L. L ⊑[RP] L. -#RP #L elim L -L // /2 width=1/ -qed. - -(* Basic_1: removed theorems 3: - csubc_clear_conf csubc_getl_conf csubc_csuba -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_ldrop.ma deleted file mode 100644 index a7c7c7a99..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_ldrop.ma +++ /dev/null @@ -1,67 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/computation/lsubc.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) - -(* Properties concerning basic local environment slicing ********************) - -(* Basic_1: was: csubc_drop_conf_O *) -(* Note: the constant 0 can not be generalized *) -lemma lsubc_ldrop_O1_trans: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ⊑[RP] K2. -#RP #L1 #L2 #H elim H -L1 -L2 -[ #X #e #H - >(ldrop_inv_atom1 … H) -H /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #X #e #H - elim (ldrop_inv_O1 … H) -H * #He #H destruct - [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=3/ - | elim (IHL12 … H) -L2 /3 width=3/ - ] -| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #X #e #H - elim (ldrop_inv_O1 … H) -H * #He #H destruct - [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=7/ - | elim (IHL12 … H) -L2 /3 width=3/ - ] -qed-. - -(* Basic_1: was: csubc_drop_conf_rev *) -lemma ldrop_lsubc_trans: ∀RR,RS,RP. - acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 ⊑[RP] K2 → - ∃∃L2. L1 ⊑[RP] L2 & ⇩[d, e] L2 ≡ K2. -#RR #RS #RP #Hacp #Hacr #L1 #K1 #d #e #H elim H -L1 -K1 -d -e -[ #d #e #X #H - >(lsubc_inv_atom1 … H) -H /2 width=3/ -| #L1 #I #V1 #X #H - elim (lsubc_inv_pair1 … H) -H * - [ #K1 #HLK1 #H destruct /3 width=3/ - | #K1 #W1 #A #HV1 #HW1 #HLK1 #H1 #H2 destruct /3 width=3/ - ] -| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 - elim (IHLK1 … HK12) -K1 /3 width=5/ -| #L1 #K1 #I #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H - elim (lsubc_inv_pair1 … H) -H * - [ #K2 #HK12 #H destruct - elim (IHLK1 … HK12) -K1 /3 width=5/ - | #K2 #W2 #A #HV2 #HW2 #HK12 #H1 #H2 destruct - elim (IHLK1 … HK12) #K #HL1K #HK2 - lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA - lapply (s7 … HA … HV2 … HLK1 HV21) -HV2 - elim (lift_total W2 d e) /4 width=9/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_ldrops.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_ldrops.ma deleted file mode 100644 index 4e26322a5..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_ldrops.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_2/computation/lsubc_ldrop.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) - -(* Properties concerning generic local environment slicing ******************) - -(* Basic_1: was: csubc_drop1_conf_rev *) -lemma ldrops_lsubc_trans: ∀RR,RS,RP. - acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀L1,K1,des. ⇩*[des] L1 ≡ K1 → ∀K2. K1 ⊑[RP] K2 → - ∃∃L2. L1 ⊑[RP] L2 & ⇩*[des] L2 ≡ K2. -#RR #RS #RP #Hacp #Hacr #L1 #K1 #des #H elim H -L1 -K1 -des -[ /2 width=3/ -| #L1 #L #K1 #des #d #e #_ #HLK1 #IHL #K2 #HK12 - elim (ldrop_lsubc_trans … Hacp Hacr … HLK1 … HK12) -Hacp -Hacr -K1 #K #HLK #HK2 - elim (IHL … HLK) -IHL -HLK /3 width=5/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_lsuba.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_lsuba.ma deleted file mode 100644 index aad454f62..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc_lsuba.ma +++ /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/static/lsuba.ma". -include "basic_2/computation/acp_aaa.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) - -(* properties concerning lenv refinement for atomic arity assignment ********) - -lemma lsubc_lsuba: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → - ∀L1,L2. L1 ⁝⊑ L2 → L1 ⊑[RP] L2. -#RR #RS #RP #H1RP #H2RP #L1 #L2 #H elim H -L1 -L2 -// /2 width=1/ /3 width=4/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs.ma deleted file mode 100644 index b7b0e1094..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs.ma +++ /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/reducibility/ltpr.ma". -include "basic_2/computation/tprs.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************) - -definition ltprs: relation lenv ≝ TC … ltpr. - -interpretation - "context-free parallel computation (environment)" - 'PRedStar L1 L2 = (ltprs L1 L2). - -(* Basic eliminators ********************************************************) - -lemma ltprs_ind: ∀L1. ∀R:predicate lenv. R L1 → - (∀L,L2. L1 ➡* L → L ➡ L2 → R L → R L2) → - ∀L2. L1 ➡* L2 → R L2. -#L1 #R #HL1 #IHL1 #L2 #HL12 -@(TC_star_ind … HL1 IHL1 … HL12) // -qed-. - -lemma ltprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 → - (∀L1,L. L1 ➡ L → L ➡* L2 → R L → R L1) → - ∀L1. L1 ➡* L2 → R L1. -#L2 #R #HL2 #IHL2 #L1 #HL12 -@(TC_star_ind_dx … HL2 IHL2 … HL12) // -qed-. - -(* Basic properties *********************************************************) - -lemma ltprs_refl: reflexive … ltprs. -/2 width=1/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma ltprs_inv_atom1: ∀L2. ⋆ ➡* L2 → L2 = ⋆. -#L2 #H @(ltprs_ind … H) -L2 // -#L #L2 #_ #HL2 #IHL1 destruct ->(ltpr_inv_atom1 … HL2) -L2 // -qed-. - -lemma ltprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ➡* L2 → - ∃∃K2,V2. K1 ➡* K2 & V1 ➡* V2 & L2 = K2. ⓑ{I} V2. -#I #K1 #L2 #V1 #H @(ltprs_ind … H) -L2 /2 width=5/ -#L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct -elim (ltpr_inv_pair1 … HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/ -qed-. - -lemma ltprs_inv_atom2: ∀L1. L1 ➡* ⋆ → L1 = ⋆. -#L1 #H @(ltprs_ind_dx … H) -L1 // -#L1 #L #HL1 #_ #IHL2 destruct ->(ltpr_inv_atom2 … HL1) -L1 // -qed-. - -lemma ltprs_inv_pair2: ∀I,L1,K2,V2. L1 ➡* K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ➡* K2 & V1 ➡* V2 & L1 = K1. ⓑ{I} V1. -#I #L1 #K2 #V2 #H @(ltprs_ind_dx … H) -L1 /2 width=5/ -#L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct -elim (ltpr_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma ltprs_fwd_length: ∀L1,L2. L1 ➡* L2 → |L1| = |L2|. -#L1 #L2 #H @(ltprs_ind … H) -L2 // -#L #L2 #_ #HL2 #IHL1 ->IHL1 -L1 >(ltpr_fwd_length … HL2) -HL2 // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_alt.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_alt.ma deleted file mode 100644 index 7d532c973..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_alt.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 "basic_2/computation/ltprs.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************) - -(* alternative definition of ltprs *) -definition ltprsa: relation lenv ≝ lpx tprs. - -interpretation - "context-free parallel computation (environment) alternative" - 'PRedStarAlt L1 L2 = (ltprsa L1 L2). - -(* Basic properties *********************************************************) - -lemma ltprs_ltprsa: ∀L1,L2. L1 ➡* L2 → L1 ➡➡* L2. -/2 width=1/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma ltprsa_ltprs: ∀L1,L2. L1 ➡➡* L2 → L1 ➡* L2. -/2 width=1/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_ldrop.ma deleted file mode 100644 index a7c320089..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_ldrop.ma +++ /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/lambda_delta/basic_2/computation/ltprs_ltprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_ltprs.ma deleted file mode 100644 index e529ee31a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/ltprs_ltprs.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_2/reducibility/ltpr_ltpr.ma". -include "basic_2/computation/ltprs.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************) - -(* Advanced properties ******************************************************) - -lemma ltprs_strip: ∀L1. ∀L:term. L ➡* L1 → ∀L2. L ➡ L2 → - ∃∃L0. L1 ➡ L0 & L2 ➡* L0. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem ltprs_conf: Confluent … ltprs. -/3 width=3/ qed. - -theorem ltprs_trans: Transitive … ltprs. -/2 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/tprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/tprs.ma deleted file mode 100644 index b094e66c1..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/tprs.ma +++ /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/reducibility/tpr.ma". - -(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************) - -(* Basic_1: includes: pr1_pr0 *) -definition tprs: relation term ≝ TC … tpr. - -interpretation "context-free parallel computation (term)" - 'PRedStar T1 T2 = (tprs T1 T2). - -(* Basic eliminators ********************************************************) - -lemma tprs_ind: ∀T1. ∀R:predicate term. R T1 → - (∀T,T2. T1 ➡* T → T ➡ T2 → R T → R T2) → - ∀T2. T1 ➡* T2 → R T2. -#T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind … HT1 IHT1 … HT12) // -qed-. - -lemma tprs_ind_dx: ∀T2. ∀R:predicate term. R T2 → - (∀T1,T. T1 ➡ T → T ➡* T2 → R T → R T1) → - ∀T1. T1 ➡* T2 → R T1. -#T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx … HT2 IHT2 … HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma tprs_refl: reflexive … tprs. -/2 width=1/ qed. - -lemma tprs_strap1: ∀T1,T,T2. T1 ➡* T → T ➡ T2 → T1 ➡* T2. -/2 width=3/ qed. - -lemma tprs_strap2: ∀T1,T,T2. T1 ➡ T → T ➡* T2 → T1 ➡* T2. -/2 width=3/ qed. - -(* Basic_1: was only: pr1_head_1 *) -lemma tprs_pair_sn: ∀I,T1,T2. T1 ➡ T2 → ∀V1,V2. V1 ➡* V2 → - ②{I} V1. T1 ➡* ②{I} V2. T2. -* [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind … H) -V2 -[1,3: /3 width=1/ -|2,4: #V #V2 #_ #HV2 #IHV1 - @(tprs_strap1 … IHV1) -IHV1 /2 width=1/ -] -qed. - -(* Basic_1: was only: pr1_head_2 *) -lemma tprs_pair_dx: ∀I,V1,V2. V1 ➡ V2 → ∀T1,T2. T1 ➡* T2 → - ②{I} V1. T1 ➡* ②{I} V2. T2. -* [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind … H) -T2 -[1,3: /3 width=1/ -|2,4: #T #T2 #_ #HT2 #IHT1 - @(tprs_strap1 … IHT1) -IHT1 /2 width=1/ -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma tprs_inv_atom1: ∀U2,k. ⋆k ➡* U2 → U2 = ⋆k. -#U2 #k #H @(tprs_ind … H) -U2 // -#U #U2 #_ #HU2 #IHU1 destruct ->(tpr_inv_atom1 … HU2) -HU2 // -qed-. - -lemma tprs_inv_cast1: ∀W1,T1,U2. ⓝW1.T1 ➡* U2 → T1 ➡* U2 ∨ - ∃∃W2,T2. W1 ➡* W2 & T1 ➡* T2 & U2 = ⓝW2.T2. -#W1 #T1 #U2 #H @(tprs_ind … H) -U2 /3 width=5/ -#U #U2 #_ #HU2 * /3 width=3/ * -#W #T #HW1 #HT1 #H destruct -elim (tpr_inv_cast1 … HU2) -HU2 /3 width=3/ * -#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/tprs_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/tprs_lift.ma deleted file mode 100644 index d0d173470..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/tprs_lift.ma +++ /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: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡* U2 → - ∃∃V2,T2. V1 ➡* V2 & T1 ➡* T2 & U2 = ⓛ{a}V2. T2. -#a #V1 #T1 #U2 #H @(tprs_ind … H) -U2 /2 width=5/ -#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -elim (tpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ -qed-. - -lemma tprs_inv_abst: ∀a,V1,V2,T1,T2. ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 → - V1 ➡* V2 ∧ T1 ➡* T2. -#a #V1 #V2 #T1 #T2 #H -elim (tprs_inv_abst1 … H) -H #V #T #HV1 #HT1 #H destruct /2 width=1/ -qed-. - -(* Relocation properties ****************************************************) - -(* Note: this was missing in basic_1 *) -lemma tprs_lift: t_liftable tprs. -/3 width=7/ qed. - -(* Note: this was missing in basic_1 *) -lemma tprs_inv_lift1: t_deliftable_sn tprs. -/3 width=3 by tpr_inv_lift1, t_deliftable_sn_TC/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/tprs_tprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/tprs_tprs.ma deleted file mode 100644 index 232244510..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/tprs_tprs.ma +++ /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: ∀T1,T. T ➡* T1 → ∀T2. T ➡ T2 → - ∃∃T0. T1 ➡ T0 & T2 ➡* T0. -/3 width=3/ qed. - -(* Main propertis ***********************************************************) - -(* Basic_1: was: pr1_confluence *) -theorem tprs_conf: Confluent … tprs. -/3 width=3/ qed. - -(* Basic_1: was: pr1_t *) -theorem tprs_trans: Transitive … tprs. -/2 width=3/ qed. - -(* Basic_1: was: pr1_comp *) -lemma tprs_pair: ∀I,V1,V2. V1 ➡* V2 → ∀T1,T2. T1 ➡* T2 → - ②{I} V1. T1 ➡* ②{I} V2. T2. -#I #V1 #V2 #H @(tprs_ind … H) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 -@(tprs_trans … (②{I}V.T2)) /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/xprs.ma deleted file mode 100644 index 854c5da27..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs.ma +++ /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/static/lsubss.ma". -include "basic_2/reducibility/xpr.ma". -(* -include "basic_2/reducibility/cnf.ma". -*) -(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************) - -definition xprs: ∀h. sd h → lenv → relation term ≝ - λh,g,L. TC … (xpr h g L). - -interpretation "extended parallel computation (term)" - 'XPRedStar h g L T1 T2 = (xprs h g L T1 T2). - -(* Basic eliminators ********************************************************) - -lemma xprs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 → - (∀T,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → R T → R T2) → - ∀T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T2. -#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind … HT1 IHT1 … HT12) // -qed-. - -lemma xprs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 → - (∀T1,T. ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → R T → R T1) → - ∀T1. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T1. -#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx … HT2 IHT2 … HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma xprs_refl: ∀h,g,L. reflexive … (xprs h g L). -/2 width=1/ qed. - -lemma xprs_strap1: ∀h,g,L,T1,T,T2. - ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2. -/2 width=3/ qed. - -lemma xprs_strap2: ∀h,g,L,T1,T,T2. - ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2. -/2 width=3/ qed. - -(* Basic inversion lemmas ***************************************************) -(* -axiom xprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U. -#L #T #U #H @(xprs_ind_dx … H) -T // -#T0 #T #H1T0 #_ #IHT #H2T0 -lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/ -qed-. -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_aaa.ma deleted file mode 100644 index 5beb8fe19..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_aaa.ma +++ /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: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •➡*[g] U → L ⊢ U ⁝ A. -#h #g #L #T #A #HT #U #H @(xprs_ind … H) -U // /2 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_cprs.ma deleted file mode 100644 index 13a4f8889..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_cprs.ma +++ /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: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2. -#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_lift.ma deleted file mode 100644 index cb151a194..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_lift.ma +++ /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: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •➡*[g] U2 → - ∃∃V2,T2. L ⊢ V1 ➡* V2 & U2 = ⓛ{a}V2. T2. -#h #g #a #L #V1 #T1 #U2 #H @(xprs_ind … H) -U2 /2 width=4/ -#U #U2 #_ #HU2 * #V #T #HV1 #H destruct -elim (xpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #_ #H destruct /3 width=4/ -qed-. - -(* Relocation properties ****************************************************) - -lemma xprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 → - ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •➡*[g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 → - ⦃h, L⦄ ⊢ U1 •➡*[g] U2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 #HT12 @(xprs_ind … HT12) -T2 -[ -HLK #T2 #HT12 - <(lift_mono … HTU1 … HT12) -T1 // -| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (xpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/ -] -qed. - -lemma xprs_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K → - ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •➡*[g] U2 → - ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •➡*[g] T2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 #HU12 @(xprs_ind … HU12) -U2 /2 width=3/ --HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 -elim (xpr_inv_lift1 … HLK … HTU … HU2) -U -HLK /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_lsubss.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_lsubss.ma deleted file mode 100644 index c883c14f3..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_lsubss.ma +++ /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: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → - ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2. -#h #g #L1 #L2 #HL12 #T1 #T2 #H @(xprs_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 -lapply (lsubss_xpr_trans … HL12 … HT2) -L2 /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_xprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_xprs.ma deleted file mode 100644 index 9593f0550..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/xprs_xprs.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 "basic_2/computation/xprs.ma". - -(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************) - -theorem xprs_trans: ∀h,g,L. transitive … (xprs h g L). -/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/conversion/cpc.ma b/matita/matita/contribs/lambda_delta/basic_2/conversion/cpc.ma deleted file mode 100644 index 5fb614a8c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/conversion/cpc.ma +++ /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.ma". - -(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************) - -definition cpc: lenv → relation term ≝ - λL,T1,T2. L ⊢ T1 ➡ T2 ∨ L ⊢ T2 ➡ T1. - -interpretation - "context-sensitive parallel conversion (term)" - 'PConv L T1 T2 = (cpc L T1 T2). - -(* Basic properties *********************************************************) - -lemma cpc_refl: ∀L. reflexive … (cpc L). -/2 width=1/ qed. - -lemma cpc_sym: ∀L. symmetric … (cpc L). -#L #T1 #T2 * /2 width=1/ -qed. - -(* Basic forward lemmas *****************************************************) - -lemma cpc_fwd_cpr: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T. -#L #T1 #T2 * /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/conversion/cpc_cpc.ma b/matita/matita/contribs/lambda_delta/basic_2/conversion/cpc_cpc.ma deleted file mode 100644 index dcea07a8b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/conversion/cpc_cpc.ma +++ /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/cpc.ma". - -(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************) - -(* Main properties **********************************************************) - -theorem cpc_conf: ∀L,T0,T1,T2. L ⊢ T0 ⬌ T1 → L ⊢ T0 ⬌ T2 → - ∃∃T. L ⊢ T1 ⬌ T & L ⊢ T2 ⬌ T. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/conversion/fpc.ma b/matita/matita/contribs/lambda_delta/basic_2/conversion/fpc.ma deleted file mode 100644 index f552d5818..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/conversion/fpc.ma +++ /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/reducibility/fpr.ma". - -(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************) - -definition fpc: bi_relation lenv term ≝ - λL1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ ∨ ⦃L2, T2⦄ ➡ ⦃L1, T1⦄. - -interpretation - "context-free parallel conversion (closure)" - 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2). - -(* Basic properties *********************************************************) - -lemma fpc_refl: bi_reflexive … fpc. -/2 width=1/ qed. - -lemma fpc_sym: bi_symmetric … fpc. -#L1 #L2 #T1 #T2 * /2 width=1/ -qed. - -(* Basic forward lemmas *****************************************************) - -lemma fpc_fwd_fpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ → - ∃∃L,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄. -#L1 #L2 #T1 #T2 * /2 width=4/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/conversion/fpc_fpc.ma b/matita/matita/contribs/lambda_delta/basic_2/conversion/fpc_fpc.ma deleted file mode 100644 index 22fc16f37..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/conversion/fpc_fpc.ma +++ /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: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ → - ∀L2,T2. ⦃L0, T0⦄ ⬌ ⦃L2, T2⦄ → - ∃∃L,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄. -/3 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/conversion/lfpc.ma b/matita/matita/contribs/lambda_delta/basic_2/conversion/lfpc.ma deleted file mode 100644 index 273873abd..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/conversion/lfpc.ma +++ /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/lfpr.ma". - -(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************) - -definition lfpc: relation lenv ≝ - λL1,L2. ⦃L1⦄ ➡ ⦃L2⦄ ∨ ⦃L2⦄ ➡ ⦃L1⦄. - -interpretation - "focalized parallel conversion (local environment)" - 'FocalizedPConv L1 L2 = (lfpc L1 L2). - -(* Basic properties *********************************************************) - -lemma lfpc_refl: ∀L. ⦃L⦄ ⬌ ⦃L⦄. -/2 width=1/ qed. - -lemma lfpc_sym: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L2⦄ ⬌ ⦃L1⦄. -#L1 #L2 * /2 width=1/ -qed. - -lemma lfpc_lfpr: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄. -#L1 #L2 * /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/conversion/lfpc_lfpc.ma b/matita/matita/contribs/lambda_delta/basic_2/conversion/lfpc_lfpc.ma deleted file mode 100644 index 69e444adb..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/conversion/lfpc_lfpc.ma +++ /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: ∀L0,L1,L2. ⦃L0⦄ ⬌ ⦃L1⦄ → ⦃L0⦄ ⬌ ⦃L2⦄ → - ∃∃L. ⦃L1⦄ ⬌ ⦃L⦄ & ⦃L2⦄ ⬌ ⦃L⦄. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma deleted file mode 100644 index 2be571525..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma +++ /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/computation/cprs.ma". -include "basic_2/computation/xprs.ma". -include "basic_2/equivalence/cpcs.ma". - -(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) - -inductive snv (h:sh) (g:sd h): lenv → predicate term ≝ -| snv_sort: ∀L,k. snv h g L (⋆k) -| snv_lref: ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → snv h g K V → snv h g L (#i) -| snv_bind: ∀a,I,L,V,T. snv h g L V → snv h g (L.ⓑ{I}V) T → snv h g L (ⓑ{a,I}V.T) -| snv_appl: ∀a,L,V,W,W0,T,U,l. snv h g L V → snv h g L T → - ⦃h, L⦄ ⊢ V •[g, l + 1] W → L ⊢ W ➡* W0 → - ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U → snv h g L (ⓐV.T) -| snv_cast: ∀L,W,T,U,l. snv h g L W → snv h g L T → - ⦃h, L⦄ ⊢ T •[g, l + 1] U → L ⊢ U ⬌* W → snv h g L (ⓝW.T) -. - -interpretation "stratified native validity (term)" - 'NativeValid h g L T = (snv h g L T). - -(* Basic inversion lemmas ***************************************************) - -fact snv_inv_lref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀i. X = #i → - ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊩ V :[g]. -#h #g #L #X * -L -X -[ #L #k #i #H destruct -| #I #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/ -| #a #I #L #V #T #_ #_ #i #H destruct -| #a #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #i #H destruct -| #L #W #T #U #l #_ #_ #_ #_ #i #H destruct -] -qed. - -lemma snv_inv_lref: ∀h,g,L,i. ⦃h, L⦄ ⊩ #i :[g] → - ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊩ V :[g]. -/2 width=3/ qed-. - -fact snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀a,I,V,T. X = ⓑ{a,I}V.T → - ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g]. -#h #g #L #X * -L -X -[ #L #k #a #I #V #T #H destruct -| #I0 #L #K #V0 #i #_ #_ #a #I #V #T #H destruct -| #b #I0 #L #V0 #T0 #HV0 #HT0 #a #I #V #T #H destruct /2 width=1/ -| #b #L #V0 #W0 #W00 #T0 #U0 #l #_ #_ #_ #_ #_ #a #I #V #T #H destruct -| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #a #I #V #T #H destruct -] -qed. - -lemma snv_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊩ ⓑ{a,I}V.T :[g] → - ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g]. -/2 width=4/ qed-. - -fact snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀V,T. X = ⓐV.T → - ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] & - ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 & - ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U. -#h #g #L #X * -L -X -[ #L #k #V #T #H destruct -| #I #L #K #V0 #i #_ #_ #V #T #H destruct -| #a #I #L #V0 #T0 #_ #_ #V #T #H destruct -| #a #L #V0 #W0 #W00 #T0 #U0 #l #HV0 #HT0 #HVW0 #HW00 #HTU0 #V #T #H destruct /2 width=8/ -| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #V #T #H destruct -] -qed. - -lemma snv_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊩ ⓐV.T :[g] → - ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] & - ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 & - ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U. -/2 width=3/ qed-. - -fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀W,T. X = ⓝW.T → - ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] & - ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W. -#h #g #L #X * -L -X -[ #L #k #W #T #H destruct -| #I #L #K #V #i #_ #_ #W #T #H destruct -| #a #I #L #V #T0 #_ #_ #W #T #H destruct -| #a #L #V #W0 #W00 #T0 #U #l #_ #_ #_ #_ #_ #W #T #H destruct -| #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HUW0 #W #T #H destruct /2 width=4/ -] -qed. - -lemma snv_inv_cast: ∀h,g,L,W,T. ⦃h, L⦄ ⊩ ⓝW.T :[g] → - ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] & - ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W. -/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_aaa.ma deleted file mode 100644 index 3d4761da8..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_aaa.ma +++ /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/computation/csn_aaa.ma". -include "basic_2/computation/xprs_aaa.ma". -include "basic_2/computation/xprs_cprs.ma". -include "basic_2/equivalence/cpcs_aaa.ma". -include "basic_2/dynamic/snv.ma". - -(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) - -(* Properties on atomic arity assignment for terms **************************) - -lemma snv_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃A. L ⊢ T ⁝ A. -#h #g #L #T #H elim H -L -T -[ /2 width=2/ -| #I #L #K #V #i #HLK #_ * /3 width=6/ -| #a * #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2/ -| #a #L #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU * #B #HV * #X #HT - lapply (xprs_aaa h g … HV W0 ?) [ /3 width=3/ ] -W #HW0 - lapply (xprs_aaa … HT … HTU) -HTU #H - elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct - lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4/ -| #L #W #T #U #l #_ #_ #HTU #HUW * #B #HW * #A #HT - lapply (aaa_cpcs_mono … HUW A … HW) -HUW /2 width=7/ -HTU #H destruct /3 width=3/ -] -qed-. - -lemma snv_csn: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → L ⊢ ⬊* T. -#h #g #L #T #H elim (snv_aaa … H) -H /2 width=2/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_lift.ma deleted file mode 100644 index 6d79ef571..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_lift.ma +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_lift.ma". -include "basic_2/equivalence/cpcs_cpcs.ma". -include "basic_2/dynamic/snv.ma". - -(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) - -(* Relocation properties ****************************************************) - -lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊩ T :[g] → ∀L,d,e. ⇩[d, e] L ≡ K → - ∀U. ⇧[d, e] T ≡ U → ⦃h, L⦄ ⊩ U :[g]. -#h #g #K #T #H elim H -K -T -[ #K #k #L #d #e #_ #X #H - >(lift_inv_sort1 … H) -X -K -d -e // -| #I #K #K0 #V #i #HK0 #_ #IHV #L #d #e #HLK #X #H - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (ldrop_trans_le … HLK … HK0 ?) -K /2 width=2/ #X #HL0 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #L0 #W #HLK0 #HVW #H destruct - /3 width=8/ - | lapply (ldrop_trans_ge … HLK … HK0 ?) -K // -Hid /3 width=8/ - ] -| #a #I #K #V #T #_ #_ #IHV #IHT #L #d #e #HLK #X #H - elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct - /4 width=4/ -| #a #K #V #V0 #V1 #T #T1 #l #_ #_ #HV0 #HV01 #HT1 #IHV #IHT #L #d #e #HLK #X #H - elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct - elim (lift_total V0 d e) #W0 #HVW0 - elim (lift_total V1 d e) #W1 #HVW1 - elim (lift_total T1 (d+1) e) #U1 #HTU1 - @(snv_appl … a … W0 … W1 … U1 l) - [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ] - @(xprs_lift … HLK … HTU … HT1) /2 width=1/ -| #K #V0 #T #V #l #_ #_ #HTV #HV0 #IHV0 #IHT #L #d #e #HLK #X #H - elim (lift_inv_flat1 … H) -H #W0 #U #HVW0 #HTU #H destruct - elim (lift_total V d e) #W #HVW - @(snv_cast … W l) [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ] -] -qed. - -lemma snv_inv_lift: ∀h,g,L,U. ⦃h, L⦄ ⊩ U :[g] → ∀K,d,e. ⇩[d, e] L ≡ K → - ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊩ T :[g]. -#h #g #L #U #H elim H -L -U -[ #L #k #K #d #e #_ #X #H - >(lift_inv_sort2 … H) -X -L -d -e // -| #I #L #L0 #W #i #HL0 #_ #IHW #K #d #e #HLK #X #H - elim (lift_inv_lref2 … H) * #Hid #H destruct - [ elim (ldrop_conf_le … HLK … HL0 ?) -L /2 width=2/ #X #HK0 #H - elim (ldrop_inv_skip1 … H ?) -H /2 width=1/ -Hid #K0 #V #HLK0 #HVW #H destruct - /3 width=8/ - | lapply (ldrop_conf_ge … HLK … HL0 ?) -L // -Hid /3 width=8/ - ] -| #a #I #L #W #U #_ #_ #IHW #IHU #K #d #e #HLK #X #H - elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=4/ -| #a #L #W #W0 #W1 #U #U1 #l #_ #_ #HW0 #HW01 #HU1 #IHW #IHU #K #d #e #HLK #X #H - elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct - elim (ssta_inv_lift1 … HW0 … HLK … HVW) -HW0 #V0 #HV0 #HVW0 - elim (cprs_inv_lift1 … HLK … HVW0 … HW01) -W0 #V1 #HVW1 #HV01 - elim (xprs_inv_lift1 … HLK … HTU … HU1) -HU1 #X #H #HTU - elim (lift_inv_bind2 … H) -H #Y #T1 #HY #HTU1 #H destruct - lapply (lift_inj … HY … HVW1) -HY #H destruct /3 width=8/ -| #L #W0 #U #W #l #_ #_ #HUW #HW0 #IHW0 #IHU #K #d #e #HLK #X #H - elim (lift_inv_flat2 … H) -H #V0 #T #HVW0 #HTU #H destruct - elim (ssta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HTV #HVW - lapply (cpcs_inv_lift … HLK … HVW … HVW0 ?) // -W /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_ssta.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_ssta.ma deleted file mode 100644 index d96994e6c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv_ssta.ma +++ /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/dynamic/snv.ma". - -(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) - -(* Properties on stratified static type assignment for terms ****************) - -lemma snv_ssta: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃∃U,l. ⦃h, L⦄ ⊢ T •[g, l] U. -#h #g #L #T #H elim H -L -T -[ #L #k elim (deg_total h g k) /3 width=3/ -| * #L #K #V #i #HLK #_ * #W #l0 #HVW - [ elim (lift_total W 0 (i+1)) /3 width=8/ - | elim (lift_total V 0 (i+1)) /3 width=8/ - ] -| #a #I #L #V #T #_ #_ #_ * /3 width=3/ -| #a #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/ -| #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *) -] -qed-. - -fact snv_ssta_conf_aux: ∀h,g,L,T. ( - ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] → - ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 → - #{L0, T0} < #{L, T} → ⦃h, L0⦄ ⊩ U0 :[g] - ) → - ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] → - ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 → - L0 = L → T0 = T → ⦃h, L0⦄ ⊩ U0 :[g]. -#h #g #L #T #IH1 #L0 #T0 * -L0 -T0 -[ -| -| -| #a #L0 #V #W #W0 #T0 #V0 #l0 #HV #HT0 #HVW #HW0 #HTV0 #X #l #H #H1 #H2 destruct - elim (ssta_inv_appl1 … H) -H #U0 #HTU0 #H destruct - lapply (IH1 … HT0 … HTU0 ?) // #HU0 - @(snv_appl … HV HU0 HVW HW0) -HV -HU0 -HVW -HW0 -| #L0 #W #T0 #W0 #l0 #_ #HT0 #_ #_ #U0 #l #H #H1 #H2 destruct -W0 - lapply (ssta_inv_cast1 … H) -H /2 width=5/ diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma deleted file mode 100644 index 9d9ceb942..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma +++ /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/conversion/cpc.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -definition cpcs: lenv → relation term ≝ - λL. TC … (cpc L). - -interpretation "context-sensitive parallel equivalence (term)" - 'PConvStar L T1 T2 = (cpcs L T1 T2). - -(* Basic eliminators ********************************************************) - -lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 → - (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) → - ∀T2. L ⊢ T1 ⬌* T2 → R T2. -#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) // -qed-. - -lemma cpcs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 → - (∀T1,T. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → R T → R T1) → - ∀T1. L ⊢ T1 ⬌* T2 → R T1. -#L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx … HT2 IHT2 … HT12) // -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: pc3_refl *) -lemma cpcs_refl: ∀L. reflexive … (cpcs L). -/2 width=1/ qed. - -(* Basic_1: was: pc3_s *) -lemma cpcs_sym: ∀L. symmetric … (cpcs L). -/3 width=1/ qed. - -lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2. -/2 width=3/ qed. - -lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -/2 width=3/ qed. - -(* Basic_1: was: pc3_pr2_r *) -lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2. -/3 width=1/ qed. - -lemma cpcs_tpr_dx: ∀L,T1,T2. T1 ➡ T2 → L ⊢ T1 ⬌* T2. -/3 width=1/ qed. - -(* Basic_1: was: pc3_pr2_x *) -lemma cpcs_cpr_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2. -/3 width=1/ qed. - -lemma cpcs_tpr_sn: ∀L,T1,T2. T2 ➡ T1 → L ⊢ T1 ⬌* T2. -/3 width=1/ qed. - -lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2. -/3 width=3/ qed. - -(* Basic_1: was: pc3_pr2_u *) -lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -/3 width=3/ qed. - -lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2. -/3 width=3/ qed. - -lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2. -/3 width=3/ qed-. - -(* Basic_1: was: pc3_pr2_u2 *) -lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -/3 width=3/ qed. - -(* Basic_1: removed theorems 9: - clear_pc3_trans pc3_ind_left - pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21 - pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0 - Basic_1: removed local theorems 6: - pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left - pc3_wcpr0_t_aux -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_aaa.ma deleted file mode 100644 index 363b8acd3..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_aaa.ma +++ /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/cprs_aaa.ma". -include "basic_2/equivalence/cpcs_cpcs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -(* Main properties about atomic arity assignment on terms *******************) - -theorem aaa_cpcs_mono: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → - ∀A1. L ⊢ T1 ⁝ A1 → ∀A2. L ⊢ T2 ⁝ A2 → - A1 = A2. -#L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2 -elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 -lapply (aaa_cprs_conf … HA1 … HT1) -T1 #HA1 -lapply (aaa_cprs_conf … HA2 … HT2) -T2 #HA2 -lapply (aaa_mono … HA1 … HA2) -L -T // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma deleted file mode 100644 index ac9de9300..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma +++ /dev/null @@ -1,204 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_lift.ma". -include "basic_2/computation/cprs_cprs.ma". -include "basic_2/conversion/cpc_cpc.ma". -include "basic_2/equivalence/cpcs_cprs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -(* Advanced inversion lemmas ************************************************) - -lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → - ∃∃T. L ⊢ T1 ➡* T & L ⊢ T2 ➡* T. -#L #T1 #T2 #H @(cpcs_ind … H) -T2 -[ /3 width=3/ -| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 - [ elim (cprs_strip … HT0 … HT2) -T #T #HT0 #HT2 - lapply (cprs_strap1 … HT10 … HT0) -T0 /2 width=3/ - | lapply (cprs_strap2 … HT2 … HT0) -T /2 width=3/ - ] -] -qed-. - -(* Basic_1: was: pc3_gen_sort *) -lemma cpcs_inv_sort: ∀L,k1,k2. L ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2. -#L #k1 #k2 #H -elim (cpcs_inv_cprs … H) -H #T #H1 ->(cprs_inv_sort1 … H1) -T #H2 -lapply (cprs_inv_sort1 … H2) -L #H destruct // -qed-. - -(* Basic_1: was: pc3_gen_sort_abst *) -lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. L ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥. -#a #L #W #T #k #H -elim (cpcs_inv_cprs … H) -H #X #H1 ->(cprs_inv_sort1 … H1) -X #H2 -elim (cprs_inv_abst1 Abst W … H2) -H2 #W0 #T0 #_ #_ #H destruct -qed-. - -(* Basic_1: was: pc3_gen_abst *) -lemma cpcs_inv_abst: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → ∀I,V. - ∧∧ L ⊢ W1 ⬌* W2 & L. ②{I}V ⊢ T1 ⬌* T2 & a1 = a2. -#a1 #a2 #L #W1 #W2 #T1 #T2 #H #I #V -elim (cpcs_inv_cprs … H) -H #T #H1 #H2 -elim (cprs_inv_abst1 I V … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct -elim (cprs_inv_abst1 I V … H2) -H2 #W #T #HW2 #HT2 #H destruct /3 width=3/ -qed-. - -(* Basic_1: was: pc3_gen_abst_shift *) -lemma cpcs_inv_abst_shift: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → ∀W. - ∧∧ L ⊢ W1 ⬌* W2 & L. ⓛW ⊢ T1 ⬌* T2 & a1 = a2. -#a1 #a2 #L #W1 #W2 #T1 #T2 #H #W -lapply (cpcs_inv_abst … H Abst W) -H // -qed. - -lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. L ⊢ ⓛ{a}W1.T1 ⬌* T → - ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2. -#a #L #W1 #T1 #T #H -elim (cpcs_inv_cprs … H) -H #X #H1 #H2 -elim (cprs_inv_abst1 Abst W1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct -@(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *) -qed-. - -lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. L ⊢ T ⬌* ⓛ{a}W1.T1 → - ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2. -/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-. - -(* Basic_1: was: pc3_gen_lift *) -lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → - ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 → - L ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12 -elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2 -elim (cprs_inv_lift1 … HLK … HTU1 … HU1) -U1 #T #HTU #HT1 -elim (cprs_inv_lift1 … HLK … HTU2 … HU2) -L -U2 #X #HXU ->(lift_inj … HXU … HTU) -X -U -d -e /2 width=3/ -qed-. - -(* Advanced properties ******************************************************) - -lemma cpr_cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2. -#L #T #T1 #T2 #HT1 #HT2 -elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/ -qed-. - -lemma cprs_cpr_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1. -#L #T #T1 #T2 #HT1 #HT2 -elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/ -qed-. - -lemma cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2. -#L #T #T1 #T2 #HT1 #HT2 -elim (cprs_conf … HT1 … HT2) /2 width=3/ -qed-. - -(* Basic_1: was only: pc3_thin_dx *) -lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 → - ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 #I -elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2 -elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *) -qed. - -lemma cpcs_flat_dx_tpr_rev: ∀L,V1,V2. V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 → - ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2. -/3 width=1/ qed. - -lemma cpcs_abst: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → - ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛ{a}V1. T1 ⬌* ⓛ{a}V2. T2. -#a #L #V1 #V2 #HV12 #V #T1 #T2 #HT12 -elim (cpcs_inv_cprs … HV12) -HV12 -elim (cpcs_inv_cprs … HT12) -HT12 -/3 width=6 by cprs_div, cprs_abst/ (**) (* /3 width=6/ is a bit slow *) -qed. - -lemma cpcs_abbr_dx: ∀a,L,V,T1,T2. L.ⓓV ⊢ T1 ⬌* T2 → L ⊢ ⓓ{a}V. T1 ⬌* ⓓ{a}V. T2. -#a #L #V #T1 #T2 #HT12 -elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *) -qed. - -lemma cpcs_bind_dx: ∀a,I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 → - L ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2. -#a * /2 width=1/ /2 width=2/ qed. - -lemma cpcs_abbr_sn: ∀a,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓓ{a}V1. T ⬌* ⓓ{a}V2. T. -#a #L #V1 #V2 #T #HV12 -elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *) -qed. - -lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T. -#a * /2 width=1/ /2 width=2/ qed. - -lemma cpcs_beta_dx: ∀a,L,V1,V2,W,T1,T2. - L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ⬌* T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ⬌* ⓓ{a}V2.T2. -#a #L #V1 #V2 #W #T1 #T2 #HV12 #HT12 -elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 -lapply (cprs_beta_dx … HV12 HT1 a) -HV12 -HT1 #HT1 -lapply (cprs_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 -@(cprs_div … HT1) /2 width=1/ -qed. - -lemma cpcs_beta_dx_tpr_rev: ∀a,L,V1,V2,W,T1,T2. - V1 ➡ V2 → L.ⓛW ⊢ T2 ⬌* T1 → - L ⊢ ⓓ{a}V2.T2 ⬌* ⓐV1.ⓛ{a}W.T1. -/4 width=1/ qed. - -(* Note: it does not hold replacing |L1| with |L2| *) -lemma cpcs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 → - ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ⬌* T2. -#L1 #T1 #T2 #HT12 -elim (cpcs_inv_cprs … HT12) -HT12 -/3 width=5 by cprs_div, cprs_lsubs_trans/ (**) (* /3 width=5/ is a bit slow *) -qed. - -(* Basic_1: was: pc3_lift *) -lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → - ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 → - K ⊢ T1 ⬌* T2 → L ⊢ U1 ⬌* U2. -#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12 -elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 -elim (lift_total T d e) #U #HTU -lapply (cprs_lift … HLK … HTU1 … HT1 … HTU) -T1 #HU1 -lapply (cprs_lift … HLK … HTU2 … HT2 … HTU) -K -T2 -T -d -e /2 width=3/ -qed. - -lemma cpcs_strip: ∀L,T1,T. L ⊢ T ⬌* T1 → ∀T2. L ⊢ T ⬌ T2 → - ∃∃T0. L ⊢ T1 ⬌ T0 & L ⊢ T2 ⬌* T0. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -(* Basic_1: was pc3_t *) -theorem cpcs_trans: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -/2 width=3/ qed. - -theorem cpcs_canc_sn: ∀L,T,T1,T2. L ⊢ T ⬌* T1 → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -/3 width=3 by cpcs_trans, cpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) - -theorem cpcs_canc_dx: ∀L,T,T1,T2. L ⊢ T1 ⬌* T → L ⊢ T2 ⬌* T → L ⊢ T1 ⬌* T2. -/3 width=3 by cpcs_trans, cpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) - -lemma cpcs_abbr1: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV1 ⊢ T1 ⬌* T2 → - L ⊢ ⓓ{a}V1. T1 ⬌* ⓓ{a}V2. T2. -#a #L #V1 #V2 #HV12 #T1 #T2 #HT12 -@(cpcs_trans … (ⓓ{a}V1.T2)) /2 width=1/ -qed. - -lemma cpcs_abbr2: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV2 ⊢ T1 ⬌* T2 → - L ⊢ ⓓ{a}V1. T1 ⬌* ⓓ{a}V2. T2. -#a #L #V1 #V2 #HV12 #T1 #T2 #HT12 -@(cpcs_trans … (ⓓ{a}V2.T1)) /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cprs.ma deleted file mode 100644 index cb1f5d76a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cprs.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 "basic_2/computation/cprs.ma". -include "basic_2/equivalence/cpcs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -(* Properties about context sensitive computation on terms ******************) - -(* Basic_1: was: pc3_pr3_r *) -lemma cpcs_cprs_dx: ∀L,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ T1 ⬌* T2. -#L #T1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /3 width=3/ -qed. - -(* Basic_1: was: pc3_pr3_x *) -lemma cpcs_cprs_sn: ∀L,T1,T2. L ⊢ T2 ➡* T1 → L ⊢ T1 ⬌* T2. -#L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /3 width=3/ -qed. - -lemma cpcs_cprs_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2. -#L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /2 width=3/ -qed. - -lemma cpcs_cprs_strap2: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -#L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /width=1/ /2 width=3/ -qed. - -lemma cpcs_cprs_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2. -#L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /2 width=3/ -qed. - -(* Basic_1: was: pc3_pr3_conf *) -lemma cpcs_cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. -#L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /width=1/ /2 width=3/ -qed. - -(* Basic_1: was: pc3_pr3_t *) -(* Basic_1: note: pc3_pr3_t should be renamed *) -lemma cprs_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2. -#L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=1/ /2 width=3/ -qed. - -lemma cprs_cpr_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2. -/3 width=5 by step, cprs_div/ qed-. - -lemma cpr_cprs_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2. -/3 width=3 by step, cprs_div/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma deleted file mode 100644 index 7012ec11e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma +++ /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: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 → - L ⊢ T2 ⬌* +ⓓV.T1. -/3 width=1/ qed. - -(* Basic_1: was only: pc3_gen_cabbr *) -lemma thin_cpcs_delift_mono: ∀L,U1,U2. L ⊢ U1 ⬌* U2 → - ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 → - ∀T2. L ⊢ ▼*[d, e] U2 ≡ T2 → K ⊢ T1 ⬌* T2. -#L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2 -elim (cpcs_inv_cprs … H) -H #U #HU1 #HU2 -elim (thin_cprs_delift_conf … HU1 … HLK … HTU1) -U1 #T #HT1 #HUT -elim (thin_cprs_delift_conf … HU2 … HLK … HTU2) -U2 -HLK #X #HT2 #H -lapply (delift_mono … H … HUT) -L #H destruct /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_ltpr.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_ltpr.ma deleted file mode 100644 index 13713e51e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_ltpr.ma +++ /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/cpr_ltpr.ma". -include "basic_2/equivalence/cpcs_cpcs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -(* Properties about context-free parallel reduction on local environments ***) - -(* Basic_1: was only: pc3_pr0_pr2_t *) -(* Basic_1: note: pc3_pr0_pr2_t should be renamed *) -lemma ltpr_cpr_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2. -#L1 #L2 #HL12 #T1 #T2 #HT12 -elim (cpr_ltpr_conf_eq … HT12 … HL12) -L1 #T #HT1 #HT2 -@(cprs_div … T) /2 width=1/ /3 width=1/ (**) (* /4 width=3/ is too long *) -qed. - -(* Basic_1: was: pc3_wcpr0_t *) -(* Basic_1: note: pc3_wcpr0_t should be renamed *) -lemma ltpr_cprs_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2. -#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 -@(cpcs_trans … IHT1) -T1 /2 width=3/ -qed. - -(* Basic_1: was: pc3_wcpr0 *) -lemma ltpr_cpcs_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2. -#L1 #L2 #HL12 #T1 #T2 #H -elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2 -@(cpcs_canc_dx … T) /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_ltpss.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_ltpss.ma deleted file mode 100644 index 43385668b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_ltpss.ma +++ /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/equivalence/cpcs_cpcs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -(* Properties concerning partial unfold on local environments ***************) - -lemma ltpss_dx_cpr_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → - ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2. -#L1 #L2 #d #e #HL12 #T1 #T2 * -lapply (ltpss_dx_weak_all … HL12) ->(ltpss_dx_fwd_length … HL12) -HL12 #HL12 #T #HT1 #HT2 -elim (ltpss_dx_tpss_conf … HT2 … HL12) -L1 #T0 #HT0 #HT20 -@(cprs_div … T0) /3 width=3/ (**) (* /4/ is too slow *) -qed. - -lemma ltpss_dx_cprs_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → - ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2. -#L1 #L2 #d #e #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 -@(cpcs_trans … IHT1) -T1 /2 width=5/ -qed. - -lemma ltpss_dx_cpcs_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → - ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2. -#L1 #L2 #d #e #HL12 #T1 #T2 #H -elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2 -@(cpcs_canc_dx … T) /2 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs.ma deleted file mode 100644 index c0e02359b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs.ma +++ /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/conversion/fpc.ma". - -(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************) - -definition fpcs: bi_relation lenv term ≝ bi_TC … fpc. - -interpretation "context-free parallel equivalence (closure)" - 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma fpcs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 → - (∀L,L2,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → R L T → R L2 T2) → - ∀L2,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L2 T2. -/3 width=7 by bi_TC_star_ind/ qed-. - -lemma fpcs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 → - (∀L1,L,T1,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → R L T → R L1 T1) → - ∀L1,T1. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L1 T1. -/3 width=7 by bi_TC_star_ind_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma fpcs_refl: bi_reflexive … fpcs. -/2 width=1/ qed. - -lemma fpcs_sym: bi_symmetric … fpcs. -/3 width=1/ qed. - -lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma fpcs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma fpcs_fpr_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=1/ qed. - -lemma fpcs_fpr_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡ ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=1/ qed. - -lemma fpcs_fpr_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → - ∀L2,T2. ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4/ qed. - -lemma fpcs_fpr_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → - ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4/ qed. - -lemma fpcs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → - ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4/ qed. - -lemma fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4/ qed-. - -lemma fpcs_fpr_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡ ⦃L1, T1⦄ → - ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_aaa.ma deleted file mode 100644 index 9f4327bff..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_aaa.ma +++ /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: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → - ∀A1. L1 ⊢ T1 ⁝ A1 → ∀A2. L2 ⊢ T2 ⁝ A2 → - A1 = A2. -#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2 -elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2 -lapply (aaa_fprs_conf … HT1 … H1) -L1 -T1 #HT1 -lapply (aaa_fprs_conf … HT2 … H2) -L2 -T2 #HT2 -lapply (aaa_mono … HT1 … HT2) -L -T // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_cpcs.ma deleted file mode 100644 index 4b51a7084..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_cpcs.ma +++ /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/fprs_cprs.ma". -include "basic_2/equivalence/cpcs_cpcs.ma". -include "basic_2/equivalence/fpcs_fprs.ma". - -(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************) - -(* Properties on context-sensitive parallel equivalence for terms ***********) - -lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄. -#L #T1 #T2 #H -elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *) -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_fpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_fpcs.ma deleted file mode 100644 index 270e8dc40..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_fpcs.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_2/computation/fprs_fprs.ma". -include "basic_2/conversion/fpc_fpc.ma". -include "basic_2/equivalence/fpcs_fprs.ma". - -(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************) - -(* Advanced inversion lemmas ************************************************) - -lemma fpcs_inv_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → - ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡* ⦃L, T⦄. -#L1 #L2 #T1 #T2 #H @(fpcs_ind … H) -L2 -T2 -[ /3 width=4/ -| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 - [ elim (fprs_strip … HT2 … HT0) -L -T #L #T #HT2 #HT0 - lapply (fprs_strap1 … HT10 … HT0) -L0 -T0 /2 width=4/ - | lapply (fprs_strap2 … HT2 … HT0) -L -T /2 width=4/ - ] -] -qed-. - -(* Advanced properties ******************************************************) - -lemma fpr_fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 -elim (fprs_strip … HT2 … HT1) /2 width=4 by fpr_fprs_div/ -qed-. - -lemma fprs_fpr_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L2, T2⦄ ⬌* ⦃L1, T1⦄. -#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 -elim (fprs_strip … HT2 … HT1) /2 width=4 by fprs_fpr_div/ -qed-. - -lemma fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 -elim (fprs_conf … HT1 … HT2) /2 width=4/ -qed-. - -lemma fpcs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ → - ∀L2,T2. ⦃L0, T0⦄ ⬌* ⦃L2, T2⦄ → - ∃∃L,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄. -/3 width=4/ qed. - -(* Main properties **********************************************************) - -theorem fpcs_trans: bi_transitive … fpcs. -/2 width=4/ qed. - -theorem fpcs_canc_sn: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ⬌* ⦃L1, T1⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) - -theorem fpcs_canc_dx: ∀L1,L2,L,T1,T2,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L2, T2⦄ ⬌* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_fprs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_fprs.ma deleted file mode 100644 index 1d3f71f9e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/fpcs_fprs.ma +++ /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: lemma 1000 *) -lemma fpcs_fprs_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /3 width=4/ -qed. - -lemma fpcs_fprs_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡* ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L2 #T1 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /3 width=4/ -qed. - -lemma fpcs_fprs_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /2 width=4/ -qed. - -lemma fpcs_fprs_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx … H) -L1 -T1 /width=1/ /2 width=4/ -qed. - -lemma fpcs_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /2 width=4/ -qed. - -lemma fpcs_fprs_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind … H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *) -qed. - -lemma fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx … H) -L2 -T2 /2 width=1/ /2 width=4/ -qed. - -lemma fprs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=7 by bi_step, fprs_div/ qed-. - -lemma fpr_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. -/3 width=4 by bi_step, fprs_div/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs.ma deleted file mode 100644 index fded17bd1..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs.ma +++ /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/conversion/lfpc.ma". - -(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************) - -definition lfpcs: relation lenv ≝ TC … lfpc. - -interpretation "focalized parallel equivalence (local environment)" - 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2). - -(* Basic eliminators ********************************************************) - -lemma lfpcs_ind: ∀L1. ∀R:predicate lenv. R L1 → - (∀L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → R L → R L2) → - ∀L2. ⦃L1⦄ ⬌* ⦃L2⦄ → R L2. -#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) // -qed-. - -lemma lfpcs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 → - (∀L1,L. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → R L → R L1) → - ∀L1. ⦃L1⦄ ⬌* ⦃L2⦄ → R L1. -#L2 #R #HL2 #IHL2 #L1 #HL12 -@(TC_star_ind_dx … HL2 IHL2 … HL12) // -qed-. - -(* Basic properties *********************************************************) - -lemma lfpcs_refl: reflexive … lfpcs. -/2 width=1/ qed. - -lemma lfprs_sym: symmetric … lfpcs. -/3 width=1/ qed. - -lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/2 width=3/ qed. - -lemma lfpcs_strap2: ∀L1,L,L2. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/2 width=3/ qed. - -lemma lfpcs_lfpr_dx: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=1/ qed. - -lemma lfpcs_lfpr_sn: ∀L1,L2. ⦃L2⦄ ➡ ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=1/ qed. - -lemma lfpcs_lfpr_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=3/ qed. - -lemma lfpcs_lfpr_strap2: ∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=3/ qed. - -lemma lfpcs_lfpr_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡ ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=3/ qed. - -lemma lfpcs_lfpr_conf: ∀L1,L. ⦃L⦄ ➡ ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_aaa.ma deleted file mode 100644 index b7cea0b7a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_aaa.ma +++ /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: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ → - ∀T,A1. L1 ⊢ T ⁝ A1 → ∀A2. L2 ⊢ T ⁝ A2 → - A1 = A2. -#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2 -elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2 -lapply (aaa_lfprs_conf … HT1 … HL1) -L1 #HT1 -lapply (aaa_lfprs_conf … HT2 … HL2) -L2 #HT2 -lapply (aaa_mono … HT1 … HT2) -L -T // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_lfpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_lfpcs.ma deleted file mode 100644 index 434068e4f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_lfpcs.ma +++ /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: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ → - ∃∃L. ⦃L1⦄ ➡* ⦃L⦄ & ⦃L2⦄ ➡* ⦃L⦄. -#L1 #L2 #H @(lfpcs_ind … H) -L2 -[ /3 width=3/ -| #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0 - [ elim (lfprs_strip … HL0 … HL2) -L #L #HL0 #HL2 - lapply (lfprs_strap1 … HL10 … HL0) -L0 /2 width=3/ - | lapply (lfprs_strap2 … HL2 … HL0) -L /2 width=3/ - ] -] -qed-. - -(* Advanced properties ******************************************************) - -lemma lfpcs_strip: ∀L,L1. ⦃L⦄ ⬌* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌ ⦃L2⦄ → - ∃∃L0. ⦃L1⦄ ⬌ ⦃L0⦄ & ⦃L2⦄ ⬌* ⦃L0⦄. -/3 width=3/ qed. - -(* Main properties **********************************************************) - -theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/2 width=3/ qed. - -theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=3 by lfpcs_trans, lfprs_sym/ qed. - -theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -/3 width=3 by lfpcs_trans, lfprs_sym/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_lfprs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_lfprs.ma deleted file mode 100644 index baf2caf27..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lfpcs_lfprs.ma +++ /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: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /3 width=3/ -qed. - -lemma lfpcs_lfprs_sn: ∀L1,L2. ⦃L2⦄ ➡* ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /3 width=3/ -qed. - -lemma lfpcs_lfprs_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L #HL1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /2 width=3/ -qed. - -lemma lfpcs_lfprs_strap2: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L #H #L2 #HL2 @(lfprs_ind_dx … H) -L1 /width=1/ /2 width=3/ -qed. - -lemma lfpcs_lfprs_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /2 width=3/ -qed. - -lemma lfpcs_lfprs_conf: ∀L1,L. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L #H #L2 #HL2 @(lfprs_ind … H) -L1 /width=1/ /2 width=3/ -qed. - -lemma lfprs_div: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. -#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /2 width=1/ /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse.ma deleted file mode 100644 index 6719c7afe..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse.ma +++ /dev/null @@ -1,114 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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.ma". -include "basic_2/computation/cprs.ma". -include "basic_2/equivalence/cpcs.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) - -(* Note: this is not transitive *) -inductive lsubse (h:sh) (g:sd h): relation lenv ≝ -| lsubse_atom: lsubse h g (⋆) (⋆) -| lsubse_pair: ∀I,L1,L2,V. lsubse h g L1 L2 → - lsubse h g (L1. ⓑ{I} V) (L2. ⓑ{I} V) -| lsubse_abbr: ∀L1,L2,V1,V2,W1,W2,l. L1 ⊢ W1 ⬌* W2 → - ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 → - lsubse h g L1 L2 → lsubse h g (L1. ⓓV1) (L2. ⓛW2) -. - -interpretation - "local environment refinement (context-sensitive parallel equivalence)" - 'CrSubEqSE h g L1 L2 = (lsubse h g L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubse_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 = ⋆ → L2 = ⋆. -#h #g #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct -] -qed-. - -lemma lsubse_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊢•⊑[g] L2 → L2 = ⋆. -/2 width=5 by lsubse_inv_atom1_aux/ qed-. - -fact lsubse_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → - ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - (∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ - ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. -#h #g #L1 #L2 * -L1 -L2 -[ #J #K1 #U1 #H destruct -| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/ -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/ -] -qed-. - -lemma lsubse_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊢•⊑[g] L2 → - (∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ - ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. -/2 width=3 by lsubse_inv_pair1_aux/ qed-. - -fact lsubse_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L2 = ⋆ → L1 = ⋆. -#h #g #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct -] -qed-. - -lemma lsubse_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊢•⊑[g] ⋆ → L1 = ⋆. -/2 width=5 by lsubse_inv_atom2_aux/ qed-. - -fact lsubse_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → - ∀I,K2,W2. L2 = K2. ⓑ{I} W2 → - (∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ - ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. -#h #g #L1 #L2 * -L1 -L2 -[ #J #K2 #U2 #H destruct -| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/ -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=10/ -] -qed-. - -lemma lsubse_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊢•⊑[g] K2. ⓑ{I} W2 → - (∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ - ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. -/2 width=3 by lsubse_inv_pair2_aux/ qed-. - -(* Basic_forward lemmas *****************************************************) - -lemma lsubse_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 ≼[0, |L1|] L2. -#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -lemma lsubse_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 ≼[0, |L2|] L2. -#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -lemma lsubse_refl: ∀h,g,L. h ⊢ L ⊢•⊑[g] L. -#h #g #L elim L -L // /2 width=1/ -qed. - -lemma lsubse_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -/3 width=5 by lsubse_fwd_lsubs2, cprs_lsubs_trans/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_cpcs.ma deleted file mode 100644 index 2cc3ce076..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_cpcs.ma +++ /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/equivalence/cpcs_cpcs.ma". -include "basic_2/equivalence/lsubse.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) - -(* Properties on context-sensitive parallel equivalence for terms ***********) - -lemma lsubse_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → - ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2. -/3 width=5 by lsubse_fwd_lsubs2, cpcs_lsubs_trans/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_ldrop.ma deleted file mode 100644 index 729f4b61f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_ldrop.ma +++ /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/equivalence/lsubse.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) - -(* Properties concerning basic local environment slicing ********************) - -(* Note: the constant 0 cannot be generalized *) -lemma lsubse_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → - ∀K1,e. ⇩[0, e] L1 ≡ K1 → - ∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & ⇩[0, e] L2 ≡ K2. -#h #g #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -] -qed-. - -(* Note: the constant 0 cannot be generalized *) -lemma lsubse_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → - ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & ⇩[0, e] L1 ≡ K1. -#h #g #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_ssta.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_ssta.ma deleted file mode 100644 index b5dc3f5ac..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/lsubse_ssta.ma +++ /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/xprs_lsubss.ma". -*) -include "basic_2/equivalence/lsubse.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) - -(* Properties on stratified native type assignment **************************) - -axiom lsubse_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g,l] U2 → - ∀L1. h ⊢ L1 ⊢•⊑[g] L2 → - ∃∃U1. ⦃h, L1⦄ ⊢ T •[g,l] U1 & L1 ⊢ U1 ⬌* U2. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csup.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csup.etc deleted file mode 100644 index dcfe086e9..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csup.etc +++ /dev/null @@ -1,157 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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 T1 ⦄ > break ⦃ L2 , break T2 ⦄ )" - non associative with precedence 45 - for @{ 'SupTerm $L1 $T1 $L2 $T2 }. - -include "basic_2/substitution/ldrop.ma". - -(* SUPCLOSURE ***************************************************************) - -inductive csup: bi_relation lenv term ≝ -| csup_lref : ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → csup L (#i) K V -| csup_bind_sn: ∀a,I,L,V,T. csup L (ⓑ{a,I}V.T) L V -| csup_bind_dx: ∀a,I,L,V,T. csup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T -| csup_flat_sn: ∀I,L,V,T. csup L (ⓕ{I}V.T) L V -| csup_flat_dx: ∀I,L,V,T. csup L (ⓕ{I}V.T) L T -. - -interpretation - "structural predecessor (closure)" - 'SupTerm L1 T1 L2 T2 = (csup L1 T1 L2 T2). - -(* Basic inversion lemmas ***************************************************) - -fact csup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ∀J. T1 = ⓪{J} → - ∃∃I,i. ⇩[0, i] L1 ≡ L2.ⓑ{I}T2 & J = LRef i. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #I #L #K #V #i #HLK #J #H destruct /2 width=4/ -| #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 csup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ > ⦃L2, T2⦄ → - ∃∃I,i. ⇩[0, i] L1 ≡ L2.ⓑ{I}T2 & J = LRef i. -/2 width=3 by csup_inv_atom1_aux/ qed-. - -fact csup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → - ∀b,J,W,U. T1 = ⓑ{b,J}W.U → - (L2 = L1 ∧ T2 = W) ∨ - (L2 = L1.ⓑ{J}W ∧ T2 = U). -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #I #L #K #V #i #_ #b #J #W #U #H destruct -| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ -| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ -| #I #L #V #T #b #J #W #U #H destruct -| #I #L #V #T #b #J #W #U #H destruct -] -qed-. - -lemma csup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ > ⦃L2, T2⦄ → - (L2 = L1 ∧ T2 = W) ∨ - (L2 = L1.ⓑ{J}W ∧ T2 = U). -/2 width=4 by csup_inv_bind1_aux/ qed-. - -fact csup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → - ∀J,W,U. T1 = ⓕ{J}W.U → - L2 = L1 ∧ (T2 = W ∨ T2 = U). -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #I #L #K #V #i #_ #J #W #U #H destruct -| #a #I #L #V #T #J #W #U #H destruct -| #a #I #L #V #T #J #W #U #H destruct -| #I #L #V #T #J #W #U #H destruct /3 width=1/ -| #I #L #V #T #J #W #U #H destruct /3 width=1/ -] -qed-. - -lemma csup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ > ⦃L2, T2⦄ → - L2 = L1 ∧ (T2 = W ∨ T2 = U). -/2 width=4 by csup_inv_flat1_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma csup_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=4 by ldrop_pair2_fwd_cw/ -qed-. - -lemma csup_fwd_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → - ∃i. ⇩[0, i] L1 ≡ L2 ∨ ⇩[0, i] L2 ≡ L1. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=2/ /4 width=2/ -#I #L1 #K1 #V1 #i #HLK1 -lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /3 width=2/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lift_csup_trans_eq: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → - ∀L,U2. ⦃L, U1⦄ > ⦃L, U2⦄ → - ∃T2. ⇧[d, e] T2 ≡ U2. -#T1 #U1 #d #e * -T1 -U1 -d -e -[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #_ #L #X #H - elim (csup_inv_bind1 … H) -H * - [ #_ #H destruct /2 width=2/ - | #H elim (discr_lpair_x_xy … H) - ] -|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H - elim (csup_inv_flat1 … H) -H #_ * #H destruct /2 width=2/ -] -#i #d #e [2,3: #_ ] #L #X #H -elim (csup_inv_atom1 … H) -H #I #j #HL #H destruct -lapply (ldrop_pair2_fwd_cw … HL X) -HL #H -elim (lt_refl_false … H) -qed-. -(* -lemma lift_csup_trans_gt: ∀L1,L2,U1,U2. ⦃L1, U1⦄ > ⦃L2, U2⦄ → - ⇩[0, 1] L2 ≡ L1 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → - ∃T2. ⇧[d + 1, e] T2 ≡ U2. -#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2 -[ #I #L1 #K1 #V #i #HLK1 #HKL1 - lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1 - lapply (ldrop_fwd_lw … HKL1) -HKL1 #HKL1 - lapply (transitive_le … HLK1 HKL1) -L1 normalize #H - - -| #a -| #a -] -#I #L1 #W1 #U1 #HL1 - - - - #X #d #e #H - lapply (ldrop_inv_refl … HL1) -HL1 -| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H - lapply (ldrop_inv_ldrop1 … HL1) - - elim (lift_inv_bind2 … H) -H #W2 #U2 #HW21 #HU21 #H destruct - - - /3 width=2/ /4 width=2/ - -*) - - - -(* Advanced inversion lemmas ************************************************) - -lemma csup_inv_lref2_be: ∀L,U,i. ⦃L, U⦄ > ⦃L, #i⦄ → - ∀T,d,e. ⇧[d, e] T ≡ U → d ≤ i → i < d + e → ⊥. -#L #U #i #H #T #d #e #HTU #Hdi #Hide -elim (lift_csup_trans_eq … HTU … H) -H -T #T #H -elim (lift_inv_lref2_be … H ? ?) // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csup_csup.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csup_csup.etc deleted file mode 100644 index 813cb969d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csup_csup.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/ldrop_ldrop.ma". -include "basic_2/substitution/csup.ma". - -(* SUPCLOSURE ***************************************************************) - -(* Advanced inversion lemmas ************************************************) - -lemma csup_inv_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → - ∀J,W,j. ⇩[0, j] L1 ≡ L2.ⓑ{J}W → T1 = #j ∧ T2 = W. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #I #L #K #V #i #HLKV #J #W #j #HLKW - elim (ldrop_conf_div … HLKV … HLKW) -L /2 width=1/ -| #a -| #a -] -#I #L #V #T #J #W #j #H -lapply (ldrop_pair2_fwd_cw … H W) -H #H -[2: lapply (transitive_lt (#{L,W}) … H) /2 width=1/ -H #H ] -elim (lt_refl_false … H) -qed-. - -(* Main forward lemmas ******************************************************) - -theorem csup_trans_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ → - ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ → - L = L0 ∨ ⦃L, T1⦄ > ⦃L, T3⦄. -#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ -[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H - lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1 - lapply (csup_fwd_cw … H) -H #H2 - lapply (transitive_lt … H1 H2) -H1 -H2 #H - elim (lt_refl_false … H) -| #a #I #L0 #V2 #T2 #T3 #HT23 - elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ - qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csupp.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csupp.etc deleted file mode 100644 index c28eaea73..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csupp.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 *) -(* *) -(**************************************************************************) - -notation "hvbox( ⦃ L1, break T1 ⦄ > + break ⦃ L2 , break T2 ⦄ )" - non associative with precedence 45 - for @{ 'SupTermPlus $L1 $T1 $L2 $T2 }. - -include "basic_2/substitution/csup.ma". - -(* PLUS-ITERATED SUPCLOSURE *************************************************) - -definition csupp: bi_relation lenv term ≝ bi_TC … csup. - -interpretation "plus-iterated structural predecessor (closure)" - 'SupTermPlus L1 T1 L2 T2 = (csupp L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma csupp_ind: ∀L1,T1. ∀R:relation2 lenv term. - (∀L2,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → R L2 T2) → - (∀L,T,L2,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → R L T → R L2 T2) → - ∀L2,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → R L2 T2. -#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H -@(bi_TC_ind … IH1 IH2 ? ? H) -qed-. - -lemma csupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. - (∀L1,T1. ⦃L1, T1⦄ > ⦃L2, T2⦄ → R L1 T1) → - (∀L1,L,T1,T. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >+ ⦃L2, T2⦄ → R L T → R L1 T1) → - ∀L1,T1. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → R L1 T1. -#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H -@(bi_TC_ind_dx … IH1 IH2 ? ? H) -qed-. - -(* Basic properties *********************************************************) - -lemma csup_csupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄. -/2 width=1/ qed. - -lemma csupp_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → - ⦃L1, T1⦄ >+ ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma csupp_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >+ ⦃L2, T2⦄ → - ⦃L1, T1⦄ >+ ⦃L2, T2⦄. -/2 width=4/ qed. - -(* Basic forward lemmas *****************************************************) - -lemma csupp_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}. -#L1 #L2 #T1 #T2 #H @(csupp_ind … H) -L2 -T2 -/3 width=3 by csup_fwd_cw, transitive_lt/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csupp_csupp.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csupp_csupp.etc deleted file mode 100644 index 5afdb68d4..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csupp_csupp.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/csupp.ma". - -(* PLUS-ITERATED SUPCLOSURE *************************************************) - -(* Main propertis ***********************************************************) - -theorem csupp_trans: bi_transitive … csupp. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups.etc deleted file mode 100644 index 7f5879426..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups.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 *) -(* *) -(**************************************************************************) - -notation "hvbox( ⦃ L1, break T1 ⦄ > * break ⦃ L2 , break T2 ⦄ )" - non associative with precedence 45 - for @{ 'SupTermStar $L1 $T1 $L2 $T2 }. - -include "basic_2/substitution/csup.ma". -include "basic_2/unfold/csupp.ma". - -(* STAR-ITERATED SUPCLOSURE *************************************************) - -definition csups: bi_relation lenv term ≝ bi_star … csup. - -interpretation "star-iterated structural predecessor (closure)" - 'SupTermStar L1 T1 L2 T2 = (csups L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma csups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 → - (∀L,L2,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → R L T → R L2 T2) → - ∀L2,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → R L2 T2. -#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H -@(bi_star_ind … IH1 IH2 ? ? H) -qed-. - -lemma csups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 → - (∀L1,L,T1,T. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >* ⦃L2, T2⦄ → R L T → R L1 T1) → - ∀L1,T1. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → R L1 T1. -#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H -@(bi_star_ind_dx … IH1 IH2 ? ? H) -qed-. - -(* Basic properties *********************************************************) - -lemma csups_refl: bi_reflexive … csups. -/2 width=1/ qed. - -lemma csupp_csups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → ⦃L1, T1⦄ >* ⦃L2, T2⦄. -/2 width=1/ qed. - -lemma csup_csups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ⦃L1, T1⦄ >* ⦃L2, T2⦄. -/2 width=1/ qed. - -lemma csups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → - ⦃L1, T1⦄ >* ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma csups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >* ⦃L2, T2⦄ → - ⦃L1, T1⦄ >* ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma csups_csupp_csupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → - ⦃L, T⦄ >+ ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma csupp_csups_csupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ → - ⦃L, T⦄ >* ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄. -/2 width=4/ qed. - -(* Basic forward lemmas *****************************************************) - -lemma csups_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}. -#L1 #L2 #T1 #T2 #H @(csups_ind … H) -L2 -T2 // -/4 width=3 by csup_fwd_cw, lt_to_le_to_lt, lt_to_le/ (**) (* slow even with trace *) -qed-. - -(* Advanced inversion lemmas for csupp **************************************) - -lemma csupp_inv_atom1_csups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ >+ ⦃L2, T2⦄ → - ∃∃I,K,V,i. ⇩[0, i] L1 ≡ K.ⓑ{I}V & - ⦃K, V⦄ >* ⦃L2, T2⦄ & J = LRef i. -#J #L1 #L2 #T2 #H @(csupp_ind … H) -L2 -T2 -[ #L2 #T2 #H - elim (csup_inv_atom1 … H) -H * #i #HL12 #H destruct /2 width=7/ -| #L #T #L2 #T2 #_ #HT2 * #I #K #V #i #HLK #HVT #H destruct /3 width=8/ -] -qed-. - -lemma csupp_inv_bind1_csups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ >+ ⦃L2, T2⦄ → - ⦃L1, W⦄ >* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ >* ⦃L2, T2⦄. -#b #J #L1 #L2 #W #U #T2 #H @(csupp_ind … H) -L2 -T2 -[ #L2 #T2 #H - elim (csup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/ -| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ -] -qed-. - -lemma csupp_inv_flat1_csups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ >+ ⦃L2, T2⦄ → - ⦃L1, W⦄ >* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ >* ⦃L2, T2⦄. -#J #L1 #L2 #W #U #T2 #H @(csupp_ind … H) -L2 -T2 -[ #L2 #T2 #H - elim (csup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/ -| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups_csups.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups_csups.etc deleted file mode 100644 index aa54d9bef..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups_csups.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/csup_csup.ma". -include "basic_2/unfold/csups.ma". - -(* STAR-ITERATED SUPCLOSURE *************************************************) - -(* Advanced forward lemmas **************************************************) - -(* -lemma csupp_strap2_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ → - ∀T3. ⦃L0, T2⦄ >+ ⦃L, T3⦄ → - L = L0 ∨ ⦃L, T1⦄ >+ ⦃L, T3⦄. -#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ -[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H - lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1 - lapply (csupp_fwd_cw … H) -H #H2 - lapply (transitive_lt … H1 H2) -H1 -H2 #H - elim (lt_refl_false … H) -| #a #I #L0 #V2 #T2 #T3 #HT23 - /3 width=5/ - - elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ - qed-. - - - - - - - - -lemma csups_strap1_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ >* ⦃L0, T2⦄ → - ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ → L = L0. -#L #L0 #T1 #T2 #H @(csups_ind_dx … H) -L -T1 // -#L1 #L #T1 #T #HL1 #_ #IHL0 #T3 #HL0 -lapply (csup_trans_fwd_refl … HL10) … HL0) -T2 -*) -lemma lift_csups_trans_aux: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → - ∀L1,L2,U2. ⦃L1, U1⦄ >* ⦃L2, U2⦄ → L1 = L2 → - ∃T2. ⇧[d, e] T2 ≡ U2. -#T1 #U1 #d #e #HTU1 #L1 #L2 #U2 #H @(csups_ind … H) -L2 -U2 /2 width=2/ -T1 -#L #L2 #U #U2 #HL1 #HL2 #IHL1 #H destruct - -* -T1 -U1 -d -e - -(* Main propertis ***********************************************************) - -theorem csups_trans: bi_transitive … csups. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ypr.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ypr.etc deleted file mode 100644 index f1510ab7e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ypr.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( h ⊢ break ⦃ L1, break T1 ⦄ • ⥸ break [ g ] break ⦃ L2 , break T2 ⦄ )" - non associative with precedence 45 - for @{ 'YPRed $h $g $L1 $T1 $L2 $T2 }. - -include "basic_2/substitution/csup.ma". -include "basic_2/reducibility/xpr.ma". - -(* HYPER PARALLEL REDUCTION ON CLOSURES *************************************) - -inductive ypr (h) (g) (L1) (T1): relation2 lenv term ≝ -| ypr_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → ypr h g L1 T1 L1 T2 -| ypr_ssta: ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] T2 → ypr h g L1 T1 L1 T2 -| ypr_csup: ∀L2,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ypr h g L1 T1 L2 T2 -. - -interpretation - "hyper parallel reduction (closure)" - 'YPRed h g L1 T1 L2 T2 = (ypr h g L1 T1 L2 T2). - -(* Basic properties *********************************************************) - -lemma ypr_refl: ∀h,g. bi_reflexive … (ypr h g). -/2 width=1/ qed. - -lemma xpr_ypr: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •➡[g] T2 → h ⊢ ⦃L, T1⦄ •⥸[g] ⦃L, T2⦄. -#h #g #L #T1 #T2 * /2 width=1/ /2 width=2/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs.etc deleted file mode 100644 index 86dc0c135..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs.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 *) -(* *) -(**************************************************************************) - -notation "hvbox( h ⊢ break ⦃ L1, break T1 ⦄ • ⥸ * break [ g ] break ⦃ L2 , break T2 ⦄ )" - non associative with precedence 45 - for @{ 'YPRedStar $h $g $L1 $T1 $L2 $T2 }. - -include "basic_2/reducibility/ypr.ma". - -(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************) - -definition yprs: ∀h. sd h → bi_relation lenv term ≝ - λh,g. bi_TC … (ypr h g). - -interpretation "hyper parallel computation (closure)" - 'YPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 → - (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ •⥸[g] ⦃L2, T2⦄ → R L T → R L2 T2) → - ∀L2,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → R L2 T2. -/3 width=7 by bi_TC_star_ind/ qed-. - -lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 → - (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ •⥸[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ •⥸*[g] ⦃L2, T2⦄ → R L T → R L1 T1) → - ∀L1,T1. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → R L1 T1. -/3 width=7 by bi_TC_star_ind_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g). -/2 width=1/ qed. - -lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L, T⦄ → - h ⊢ ⦃L, T⦄ •⥸[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄. -/2 width=4/ qed. - -lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ •⥸[g] ⦃L, T⦄ → - h ⊢ ⦃L, T⦄ •⥸*[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_csups.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_csups.etc deleted file mode 100644 index 08c939d8d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_csups.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/unfold/csups.ma". -include "basic_2/computation/yprs.ma". - -(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************) - -(* Properties on iterated supclosure ****************************************) - -lemma csups_yprs: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → - h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄. -#h #g #L1 #L2 #T1 #T2 #H @(csups_ind … H) -L2 -T2 /3 width=1/ /3 width=4/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_xprs.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_xprs.etc deleted file mode 100644 index 2feb88a2f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_xprs.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/computation/xprs_cprs.ma". -include "basic_2/computation/yprs.ma". - -(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************) - -(* Properties on extended parallel computation for terms ********************) - -lemma xprs_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → - h ⊢ ⦃L, T1⦄ •⥸*[g] ⦃L, T2⦄. -#h #g #L #T1 #T2 #H @(xprs_ind … H) -T2 // /3 width=4/ -qed. - -lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ •⥸*[g] ⦃L, T2⦄. -/3 width=1/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_yprs.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_yprs.etc deleted file mode 100644 index d737dd817..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/yprs_yprs.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/yprs.ma". - -(* HYPER PARALLEL COMPUTATION ON TERMS **************************************) - -theorem yprs_trans: ∀h,g. bi_transitive … (yprs h g). -/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ysteps.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ysteps.etc deleted file mode 100644 index 149e7895b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ysteps.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 *) -(* *) -(**************************************************************************) - -notation "hvbox( h ⊢ break ⦃ L1, break T1 ⦄ • ⭃ * break [ g ] break ⦃ L2 , break T2 ⦄ )" - non associative with precedence 45 - for @{ 'YPRedStepStar $h $g $L1 $T1 $L2 $T2 }. - -include "basic_2/substitution/csup.ma". -include "basic_2/computation/yprs.ma". - -(* ITERATED STEP OF HYPER PARALLEL COMPUTATION ON CLOSURES ******************) - -inductive ysteps (h) (g) (L1) (T1) (L2) (T2): Prop ≝ -| ysteps_intro: h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → (L1 = L2 → T1 = T2 → ⊥) → - ysteps h g L1 T1 L2 T2 -. - -interpretation "iterated step of hyper parallel computation (closure)" - 'YPRedStepStar h g L1 T1 L2 T2 = (ysteps h g L1 T1 L2 T2). - -(* Basic properties *********************************************************) - -lemma ssta_ysteps: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → - h ⊢ ⦃L, T⦄ •⭃*[g] ⦃L, U⦄. -#h #g #L #T #U #l #HTU -@ysteps_intro /3 width=2/ #_ #H destruct -elim (ssta_inv_refl … HTU) -qed. - -lemma csup_ysteps: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → - h ⊢ ⦃L1, T1⦄ •⭃*[g] ⦃L2, T2⦄. -#h #g #L1 #L2 #T1 #T2 #H -lapply (csup_fwd_cw … H) #H1 -@ysteps_intro /3 width=1/ -H #H2 #H3 destruct -elim (lt_refl_false … H1) -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ysteps_csups.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ysteps_csups.etc deleted file mode 100644 index 2e48f396d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/csup/ysteps_csups.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/computation/yprs_csups.ma". -include "basic_2/computation/ysteps.ma". - -(* ITERATED STEP OF HYPER PARALLEL COMPUTATION ON CLOSURES ******************) - -(* Properties on iterated supclosure ****************************************) - -lemma csups_ysteps: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → - h ⊢ ⦃L1, T1⦄ •⭃*[g] ⦃L2, T2⦄. -#h #g #L1 #L2 #T1 #T2 #H -lapply (csups_fwd_cw … H) #H1 -@ysteps_intro /2 width=1/ -H #H2 #H3 destruct -elim (lt_refl_false … H1) -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/hod/ntas.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/hod/ntas.etc deleted file mode 100644 index 8cfaa343b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/hod/ntas.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 *) -(* *) -(**************************************************************************) - -notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 : : * break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeTypeStarAlt $h $L $T1 $T2 }. - -include "basic_2/dynamic/nta.ma". - -(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************) - -definition ntas: sh → lenv → relation term ≝ - λh,L. star … (nta h L). - -interpretation "higher order native type assignment (term)" - 'NativeTypeStar h L T U = (ntas h L T U). - -(* Basic eliminators ********************************************************) -(* -lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 → - (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) → - ∀T2. L ⊢ T1 ➡* T2 → R T2. -#L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind … HT1 IHT1 … HT12) // -qed-. -*) -axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 → - (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) → - ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1. -(* -#h #L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(star_ind_dx … HT2 IHT2 … HT12) // -qed-. -*) -(* Basic properties *********************************************************) - -lemma ntas_refl: ∀h,L,T. ⦃h, L⦄ ⊢ T :* T. -// qed. - -lemma ntas_strap1: ∀h,L,T1,T,T2. - ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2. -/2 width=3/ qed. - -lemma ntas_strap2: ∀h,L,T1,T,T2. - ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2. -/2 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/hod/ntas_lift.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/hod/ntas_lift.etc deleted file mode 100644 index 1adced79d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/hod/ntas_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/dynamic/nta_lift.ma". -include "basic_2/hod/ntas.ma". - -(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************) - -(* Advanced properties on native type assignment for terms ******************) - -lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U → - ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U. -#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/ -qed. - -axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U → - ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z. -(* REQUIRES SUBJECT CONVERSION -#h #L #T #W #Y #H @(ntas_ind_dx … H) -T -[ #U #HYU - elim (nta_fwd_correct … HYU) #U0 #HU0 - elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U -*) - -(* Advanced inversion lemmas on native type assignment for terms ************) - -fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X → - ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & - L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. -#h #L #Z #U #H elim H -L -Z -U -[ #L #k #X #Y #H destruct -| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct -| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct -| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct -| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/ -| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct - elim (IHUW U Y ?) -IHUW // /3 width=9/ -| #L #Z #U #_ #_ #X #Y #H destruct -| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct - elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT - lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/ -] -qed. - -(* Basic_1: was only: ty3_gen_appl *) -lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U → - ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & - L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. -/2 width=3/ qed-. - -axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U → - ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W & - L ⊢ ⓐZ.ⓛY.W ⬌* U. -(* REQUIRES SUBJECT REDUCTION -#h #L #Z #Y #X #U #H -elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT -elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V -lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U -@(ex3_1_intro … HX0U) /2 width=2/ -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lcpcs.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lcpcs.etc deleted file mode 100644 index d815739fb..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lcpcs.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( L1 ⊢ ⬌* break term 46 L2 )" - non associative with precedence 45 - for @{ 'CPConvStar $L1 $L2 }. - -include "basic_2/grammar/lenv_px_sn.ma". -include "basic_2/equivalence/cpcs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************) - -definition lcpcs: relation lenv ≝ lpx_sn … cpcs. - -interpretation "context-sensitive parallel equivalence (local environment)" - 'CPConvStar L1 L2 = (lcpcs L1 L2). - -(* Basic inversion lemmas ***************************************************) - -lemma lcpcs_inv_atom1: ∀L2. ⋆ ⊢ ⬌* L2 → L2 = ⋆. -/2 width=2 by lpx_sn_inv_atom1/ qed-. - -lemma lcpcs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ⬌* L2 → - ∃∃K2,V2. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L2 = K2. ⓑ{I} V2. -/2 width=1 by lpx_sn_inv_pair1/ qed-. - -lemma lcpcs_inv_atom2: ∀L1. L1 ⊢ ⬌* ⋆ → L1 = ⋆. -/2 width=2 by lpx_sn_inv_atom2/ qed-. - -lemma lcpcs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ⬌* K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L1 = K1. ⓑ{I} V1. -/2 width=1 by lpx_sn_inv_pair2/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lcpcs_fwd_length: ∀L1,L2. L1 ⊢ ⬌* L2 → |L1| = |L2|. -/2 width=2 by lpx_sn_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lcpcs_ltpr.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lcpcs_ltpr.etc deleted file mode 100644 index ecc6be867..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lcpcs_ltpr.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/ltpr.ma". -include "basic_2/equivalence/lcpcs.ma". - -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************) - -(* Properties on context-free parallel reduction for local environments *****) - -lemma ltpr_lcpcs: ∀L1,L2. L1 ➡ L2 → L1 ⊢ ⬌* L2. -#L1 #L2 #H elim H -L1 -L2 // /4 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lenv_px_sn.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lenv_px_sn.etc deleted file mode 100644 index fddab0332..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lenv_px/lenv_px_sn.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/grammar/lenv_length.ma". - -(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **********) - -inductive lpx_sn (R:lenv→relation term): relation lenv ≝ -| lpx_sn_stom: lpx_sn R (⋆) (⋆) -| lpx_sn_pair: ∀I,K1,K2,V1,V2. - lpx_sn R K1 K2 → R K1 V1 V2 → lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) -. - -(* Basic inversion lemmas ***************************************************) - -fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆. -/2 width=4 by lpx_sn_inv_atom1_aux/ qed-. - -fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2. -#R #L1 #L2 * -L1 -L2 -[ #J #K1 #V1 #H destruct -| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/ -] -qed-. - -lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 → - ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2. -/2 width=3 by lpx_sn_inv_pair1_aux/ qed-. - -fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆. -/2 width=4 by lpx_sn_inv_atom2_aux/ qed-. - -fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1. -#R #L1 #L2 * -L1 -L2 -[ #J #K2 #V2 #H destruct -| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/ -] -qed-. - -lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) → - ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1. -/2 width=3 by lpx_sn_inv_pair2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|. -#R #L1 #L2 #H elim H -L1 -L2 normalize // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubn/lsubn_lsubn.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lsubn/lsubn_lsubn.etc deleted file mode 100644 index 9ef3dda88..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/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: ∀h,L1,L. h ⊢ L1 :⊑ L → ∀L2. h ⊢ L :⊑ L2 → h ⊢ L1 :⊑ L2. -#h #L1 #L #H elim H -L1 -L -[ #X #H >(lsubn_inv_atom1 … H) -H // -| #I #L1 #L #V #HL1 #H1W #IHL1 #X #H - elim (lsubn_inv_pair1 … H) -H * #L2 - [ #HL2 #H #H2W destruct /4 width=1/ - | #W #H1VW #H2VW #HL2 #H1 #H2 destruct /3 width=3/ - ] -| #L1 #L #V1 #W1 #H1VW1 #H2VW1 #HL1 #IHL1 #X #H - elim (lsubn_inv_pair1 … H) -H * #L2 - [ #HL2 #H #HW destruct /3 width=1/ - | #V #_ #_ #_ #_ #H destruct - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv.etc deleted file mode 100644 index 25122262a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv.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 *) -(* *) -(**************************************************************************) - -include "basic_2/dynamic/snv.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) - -(* Note: this is not transitive *) -inductive lsubsv (h:sh) (g:sd h): relation lenv ≝ -| lsubsv_atom: lsubsv h g (⋆) (⋆) -| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 → - lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V) -| lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → L1 ⊢ W2 ⬌* W1 → - ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 → - lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2) -. - -interpretation - "local environment refinement (stratified native validity)" - 'CrSubEqV h g L1 L2 = (lsubsv h g L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ → L2 = ⋆. -#h #g #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct -] -qed-. - -lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆. -/2 width=5 by lsubsv_inv_atom1_aux/ qed-. - -fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ - ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. -#h #g #L1 #L2 * -L1 -L2 -[ #J #K1 #U1 #H destruct -| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/ -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/ -] -qed-. - -lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 → - (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ - ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. -/2 width=3 by lsubsv_inv_pair1_aux/ qed-. - -fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆. -#h #g #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct -] -qed-. - -lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆. -/2 width=5 by lsubsv_inv_atom2_aux/ qed-. - -fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀I,K2,W2. L2 = K2. ⓑ{I} W2 → - (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ - ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. -#h #g #L1 #L2 * -L1 -L2 -[ #J #K2 #U2 #H destruct -| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/ -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/ -] -qed-. - -lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 → - (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ - ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & - K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. -/2 width=3 by lsubsv_inv_pair2_aux/ qed-. - -(* Basic_forward lemmas *****************************************************) - -lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2. -#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2. -#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -lemma lsubsv_refl: ∀h,g,L. h ⊢ L ⊩:⊑[g] L. -#h #g #L elim L -L // /2 width=1/ -qed. - -lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -/3 width=5 by lsubsv_fwd_lsubs2, cprs_lsubs_trans/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_cpcs.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_cpcs.etc deleted file mode 100644 index 87a72e0fd..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_cpcs.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/equivalence/cpcs_cpcs.ma". -include "basic_2/dynamic/lsubsv.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) - -(* Properties on context-sensitive parallel equivalence for terms ***********) - -lemma lsubsv_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2. -/3 width=5 by lsubsv_fwd_lsubs2, cpcs_lsubs_trans/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_ldrop.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_ldrop.etc deleted file mode 100644 index 2c3381f86..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_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/lsubsv.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) - -(* Properties concerning basic local environment slicing ********************) - -(* Note: the constant 0 cannot be generalized *) -lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀K1,e. ⇩[0, e] L1 ≡ K1 → - ∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & ⇩[0, e] L2 ≡ K2. -#h #g #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -] -qed. - -(* Note: the constant 0 cannot be generalized *) -lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & ⇩[0, e] L1 ≡ K1. -#h #g #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_snv.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_snv.etc deleted file mode 100644 index e5bd9951e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_snv.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/dynamic/lsubsv_ldrop.ma". -include "basic_2/dynamic/lsubsv_ssta.ma". -include "basic_2/dynamic/lsubsv_cpcs.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) - -(* Properties concerning stratified native validity *************************) - -axiom lsubsv_xprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → - ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2. -(* -/3 width=3 by lsubsv_fwd_lsubss, lsubss_xprs_trans/ qed-. -*) -axiom lsubsv_snv_trans: ∀h,g,L2,T. ⦃h, L2⦄ ⊩ T :[g] → - ∀L1. h ⊢ L1 ⊩:⊑[g] L2 → ⦃h, L1⦄ ⊩ T :[g]. -(* -#h #g #L2 #T #H elim H -L2 -T // -[ #I2 #L2 #K2 #V2 #i #HLK2 #_ #IHV2 #L1 #HL12 - elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 - elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -IHV2 ] - [ #HK12 #H destruct /3 width=5/ - | #V1 #l #HV1 #_ #_ #_ #H destruct /2 width=5/ - ] -| #a #I #L2 #V #T #_ #_ #IHV #IHT #L1 #HL12 /4 width=1/ -| #a #L2 #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU #IHV #IHT #L1 #HL12 - lapply (IHV … HL12) -IHV #HV - lapply (IHT … HL12) -IHT #HT - lapply (lsubsv_ssta_trans … HVW … HL12) -HVW #HVW - lapply (lsubsv_cprs_trans … HL12 … HW0) -HW0 #HW0 - lapply (lsubsv_xprs_trans … HL12 … HTU) -HL12 -HTU /2 width=8/ -| #L2 #W #T #U #l #_ #_ #HTU #HWU #IHW #IHT #L1 #HL12 - lapply (IHW … HL12) -IHW #HW - lapply (IHT … HL12) -IHT #HT - lapply (lsubsv_ssta_trans … HTU … HL12) -HTU #HTU - lapply (lsubsv_cpcs_trans … HL12 … HWU) -HL12 -HWU /2 width=4/ -] -qed-. -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_ssta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_ssta.etc deleted file mode 100644 index 1e5b5fddd..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/lsubsv/lsubsv_ssta.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/xprs_lsubss.ma". -include "basic_2/dynamic/lsubsv.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) - -(* Properties on stratified native type assignment **************************) - -axiom lsubsv_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g,l] U2 → - ∀L1. h ⊢ L1 ⊩:⊑[g] L2 → - ∃∃U1. L1 ⊢ U2 ⬌* U1 & ⦃h, L1⦄ ⊢ T •[g,l] U1. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn.etc deleted file mode 100644 index c4359c35f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/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 ⊢ break term 46 L1 : ⊑ break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqN $h $L1 $L2 }. - -notation "hvbox( h ⊢ break term 46 L1 : : ⊑ break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqNAlt $h $L1 $L2 }. - -include "basic_2/dynamic/nta.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************) - -(* Note: may not be transitive *) -inductive lsubn (h:sh): relation lenv ≝ -| lsubn_atom: lsubn h (⋆) (⋆) -| lsubn_pair: ∀I,L1,L2,W. lsubn h L1 L2 → lsubn h (L1. ⓑ{I} W) (L2. ⓑ{I} W) -| lsubn_abbr: ∀L1,L2,V,W. ⦃h, L1⦄ ⊢ V : W → ⦃h, L2⦄ ⊢ V : W → - lsubn h L1 L2 → lsubn h (L1. ⓓV) (L2. ⓛW) -. - -interpretation - "local environment refinement (native type assigment)" - 'CrSubEqN h L1 L2 = (lsubn h L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 = ⋆ → L2 = ⋆. -#h #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #_ #_ #_ #H destruct -] -qed. - -lemma lsubn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑ L2 → L2 = ⋆. -/2 width=4/ qed-. - -fact lsubn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V → - (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & - h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr. -#h #L1 #L2 * -L1 -L2 -[ #I #K1 #V #H destruct -| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/ -| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/ -] -qed. - -lemma lsubn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑ L2 → - (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & - h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr. -/2 width=3/ qed-. - -fact lsubn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L2 = ⋆ → L1 = ⋆. -#h #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #_ #_ #_ #H destruct -] -qed. - -lemma lsubc_inv_atom2: ∀h,L1. h ⊢ L1 :⊑ ⋆ → L1 = ⋆. -/2 width=4/ qed-. - -fact lsubn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W → - (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & - h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst. -#h #L1 #L2 * -L1 -L2 -[ #I #K2 #W #H destruct -| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/ -| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/ -] -qed. - -(* Basic_1: was: csubt_gen_bind *) -lemma lsubn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑ K2. ⓑ{I} W → - (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & - h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst. -/2 width=3/ qed-. - -(* Basic_forward lemmas *****************************************************) - -lemma lsubn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L1|] L2. -#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -lemma lsubn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L2|] L2. -#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: csubt_refl *) -lemma lsubn_refl: ∀h,L. h ⊢ L :⊑ L. -#h #L elim L -L // /2 width=1/ -qed. - -(* Basic_1: removed theorems 6: - csubt_gen_flat csubt_drop_flat csubt_clear_conf - csubt_getl_abbr csubt_getl_abst csubt_ty3_ld -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn_cpcs.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn_cpcs.etc deleted file mode 100644 index 5f610bc96..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/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: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → - ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2. -/3 width=4 by lsubn_fwd_lsubs2, cpr_lsubs_trans/ qed. - -lemma cprs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -/3 width=4 by lsubn_fwd_lsubs2, cprs_lsubs_trans/ qed. - -(* Basic_1: was: csubt_pc3 *) -lemma cpcs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → - ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2. -/3 width=4 by lsubn_fwd_lsubs2, cpcs_lsubs_trans/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn_ldrop.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn_ldrop.etc deleted file mode 100644 index a16fff618..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/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: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 → - ∃∃K2. h ⊢ K1 :⊑ K2 & ⇩[0, e] L2 ≡ K2. -#h #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -] -qed. - -(* Note: the constant 0 cannot be generalized *) -(* Basic_1: was only: csubt_drop_abbr csubt_drop_abst *) -lemma lsubn_ldrop_O1_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. h ⊢ K1 :⊑ K2 & ⇩[0, e] L1 ≡ K1. -#h #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn_nta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/lsubn_nta.etc deleted file mode 100644 index 5832b00b6..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/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: ∀h,L2,T,U. ⦃h, L2⦄ ⊢ T : U → ∀L1. h ⊢ L1 :⊑ L2 → - ⦃h, L1⦄ ⊢ T : U. -#h #L2 #T #U #H elim H -L2 -T -U -[ // -| #L2 #K2 #V2 #W2 #U2 #i #HLK2 #_ #WU2 #IHVW2 #L1 #HL12 - elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 - elim (lsubn_inv_pair2 … H) -H * #K1 - [ #HK12 #H destruct /3 width=6/ - | #V1 #_ #_ #_ #_ #H destruct - ] -| #L2 #K2 #W2 #V2 #U2 #i #HLK2 #_ #HWU2 #IHWV2 #L1 #HL12 - elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 - elim (lsubn_inv_pair2 … H) -H * #K1 [ | -IHWV2 ] - [ #HK12 #H destruct /3 width=6/ - | #V1 #H1V1W2 #_ #_ #H #_ destruct /2 width=6/ - ] -| /4 width=2/ -| /3 width=1/ -| /3 width=2/ -| /3 width=1/ -| /4 width=6/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta.etc deleted file mode 100644 index fa4a8ed8f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta.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/static/sh.ma". -include "basic_2/equivalence/cpcs.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -inductive nta (h:sh): lenv → relation term ≝ -| nta_sort: ∀L,k. nta h L (⋆k) (⋆(next h k)) -| nta_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → nta h K V W → - ⇧[0, i + 1] W ≡ U → nta h L (#i) U -| nta_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → nta h K W V → - ⇧[0, i + 1] W ≡ U → nta h L (#i) U -| nta_bind: ∀I,L,V,W,T,U. nta h L V W → nta h (L. ⓑ{I} V) T U → - nta h L (ⓑ{I}V.T) (ⓑ{I}V.U) -| nta_appl: ∀L,V,W,T,U. nta h L V W → nta h L (ⓛW.T) (ⓛW.U) → - nta h L (ⓐV.ⓛW.T) (ⓐV.ⓛW.U) -| nta_pure: ∀L,V,W,T,U. nta h L T U → nta h L (ⓐV.U) W → - nta h L (ⓐV.T) (ⓐV.U) -| nta_cast: ∀L,T,U. nta h L T U → nta h L (ⓝU. T) U -| nta_conv: ∀L,T,U1,U2,V2. nta h L T U1 → L ⊢ U1 ⬌* U2 → nta h L U2 V2 → - nta h L T U2 -. - -interpretation "native type assignment (term)" - 'NativeType h L T U = (nta h L T U). - -(* Basic properties *********************************************************) - -(* Basic_1: was: ty3_cast *) -lemma nta_cast_old: ∀h,L,W,T,U. - ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ U : W → ⦃h, L⦄ ⊢ ⓝU.T : ⓝW.U. -/4 width=3/ qed. - -(* Basic_1: was: ty3_typecheck *) -lemma nta_typecheck: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃T0. ⦃h, L⦄ ⊢ ⓝU.T : T0. -/3 width=2/ qed. - -(* Basic_1: removed theorems 4: - ty3_getl_subst0 ty3_fsubst0 ty3_csubst0 ty3_subst0 -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_aaa.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_aaa.etc deleted file mode 100644 index 962856983..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_aaa.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/computation/csn_aaa.ma". -include "basic_2/equivalence/lcpcs_aaa.ma". -include "basic_2/dynamic/nta.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -(* Forward lemmas on atomic arity assignment for terms **********************) - -lemma nta_fwd_aaa: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃∃A. L ⊢ T ⁝ A & L ⊢ U ⁝ A. -#h #L #T #U #H elim H -L -T -U -[ /2 width=3/ -| #L #K #V #W #U #i #HLK #_ #HWU * #B #HV #HW - lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/ -| #L #K #W #V #U #i #HLK #_ #HWU * #B #HW #_ -V - lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/ -| * #L #V #W #T #U #_ #_ * #B #HV #HW * #A #HT #HU - [ /3 width=3/ | /3 width=5/ ] -| #L #V #W #T #U #_ #_ * #B #HV #HW * #X #H1 #H2 - elim (aaa_inv_abst … H1) -H1 #B1 #A1 #HW1 #HT #H destruct - elim (aaa_inv_abst … H2) -H2 #B2 #A #_ #HU #H destruct - lapply (aaa_mono … HW1 … HW) -HW1 #H destruct /4 width=5/ -| #L #V #W #T #U #_ #_ * #X #HT #HUX * #A #H #_ -W - elim (aaa_inv_appl … H) -H #B #HV #HUA - lapply (aaa_mono … HUA … HUX) -HUX #H destruct /3 width=5/ -| #L #T #U #_ * #A #HT #HU /3 width=3/ -| #L #T #U1 #U2 #V2 #_ #HU12 #_ * #X #HT #HU1 * #A #HU2 #_ - lapply (aaa_cpcs_mono … HU12 … HU1 … HU2) -U1 #H destruct /2 width=3/ -] -qed-. - -(* Note: this is the stong normalization property *) -(* Basic_1: was only: ty3_sn3 *) -theorem nta_fwd_csn: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → L ⊢ ⬇* T ∧ L ⊢ ⬇* U. -#h #L #T #U #H elim (nta_fwd_aaa … H) -H /3 width=2/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_alt.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_alt.etc deleted file mode 100644 index 8cbd59518..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_alt.etc +++ /dev/null @@ -1,190 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/nta.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -(* alternative definition of nta *) -inductive ntaa (h:sh): lenv → relation term ≝ -| ntaa_sort: ∀L,k. ntaa h L (⋆k) (⋆(next h k)) -| ntaa_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → ntaa h K V W → - ⇧[0, i + 1] W ≡ U → ntaa h L (#i) U -| ntaa_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → ntaa h K W V → - ⇧[0, i + 1] W ≡ U → ntaa h L (#i) U -| ntaa_bind: ∀I,L,V,W,T,U. ntaa h L V W → ntaa h (L. ⓑ{I} V) T U → - ntaa h L (ⓑ{I}V.T) (ⓑ{I}V.U) -| ntaa_appl: ∀L,V,W,T,U. ntaa h L V W → ntaa h L (ⓛW.T) (ⓛW.U) → - ntaa h L (ⓐV.ⓛW.T) (ⓐV.ⓛW.U) -| ntaa_pure: ∀L,V,W,T,U. ntaa h L T U → ntaa h L (ⓐV.U) W → - ntaa h L (ⓐV.T) (ⓐV.U) -| ntaa_cast: ∀L,T,U,W. ntaa h L T U → ntaa h L U W → ntaa h L (ⓝU. T) U -| ntaa_conv: ∀L,T,U1,U2,V2. ntaa h L T U1 → L ⊢ U1 ⬌* U2 → ntaa h L U2 V2 → - ntaa h L T U2 -. - -interpretation "native type assignment (term) alternative" - 'NativeTypeAlt h L T U = (ntaa h L T U). - -(* Advanced inversion lemmas ************************************************) - -fact ntaa_inv_bind1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T :: U → ∀J,X,Y. T = ⓑ{J}Y.X → - ∃∃Z1,Z2. ⦃h, L⦄ ⊢ Y :: Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :: Z2 & - L ⊢ ⓑ{J}Y.Z2 ⬌* U. -#h #L #T #U #H elim H -L -T -U -[ #L #k #J #X #Y #H destruct -| #L #K #V #W #U #i #_ #_ #_ #_ #J #X #Y #H destruct -| #L #K #W #V #U #i #_ #_ #_ #_ #J #X #Y #H destruct -| #I #L #V #W #T #U #HVW #HTU #_ #_ #J #X #Y #H destruct /2 width=3/ -| #L #V #W #T #U #_ #_ #_ #_ #J #X #Y #H destruct -| #L #V #W #T #U #_ #_ #_ #_ #J #X #Y #H destruct -| #L #T #U #W #_ #_ #_ #_ #J #X #Y #H destruct -| #L #T #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #J #X #Y #H destruct - elim (IHTU1 ????) -IHTU1 [5: // |2,3,4: skip ] #Z1 #Z2 #HZ1 #HZ2 #HU1 - lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/ -] -qed. - -lemma ntaa_inv_bind1: ∀h,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X :: U → - ∃∃Z1,Z2. ⦃h, L⦄ ⊢ Y :: Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :: Z2 & - L ⊢ ⓑ{J}Y.Z2 ⬌* U. -/2 width=3/ qed-. - -lemma ntaa_nta: ∀h,L,T,U. ⦃h, L⦄ ⊢ T :: U → ⦃h, L⦄ ⊢ T : U. -#h #L #T #U #H elim H -L -T -U -// /2 width=1/ /2 width=2/ /2 width=3/ /2 width=6/ -qed-. - -(* Properties on relocation *************************************************) - -lemma ntaa_lift: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 :: U1 → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → - ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 :: U2. -#h #L1 #T1 #U1 #H elim H -L1 -T1 -U1 -[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2 - >(lift_inv_sort1 … H1) -X1 - >(lift_inv_sort1 … H2) -X2 // -| #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 - elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct - /3 width=8/ - | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 - lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ - ] -| #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (cprs_inv_sort1 … H) -H // -| -| -| -| -| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #T2 #H - elim (cprs_inv_appl1 … H) -H * - [ #V2 #T0 #HV12 #HT10 #H destruct - elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U - @(nta_conv … (ⓐV2.U1)) (* /2 width=1/*) [ /4 width=2/] (**) (* explicit constructor, /5 width=5/ is too slow *) - | #V2 #W2 #T0 #HV12 #HT10 #HT02 - lapply (IHTU1 … HL12 (ⓛW2.T0) ?) -IHTU1 /2 width=1/ -HT10 #H - elim (nta_inv_bind1 … H) -H #W #U0 #HW2 #HTU0 #HU01 - elim (cpcs_inv_abst1 … HU01) -HU01 #W #U #HU1 #HU0 - lapply (IHUW1 … HL12 (ⓐV2.ⓛW.U) ?) -IHUW1 -HL12 /2 width=1/ -HV12 #H - - - - elim (nta_fwd_pure1 … H) -H #W0 #U2 #HVU2 #H #HW01 - elim (nta_inv_bind1 … H) -H #W3 #U3 #HW3 #HU3 #H - elim (cpcs_inv_abst1 … H) -H #W4 #U4 -*) -(* -axiom nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → - ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. -#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U -[ #L1 #k #L2 #_ #T2 #H - >(tpr_inv_atom1 … H) -H // -| #L1 #K1 #V1 #W #U #i #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H - >(tpr_inv_atom1 … H) -T2 - elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H - elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/ -| #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #HL12 #T2 #H - >(tpr_inv_atom1 … H) -T2 - elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H - elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK - elim (lift_total V1 0 (i+1)) #W #HW - lapply (nta_lift h … HLK … HWU1 … HW) /2 width=1/ -HLK -HW - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpr_lift … HW12 … HWU1 … HWU2) -HWU1 #HU12 - @(nta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /3 width=6/ is too slow *) -| #I #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H - elim (tpr_inv_bind1 … H) -H * - [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct - lapply (IHVW1 … HL12 … HV12) #HV2W1 - lapply (IHVW1 L2 … V1 ?) // -IHVW1 #HWV1 - lapply (IHTU1 (L2.ⓑ{I}V2) … HT10) -HT10 /2 width=1/ #HT0U1 - lapply (IHTU1 (L2.ⓑ{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H - lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02 - lapply (nta_tps_conf … HT0U1 … HT02) -T0 #HT2U1 - elim (nta_fwd_correct … H) -H #U2 #HU12 - @(nta_conv … (ⓑ{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | #T #HT1 #HTX #H destruct - lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HVW1 - elim (lift_total X 0 1) #Y #HXY - lapply (tpr_lift … HTX … HT1 … HXY) -T #H - lapply (IHTU1 (L2.ⓓV1) … H) -T1 /2 width=1/ -L1 #H - elim (nta_fwd_correct … H) #T1 #HUT1 - elim (nta_thin_conf … H L2 0 (0+1) ? ?) -H /2 width=1/ /3 width=1/ #T #U #HTU #H - normalize in ⊢ (??%??? → ?); #HU1 - lapply (delift_inv_lift1_eq … H L2 … HXY) -Y /2 width=1/ #H destruct - @(nta_conv … U) // /2 width=2/ - ] -| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H - elim (tpr_inv_appl1 … H) -H * - [ #V2 #Y #HV12 #HY #H destruct - elim (tpr_inv_abst1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct - lapply (IHTU1 L2 ? (ⓛW1.T1) ?) // #H - elim (nta_fwd_correct … H) -H #X #H - elim (nta_inv_bind1 … H) -H #W #U #HW #HU #_ - @(nta_conv … (ⓐV2.ⓛW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) - | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - lapply (IHVW1 … HL12 … HV12) #HVW2 - lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HV1W2 - lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1 - elim (nta_fwd_correct … H1) #T #H2 - elim (nta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H - elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21 - elim (nta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0 - lapply (lsubn_nta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2 - @(nta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *) - | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct - ] -| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #X #H - elim (tpr_inv_appl1 … H) -H * - [ #V2 #T2 #HV12 #HT12 #H destruct - elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U - @(nta_conv … (ⓐV2.U1)) /2 width=1/ /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) - | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 /2 width=1/ -T0 #H - elim (nta_inv_bind1 … H) -H #W #U2 #HW2 #HTU2 #HU21 - lapply (IHUW1 … HL12 (ⓐV2.U1) ?) -IHUW1 -HL12 /2 width=1/ #H - elim (nta_inv_pure1 … H) -H #V0 #U0 #U #HV20 #HU10 #HU0W1 #HU0 - @(nta_conv … (ⓓV2.U2)) - [2: @nta_bind // - @(lsubn_nta_trans … HTU2) @lsubn_abbr // -(* - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 - lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ -*) -*) -(* -axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y → - ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W. - -*) -(* SEGMENT 2 -| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct - lapply (cpr_tpss … HU12) /4 width=4/ -| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 - @(nta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) -] -qed. -*) - -(* SEGMENT 3 -fact nta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 → - ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. - - - | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct - elim (nta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 - lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 - lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0 - lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2 - @(nta_abbr … HW2) -HW2 - @(nta_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) - ] -| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_cast1 … H) -H - [ * #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2 - lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ - | -HV1 #HT1X - lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/ - ] -] -qed. - -/2 width=9/ qed. - -axiom nta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A. -/2 width=5/ qed. - -axiom nta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A. -/2 width=5/ qed. -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_ltpss.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_ltpss.etc deleted file mode 100644 index 828fd82e0..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_ltpss.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/equivalence/cpcs_ltpss.ma". -include "basic_2/dynamic/nta_nta.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -(* Properties about parallel unfold *****************************************) - -lemma nta_ltpss_tpss_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 : U. -#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U -[ #L1 #k #L2 #d #e #_ #T2 #H - >(tpss_inv_sort1 … H) -H // -| #L1 #K1 #V1 #W #U #i #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H - [ #H destruct - elim (lt_or_ge i d) #Hdi - [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct - /3 width=7/ - | elim (lt_or_ge i (d + e)) #Hide [ | -Hdi ] - [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct - /3 width=7/ - | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=7/ - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 - elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct - lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2 - lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=9/ - ] -| #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ] - [ #H destruct - elim (lift_total V1 0 (i+1)) #W #HW - elim (lt_or_ge i d) #Hdi [ -HWV1 ] - [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK - lapply (nta_lift h … HLK … HWU1 … HW) /2 width=4/ -HW - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 - lapply (cpr_tpss … HU12) -HU12 #HU12 - @(nta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -HW -Hdi ] - [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK - lapply (nta_lift h … HLK … HWU1 … HW) /2 width=4/ -HW - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 - lapply (cpr_tpss … HU12) -HU12 #HU12 - @(nta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=6/ - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_ - elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct - lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct - ] -| #I #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HV - lapply (IHTU1 (L2.ⓑ{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H - elim (nta_fwd_correct … H) -H #U2 #HU12 - @(nta_conv … (ⓑ{I}V2.U1)) /3 width=2/ /3 width=4/ /4 width=4/ (**) (* explicit constructor, /5 width=6/ is too slow *) -| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct - elim (tpss_inv_bind1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HV - lapply (IHTU1 L2 d e ? (ⓛW1.T1) ?) // #H - elim (nta_fwd_correct … H) -H #X #H - elim (nta_inv_bind1 … H) -H #W #U #HW #HU #_ - @(nta_conv … (ⓐV2.ⓛW1.U1)) /3 width=2/ /3 width=4/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) -| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HV - elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=4/ ] #U #HU - @(nta_conv … (ⓐV2.U1)) // /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) -| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct - lapply (cpr_tpss … HU12) /4 width=4/ -| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 - @(nta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) -] -qed. - -lemma nta_ltpss_tps_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → ⦃h, L2⦄ ⊢ T2 : U. -/3 width=7/ qed. - -lemma nta_ltpss_conf: ∀h,L1,T,U. ⦃h, L1⦄ ⊢ T : U → - ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T : U. -/2 width=7/ qed. - -lemma nta_tpss_conf: ∀h,L,T1,U. ⦃h, L⦄ ⊢ T1 : U → - ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 : U. -/2 width=7/ qed. - -lemma nta_tps_conf: ∀h,L,T1,U. ⦃h, L⦄ ⊢ T1 : U → - ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 : U. -/2 width=7/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_nta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_nta.etc deleted file mode 100644 index 05eb6c55d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_nta.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/dynamic/nta_lift.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: ty3_unique *) -theorem nta_mono: ∀h,L,T,U1. ⦃h, L⦄ ⊢ T : U1 → ∀U2. ⦃h, L⦄ ⊢ T : U2 → - L ⊢ U1 ⬌* U2. -#h #L #T #U1 #H elim H -L -T -U1 -[ #L #k #X #H - lapply (nta_inv_sort1 … H) -H // -| #L #K #V #W11 #W12 #i #HLK #_ #HW112 #IHVW11 #X #H - elim (nta_inv_lref1 … H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX - lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK - @(cpcs_trans … HX) -X /3 width=9 by cpcs_lift/ (**) (* to slow without trace *) -| #L #K #W #V1 #V #i #HLK #_ #HWV #_ #X #H - elim (nta_inv_lref1 … H) -H * #K0 #W0 #V2 #V0 #HLK0 #_ #HWV0 #HX - lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H destruct - lapply (lift_mono … HWV0 … HWV) -HWV0 -HWV #H destruct // -| #I #L #V #W1 #T #U1 #_ #_ #_ #IHTU1 #X #H - elim (nta_inv_bind1 … H) -H #W2 #U2 #_ #HTU2 #H - @(cpcs_trans … H) -X /3 width=1/ -| #L #V #W1 #T #U1 #_ #_ #_ #IHTU1 #X #H - elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H - @(cpcs_trans … H) -X /3 width=1/ -| #L #V #W1 #T #U1 #_ #_ #IHTU1 #_ #X #H - elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H - @(cpcs_trans … H) -X /3 width=1/ -| #L #T #U1 #_ #_ #X #H - elim (nta_inv_cast1 … H) -H // -| #L #T #U11 #U12 #V12 #_ #HU112 #_ #IHTU11 #_ #U2 #HTU2 - @(cpcs_canc_sn … HU112) -U12 /2 width=1/ -] -qed-. - -(* Advanced properties ******************************************************) - -lemma nta_cast_alt: ∀h,L,T,W,U. ⦃h, L⦄ ⊢ T : W → ⦃h, L⦄ ⊢ T : U → - ⦃h, L⦄ ⊢ ⓝW.T : U. -#h #L #T #W #U #HTW #HTU -lapply (nta_mono … HTW … HTU) #HWU -elim (nta_fwd_correct … HTU) -HTU /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_sta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_sta.etc deleted file mode 100644 index 6268b98b1..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_sta.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/static/sta.ma". -include "basic_2/equivalence/cpcs_cpcs.ma". -include "basic_2/dynamic/nta.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -(* Properties on static type assignment *************************************) - -lemma nta_fwd_sta: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → - ∃∃U0. ⦃h, L⦄ ⊢ T • U0 & L ⊢ U0 ⬌* U. -#h #L #T #U #H elim H -L -T -U -[ /2 width=3/ -| #L #K #V #W1 #V1 #i #HLK #_ #HWV1 * #W0 #HVW0 #HW01 - elim (lift_total W0 0 (i+1)) #V0 #HWV0 - lapply (ldrop_fwd_ldrop2 … HLK) #HLK0 - lapply (cpcs_lift … HLK0 … HWV0 … HWV1 HW01) -HLK0 -HWV1 -HW01 /3 width=6/ -| #L #K #W #V1 #W1 #i #HLK #HWV1 #HW1 * /3 width=6/ -| #I #L #V #W #T #U #_ #_ * #W0 #_ #_ * /3 width=3/ -| #L #V #W #T #U #_ #_ * #W0 #_ #HW0 * #X #H #HX - elim (sta_inv_bind1 … H) -H #U0 #HTU0 #H destruct - @(ex2_1_intro … (ⓐV.ⓛW.U0)) /2 width=1/ /3 width=1/ -| #L #V #W #T #U #_ #_ * #U0 #HTU0 #HU0 #_ -W - @(ex2_1_intro … (ⓐV.U0)) /2 width=1/ -| #L #T #U #HTU * #U0 #HTU0 #HU0 /3 width=3/ -| #L #T #U1 #U2 #V2 #_ #HU12 #_ * #U0 #HTU0 #HU01 #_ - lapply (cpcs_trans … HU01 … HU12) -U1 /2 width=3/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_thin.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_thin.etc deleted file mode 100644 index f927f841a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/nta/nta_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/nta_lift.ma". - -(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) - -(* Properties on basic local environment thinning ***************************) - -(* Note: this is known as the substitution lemma *) -(* Basic_1: was only: ty3_gen_cabbr *) -lemma nta_thin_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 : U1 → - ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 → - ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 : U2 & - L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2. -#h #L1 #T1 #U1 #H elim H -L1 -T1 -U1 -[ /2 width=5/ -| #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12 - elim (lt_or_ge i d) #Hdi [ -HVW1 ] - [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H - lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1 - elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2 - elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12 - lapply (delift_mono … H … HV12) -H -HV12 #H destruct - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1 - lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 // - >minus_plus minus_plus #HU1 - lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2 - lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/ - | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei - lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1 - elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W - commutative_plus minus_plus commutative_plus 0 & L ⊢ U0 ⬌* U - ). -#h #L #T #U #l #H elim H -L -T -U -l -[ #L #k #j #H destruct -| #L #K #V #W #U #i #l #HLK #HVW #HWU #_ #j #H destruct /3 width=8/ -| #L #K #W #V #U #i #l #HLK #HWV #HWU #_ #j #H destruct /3 width=8/ -| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct -| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct -| #L #V #T #U #W #l #_ #_ #_ #_ #j #H destruct -| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #j #H destruct -| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #j #H destruct - elim (IHTU1 ??) -IHTU1 [4: // |2: skip ] * #K #V #W #U0 #HLK #HVW #HWU0 [2: #H ] #HU01 - lapply (cpcs_trans … HU01 … HU12) -U1 /3 width=8/ -] -qed. - -lemma snta_inv_lref1: ∀h,L,U,i,l. ⦃h, L⦄ ⊢ #i :[l] U → - (∃∃K,V,W,U0. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V :[l] W & - ⇧[0, i + 1] W ≡ U0 & L ⊢ U0 ⬌* U - ) ∨ - (∃∃K,W,V,U0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W :[l-1] V & - ⇧[0, i + 1] W ≡ U0 & l > 0 & L ⊢ U0 ⬌* U - ). -/2 width=3/ qed-. - -fact snta_inv_bind1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀J,X,Y. T = ⓑ{J}Y.X → - ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 & - ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 & - L ⊢ ⓑ{J}Y.Z2 ⬌* U. -#h #L #T #U #l #H elim H -L -T -U -l -[ #L #k #J #X #Y #H destruct -| #L #K #V #W #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct -| #L #K #W #V #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct -| #I #L #V #W #T #U #l1 #l2 #HVW #HTU #_ #_ #J #X #Y #H destruct /2 width=3/ -| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct -| #L #V #T #U #W #l #_ #_ #_ #_ #J #X #Y #H destruct -| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct -| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #J #X #Y #H destruct - elim (IHTU1 ????) -IHTU1 [5: // |2,3,4: skip ] #Z1 #Z2 #l0 #HZ1 #HZ2 #HU1 - lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/ -] -qed. - -lemma snta_inv_bind1: ∀h,J,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{J}Y.X :[l] U → - ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 & - L ⊢ ⓑ{J}Y.Z2 ⬌* U. -/2 width=3/ qed-. - -fact snta_inv_cast1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀X,Y. T = ⓝY.X → - ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U. -#h #L #T #U #l #H elim H -L -T -U -l -[ #L #k #X #Y #H destruct -| #L #K #V #W #U #i #l #_ #_ #_ #_ #X #Y #H destruct -| #L #K #W #V #U #i #l #_ #_ #_ #_ #X #Y #H destruct -| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct -| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct -| #L #V #T #U #W #l #_ #_ #_ #_ #X #Y #H destruct -| #L #T #U #W #l1 #l2 #HTU #_ #_ #_ #X #Y #H destruct /2 width=1/ -| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct - elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #HXY #HU1 - lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=1/ -] -qed. - -lemma snta_inv_cast1: ∀h,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X :[l] U → - ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U. -/2 width=3/ qed-. - -(* Properties on relocation *************************************************) - -lemma snta_lift: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 → - ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → - ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 → - ⦃h, L2⦄ ⊢ T2 :[l] U2. -#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l -[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2 - >(lift_inv_sort1 … H1) -X1 - >(lift_inv_sort1 … H2) -X2 // -| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 - elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct - /3 width=8/ - | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 - lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ - ] -| #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (tpr_inv_atom1 … H) -H // -| #L1 #K1 #V1 #W #U #i1 #l #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H #Hl -IH - >(tpr_inv_atom1 … H) -T2 - elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H - elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/ -| #L1 #K1 #W1 #V1 #U1 #i1 #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #HL12 #T2 #H #Hl -IH -(* - >(tpr_inv_atom1 … H) -T2 - elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H - elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK - elim (lift_total V1 0 (i+1)) #W #HW - lapply (snta_lift h … HLK … HWU1 … HW) /2 width=1/ -HLK -HW - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpr_lift … HW12 … HWU1 … HWU2) -HWU1 #HU12 - @(snta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /3 width=6/ is too slow *) -*) -| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH -(* - elim (tpr_inv_bind1 … H) -H * - [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct - lapply (IHVW1 … HL12 … HV12) #HV2W1 - lapply (IHVW1 L2 … V1 ?) // -IHVW1 #HWV1 - lapply (IHTU1 (L2.ⓑ{I}V2) … HT1) -HT1 /2 width=1/ #HTU1 - lapply (IHTU1 (L2.ⓑ{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H - lapply (tps_lsubs_trans … HT2 (L2.ⓑ{I}V2) ?) -HT2 /2 width=1/ #HT2 - lapply (snta_tps_conf … HTU1 … HT2) -T #HT2U1 - elim (snta_fwd_correct … H) -H #U2 #HU12 - @(snta_conv … (ⓑ{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | #T #HT1 #HTX #H destruct - lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HVW1 - lapply (IHTU1 (L2.ⓓV1) … HT1) -T1 /2 width=1/ -L1 #H - elim (snta_fwd_correct … H) #T1 #HUT1 - elim (snta_ldrop_conf … H L2 0 1 ? ?) -H // /2 width=1/ #T0 #U0 #HTU0 #H #HU10 - lapply (delift_inv_lift1_eq … H L2 … HTX) -H -HTX /2 width=1/ #H destruct - @(snta_conv … HTU0) /2 width=2/ - ] -*) -| #L1 #V1 #W11 #W2 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH -(* - elim (tpr_inv_appl1 … H) -H * - [ #V2 #Y #HV12 #HY #H destruct - elim (tpr_inv_abst1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct - lapply (IHTU1 L2 ? (ⓛW1.T1) ?) // #H - elim (snta_fwd_correct … H) -H #X #H - elim (snta_inv_bind1 … H) -H #W #U #HW #HU #_ - @(snta_conv … (ⓐV2.ⓛW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) - | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - lapply (IHVW1 … HL12 … HV12) #HVW2 - lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HV1W2 - lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1 - elim (snta_fwd_correct … H1) #T #H2 - elim (snta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H - elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21 - elim (snta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0 - lapply (lsubsn_snta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2 - @(snta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *) - | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct - ] -*) -| #L1 #V1 #T1 #U1 #W1 #l #_ #HUW1 #IHTU1 #_ #L2 #HL12 #X #H #Hl - elim (tpr_inv_appl1 … H) -H * - [ #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpr … HV12 L2) #HV - elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=6/ ] #U - @(snta_conv … (ⓐV2.U1)) /2 width=1/ -HV12 /4 width=8 by snta_pure, cprs_flat_dx/ (**) (* explicit constructor, /4 width=8/ is too slow without trace *) - | #V2 #W0 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - lapply (IHTU1 … HL12 (ⓛW0.T2) ? ?) -IHTU1 // /2 width=1/ -T0 #H1 - lapply (IH … (ⓐV2.U1) … HUW1 HL12 ?) // /3 width=1/ #H2 - lapply (snta_pure … H1 H2) -H2 #H - elim (snta_inv_bind1 … H1) -H1 #V0 #U2 #l1 #HWV0 #HTU2 #HU21 - @(snta_conv … (ⓓV2.U2)) (**) (* explicit constructor *) - [2: -(* - @snta_bind /3 width=2/ /3 width=6/ (**) (* /4 width=6/ is a bit slow *) -*) - |3: @(cpcs_cpr_conf … (ⓐV1.ⓛW0.U2)) /2 width=1/ - |4: /2 width=5/ - | skip - ] -(* - elim (snta_fwd_pure1 … H) -H #T1 #W2 #HVW2 #HUT1 #HTW1 - - elim (cpcs_inv_abst1 … HU21) #W3 #U3 #HU13 #H - elim (cprs_inv_abst … H Abst W0) -H #HW03 #_ - elim (pippo … IH … HUW1 ? V2 W3 U3 HL12 ? ?) -IH -HUW1 -HL12 // /3 width=1/ -HU13 #l2 #HV2W3 - lapply (snta_conv h L2 V2 W3 W0 V0 (l1+1) ? ? ?) /2 width=1/ -HV2W3 -HW03 -HWV0 #HV2W0 -*) -(* SEGMENT 1.5 - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 - lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ - -axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y → - ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W. - -*) -(* SEGMENT 2 -| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct - lapply (cpr_tpss … HU12) /4 width=4/ -| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 - @(snta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) -] -qed. -*) - -(* SEGMENT 3 -fact snta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 → - ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. - - - | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct - elim (snta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 - lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 - lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0 - lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2 - @(snta_abbr … HW2) -HW2 - @(snta_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) - ] -| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_cast1 … H) -H - [ * #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2 - lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ - | -HV1 #HT1X - lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/ - ] -] -qed. - -lemma snta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → - ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. - -/2 width=9/ qed. - -axiom snta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A. -/2 width=5/ qed. - -axiom snta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A. -/2 width=5/ qed. -*) -*)*) \ No newline at end of file diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/snta/snta_ltpss.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/snta/snta_ltpss.etc deleted file mode 100644 index 0e5f3930e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/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: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U. -#h #L1 #T1 #U #l #H elim H -L1 -T1 -U -l -[ #L1 #k #L2 #d #e #_ #T2 #H - >(tpss_inv_sort1 … H) -H // -| #L1 #K1 #V1 #W #U #i #l #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H - [ #H destruct - elim (lt_or_ge i d) #Hdi - [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct - /3 width=7/ - | elim (lt_or_ge i (d + e)) #Hide [ | -Hdi ] - [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct - /3 width=7/ - | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=7/ - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 - elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct - lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2 - lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=9/ - ] -| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ] - [ #H destruct - elim (lift_total V1 0 (i+1)) #W #HW - elim (lt_or_ge i d) #Hdi [ -HWV1 ] - [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK - lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 - lapply (cpr_tpss … HU12) -HU12 #HU12 - @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -HW -Hdi ] - [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) #HLK - lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 - lapply (cpr_tpss … HU12) -HU12 #HU12 - @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) - | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=6/ - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_ - elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct - lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct - ] -| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HV - lapply (IHTU1 (L2.ⓑ{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H - elim (snta_fwd_correct … H) -H #U2 #HU12 - @(snta_conv … (ⓑ{I}V2.U1)) /3 width=2/ /3 width=4/ /4 width=4/ (**) (* explicit constructor, /5 width=6/ is too slow *) -| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct - elim (tpss_inv_bind1 … HY) -HY #W21 #T2 #HW121 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HVV12 - lapply (IHTU1 L2 d e ? (ⓛW21.T2) ?) -IHTU1 // /2 width=1/ -HW121 -HT12 #H0 - elim (snta_fwd_correct … H0) #X #H - elim (snta_inv_bind1 … H) -H #W #U #l0 #HW #HU #_ - @(snta_conv … (ⓐV2.ⓛW12.U1)) /3 width=2/ /3 width=4/ /3 width=5/ (**) (* explicit constructor, /4 width=5/ is too slow *) -| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HV - elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=4/ ] #U - @(snta_conv … (ⓐV2.U1)) /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) -| #L1 #T1 #U1 #W1 #l1 #l2 #HTU1 #HUW1 #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (snta_fwd_correct … HTU1) -HTU1 #U #H - elim (snta_mono … HUW1 … H) -HUW1 -H #H #_ -U destruct - elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct - lapply (cpr_tpss … HU12) #HU /4 width=4/ -| #L1 #T1 #U11 #U12 #U #l #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 - @(snta_conv … U11) /2 width=5/ (**) (* explicit constructor, /3 width=7/ is too slow *) -] -qed. - -lemma snta_ltpss_tps_conf: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U. -/3 width=7/ qed. - -lemma snta_ltpss_conf: ∀h,L1,T,U,l. ⦃h, L1⦄ ⊢ T :[l] U → - ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T :[l] U. -/2 width=7/ qed. - -lemma snta_tpss_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U → - ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U. -/2 width=7/ qed. - -lemma snta_tps_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U → - ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U. -/2 width=7/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/snta/snta_snta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/snta/snta_snta.etc deleted file mode 100644 index db71e1192..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/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: ∀h,L,T,U1,l1. ⦃h, L⦄ ⊢ T :[l1] U1 → - ∀U2,l2. ⦃h, L⦄ ⊢ T :[l2] U2 → l1 = l2 ∧ L ⊢ U1 ⬌* U2. -#h #L #T #U1 #l1 #H elim H -L -T -U1 -l1 -[ #L #k #X #l2 #H - lapply (snta_inv_sort1 … H) -H * /2 width=1/ -| #L #K #V #W11 #W12 #i #l1 #HLK #_ #HW112 #IHVW11 #X #l2 #H - elim (snta_inv_lref1 … H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX - lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK - elim (IHVW11 … HVW21) -IHVW11 -HVW21 #Hl12 #HW121 - lapply (cpcs_lift … HLK … HW112 … HW212 ?) // -K -W11 -W21 /3 width=3/ -| #L #K #W #V1 #V #i #l1 #HLK #_ #HWV #IHWV1 #X #l2 #H - elim (snta_inv_lref1 … H) -H * #K0 #W0 #V2 #V0 #HLK0 #HW0V2 #HWV0 [2: #HL2 ] #HX - lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H destruct - lapply (lift_mono … HWV0 … HWV) -HWV0 -HWV #H destruct - elim (IHWV1 … HW0V2) -IHWV1 -HW0V2 /3 width=1/ -| #I #L #V #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H - elim (snta_inv_bind1 … H) -H #W2 #U2 #l20 #_ #HTU2 #H - elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12 - lapply (cpcs_trans … (ⓑ{I}V.U1) … H) -H /2 width=1/ -| #L #V #W #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H - elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H - elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12 - lapply (cpcs_trans … (ⓐV.ⓛW1.U1) … H) -H /2 width=1/ -| #L #V #T #U1 #W1 #l1 #_ #_ #IHTU1 #_ #X #l2 #H - elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H - elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12 - lapply (cpcs_trans … (ⓐV.U1) … H) -H /2 width=1/ -| #L #T #U1 #W1 #l10 #l1 #_ #_ #IHTU1 #_ #X #l2 #H - elim (snta_inv_cast1 … H) -H #HTU1 - elim (IHTU1 … HTU1) -IHTU1 -HTU1 /2 width=1/ -| #L #T #U11 #U12 #V12 #l1 #_ #HU112 #_ #IHTU11 #_ #U2 #l2 #HTU2 - elim (IHTU11 … HTU2) -IHTU11 -HTU2 #Hl12 #H - lapply (cpcs_canc_sn … HU112 … H) -U11 /2 width=1/ -] -qed-. - -(* Advanced properties ******************************************************) - -lemma snta_cast_alt: ∀h,L,T,W,U,l. ⦃h, L⦄ ⊢ T :[l] W → ⦃h, L⦄ ⊢ T :[l] U → - ⦃h, L⦄ ⊢ ⓝW.T :[l] U. -#h #L #T #W #U #l #HTW #HTU -elim (snta_mono … HTW … HTU) #_ #HWU -elim (snta_fwd_correct … HTU) -HTU /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/snta/snta_thin.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/snta/snta_thin.etc deleted file mode 100644 index ceb5375bf..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/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: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 → - ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 → - ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 :[l] U2 & - L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2. -#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l -[ /2 width=5/ -| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12 - elim (lt_or_ge i d) #Hdi [ -HVW1 ] - [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H - lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1 - elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2 - elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12 - lapply (delift_mono … H … HV12) -H -HV12 #H destruct - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1 - lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 // - >minus_plus minus_plus #HU1 - lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2 - lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/ - | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei - lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1 - elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W - commutative_plus minus_plus commutative_plus (deg_mono … Hkl HkO) -g -l // -| #T0 #U0 #l0 #HTU0 #_ #IHU0 #k #H #l #Hkl destruct - elim (ssta_inv_sort1 … HTU0) -L #HkS #H destruct - lapply (deg_mono … Hkl HkS) -Hkl #H destruct - >(IHU0 (next h k) ? l0) -IHU0 // /2 width=1/ >iter_SO >iter_n_Sm // -] -qed. - -lemma sstas_inv_sort1: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k •*[g] U → ∀l. deg h g k l → - U = ⋆((next h)^l k). -/2 width=6/ qed-. - -fact sstas_inv_bind1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → - ∀J,X,Y. T = ⓑ{J}Y.X → - ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z. -#h #g #L #T #U #H @(sstas_ind_alt … H) -T -[ #U0 #HU0 #J #X #Y #H destruct - elim (ssta_inv_bind1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/ -| #T0 #U0 #l #HTU0 #_ #IHU0 #J #X #Y #H destruct - elim (ssta_inv_bind1 … HTU0) -HTU0 #X0 #HX0 #H destruct - elim (IHU0 J X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/ -] -qed. - -lemma sstas_inv_bind1: ∀h,g,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X •*[g] U → - ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z. -/2 width=3/ qed-. - -fact sstas_inv_appl1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀X,Y. T = ⓐY.X → - ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z. -#h #g #L #T #U #H @(sstas_ind_alt … H) -T -[ #U0 #HU0 #X #Y #H destruct - elim (ssta_inv_appl1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/ -| #T0 #U0 #l #HTU0 #_ #IHU0 #X #Y #H destruct - elim (ssta_inv_appl1 … HTU0) -HTU0 #X0 #HX0 #H destruct - elim (IHU0 X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/ -] -qed. - -lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U → - ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z. -/2 width=3/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma sstas_fwd_correct: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → - ∃∃W. ⦃h, L⦄ ⊢ U •[g, 0] W & ⦃h, L⦄ ⊢ U •*[g] U. -#h #g #L #T #U #H @(sstas_ind_alt … H) -T /2 width=1/ /3 width=2/ -qed-. - -(* Basic_1: removed theorems 7: - sty1_bind sty1_abbr sty1_appl sty1_cast2 - sty1_lift sty1_correct sty1_trans -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_lift.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_lift.etc deleted file mode 100644 index 838c7b6d4..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_lift.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/static/ssta_lift.ma". -include "basic_2/unwind/sstas.ma". - -(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************) - -(* Advanced properties ******************************************************) - -lemma sstas_total_S: ∀h,g,L,l,T,U. ⦃h, L⦄ ⊢ T•[g, l + 1]U → - ∃∃W. ⦃h, L⦄ ⊢ T •*[g] W & ⦃h, L⦄ ⊢ U •*[g] W. -#h #g #L #l @(nat_ind_plus … l) -l -[ #T #U #HTU - elim (ssta_fwd_correct … HTU) /4 width=4/ -| #l #IHl #T #U #HTU - elim (ssta_fwd_correct … HTU) (lift_mono … HX … HU12) -X - elim (lift_total T1 d e) /3 width=10/ -| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL21 #T2 #HT02 #U2 #HU12 - elim (lift_total U0 d e) /3 width=10/ -] -qed. - -lemma sstas_inv_lift1: ∀h,g,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 → - ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 → - ∃∃U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 & ⇧[d, e] U1 ≡ U2. -#h #g #L2 #T2 #U2 #H @(sstas_ind_alt … H) -T2 -[ #T2 #HUT2 #L1 #d #e #HL21 #U1 #HU12 - elim (ssta_inv_lift1 … HUT2 … HL21 … HU12) -HUT2 -HL21 /3 width=3/ -| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L1 #d #e #HL21 #U1 #HU12 - elim (ssta_inv_lift1 … HTU0 … HL21 … HU12) -HTU0 -HU12 #U #HU1 #HU0 - elim (IHU01 … HL21 … HU0) -IHU01 -HL21 -U0 /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_ltpss.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_ltpss.etc deleted file mode 100644 index e0aa94207..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_ltpss.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/static/ssta_ltpss.ma". -include "basic_2/unwind/sstas.ma". - -(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************) - -(* Properties about parallel unfold *****************************************) - -lemma sstas_ltpss_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → - ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & - L2 ⊢ U1 ▶* [d, e] U2. -#h #g #L1 #T1 #U1 #H @(sstas_ind_alt … H) -T1 -[ #T1 #HUT1 #L2 #d #e #HL12 #U2 #HU12 - elim (ssta_ltpss_tpss_conf … HUT1 … HL12 … HU12) -HUT1 -HL12 /3 width=3/ -| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0 - elim (ssta_ltpss_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0 - elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/ -] -qed. - -lemma sstas_ltpss_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → - ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2. -/3 width=5/ qed. - -lemma sstas_ltpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2. -/2 width=5/ qed. - -lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 → - ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → - ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2. -/2 width=5/ qed. - -lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 → - ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → - ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2. -/2 width=5/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_sstas.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_sstas.etc deleted file mode 100644 index 2f7e261b4..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/sstas/sstas_sstas.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/unfold/delift_lift.ma". -include "basic_2/static/ssta_ssta.ma". -include "basic_2/unwind/sstas_lift.ma". - -(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************) - -(* Advanced inversion lemmas ************************************************) - -lemma sstas_inv_O: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → - ∀T0. ⦃h, L⦄ ⊢ T •[g , 0] T0 → U = T. -#h #g #L #T #U #H @(sstas_ind_alt … H) -T // -#T0 #U0 #l0 #HTU0 #_ #_ #T1 #HT01 -elim (ssta_mono … HTU0 … HT01) (sstas_inv_O … HU12 … HUT1) -h -L -T1 -U2 // -| #T0 #U0 #l0 #HTU0 #_ #IHU01 #U2 #HU12 - lapply (sstas_inv_S … HU12 … HTU0) -T0 -l0 /2 width=1/ -] -qed-. - -(* More advancd inversion lemmas ********************************************) - -fact sstas_inv_lref1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀j. T = #j → - ∃∃I,K,V,W. ⇩[0, j] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V •*[g] W & - L ⊢ ▼*[0, j + 1] U ≡ W. -#h #g #L #T #U #H @(sstas_ind_alt … H) -T -[ #T #HUT #j #H destruct - elim (ssta_inv_lref1 … HUT) -HUT * #K #V #W [2: #l] #HLK #HVW #HVT - [ (sstas_mono … HWX … HWT) -X -W /3 width=7/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/sta/sta.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/sta/sta.etc deleted file mode 100644 index 20302c623..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/sta/sta.etc +++ /dev/null @@ -1,128 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/ldrop.ma". -include "basic_2/static/sh.ma". - -(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************) - -inductive sta (h:sh): lenv → relation term ≝ -| sta_sort: ∀L,k. sta h L (⋆k) (⋆(next h k)) -| sta_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → sta h K V W → - ⇧[0, i + 1] W ≡ U → sta h L (#i) U -| sta_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → sta h K W V → - ⇧[0, i + 1] W ≡ U → sta h L (#i) U -| sta_bind: ∀I,L,V,T,U. sta h (L. ⓑ{I} V) T U → - sta h L (ⓑ{I}V.T) (ⓑ{I}V.U) -| sta_appl: ∀L,V,T,U. sta h L T U → - sta h L (ⓐV.T) (ⓐV.U) -| sta_cast: ∀L,W,T,U. sta h L T U → sta h L (ⓝW. T) U -. - -interpretation "static type assignment (term)" - 'StaticType h L T U = (sta h L T U). - -(* Basic inversion lemmas ************************************************) - -fact sta_inv_sort1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀k0. T = ⋆k0 → - U = ⋆(next h k0). -#h #L #T #U * -L -T -U -[ #L #k #k0 #H destruct // -| #L #K #V #W #U #i #_ #_ #_ #k0 #H destruct -| #L #K #W #V #U #i #_ #_ #_ #k0 #H destruct -| #I #L #V #T #U #_ #k0 #H destruct -| #L #V #T #U #_ #k0 #H destruct -| #L #W #T #U #_ #k0 #H destruct -qed. - -(* Basic_1: was: sty0_gen_sort *) -lemma sta_inv_sort1: ∀h,L,U,k. ⦃h, L⦄ ⊢ ⋆k • U → U = ⋆(next h k). -/2 width=4/ qed-. - -fact sta_inv_lref1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀j. T = #j → - (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V • W & - ⇧[0, j + 1] W ≡ U - ) ∨ - (∃∃K,W,V. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W • V & - ⇧[0, j + 1] W ≡ U - ). -#h #L #T #U * -L -T -U -[ #L #k #j #H destruct -| #L #K #V #W #U #i #HLK #HVW #HWU #j #H destruct /3 width=6/ -| #L #K #W #V #U #i #HLK #HWV #HWU #j #H destruct /3 width=6/ -| #I #L #V #T #U #_ #j #H destruct -| #L #V #T #U #_ #j #H destruct -| #L #W #T #U #_ #j #H destruct -] -qed. - -(* Basic_1: was sty0_gen_lref *) -lemma sta_inv_lref1: ∀h,L,U,i. ⦃h, L⦄ ⊢ #i • U → - (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V • W & - ⇧[0, i + 1] W ≡ U - ) ∨ - (∃∃K,W,V. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W • V & - ⇧[0, i + 1] W ≡ U - ). -/2 width=3/ qed-. - -fact sta_inv_bind1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀J,X,Y. T = ⓑ{J}Y.X → - ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X • Z & U = ⓑ{J}Y.Z. -#h #L #T #U * -L -T -U -[ #L #k #J #X #Y #H destruct -| #L #K #V #W #U #i #_ #_ #_ #J #X #Y #H destruct -| #L #K #W #V #U #i #_ #_ #_ #J #X #Y #H destruct -| #I #L #V #T #U #HTU #J #X #Y #H destruct /2 width=3/ -| #L #V #T #U #_ #J #X #Y #H destruct -| #L #W #T #U #_ #J #X #Y #H destruct -] -qed. - -(* Basic_1: was: sty0_gen_bind *) -lemma sta_inv_bind1: ∀h,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X • U → - ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X • Z & U = ⓑ{J}Y.Z. -/2 width=3/ qed-. - -fact sta_inv_appl1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀X,Y. T = ⓐY.X → - ∃∃Z. ⦃h, L⦄ ⊢ X • Z & U = ⓐY.Z. -#h #L #T #U * -L -T -U -[ #L #k #X #Y #H destruct -| #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct -| #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct -| #I #L #V #T #U #_ #X #Y #H destruct -| #L #V #T #U #HTU #X #Y #H destruct /2 width=3/ -| #L #W #T #U #_ #X #Y #H destruct -] -qed. - -(* Basic_1: was: sty0_gen_appl *) -lemma sta_inv_appl1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X • U → - ∃∃Z. ⦃h, L⦄ ⊢ X • Z & U = ⓐY.Z. -/2 width=3/ qed-. - -fact sta_inv_cast1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀X,Y. T = ⓝY.X → - ⦃h, L⦄ ⊢ X • U. -#h #L #T #U * -L -T -U -[ #L #k #X #Y #H destruct -| #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct -| #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct -| #I #L #V #T #U #_ #X #Y #H destruct -| #L #V #T #U #_ #X #Y #H destruct -| #L #W #T #U #HTU #X #Y #H destruct // -] -qed. - -(* Basic_1: was: sty0_gen_cast *) -lemma sta_inv_cast1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓝY.X • U → ⦃h, L⦄ ⊢ X • U. -/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/sta/sta_lift.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/sta/sta_lift.etc deleted file mode 100644 index c9fbda014..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/sta/sta_lift.etc +++ /dev/null @@ -1,120 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/static/sta.ma". - -(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************) - -(* Properties on relocation *************************************************) - -(* Basic_1: was: sty0_lift *) -lemma sta_lift: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → - ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 • U2. -#h #L1 #T1 #U1 #H elim H -L1 -T1 -U1 -[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2 - >(lift_inv_sort1 … H1) -X1 - >(lift_inv_sort1 … H2) -X2 // -| #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 - elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct - /3 width=8/ - | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 - lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ - ] -| #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (lift_inv_sort2 … H) -X /2 width=3/ -| #L2 #K2 #V2 #W2 #W #i #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ] - [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 - elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HVW1 #HW12 - elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ - | minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ - | (tpss_inv_sort1 … H) -H /2 width=3/ -| #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H [ | -HVW1 ] - [ #H destruct - elim (lt_or_ge i d) #Hdi [ -HVW1 | ] - [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 - lapply (ldrop_fwd_ldrop2 … HLK2) #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 - >minus_plus minus_plus #H - lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H - lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H - lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus (sta_inv_sort1 … H) -X // -| #L #K #V #W #U1 #i #HLK #_ #HWU1 #IHVW #U2 #H - elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HW0U2 - lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct - lapply (IHVW … HVW0) -IHVW -HVW0 #H destruct - >(lift_mono … HWU1 … HW0U2) -W0 -U1 // -| #L #K #W #V #U1 #i #HLK #_ #HWU1 #IHWV #U2 #H - elim (sta_inv_lref1 … H) -H * #K0 #W0 #V0 #HLK0 #HWV0 #HV0U2 - lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct - lapply (IHWV … HWV0) -IHWV -HWV0 #H destruct - >(lift_mono … HWU1 … HV0U2) -W -U1 // -| #I #L #V #T #U1 #_ #IHTU1 #X #H - elim (sta_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1/ -| #L #V #T #U1 #_ #IHTU1 #X #H - elim (sta_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1/ -| #L #W #T #U1 #_ #IHTU1 #U2 #H - lapply (sta_inv_cast1 … H) -H /2 width=1/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/etc/top/lenv_top.etc b/matita/matita/contribs/lambda_delta/basic_2/etc/top/lenv_top.etc deleted file mode 100644 index ab90cebe7..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/etc/top/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 𝟙 break term 46 T2 )" - non associative with precedence 45 - for @{ 'RTop $T1 $T2 }. - -include "basic_2/grammar/lenv_px.ma". - -(* POINTWISE EXTENSION OF TOP RELATION FOR TERMS ****************************) - -definition ttop: relation term ≝ λT1,T2. True. - -definition ltop: relation lenv ≝ lpx ttop. - -interpretation - "top reduction (environment)" - 'RTop L1 L2 = (ltop L1 L2). - -(* Basic properties *********************************************************) - -lemma ltop_refl: reflexive … ltop. -/2 width=1/ qed. - -lemma ltop_sym: symmetric … ltop. -/2 width=1/ qed. - -lemma ltop_trans: transitive … ltop. -/2 width=3/ qed. - -lemma ltop_append: ∀K1,K2. K1 𝟙 K2 → ∀L1,L2. L1 𝟙 L2 → L1 @@ K1 𝟙 L2 @@ K2. -/2 width=1/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma ltop_inv_atom1: ∀L2. ⋆ 𝟙 L2 → L2 = ⋆. -/2 width=2 by lpx_inv_atom1/ qed-. - -lemma ltop_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 𝟙 L2 → - ∃∃K2,V2. K1 𝟙 K2 & L2 = K2. ⓑ{I} V2. -#K1 #I #V1 #L2 #H -elim (lpx_inv_pair1 … H) -H /2 width=4/ -qed-. - -lemma ltop_inv_atom2: ∀L1. L1 𝟙 ⋆ → L1 = ⋆. -/2 width=2 by lpx_inv_atom2/ qed-. - -lemma ltop_inv_pair2: ∀L1,K2,I,V2. L1 𝟙 K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 𝟙 K2 & L1 = K1. ⓑ{I} V1. -#L1 #K2 #I #V2 #H -elim (lpx_inv_pair2 … H) -H /2 width=4/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma ltop_fwd_length: ∀L1,L2. L1 𝟙 L2 → |L1| = |L2|. -/2 width=2 by lpx_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/aarity.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/aarity.ma deleted file mode 100644 index 7489da188..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/aarity.ma +++ /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 *) -(* *) -(**************************************************************************) - -(* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES - * Suggested invocation to start formal specifications with: - * - Patience on me to gain peace and perfection! - - *) - -include "ground_2/star.ma". -include "basic_2/notation.ma". - -(* ATOMIC ARITY *************************************************************) - -inductive aarity: Type[0] ≝ - | AAtom: aarity (* atomic aarity construction *) - | APair: aarity → aarity → aarity (* binary aarity construction *) -. - -interpretation "aarity construction (atomic)" - 'Item0 = AAtom. - -interpretation "aarity construction (binary)" - 'SnItem2 A1 A2 = (APair A1 A2). - -(* Basic inversion lemmas ***************************************************) - -lemma discr_apair_xy_x: ∀A,B. ②B. A = B → ⊥. -#A #B elim B -B -[ #H destruct -| #Y #X #IHY #_ #H destruct - -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *) - /2 width=1/ -] -qed-. - -lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → ⊥. -#B #A elim A -A -[ #H destruct -| #Y #X #_ #IHX #H destruct - -H (**) (* destruct: the destucted equality is not erased *) - /2 width=1/ -] -qed-. - -(* Basic properties *********************************************************) - -lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2). -#A1 elim A1 -A1 -[ #A2 elim A2 -A2 /2 width=1/ - #B2 #A2 #_ #_ @or_intror #H destruct -| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2 - [ -IHB1 -IHA1 @or_intror #H destruct - | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1 - [ #H destruct elim (IHA1 A2) -IHA1 - [ #H destruct /2 width=1/ - | #HA12 @or_intror #H destruct /2 width=1/ - ] - | -IHA1 #HB12 @or_intror #H destruct /2 width=1/ - ] - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/cl_shift.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/cl_shift.ma deleted file mode 100644 index bbdc8e7d0..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/cl_shift.ma +++ /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/lenv_append.ma". - -(* SHIFT OF A CLOSURE *******************************************************) - -let rec shift L T on L ≝ match L with -[ LAtom ⇒ T -| LPair L I V ⇒ shift L (-ⓑ{I} V. T) -]. - -interpretation "shift (closure)" 'Append L T = (shift L T). - -(* Basic properties *********************************************************) - -lemma shift_append_assoc: ∀L,K. ∀T:term. (L @@ K) @@ T = L @@ K @@ T. -#L #K elim K -K // normalize // -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma shift_inj: ∀L1,L2. ∀T1,T2:term. L1 @@ T1 = L2 @@ T2 → |L1| = |L2| → - L1 = L2 ∧ T1 = T2. -#L1 elim L1 -L1 -[ * normalize /2 width=1/ - #L2 #I2 #V2 #T1 #T2 #_ append_length in H2; #H - elim (plus_xySz_x_false … H) -| #K1 #I1 #V1 #IH * normalize - [ #L1 #L2 #H1 #H2 destruct - normalize in H2; >append_length in H2; #H - elim (plus_xySz_x_false … (sym_eq … H)) - | #K2 #I2 #V2 #L1 #L2 #H1 #H2 - elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *) - elim (IH … H1 ?) -IH -H1 // -H2 /2 width=1/ - ] -] -qed-. - -lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆. -#L #K #H -elim (append_inj_dx … (⋆) … H ?) // -qed-. - -lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V. -#I #L #K #V #H -elim (append_inj_dx … (⋆.ⓑ{I}V) … H ?) // -qed-. - -lemma length_inv_pos_dx_append: ∀d,L. |L| = d + 1 → - ∃∃I,K,V. |K| = d & L = ⋆.ⓑ{I}V @@ K. -#d @(nat_ind_plus … d) -d -[ #L #H - elim (length_inv_pos_dx … H) -H #I #K #V #H - >(length_inv_zero_dx … H) -H #H destruct - @ex2_3_intro [4: /2 width=2/ |5: // |1,2,3: skip ] (**) (* /3/ does not work *) -| #d #IHd #L #H - elim (length_inv_pos_dx … H) -H #I #K #V #H - elim (IHd … H) -IHd -H #I0 #K0 #V0 #H1 #H2 #H3 destruct - @(ex2_3_intro … (K0.ⓑ{I}V)) // -] -qed-. - -(* Basic_eliminators ********************************************************) - -fact lenv_ind_dx_aux: ∀R:predicate lenv. R ⋆ → - (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) → - ∀d,L. |L| = d → R L. -#R #Hatom #Hpair #d @(nat_ind_plus … d) -d -[ #L #H >(length_inv_zero_dx … H) -H // -| #d #IH #L #H - elim (length_inv_pos_dx_append … H) -H #I #K #V #H1 #H2 destruct /3 width=1/ -] -qed-. - -lemma lenv_ind_dx: ∀R:predicate lenv. R ⋆ → - (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) → - ∀L. R L. -/3 width=2 by lenv_ind_dx_aux/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma length_inv_pos_sn_append: ∀d,L. 1 + d = |L| → - ∃∃I,K,V. d = |K| & L = ⋆. ⓑ{I}V @@ K. -#d >commutative_plus @(nat_ind_plus … d) -d -[ #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct - >(length_inv_zero_sn … H1) -K - @(ex2_3_intro … (⋆)) // (**) (* explicit constructor *) -| #d #IHd #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct - >H1 in IHd; -H1 #IHd - elim (IHd K ?) -IHd // #J #L #W #H1 #H2 destruct - @(ex2_3_intro … (L.ⓑ{I}V)) // (**) (* explicit constructor *) - >append_length /2 width=1/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_length.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_length.ma deleted file mode 100644 index faf6de02d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_length.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 "basic_2/grammar/lenv.ma". - -(* LENGTH OF A LOCAL ENVIRONMENT ********************************************) - -let rec length L ≝ match L with -[ LAtom ⇒ 0 -| LPair L _ _ ⇒ length L + 1 -]. - -interpretation "length (local environment)" 'card L = (length L). - -(* Basic inversion lemmas ***************************************************) - -lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆. -* // #L #I #V normalize (lpx_inv_atom1 … H) -X /2 width=3/ -| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H - elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct - elim (IHK01 … HK02) -K0 #K #HK1 #HK2 - elim (HR … HV01 … HV02) -HR -V0 /3 width=5/ -] -qed. - -lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2. -#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/ -qed. - -lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L → - ∀L2. lpx R L L2 → lpx (TC … R) L1 L2. -#R #L1 #L #H elim H -L /2 width=1/ -#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H -elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ -qed. - -lemma TC_lpx_pair_dx: ∀R. reflexive ? R → - ∀I,K,V1,V2. TC … R V1 V2 → - TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2). -#R #HR #I #K #V1 #V2 #H elim H -V2 -/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *) -qed. - -lemma TC_lpx_pair_sn: ∀R. reflexive ? R → - ∀I,V,K1,K2. TC … (lpx R) K1 K2 → - TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V). -#R #HR #I #V #K1 #K2 #H elim H -K2 -/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *) -qed. - -lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2. -#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/ -qed. - -lemma lpx_inv_TC: ∀R. reflexive ? R → - ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2. -#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/ -qed. - -lemma lpx_append: ∀R,K1,K2. lpx R K1 K2 → ∀L1,L2. lpx R L1 L2 → - lpx R (L1 @@ K1) (L2 @@ K2). -#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px_bi.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px_bi.ma deleted file mode 100644 index 931d075ab..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px_bi.ma +++ /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 ≝ -| lpx_bi_stom: lpx_bi R (⋆) (⋆) -| lpx_bi_pair: ∀I,K1,K2,V1,V2. - lpx_bi R K1 K2 → R K1 V1 K2 V2 → - lpx_bi R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) -. - -(* Basic inversion lemmas ***************************************************) - -fact lpx_bi_inv_atom1_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L1 = ⋆ → L2 = ⋆. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_bi_inv_atom1: ∀R,L2. lpx_bi R (⋆) L2 → L2 = ⋆. -/2 width=4 by lpx_bi_inv_atom1_aux/ qed-. - -fact lpx_bi_inv_pair1_aux: ∀R,L1,L2. lpx_bi R L1 L2 → - ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. lpx_bi R K1 K2 & - R K1 V1 K2 V2 & L2 = K2. ⓑ{I} V2. -#R #L1 #L2 * -L1 -L2 -[ #J #K1 #V1 #H destruct -| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/ -] -qed-. - -lemma lpx_bi_inv_pair1: ∀R,I,K1,V1,L2. lpx_bi R (K1. ⓑ{I} V1) L2 → - ∃∃K2,V2. lpx_bi R K1 K2 & R K1 V1 K2 V2 & - L2 = K2. ⓑ{I} V2. -/2 width=3 by lpx_bi_inv_pair1_aux/ qed-. - -fact lpx_bi_inv_atom2_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L2 = ⋆ → L1 = ⋆. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_bi_inv_atom2: ∀R,L1. lpx_bi R L1 (⋆) → L1 = ⋆. -/2 width=4 by lpx_bi_inv_atom2_aux/ qed-. - -fact lpx_bi_inv_pair2_aux: ∀R,L1,L2. lpx_bi R L1 L2 → - ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & - L1 = K1. ⓑ{I} V1. -#R #L1 #L2 * -L1 -L2 -[ #J #K2 #V2 #H destruct -| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/ -] -qed-. - -lemma lpx_bi_inv_pair2: ∀R,I,L1,K2,V2. lpx_bi R L1 (K2. ⓑ{I} V2) → - ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & - L1 = K1. ⓑ{I} V1. -/2 width=3 by lpx_bi_inv_pair2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lpx_bi_fwd_length: ∀R,L1,L2. lpx_bi R L1 L2 → |L1| = |L2|. -#R #L1 #L2 #H elim H -L1 -L2 normalize // -qed-. - -(* Basic properties *********************************************************) - -lemma lpx_bi_refl: ∀R. bi_reflexive ? ? R → reflexive … (lpx_bi R). -#R #HR #L elim L -L // /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_weight.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_weight.ma deleted file mode 100644 index 59e2e6172..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_weight.ma +++ /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/grammar/term_weight.ma". -include "basic_2/grammar/lenv.ma". - -(* WEIGHT OF A LOCAL ENVIRONMENT ********************************************) - -let rec lw L ≝ match L with -[ LAtom ⇒ 0 -| LPair L _ V ⇒ lw L + #{V} -]. - -interpretation "weight (local environment)" 'Weight L = (lw L). - -(* Basic properties *********************************************************) - -lemma lw_pair: ∀I,L,V. #{L} < #{(L.ⓑ{I}V)}. -/3 width=1/ qed. - -(* Basic eliminators ********************************************************) - -axiom lw_ind: ∀R:predicate lenv. - (∀L2. (∀L1. #{L1} < #{L2} → R L1) → R L2) → - ∀L. R L. - -(* Basic_1: removed theorems 2: clt_cong clt_head clt_thead *) -(* Basic_1: note: clt_thead should be renamed clt_ctail *) diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/term.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/term.ma deleted file mode 100644 index 5ccb5e4ac..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/term.ma +++ /dev/null @@ -1,135 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/item.ma". - -(* TERMS ********************************************************************) - -(* terms *) -inductive term: Type[0] ≝ - | TAtom: item0 → term (* atomic item construction *) - | TPair: item2 → term → term → term (* binary item construction *) -. - -interpretation "term construction (atomic)" - 'Item0 I = (TAtom I). - -interpretation "term construction (binary)" - 'SnItem2 I T1 T2 = (TPair I T1 T2). - -interpretation "term binding construction (binary)" - 'SnBind2 a I T1 T2 = (TPair (Bind2 a I) T1 T2). - -interpretation "term positive binding construction (binary)" - 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2). - -interpretation "term negative binding construction (binary)" - 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2). - -interpretation "term flat construction (binary)" - 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2). - -interpretation "sort (term)" - 'Star k = (TAtom (Sort k)). - -interpretation "local reference (term)" - 'LRef i = (TAtom (LRef i)). - -interpretation "global reference (term)" - 'GRef p = (TAtom (GRef p)). - -interpretation "abbreviation (term)" - 'SnAbbr a T1 T2 = (TPair (Bind2 a Abbr) T1 T2). - -interpretation "positive abbreviation (term)" - 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2). - -interpretation "negative abbreviation (term)" - 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2). - -interpretation "abstraction (term)" - 'SnAbst a T1 T2 = (TPair (Bind2 a Abst) T1 T2). - -interpretation "positive abstraction (term)" - 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2). - -interpretation "negative abstraction (term)" - 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2). - -interpretation "application (term)" - 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2). - -interpretation "native type annotation (term)" - 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2). - -(* Basic properties *********************************************************) - -(* Basic_1: was: term_dec *) -axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2). - -(* Basic inversion lemmas ***************************************************) - -lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → ⊥. -#I #T #V elim V -V -[ #J #H destruct -| #J #W #U #IHW #_ #H destruct - -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *) - /2 width=1/ -] -qed-. - -(* Basic_1: was: thead_x_y_y *) -lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → ⊥. -#I #V #T elim T -T -[ #J #H destruct -| #J #W #U #_ #IHU #H destruct - -H (**) (* destruct: the destucted equality is not erased *) - /2 width=1/ -] -qed-. - -lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2. - (②{I} V1. T1 = ②{I} V2. T2 → ⊥) → - (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)). -#I #V1 #T1 #V2 #T2 #H -elim (term_eq_dec V1 V2) /3 width=1/ #HV12 destruct -@or_intror @conj // #HT12 destruct /2 width=1/ -qed-. - -lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2. - (②{I} V1. T1 = ②{I} V2. T2 → ⊥) → - (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)). -#I #V1 #T1 #V2 #T2 #H -elim (term_eq_dec T1 T2) /3 width=1/ #HT12 destruct -@or_intror @conj // #HT12 destruct /2 width=1/ -qed-. - -lemma eq_false_inv_beta: ∀a,V1,V2,W1,W2,T1,T2. - (ⓐV1. ⓛ{a}W1. T1 = ⓐV2. ⓛ{a}W2 .T2 → ⊥) → - (W1 = W2 → ⊥) ∨ - (W1 = W2 ∧ (ⓓ{a}V1. T1 = ⓓ{a}V2. T2 → ⊥)). -#a #V1 #V2 #W1 #W2 #T1 #T2 #H -elim (eq_false_inv_tpair_sn … H) -H -[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/ - #H destruct @or_intror @conj // #H destruct /2 width=1/ -| * #H1 #H2 destruct - elim (eq_false_inv_tpair_sn … H2) -H2 /3 width=1/ - * #H #HT12 destruct - @or_intror @conj // #H destruct /2 width=1/ -] -qed. - -(* Basic_1: removed theorems 3: - not_void_abst not_abbr_void not_abst_void -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/term_simple.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/term_simple.ma deleted file mode 100644 index 328dc55a6..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/term_simple.ma +++ /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/grammar/term.ma". - -(* SIMPLE (NEUTRAL) TERMS ***************************************************) - -inductive simple: predicate term ≝ - | simple_atom: ∀I. simple (⓪{I}) - | simple_flat: ∀I,V,T. simple (ⓕ{I} V. T) -. - -interpretation "simple (term)" 'Simple T = (simple T). - -(* Basic inversion lemmas ***************************************************) -(* -lemma mt: ∀R1,R2:Prop. (R1 → R2) → (R2 → ⊥) → R1 → ⊥. -/3 width=1/ qed. -*) -fact simple_inv_bind_aux: ∀T. 𝐒⦃T⦄ → ∀a,J,W,U. T = ⓑ{a,J} W. U → ⊥. -#T * -T -[ #I #a #J #W #U #H destruct -| #I #V #T #a #J #W #U #H destruct -] -qed. - -lemma simple_inv_bind: ∀a,I,V,T. 𝐒⦃ⓑ{a,I} V. T⦄ → ⊥. -/2 width=7/ qed-. (**) (* auto fails if mt is enabled *) - -lemma simple_inv_pair: ∀I,V,T. 𝐒⦃②{I}V.T⦄ → ∃J. I = Flat2 J. -* /2 width=2/ #a #I #V #T #H -elim (simple_inv_bind … H) -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/term_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/term_vector.ma deleted file mode 100644 index 7169d1af4..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/term_vector.ma +++ /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 "ground_2/list.ma". -include "basic_2/grammar/term_simple.ma". - -(* TERMS ********************************************************************) - -let rec applv Vs T on Vs ≝ - match Vs with - [ nil ⇒ T - | cons hd tl ⇒ ⓐhd. (applv tl T) - ]. - -interpretation "application o vevtor (term)" - 'SnApplV Vs T = (applv Vs T). - -(* properties concerning simple terms ***************************************) - -lemma applv_simple: ∀T,Vs. 𝐒⦃T⦄ → 𝐒⦃ⒶVs.T⦄. -#T * // -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/term_weight.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/term_weight.ma deleted file mode 100644 index d8f39a3a1..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/term_weight.ma +++ /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/grammar/term.ma". - -(* WEIGHT OF A TERM *********************************************************) - -let rec tw T ≝ match T with -[ TAtom _ ⇒ 1 -| TPair _ V T ⇒ tw V + tw T + 1 -]. - -interpretation "weight (term)" 'Weight T = (tw T). - -(* Basic properties *********************************************************) - -(* Basic_1: was: tweight_lt *) -lemma tw_pos: ∀T. 1 ≤ #{T}. -#T elim T -T // -qed. - -(* Basic eliminators ********************************************************) - -axiom tw_ind: ∀R:predicate term. - (∀T2. (∀T1. #{T1} < #{T2} → R T1) → R T2) → - ∀T. R T. - -(* Basic_1: removed theorems 11: - wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O - weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind - removed local theorems 1: q_ind -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/tshf.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/tshf.ma deleted file mode 100644 index a8873c18b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/tshf.ma +++ /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/grammar/term_simple.ma". - -(* SAME HEAD TERM FORMS *****************************************************) - -inductive tshf: relation term ≝ - | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I}) - | tshf_abbr: ∀V1,V2,T1,T2. tshf (-ⓓV1. T1) (-ⓓV2. T2) - | tshf_abst: ∀a,V1,V2,T1,T2. tshf (ⓛ{a}V1. T1) (ⓛ{a}V2. T2) - | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄ → - tshf (ⓐV1. T1) (ⓐV2. T2) -. - -interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2). - -(* Basic properties *********************************************************) - -lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1. -#T1 #T2 #H elim H -T1 -T2 /2 width=1/ -qed. - -lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2. -#T1 #T2 #H elim H -T1 -T2 // /2 width=1/ -qed. - -lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1. -/3 width=2/ qed. - -lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. -#T1 #T2 #H elim H -T1 -T2 // -[ #V1 #V2 #T1 #T2 #H - elim (simple_inv_bind … H) -| #a #V1 #V2 #T1 #T2 #H - elim (simple_inv_bind … H) -] -qed. (**) (* remove from index *) - -lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. -/3 width=3/ qed-. - -(* Basic inversion lemmas ***************************************************) - -fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀a,I,W1,U1. T1 = ⓑ{a,I}W1.U1 → - ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 & - (Bind2 a I = Bind2 false Abbr ∨ I = Abst). -#T1 #T2 * -T1 -T2 -[ #J #a #I #W1 #U1 #H destruct -| #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/ -| #b #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/ -| #V1 #V2 #T1 #T2 #_ #_ #_ #a #I #W1 #U1 #H destruct -] -qed. - -lemma tshf_inv_bind1: ∀a,I,W1,U1,T2. ⓑ{a,I}W1.U1 ≈ T2 → - ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 & - (Bind2 a I = Bind2 false Abbr ∨ I = Abst). -/2 width=5/ qed-. - -fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 → - ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ & - I = Appl & T2 = ⓐW2. U2. -#T1 #T2 * -T1 -T2 -[ #J #I #W1 #U1 #H destruct -| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct -| #a #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct -| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/ -] -qed. - -lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 → - ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ & - I = Appl & T2 = ⓐW2. U2. -/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc.ma deleted file mode 100644 index 78a9b4987..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc.ma +++ /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/grammar/term_simple.ma". - -(* SAME TOP TERM CONSTRUCTOR ************************************************) - -inductive tstc: relation term ≝ - | tstc_atom: ∀I. tstc (⓪{I}) (⓪{I}) - | tstc_pair: ∀I,V1,V2,T1,T2. tstc (②{I} V1. T1) (②{I} V2. T2) -. - -interpretation "same top constructor (term)" 'Iso T1 T2 = (tstc T1 T2). - -(* Basic inversion lemmas ***************************************************) - -fact tstc_inv_atom1_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T1 = ⓪{I} → T2 = ⓪{I}. -#T1 #T2 * -T1 -T2 // -#J #V1 #V2 #T1 #T2 #I #H destruct -qed. - -(* Basic_1: was: iso_gen_sort iso_gen_lref *) -lemma tstc_inv_atom1: ∀I,T2. ⓪{I} ≃ T2 → T2 = ⓪{I}. -/2 width=3/ qed-. - -fact tstc_inv_pair1_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W1,U1. T1 = ②{I}W1.U1 → - ∃∃W2,U2. T2 = ②{I}W2. U2. -#T1 #T2 * -T1 -T2 -[ #J #I #W1 #U1 #H destruct -| #J #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/ -] -qed. - -(* Basic_1: was: iso_gen_head *) -lemma tstc_inv_pair1: ∀I,W1,U1,T2. ②{I}W1.U1 ≃ T2 → - ∃∃W2,U2. T2 = ②{I}W2. U2. -/2 width=5/ qed-. - -fact tstc_inv_atom2_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T2 = ⓪{I} → T1 = ⓪{I}. -#T1 #T2 * -T1 -T2 // -#J #V1 #V2 #T1 #T2 #I #H destruct -qed. - -lemma tstc_inv_atom2: ∀I,T1. T1 ≃ ⓪{I} → T1 = ⓪{I}. -/2 width=3/ qed-. - -fact tstc_inv_pair2_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W2,U2. T2 = ②{I}W2.U2 → - ∃∃W1,U1. T1 = ②{I}W1. U1. -#T1 #T2 * -T1 -T2 -[ #J #I #W2 #U2 #H destruct -| #J #V1 #V2 #T1 #T2 #I #W2 #U2 #H destruct /2 width=3/ -] -qed. - -lemma tstc_inv_pair2: ∀I,T1,W2,U2. T1 ≃ ②{I}W2.U2 → - ∃∃W1,U1. T1 = ②{I}W1. U1. -/2 width=5/ qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: iso_refl *) -lemma tstc_refl: ∀T. T ≃ T. -#T elim T -T // -qed. - -lemma tstc_sym: ∀T1,T2. T1 ≃ T2 → T2 ≃ T1. -#T1 #T2 #H elim H -T1 -T2 // -qed. - -lemma tstc_dec: ∀T1,T2. Decidable (T1 ≃ T2). -* #I1 [2: #V1 #T1 ] * #I2 [2,4: #V2 #T2 ] -[ elim (item2_eq_dec I1 I2) #HI12 - [ destruct /2 width=1/ - | @or_intror #H - elim (tstc_inv_pair1 … H) -H #V #T #H destruct /2 width=1/ - ] -| @or_intror #H - lapply (tstc_inv_atom1 … H) -H #H destruct -| @or_intror #H - lapply (tstc_inv_atom2 … H) -H #H destruct -| elim (item0_eq_dec I1 I2) #HI12 - [ destruct /2 width=1/ - | @or_intror #H - lapply (tstc_inv_atom2 … H) -H #H destruct /2 width=1/ - ] -] -qed. - -lemma simple_tstc_repl_dx: ∀T1,T2. T1 ≃ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. -#T1 #T2 * -T1 -T2 // -#I #V1 #V2 #T1 #T2 #H -elim (simple_inv_pair … H) -H #J #H destruct // -qed. (**) (* remove from index *) - -lemma simple_tstc_repl_sn: ∀T1,T2. T1 ≃ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. -/3 width=3/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc_tstc.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc_tstc.ma deleted file mode 100644 index df6fe3729..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc_tstc.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_2/grammar/tstc.ma". - -(* SAME TOP TERM CONSTRUCTOR ************************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: iso_trans *) -theorem tstc_trans: ∀T1,T. T1 ≃ T → ∀T2. T ≃ T2 → T1 ≃ T2. -#T1 #T * -T1 -T // -#I #V1 #V #T1 #T #X #H -elim (tstc_inv_pair1 … H) -H #V2 #T2 #H destruct // -qed. - -theorem tstc_canc_sn: ∀T,T1. T ≃ T1 → ∀T2. T ≃ T2 → T1 ≃ T2. -/3 width=3/ qed. - -theorem tstc_canc_dx: ∀T1,T. T1 ≃ T → ∀T2. T2 ≃ T → T1 ≃ T2. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc_vector.ma deleted file mode 100644 index 1e35292ef..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/tstc_vector.ma +++ /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/grammar/term_vector.ma". -include "basic_2/grammar/tstc.ma". - -(* SAME TOP TERM CONSTRUCTOR ************************************************) - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *) -lemma tstc_inv_bind_appls_simple: ∀a,I,Vs,V2,T1,T2. ⒶVs.T1 ≃ ⓑ{a,I} V2. T2 → - 𝐒⦃T1⦄ → ⊥. -#a #I #Vs #V2 #T1 #T2 #H -elim (tstc_inv_pair2 … H) -H #V0 #T0 -elim Vs -Vs normalize -[ #H destruct #H - @(simple_inv_bind … H) -| #V #Vs #_ #H destruct -] -qed. - diff --git a/matita/matita/contribs/lambda_delta/basic_2/names.txt b/matita/matita/contribs/lambda_delta/basic_2/names.txt deleted file mode 100644 index 6c34328b4..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/names.txt +++ /dev/null @@ -1,48 +0,0 @@ -NAMING CONVENTIONS FOR METAVARIABLES - -A,B : arity -C,D : candidate of reducibility -E,F : RTM environment -G : global environment -H : reserved: transient premise -IH : reserved: inductive premise -I,J : item -K,L : local environment -M,N : reserved: future use -O,P,Q : -R : generic predicate (relation) -S : RTM stack -T,U,V,W: term -X,Y,Z : reserved: transient objet denoted by a capital letter - -a,b : binder polarity -c : reserved: future use (lambda_delta 3) -d : relocation depth -e : relocation height -f : -g : sort degree parameter -h : sort hierarchy parameter -i,j : local reference position index (de Bruijn's) -k : sort index -l : term degree -m,n : reserved: future use -o : -p,q : global reference position -r,s : -t,u : local reference position level (de Bruijn's) -v,w : -x,y,z : reserved: transient objet denoted by a small letter - -NAMING CONVENTIONS FOR CONSTRUCTORS - -0: atomic -2: binary - -A: application to vector - -a: application -b: binder -d: abbreviation -f: flat -l: abstraction -n: native type annotation diff --git a/matita/matita/contribs/lambda_delta/basic_2/notation.ma b/matita/matita/contribs/lambda_delta/basic_2/notation.ma deleted file mode 100644 index 3ff8f21e5..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/notation.ma +++ /dev/null @@ -1,475 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -(* Grammar ******************************************************************) - -notation "⓪" - non associative with precedence 90 - for @{ 'Item0 }. - -notation "hvbox( ⓪ { term 46 I } )" - non associative with precedence 90 - for @{ 'Item0 $I }. - -notation "⋆" - non associative with precedence 90 - for @{ 'Star }. - -notation "hvbox( ⋆ term 90 k )" - non associative with precedence 90 - for @{ 'Star $k }. - -notation "hvbox( # term 90 i )" - non associative with precedence 90 - for @{ 'LRef $i }. - -notation "hvbox( § term 90 p )" - non associative with precedence 90 - for @{ 'GRef $p }. - -notation "hvbox( ② term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnItem2 $T1 $T }. - -notation "hvbox( ② { term 46 I } break term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnItem2 $I $T1 $T }. - -notation "hvbox( ⓑ { term 46 a , term 46 I } break term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnBind2 $a $I $T1 $T }. - -notation "hvbox( + ⓑ { term 46 I } break term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnBind2Pos $I $T1 $T }. - -notation "hvbox( - ⓑ { term 46 I } break term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnBind2Neg $I $T1 $T }. - -notation "hvbox( ⓕ { term 46 I } break term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnFlat2 $I $T1 $T }. - -notation "hvbox( ⓓ { term 46 a } term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAbbr $a $T1 $T2 }. - -notation "hvbox( + ⓓ term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAbbrPos $T1 $T2 }. - -notation "hvbox( - ⓓ term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAbbrNeg $T1 $T2 }. - -notation "hvbox( ⓛ { term 46 a } term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAbst $a $T1 $T2 }. - -notation "hvbox( + ⓛ term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAbstPos $T1 $T2 }. - -notation "hvbox( - ⓛ term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAbstNeg $T1 $T2 }. - -notation "hvbox( ⓐ term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnAppl $T1 $T2 }. - -notation "hvbox( ⓝ term 55 T1 . break term 55 T2 )" - non associative with precedence 55 - for @{ 'SnCast $T1 $T2 }. - -notation "hvbox( Ⓐ term 55 T1 . break term 55 T )" - non associative with precedence 55 - for @{ 'SnApplV $T1 $T }. - -notation > "hvbox( T . break ②{ term 46 I } break term 47 T1 )" - non associative with precedence 46 - for @{ 'DxBind2 $T $I $T1 }. - -notation "hvbox( T . break ⓑ { term 46 I } break term 48 T1 )" - non associative with precedence 47 - for @{ 'DxBind2 $T $I $T1 }. - -notation "hvbox( T1 . break ⓓ T2 )" - left associative with precedence 48 - for @{ 'DxAbbr $T1 $T2 }. - -notation "hvbox( T1 . break ⓛ T2 )" - left associative with precedence 49 - for @{ 'DxAbst $T1 $T2 }. - -notation "hvbox( T . break ④ { term 46 I } break { term 46 T1 , break term 46 T2 , break term 46 T3 } )" - non associative with precedence 50 - for @{ 'DxItem4 $T $I $T1 $T2 $T3 }. - -notation "hvbox( # { term 46 x } )" - non associative with precedence 90 - for @{ 'Weight $x }. - -notation "hvbox( # { term 46 x , break term 46 y } )" - non associative with precedence 90 - for @{ 'Weight $x $y }. - -notation "hvbox( 𝐒 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'Simple $T }. - -notation "hvbox( L ⊢ break term 46 T1 ≈ break term 46 T2 )" - non associative with precedence 45 - for @{ 'Hom $L $T1 $T2 }. - -notation "hvbox( T1 ≃ break term 46 T2 )" - non associative with precedence 45 - for @{ 'Iso $T1 $T2 }. - -(* Substitution *************************************************************) - -notation "hvbox( ⇧ [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'RLift $d $e $T1 $T2 }. - -notation "hvbox( T1 break ≼ [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'SubEq $T1 $d $e $T2 }. - -notation "hvbox( ≽ [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'SubEqBottom $d $e $T2 }. - -notation "hvbox( ⇩ [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'RDrop $e $L1 $L2 }. - -notation "hvbox( ⇩ [ term 46 d , break term 46 e ] break term 46 L1 ≡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'RDrop $d $e $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'RestSupTerm $L1 $T1 $L2 $T2 }. - -notation "hvbox( L ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )" - non associative with precedence 45 - for @{ 'ICM $L $T $k }. - -notation "hvbox( L ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubst $L $T1 $d $e $T2 }. - -(* Unfold *******************************************************************) - -notation "hvbox( @ ⦃ term 46 T1 , break term 46 f ⦄ ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'RAt $T1 $f $T2 }. - -notation "hvbox( T1 ▭ break term 46 T2 ≡ break term 46 T )" - non associative with precedence 45 - for @{ 'RMinus $T1 $T2 $T }. - -notation "hvbox( ⇧ * [ term 46 e ] break term 46 T1 ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'RLiftStar $e $T1 $T2 }. - -notation "hvbox( ⇩ * [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'RDropStar $e $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'RestSupTermPlus $L1 $T1 $L2 $T2 }. - -notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'RestSupTermStar $L1 $T1 $L2 $T2 }. - -notation "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStar $T1 $d $e $T2 }. - -notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStar $L $T1 $d $e $T2 }. - -notation "hvbox( L ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStarAlt $L $T1 $d $e $T2 }. - -notation "hvbox( T1 break ⊢ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStarSn $T1 $d $e $T2 }. - -notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }. - -notation "hvbox( ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'TSubst $T1 $d $e $T2 }. - -notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'TSubst $L $T1 $d $e $T2 }. - -notation "hvbox( ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'TSubstAlt $T1 $d $e $T2 }. - -notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'TSubstAlt $L $T1 $d $e $T2 }. - -(* Static typing ************************************************************) - -notation "hvbox( L ⊢ break term 46 T ⁝ break term 46 A )" - non associative with precedence 45 - for @{ 'AtomicArity $L $T $A }. - -notation "hvbox( T1 ⁝ ⊑ break term 46 T2 )" - non associative with precedence 45 - for @{ 'CrSubEqA $T1 $T2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )" - non associative with precedence 45 - for @{ 'BinaryArity $h $L $T $A }. - -notation "hvbox( h ⊢ break term 46 L1 ÷ ⊑ break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqB $h $L1 $L2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • break [ term 46 g , break term 46 l ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'StaticType $h $g $l $L $T1 $T2 }. - -notation "hvbox( h ⊢ break term 46 L1 • ⊑ [ term 46 g ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqS $h $g $L1 $L2 }. - -(* Unwind *******************************************************************) - -notation "hvbox( L1 ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'Unwind $L1 $T $L2 }. - -(* Reducibility *************************************************************) - -notation "hvbox( L ⊢ break 𝐑 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'Reducible $L $T }. - -notation "hvbox( L ⊢ break 𝐈 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'NotReducible $L $T }. - -notation "hvbox( L ⊢ break 𝐍 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'Normal $L $T }. - -(* this might be removed *) -notation "hvbox( 𝐇𝐑 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'HdReducible $T }. - -(* this might be removed *) -notation "hvbox( L ⊢ break 𝐇𝐑 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'HdReducible $L $T }. - -(* this might be removed *) -notation "hvbox( 𝐇𝐈 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'NotHdReducible $T }. - -(* this might be removed *) -notation "hvbox( L ⊢ break 𝐇𝐈 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'NotHdReducible $L $T }. - -(* this might be removed *) -notation "hvbox( 𝐇𝐍 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'HdNormal $T }. - -(* this might be removed *) -notation "hvbox( L ⊢ break 𝐇𝐍 ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'HdNormal $L $T }. - -notation "hvbox( T1 ➡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'PRed $T1 $T2 }. - -notation "hvbox( L ⊢ break term 46 T1 ➡ break term 46 T2 )" - non associative with precedence 45 - for @{ 'PRed $L $T1 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ➡ break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRed $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }. - -notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRedAlt $L1 $L2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • ➡ break [ term 46 g ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'XPRed $h $g $L $T1 $T2 }. - -(* Computation **************************************************************) - -notation "hvbox( T1 ➡ * break term 46 T2 )" - non associative with precedence 45 - for @{ 'PRedStar $T1 $T2 }. - -notation "hvbox( L ⊢ break term 46 T1 ➡ * break term 46 T2 )" - non associative with precedence 45 - for @{ 'PRedStar $L $T1 $T2 }. - -notation "hvbox( T1 ➡ ➡ * break term 46 T2 )" - non associative with precedence 45 - for @{ 'PRedStarAlt $T1 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ➡ * break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRedStar $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ * break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRedStarAlt $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPRedStarAlt $L1 $T1 $L2 $T2 }. - -notation "hvbox( L ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ Tterm 46 2 ⦄ )" - non associative with precedence 45 - for @{ 'PEval $L $T1 $T2 }. - -notation "hvbox( ⬊ * term 46 T )" - non associative with precedence 45 - for @{ 'SN $T }. - -notation "hvbox( L ⊢ ⬊ * break term 46 T )" - non associative with precedence 45 - for @{ 'SN $L $T }. - -notation "hvbox( L ⊢ ⬊ ⬊ * break term 46 T )" - non associative with precedence 45 - for @{ 'SNAlt $L $T }. - -notation "hvbox( ⦃ term 46 L, break term 46 T ⦄ ϵ break [ term 46 R ] break 〚term 46 A 〛 )" - non associative with precedence 45 - for @{ 'InEInt $R $L $T $A }. - -notation "hvbox( T1 ⊑ break [ term 46 R ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'CrSubEq $T1 $R $T2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • ➡ * break [ term 46 g ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'XPRedStar $h $g $L $T1 $T2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • ⬊ * break [ term 46 g ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'XSN $h $g $L $T }. - -(* Conversion ***************************************************************) - -notation "hvbox( L ⊢ break term 46 T1 ⬌ break term 46 T2 )" - non associative with precedence 45 - for @{ 'PConv $L $T1 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConv $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConvAlt $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }. - -(* Equivalence **************************************************************) - -notation "hvbox( L ⊢ break term 46 T1 ⬌* break term 46 T2 )" - non associative with precedence 45 - for @{ 'PConvStar $L $T1 $T2 }. - -notation "hvbox( h ⊢ break term 46 L1 ⊢ • ⊑ [ term 46 g ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqSE $h $g $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ * break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConvStar $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }. - -notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConvStarAlt $L1 $L2 }. - -notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }. - -(* Dynamic typing ***********************************************************) - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊩ break term 46 T : break [ term 46 g ] )" - non associative with precedence 45 - for @{ 'NativeValid $h $g $L $T }. - -notation "hvbox( h ⊢ break term 46 L1 ⊩ : ⊑ [ term 46 g ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'CrSubEqV $h $g $L1 $L2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeType $h $L $T1 $T2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeTypeAlt $h $L $T1 $T2 }. - -(* Higher order dynamic typing **********************************************) - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeTypeStar $h $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr.ma deleted file mode 100644 index 95c01d19f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr.ma +++ /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/reducibility/cpr.ma". -include "basic_2/reducibility/fpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************) - -definition cfpr: lenv → bi_relation lenv term ≝ - λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2. - -interpretation - "context-sensitive parallel reduction (closure)" - 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2). - -(* Basic properties *********************************************************) - -lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L). -/2 width=1/ qed. - -lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄. -#L1 #L2 #T1 #T2 * /3 width=1/ -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆. -#L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/ -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ → - ∃∃K2,V2. V1 ➡ V2 & - ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ & - L2 = ⋆.ⓑ{I}V2@@K2. -#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H -elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct ->shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H -elim (tpr_inv_bind1 … H) -H * -[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/ -| #T0 #_ #_ #H destruct -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_aaa.ma deleted file mode 100644 index 4c9b8ac97..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_aaa.ma +++ /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: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → - ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A. -#L1 #T1 #A #HT1 #L2 #T2 #H -elim (fpr_inv_all … H) -H -/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_cfpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_cfpr.ma deleted file mode 100644 index f442be28b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_cfpr.ma +++ /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: ∀L. bi_confluent … (cfpr L). -#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02 -elim (cpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2 -elim (cpr_fwd_shift1 … H1) #L0 #T0 #HL10 #H destruct /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_cpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_cpr.ma deleted file mode 100644 index a46c9776b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_cpr.ma +++ /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/unfold/ltpss_sn_alt.ma". -include "basic_2/reducibility/cpr_tpss.ma". -include "basic_2/reducibility/cpr_cpr.ma". -include "basic_2/reducibility/cfpr_ltpss.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************) - -(* Advanced properties ******************************************************) - -lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 → - L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄. -#L1 #L #H elim H -L1 -L -[ #L2 #T1 #T2 #HT12 #HL2 - lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct - lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/ -| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H - elim (ltpss_sn_inv_tpss21 … H ?) -H // append_length >append_length #H - lapply (injective_plus_r … H) -H #H - @(ex3_1_intro … (⋆.ⓑ{I}V@@Y)) append_length H -H >commutative_plus /3 width=1/ -] -qed-. - -lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → - ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2. -#L1 #L2 #T1 #T2 #H -lapply (fpr_cfpr … H) -H #H -elim (cfpr_inv_all … H) -H /2 width=4/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_ltpss.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_ltpss.ma deleted file mode 100644 index 424911376..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cfpr_ltpss.ma +++ /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.ma". -include "basic_2/reducibility/cfpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************) - -(* Advanced inversion lemmas ************************************************) - -lemma cfpr_inv_pair1: ∀I,L,K1,L2,V1,T1,T2. L ⊢ ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ → - ∃∃K2,V,V2. V1 ➡ V & L ⊢ V ▶* [0, |L|] V2 & - L.ⓑ{I}V ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ & - L2 = ⋆.ⓑ{I}V2@@K2. -* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H -elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct ->shift_append_assoc >shift_append_assoc normalize in ⊢ (??%%→?); #H -[ elim (cpr_inv_abbr1 … H) -H * - [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/ - | #T0 #_ #_ #H destruct - ] -| elim (cpr_inv_abst1 … H Abst V2) -H - #V #T * #V0 #HV10 #HV0 #HT1 #H destruct - lapply (ltpss_sn_cpr_trans (L.ⓛV0) … 0 (|L|+1) … HT1) -HT1 /2 width=1/ #HT12 - /3 width=7/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cif.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cif.ma deleted file mode 100644 index 0ea9d519b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cif.ma +++ /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/reducibility/crf.ma". - -(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************) - -definition cif: lenv → predicate term ≝ λL,T. L ⊢ 𝐑⦃T⦄ → ⊥. - -interpretation "context-sensitive irreducibility (term)" - 'NotReducible L T = (cif L T). - -(* Basic inversion lemmas ***************************************************) - -lemma cif_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐈⦃#i⦄ → ⊥. -/3 width=3/ qed-. - -lemma cif_inv_ri2: ∀I,L,V,T. ri2 I → L ⊢ 𝐈⦃②{I}V.T⦄ → ⊥. -/3 width=1/ qed-. - -lemma cif_inv_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ → - L ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄. -/4 width=1/ qed-. - -lemma cif_inv_bind: ∀a,I,L,V,T. L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ → - ∧∧ L ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I. -#a * [ elim a -a ] -[ #L #V #T #H elim (H ?) -H /3 width=1/ -|*: #L #V #T #H elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=1/ -] -qed-. - -lemma cif_inv_appl: ∀L,V,T. L ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄. -#L #V #T #HVT @and3_intro /3 width=1/ -generalize in match HVT; -HVT elim T -T // -* // #a * #U #T #_ #_ #H elim (H ?) -H /2 width=1/ -qed-. - -lemma cif_inv_flat: ∀I,L,V,T. L ⊢ 𝐈⦃ⓕ{I}V.T⦄ → - ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl. -* #L #V #T #H -[ elim (cif_inv_appl … H) -H /2 width=1/ -| elim (cif_inv_ri2 … H) -H /2 width=1/ -] -qed-. - -(* Basic properties *********************************************************) - -lemma tif_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄. -/2 width=2 by trf_inv_atom/ qed. - -lemma cif_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄. -#a #I #L #V #T #HI #HV #HT #H -elim (crf_inv_ib2 … HI H) -HI -H /2 width=1/ -qed. - -lemma cif_appl: ∀L,V,T. L ⊢ 𝐈⦃V⦄ → L ⊢ 𝐈⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐈⦃ⓐV.T⦄. -#L #V #T #HV #HT #H1 #H2 -elim (crf_inv_appl … H2) -H2 /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cif_append.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cif_append.ma deleted file mode 100644 index 45fd178cf..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cif_append.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 "basic_2/reducibility/crf_append.ma". -include "basic_2/reducibility/cif.ma". - -(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************) - -(* Advanved properties ******************************************************) - -lemma cif_labst_last: ∀L,T,W. L ⊢ 𝐈⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄. -/3 width=2 by crf_inv_labst_last/ qed. - -lemma cif_tif: ∀T,W. ⋆ ⊢ 𝐈⦃T⦄ → ⋆.ⓛW ⊢ 𝐈⦃T⦄. -/3 width=2 by crf_inv_trf/ qed. - -(* Advanced inversion lemmas ************************************************) - -lemma cif_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃T⦄. -/3 width=1/ qed-. - -lemma cif_inv_tif: ∀T,W. ⋆.ⓛW ⊢ 𝐈⦃T⦄ → ⋆ ⊢ 𝐈⦃T⦄. -/3 width=1/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf.ma deleted file mode 100644 index 02bbcf87a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf.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_2/reducibility/cpr.ma". - -(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************) - -definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …). - -interpretation - "context-sensitive normality (term)" - 'Normal L T = (cnf L T). - -(* Basic inversion lemmas ***************************************************) - -lemma cnf_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄. -#L #V1 #T1 #HVT1 @and3_intro -[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct // -| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct // -| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H - [ elim (lift_total V1 0 1) #V2 #HV12 - lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3/ -HV12 #H destruct - | lapply (H (ⓓ{a}V1.U1) ?) -H /3 width=1/ #H destruct -] -qed-. - -lemma cnf_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥. -#L #V #T #H elim (is_lift_dec T 0 1) -[ * #U #HTU - lapply (H U ?) -H /3 width=3 by cpr_tpr, tpr_zeta/ #H destruct (**) (* auto too slow without trace *) - elim (lift_inv_pair_xy_y … HTU) -| #HT - elim (tps_full (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12 - lapply (H (+ⓓV.T2) ?) -H /3 width=3 by cpr_tpr, tpr_delta/ -HT2 #H destruct /3 width=2/ (**) (* auto too slow without trace *) -] -qed. - -lemma cnf_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥. -#L #V #T #H lapply (H T ?) -H /2 width=1/ #H -@discr_tpair_xy_y // -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: nf2_sort *) -lemma cnf_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄. -#L #k #X #H ->(cpr_inv_sort1 … H) // -qed. - -(* Basic_1: was: nf2_dec *) -axiom cnf_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨ - ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥). - -(* Basic_1: removed theorems 1: nf2_abst_shift *) diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf_cif.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf_cif.ma deleted file mode 100644 index 51ce95aa0..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf_cif.ma +++ /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: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2. -#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e -[ // -| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e - elim (cif_inv_delta … HLK ?) // -| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H - elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct - lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/ -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H - elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/ -] -qed. - -lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2. -#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 #HT1 -lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/ -qed. - -lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2. -#T1 #T2 #H elim H -T1 -T2 -[ // -| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H - [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_ - >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // - | elim (cif_inv_ri2 … H) /2 width=1/ - ] -| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H - elim (cif_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H - [ lapply (tps_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 - elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct - lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct - lapply (IHT1 … HT1) -IHT1 #H destruct - lapply (tps_cif_eq … HT2 ?) -HT2 // - | <(tps_inv_refl_SO2 … HT2 ?) -HT2 // - elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/ - ] -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H - elim (cif_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -| #V1 #T1 #T #T2 #_ #_ #_ #L #H - elim (cif_inv_ri2 … H) /2 width=1/ -| #V1 #T1 #T2 #_ #_ #L #H - elim (cif_inv_ri2 … H) /2 width=1/ -] -qed. - -lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2. -#L #T1 #T2 * #T0 #HT10 #HT02 #HT1 -lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/ -qed. - -theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄. -/3 width=3/ qed. - -(* Note: this property is unusual *) -lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥. -#L #T #H elim H -L -T -[ #L #K #V #i #HLK #H - elim (cnf_inv_delta … HLK H) -| #L #V #T #_ #IHV #H - elim (cnf_inv_appl … H) -H /2 width=1/ -| #L #V #T #_ #IHT #H - elim (cnf_inv_appl … H) -H /2 width=1/ -| #I #L #V #T * #H1 #H2 destruct - [ elim (cnf_inv_zeta … H2) - | elim (cnf_inv_tau … H2) - ] -|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct - [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/ - |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/ - ] -| #a #L #V #W #T #H - elim (cnf_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -| #a #L #V #W #T #H - elim (cnf_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -] -qed. - -theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf_lift.ma deleted file mode 100644 index 0e1a8551f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf_lift.ma +++ /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/reducibility/cpr_lift.ma". -include "basic_2/reducibility/cpr_cpr.ma". -include "basic_2/reducibility/cnf.ma". - -(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************) - -(* Advanced inversion lemmas ************************************************) - -lemma cnf_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐍⦃#i⦄ → ⊥. -#L #K #V #i #HLK #H -elim (lift_total V 0 (i+1)) #W #HVW -lapply (H W ?) -H [ /3 width=6/ ] -HLK #H destruct -elim (lift_inv_lref2_be … HVW ? ?) -HVW // -qed-. - -lemma cnf_inv_abst: ∀a,L,V,T. L ⊢ 𝐍⦃ⓛ{a}V.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄. -#a #L #V1 #T1 #HVT1 @conj -[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct // -| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct // -] -qed-. - -lemma cnf_inv_abbr: ∀L,V,T. L ⊢ 𝐍⦃-ⓓV.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄. -#L #V1 #T1 #HVT1 @conj -[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct // -| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct // -] -qed-. - -(* Advanced properties ******************************************************) - -(* Basic_1: was only: nf2_csort_lref *) -lemma cnf_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍⦃#i⦄. -#L #i #HLK #X #H -elim (cpr_inv_lref1 … H) // * -#K0 #V0 #V1 #HLK0 #_ #_ #_ -lapply (ldrop_mono … HLK … HLK0) -L #H destruct -qed. - -(* Basic_1: was: nf2_lref_abst *) -lemma cnf_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍⦃#i⦄. -#L #K #V #i #HLK #X #H -elim (cpr_inv_lref1 … H) // * -#K0 #V0 #V1 #HLK0 #_ #_ #_ -lapply (ldrop_mono … HLK … HLK0) -L #H destruct -qed. - -(* Basic_1: was: nf2_abst *) -lemma cnf_abst: ∀a,I,L,V,W,T. L ⊢ 𝐍⦃W⦄ → L. ⓑ{I} V ⊢ 𝐍⦃T⦄ → L ⊢ 𝐍⦃ⓛ{a}W.T⦄. -#a #I #L #V #W #T #HW #HT #X #H -elim (cpr_inv_abst1 … H I V) -H #W0 #T0 #HW0 #HT0 #H destruct ->(HW … HW0) -W0 >(HT … HT0) -T0 // -qed. - -(* Basic_1: was only: nf2_appl_lref *) -lemma cnf_appl_simple: ∀L,V,T. L ⊢ 𝐍⦃V⦄ → L ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐍⦃ⓐV.T⦄. -#L #V #T #HV #HT #HS #X #H -elim (cpr_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct ->(HV … HV0) -V0 >(HT … HT0) -T0 // -qed. - -(* Relocation properties ****************************************************) - -(* Basic_1: was: nf2_lift *) -lemma cnf_lift: ∀L0,L,T,T0,d,e. - L ⊢ 𝐍⦃T⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍⦃T0⦄. -#L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H -elim (cpr_inv_lift1 … HL0 … HT0 … H) -L0 #T1 #HT10 #HT1 -<(HLT … HT1) in HT0; -L #HT0 ->(lift_mono … HT10 … HT0) -T1 -X // -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr.ma deleted file mode 100644 index 22fbfc148..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr.ma +++ /dev/null @@ -1,103 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/unfold/tpss.ma". -include "basic_2/reducibility/tpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) - -(* Basic_1: includes: pr2_delta1 *) -definition cpr: lenv → relation term ≝ - λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2. - -interpretation - "context-sensitive parallel reduction (term)" - 'PRed L T1 T2 = (cpr L T1 T2). - -(* Basic properties *********************************************************) - -lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2. -/4 width=3/ qed-. - -(* Basic_1: was by definition: pr2_free *) -lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2. -/2 width=3/ qed. - -lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2. -/3 width=5/ qed. - -lemma cpr_refl: ∀L,T. L ⊢ T ➡ T. -/2 width=1/ qed. - -(* Note: new property *) -(* Basic_1: was only: pr2_thin_dx *) -lemma cpr_flat: ∀I,L,V1,V2,T1,T2. - L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2. -#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/ -qed. - -lemma cpr_cast: ∀L,V,T1,T2. - L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2. -#L #V #T1 #T2 * /3 width=3/ -qed. - -(* Note: it does not hold replacing |L1| with |L2| *) -(* Basic_1: was only: pr2_change *) -lemma cpr_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → - ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2. -#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12 -lapply (tpss_lsubs_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/ -qed. - -(* Basic inversion lemmas ***************************************************) - -(* Basic_1: was: pr2_gen_csort *) -lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2. -#T1 #T2 * #T #HT normalize #HT2 -<(tpss_inv_refl_O2 … HT2) -HT2 // -qed-. - -(* Basic_1: was: pr2_gen_sort *) -lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k. -#L #T2 #k * #X #H ->(tpr_inv_atom1 … H) -H #H ->(tpss_inv_sort1 … H) -H // -qed-. - -(* Basic_1: was: pr2_gen_cast *) -lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → ( - ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 & - U2 = ⓝV2. T2 - ) ∨ L ⊢ T1 ➡ U2. -#L #V1 #T1 #U2 * #X #H #HU2 -elim (tpr_inv_cast1 … H) -H /3 width=3/ -* #V #T #HV1 #HT1 #H destruct -elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T → - ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. -#L #L1 #T1 #T * #X #H1 #H2 -elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct -elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/ -qed-. - -(* Basic_1: removed theorems 6: - pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans - pr2_gen_ctail pr2_ctail - Basic_1: removed local theorems 3: - pr2_free_free pr2_free_delta pr2_delta_delta -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_aaa.ma deleted file mode 100644 index 7c1273334..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_aaa.ma +++ /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: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A. -#L #T1 #A #HT1 #T2 * /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_cpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_cpr.ma deleted file mode 100644 index 8d2eb1f1d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_cpr.ma +++ /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: ∀a,I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 → - L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2. -#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12 -@ex2_1_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *) -qed. - -(* Basic_1: was only: pr2_gen_cbind *) -lemma cpr_bind_dx: ∀a,I,L,V1,V2,T1,T2. V1 ➡ V2 → L. ⓑ{I} V2 ⊢ T1 ➡ T2 → - L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2. -#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2 -elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 (tpr_inv_atom1 … H) -H #H -elim (tpss_inv_lref1 … H) -H /2 width=1/ -* /3 width=6/ -qed-. - -(* Basic_1: was pr2_gen_abbr *) -lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡ U2 → - (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 & - L. ⓓV ⊢ T1 ➡ T2 & - U2 = ⓓ{a}V2. T2 - ) ∨ - ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2 & a = true. -#a #L #V1 #T1 #Y * #X #H1 #H2 -elim (tpr_inv_abbr1 … H1) -H1 * -[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct - elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct - lapply (tps_lsubs_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0 - lapply (tps_weak_all … HT0) -HT0 #HT0 - lapply (tpss_lsubs_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02 - lapply (tpss_weak_all … HT02) -HT02 #HT02 - lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/ -| #T2 #HT12 #HXT2 #H destruct - elim (lift_total Y 0 1) #Z #HYZ - lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H - lapply (cpr_intro … HT12 … H) -T2 /3 width=3/ -] -qed-. - -(* Basic_1: was: pr2_gen_abst *) -lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡ U2 → ∀I,W. - ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2. T2. -#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W -elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct -elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct -lapply (tpss_lsubs_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/ -qed-. - -(* Basic_1: was pr2_gen_appl *) -lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 → - ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 & - U2 = ⓐV2. T2 - | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 & - U0 = ⓛ{a}W. T1 & - U2 = ⓓ{a}V2. T2 - | ∃∃a,V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 & - ⇧[0,1] V2 ≡ V & - U0 = ⓓ{a}W1. T1 & - U2 = ⓓ{a}W2. ⓐV. T2. -#L #V1 #U0 #Y * #X #H1 #H2 -elim (tpr_inv_appl1 … H1) -H1 * -[ #V #U #HV1 #HU0 #H destruct - elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/ -| #a #V #W #T0 #T #HV1 #HT0 #H #H1 destruct - elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct - lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=9/ -| #a #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct - elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct - elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct - elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V (ltpr_fwd_length … HL12) in HT2; #HT2 -elim (tpr_tpss_ltpr … HL12 … HT2) -L1 /3 width=3/ -qed. - -lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → - ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 → - ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2. -#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 -elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2 -elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2 -lapply (tpss_weak_all … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *) -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_ltpss.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_ltpss.ma deleted file mode 100644 index b728d9dc2..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_ltpss.ma +++ /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/unfold/ltpss_sn_ltpss_sn.ma". -include "basic_2/reducibility/cpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) - -(* Properties concerning partial unfold on local environments ***************) - -lemma ltpss_sn_cpr_trans: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → - ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2. -#L1 #L2 #d #e #HL12 #T1 #T2 * -lapply (ltpss_sn_weak_all … HL12) -<(ltpss_sn_fwd_length … HL12) -HL12 /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_tpss.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_tpss.ma deleted file mode 100644 index e5afe8737..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_tpss.ma +++ /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/unfold/tpss_tpss.ma". -include "basic_2/reducibility/cpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) - -(* Properties on partial unfold for terms ***********************************) - -lemma cpr_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡ T → - ∀T2. L ⊢ T ▶* [O, |L|] T2 → L ⊢ T1 ➡ T2. -#L #T1 #T * #T0 #HT10 #HT0 #T2 #HT2 -lapply (tpss_trans_eq … HT0 HT2) -T /2 width=3/ -qed. - -lemma cpr_tps_trans: ∀L,T1,T. L ⊢ T1 ➡ T → - ∀T2. L ⊢ T ▶ [O, |L|] T2 → L ⊢ T1 ➡ T2. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/crf.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/crf.ma deleted file mode 100644 index 5015f033c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/crf.ma +++ /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/substitution/ldrop.ma". - -(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************) - -(* reducible binary items *) -definition ri2: item2 → Prop ≝ - λI. I = Bind2 true Abbr ∨ I = Flat2 Cast. - -(* irreducible binary binders *) -definition ib2: bool → bind2 → Prop ≝ - λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr. - -(* reducible terms *) -inductive crf: lenv → predicate term ≝ -| crf_delta : ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → crf L (#i) -| crf_appl_sn: ∀L,V,T. crf L V → crf L (ⓐV. T) -| crf_appl_dx: ∀L,V,T. crf L T → crf L (ⓐV. T) -| crf_ri2 : ∀I,L,V,T. ri2 I → crf L (②{I}V. T) -| crf_ib2_sn : ∀a,I,L,V,T. ib2 a I → crf L V → crf L (ⓑ{a,I}V. T) -| crf_ib2_dx : ∀a,I,L,V,T. ib2 a I → crf (L.ⓑ{I}V) T → crf L (ⓑ{a,I}V. T) -| crf_beta : ∀a,L,V,W,T. crf L (ⓐV. ⓛ{a}W. T) -| crf_theta : ∀a,L,V,W,T. crf L (ⓐV. ⓓ{a}W. T) -. - -interpretation - "context-sensitive reducibility (term)" - 'Reducible L T = (crf L T). - -(* Basic inversion lemmas ***************************************************) - -fact trf_inv_atom_aux: ∀I,L,T. L ⊢ 𝐑⦃T⦄ → L = ⋆ → T = ⓪{I} → ⊥. -#I #L #T * -L -T -[ #L #K #V #i #HLK #H1 #H2 destruct - lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct -| #L #V #T #_ #_ #H destruct -| #L #V #T #_ #_ #H destruct -| #J #L #V #T #_ #_ #H destruct -| #a #J #L #V #T #_ #_ #_ #H destruct -| #a #J #L #V #T #_ #_ #_ #H destruct -| #a #L #V #W #T #_ #H destruct -| #a #L #V #W #T #_ #H destruct -] -qed. - -lemma trf_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥. -/2 width=6/ qed-. - -fact trf_inv_lref_aux: ∀L,T. L ⊢ 𝐑⦃T⦄ → ∀i. T = #i → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV. -#L #T * -L -T -[ #L #K #V #j #HLK #i #H destruct /2 width=3/ -| #L #V #T #_ #i #H destruct -| #L #V #T #_ #i #H destruct -| #J #L #V #T #_ #i #H destruct -| #a #J #L #V #T #_ #_ #i #H destruct -| #a #J #L #V #T #_ #_ #i #H destruct -| #a #L #V #W #T #i #H destruct -| #a #L #V #W #T #i #H destruct -] -qed. - -lemma trf_inv_lref: ∀L,i. L ⊢ 𝐑⦃#i⦄ → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV. -/2 width=3/ qed-. - -fact crf_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → L ⊢ 𝐑⦃T⦄ → T = ⓑ{a,I}W. U → - L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃U⦄. -#a #I #L #W #U #T #HI * -T -[ #L #K #V #i #_ #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #J #L #V #T #H1 #H2 destruct - elim H1 -H1 #H destruct - elim HI -HI #H destruct -| #b #J #L #V #T #_ #HV #H destruct /2 width=1/ -| #b #J #L #V #T #_ #HT #H destruct /2 width=1/ -| #b #L #V #W #T #H destruct -| #b #L #V #W #T #H destruct -] -qed. - -lemma crf_inv_ib2: ∀a,I,L,W,T. ib2 a I → L ⊢ 𝐑⦃ⓑ{a,I}W.T⦄ → - L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃T⦄. -/2 width=5/ qed-. - -fact crf_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW. U → - ∨∨ L ⊢ 𝐑⦃W⦄ | L ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥). -#L #W #U #T * -T -[ #L #K #V #i #_ #H destruct -| #L #V #T #HV #H destruct /2 width=1/ -| #L #V #T #HT #H destruct /2 width=1/ -| #J #L #V #T #H1 #H2 destruct - elim H1 -H1 #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W0 #T #H destruct - @or3_intro2 #H elim (simple_inv_bind … H) -| #a #L #V #W0 #T #H destruct - @or3_intro2 #H elim (simple_inv_bind … H) -] -qed. - -lemma crf_inv_appl: ∀L,V,T. L ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ L ⊢ 𝐑⦃V⦄ | L ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥). -/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/crf_append.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/crf_append.ma deleted file mode 100644 index f50b97e95..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/crf_append.ma +++ /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/substitution/ldrop_append.ma". -include "basic_2/reducibility/crf.ma". - -(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************) - -(* Advanved properties ******************************************************) - -lemma crf_labst_last: ∀L,T,W. L ⊢ 𝐑⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄. -#L #T #W #H elim H -L -T /2 width=1/ -#L #K #V #i #HLK -lapply (ldrop_fwd_ldrop2_length … HLK) #Hi -lapply (ldrop_O1_append_sn_le … HLK … (⋆.ⓛW)) -HLK /2 width=2/ -Hi /2 width=3/ -qed. - -lemma crf_trf: ∀T,W. ⋆ ⊢ 𝐑⦃T⦄ → ⋆.ⓛW ⊢ 𝐑⦃T⦄. -#T #W #H lapply (crf_labst_last … W H) // -qed. - -(* Advanced inversion lemmas ************************************************) - -fact crf_inv_labst_last_aux: ∀L1,T,W. L1 ⊢ 𝐑⦃T⦄ → - ∀L2. L1 = ⋆.ⓛW @@ L2 → L2 ⊢ 𝐑⦃T⦄. -#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_ldrop2_length … HLK1) - >append_length >commutative_plus normalize in ⊢ (??% → ?); #H - elim (le_to_or_lt_eq i (|L2|) ?) /2 width=1/ -H #Hi destruct - [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2 - lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi - normalize #H destruct /2 width=3/ - | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // HL01 #HL12 #HT02 -elim (tpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2 -elim (tpr_fwd_shift1 … H1) #L #T #HL1 #H destruct /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr.ma deleted file mode 100644 index 5e1ba7992..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr.ma +++ /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/unfold/ltpss_sn.ma". -include "basic_2/reducibility/ltpr.ma". - -(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************) - -definition lfpr: relation lenv ≝ - λL1,L2. ∃∃L. L1 ➡ L & L ⊢ ▶* [0, |L|] L2 -. - -interpretation - "focalized parallel reduction (environment)" - 'FocalizedPRed L1 L2 = (lfpr L1 L2). - -(* Basic properties *********************************************************) - -(* Note: lemma 250 *) -lemma lfpr_refl: ∀L. ⦃L⦄ ➡ ⦃L⦄. -/2 width=3/ qed. - -lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄. -/3 width=5/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆. -#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_sn_inv_atom1 … HL2) -HL2 // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_aaa.ma deleted file mode 100644 index 6f6c49df3..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_aaa.ma +++ /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: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → L2 ⊢ T ⁝ A. -#L1 #T #A #HT #L2 * /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_alt.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_alt.ma deleted file mode 100644 index adff7ad07..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_alt.ma +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_bi.ma". -include "basic_2/reducibility/fpr_cpr.ma". -include "basic_2/reducibility/lfpr_fpr.ma". - -(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************) - -(* alternative definition *) -definition lfpra: relation lenv ≝ lpx_bi fpr. - -interpretation - "focalized parallel reduction (environment) alternative" - 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2). - -(* Basic properties *********************************************************) - -lemma lfpra_refl: reflexive … lfpra. -/2 width=1/ qed. - -lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄. -#L1 elim L1 -L1 -[ #L2 #T1 #T2 #H - elim (fpr_inv_atom1 … H) -H #_ #H destruct // -| #L1 #I #V1 #IH #L2 #T1 #T2 #H - elim (fpr_inv_pair1 … H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/ -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆. -/2 width=2 by lpx_bi_inv_atom1/ qed-. - -lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ → - ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & - L2 = K2. ⓑ{I} V2. -/2 width=1 by lpx_bi_inv_pair1/ qed-. - -lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆. -/2 width=2 by lpx_bi_inv_atom2/ qed-. - -lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ → - ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & - L1 = K1. ⓑ{I} V1. -/2 width=1 by lpx_bi_inv_pair2/ qed-. - -lemma lfpra_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ∀T.⦃L1, T⦄ ➡ ⦃L2, T⦄. -#L1 #L2 * -L1 -L2 // /3 width=1/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|. -/2 width=2 by lpx_bi_fwd_length/ qed-. - -(* Main properties **********************************************************) - -theorem lfpr_lfpra: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄. -#L1 #L2 #H -lapply (lfpr_inv_fpr … H (⋆0)) -H /2 width=3/ -qed. - -theorem lfpra_lfpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ⦃L1⦄ ➡ ⦃L2⦄. -#L1 #L2 #H -lapply (lfpra_inv_fpr … H (⋆0)) -H /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_cpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_cpr.ma deleted file mode 100644 index 2a40f58bd..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_cpr.ma +++ /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/ltpss_sn_ltpss_sn.ma". -include "basic_2/reducibility/cpr.ma". -include "basic_2/reducibility/lfpr.ma". - -(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************) - -(* Advanced properties ****************************************************) - -lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 → - ∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄. -#L1 #L2 * #L #HL1 #HL2 #V1 #V2 * -<(ltpss_sn_fwd_length … HL2) #V #HV1 #HV2 #I -lapply (ltpss_sn_tpss_trans_eq … HV2 … HL2) -HV2 #V2 -@(ex2_1_intro … (L.ⓑ{I}V)) /2 width=1/ (**) (* explicit constructor *) -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_fpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_fpr.ma deleted file mode 100644 index 9a226be58..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_fpr.ma +++ /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 ***********************) - -(* Inversion lemmas on context-free parallel reduction for closures *********) - -lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄. -#L1 #L2 #T1 #T2 #H -elim (fpr_inv_all … H) -H /2 width=3/ -qed. - -(* Inversion lemmas on context-free parallel reduction for closures *********) - -lemma lfpr_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡ ⦃L2, T⦄. -#L1 #L2 * /2 width=4/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_lfpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_lfpr.ma deleted file mode 100644 index 7031e792b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lfpr_lfpr.ma +++ /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: ∀L0,L1,L2. ⦃L0⦄ ➡ ⦃L1⦄ → ⦃L0⦄ ➡ ⦃L2⦄ → - ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄. -#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2 -lapply (ltpr_fwd_length … HK01) #H ->(ltpr_fwd_length … HK02) in H; #H -elim (ltpr_conf … HK01 … HK02) -K0 #K #HK1 #HK2 -lapply (ltpss_sn_fwd_length … HKL1) #H1 -lapply (ltpss_sn_fwd_length … HKL2) #H2 ->H1 in HKL1 H; -H1 #HKL1 ->H2 in HKL2; -H2 #HKL2 #H -elim (ltpr_ltpss_sn_conf … HKL1 … HK1) -K1 #K1 #HK1 #HLK1 -elim (ltpr_ltpss_sn_conf … HKL2 … HK2) -K2 #K2 #HK2 #HLK2 -elim (ltpss_sn_conf … HK1 … HK2) -K #K #HK1 #HK2 -lapply (ltpr_fwd_length … HLK1) #H1 -lapply (ltpr_fwd_length … HLK2) #H2 -/3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr.ma deleted file mode 100644 index a910ea7df..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr.ma +++ /dev/null @@ -1,67 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/grammar/lenv_px.ma". -include "basic_2/reducibility/tpr.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) - -definition ltpr: relation lenv ≝ lpx tpr. - -interpretation - "context-free parallel reduction (environment)" - 'PRed L1 L2 = (ltpr L1 L2). - -(* Basic properties *********************************************************) - -lemma ltpr_refl: reflexive … ltpr. -/2 width=1/ qed. - -lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2. -/2 width=1/ qed. - -(* Basic inversion lemmas ***************************************************) - -(* Basic_1: was: wcpr0_gen_sort *) -lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆. -/2 width=2 by lpx_inv_atom1/ qed-. - -(* Basic_1: was: wcpr0_gen_head *) -lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 → - ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2. -/2 width=1 by lpx_inv_pair1/ qed-. - -lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆. -/2 width=2 by lpx_inv_atom2/ qed-. - -lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1. -/2 width=1 by lpx_inv_pair2/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|. -/2 width=2 by lpx_fwd_length/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L → - ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2. -/2 width=1 by lpx_inv_append1/ qed-. - -lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 → - ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1. -/2 width=1 by lpx_inv_append2/ qed-. - -(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *) diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_aaa.ma deleted file mode 100644 index b45dbb99e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_aaa.ma +++ /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/static/aaa_ltpss_dx.ma". -include "basic_2/static/lsuba_aaa.ma". -include "basic_2/reducibility/ltpr_ldrop.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) - -(* Properties about atomic arity assignment on terms ************************) - -fact aaa_ltpr_tpr_conf_aux: ∀L,T,L1,T1,A. L1 ⊢ T1 ⁝ A → L = L1 → T = T1 → - ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A. -#L #T @(fw_ind … L T) -L -T #L #T #IH -#L1 #T1 #A * -L1 -T1 -A -[ -IH #L1 #k #H1 #H2 #L2 #_ #T2 #H destruct - >(tpr_inv_atom1 … H) -H // -| #I #L1 #K1 #V1 #B #i #HLK1 #HK1 #H1 #H2 #L2 #HL12 #T2 #H destruct - >(tpr_inv_atom1 … H) -T2 - lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1 - elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2 - elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct - lapply (IH … HKV1 … HK1 … HK12 … HV12) // -L1 -K1 -V1 /2 width=5/ -| #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_abbr1 … H) -H * - [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct - lapply (tps_lsubs_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2 - lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB - lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/ - | -B #T #HT1 #HXT #H destruct - lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA - @(aaa_inv_lift … HA … HXT) /2 width=1/ - ] -| #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB - lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/ -| #L1 #V1 #T1 #B #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_appl1 … H) -H * - [ #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ /2 width=3/ - | #a #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - elim (aaa_inv_abst … HT1) -HT1 #B0 #A0 #HB0 #HA0 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 - lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -T0 -L1 -V1 /4 width=7/ - | #a #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct - elim (aaa_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 - lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 - lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0 - lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -V1 -T0 -L1 -W0 #HT2 - @(aaa_abbr … HW2) -HW2 - @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) - ] -| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_cast1 … H) -H - [ * #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2 - lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ - | -HV1 #HT1X - lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/ - ] -] -qed. - -lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 → - ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A. -/2 width=9/ qed. - -lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A. -/2 width=5/ qed. - -lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A. -/2 width=5/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ldrop.ma deleted file mode 100644 index 945279795..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ldrop.ma +++ /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/substitution/ldrop_lpx.ma". -include "basic_2/reducibility/tpr_lift.ma". -include "basic_2/reducibility/ltpr.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) - -(* 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. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpr.ma deleted file mode 100644 index 4a27a6e70..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpr.ma +++ /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: ∀L0:lenv. ∀L1. L0 ➡ L1 → ∀L2. L0 ➡ L2 → - ∃∃L. L1 ➡ L & L2 ➡ L. -#L0 #L1 #H elim H -L0 -L1 /2 width=3/ -#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H -elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct -elim (IHK01 … HK02) -K0 #K #HK1 #HK2 -elim (tpr_conf … HV01 HV02) -V0 /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpss_dx.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpss_dx.ma deleted file mode 100644 index cee1cb49e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpss_dx.ma +++ /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/tpr_tpss.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) - -(* Properties concerning dx parallel unfold on local environments ***********) - -lemma ltpr_ltpss_dx_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 → - ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2. -#L1 #K1 #d #e #H elim H -L1 -K1 -d -e -[ /2 width=3/ -| #L1 #I #V1 #X #H - elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ -| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H - elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12 - elim (tpr_tpss_ltpr … HK12 … HV12 … HVW1) -V1 /3 width=5/ -| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H - elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12 - elim (tpr_tpss_ltpr … HK12 … HV12 … HVW1) -V1 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpss_sn.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpss_sn.ma deleted file mode 100644 index 823762356..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_ltpss_sn.ma +++ /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: ∀L1,K1,d,e. L1 ⊢ ▶* [d, e] K1 → ∀L2. L1 ➡ L2 → - ∃∃K2. L2 ⊢ ▶* [d, e] K2 & K1 ➡ K2. -#L1 #K1 #d #e #H -lapply (ltpss_sn_ltpssa … H) -H #H -@(ltpssa_ind … H) -K1 /2 width=3/ -#K #K1 #_ #HK1 #IHK #L2 #HL12 -elim (IHK … HL12) -L1 #K2 #HLK2 #HK2 -elim (ltpr_ltpss_dx_conf … HK1 … HK2) -K /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_tps.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_tps.ma deleted file mode 100644 index 75792eef0..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_tps.ma +++ /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/reducibility/ltpr_ldrop.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) - -(* Properties concerning parallel substitution on terms *********************) - -lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 → - ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2. -#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e -[ /2 width=3/ -| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12 - elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H - elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2 - elim (lift_total V1 0 (i+1)) #W1 #HVW1 - lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/ -| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 - elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2 - elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/ -| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 - elim (IHV12 … HL12) -IHV12 - elim (IHT12 … HL12) -L2 /3 width=5/ -] -qed. - -lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 → - ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T. -#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e -[ /2 width=3/ -| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12 - elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2 - elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1 - elim (lift_total V2 0 (i+1)) #W2 #HVW2 - lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/ -| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 - elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2 - elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/ -| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 - elim (IHV12 … HL12) -IHV12 - elim (IHT12 … HL12) -L1 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/thnf.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/thnf.ma deleted file mode 100644 index ab864268f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/thnf.ma +++ /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/tshf.ma". -include "basic_2/reducibility/tpr.ma". - -(* CONTEXT-FREE WEAK HEAD NORMAL TERMS **************************************) - -definition thnf: predicate term ≝ NF … tpr tshf. - -interpretation - "context-free head normality (term)" - 'HdNormal T = (thnf T). - -(* Basic inversion lemmas ***************************************************) - -lemma thnf_inv_tshf: ∀T. 𝐇𝐍⦃T⦄ → T ≈ T. -normalize /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -lemma tpr_tshf: ∀T1,T2. T1 ➡ T2 → T1 ≈ T1 → T1 ≈ T2. -#T1 #T2 #H elim H -T1 -T2 // -[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H - elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct - lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2 - lapply (simple_tshf_repl_dx … HUT2 HT1) /2 width=1/ -| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H - elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H - elim (simple_inv_bind … H) -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H - elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct // -| #a #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H - elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H - elim (simple_inv_bind … H) -| #V #T #T1 #T2 #_ #_ #_ #H - elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct -| #V #T1 #T2 #_ #_ #H - elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct -] -qed. - -lemma thnf_tshf: ∀T. T ≈ T → 𝐇𝐍⦃T⦄. -/3 width=1/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr.ma deleted file mode 100644 index 957b5f3a8..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr.ma +++ /dev/null @@ -1,229 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) - -(* Basic_1: includes: pr0_delta1 *) -inductive tpr: relation term ≝ -| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I}) -| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 → - tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) -| tpr_beta : ∀a,V1,V2,W,T1,T2. - tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2) -| tpr_delta: ∀a,I,V1,V2,T1,T,T2. - tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 → - tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) -| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2. - tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 → - tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2) -| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2 -| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2 -. - -interpretation - "context-free parallel reduction (term)" - 'PRed T1 T2 = (tpr T1 T2). - -(* Basic properties *********************************************************) - -lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2. -/2 width=3/ qed. - -(* Basic_1: was by definition: pr0_refl *) -lemma tpr_refl: reflexive … tpr. -#T elim T -T // -#I elim I -I /2 width=1/ -qed. - -(* Basic inversion lemmas ***************************************************) - -fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}. -#U1 #U2 * -U1 -U2 -[ // -| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct -| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct -| #V #T1 #T #T2 #_ #_ #k #H destruct -| #V #T1 #T2 #_ #k #H destruct -] -qed. - -(* Basic_1: was: pr0_gen_sort pr0_gen_lref *) -lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}. -/2 width=3/ qed-. - -fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → - (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T & - ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 & - U2 = ⓑ{a,I} V2. T2 - ) ∨ - ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr. -#U1 #U2 * -U1 -U2 -[ #J #a #I #V #T #H destruct -| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct -| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct -| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/ -| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct -| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/ -| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct -] -qed. - -lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 → - (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T & - ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 & - U2 = ⓑ{a,I} V2. T2 - ) ∨ - ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr. -/2 width=3/ qed-. - -(* Basic_1: was pr0_gen_abbr *) -lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 → - (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T & - ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 & - U2 = ⓓ{a}V2. T2 - ) ∨ - ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true. -#a #V1 #T1 #U2 #H -elim (tpr_inv_bind1 … H) -H * /3 width=7/ -qed-. - -fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 → - ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 & - U2 = ⓕ{I} V2. T2 - | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 & - U0 = ⓛ{a}W. T1 & - U2 = ⓓ{a}V2. T2 & I = Appl - | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 & - ⇧[0,1] V2 ≡ V & - U0 = ⓓ{a}W1. T1 & - U2 = ⓓ{a}W2. ⓐV. T2 & - I = Appl - | (U0 ➡ U2 ∧ I = Cast). -#U1 #U2 * -U1 -U2 -[ #I #J #V #T #H destruct -| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/ -| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/ -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/ -| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct -| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/ -] -qed. - -lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 → - ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 & - U2 = ⓕ{I} V2. T2 - | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 & - U0 = ⓛ{a}W. T1 & - U2 = ⓓ{a}V2. T2 & I = Appl - | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 & - ⇧[0,1] V2 ≡ V & - U0 = ⓓ{a}W1. T1 & - U2 = ⓓ{a}W2. ⓐV. T2 & - I = Appl - | (U0 ➡ U2 ∧ I = Cast). -/2 width=3/ qed-. - -(* Basic_1: was pr0_gen_appl *) -lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 → - ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 & - U2 = ⓐV2. T2 - | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 & - U0 = ⓛ{a}W. T1 & - U2 = ⓓ{a}V2. T2 - | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 & - ⇧[0,1] V2 ≡ V & - U0 = ⓓ{a}W1. T1 & - U2 = ⓓ{a}W2. ⓐV. T2. -#V1 #U0 #U2 #H -elim (tpr_inv_flat1 … H) -H * -/3 width=5/ /3 width=9/ /3 width=13/ -#_ #H destruct -qed-. - -(* Note: the main property of simple terms *) -lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ → - ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & - U = ⓐV2. T2. -#V1 #T1 #U #H #HT1 -elim (tpr_inv_appl1 … H) -H * -[ /2 width=5/ -| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct - elim (simple_inv_bind … HT1) -| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct - elim (simple_inv_bind … HT1) -] -qed-. - -(* Basic_1: was: pr0_gen_cast *) -lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 → - (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2) - ∨ T1 ➡ U2. -#V1 #T1 #U2 #H -elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2 -[ #_ #_ #_ #_ #H destruct -| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct -] -qed-. - -fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i → - ∨∨ T1 = #i - | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T - | ∃∃V,T. T ➡ #i & T1 = ⓝV. T. -#T1 #T2 * -T1 -T2 -[ #I #i #H destruct /2 width=1/ -| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct -| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct -| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct - lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/ -| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/ -] -qed. - -lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i → - ∨∨ T1 = #i - | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T - | ∃∃V,T. T ➡ #i & T1 = ⓝV. T. -/2 width=3/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T → - ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. -#L1 @(lenv_ind_dx … L1) -L1 normalize -[ #T1 #T #HT1 - @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *) -| #I #L1 #V1 #IH #T1 #X - >shift_append_assoc normalize #H - elim (tpr_inv_bind1 … H) -H * - [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct - elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct - elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct - >append_length >HL1 >HL2 -L1 -L - @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) - | #T #_ #_ #H destruct - ] -] -qed-. - -(* Basic_1: removed theorems 3: - pr0_subst0_back pr0_subst0_fwd pr0_subst0 - Basic_1: removed local theorems: 1: pr0_delta_tau -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_delift.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_delift.ma deleted file mode 100644 index 99e621d15..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_delift.ma +++ /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/unfold/delift.ma". -include "basic_2/reducibility/tpr_tpss.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) - -(* Properties on inverse basic term relocation ******************************) - -lemma tpr_delift_conf: ∀U1,U2. U1 ➡ U2 → ∀L,T1,d,e. L ⊢ ▼*[d, e] U1 ≡ T1 → - ∃∃T2. T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2. -#U1 #U2 #HU12 #L #T1 #d #e * #W1 #HUW1 #HTW1 -elim (tpr_tpss_conf … HU12 … HUW1) -U1 #U1 #HWU1 #HU21 -elim (tpr_inv_lift1 … HWU1 … HTW1) -W1 /3 width=5/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_lift.ma deleted file mode 100644 index b4d76066a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_lift.ma +++ /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/substitution/tps_lift.ma". -include "basic_2/reducibility/tpr.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) - -(* Relocation properties ****************************************************) - -(* Basic_1: was: pr0_lift *) -lemma tpr_lift: t_liftable tpr. -#T1 #T2 #H elim H -T1 -T2 -[ * #i #U1 #d #e #HU1 #U2 #HU2 - lapply (lift_mono … HU1 … HU2) -HU1 #H destruct - [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct // - | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct // - | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct // - ] -| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 - elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct - elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/ -| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 - elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct - elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct - elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/ -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2 - elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct - elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct - elim (lift_total T (d + 1) e) #U #HTU - @tpr_delta - [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *) -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 - elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct - elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct - elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct - elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct - elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/ -| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2 - elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V - elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/ -| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2 - elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/ -] -qed. - -(* Basic_1: was: pr0_gen_lift *) -lemma tpr_inv_lift1: t_deliftable_sn tpr. -#T1 #T2 #H elim H -T1 -T2 -[ * #i #X #d #e #HX - [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/ - | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/ - ] -| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX - elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct - elim (IHV12 … HV01) -V1 - elim (IHT12 … HT01) -T1 /3 width=5/ -| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX - elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct - elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct - elim (IHV12 … HV01) -V1 - elim (IHT12 … HT01) -T1 /3 width=5/ -| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX - elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct - elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12 - elim (IHT1 … HUT1) -T1 #U #HUT #HU1 - elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2 - @ex2_1_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *) -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX - elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct - elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct - elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03 - elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03 - elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03 - elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2 - @ex2_1_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *) -| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX - elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct - elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31 - elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/ -| #V #T1 #T2 #_ #IHT12 #X #d #e #HX - elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct - elim (IHT12 … HT01) -T1 /3 width=3/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -fact tpr_inv_abst1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,V1,T1. U1 = ⓛ{a}V1. T1 → - ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2. -#U1 #U2 * -U1 -U2 -[ #I #a #V #T #H destruct -| #I #V1 #V2 #T1 #T2 #_ #_ #a #V #T #H destruct -| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #V #T #H destruct -| #b #I #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #V0 #T0 #H destruct - <(tps_inv_refl_SO2 … HT2 ? ? ?) -T2 /2 width=5/ -| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #V0 #T0 #H destruct -| #V #T1 #T #T2 #_ #_ #a #V0 #T0 #H destruct -| #V #T1 #T2 #_ #a #V0 #T0 #H destruct -] -qed. - -(* Basic_1: was pr0_gen_abst *) -lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 → - ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2. -/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tpr.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tpr.ma deleted file mode 100644 index 1522d00c0..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tpr.ma +++ /dev/null @@ -1,283 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_tpss.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) - -(* Confluence lemmas ********************************************************) - -fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X. -/2 width=3/ qed. - -fact tpr_conf_flat_flat: - ∀I,V0,V1,T0,T1,V2,T2. ( - ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 → - ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0. -#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 -elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2 -elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/ -qed. - -fact tpr_conf_flat_beta: - ∀a,V0,V1,T1,V2,W0,U0,T2. ( - ∀X0:term. #{X0} < #{V0} + (#{W0} + #{U0} + 1) + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → V0 ➡ V2 → - U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 → - ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X. -#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H -elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct -elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/ -qed. - -(* Basic-1: was: - pr0_cong_upsilon_refl pr0_cong_upsilon_zeta - pr0_cong_upsilon_cong pr0_cong_upsilon_delta -*) -fact tpr_conf_flat_theta: - ∀a,V0,V1,T1,V2,V,W0,W2,U0,U2. ( - ∀X0:term. #{X0} < #{V0} + (#{W0} + #{U0} + 1) + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V → - W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 → - ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X. -#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H -elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2 -elim (lift_total VV 0 1) #VVV #HVV -lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV -elim (tpr_inv_abbr1 … H) -H * -(* case 1: delta *) -[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct - elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2 - elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2 - elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1 - @ex2_1_intro - [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ] - |1:skip - |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/ - ] (**) (* /5 width=14/ is too slow *) -(* case 3: zeta *) -| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1 - elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2 - elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct - @(ex2_1_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *) -] -qed. - -fact tpr_conf_flat_cast: - ∀X2,V0,V1,T0,T1. ( - ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 → - ∃∃X. ⓝV1. T1 ➡ X & X2 ➡ X. -#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02 -elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/ -qed. - -fact tpr_conf_beta_beta: - ∀a. ∀W0:term. ∀V0,V1,T0,T1,V2,T2. ( - ∀X0:term. #{X0} < #{V0} + (#{W0} + #{T0} + 1) + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 → - ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X. -#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 -elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ -elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/ -qed. - -(* Basic_1: was: pr0_cong_delta pr0_delta_delta *) -fact tpr_conf_delta_delta: - ∀a,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. ( - ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 → - ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 → - ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 → - ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X. -#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2 -elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2 -elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2 -elim (tpr_tps_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1 -elim (tpr_tps_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2 -elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2 -@ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *) -qed. - -fact tpr_conf_delta_zeta: - ∀X2,V0,V1,T0,T1,TT1,T2. ( - ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 → - T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 → - ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X. -#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2 -elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2 -elim (tpr_tps_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT -elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2 -lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /3 width=3/ -qed. - -(* Basic_1: was: pr0_upsilon_upsilon *) -fact tpr_conf_theta_theta: - ∀a,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. ( - ∀X0:term. #{X0} < #{V0} + (#{W0} + #{T0} + 1) + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 → - ⇧[O, 1] V1 ≡ VV1 → ⇧[O, 1] V2 ≡ VV2 → - ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X. -#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2 -elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2 -elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2 -elim (lift_total V 0 1) #VV #HVV -lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1 -lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2 -@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *) -qed. - -fact tpr_conf_zeta_zeta: - ∀V0:term. ∀X2,TT0,T0,T1,TT2. ( - ∀X0:term. #{X0} < #{V0} + #{TT0} + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 → - TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 → - ∃∃X. T1 ➡ X & X2 ➡ X. -#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2 -elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2 -elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10 -elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2 -lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/ -qed. - -fact tpr_conf_tau_tau: - ∀V0,T0:term. ∀X2,T1. ( - ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - T0 ➡ T1 → T0 ➡ X2 → - ∃∃X. T1 ➡ X & X2 ➡ X. -#V0 #T0 #X2 #T1 #IH #HT01 #HT02 -elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/ -qed. - -(* Confluence ***************************************************************) - -fact tpr_conf_aux: - ∀Y0:term. ( - ∀X0:term. #{X0} < #{Y0} → - ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → - ∃∃X. X1 ➡ X & X2 ➡ X - ) → - ∀X0,X1,X2. X0 ➡ X1 → X0 ➡ X2 → X0 = Y0 → - ∃∃X. X1 ➡ X & X2 ➡ X. -#Y0 #IH #X0 #X1 #X2 * -X0 -X1 -[ #I1 #H1 #H2 destruct - lapply (tpr_inv_atom1 … H1) -H1 -(* case 1: atom, atom *) - #H1 destruct // -| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct - elim (tpr_inv_flat1 … H1) -H1 * -(* case 2: flat, flat *) - [ #V2 #T2 #HV02 #HT02 #H destruct - /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *) -(* case 3: flat, beta *) - | #b #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct - /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *) -(* case 4: flat, theta *) - | #b #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct - /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *) -(* case 5: flat, tau *) - | #HT02 #H destruct - /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *) - ] -| #a #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct - elim (tpr_inv_appl1 … H1) -H1 * -(* case 6: beta, flat (repeated) *) - [ #V2 #T2 #HV02 #HT02 #H destruct - @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/ -(* case 7: beta, beta *) - | #b #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct - /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *) -(* case 8, beta, theta (excluded) *) - | #b #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct - ] -| #a #I1 #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct - elim (tpr_inv_bind1 … H1) -H1 * -(* case 9: delta, delta *) - [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct - /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *) -(* case 10: delta, zeta *) - | #T2 #HT20 #HTX2 #H1 #H2 destruct - /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *) - ] -| #a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct - elim (tpr_inv_appl1 … H1) -H1 * -(* case 11: theta, flat (repeated) *) - [ #V2 #T2 #HV02 #HT02 #H destruct - @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/ -(* case 12: theta, beta (repeated) *) - | #b #V2 #WW0 #TT0 #T2 #_ #_ #H destruct -(* case 13: theta, theta *) - | #b #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct - /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *) - ] -| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct - elim (tpr_inv_abbr1 … H1) -H1 * -(* case 14: zeta, delta (repeated) *) - [ #V2 #TT2 #T2 #HV02 #HTT02 #HTT2 #H destruct - @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/ -(* case 15: zeta, zeta *) - | #TT2 #HTT02 #HXTT2 - /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *) - ] -| #V0 #T0 #T1 #HT01 #H1 #H2 destruct - elim (tpr_inv_cast1 … H1) -H1 -(* case 16: tau, flat (repeated) *) - [ * #V2 #T2 #HV02 #HT02 #H destruct - @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/ -(* case 17: tau, tau *) - | #HT02 - /3 width=5 by tpr_conf_tau_tau/ - ] -] -qed. - -(* Basic_1: was: pr0_confluence *) -theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 → - ∃∃T. T1 ➡ T & T2 ➡ T. -#T @(tw_ind … T) -T /3 width=6 by tpr_conf_aux/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tps.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tps.ma deleted file mode 100644 index 12cf13c0f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tps.ma +++ /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 TERMS *********************************) - -(* Properties on parallel substitution for terms ****************************) - -(* Basic_1: was: pr0_subst1_fwd *) -lemma ltpr_tpr_conf: ∀L1,T,U1,d,e. L1 ⊢ T ▶ [d, e] U1 → ∀L2. L1 ➡ L2 → - ∃∃U2. U1 ➡ U2 & L2 ⊢ T ▶ [d, e] U2. -#L1 #T #U1 #d #e #H elim H -L1 -T -U1 -d -e -[ /2 width=3/ -| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12 - elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2 - elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1 - elim (lift_total V2 0 (i+1)) #W2 #HVW2 - lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=6/ -| #L1 #a #I #V #W1 #T #U1 #d #e #_ #_ #IHV #IHT #L2 #HL12 - elim (IHV … HL12) -IHV #W2 #HW12 - elim (IHT (L2.ⓑ{I}W2) ?) -IHT /2 width=1/ -L1 /3 width=5/ -| #L1 #I #V #W1 #T #U1 #d #e #_ #_ #IHV #IHT #L2 #HL12 - elim (IHV … HL12) -IHV - elim (IHT … HL12) -IHT -HL12 /3 width=5/ -] -qed. - -(* Basic_1: was: pr0_subst1_back *) -lemma ltpr_tps_trans: ∀L2,T,U2,d,e. L2 ⊢ T ▶ [d, e] U2 → ∀L1. L1 ➡ L2 → - ∃∃U1. U1 ➡ U2 & L1 ⊢ T ▶ [d, e] U1. -#L2 #T #U2 #d #e #H elim H -L2 -T -U2 -d -e -[ /2 width=3/ -| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12 - elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H - elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2 - elim (lift_total V1 0 (i+1)) #W1 #HVW1 - lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=6/ -| #L2 #a #I #V #W2 #T #U2 #d #e #_ #_ #IHV #IHT #L1 #HL12 - elim (IHV … HL12) -IHV #W1 #HW12 - elim (IHT (L1.ⓑ{I}W1) ?) -IHT /2 width=1/ -L2 /3 width=5/ -| #L2 #I #V #W2 #T #U2 #d #e #_ #_ #IHV #IHT #L1 #HL12 - elim (IHV … HL12) -IHV - elim (IHT … HL12) -IHT -HL12 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tpss.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tpss.ma deleted file mode 100644 index e1ead4e44..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tpss.ma +++ /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/tpr_tps.ma". - -(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) - -(* Unfold properties ********************************************************) - -(* Basic_1: was: pr0_subst1 *) -lemma tpr_tps_ltpr: ∀T1,T2. T1 ➡ T2 → - ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 → - ∀L2. L1 ➡ L2 → - ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2. -#T1 #T2 #H elim H -T1 -T2 -[ #I #L1 #d #e #U1 #H #L2 #HL12 - elim (ltpr_tpr_conf … H … HL12) -L1 /3 width=3/ -| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 - elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct - elim (IHV12 … HVW1 … HL12) -V1 - elim (IHT12 … HTU1 … HL12) -T1 -HL12 /3 width=5/ -| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 - elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct - elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct - elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2 - elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 - lapply (tpss_lsubs_trans … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/ -| #a #I #V1 #V2 #T1 #T #T2 #HV12 #_ #HT2 #IHV12 #IHT1 #L1 #d #e #X #H #L2 #HL12 - elim (tps_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct - elim (IHV12 … HVW1 … HL12) -V1 #W2 #HW12 #HVW2 - elim (IHT1 … HTU1 (L2. ⓑ{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU - elim (tpss_strip_neq … HTU … HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2 - lapply (tps_lsubs_trans … HU2 (L2. ⓑ{I} V2) ?) -HU2 /2 width=1/ #HU2 - elim (ltpss_dx_tps_conf … HU2 (L2. ⓑ{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23 - lapply (tps_lsubs_trans … HU3 (⋆. ⓑ{I} W2) ?) -HU3 /2 width=1/ #HU3 - lapply (tpss_lsubs_trans … HU23 (L2. ⓑ{I} W2) ?) -HU23 /2 width=1/ #HU23 - lapply (tpss_trans_eq … HTU2 … HU23) -U2 /3 width=5/ -| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12 - elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct - elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct - elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2 - elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2 - elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 - elim (lift_total VV2 0 1) #VV #H2VV - lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV - @ex2_1_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *) -| #V #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #e #X #H #L2 #HL12 - elim (tps_inv_bind1 … H) -H #W #U1 #_ #HTU1 #H destruct -V - elim (IHT1 … HTU1 (L2.ⓓW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU - elim (tpss_inv_lift1_ge … HTU L2 … HT2 ?) -T (aaa_inv_sort … H) -H // -| #I1 #L #K1 #V1 #B #i #HLK1 #_ #IHA1 #A2 #H - elim (aaa_inv_lref … H) -H #I2 #K2 #V2 #HLK2 #HA2 - lapply (ldrop_mono … HLK1 … HLK2) -L #H destruct /2 width=1/ -| #a #L #V #T #B1 #A1 #_ #_ #_ #IHA1 #A2 #H - elim (aaa_inv_abbr … H) -H /2 width=1/ -| #a #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H - elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1/ -| #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H - elim (aaa_inv_appl … H) -H #B2 #_ #HA2 - lapply (IHA1 … HA2) -L #H destruct // -| #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H - elim (aaa_inv_cast … H) -H /2 width=1/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/static/aaa_lift.ma deleted file mode 100644 index b40f1a9b1..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_lift.ma +++ /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 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/ldrop_ldrop.ma". -include "basic_2/static/aaa.ma". - -(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) - -(* Properties concerning basic relocation ***********************************) - -lemma aaa_lift: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → - ∀T2. ⇧[d, e] T1 ≡ T2 → L2 ⊢ T2 ⁝ A. -#L1 #T1 #A #H elim H -L1 -T1 -A -[ #L1 #k #L2 #d #e #_ #T2 #H - >(lift_inv_sort1 … H) -H // -| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #d #e #HL21 #T2 #H - elim (lift_inv_lref1 … H) -H * #Hid #H destruct - [ elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct - /3 width=8/ - | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ - ] -| #a #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H - elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - /4 width=4/ -| #a #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H - elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - /4 width=4/ -| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H - elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - /3 width=4/ -| #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #d #e #HL21 #X #H - elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - /3 width=4/ -] -qed. - -lemma aaa_inv_lift: ∀L2,T2,A. L2 ⊢ T2 ⁝ A → ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → - ∀T1. ⇧[d, e] T1 ≡ T2 → L1 ⊢ T1 ⁝ A. -#L2 #T2 #A #H elim H -L2 -T2 -A -[ #L2 #k #L1 #d #e #_ #T1 #H - >(lift_inv_sort2 … H) -H // -| #I #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #d #e #HL21 #T1 #H - elim (lift_inv_lref2 … H) -H * #Hid #H destruct - [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // -Hid /3 width=8/ - | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // -Hid /3 width=8/ - ] -| #a #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H - elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - /4 width=4/ -| #a #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H - elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - /4 width=4/ -| #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H - elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - /3 width=4/ -| #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #d #e #HL21 #X #H - elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_lifts.ma b/matita/matita/contribs/lambda_delta/basic_2/static/aaa_lifts.ma deleted file mode 100644 index 7514f6dc5..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_lifts.ma +++ /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/unfold/ldrops.ma". -include "basic_2/static/aaa_lift.ma". - -(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) - -(* Properties concerning generic relocation *********************************) - -lemma aaa_lifts: ∀L1,L2,T2,A,des. ⇩*[des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 → - L1 ⊢ T1 ⁝ A → L2 ⊢ T2 ⁝ A. -#L1 #L2 #T2 #A #des #H elim H -L1 -L2 -des -[ #L #T1 #H #HT1 - <(lifts_inv_nil … H) -H // -| #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1 - elim (lifts_inv_cons … H) -H /3 width=9/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_ltpss_dx.ma b/matita/matita/contribs/lambda_delta/basic_2/static/aaa_ltpss_dx.ma deleted file mode 100644 index 2f2d07360..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_ltpss_dx.ma +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/unfold/tpss_tpss.ma". -include "basic_2/unfold/ltpss_dx_ldrop.ma". -include "basic_2/static/aaa_lift.ma". - -(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) - -(* Properties about dx parallel unfold **************************************) - -(* Note: lemma 500 *) -lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A. -#L1 #T1 #A #H elim H -L1 -T1 -A -[ #L1 #k #L2 #d #e #_ #T2 #H - >(tpss_inv_sort1 … H) -H // -| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H - [ #H destruct - elim (lt_or_ge i d) #Hdi - [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct - /3 width=8 by aaa_lref/ (**) (* too slow without trace *) - | elim (lt_or_ge i (d + e)) #Hide - [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct - /3 width=8 by aaa_lref/ (**) (* too slow without trace *) - | -Hdi - lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide - /3 width=8 by aaa_lref/ (**) (* too slow without trace *) - ] - ] - | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 - elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 - elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct - lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2 - lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/ - ] -| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/ -| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/ -| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ -| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ -] -qed. - -lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → - ∀L2,d,e. L1 ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A. -/3 width=7/ qed. - -lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → - ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A. -/2 width=7/ qed. - -lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → - ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A. -/2 width=7/ qed. - -lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → - ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A. -/2 width=7/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_ltpss_sn.ma b/matita/matita/contribs/lambda_delta/basic_2/static/aaa_ltpss_sn.ma deleted file mode 100644 index 4f2a44827..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/aaa_ltpss_sn.ma +++ /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: ∀L1,T,A. L1 ⊢ T ⁝ A → - ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A. -#L1 #T #A #HT #L2 #d #e #HL12 -lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12 -@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/ -qed. - -lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → - ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A. -/3 width=5/ qed. - -lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → - ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → - ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A. -/3 width=5/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsuba.ma deleted file mode 100644 index aa4800fd5..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba.ma +++ /dev/null @@ -1,92 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) - -inductive lsuba: relation lenv ≝ -| lsuba_atom: lsuba (⋆) (⋆) -| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V) -| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ⁝ A → L2 ⊢ W ⁝ A → - lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW) -. - -interpretation - "local environment refinement (atomic arity assigment)" - 'CrSubEqA L1 L2 = (lsuba L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆. -#L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #A #_ #_ #_ #H destruct -] -qed. - -lemma lsuba_inv_atom1: ∀L2. ⋆ ⁝⊑ L2 → L2 = ⋆. -/2 width=3/ qed-. - -fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V → - (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & - L2 = K2. ⓛW & I = Abbr. -#L1 #L2 * -L1 -L2 -[ #I #K1 #V #H destruct -| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/ -| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/ -] -qed. - -lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ⁝⊑ L2 → - (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & - L2 = K2. ⓛW & I = Abbr. -/2 width=3/ qed-. - -fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆. -#L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #A #_ #_ #_ #H destruct -] -qed. - -lemma lsubc_inv_atom2: ∀L1. L1 ⁝⊑ ⋆ → L1 = ⋆. -/2 width=3/ qed-. - -fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W → - (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & - L1 = K1. ⓓV & I = Abst. -#L1 #L2 * -L1 -L2 -[ #I #K2 #W #H destruct -| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/ -| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/ -] -qed. - -lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ⁝⊑ K2. ⓑ{I} W → - (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & - L1 = K1. ⓓV & I = Abst. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -lemma lsuba_refl: ∀L. L ⁝⊑ L. -#L elim L -L // /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_aaa.ma deleted file mode 100644 index 66e802aae..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_aaa.ma +++ /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/static/aaa_aaa.ma". -include "basic_2/static/lsuba_ldrop.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) - -(* Properties concerning atomic arity assignment ****************************) - -lemma lsuba_aaa_conf: ∀L1,V,A. L1 ⊢ V ⁝ A → ∀L2. L1 ⁝⊑ L2 → L2 ⊢ V ⁝ A. -#L1 #V #A #H elim H -L1 -V -A -[ // -| #I #L1 #K1 #V1 #B #i #HLK1 #HV1B #IHV1 #L2 #HL12 - elim (lsuba_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2 - elim (lsuba_inv_pair1 … H) -H * #K2 - [ #HK12 #H destruct /3 width=5/ - | #V2 #A1 #HV1A1 #HV2 #_ #H1 #H2 destruct - >(aaa_mono … HV1B … HV1A1) -B -HV1A1 /2 width=5/ - ] -| /4 width=2/ -| /4 width=1/ -| /3 width=3/ -| /3 width=1/ -] -qed. - -lemma lsuba_aaa_trans: ∀L2,V,A. L2 ⊢ V ⁝ A → ∀L1. L1 ⁝⊑ L2 → L1 ⊢ V ⁝ A. -#L2 #V #A #H elim H -L2 -V -A -[ // -| #I #L2 #K2 #V2 #B #i #HLK2 #HV2B #IHV2 #L1 #HL12 - elim (lsuba_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 - elim (lsuba_inv_pair2 … H) -H * #K1 - [ #HK12 #H destruct /3 width=5/ - | #V1 #A1 #HV1 #HV2A1 #_ #H1 #H2 destruct - >(aaa_mono … HV2B … HV2A1) -B -HV2A1 /2 width=5/ - ] -| /4 width=2/ -| /4 width=1/ -| /3 width=3/ -| /3 width=1/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_ldrop.ma deleted file mode 100644 index 247a8b221..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_ldrop.ma +++ /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/static/lsuba.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) - -(* Properties concerning basic local environment slicing ********************) - -(* Note: the constant 0 cannot be generalized *) -lemma lsuba_ldrop_O1_conf: ∀L1,L2. L1 ⁝⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 → - ∃∃K2. K1 ⁝⊑ K2 & ⇩[0, e] L2 ≡ K2. -#L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -] -qed-. - -(* Note: the constant 0 cannot be generalized *) -lemma lsuba_ldrop_O1_trans: ∀L1,L2. L1 ⁝⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. K1 ⁝⊑ K2 & ⇩[0, e] L1 ≡ K1. -#L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_lsuba.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_lsuba.ma deleted file mode 100644 index 5d64516a5..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsuba_lsuba.ma +++ /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/static/lsuba_aaa.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) - -(* Main properties **********************************************************) - -theorem lsuba_trans: ∀L1,L. L1 ⁝⊑ L → ∀L2. L ⁝⊑ L2 → L1 ⁝⊑ L2. -#L1 #L #H elim H -L1 -L -[ #X #H >(lsuba_inv_atom1 … H) -H // -| #I #L1 #L #V #HL1 #IHL1 #X #H - elim (lsuba_inv_pair1 … H) -H * #L2 - [ #HL2 #H destruct /3 width=1/ - | #V #A #HLV #HL2V #HL2 #H1 #H2 destruct /3 width=3/ - ] -| #L1 #L #V1 #W #A1 #HV1 #HW #HL1 #IHL1 #X #H - elim (lsuba_inv_pair1 … H) -H * #L2 - [ #HL2 #H destruct /3 width=5/ - | #V #A2 #_ #_ #_ #_ #H destruct - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsubss.ma deleted file mode 100644 index 356d7fe11..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss.ma +++ /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/static/ssta.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******) - -(* Note: may not be transitive *) -inductive lsubss (h:sh) (g:sd h): relation lenv ≝ -| lsubss_atom: lsubss h g (⋆) (⋆) -| lsubss_pair: ∀I,L1,L2,W. lsubss h g L1 L2 → - lsubss h g (L1. ⓑ{I} W) (L2. ⓑ{I} W) -| lsubss_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V •[g, l+1] W → ⦃h, L2⦄ ⊢ V •[g, l+1] W → - lsubss h g L1 L2 → lsubss h g (L1. ⓓV) (L2. ⓛW) -. - -interpretation - "local environment refinement (stratified static type assigment)" - 'CrSubEqS h g L1 L2 = (lsubss h g L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lsubss_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 = ⋆ → L2 = ⋆. -#h #g #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #l #_ #_ #_ #H destruct -] -qed. - -lemma lsubss_inv_atom1: ∀h,g,L2. h ⊢ ⋆ •⊑[g] L2 → L2 = ⋆. -/2 width=5/ qed-. - -fact lsubss_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → - ∀I,K1,V. L1 = K1. ⓑ{I} V → - (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & - h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr. -#h #g #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 lsubss_inv_pair1: ∀h,g,I,K1,L2,V. h ⊢ K1. ⓑ{I} V •⊑[g] L2 → - (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨ - ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & - h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr. -/2 width=3/ qed-. - -fact lsubss_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L2 = ⋆ → L1 = ⋆. -#h #g #L1 #L2 * -L1 -L2 -[ // -| #I #L1 #L2 #V #_ #H destruct -| #L1 #L2 #V #W #l #_ #_ #_ #H destruct -] -qed. - -lemma lsubss_inv_atom2: ∀h,g,L1. h ⊢ L1 •⊑[g] ⋆ → L1 = ⋆. -/2 width=5/ qed-. - -fact lsubss_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → - ∀I,K2,W. L2 = K2. ⓑ{I} W → - (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & - h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst. -#h #g #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 lsubss_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 •⊑[g] K2. ⓑ{I} W → - (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨ - ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & - h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst. -/2 width=3/ qed-. - -(* Basic_forward lemmas *****************************************************) - -lemma lsubss_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L1|] L2. -#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -lemma lsubss_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L2|] L2. -#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ -qed-. - -(* Basic properties *********************************************************) - -lemma lsubss_refl: ∀h,g,L. h ⊢ L •⊑[g] L. -#h #g #L elim L -L // /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_ldrop.ma deleted file mode 100644 index 82ede6149..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_ldrop.ma +++ /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/static/lsubss.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******) - -(* Properties concerning basic local environment slicing ********************) - -(* Note: the constant 0 cannot be generalized *) -lemma lsubss_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → - ∀K1,e. ⇩[0, e] L1 ≡ K1 → - ∃∃K2. h ⊢ K1 •⊑[g] K2 & ⇩[0, e] L2 ≡ K2. -#h #g #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K1 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK1 - [ destruct - elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK1) -L1 /3 width=3/ - ] -] -qed. - -(* Note: the constant 0 cannot be generalized *) -lemma lsubss_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → - ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. h ⊢ K1 •⊑[g] K2 & ⇩[0, e] L1 ≡ K1. -#h #g #L1 #L2 #H elim H -L1 -L2 -[ /2 width=3/ -| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #HLK2 - [ destruct - elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H - <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ - | elim (IHL12 … HLK2) -L2 /3 width=3/ - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_lsubss.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_lsubss.ma deleted file mode 100644 index d9f9496ba..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_lsubss.ma +++ /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/static/lsubss_ssta.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STATIC TYPE ASSIGNMENT ******************) - -(* Main properties **********************************************************) - -theorem lsubss_trans: ∀h,g,L1,L. h ⊢ L1 •⊑[g] L → ∀L2. h ⊢ L •⊑[g] L2 → - h ⊢ L1 •⊑[g] L2. -#h #g #L1 #L #H elim H -L1 -L -[ #X #H >(lsubss_inv_atom1 … H) -H // -| #I #L1 #L #W #HL1 #IHL1 #X #H - elim (lsubss_inv_pair1 … H) -H * #L2 - [ #HL2 #H destruct /3 width=1/ - | #V #l #H1WV #H2WV #HL2 #H1 #H2 destruct /3 width=3/ - ] -| #L1 #L #V1 #W1 #l #H1VW1 #H2VW1 #HL1 #IHL1 #X #H - elim (lsubss_inv_pair1 … H) -H * #L2 - [ #HL2 #H destruct /3 width=5/ - | #V #l0 #_ #_ #_ #_ #H destruct - ] -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_ssta.ma b/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_ssta.ma deleted file mode 100644 index f9c628921..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/lsubss_ssta.ma +++ /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/static/ssta_lift.ma". -include "basic_2/static/ssta_ssta.ma". -include "basic_2/static/lsubss_ldrop.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******) - -(* Properties concerning stratified native type assignment ******************) - -lemma lsubss_ssta_trans: ∀h,g,L2,T,U,l. ⦃h, L2⦄ ⊢ T •[g,l] U → - ∀L1. h ⊢ L1 •⊑[g] L2 → ⦃h, L1⦄ ⊢ T •[g,l] U. -#h #g #L2 #T #U #l #H elim H -L2 -T -U -l -[ /2 width=1/ -| #L2 #K2 #V2 #W2 #U2 #i #l #HLK2 #_ #HWU2 #IHVW2 #L1 #HL12 - elim (lsubss_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 - elim (lsubss_inv_pair2 … H) -H * #K1 [ | -HWU2 -IHVW2 -HLK1 ] - [ #HK12 #H destruct /3 width=6/ - | #V1 #l0 #_ #_ #_ #_ #H destruct - ] -| #L2 #K2 #W2 #V2 #U2 #i #l #HLK2 #HWV2 #HWU2 #IHWV2 #L1 #HL12 - elim (lsubss_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 - elim (lsubss_inv_pair2 … H) -H * #K1 [ -HWV2 | -IHWV2 ] - [ #HK12 #H destruct /3 width=6/ - | #V1 #l0 #H1 #H2 #_ #H #_ destruct - elim (ssta_fwd_correct … H2) -H2 #V #H - elim (ssta_mono … HWV2 … H) -HWV2 -H /2 width=6/ - ] -| /4 width=1/ -| /3 width=1/ -| /3 width=1/ -] -qed. - -lemma lsubss_ssta_conf: ∀h,g,L1,T,U,l. ⦃h, L1⦄ ⊢ T •[g,l] U → - ∀L2. h ⊢ L1 •⊑[g] L2 → ⦃h, L2⦄ ⊢ T •[g,l] U. -#h #g #L1 #T #U #l #H elim H -L1 -T -U -l -[ /2 width=1/ -| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #HL12 - elim (lsubss_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2 - elim (lsubss_inv_pair1 … H) -H * #K2 [ -HVW1 | -IHVW1 ] - [ #HK12 #H destruct /3 width=6/ - | #V2 #l0 #H1 #H2 #_ #H #_ destruct - elim (ssta_mono … HVW1 … H1) -HVW1 -H1 #H1 #H2 destruct - elim (ssta_fwd_correct … H2) -H2 /2 width=6/ - ] -| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #_ #HWU1 #IHWV1 #L2 #HL12 - elim (lsubss_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2 - elim (lsubss_inv_pair1 … H) -H * #K2 [ | -HWU1 -IHWV1 -HLK2 ] - [ #HK12 #H destruct /3 width=6/ - | #V2 #l0 #_ #_ #_ #_ #H destruct - ] -| /4 width=1/ -| /3 width=1/ -| /3 width=1/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/static/sd.ma b/matita/matita/contribs/lambda_delta/basic_2/static/sd.ma deleted file mode 100644 index 63143b19b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/static/sd.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 "basic_2/static/sh.ma". - -(* SORT DEGREE **************************************************************) - -(* sort degree specification *) -record sd (h:sh): Type[0] ≝ { - deg : relation nat; (* degree of the sort *) - deg_total: ∀k. ∃l. deg k l; (* functional relation axioms *) - deg_mono : ∀k,l1,l2. deg k l1 → deg k l2 → l1 = l2; - deg_next : ∀k,l. deg k l → deg (next h k) (l - 1) (* compatibility condition *) -}. - -(* Notable specifications ***************************************************) - -definition deg_O: relation nat ≝ λk,l. l = 0. - -definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O …. -// /2 width=1/ /2 width=2/ qed. - -inductive deg_SO (h:sh) (k:nat) (k0:nat): predicate nat ≝ -| deg_SO_pos : ∀l0. (next h)^l0 k0 = k → deg_SO h k k0 (l0 + 1) -| deg_SO_zero: ((∃l0. (next h)^l0 k0 = k) → ⊥) → deg_SO h k k0 0 -. - -fact deg_SO_inv_pos_aux: ∀h,k,k0,l0. deg_SO h k k0 l0 → ∀l. l0 = l + 1 → - (next h)^l k0 = k. -#h #k #k0 #l0 * -l0 -[ #l0 #Hl0 #l #H - lapply (injective_plus_l … H) -H #H destruct // -| #_ #l0 H -H #H - lapply (transitive_lt … H HK12) -k1 #H1 - lapply (nexts_le h k2 l) #H2 - lapply (le_to_lt_to_lt … H2 H1) -h -l #H - elim (lt_refl_false … H) -qed. - -definition sd_SO: ∀h. nat → sd h ≝ λh,k. mk_sd h (deg_SO h k) …. -[ #k0 - lapply (nexts_dec h k0 k) * [ * /3 width=2/ | /4 width=2/ ] -| #K0 #l1 #l2 * [ #l01 ] #H1 * [1,3: #l02 ] #H2 // - [ < H2 in H1; -H2 #H - lapply (nexts_inj … H) -H #H destruct // - | elim (H1 ?) /2 width=2/ - | elim (H2 ?) /2 width=2/ - ] -| #k0 #l0 * - [ #l #H destruct elim l -l normalize /2 width=1/ - | #H1 @deg_SO_zero * #l #H2 destruct - @H1 -H1 @(ex_intro … (S l)) /2 width=1/ (**) (* explicit constructor *) - ] -] -qed. - -let rec sd_l (h:sh) (k:nat) (l:nat) on l : sd h ≝ - match l with - [ O ⇒ sd_O h - | S l ⇒ match l with - [ O ⇒ sd_SO h k - | _ ⇒ sd_l h (next h k) l - ] - ]. - -(* Basic properties *********************************************************) - -lemma deg_pred: ∀h,g,k,l. deg h g (next h k) (l + 1) → deg h g k (l + 2). -#h #g #k #l #H1 -elim (deg_total h g k) #l0 #H0 -lapply (deg_next … H0) #H2 -lapply (deg_mono … H1 H2) -H1 -H2 #H -<(associative_plus l 1 1) >H (lift_inv_sort1 … H1) -X1 - >(lift_inv_sort1 … H2) -X2 /2 width=1/ -| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 - elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct - /3 width=8/ - | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 - lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ - ] -| #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hid #H destruct - [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (lift_inv_sort2 … H) -X /3 width=3/ -| #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ] - [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 - elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HVW1 #HW12 - elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ - | minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ - | (tpss_inv_sort1 … H) -H /3 width=3/ -| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H - elim (tpss_inv_lref1 … H) -H [ | -HVW1 ] - [ #H destruct - elim (lt_or_ge i d) #Hdi [ -HVW1 | ] - [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 - elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct - elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 - lapply (ldrop_fwd_ldrop2 … HLK2) #H - elim (lift_total W2 0 (i+1)) #U2 #HWU2 - lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 - >minus_plus minus_plus #H - lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H - lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H - lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus (deg_mono … Hkl2 … Hkl) -g -L -l2 /2 width=1/ -| #L #K #V #W #U1 #i #l1 #HLK #_ #HWU1 #IHVW #U2 #l2 #H - elim (ssta_inv_lref1 … H) -H * #K0 #V0 #W0 [2: #l0] #HLK0 #HVW0 #HW0U2 - lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct - lapply (IHVW … HVW0) -IHVW -HVW0 * #H1 #H2 destruct - >(lift_mono … HWU1 … HW0U2) -W0 -U1 /2 width=1/ -| #L #K #W #V #U1 #i #l1 #HLK #_ #HWU1 #IHWV #U2 #l2 #H - elim (ssta_inv_lref1 … H) -H * #K0 #W0 #V0 [2: #l0 ] #HLK0 #HWV0 #HV0U2 - lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct - lapply (IHWV … HWV0) -IHWV -HWV0 * #H1 #H2 destruct - >(lift_mono … HWU1 … HV0U2) -W -U1 /2 width=1/ -| #a #I #L #V #T #U1 #l1 #_ #IHTU1 #X #l2 #H - elim (ssta_inv_bind1 … H) -H #U2 #HTU2 #H destruct - elim (IHTU1 … HTU2) -T /3 width=1/ -| #L #V #T #U1 #l1 #_ #IHTU1 #X #l2 #H - elim (ssta_inv_appl1 … H) -H #U2 #HTU2 #H destruct - elim (IHTU1 … HTU2) -T /3 width=1/ -| #L #W1 #T #U1 #l1 #_ #IHTU1 #U2 #l2 #H - lapply (ssta_inv_cast1 … H) -H #HTU2 - elim (IHTU1 … HTU2) -T /2 width=1/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/frsup.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/frsup.ma deleted file mode 100644 index 31d6c9fee..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/frsup.ma +++ /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/grammar/cl_weight.ma". -include "basic_2/substitution/lift.ma". - -(* RESTRICTED SUPCLOSURE ****************************************************) - -inductive frsup: bi_relation lenv term ≝ -| frsup_bind_sn: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) L V -| frsup_bind_dx: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T -| frsup_flat_sn: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L V -| frsup_flat_dx: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L T -. - -interpretation - "restricted structural predecessor (closure)" - 'RestSupTerm L1 T1 L2 T2 = (frsup L1 T1 L2 T2). - -(* Basic inversion lemmas ***************************************************) - -fact frsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → - ∀J. T1 = ⓪{J} → ⊥. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #a #I #L #V #T #J #H destruct -| #a #I #L #V #T #J #H destruct -| #I #L #V #T #J #H destruct -| #I #L #V #T #J #H destruct -] -qed-. - -lemma frsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁ ⦃L2, T2⦄ → ⊥. -/2 width=7 by frsup_inv_atom1_aux/ qed-. - -fact frsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → - ∀b,J,W,U. T1 = ⓑ{b,J}W.U → - (L2 = L1 ∧ T2 = W) ∨ - (L2 = L1.ⓑ{J}W ∧ T2 = U). -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ -| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ -| #I #L #V #T #b #J #W #U #H destruct -| #I #L #V #T #b #J #W #U #H destruct -] -qed-. - -lemma frsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁ ⦃L2, T2⦄ → - (L2 = L1 ∧ T2 = W) ∨ - (L2 = L1.ⓑ{J}W ∧ T2 = U). -/2 width=4 by frsup_inv_bind1_aux/ qed-. - -fact frsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → - ∀J,W,U. T1 = ⓕ{J}W.U → - L2 = L1 ∧ (T2 = W ∨ T2 = U). -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #a #I #L #V #T #J #W #U #H destruct -| #a #I #L #V #T #J #W #U #H destruct -| #I #L #V #T #J #W #U #H destruct /3 width=1/ -| #I #L #V #T #J #W #U #H destruct /3 width=1/ -] -qed-. - -lemma frsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁ ⦃L2, T2⦄ → - L2 = L1 ∧ (T2 = W ∨ T2 = U). -/2 width=4 by frsup_inv_flat1_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma frsup_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ -qed-. - -lemma frsup_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → #{L1} ≤ #{L2}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ -qed-. - -lemma frsup_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → #{T2} < #{T1}. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/ -qed-. - -lemma frsup_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L. -#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 -[ #a -| #a #I #L #V #_ @(ex_intro … (⋆.ⓑ{I}V)) // -] -#I #L #V #T @(ex_intro … (⋆)) // -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lift_frsup_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → - ∀L,K,U2. ⦃L, U1⦄ ⧁ ⦃L @@ K, U2⦄ → - ∃T2. ⇧[d + |K|, e] T2 ≡ U2. -#T1 #U1 #d #e * -T1 -U1 -d -e -[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HTU1 #L #K #X #H - elim (frsup_inv_bind1 … H) -H * - [ -HTU1 #H1 #H2 destruct - >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/ - | -HVW1 #H1 #H2 destruct - >(append_inv_pair_dx … H1) -L -K normalize /2 width=2/ - ] -|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #K #X #H - elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct - >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/ -] -#i #d #e [2,3: #_ ] #L #K #X #H -elim (frsup_inv_atom1 … H) -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop.ma deleted file mode 100644 index 218389e1c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop.ma +++ /dev/null @@ -1,80 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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". - -(* GLOBAL ENVIRONMENT SLICING ***********************************************) - -inductive gdrop (e:nat): relation genv ≝ -| gdrop_gt: ∀G. |G| ≤ e → gdrop e G (⋆) -| gdrop_eq: ∀G. |G| = e + 1 → gdrop e G G -| gdrop_lt: ∀I,G1,G2,V. e < |G1| → gdrop e G1 G2 → gdrop e (G1. ⓑ{I} V) G2 -. - -interpretation "global slicing" - 'RDrop e G1 G2 = (gdrop e G1 G2). - -(* basic inversion lemmas ***************************************************) - -lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆. -#G1 #G2 #e * -G1 -G2 // -[ #G #H >H -H >commutative_plus #H - lapply (le_plus_to_le_r … 0 H) -H #H - lapply (le_n_O_to_eq … H) -H #H destruct -| #I #G1 #G2 #V #H1 #_ #H2 - lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H - lapply (lt_plus_to_lt_l … 0 H) -H #H - elim (lt_zero_false … H) -] -qed-. - -lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2. -#G1 #G2 #e * -G1 -G2 // -[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H - lapply (le_plus_to_le_r … 0 H) -H #H - lapply (le_n_O_to_eq … H) -H #H destruct -| #I #G1 #G2 #V #H1 #_ normalize #H2 - <(injective_plus_l … H2) in H1; -H2 #H - elim (lt_refl_false … H) -] -qed-. - -fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V → - e < |G1| → ⇩[e] G1 ≡ G2. -#I #G #G1 #G2 #V #e * -G -G2 -[ #G #H1 #H destruct #H2 - lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H - lapply (lt_plus_to_lt_l … 0 H) -H #H - elim (lt_zero_false … H) -| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H - elim (lt_refl_false … H) -| #J #G #G2 #W #_ #HG2 #H destruct // -] -qed. - -lemma gdrop_inv_lt: ∀I,G1,G2,V,e. - ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2. -/2 width=5/ qed-. - -(* Basic properties *********************************************************) - -lemma gdrop_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2. -#e #G1 elim G1 -G1 /3 width=2/ -#I #V #G1 * #G2 #HG12 -elim (lt_or_eq_or_gt e (|G1|)) #He -[ /3 width=2/ -| destruct /3 width=2/ -| @ex_intro [2: @gdrop_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *) -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop_gdrop.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop_gdrop.ma deleted file mode 100644 index 0bc1a40d5..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop_gdrop.ma +++ /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/gdrop.ma". - -(* GLOBAL ENVIRONMENT SLICING ***********************************************) - -(* Main properties **********************************************************) - -theorem gdrop_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2. -#G #G1 #e #H elim H -G -G1 -[ #G #He #G2 #H - >(gdrop_inv_gt … H He) -H -He // -| #G #He #G2 #H - >(gdrop_inv_eq … H He) -H -He // -| #I #G #G1 #V #He #_ #IHG1 #G2 #H - lapply (gdrop_inv_lt … H He) -H -He /2 width=1/ -] -qed-. - -lemma gdrop_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2). -#G1 #G2 #e -elim (gdrop_total e G1) #G #HG1 -elim (genv_eq_dec G G2) #HG2 -[ destruct /2 width=1/ -| @or_intror #HG12 - lapply (gdrop_mono … HG1 … HG12) -HG1 -HG12 /2 width=1/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma deleted file mode 100644 index 9511648aa..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma +++ /dev/null @@ -1,313 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/cl_weight.ma". -include "basic_2/substitution/lift.ma". -include "basic_2/substitution/lsubs.ma". - -(* LOCAL ENVIRONMENT SLICING ************************************************) - -(* Basic_1: includes: drop_skip_bind *) -inductive ldrop: nat → nat → relation lenv ≝ -| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆) -| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) -| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2 -| ldrop_skip : ∀L1,L2,I,V1,V2,d,e. - ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 → - ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) -. - -interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2). - -definition l_liftable: (lenv → relation term) → Prop ≝ - λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K → - ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2. - -definition l_deliftable_sn: (lenv → relation term) → Prop ≝ - λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K → - ∀T1. ⇧[d, e] T1 ≡ U1 → - ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2. - -definition dropable_sn: relation lenv → Prop ≝ - λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 → - ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2. - -definition dedropable_sn: relation lenv → Prop ≝ - λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 → - ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2. - -definition dropable_dx: relation lenv → Prop ≝ - λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → - ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2. - -(* Basic inversion lemmas ***************************************************) - -fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| // -| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -(* Basic_1: was: drop_gen_refl *) -lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2. -/2 width=5/ qed-. - -fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ → - L2 = ⋆. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| #L #I #V #H destruct -| #L1 #L2 #I #V #e #_ #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct -] -qed. - -(* Basic_1: was: drop_gen_sort *) -lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆. -/2 width=5/ qed-. - -fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → - ∀K,I,V. L1 = K. ⓑ{I} V → - (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ - (0 < e ∧ ⇩[d, e - 1] K ≡ L2). -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #K #I #V #H destruct -| #L #I #V #_ #K #J #W #HX destruct /3 width=1/ -| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 → - (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ - (0 < e ∧ ⇩[0, e - 1] K ≡ L2). -/2 width=3/ qed-. - -lemma ldrop_inv_pair1: ∀K,I,V,L2. ⇩[0, 0] K. ⓑ{I} V ≡ L2 → L2 = K. ⓑ{I} V. -#K #I #V #L2 #H -elim (ldrop_inv_O1 … H) -H * // #H destruct -elim (lt_refl_false … H) -qed-. - -(* Basic_1: was: drop_gen_drop *) -lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2. - ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2. -#e #K #I #V #L2 #H #He -elim (ldrop_inv_O1 … H) -H * // #H destruct -elim (lt_refl_false … He) -qed-. - -fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d → - ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 & - ⇧[d - 1, e] V2 ≡ V1 & - L2 = K2. ⓑ{I} V2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K #V #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) -| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/ -] -qed. - -(* Basic_1: was: drop_gen_skip_l *) -lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d → - ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 & - ⇧[d - 1, e] V2 ≡ V1 & - L2 = K2. ⓑ{I} V2. -/2 width=3/ qed-. - -fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d → - ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & - ⇧[d - 1, e] V2 ≡ V1 & - L1 = K1. ⓑ{I} V1. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K #V #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) -| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/ -] -qed. - -(* Basic_1: was: drop_gen_skip_r *) -lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d → - ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 & - L1 = K1. ⓑ{I} V1. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was by definition: drop_refl *) -lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L. -#L elim L -L // -qed. - -lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e. - ⇩[0, e - 1] L1 ≡ L2 → 0 < e → ⇩[0, e] L1. ⓑ{I} V ≡ L2. -#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ -qed. - -lemma ldrop_skip_lt: ∀L1,L2,I,V1,V2,d,e. - ⇩[d - 1, e] L1 ≡ L2 → ⇧[d - 1, e] V2 ≡ V1 → 0 < d → - ⇩[d, e] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2. -#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/ -qed. - -lemma ldrop_O1_le: ∀i,L. i ≤ |L| → ∃K. ⇩[0, i] L ≡ K. -#i @(nat_ind_plus … i) -i /2 width=2/ -#i #IHi * -[ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct -| #L #I #V normalize #H - elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/ -] -qed. - -lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V. -#L elim L -L -[ #i #H elim (lt_zero_false … H) -| #L #I #V #IHL #i @(nat_ind_plus … i) -i /2 width=4/ - #i #_ normalize #H - elim (IHL i ? ) -IHL /2 width=1/ -H /3 width=4/ -] -qed. - -lemma ldrop_lsubs_ldrop2_abbr: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → - ∀K2,V,i. ⇩[0, i] L2 ≡ K2. ⓓV → - d ≤ i → i < d + e → - ∃∃K1. K1 ≼ [0, d + e - i - 1] K2 & - ⇩[0, i] L1 ≡ K1. ⓓV. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e -[ #d #e #K1 #V #i #H - lapply (ldrop_inv_atom1 … H) -H #H destruct -| #L1 #L2 #K1 #V #i #_ #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie - elim (ldrop_inv_O1 … H) -H * #Hi #HLK1 - [ -IHL12 -Hie destruct - minus_minus_comm >arith_b1 // /4 width=3/ - ] -| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie - elim (ldrop_inv_O1 … H) -H * #Hi #HLK1 - [ -IHL12 -Hie -Hi destruct - | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/ - ] -| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide - elim (le_inv_plus_l … Hdi) #Hdim #Hi - lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1 - elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/ -] -qed. - -lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R). -#R #HR #L1 #K1 #d #e #HLK1 #L2 #H elim H -L2 -[ #L2 #HL12 - elim (HR … HLK1 … HL12) -HR -L1 /3 width=3/ -| #L #L2 #_ #HL2 * #K #HK1 #HLK - elim (HR … HLK … HL2) -HR -L /3 width=3/ -] -qed. - -lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R). -#R #HR #L1 #K1 #d #e #HLK1 #K2 #H elim H -K2 -[ #K2 #HK12 - elim (HR … HLK1 … HK12) -HR -K1 /3 width=3/ -| #K #K2 #_ #HK2 * #L #HL1 #HLK - elim (HR … HLK … HK2) -HR -K /3 width=3/ -] -qed. - -lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R). -#R #HR #L1 #L2 #H elim H -L2 -[ #L2 #HL12 #K2 #e #HLK2 - elim (HR … HL12 … HLK2) -HR -L2 /3 width=3/ -| #L #L2 #_ #HL2 #IHL1 #K2 #e #HLK2 - elim (HR … HL2 … HLK2) -HR -L2 #K #HLK #HK2 - elim (IHL1 … HLK) -L /3 width=5/ -] -qed. - -(* Basic forvard lemmas *****************************************************) - -(* Basic_1: was: drop_S *) -lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 → - ⇩[O, e + 1] L1 ≡ K2. -#L1 elim L1 -L1 -[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct -| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #H - [ -IHL1 destruct /2 width=1/ - | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/ - ] -] -qed-. - -lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → #{L2} ≤ #{L1}. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize -[ /2 width=3/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 - >(tw_lift … HV21) -HV21 /2 width=1/ -] -qed-. - -lemma ldrop_pair2_fwd_fw: ∀I,L,K,V,d,e. ⇩[d, e] L ≡ K. ⓑ{I} V → - ∀T. #{K, V} < #{L, T}. -#I #L #K #V #d #e #H #T -lapply (ldrop_fwd_lw … H) -H #H -@(le_to_lt_to_lt … H) -H /3 width=1/ -qed-. - -lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e. - ⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|. -#L1 elim L1 -L1 -[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct -| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #H - [ -IHL1 destruct // - | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/ - ] -] -qed-. - -lemma ldrop_fwd_O1_length: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → |L2| = |L1| - e. -#L1 elim L1 -L1 -[ #L2 #e #H >(ldrop_inv_atom1 … H) -H // -| #K1 #I1 #V1 #IHL1 #L2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #H - [ -IHL1 destruct // - | lapply (IHL1 … H) -IHL1 -H #H >H -H normalize - >minus_le_minus_minus_comm // - ] -] -qed-. - -(* Basic_1: removed theorems 50: - drop_ctail drop_skip_flat - cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf - drop_clear drop_clear_O drop_clear_S - clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r - clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle - getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans - getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt - getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev - drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge - getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O - getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le - getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_append.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_append.ma deleted file mode 100644 index 359d39c80..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_append.ma +++ /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/ldrop.ma". - -(* DROPPING *****************************************************************) - -(* Properties on append for local environments ******************************) - -fact ldrop_O1_append_sn_le_aux: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → - d = 0 → e ≤ |L1| → - ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize // /4 width=1/ -#d #e #_ #H #L -d -lapply (le_n_O_to_eq … H) -H // -qed-. - -lemma ldrop_O1_append_sn_le: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → e ≤ |L1| → - ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2. -/2 width=3 by ldrop_O1_append_sn_le_aux/ qed. - -(* Inversion lemmas on append for local environments ************************) - -lemma ldrop_O1_inv_append1_ge: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K → - |L2| ≤ e → ⇩[0, e - |L2|] L1 ≡ K. -#K #L1 #L2 elim L2 -L2 normalize // -#L2 #I #V #IHL2 #e #H #H1e -elim (ldrop_inv_O1 … H) -H * #H2e #HL12 destruct -[ lapply (le_n_O_to_eq … H1e) -H1e -IHL2 - >commutative_plus normalize #H destruct -| minus_minus_comm /3 width=1/ -] -qed-. - -lemma ldrop_O1_inv_append1_le: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K → e ≤ |L2| → - ∀K2. ⇩[0, e] L2 ≡ K2 → K = L1 @@ K2. -#K #L1 #L2 elim L2 -L2 normalize -[ #e #H1 #H2 #K2 #H3 - lapply (le_n_O_to_eq … H2) -H2 #H2 - lapply (ldrop_inv_atom1 … H3) -H3 #H3 destruct - >(ldrop_inv_refl … H1) -H1 // -| #L2 #I #V #IHL2 #e @(nat_ind_plus … e) -e [ -IHL2 ] - [ #H1 #_ #K2 #H2 - lapply (ldrop_inv_refl … H1) -H1 #H1 - lapply (ldrop_inv_refl … H2) -H2 #H2 destruct // - | #e #_ #H1 #H1e #K2 #H2 - lapply (ldrop_inv_ldrop1 … H1 ?) -H1 // - lapply (ldrop_inv_ldrop1 … H2 ?) -H2 // /3 width=4/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_ldrop.ma deleted file mode 100644 index 07d9c53e4..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_ldrop.ma +++ /dev/null @@ -1,176 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/lift_lift.ma". -include "basic_2/substitution/ldrop.ma". - -(* DROPPING *****************************************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: drop_mono *) -theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 → - ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2. -#d #e #L #L1 #H elim H -d -e -L -L1 -[ #d #e #L2 #H - >(ldrop_inv_atom1 … H) -L2 // -| #K #I #V #L2 #HL12 - <(ldrop_inv_refl … HL12) -L2 // -| #L #K #I #V #e #_ #IHLK #L2 #H - lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/ -| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H - elim (ldrop_inv_skip1 … H ?) -H // (lift_inj … HVT1 … HVT2) -HVT1 -HVT2 - >(IHLK1 … HLK2) -IHLK1 -HLK2 // -] -qed-. - -(* Basic_1: was: drop_conf_ge *) -theorem ldrop_conf_ge: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 → - ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 + e1 ≤ e2 → - ⇩[0, e2 - e1] L1 ≡ L2. -#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1 -[ #d #e #e2 #L2 #H - >(ldrop_inv_atom1 … H) -L2 // -| // -| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2 - minus_minus_comm /3 width=1/ -| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2 - lapply (transitive_le 1 … Hdee2) // #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2 - lapply (transitive_le (1 + e) … Hdee2) // #Hee2 - @ldrop_ldrop_lt >minus_minus_comm /3 width=1/ (**) (* explicit constructor *) -] -qed. - -(* Note: apparently this was missing in basic_1 *) -theorem ldrop_conf_be: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → - ∃∃L. ⇩[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L. -#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq … He2) -He2 #H destruct - lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ -| normalize #L0 #K0 #I #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HL20 - [ -IHLK0 -He21 destruct plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - minus_le_minus_minus_comm // /3 width=3/ - ] -] -qed. - -(* Basic_1: was: drop_trans_ge *) -theorem ldrop_trans_ge: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L → - ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2. -#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L -[ #d #e #e2 #L2 #H - >(ldrop_inv_atom1 … H) -H -L2 // -| // -| /3 width=1/ -| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2 - lapply (lt_to_le_to_lt 0 … Hde2) // #He2 - lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HL2 - @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2 width=1/ (**) (* explicit constructor *) -] -qed. - -(* Basic_1: was: drop_trans_le *) -theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L → - ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 → - ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2. -#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L -[ #d #e #e2 #L2 #H - >(ldrop_inv_atom1 … H) -L2 /2 width=3/ -| #K #I #V #e2 #L2 #HL2 #H - lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/ -| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H - lapply (le_n_O_to_eq … H) -H #H destruct - elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0 - lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d - elim (ldrop_inv_O1 … H) -H * - [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/ - | -HL12 -HV12 #He2 #HL2 - elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ] - ] -] -qed. - -(* Basic_1: was: drop_conf_rev *) -axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L → - ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1. - -(* Basic_1: was: drop_conf_lt *) -lemma ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 → - ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 → - e2 < d1 → let d ≝ d1 - e2 - 1 in - ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 & - ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2. -#d1 #e1 #L #L1 #H1 #e2 #K2 #I #V2 #H2 #He2d1 -elim (ldrop_conf_le … H1 … H2 ?) -L [2: /2 width=2/] #K #HL1K #HK2 -elim (ldrop_inv_skip1 … HK2 ?) -HK2 [2: /2 width=1/] #K1 #V1 #HK21 #HV12 #H destruct /2 width=5/ -qed. - -lemma ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L. - ⇩[d1, e1] L1 ≡ L → ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → - ⇩[0, e2 + e1] L1 ≡ L2. -#e1 #e1 #e2 >commutative_plus /2 width=5/ -qed. - -lemma ldrop_conf_div: ∀I1,L,K,V1,e1. ⇩[0, e1] L ≡ K. ⓑ{I1} V1 → - ∀I2,V2,e2. ⇩[0, e2] L ≡ K. ⓑ{I2} V2 → - ∧∧ e1 = e2 & I1 = I2 & V1 = V2. -#I1 #L #K #V1 #e1 #HLK1 #I2 #V2 #e2 #HLK2 -elim (le_or_ge e1 e2) #He -[ lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -| lapply (ldrop_conf_ge … HLK2 … HLK1 ?) -] -HLK1 -HLK2 // #HK -lapply (ldrop_fwd_O1_length … HK) #H -elim (discr_minus_x_xy … H) -H -[1,3: normalize H in HK; #HK -lapply (ldrop_inv_refl … HK) -HK #H destruct -lapply (inv_eq_minus_O … H) -H /3 width=1/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_lpx.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_lpx.ma deleted file mode 100644 index 2605b921c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_lpx.ma +++ /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/substitution/ldrop.ma". - -(* DROPPING *****************************************************************) - -(* Properties on pointwise extension ****************************************) - -lemma lpx_deliftable_dropable: ∀R. t_deliftable_sn R → dropable_sn (lpx R). -#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e -[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/ -| #K1 #I #V1 #X #H - elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ -| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H - elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (IHLK1 … HL12) -L1 /3 width=3/ -| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H - elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct - elim (HR … HV12 … HWV1) -V1 - elim (IHLK1 … HL12) -L1 /3 width=5/ -] -qed. - -lemma lpx_liftable_dedropable: ∀R. reflexive ? R → - t_liftable R → dedropable_sn (lpx R). -#R #H1R #H2R #L1 #K1 #d #e #H elim H -L1 -K1 -d -e -[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/ -| #K1 #I #V1 #X #H - elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/ -| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 - elim (IHLK1 … HK12) -K1 /3 width=5/ -| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H - elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct - elim (lift_total W2 d e) #V2 #HWV2 - lapply (H2R … HW12 … HWV1 … HWV2) -W1 - elim (IHLK1 … HK12) -K1 /3 width=5/ -] -qed. - -fact lpx_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx R L1 L2 → - d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx R K1 K2. -#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e -[ #d #e #X #H >(lpx_inv_atom2 … H) -H /2 width=3/ -| #K2 #I #V2 #X #H - elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/ -| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_ - elim (lpx_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct - elim (IHLK2 … HL12 ?) -L2 // /3 width=3/ -| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_ - >commutative_plus normalize #H destruct -] -qed. - -lemma ltpr_dropable: ∀R. dropable_dx (lpx R). -/2 width=5/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma deleted file mode 100644 index 78a15d70d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma +++ /dev/null @@ -1,92 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/lsubs_sfr.ma". -include "basic_2/substitution/ldrop_ldrop.ma". - -(* DROPPING *****************************************************************) - -(* Inversion lemmas about local env. full refinement for substitution *******) - -(* Note: ldrop_ldrop not needed *) -lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≽ [d, e] L → - d ≤ i → i < d + e → I = Abbr. -#I #L elim L -L -[ #K #V #i #H - lapply (ldrop_inv_atom1 … H) -H #H destruct -| #L #J #W #IHL #K #V #i #H - elim (ldrop_inv_O1 … H) -H * - [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct - lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct - lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H - elim (lsubs_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct // - | #Hi #HLK #d @(nat_ind_plus … d) -d - [ #e #H #_ #Hide - elim (sfr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct - @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/ - | #d #_ #e #H #Hdi #Hide - lapply (sfr_inv_skip … H ?) -H // #HL - @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/ - ] - ] -] -qed-. - -(* Properties about local env. full refinement for substitution *************) - -(* Note: ldrop_ldrop not needed *) -lemma sfr_ldrop: ∀L,d,e. - (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) → - ≽ [d, e] L. -#L elim L -L // -#L #I #V #IHL #d @(nat_ind_plus … d) -d -[ #e @(nat_ind_plus … e) -e // - #e #_ #HH - >(HH I L V 0 ? ? ?) // /5 width=6/ -| /5 width=6/ -] -qed. - -lemma sfr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 → - dd + ee ≤ d → ≽ [dd, ee] L2. -#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee -@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2 -lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid -elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H -elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct -@(sfr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 // -qed. - -lemma sfr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → - ∀dd,ee. ≽ [dd, ee] L1 → - dd ≤ d + e → d + e ≤ dd + ee → - ≽ [d, dd + ee - d - e] L2. -#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee -@sfr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2 -lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie ->commutative_plus in Hiddee; >minus_minus_comm commutative_plus // -Hddie /2 width=1/ -qed. - -lemma sfr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 → - d + e ≤ dd → ≽ [dd - e, ee] L2. -#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee -@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2 -elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd ->plus_minus in Hiddee; // #Hiddee -lapply (transitive_le … Hdde Hddi) -Hdde #Hid -lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2 -@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lift.ma deleted file mode 100644 index 36c353ba9..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift.ma +++ /dev/null @@ -1,402 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_weight.ma". -include "basic_2/grammar/term_simple.ma". - -(* BASIC TERM RELOCATION ****************************************************) - -(* Basic_1: includes: - lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat -*) -inductive lift: nat → nat → relation term ≝ -| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k) -| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i) -| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e)) -| lift_gref : ∀p,d,e. lift d e (§p) (§p) -| lift_bind : ∀a,I,V1,V2,T1,T2,d,e. - lift d e V1 V2 → lift (d + 1) e T1 T2 → - lift d e (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) -| lift_flat : ∀I,V1,V2,T1,T2,d,e. - lift d e V1 V2 → lift d e T1 T2 → - lift d e (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) -. - -interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2). - -definition t_liftable: relation term → Prop ≝ - λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 → - ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2. - -definition t_deliftable_sn: relation term → Prop ≝ - λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → - ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2. - -(* Basic inversion lemmas ***************************************************) - -fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2. -#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/ -qed. - -lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2. -/2 width=4/ qed-. - -fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k. -#d #e #T1 #T2 * -d -e -T1 -T2 // -[ #i #d #e #_ #k #H destruct -| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -] -qed. - -lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k. -/2 width=5/ qed-. - -fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i → - (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). -#d #e #T1 #T2 * -d -e -T1 -T2 -[ #k #d #e #i #H destruct -| #j #d #e #Hj #i #Hi destruct /3 width=1/ -| #j #d #e #Hj #i #Hi destruct /3 width=1/ -| #p #d #e #i #H destruct -| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct -] -qed. - -lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → - (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). -/2 width=3/ qed-. - -lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i. -#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // -#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd -elim (lt_refl_false … Hdd) -qed-. - -lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e). -#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // -#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd -elim (lt_refl_false … Hdd) -qed-. - -fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p. -#d #e #T1 #T2 * -d -e -T1 -T2 // -[ #i #d #e #_ #k #H destruct -| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -] -qed. - -lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p. -/2 width=5/ qed-. - -fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → - ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 → - ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 & - T2 = ⓑ{a,I} V2. U2. -#d #e #T1 #T2 * -d -e -T1 -T2 -[ #k #d #e #a #I #V1 #U1 #H destruct -| #i #d #e #_ #a #I #V1 #U1 #H destruct -| #i #d #e #_ #a #I #V1 #U1 #H destruct -| #p #d #e #a #I #V1 #U1 #H destruct -| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/ -| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct -] -qed. - -lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⇧[d,e] ⓑ{a,I} V1. U1 ≡ T2 → - ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 & - T2 = ⓑ{a,I} V2. U2. -/2 width=3/ qed-. - -fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → - ∀I,V1,U1. T1 = ⓕ{I} V1.U1 → - ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 & - T2 = ⓕ{I} V2. U2. -#d #e #T1 #T2 * -d -e -T1 -T2 -[ #k #d #e #I #V1 #U1 #H destruct -| #i #d #e #_ #I #V1 #U1 #H destruct -| #i #d #e #_ #I #V1 #U1 #H destruct -| #p #d #e #I #V1 #U1 #H destruct -| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct -| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/ -] -qed. - -lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] ⓕ{I} V1. U1 ≡ T2 → - ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 & - T2 = ⓕ{I} V2. U2. -/2 width=3/ qed-. - -fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k. -#d #e #T1 #T2 * -d -e -T1 -T2 // -[ #i #d #e #_ #k #H destruct -| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -] -qed. - -(* Basic_1: was: lift_gen_sort *) -lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k. -/2 width=5/ qed-. - -fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i → - (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). -#d #e #T1 #T2 * -d -e -T1 -T2 -[ #k #d #e #i #H destruct -| #j #d #e #Hj #i #Hi destruct /3 width=1/ -| #j #d #e #Hj #i #Hi destruct (plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/ -qed. - -lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i. -/2 width=1/ qed-. - -(* Basic_1: was: lift_r *) -lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T. -#T elim T -T -[ * #i // #d elim (lt_or_ge i d) /2 width=1/ -| * /2 width=1/ -] -qed. - -lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2. -#T1 elim T1 -T1 -[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/ -| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e - elim (IHV1 d e) -IHV1 #V2 #HV12 - [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/ - | elim (IHT1 d e) -IHT1 /3 width=2/ - ] -] -qed. - -(* Basic_1: was: lift_free (right to left) *) -lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 → - ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 → - ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2. -#d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2 -[ /3 width=3/ -| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_ - lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/ -| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12 - lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21 - >(plus_minus_m_m e2 e1 ?) // /3 width=3/ -| /3 width=3/ -| #a #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 - elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b - elim (IHT (d2+1) … ? ? He12) /2 width=1/ /3 width=5/ -| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 - elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b - elim (IHT d2 … ? ? He12) // /3 width=5/ -] -qed. - -(* Basic_1: was only: dnf_dec2 dnf_dec *) -lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2). -#T1 elim T1 -T1 -[ * [1,3: /3 width=2/ ] #i #d #e - elim (lt_dec i d) #Hid - [ /4 width=2/ - | lapply (false_lt_to_le … Hid) -Hid #Hid - elim (lt_dec i (d + e)) #Hide - [ @or_intror * #T1 #H - elim (lift_inv_lref2_be … H Hid Hide) - | lapply (false_lt_to_le … Hide) -Hide /4 width=2/ - ] - ] -| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e - [ elim (IHV2 d e) -IHV2 - [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2 - [ * #T1 #HT12 @or_introl /3 width=2/ - | -V1 #HT2 @or_intror * #X #H - elim (lift_inv_bind2 … H) -H /3 width=2/ - ] - | -IHT2 #HV2 @or_intror * #X #H - elim (lift_inv_bind2 … H) -H /3 width=2/ - ] - | elim (IHV2 d e) -IHV2 - [ * #V1 #HV12 elim (IHT2 d e) -IHT2 - [ * #T1 #HT12 /4 width=2/ - | -V1 #HT2 @or_intror * #X #H - elim (lift_inv_flat2 … H) -H /3 width=2/ - ] - | -IHT2 #HV2 @or_intror * #X #H - elim (lift_inv_flat2 … H) -H /3 width=2/ - ] - ] -] -qed. - -lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R). -#R #HR #T1 #T2 #H elim H -T2 -[ /3 width=7/ -| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2 - elim (lift_total T d e) /3 width=9/ -] -qed. - -lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R). -#R #HR #U1 #U2 #H elim H -U2 -[ #U2 #HU12 #T1 #d #e #HTU1 - elim (HR … HU12 … HTU1) -U1 /3 width=3/ -| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1 - elim (IHU1 … HTU1) -U1 #T #HTU #HT1 - elim (HR … HU2 … HTU) -U /3 width=5/ -] -qed-. - -(* Basic_1: removed theorems 7: - lift_head lift_gen_head - lift_weight_map lift_weight lift_weight_add lift_weight_add_O - lift_tlt_dx -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma deleted file mode 100644 index 3e18bff32..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma +++ /dev/null @@ -1,217 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/lift.ma". - -(* BASIC TERM RELOCATION ****************************************************) - -(* Main properies ***********************************************************) - -(* Basic_1: was: lift_inj *) -theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2. -#d #e #T1 #U #H elim H -d -e -T1 -U -[ #k #d #e #X #HX - lapply (lift_inv_sort2 … HX) -HX // -| #i #d #e #Hid #X #HX - lapply (lift_inv_lref2_lt … HX ?) -HX // -| #i #d #e #Hdi #X #HX - lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/ -| #p #d #e #X #HX - lapply (lift_inv_gref2 … HX) -HX // -| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ -] -qed-. - -(* Basic_1: was: lift_gen_lift *) -theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T → - d1 ≤ d2 → - ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1. -#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T -[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 - lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/ -| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12 - lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 - lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/ -| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12 - elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct - [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/ - | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H - elim (le_inv_plus_l … H) -H #Hide2 #He2i - lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12 - >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i - /4 width=3/ - ] -| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 - lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/ -| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 - lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct - elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1 - >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /3 width=5/ -| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 - lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct - elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1 - elim (IHU … HU2 ?) // /3 width=5/ -] -qed. - -(* Note: apparently this was missing in basic_1 *) -theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → - ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T → - e ≤ e1 → e1 ≤ e + e2 → - ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1. -#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T -[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/ -| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2 - >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ] -| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2 - elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2 - [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/ - | >(lift_inv_lref2_ge … H ?) -H // - lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i - elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1 - @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12 - @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ] - ] -| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/ -| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2 - elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct - elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2 - elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/ -| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2 - elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct - elim (IHV1 … HV2 ? ?) -V // - elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/ -] -qed. - -theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2. -#d #e #T #U1 #H elim H -d -e -T -U1 -[ #k #d #e #X #HX - lapply (lift_inv_sort1 … HX) -HX // -| #i #d #e #Hid #X #HX - lapply (lift_inv_lref1_lt … HX ?) -HX // -| #i #d #e #Hdi #X #HX - lapply (lift_inv_lref1_ge … HX ?) -HX // -| #p #d #e #X #HX - lapply (lift_inv_gref1 … HX) -HX // -| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ -] -qed-. - -(* Basic_1: was: lift_free (left to right) *) -theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → - d1 ≤ d2 → d2 ≤ d1 + e1 → ⇧[d1, e1 + e2] T1 ≡ T2. -#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T -[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ - >(lift_inv_sort1 … HT2) -HT2 // -| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_ - lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 - lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1/ -| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21 - lapply (lift_inv_lref1_ge … HT2 ?) -HT2 - [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/ - | -Hd21 /2 width=1/ - ] -| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ - >(lift_inv_gref1 … HT2) -HT2 // -| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 - elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct - lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10 - lapply (IHT12 … HT20 ? ?) /2 width=1/ -| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 - elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct - lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10 - lapply (IHT12 … HT20 ? ?) // /2 width=1/ -] -qed. - -(* Basic_1: was: lift_d (right to left) *) -theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d2 ≤ d1 → - ∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T0 ≡ T2. -#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T -[ #k #d1 #e1 #d2 #e2 #X #HX #_ - >(lift_inv_sort1 … HX) -HX /2 width=3/ -| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ - lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2 - elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /3 width=3/ /4 width=3/ -| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21 - lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2 - lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct - >plus_plus_comm_23 /4 width=3/ -| #p #d1 #e1 #d2 #e2 #X #HX #_ - >(lift_inv_gref1 … HX) -HX /2 width=3/ -| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 - elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct - elim (IHV12 … HV20 ?) -IHV12 -HV20 // - elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /3 width=5/ -| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 - elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct - elim (IHV12 … HV20 ?) -IHV12 -HV20 // - elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/ -] -qed. - -(* Basic_1: was: lift_d (left to right) *) -theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → - ∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[d1, e1] T0 ≡ T2. -#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T -[ #k #d1 #e1 #d2 #e2 #X #HX #_ - >(lift_inv_sort1 … HX) -HX /2 width=3/ -| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded - lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e - lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1/ #Hid2e - lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2 - lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/ -| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ - elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/ -| #p #d1 #e1 #d2 #e2 #X #HX #_ - >(lift_inv_gref1 … HX) -HX /2 width=3/ -| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded - elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct - elim (IHV12 … HV20 ?) -IHV12 -HV20 // - elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T - (lift_mono … H … HT1) -T // -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift_vector.ma deleted file mode 100644 index cdc11129d..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift_vector.ma +++ /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/substitution/lift_lift.ma". -include "basic_2/substitution/lift_vector.ma". - -(* BASIC TERM VECTOR RELOCATION *********************************************) - -(* Main properies ***********************************************************) - -theorem liftv_mono: ∀Ts,U1s,d,e. ⇧[d,e] Ts ≡ U1s → - ∀U2s:list term. ⇧[d,e] Ts ≡ U2s → U1s = U2s. -#Ts #U1s #d #e #H elim H -Ts -U1s -[ #U2s #H >(liftv_inv_nil1 … H) -H // -| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct - elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct - >(lift_mono … HTU1 … HTU2) -T /3 width=1/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_vector.ma deleted file mode 100644 index 35ecb6535..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_vector.ma +++ /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/grammar/term_vector.ma". -include "basic_2/substitution/lift.ma". - -(* BASIC TERM VECTOR RELOCATION *********************************************) - -inductive liftv (d,e:nat) : relation (list term) ≝ -| liftv_nil : liftv d e ◊ ◊ -| liftv_cons: ∀T1s,T2s,T1,T2. - ⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s → - liftv d e (T1 @ T1s) (T2 @ T2s) -. - -interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s). - -(* Basic inversion lemmas ***************************************************) - -fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊. -#T1s #T2s #d #e * -T1s -T2s // -#T1s #T2s #T1 #T2 #_ #_ #H destruct -qed. - -lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊. -/2 width=5/ qed-. - -fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → - ∀U1,U1s. T1s = U1 @ U1s → - ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s & - T2s = U2 @ U2s. -#T1s #T2s #d #e * -T1s -T2s -[ #U1 #U1s #H destruct -| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5/ -] -qed. - -lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 @ U1s ≡ T2s → - ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s & - T2s = U2 @ U2s. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s. -#d #e #T1s elim T1s -T1s -[ /2 width=2/ -| #T1 #T1s * #T2s #HT12s - elim (lift_total T1 d e) /3 width=2/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lsubs.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lsubs.ma deleted file mode 100644 index f27883b02..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lsubs.ma +++ /dev/null @@ -1,194 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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". - -(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************) - -inductive lsubs: nat → nat → relation lenv ≝ -| lsubs_sort: ∀d,e. lsubs d e (⋆) (⋆) -| lsubs_OO: ∀L1,L2. lsubs 0 0 L1 L2 -| lsubs_abbr: ∀L1,L2,V,e. lsubs 0 e L1 L2 → - lsubs 0 (e + 1) (L1. ⓓV) (L2.ⓓV) -| lsubs_abst: ∀L1,L2,I,V1,V2,e. lsubs 0 e L1 L2 → - lsubs 0 (e + 1) (L1. ⓑ{I}V1) (L2. ⓛV2) -| lsubs_skip: ∀L1,L2,I1,I2,V1,V2,d,e. - lsubs d e L1 L2 → lsubs (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2) -. - -interpretation - "local environment refinement (substitution)" - 'SubEq L1 d e L2 = (lsubs d e L1 L2). - -definition lsubs_trans: ∀S. (lenv → relation S) → Prop ≝ λS,R. - ∀L2,s1,s2. R L2 s1 s2 → - ∀L1,d,e. L1 ≼ [d, e] L2 → R L1 s1 s2. - -(* Basic properties *********************************************************) - -lemma lsubs_bind_eq: ∀L1,L2,e. L1 ≼ [0, e] L2 → ∀I,V. - L1. ⓑ{I} V ≼ [0, e + 1] L2.ⓑ{I} V. -#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/ -qed. - -lemma lsubs_abbr_lt: ∀L1,L2,V,e. L1 ≼ [0, e - 1] L2 → 0 < e → - L1. ⓓV ≼ [0, e] L2.ⓓV. -#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ -qed. - -lemma lsubs_abst_lt: ∀L1,L2,I,V1,V2,e. L1 ≼ [0, e - 1] L2 → 0 < e → - L1. ⓑ{I}V1 ≼ [0, e] L2. ⓛV2. -#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ -qed. - -lemma lsubs_skip_lt: ∀L1,L2,d,e. L1 ≼ [d - 1, e] L2 → 0 < d → - ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 ≼ [d, e] L2. ⓑ{I2} V2. -#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/ -qed. - -lemma lsubs_bind_lt: ∀I,L1,L2,V,e. L1 ≼ [0, e - 1] L2 → 0 < e → - L1. ⓓV ≼ [0, e] L2. ⓑ{I}V. -* /2 width=1/ qed. - -lemma lsubs_refl: ∀d,e,L. L ≼ [d, e] L. -#d elim d -d -[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/ -| #d #IHd #e #L elim L -L // /2 width=1/ -] -qed. - -lemma TC_lsubs_trans: ∀S,R. lsubs_trans S R → lsubs_trans S (λL. (TC … (R L))). -#S #R #HR #L1 #s1 #s2 #H elim H -s2 -[ /3 width=5/ -| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12 - lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/ -] -qed. - -(* Basic inversion lemmas ***************************************************) - -fact lsubs_inv_atom1_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → L1 = ⋆ → - L2 = ⋆ ∨ (d = 0 ∧ e = 0). -#L1 #L2 #d #e * -L1 -L2 -d -e -[ /2 width=1/ -| /3 width=1/ -| #L1 #L2 #W #e #_ #H destruct -| #L1 #L2 #I #W1 #W2 #e #_ #H destruct -| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct -] -qed. - -lemma lsubs_inv_atom1: ∀L2,d,e. ⋆ ≼ [d, e] L2 → - L2 = ⋆ ∨ (d = 0 ∧ e = 0). -/2 width=3/ qed-. - -fact lsubs_inv_skip1_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → - ∀I1,K1,V1. L1 = K1.ⓑ{I1}V1 → 0 < d → - ∃∃I2,K2,V2. K1 ≼ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2. -#L1 #L2 #d #e * -L1 -L2 -d -e -[ #d #e #I1 #K1 #V1 #H destruct -| #L1 #L2 #I1 #K1 #V1 #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ -] -qed. - -lemma lsubs_inv_skip1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ≼ [d, e] L2 → 0 < d → - ∃∃I2,K2,V2. K1 ≼ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2. -/2 width=5/ qed-. - -fact lsubs_inv_atom2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → L2 = ⋆ → - L1 = ⋆ ∨ (d = 0 ∧ e = 0). -#L1 #L2 #d #e * -L1 -L2 -d -e -[ /2 width=1/ -| /3 width=1/ -| #L1 #L2 #W #e #_ #H destruct -| #L1 #L2 #I #W1 #W2 #e #_ #H destruct -| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct -] -qed. - -lemma lsubs_inv_atom2: ∀L1,d,e. L1 ≼ [d, e] ⋆ → - L1 = ⋆ ∨ (d = 0 ∧ e = 0). -/2 width=3/ qed-. - -fact lsubs_inv_abbr2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → - ∀K2,V. L2 = K2.ⓓV → d = 0 → 0 < e → - ∃∃K1. K1 ≼ [0, e - 1] K2 & L1 = K1.ⓓV. -#L1 #L2 #d #e * -L1 -L2 -d -e -[ #d #e #K1 #V #H destruct -| #L1 #L2 #K1 #V #_ #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #W #e #HL12 #K1 #V #H #_ #_ destruct /2 width=3/ -| #L1 #L2 #I #W1 #W2 #e #_ #K1 #V #H destruct -| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #K1 #V #_ >commutative_plus normalize #H destruct -] -qed. - -lemma lsubs_inv_abbr2: ∀L1,K2,V,e. L1 ≼ [0, e] K2.ⓓV → 0 < e → - ∃∃K1. K1 ≼ [0, e - 1] K2 & L1 = K1.ⓓV. -/2 width=5/ qed-. - -fact lsubs_inv_skip2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → - ∀I2,K2,V2. L2 = K2.ⓑ{I2}V2 → 0 < d → - ∃∃I1,K1,V1. K1 ≼ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1. -#L1 #L2 #d #e * -L1 -L2 -d -e -[ #d #e #I1 #K1 #V1 #H destruct -| #L1 #L2 #I1 #K1 #V1 #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ -] -qed. - -lemma lsubs_inv_skip2: ∀I2,L1,K2,V2,d,e. L1 ≼ [d, e] K2.ⓑ{I2}V2 → 0 < d → - ∃∃I1,K1,V1. K1 ≼ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1. -/2 width=5/ qed-. - -(* Basic forward lemmas *****************************************************) - -fact lsubs_fwd_length_full1_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → - d = 0 → e = |L1| → |L1| ≤ |L2|. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize -[ // -| /2 width=1/ -| /3 width=1/ -| /3 width=1/ -| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma lsubs_fwd_length_full1: ∀L1,L2. L1 ≼ [0, |L1|] L2 → |L1| ≤ |L2|. -/2 width=5/ qed-. - -fact lsubs_fwd_length_full2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → - d = 0 → e = |L2| → |L2| ≤ |L1|. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize -[ // -| /2 width=1/ -| /3 width=1/ -| /3 width=1/ -| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma lsubs_fwd_length_full2: ∀L1,L2. L1 ≼ [0, |L2|] L2 → |L2| ≤ |L1|. -/2 width=5/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lsubs_sfr.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lsubs_sfr.ma deleted file mode 100644 index b71f25e51..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lsubs_sfr.ma +++ /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/substitution/lsubs.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************) - -(* bottom element of the refinement *) -definition sfr: nat → nat → predicate lenv ≝ - λd,e. NF_sn … (lsubs d e) (lsubs d e …). - -interpretation - "local environment full refinement (substitution)" - 'SubEqBottom d e L = (sfr d e L). - -(* Basic properties *********************************************************) - -lemma sfr_atom: ∀d,e. ≽ [d, e] ⋆. -#d #e #L #H -elim (lsubs_inv_atom2 … H) -H -[ #H destruct // -| * #H1 #H2 destruct // -] -qed. - -lemma sfr_OO: ∀L. ≽ [0, 0] L. -// qed. - -lemma sfr_abbr: ∀L,V,e. ≽ [0, e] L → ≽ [0, e + 1] L.ⓓV. -#L #V #e #HL #K #H -elim (lsubs_inv_abbr2 … H ?) -H // (plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/ - | -Hdi -Hdj #Hid - generalize in match Hide; -Hide (**) (* rewriting in the premises, rewrites in the goal too *) - >(plus_minus_m_m … Hjde) in ⊢ (% → ?); -Hjde /4 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_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/ -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide - elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 // - -Hdi -Hide /3 width=5/ -] -qed. - -lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → - ∀i. d ≤ i → i ≤ d + e → - ∃∃T. L ⊢ T1 ▶ [i, d + e - i] T & - L ⊢ T ▶ [d, i - d] T2. -#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e -[ /2 width=3/ -| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde - elim (lt_or_ge i j) - [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /4 width=4/ - | -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_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/ -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide - elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 // - -Hdi -Hide /3 width=5/ -] -qed. - -lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 → - ∀L. L @@ K ⊢ T1 ▶ [d, e] T2. -#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/ -#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L -lapply (ldrop_fwd_ldrop2_length … HK0) #H -@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *) -@(ldrop_O1_append_sn_le … HK0) /2 width=2/ -qed. - -(* Basic inversion lemmas ***************************************************) - -fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} → - T2 = ⓪{I} ∨ - ∃∃K,V,i. d ≤ i & i < d + e & - ⇩[O, i] L ≡ K. ⓓV & - ⇧[O, i + 1] V ≡ T2 & - I = LRef i. -#L #T1 #T2 #d #e * -L -T1 -T2 -d -e -[ #L #I #d #e #J #H destruct /2 width=1/ -| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/ -| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct -] -qed. - -lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶ [d, e] T2 → - T2 = ⓪{I} ∨ - ∃∃K,V,i. d ≤ i & i < d + e & - ⇩[O, i] L ≡ K. ⓓV & - ⇧[O, i + 1] V ≡ T2 & - I = LRef i. -/2 width=3/ qed-. - - -(* Basic_1: was: subst1_gen_sort *) -lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶ [d, e] T2 → T2 = ⋆k. -#L #T2 #k #d #e #H -elim (tps_inv_atom1 … H) -H // -* #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -(* Basic_1: was: subst1_gen_lref *) -lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶ [d, e] T2 → - T2 = #i ∨ - ∃∃K,V. d ≤ i & i < d + e & - ⇩[O, i] L ≡ K. ⓓV & - ⇧[O, i + 1] V ≡ T2. -#L #T2 #i #d #e #H -elim (tps_inv_atom1 … H) -H /2 width=1/ -* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/ -qed-. - -lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶ [d, e] T2 → T2 = §p. -#L #T2 #p #d #e #H -elim (tps_inv_atom1 … H) -H // -* #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 → - ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → - ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & - L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 & - U2 = ⓑ{a,I} V2. T2. -#d #e #L #U1 #U2 * -d -e -L -U1 -U2 -[ #L #k #d #e #a #I #V1 #T1 #H destruct -| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct -| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/ -| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct -] -qed. - -lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶ [d, e] U2 → - ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & - L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 & - U2 = ⓑ{a,I} V2. T2. -/2 width=3/ qed-. - -fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 → - ∀I,V1,T1. U1 = ⓕ{I} V1. T1 → - ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 & - U2 = ⓕ{I} V2. T2. -#d #e #L #U1 #U2 * -d -e -L -U1 -U2 -[ #L #k #d #e #I #V1 #T1 #H destruct -| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct -| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct -| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/ -] -qed. - -lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶ [d, e] U2 → - ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 & - U2 = ⓕ{I} V2. T2. -/2 width=3/ qed-. - -fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → e = 0 → T1 = T2. -#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e -[ // -| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct - lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide shift_append_assoc normalize #H - elim (tps_inv_bind1 … H) -H - #V0 #T0 #_ #HT10 #H destruct - elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) -] -qed-. - -(* Basic_1: removed theorems 25: - subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt - subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans - subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s - subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt - subst0_confluence_neq subst0_confluence_eq subst0_tlt_head - subst0_confluence_lift subst0_tlt - subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma deleted file mode 100644 index 5ffc94922..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_lift.ma +++ /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: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 → - ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2. -#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1 -[ // -| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct - elim (lt_or_ge i (d+1)) #HiSd - [ -Hide1 -HV0 - lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct - lapply (ldrop_mono … HLK0 … HLK) #H destruct - | -V -Hdi /2 width=4/ - ] -| /4 width=3/ -| /3 width=3/ -] -qed. - -lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 → - ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2. -/2 width=3/ qed-. - -lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 → - ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2. -#L #T1 #T2 #d #HT12 #K #V #HLK -lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12 -lapply (tps_inv_refl_O2 … HT12) -HT12 // -qed-. - -(* Relocation properties ****************************************************) - -(* Basic_1: was: subst1_lift_lt *) -lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 → - ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K → - ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 → - dt + et ≤ d → - L ⊢ U1 ▶ [dt, et] U2. -#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et -[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ - >(lift_mono … H1 … H2) -H1 -H2 // -| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd - lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid - lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct - elim (lift_trans_ge … HVW … HWU2 ?) -W // (lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/ -| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd - elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *) -| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd - elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ -] -qed. - -lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 → - ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K → - ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 → - dt ≤ d → d ≤ dt + et → - L ⊢ U1 ▶ [dt, et + e] U2. -#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et -[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_ - >(lift_mono … H1 … H2) -H1 -H2 // -| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_ - elim (lift_inv_lref1 … H) -H * #Hid #H destruct - [ -Hdtd - lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete - elim (lift_trans_ge … HVW … HWU2 ?) -W // (lift_mono … HVY … HVW) -V #H destruct /2 width=4/ - | -Hdti - lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti - lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/ - ] -| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet - elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ] - ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *) -| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd - elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ -] -qed. - -(* Basic_1: was: subst1_lift_ge *) -lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 → - ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K → - ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 → - d ≤ dt → - L ⊢ U1 ▶ [dt + e, et] U2. -#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et -[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ - >(lift_mono … H1 … H2) -H1 -H2 // -| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt - lapply (transitive_le … Hddt … Hdti) -Hddt #Hid - lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct - lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 - lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/ -| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt - elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *) -| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt - elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/ -] -qed. - -(* Basic_1: was: subst1_gen_lift_lt *) -lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt + et ≤ d → - ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et -[ #L * #i #dt #et #K #d #e #_ #T1 #H #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/ - | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/ - ] -| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd - lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid - lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct - elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV - elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus 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 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct - elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2 - elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *) - /3 width=5/ -| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet - elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct - elim (IHV12 … HLK … HWV1 ? ?) -V1 // - elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/ -] -qed. - -(* Basic_1: was: subst1_gen_lift_ge *) -lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d + e ≤ dt → - ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et -[ #L * #i #dt #et #K #d #e #_ #T1 #H #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/ - | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/ - | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/ - ] -| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt - lapply (transitive_le … Hdedt … Hdti) #Hdei - elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt - elim (le_inv_plus_l … Hdei) #Hdie #Hei - lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct - lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV - elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie - #V0 #HV10 >plus_minus // plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *) -| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd - elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct - elim (le_inv_plus_l … Hdetd) #_ #Hedt - elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2 - elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ] - IHV12 // >IHT12 // -| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct - >IHV12 // >IHT12 // -] -qed. -(* - Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?) - (subst0 i v t1 (lift h d u2)) -> - (le (plus d h) i) -> - (EX u1 | (subst0 (minus i h) v u1 u2) & - t1 = (lift h d u1) - ). - - - Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?) - (subst0 i v t1 (lift h d u2)) -> - (le d i) -> (lt i (plus d h)) -> - (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)). -*) -lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → - ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet -elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 -lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 -lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU -lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus (lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/ - ] -| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX - elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02 - elim (IHV01 … HV02) -V0 #V #HV1 #HV2 - elim (IHT01 … HT02) -T0 #T #HT1 #HT2 - lapply (tps_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubs_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/ -| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX - elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02) -V0 - elim (IHT01 … HT02) -T0 /3 width=5/ -] -qed. - -(* Basic_1: was: subst1_confluence_neq *) -theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 → - ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 → - (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) → - ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T. -#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1 -[ /2 width=3/ -| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2 - elim (tps_inv_lref1 … H1) -H1 - [ #H destruct /4 width=4/ - | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded - [ -Hd1 -Hde2 - lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H - lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H - elim (lt_refl_false … H) - | -Hd2 -Hde1 - lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H - lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H - elim (lt_refl_false … H) - ] - ] -| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H - elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2 - elim (IHT01 … HT02 ?) -T0 - [ -H #T #HT1 #HT2 - lapply (tps_lsubs_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubs_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/ - | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H - [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *) - ] -| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H - elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02 H) -V0 - elim (IHT01 … HT02 H) -T0 -H /3 width=5/ -] -qed. - -(* Note: the constant 1 comes from tps_subst *) -(* Basic_1: was: subst1_trans *) -theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 → - ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e → - L ⊢ T1 ▶ [d, e] T2. -#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e -[ #L #I #d #e #T2 #H #He - elim (tps_inv_atom1 … H) -H - [ #H destruct // - | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct - lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/ - ] -| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He - lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2 - <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/ -| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He - elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct - lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02 - lapply (IHT10 … HT02 He) -T0 #HT12 - lapply (tps_lsubs_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/ -| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He - elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/ -] -qed. - -theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 → - ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 → - ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2. -#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1 -[ /2 width=3/ -| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1 - lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1 - lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2 - <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4 width=4/ -| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 - elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02 - elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V - elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2 - lapply (tps_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubs_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/ -| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 - elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV10 … HV02 ?) -V0 // - elim (IHT10 … HT02 ?) -T0 // /3 width=6/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift.ma deleted file mode 100644 index e8ac23dae..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift.ma +++ /dev/null @@ -1,108 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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". - -(* INVERSE BASIC TERM RELOCATION *******************************************) - -definition delift: nat → nat → lenv → relation term ≝ - λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T. - -interpretation "inverse basic relocation (term)" - 'TSubst L T1 d e T2 = (delift d e L T1 T2). - -(* Basic properties *********************************************************) - -lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 → - ∀L. L ⊢ ▼*[d, e] T2 ≡ T1. -/2 width=3/ qed. - -lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T. -/2 width=3/ qed. - -lemma delift_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 → - ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2. -#L1 #T1 #T2 #d #e * /3 width=3/ -qed. - -lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k. -/2 width=3/ qed. - -lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i. -/3 width=3/ qed. - -lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e). -/3 width=3/ qed. - -lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p. -/2 width=3/ qed. - -lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e. - L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 → - L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2. -#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2 -lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/ -qed. - -lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e. - L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 → - L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2. -#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/ -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k. -#L #U2 #d #e #k * #U #HU ->(tpss_inv_sort1 … HU) -HU #HU2 ->(lift_inv_sort2 … HU2) -HU2 // -qed-. - -lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p. -#L #U #d #e #p * #U #HU ->(tpss_inv_gref1 … HU) -HU #HU2 ->(lift_inv_gref2 … HU2) -HU2 // -qed-. - -lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 → - ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 & - L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 & - U2 = ⓑ{a,I} V2. T2. -#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 -elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct -elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2 -lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ -qed-. - -lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 → - ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 & - L ⊢ ▼*[d, e] T1 ≡ T2 & - U2 = ⓕ{I} V2. T2. -#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 -elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct -elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/ -qed-. - -lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2. -#L #T1 #T2 #d * #T #HT1 ->(tpss_inv_refl_O2 … HT1) -HT1 #HT2 ->(lift_inv_refl_O2 … HT2) -HT2 // -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → #{T1} ≤ #{T2}. -#L #T1 #T2 #d #e * #T #HT1 #HT2 ->(tw_lift … HT2) -T2 /2 width=4 by tpss_fwd_tw / -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_alt.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_alt.ma deleted file mode 100644 index 9a3eb1b7c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_alt.ma +++ /dev/null @@ -1,100 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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". - -(* INVERSE BASIC TERM RELOCATION *******************************************) - -(* alternative definition of inverse basic term relocation *) -inductive delifta: nat → nat → lenv → relation term ≝ -| delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k) -| delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i) -| delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → - ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 → - ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2 -| delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e)) -| delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p) -| delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e. - delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 → - delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) -| delifta_flat : ∀L,I,V1,V2,T1,T2,d,e. - delifta d e L V1 V2 → delifta d e L T1 T2 → - delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) -. - -interpretation "inverse basic relocation (term) alternative" - 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2). - -(* Basic properties *********************************************************) - -lemma delifta_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 → - ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2. -#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/ -[ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 - elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ -| /4 width=1/ -| /3 width=1/ -] -qed. - -lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2. -#L #T1 @(fw_ind … L T1) -L -T1 #L #T1 elim T1 -T1 -[ * #i #IH #T2 #d #e #H - [ >(delift_inv_sort1 … H) -H // - | elim (delift_inv_lref1 … H) -H * /2 width=1/ - #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2 - lapply (ldrop_pair2_fwd_fw … HLK) #H - lapply (IH … HV12) // -H /2 width=6/ - | >(delift_inv_gref1 … H) -H // - ] -| * [ #a ] #I #V1 #T1 #_ #_ #IH #X #d #e #H - [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12 - lapply (IH … HV12) -HV12 // #HV12 - lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12 - lapply (delifta_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ - | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HV12) -HV12 // - lapply (IH … HT12) -IH -HT12 // /2 width=1/ - ] -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2. -#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/ -qed-. - -lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term. - (∀L,d,e,k. R d e L (⋆k) (⋆k)) → - (∀L,d,e,i. i < d → R d e L (#i) (#i)) → - (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → - ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 → - ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2 - ) → - (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) → - (∀L,d,e,p. R d e L (§p) (§p)) → - (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 → - L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 → - R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) - ) → - (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 → - L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 → - R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) - ) → - ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2. -#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e -// /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_delift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_delift.ma deleted file mode 100644 index a5c563565..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_delift.ma +++ /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: ∀L,T,T1,T2,d,e. - L ⊢ ▼*[d, e] T ≡ T1 → L ⊢ ▼*[d, e] T ≡ T2 → T1 = T2. -#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2 -elim (tpss_conf_eq … H1TU1 … H1TU2) -T #U #HU1 #HU2 -lapply (tpss_inv_lift1_eq … HU1 … H2TU1) -HU1 #H destruct -lapply (tpss_inv_lift1_eq … HU2 … H2TU2) -HU2 #H destruct -lapply (lift_inj … H2TU1 … H2TU2) // -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma deleted file mode 100644 index 01ee6108e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma +++ /dev/null @@ -1,167 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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_sfr.ma". -include "basic_2/unfold/tpss_lift.ma". -include "basic_2/unfold/delift.ma". - -(* INVERSE BASIC TERM RELOCATION *******************************************) - -(* Advanced properties ******************************************************) - -lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e → - ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 → - ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2. -#L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2 -elim (lift_total V 0 (i+1)) #U #HVU -lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus commutative_plus in ⊢ (??%??→?); 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 #d #e #Hde #HL #H destruct - elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12 - elim (IH (L.ⓑ{I}V1) T1 ? ? (d+1) e ? ? ?) -IH [3,6: // |2: skip |4,5: /2 width=1/ ] -Hde -HL #T2 #HT12 - lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/ -| #I #V1 #T1 #d #e #Hde #HL #H destruct - elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12 - elim (IH … T1 … Hde HL ?) -IH -Hde -HL [3,4: // |2: skip ] /3 width=2/ -] -qed. - -lemma sfr_delift: ∀L,T1,d,e. d + e ≤ |L| → ≽ [d, e] L → - ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2. -/2 width=2/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i. -#L #U2 #i #d #e * #U #HU #HU2 #Hid -elim (tpss_inv_lref1 … HU) -HU -[ #H destruct >(lift_inv_lref2_lt … HU2) // -| * #K #V1 #V2 #Hdi - lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi - elim (lt_refl_false … Hi) -] -qed-. - -lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 → - d ≤ i → i < d + e → - ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 & - K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 & - ⇧[0, d] V2 ≡ U2. -#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide -elim (tpss_inv_lref1 … HU) -HU -[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) // -| * #K #V1 #V #_ #_ #HLK #HV1 #HVU - elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/ -] -qed-. - -lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → - d + e ≤ i → U2 = #(i - e). -#L #U2 #i #d #e * #U #HU #HU2 #Hdei -elim (tpss_inv_lref1 … HU) -HU -[ #H destruct >(lift_inv_lref2_ge … HU2) // -| * #K #V1 #V2 #_ #Hide - lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi - elim (lt_refl_false … Hi) -] -qed-. - -lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → - ∨∨ (i < d ∧ U2 = #i) - | (∃∃K,V1,V2. d ≤ i & i < d + e & - ⇩[0, i] L ≡ K. ⓓV1 & - K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 & - ⇧[0, d] V2 ≡ U2 - ) - | (d + e ≤ i ∧ U2 = #(i - e)). -#L #U2 #i #d #e #H -elim (lt_or_ge i d) #Hdi -[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/ -| elim (lt_or_ge i (d+e)) #Hide - [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/ - | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/ - ] -] -qed-. - -(* Properties on basic term relocation **************************************) - -lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 → - ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K → - ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 → - L ⊢ ▼*[dt, et] U1 ≡ U2. -#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2 -elim (lift_total T d e) #U #HTU -lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1 -elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU ->(lift_mono … HTU2 … HT2) -T2 /2 width=3/ -qed. - -lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 → - ∀L,U1,d,e. dt ≤ d → d ≤ dt + et → - ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → - L ⊢ ▼*[dt, et + e] U1 ≡ T2. -#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 -elim (lift_total T d e) #U #HTU -lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1 -lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/ -qed. - -lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 → - ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K → - ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → - L ⊢ ▼*[dt + e, et] U1 ≡ U2. -#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2 -elim (lift_total T d e) #U #HTU -lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1 -elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU ->(lift_mono … HTU2 … HT2) -T2 /2 width=3/ -qed. - -lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 → - ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2. -#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1 -lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct -lapply (lift_inj … HTU1 … HTU2) -U2 // -qed-. - -lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T → - ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e → - L ⊢ ▼*[d, e] T1 ≡ T2. -#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide -lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10 -lapply (lift_trans_be … HT2 … HT0 ? ?) -T // ->commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?); -append_assoc #H -elim (append_inj_dx … H ?) -H // #_ #H destruct -(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/ -| /2 width=5 by lift_frsupp_trans/ -] -qed-. - -(* Advanced inversion lemmas for frsupp **************************************) - -lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥. -#J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 // -#L2 #T2 #H elim (frsup_inv_atom1 … H) -qed-. - -lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ → - ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄. -#b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2 -[ #L2 #T2 #H - elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/ -| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ -] -qed-. - -lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ → - ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄. -#J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2 -[ #L2 #T2 #H - elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/ -| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/frsups_frsups.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/frsups_frsups.ma deleted file mode 100644 index e7b7de26e..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/frsups_frsups.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 "basic_2/unfold/frsups.ma". - -(* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************) - -(* Main propertis ***********************************************************) - -theorem frsups_trans: bi_transitive … frsups. -/2 width=4/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2.ma deleted file mode 100644 index 562b79530..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2.ma +++ /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/grammar/term_vector.ma". - -(* GENERIC RELOCATION WITH PAIRS ********************************************) - -inductive at: list2 nat nat → relation nat ≝ -| at_nil: ∀i. at ⟠ i i -| at_lt : ∀des,d,e,i1,i2. i1 < d → - at des i1 i2 → at ({d, e} @ des) i1 i2 -| at_ge : ∀des,d,e,i1,i2. d ≤ i1 → - at des (i1 + e) i2 → at ({d, e} @ des) i1 i2 -. - -interpretation "application (generic relocation with pairs)" - 'RAt i1 des i2 = (at des i1 i2). - -(* Basic inversion lemmas ***************************************************) - -fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ⟠ → i1 = i2. -#des #i1 #i2 * -des -i1 -i2 -[ // -| #des #d #e #i1 #i2 #_ #_ #H destruct -| #des #d #e #i1 #i2 #_ #_ #H destruct -] -qed. - -lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2. -/2 width=3/ qed-. - -fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → - ∀d,e,des0. des = {d, e} @ des0 → - i1 < d ∧ @⦃i1, des0⦄ ≡ i2 ∨ - d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2. -#des #i1 #i2 * -des -i1 -i2 -[ #i #d #e #des #H destruct -| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/ -| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/ -] -qed. - -lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 → - i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨ - d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2. -/2 width=3/ qed-. - -lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 → - i1 < d → @⦃i1, des⦄ ≡ i2. -#des #d #e #i1 #e2 #H -elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d -lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd -elim (lt_refl_false … Hd) -qed-. - -lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 → - d ≤ i1 → @⦃i1 + e, des⦄ ≡ i2. -#des #d #e #i1 #e2 #H -elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1 -lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd -elim (lt_refl_false … Hd) -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_gr2.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_gr2.ma deleted file mode 100644 index 20ce856d6..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_gr2.ma +++ /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/gr2.ma". - -(* GENERIC RELOCATION WITH PAIRS ********************************************) - -(* Main properties **********************************************************) - -theorem at_mono: ∀des,i,i1. @⦃i, des⦄ ≡ i1 → ∀i2. @⦃i, des⦄ ≡ i2 → i1 = i2. -#des #i #i1 #H elim H -des -i -i1 -[ #i #x #H <(at_inv_nil … H) -x // -| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H - lapply (at_inv_cons_lt … H Hid) -H -Hid /2 width=1/ -| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H - lapply (at_inv_cons_ge … H Hdi) -H -Hdi /2 width=1/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_minus.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_minus.ma deleted file mode 100644 index 6138548cd..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_minus.ma +++ /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 *) -(* *) -(**************************************************************************) - -include "basic_2/unfold/gr2.ma". - -(* GENERIC RELOCATION WITH PAIRS ********************************************) - -inductive minuss: nat → relation (list2 nat nat) ≝ -| minuss_nil: ∀i. minuss i ⟠ ⟠ -| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 → - minuss i ({d, e} @ des1) ({d - i, e} @ des2) -| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 → - minuss i ({d, e} @ des1) des2 -. - -interpretation "minus (generic relocation with pairs)" - 'RMinus des1 i des2 = (minuss i des1 des2). - -(* Basic inversion lemmas ***************************************************) - -fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ → des2 = ⟠. -#des1 #des2 #i * -des1 -des2 -i -[ // -| #des1 #des2 #d #e #i #_ #_ #H destruct -| #des1 #des2 #d #e #i #_ #_ #H destruct -] -qed. - -lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠. -/2 width=4/ qed-. - -fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → - ∀d,e,des. des1 = {d, e} @ des → - d ≤ i ∧ des ▭ e + i ≡ des2 ∨ - ∃∃des0. i < d & des ▭ i ≡ des0 & - des2 = {d - i, e} @ des0. -#des1 #des2 #i * -des1 -des2 -i -[ #i #d #e #des #H destruct -| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3/ -| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1/ -] -qed. - -lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 → - d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨ - ∃∃des. i < d & des1 ▭ i ≡ des & - des2 = {d - i, e} @ des. -/2 width=3/ qed-. - -lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 → - d ≤ i → des1 ▭ e + i ≡ des2. -#des1 #des2 #d #e #i #H -elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi -lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi -elim (lt_refl_false … Hi) -qed-. - -lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 → - i < d → - ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} @ des. -#des1 #des2 #d #e #i #H -elim (minuss_inv_cons1 … H) -H * /2 width=3/ #Hdi #_ #Hid -lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi -elim (lt_refl_false … Hi) -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_plus.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_plus.ma deleted file mode 100644 index bd8d1a9be..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/gr2_plus.ma +++ /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/gr2.ma". - -(* GENERIC RELOCATION WITH PAIRS ********************************************) - -let rec pluss (des:list2 nat nat) (i:nat) on des ≝ match des with -[ nil2 ⇒ ⟠ -| cons2 d e des ⇒ {d + i, e} @ pluss des i -]. - -interpretation "plus (generic relocation with pairs)" - 'plus x y = (pluss x y). - -(* Basic inversion lemmas ***************************************************) - -lemma pluss_inv_nil2: ∀i,des. des + i = ⟠ → des = ⟠. -#i * // normalize -#d #e #des #H destruct -qed. - -lemma pluss_inv_cons2: ∀i,d,e,des2,des. des + i = {d, e} @ des2 → - ∃∃des1. des1 + i = des2 & des = {d - i, e} @ des1. -#i #d #e #des2 * normalize -[ #H destruct -| #d1 #e1 #des1 #H destruct /2 width=3/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops.ma deleted file mode 100644 index b899bd273..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops.ma +++ /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/substitution/ldrop.ma". -include "basic_2/unfold/gr2_minus.ma". -include "basic_2/unfold/lifts.ma". - -(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************) - -inductive ldrops: list2 nat nat → relation lenv ≝ -| ldrops_nil : ∀L. ldrops ⟠ L L -| ldrops_cons: ∀L1,L,L2,des,d,e. - ldrops des L1 L → ⇩[d,e] L ≡ L2 → ldrops ({d, e} @ des) L1 L2 -. - -interpretation "generic local environment slicing" - 'RDropStar des T1 T2 = (ldrops des T1 T2). - -(* Basic inversion lemmas ***************************************************) - -fact ldrops_inv_nil_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → des = ⟠ → L1 = L2. -#L1 #L2 #des * -L1 -L2 -des // -#L1 #L #L2 #d #e #des #_ #_ #H destruct -qed. - -(* Basic_1: was: drop1_gen_pnil *) -lemma ldrops_inv_nil: ∀L1,L2. ⇩*[⟠] L1 ≡ L2 → L1 = L2. -/2 width=3/ qed-. - -fact ldrops_inv_cons_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → - ∀d,e,tl. des = {d, e} @ tl → - ∃∃L. ⇩*[tl] L1 ≡ L & ⇩[d, e] L ≡ L2. -#L1 #L2 #des * -L1 -L2 -des -[ #L #d #e #tl #H destruct -| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct - /2 width=3/ -qed. - -(* Basic_1: was: drop1_gen_pcons *) -lemma ldrops_inv_cons: ∀L1,L2,d,e,des. ⇩*[{d, e} @ des] L1 ≡ L2 → - ∃∃L. ⇩*[des] L1 ≡ L & ⇩[d, e] L ≡ L2. -/2 width=3/ qed-. - -lemma ldrops_inv_skip2: ∀I,des,i,des2. des ▭ i ≡ des2 → - ∀L1,K2,V2. ⇩*[des2] L1 ≡ K2. ⓑ{I} V2 → - ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 & - ⇩*[des1] K1 ≡ K2 & - ⇧*[des1] V2 ≡ V1 & - L1 = K1. ⓑ{I} V1. -#I #des #i #des2 #H elim H -des -i -des2 -[ #i #L1 #K2 #V2 #H - >(ldrops_inv_nil … H) -L1 /2 width=7/ -| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H - elim (ldrops_inv_cons … H) -H #L #HL1 #H - elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ #K #V >minus_plus #HK2 #HV2 #H destruct - elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct - @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7/ | skip ] - normalize >plus_minus // @minuss_lt // /2 width=1/ (**) (* explicit constructors, /3 width=1/ is a bit slow *) -| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H - elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct - /4 width=7/ -] -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: drop1_skip_bind *) -lemma ldrops_skip: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 → - ∀I. ⇩*[des + 1] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2. -#L1 #L2 #des #H elim H -L1 -L2 -des -[ #L #V1 #V2 #HV12 #I - >(lifts_inv_nil … HV12) -HV12 // -| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I - elim (lifts_inv_cons … H) -H /3 width=5/ -]. -qed. - -(* Basic_1: removed theorems 1: drop1_getl_trans -*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops_ldrop.ma deleted file mode 100644 index 6ca2f73df..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops_ldrop.ma +++ /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/substitution/ldrop_ldrop.ma". -include "basic_2/unfold/ldrops.ma". - -(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************) - -(* Properties concerning basic local environment slicing ********************) - -lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[des] L1 ≡ L → ∀L2,i. ⇩[0, i] L ≡ L2 → - ∃∃L0,des0,i0. ⇩[0, i0] L1 ≡ L0 & ⇩*[des0] L0 ≡ L2 & - @⦃i, des⦄ ≡ i0 & des ▭ i ≡ des0. -#L1 #L #des #H elim H -L1 -L -des -[ /2 width=7/ -| #L1 #L3 #L #des3 #d #e #_ #HL3 #IHL13 #L2 #i #HL2 - elim (lt_or_ge i d) #Hid - [ elim (ldrop_trans_le … HL3 … HL2 ?) -L /2 width=2/ #L #HL3 #HL2 - elim (IHL13 … HL3) -L3 /3 width=7/ - | lapply (ldrop_trans_ge … HL3 … HL2 ?) -L // #HL32 - elim (IHL13 … HL32) -L3 /3 width=7/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops_ldrops.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops_ldrops.ma deleted file mode 100644 index 7709561a2..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/ldrops_ldrops.ma +++ /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/unfold/ldrops_ldrop.ma". - -(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: drop1_trans *) -theorem ldrops_trans: ∀L,L2,des2. ⇩*[des2] L ≡ L2 → ∀L1,des1. ⇩*[des1] L1 ≡ L → - ⇩*[des2 @@ des1] L1 ≡ L2. -#L #L2 #des2 #H elim H -L -L2 -des2 // /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts.ma deleted file mode 100644 index 40158acbe..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts.ma +++ /dev/null @@ -1,150 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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/lift.ma". -include "basic_2/unfold/gr2_plus.ma". - -(* GENERIC TERM RELOCATION **************************************************) - -inductive lifts: list2 nat nat → relation term ≝ -| lifts_nil : ∀T. lifts ⟠ T T -| lifts_cons: ∀T1,T,T2,des,d,e. - ⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} @ des) T1 T2 -. - -interpretation "generic relocation (term)" - 'RLiftStar des T1 T2 = (lifts des T1 T2). - -(* Basic inversion lemmas ***************************************************) - -fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2. -#T1 #T2 #des * -T1 -T2 -des // -#T1 #T #T2 #d #e #des #_ #_ #H destruct -qed. - -lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2. -/2 width=3/ qed-. - -fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → - ∀d,e,tl. des = {d, e} @ tl → - ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2. -#T1 #T2 #des * -T1 -T2 -des -[ #T #d #e #tl #H destruct -| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct - /2 width=3/ -qed. - -lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} @ des] T1 ≡ T2 → - ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2. -/2 width=3/ qed-. - -(* Basic_1: was: lift1_sort *) -lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k. -#T2 #k #des elim des -des -[ #H <(lifts_inv_nil … H) -H // -| #d #e #des #IH #H - elim (lifts_inv_cons … H) -H #X #H - >(lift_inv_sort1 … H) -H /2 width=1/ -] -qed-. - -(* Basic_1: was: lift1_lref *) -lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 → - ∃∃i2. @⦃i1, des⦄ ≡ i2 & T2 = #i2. -#T2 #des elim des -des -[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3/ -| #d #e #des #IH #i1 #H - elim (lifts_inv_cons … H) -H #X #H1 #H2 - elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct - elim (IH … H2) -IH -H2 /3 width=3/ -] -qed-. - -lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p. -#T2 #p #des elim des -des -[ #H <(lifts_inv_nil … H) -H // -| #d #e #des #IH #H - elim (lifts_inv_cons … H) -H #X #H - >(lift_inv_gref1 … H) -H /2 width=1/ -] -qed-. - -(* Basic_1: was: lift1_bind *) -lemma lifts_inv_bind1: ∀a,I,T2,des,V1,U1. ⇧*[des] ⓑ{a,I} V1. U1 ≡ T2 → - ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 & - T2 = ⓑ{a,I} V2. U2. -#a #I #T2 #des elim des -des -[ #V1 #U1 #H - <(lifts_inv_nil … H) -H /2 width=5/ -| #d #e #des #IHdes #V1 #U1 #H - elim (lifts_inv_cons … H) -H #X #H #HT2 - elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct - elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct - /3 width=5/ -] -qed-. - -(* Basic_1: was: lift1_flat *) -lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 → - ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 & - T2 = ⓕ{I} V2. U2. -#I #T2 #des elim des -des -[ #V1 #U1 #H - <(lifts_inv_nil … H) -H /2 width=5/ -| #d #e #des #IHdes #V1 #U1 #H - elim (lifts_inv_cons … H) -H #X #H #HT2 - elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct - elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct - /3 width=5/ -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. -#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_dx/ -qed-. - -lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. -#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_sn/ -qed-. - -(* Basic properties *********************************************************) - -lemma lifts_bind: ∀a,I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 → - ∀T1. ⇧*[des + 1] T1 ≡ T2 → - ⇧*[des] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2. -#a #I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des -[ #V #T1 #H >(lifts_inv_nil … H) -H // -| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H - elim (lifts_inv_cons … H) -H /3 width=3/ -] -qed. - -lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 → - ∀T1. ⇧*[des] T1 ≡ T2 → - ⇧*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2. -#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des -[ #V #T1 #H >(lifts_inv_nil … H) -H // -| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H - elim (lifts_inv_cons … H) -H /3 width=3/ -] -qed. - -lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2. -#des elim des -des /2 width=2/ -#d #e #des #IH #T1 -elim (lift_total T1 d e) #T #HT1 -elim (IH T) -IH /3 width=4/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_lift.ma deleted file mode 100644 index 6ad3ff015..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_lift.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 "basic_2/substitution/lift_lift.ma". -include "basic_2/unfold/gr2_minus.ma". -include "basic_2/unfold/lifts.ma". - -(* GENERIC TERM RELOCATION **************************************************) - -(* Properties concerning basic term relocation ******************************) - -(* Basic_1: was: lift1_xhg (right to left) *) -lemma lifts_lift_trans_le: ∀T1,T,des. ⇧*[des] T1 ≡ T → ∀T2. ⇧[0, 1] T ≡ T2 → - ∃∃T0. ⇧[0, 1] T1 ≡ T0 & ⇧*[des + 1] T0 ≡ T2. -#T1 #T #des #H elim H -T1 -T -des -[ /2 width=3/ -| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2 - elim (IHT13 … HT2) -T #T #HT3 #HT2 - elim (lift_trans_le … HT13 … HT3 ?) -T3 // /3 width=5/ -] -qed-. - -(* Basic_1: was: lift1_free (right to left) *) -lemma lifts_lift_trans: ∀des,i,i0. @⦃i, des⦄ ≡ i0 → - ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 → - ∀T1,T0. ⇧*[des0] T1 ≡ T0 → - ∀T2. ⇧[O, i0 + 1] T0 ≡ T2 → - ∃∃T. ⇧[0, i + 1] T1 ≡ T & ⇧*[des] T ≡ T2. -#des elim des -des normalize -[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2 - <(at_inv_nil … H1) -x #HT02 - lapply (minuss_inv_nil1 … H2) -H2 #H - >(pluss_inv_nil2 … H) in HT10; -des0 #H - >(lifts_inv_nil … H) -T1 /2 width=3/ -| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02 - elim (at_inv_cons … H1) -H1 * #Hid #Hi0 - [ elim (minuss_inv_cons1_lt … H2 ?) -H2 [2: /2 width=1/ ] #des1 #Hdes1 minus_plus #HT1 #HT0 - elim (IHdes … Hi0 … Hdes1 … HT0 … HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02 - elim (lift_trans_le … HT1 … HT0 ?) -T /2 width=1/ #T #HT1 commutative_plus in Hi0; #Hi0 - lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] (liftv_inv_nil1 … H) -T1s /2 width=3/ -| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H - elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct - elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s - elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5/ -] -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_lifts.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_lifts.ma deleted file mode 100644 index 72948f04b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_lifts.ma +++ /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/unfold/lifts_lift.ma". - -(* GENERIC RELOCATION *******************************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: lift1_lift1 (left to right) *) -theorem lifts_trans: ∀T1,T,des1. ⇧*[des1] T1 ≡ T → ∀T2:term. ∀des2. ⇧*[des2] T ≡ T2 → - ⇧*[des1 @@ des2] T1 ≡ T2. -#T1 #T #des1 #H elim H -T1 -T -des1 // /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_vector.ma deleted file mode 100644 index 9ea173a56..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/lifts_vector.ma +++ /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/substitution/lift_vector.ma". -include "basic_2/unfold/lifts.ma". - -(* GENERIC TERM VECTOR RELOCATION *******************************************) - -inductive liftsv (des:list2 nat nat) : relation (list term) ≝ -| liftsv_nil : liftsv des ◊ ◊ -| liftsv_cons: ∀T1s,T2s,T1,T2. - ⇧*[des] T1 ≡ T2 → liftsv des T1s T2s → - liftsv des (T1 @ T1s) (T2 @ T2s) -. - -interpretation "generic relocation (vector)" - 'RLiftStar des T1s T2s = (liftsv des T1s T2s). - -(* Basic inversion lemmas ***************************************************) - -(* Basic_1: was: lifts1_flat (left to right) *) -lemma lifts_inv_applv1: ∀V1s,U1,T2,des. ⇧*[des] Ⓐ V1s. U1 ≡ T2 → - ∃∃V2s,U2. ⇧*[des] V1s ≡ V2s & ⇧*[des] U1 ≡ U2 & - T2 = Ⓐ V2s. U2. -#V1s elim V1s -V1s normalize -[ #T1 #T2 #des #HT12 - @(ex3_2_intro) [3,4: // |1,2: skip | // ] (**) (* explicit constructor *) -| #V1 #V1s #IHV1s #T1 #X #des #H - elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct - elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct - @(ex3_2_intro) [4: // |3: /2 width=2/ |1,2: skip | // ] (**) (* explicit constructor *) -] -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: lifts1_flat (right to left) *) -lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s → - ∀T1,T2. ⇧*[des] T1 ≡ T2 → - ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2. -#V1s #V2s #des #H elim H -V1s -V2s // /3 width=1/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_dx.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_dx.ma deleted file mode 100644 index 6ba09b962..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_dx.ma +++ /dev/null @@ -1,274 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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". - -(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************) - -(* Basic_1: includes: csubst1_bind *) -inductive ltpss_dx: nat → nat → relation lenv ≝ -| ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆) -| ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) -| ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e. - ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 → - ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) -| ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e. - ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 → - ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) -. - -interpretation "parallel unfold (local environment, dx variant)" - 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2. -#d #e #L1 #L2 #H elim H -d -e -L1 -L2 // -[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct - >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) // -] -qed. - -lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2. -/2 width=4/ qed-. - -fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2. - L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| #L #I #V #H destruct -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct -] -qed. - -lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e → - ∀K1,I,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. K1 ▶* [0, e - 1] K2 & - K2 ⊢ V1 ▶* [0, e - 1] V2 & - L2 = K2. ⓑ{I} V2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #_ #K1 #I #V1 #H destruct -| #L1 #I #V #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e → - ∃∃K2,V2. K1 ▶* [0, e - 1] K2 & - K2 ⊢ V1 ▶* [0, e - 1] V2 & - L2 = K2. ⓑ{I} V2. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d → - ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. K1 ▶* [d - 1, e] K2 & - K2 ⊢ V1 ▶* [d - 1, e] V2 & - L2 = K2. ⓑ{I} V2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K1 #V1 #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d → - ∃∃K2,V2. K1 ▶* [d - 1, e] K2 & - K2 ⊢ V1 ▶* [d - 1, e] V2 & - L2 = K2. ⓑ{I} V2. -/2 width=3/ qed-. - -fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2. - L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| #L #I #V #H destruct -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct -] -qed. - -lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e → - ∀K2,I,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ▶* [0, e - 1] K2 & - K2 ⊢ V1 ▶* [0, e - 1] V2 & - L1 = K1. ⓑ{I} V1. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #_ #K1 #I #V1 #H destruct -| #L1 #I #V #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e → - ∃∃K1,V1. K1 ▶* [0, e - 1] K2 & - K2 ⊢ V1 ▶* [0, e - 1] V2 & - L1 = K1. ⓑ{I} V1. -/2 width=5/ qed-. - -fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d → - ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ▶* [d - 1, e] K2 & - K2 ⊢ V1 ▶* [d - 1, e] V2 & - L1 = K1. ⓑ{I} V1. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K2 #V2 #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d → - ∃∃K1,V1. K1 ▶* [d - 1, e] K2 & - K2 ⊢ V1 ▶* [d - 1, e] V2 & - L1 = K1. ⓑ{I} V1. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e. - L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 → - L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e. - L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 → - L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e. - L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 → - 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2. -#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He ->(plus_minus_m_m e 1) /2 width=1/ -qed. - -lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e. - L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 → - 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2. -#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd ->(plus_minus_m_m d 1) /2 width=1/ -qed. - -lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e. - L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 → - 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e. - L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 → - 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2. -/3 width=1/ qed. - -(* Basic_1: was by definition: csubst1_refl *) -lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L. -#L elim L -L // -#L #I #V #IHL * /2 width=1/ * /2 width=1/ -qed. - -lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 → - ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2. -#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 // -[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2 - lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2; - lapply (lt_to_le_to_lt 0 … Hde2) // #He2 - lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/ -| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12 - >plus_plus_comm_23 in Hde12; #Hde12 - elim (le_to_or_lt_eq 0 d2 ?) // #H destruct - [ lapply (le_plus_to_minus_r … Hde12) -Hde12 plus_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 … H) -H // -| // -| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 - elim (le_inv_plus_l … He12) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ -| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ -] -qed. - -lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → - d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2. -#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H // -| // -| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 - elim (le_inv_plus_l … He12) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/ -| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ -] -qed. - -lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → - ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L. -#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq … He2) -He2 #H destruct - lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ -| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - (ldrop_inv_atom1 … H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq … He2) -He2 #H destruct - lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ -| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - (ldrop_inv_atom1 … H) -H /2 width=3/ -| /2 width=3/ -| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2 - lapply (le_n_O_to_eq … He2) -He2 #He2 destruct - lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ -| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 … H) -H /2 width=3/ -| /2 width=3/ -| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2 - lapply (le_n_O_to_eq … He2) -He2 #He2 destruct - lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ -| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK10 -He2d1 destruct minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 - elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/ -| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 - elim (IHVW2 … HL01) -IHVW2 - elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/ -] -qed. - -lemma ltpss_dx_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 → - ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 → - L1 ⊢ T2 ▶* [d2, e2] U2. -#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 // -#U #U2 #_ #HU2 #IHU -lapply (ltpss_dx_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/ -qed. - -(* Basic_1: was: subst1_subst1 *) -lemma ltpss_dx_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 → - ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → - ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T & - L0 ⊢ T ▶* [d1, e1] U2. -#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2 -[ /2 width=3/ -| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 - elim (lt_or_ge i2 d1) #Hi2d1 - [ elim (ltpss_dx_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1 - elim (ltpss_dx_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H - elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 - lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 - elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/ -| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 - elim (IHVW2 … HL10) -IHVW2 - elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_sn.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_sn.ma deleted file mode 100644 index 0d13a5a3f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_sn.ma +++ /dev/null @@ -1,255 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| 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". - -(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************) - -inductive ltpss_sn: nat → nat → relation lenv ≝ -| ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆) -| ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) -| ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e. - ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 → - ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) -| ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e. - ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 → - ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) -. - -interpretation "parallel unfold (local environment, sn variant)" - 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2. -#d #e #L1 #L2 #H elim H -d -e -L1 -L2 // -[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct - >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) // -] -qed. - -lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2. -/2 width=4/ qed-. - -fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2. - L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| #L #I #V #H destruct -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct -] -qed. - -lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e → - ∀K1,I,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 & - K1 ⊢ V1 ▶* [0, e - 1] V2 & - L2 = K2. ⓑ{I} V2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #_ #K1 #I #V1 #H destruct -| #L1 #I #V #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e → - ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 & - K1 ⊢ V1 ▶* [0, e - 1] V2 & - L2 = K2. ⓑ{I} V2. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d → - ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 & - K1 ⊢ V1 ▶* [d - 1, e] V2 & - L2 = K2. ⓑ{I} V2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K1 #V1 #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d → - ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 & - K1 ⊢ V1 ▶* [d - 1, e] V2 & - L2 = K2. ⓑ{I} V2. -/2 width=3/ qed-. - -fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2. - L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| #L #I #V #H destruct -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct -] -qed. - -lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e → - ∀K2,I,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 & - K1 ⊢ V1 ▶* [0, e - 1] V2 & - L1 = K1. ⓑ{I} V1. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #_ #K1 #I #V1 #H destruct -| #L1 #I #V #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e → - ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 & - K1 ⊢ V1 ▶* [0, e - 1] V2 & - L1 = K1. ⓑ{I} V1. -/2 width=5/ qed-. - -fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d → - ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 & - K1 ⊢ V1 ▶* [d - 1, e] V2 & - L1 = K1. ⓑ{I} V1. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K2 #V2 #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) -| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ -] -qed. - -lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d → - ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 & - K1 ⊢ V1 ▶* [d - 1, e] V2 & - L1 = K1. ⓑ{I} V1. -/2 width=3/ qed-. - -(* Basic properties *********************************************************) - -lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e. - L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 → - L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e. - L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 → - L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e. - L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 → - 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2. -#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He ->(plus_minus_m_m e 1) /2 width=1/ -qed. - -lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e. - L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 → - 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2. -#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd ->(plus_minus_m_m d 1) /2 width=1/ -qed. - -lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e. - L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 → - 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e. - L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 → - 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2. -/3 width=1/ qed. - -lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L. -#L elim L -L // -#L #I #V #IHL * /2 width=1/ * /2 width=1/ -qed. - -lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 → - ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2. -#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 // -[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2 - lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2; - lapply (lt_to_le_to_lt 0 … Hde2) // #He2 - lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/ -| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12 - >plus_plus_comm_23 in Hde12; #Hde12 - elim (le_to_or_lt_eq 0 d2 ?) // #H destruct - [ lapply (le_plus_to_minus_r … Hde12) -Hde12 plus_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 … H) -H // -| // -| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 - elim (le_inv_plus_l … He12) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ -| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ -] -qed. - -lemma ltpss_sn_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → - d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2. -#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H // -| // -| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 - elim (le_inv_plus_l … He12) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/ -| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 - lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ -] -qed. - -lemma ltpss_sn_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → - ∃∃L. L2 ⊢ ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L. -#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 -[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq … He2) -He2 #H destruct - lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ -| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - (ldrop_inv_atom1 … H) -H /2 width=3/ -| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 - lapply (le_n_O_to_eq … He2) -He2 #H destruct - lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ -| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 - elim (le_inv_plus_l … Hd1e2) #_ #He2 - (ldrop_inv_atom1 … H) -H /2 width=3/ -| /2 width=3/ -| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2 - lapply (le_n_O_to_eq … He2) -He2 #He2 destruct - lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ -| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 … H) -H /2 width=3/ -| /2 width=3/ -| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2 - lapply (le_n_O_to_eq … He2) -He2 #He2 destruct - lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ -| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1 - lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 - [ -IHK10 -He2d1 destruct shift_append_assoc #H - elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct - append_length minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 - elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H - lapply (tpss_lsubs_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/ -| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 - elim (IHVW2 … HL01) -IHVW2 - elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/ -] -qed. - -lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 → - ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 → - L1 ⊢ T2 ▶* [d2, e2] U2. -#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 // -#U #U2 #_ #HU2 #IHU -lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/ -qed. - -lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 → - ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → - ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T & - L1 ⊢ T ▶* [d1, e1] U2. -#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2 -[ /2 width=3/ -| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 - elim (lt_or_ge i2 d1) #Hi2d1 - [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1 - elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct - lapply (ldrop_fwd_ldrop2 … HLK1) #H - elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 - lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ - | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/ - ] - ] -| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 - elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2 - elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H - lapply (tpss_lsubs_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/ -| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 - elim (IHVW2 … HL10) -IHVW2 - elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/thin.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/thin.ma deleted file mode 100644 index 65fb76fe0..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/thin.ma +++ /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.ma". - -(* BASIC LOCAL ENVIRONMENT THINNING *****************************************) - -definition thin: nat → nat → relation lenv ≝ - λd,e,L1,L2. ∃∃L. L1 ⊢ ▶* [d, e] L & ⇩[d, e] L ≡ L2. - -interpretation "basic thinning (local environment)" - 'TSubst L1 d e L2 = (thin d e L1 L2). - -(* Basic properties *********************************************************) - -lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ▼*[d, e] L1 ≡ L2. -/2 width=3/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma thin_inv_thin1: ∀I,K1,V1,L2,e. ▼*[0, e] K1.ⓑ{I} V1 ≡ L2 → 0 < e → - ▼*[0, e - 1] K1 ≡ L2. -#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e -elim (ltpss_sn_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct -lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/thin_delift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/thin_delift.ma deleted file mode 100644 index b5ffc5e4f..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/thin_delift.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_2/unfold/delift_tpss.ma". -include "basic_2/unfold/delift_ltpss.ma". -include "basic_2/unfold/thin.ma". - -(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************) - -(* Inversion lemmas on inverse basic term relocation ************************) - -lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d → - ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 & - K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 & - L2 = K2. ⓑ{I} V2. -#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e -elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct -elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/ -qed-. - -(* Properties on inverse basic term relocation ******************************) - -lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 → - ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2. -#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I -elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0 -lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct -lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/ -qed. - -lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 → - ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → - ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd → - ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & - L ⊢ ▼*[dd, ee] U2 ≡ T2. -#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd -lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1 -elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2 -elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT -lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT -lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/ -qed. - -lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 → - ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → - ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd → - ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & - L ⊢ ▼*[dd, ee] U2 ≡ T2. -/3 width=3/ qed. - -lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 → - ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → - ∀K. ▼*[dd, ee] L ≡ K → - d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee → - ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 & - L ⊢ ▼*[dd, ee] U2 ≡ T2. -#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee -lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1 -elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2 -elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT -lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT -lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/ -qed. - -lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 → - ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → - ∀K. ▼*[dd, ee] L ≡ K → - d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee → - ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 & - L ⊢ ▼*[dd, ee] U2 ≡ T2. -/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *) - -lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 → - ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → - ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e → - ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 & - L ⊢ ▼*[dd, ee] U2 ≡ T2. -#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee -lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1 -elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2 -elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT -lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT -lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/ -qed. - -lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 → - ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → - ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e → - ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 & - L ⊢ ▼*[dd, ee] U2 ≡ T2. -/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/thin_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/thin_ldrop.ma deleted file mode 100644 index 498660e1c..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/thin_ldrop.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 "basic_2/substitution/ldrop_ldrop.ma". -include "basic_2/unfold/ltpss_sn_ldrop.ma". -include "basic_2/unfold/thin.ma". - -(* BASIC LOCAL ENVIRONMENT THINNING *****************************************) - -(* Properties on local environment slicing **********************************) - -lemma thin_ldrop_conf_ge: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → - d1 + e1 ≤ e2 → ⇩[0, e2 - e1] L1 ≡ L2. -#L0 #L1 #d1 #e1 * /3 width=8 by ltpss_sn_ldrop_conf_ge, ldrop_conf_ge/ -qed. - -lemma thin_ldrop_conf_be: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → - ∃∃L. ▼*[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L. -#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #Hd1e2 #He2de1 -elim (ltpss_sn_ldrop_conf_be … HL0 … HL02 ? ?) -L0 // #L0 #HL20 #HL0 -elim (ldrop_conf_be … HL1 … HL0 ? ?) -L // -Hd1e2 -He2de1 /3 width=3/ -qed. - -lemma thin_ldrop_conf_le: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 → - ∃∃L. ▼*[d1 - e2, e1] L2 ≡ L & ⇩[0, e2] L1 ≡ L. -#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #He2d1 -elim (ltpss_sn_ldrop_conf_le … HL0 … HL02 ?) -L0 // #L0 #HL20 #HL0 -elim (ldrop_conf_le … HL1 … HL0 ?) -L // -He2d1 /3 width=3/ -qed. - -lemma thin_ldrop_trans_ge: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → - d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2. -#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #Hd1e2 -lapply (ldrop_trans_ge … HL0 … HL02 ?) -L0 // #HL2 -lapply (ltpss_sn_ldrop_trans_ge … HL1 … HL2 ?) -L // /2 width=1/ -qed. - -lemma thin_ldrop_trans_le: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 → - ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 → - ∃∃L. ▼*[d1 - e2, e1] L ≡ L2 & ⇩[0, e2] L1 ≡ L. -#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #He2d1 -elim (ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL0 #HL02 -elim (ltpss_sn_ldrop_trans_le … HL1 … HL0 He2d1) -L -He2d1 /3 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma deleted file mode 100644 index 93916208b..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.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 "basic_2/substitution/tps.ma". - -(* PARTIAL UNFOLD ON TERMS **************************************************) - -definition tpss: nat → nat → lenv → relation term ≝ - λd,e,L. TC … (tps d e L). - -interpretation "partial unfold (term)" - 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2). - -(* Basic eliminators ********************************************************) - -lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 → - (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) → - ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2. -#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind … HT1 IHT1 … HT12) // -qed-. - -lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 → - (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) → - ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1. -#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx … HT2 IHT2 … HT12) // -qed-. - -(* Basic properties *********************************************************) - -lemma tpss_strap1: ∀L,T1,T,T2,d,e. - L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. -/2 width=3/ qed. - -lemma tpss_strap2: ∀L,T1,T,T2,d,e. - L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. -/2 width=3/ qed. - -lemma tpss_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 → - ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2. -/3 width=3/ qed. - -lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T. -/2 width=1/ qed. - -lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 → - ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 → - L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2. -#L #V1 #V2 #d #e #HV12 elim HV12 -V2 -[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2 - [ /3 width=5/ - | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) - ] -| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12 - lapply (tpss_lsubs_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12 - lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) -] -qed. - -lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e. - L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 → - L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2. -#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2 -[ #V2 #HV12 #HT12 elim HT12 -T2 - [ /3 width=1/ - | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) - ] -| #V #V2 #_ #HV12 #IHV #HT12 - lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) -] -qed. - -lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 → - ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → - L ⊢ T1 ▶* [d2, e2] T2. -#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2 -[ // -| #T #T2 #_ #HT12 #IHT - lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/ -] -qed. - -lemma tpss_weak_top: ∀L,T1,T2,d,e. - L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2. -#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 -[ // -| #T #T2 #_ #HT12 #IHT - lapply (tps_weak_top … HT12) -HT12 /2 width=3/ -] -qed. - -lemma tpss_weak_all: ∀L,T1,T2,d,e. - L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2. -#L #T1 #T2 #d #e #HT12 -lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12 -lapply (tpss_weak_top … HT12) // -qed. - -lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 → - ∀L. L @@ K ⊢ T1 ▶* [d, e] T2. -#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/ -qed. - -(* Basic inversion lemmas ***************************************************) - -(* Note: this can be derived from tpss_inv_atom1 *) -lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k. -#L #T2 #k #d #e #H @(tpss_ind … H) -T2 -[ // -| #T #T2 #_ #HT2 #IHT destruct - >(tps_inv_sort1 … HT2) -HT2 // -] -qed-. - -(* Note: this can be derived from tpss_inv_atom1 *) -lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p. -#L #T2 #p #d #e #H @(tpss_ind … H) -T2 -[ // -| #T #T2 #_ #HT2 #IHT destruct - >(tps_inv_gref1 … HT2) -HT2 // -] -qed-. - -lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 → - ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & - L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 & - U2 = ⓑ{a,I} V2. T2. -#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2 -[ /2 width=5/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H - lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ -] -qed-. - -lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 → - ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 & - U2 = ⓕ{I} V2. T2. -#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2 -[ /2 width=5/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/ -] -qed-. - -lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2. -#L #T1 #T2 #d #H @(tpss_ind … H) -T2 -[ // -| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 // -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #{T1} ≤ #{T2}. -#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 -lapply (tps_fwd_tw … HT2) -HT2 #HT2 -@(transitive_le … IHT1) // -qed-. - -lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T → - ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. -#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T -[ /2 width=4/ -| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct - elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/ -] -qed-. - \ No newline at end of file diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_alt.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_alt.ma deleted file mode 100644 index ae1dcf624..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_alt.ma +++ /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/tpss_lift.ma". - -(* PARALLEL UNFOLD ON TERMS *************************************************) - -(* alternative definition of tpss *) -inductive tpssa: nat → nat → lenv → relation term ≝ -| tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I}) -| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → - ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 → - ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2 -| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e. - tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 → - tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) -| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e. - tpssa d e L V1 V2 → tpssa d e L T1 T2 → - tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) -. - -interpretation "parallel unfold (term) alternative" - 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2). - -(* Basic properties *********************************************************) - -lemma tpssa_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 → - ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2. -#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e -[ // -| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 - elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ -| /4 width=1/ -| /3 width=1/ -] -qed. - -lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T. -#T elim T -T // -#I elim I -I /2 width=1/ -qed. - -lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T → - ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2. -#L #T1 #T #d #e #H elim H -L -T1 -T -d -e -[ #L #I #d #e #X #H - elim (tps_inv_atom1 … H) -H // * /2 width=6/ -| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H - lapply (ldrop_fwd_ldrop2 … HLK) #H0LK - lapply (tps_weak … H 0 (d+e) ? ?) -H // #H - elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/ -| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H - elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct - lapply (tps_lsubs_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2 - lapply (IHV1 … HV2) -IHV1 -HV2 #HV12 - lapply (IHT1 … HT2) -IHT1 -HT2 #HT12 - lapply (tpssa_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ -| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H - elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/ -] -qed. - -lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2. -#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/ -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. -#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/ -qed-. - -lemma tpss_ind_alt: ∀R:ℕ→ℕ→lenv→relation term. - (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) → - (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → - ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 → - ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2 - ) → - (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 → - L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 → - R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) - ) → - (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 → - L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 → - R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) - ) → - ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2. -#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e -// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/ -qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_lift.ma deleted file mode 100644 index a68f86e32..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_lift.ma +++ /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: ∀L,K,V,U1,i,d,e. - d ≤ i → i < d + e → - ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 → - ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2. -#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1 -[ /3 width=4/ -| #U #U1 #_ #HU1 #IHU #U2 #HU12 - elim (lift_total U 0 (i+1)) #U0 #HU0 - lapply (IHU … HU0) -IHU #H - lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK - lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02 - lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ] -] -qed. - -(* Advanced inverion lemmas *************************************************) - -lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 → - T2 = ⓪{I} ∨ - ∃∃K,V1,V2,i. d ≤ i & i < d + e & - ⇩[O, i] L ≡ K. ⓓV1 & - K ⊢ V1 ▶* [0, d + e - i - 1] V2 & - ⇧[O, i + 1] V2 ≡ T2 & - I = LRef i. -#L #T2 #I #d #e #H @(tpss_ind … H) -T2 -[ /2 width=1/ -| #T #T2 #_ #HT2 * - [ #H destruct - elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ] - | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI - lapply (ldrop_fwd_ldrop2 … HLK) #H - elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 (lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ -] -qed. - -lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 → - ∀L,U1,d,e. dt ≤ d → d ≤ dt + et → - ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → - ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2. -#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ -] -qed. - -lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 → - ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K → - ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → - L ⊢ U1 ▶* [dt + e, et] U2. -#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T d e) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ -] -qed. - -lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt + et ≤ d → - ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt ≤ d → d + e ≤ dt + et → - ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d + e ≤ dt → - ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e. - L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2. -#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 // -#U #U2 #_ #HU2 #IHU destruct -<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 // -qed. - -lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → - ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 & - ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt ≤ d → dt + et ≤ d + e → - ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. - -lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt ≤ d → d ≤ dt + et → dt + et ≤ d + e → - ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2 -[ /2 width=3/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_tpss.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_tpss.ma deleted file mode 100644 index 3f41b0083..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss_tpss.ma +++ /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: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2. -#L #T1 #T2 #d #H @(tpss_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 -lapply (tps_trans_ge … IHT1 … HT2 ?) // -qed-. - -(* Advanced properties ******************************************************) - -lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 → - ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → - ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T. -/3 width=3/ qed. - -lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 → - ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 → - (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) → - ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T. -/3 width=3/ qed. - -lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 → - ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 → - ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2. -/3 width=3/ qed. - -lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 → - ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 → - ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2. -/3 width=3/ qed. - -lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → - ∀i. d ≤ i → i ≤ d + e → - ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2. -#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2 -[ /2 width=3/ -| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 - elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02 - elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ] - /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *) -] -qed. - -lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → - ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 & - ⇧[d, e] T2 ≡ U2. -#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet -elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 -lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 -lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // minus_minus_comm >minus_le_minus_minus_comm // -qed. - -lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b. -#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1/ -qed. - -lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b. -/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed. - -lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b → - a1 - c1 + a2 - (b - c1) = a1 + a2 - b. -#a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/ -qed. - -(* Inversion & forward lemmas ***********************************************) - -axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2). - -axiom lt_dec: ∀n1,n2. Decidable (n1 < n2). - -lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m. -#m #n elim (lt_or_ge m n) /2 width=1/ -#H elim H -m /2 width=1/ -#m #Hm * #H /2 width=1/ /3 width=1/ -qed-. - -lemma lt_refl_false: ∀n. n < n → ⊥. -#n #H elim (lt_to_not_eq … H) -H /2 width=1/ -qed-. - -lemma lt_zero_false: ∀n. n < 0 → ⊥. -#n #H elim (lt_to_not_le … H) -H /2 width=1/ -qed-. - -lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x. -#x #y #H elim (decidable_lt x y) /2 width=1/ -#Hxy elim (H Hxy) -qed-. - -lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥. -#y #z #x elim x -x -[ #H lapply (le_n_O_to_eq … H) -H - commutative_plus // -qed. - -lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b). -#B #f #b #l elim l -l normalize // -qed. - -(* Trichotomy operator ******************************************************) - -(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *) -let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝ - match n1 with - [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ] - | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ] - ]. - -lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1. -#A #a1 #a2 #a3 #n2 elim n2 -n2 -[ #n1 #H elim (lt_zero_false … H) -| #n2 #IH #n1 elim n1 -n1 // /3 width=1/ -] -qed. - -lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2. -#A #a1 #a2 #a3 #n elim n -n normalize // -qed. - -lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3. -#A #a1 #a2 #a3 #n1 elim n1 -n1 -[ #n2 #H elim (lt_zero_false … H) -| #n1 #IH #n2 elim n2 -n2 // /3 width=1/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/ground_2/list.ma b/matita/matita/contribs/lambda_delta/ground_2/list.ma deleted file mode 100644 index 9a5ac0aeb..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/list.ma +++ /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 "ground_2/arith.ma". - -(* LISTS ********************************************************************) - -inductive list (A:Type[0]) : Type[0] := - | nil : list A - | cons: A → list A → list A. - -interpretation "nil (list)" 'Nil = (nil ?). - -interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl). - -let rec all A (R:predicate A) (l:list A) on l ≝ - match l with - [ nil ⇒ ⊤ - | cons hd tl ⇒ R hd ∧ all A R tl - ]. - -inductive list2 (A1,A2:Type[0]) : Type[0] := - | nil2 : list2 A1 A2 - | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2. - -interpretation "nil (list of pairs)" 'Nil2 = (nil2 ? ?). - -interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl). - -let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with -[ nil2 ⇒ l2 -| cons2 a1 a2 tl ⇒ {a1, a2} @ append2 A1 A2 tl l2 -]. - -interpretation "append (list of pairs)" - 'Append l1 l2 = (append2 ? ? l1 l2). - -let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with -[ nil2 ⇒ 0 -| cons2 _ _ l ⇒ length2 A1 A2 l + 1 -]. - -interpretation "length (list of pairs)" - 'card l = (length2 ? ? l). diff --git a/matita/matita/contribs/lambda_delta/ground_2/notation.ma b/matita/matita/contribs/lambda_delta/ground_2/notation.ma deleted file mode 100644 index 4ac2e6e64..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/notation.ma +++ /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 *) -(* *) -(**************************************************************************) - -(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) - -(* Logic ********************************************************************) - -notation "⊥" - non associative with precedence 90 - for @{'false}. - -notation "⊤" - non associative with precedence 90 - for @{'true}. - -(* Lists ********************************************************************) - -notation "◊" - non associative with precedence 90 - for @{'Nil}. - -notation "hvbox( hd @ break tl )" - right associative with precedence 47 - for @{'Cons $hd $tl}. - -notation "hvbox( l1 @@ break l2 )" - right associative with precedence 47 - for @{'Append $l1 $l2 }. - -notation "⟠" - non associative with precedence 90 - for @{'Nil2}. - -notation "hvbox( { hd1 , break hd2 } @ break tl )" - non associative with precedence 47 - for @{'Cons $hd1 $hd2 $tl}. diff --git a/matita/matita/contribs/lambda_delta/ground_2/star.ma b/matita/matita/contribs/lambda_delta/ground_2/star.ma deleted file mode 100644 index 1e46a48c6..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/star.ma +++ /dev/null @@ -1,158 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM 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/star.ma". -include "ground_2/xoa_props.ma". -include "ground_2/notation.ma". - -(* PROPERTIES OF RELATIONS **************************************************) - -definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥). - -definition Confluent: ∀A. ∀R: relation A. Prop ≝ λA,R. - ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → - ∃∃a. R a1 a & R a2 a. - -definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R. - ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2. - -definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. - ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & R1 a2 a. - -definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. - ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & R1 a a2. - -definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R. - ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 → - ∃∃a,b. R a1 b1 a b & R a2 b2 a b. - -lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 → - ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & TC … R1 a2 a. -#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 -[ #a1 #Ha01 #a2 #Ha02 - elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/ -| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 - elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 - elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/ -] -qed. - -lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 → - ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 → - ∃∃a. TC … R2 a1 a & R1 a2 a. -#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 -[ #a2 #Ha02 #a1 #Ha01 - elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/ -| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01 - elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0 - elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/ -] -qed. - -lemma TC_confluent2: ∀A,R1,R2. - confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2). -#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 -[ #a1 #Ha01 #a2 #Ha02 - elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/ -| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 - elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 - elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/ -] -qed. - -lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 → - ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & TC … R1 a a2. -#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 -[ #a0 #Ha10 #a2 #Ha02 - elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/ -| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 - elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02 - elim (IHa … Ha0) -a /4 width=3/ -] -qed. - -lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 → - ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 → - ∃∃a. TC … R2 a1 a & R1 a a2. -#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 -[ #a2 #Ha02 #a1 #Ha10 - elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/ -| #a #a2 #_ #Ha02 #IHa #a1 #Ha10 - elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0 - elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/ -] -qed. - -lemma TC_transitive2: ∀A,R1,R2. - transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2). -#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 -[ #a0 #Ha10 #a2 #Ha02 - elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3/ -| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 - elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02 - elim (IHa … Ha0) -a /4 width=3/ -] -qed. - -definition NF: ∀A. relation A → relation A → predicate A ≝ - λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1. - -inductive SN (A) (R,S:relation A): predicate A ≝ -| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1 -. - -lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a. -#A #R #S #a1 #Ha1 -@SN_intro #a2 #HRa12 #HSa12 -elim (HSa12 ?) -HSa12 /2 width=1/ -qed. - -definition NF_sn: ∀A. relation A → relation A → predicate A ≝ - λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1. - -inductive SN_sn (A) (R,S:relation A): predicate A ≝ -| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2 -. - -lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a. -#A #R #S #a2 #Ha2 -@SN_sn_intro #a1 #HRa12 #HSa12 -elim (HSa12 ?) -HSa12 /2 width=1/ -qed. - -lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R → - ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 → - ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b. -#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2 -[ #a2 #b2 #H02 - elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4/ -| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2 - elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4/ -] -qed. - -lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R → - bi_confluent A B (bi_TC … R). -#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1 -[ #a1 #b1 #H01 #a2 #b2 #H02 - elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4/ -| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02 - elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20 - elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7/ -] -qed. diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml b/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml deleted file mode 100644 index c6a00c160..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml +++ /dev/null @@ -1,45 +0,0 @@ - - -
- $(MATITA_RT_BASE_DIR) - -
-
- contribs/lambda_delta/ground_2/ - xoa - xoa_notation - basics/pts.ma - 1 2 - 1 3 - 2 1 - 2 2 - 2 3 - 3 1 - 3 2 - 3 3 - 3 4 - 4 1 - 4 2 - 4 3 - 4 4 - 4 5 - 5 2 - 5 3 - 5 4 - 5 5 - 6 4 - 6 5 - 6 6 - 6 7 - 7 7 - 3 - 4 - 3 - 4 -
- diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa.ma b/matita/matita/contribs/lambda_delta/ground_2/xoa.ma deleted file mode 100644 index ac4c8f9f7..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa.ma +++ /dev/null @@ -1,239 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The 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". - -(* multiple existental quantifier (1, 2) *) - -inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝ - | ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ? -. - -interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0). - -(* multiple existental quantifier (1, 3) *) - -inductive ex1_3 (A0,A1,A2:Type[0]) (P0:A0→A1→A2→Prop) : Prop ≝ - | ex1_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → ex1_3 ? ? ? ? -. - -interpretation "multiple existental quantifier (1, 3)" 'Ex P0 = (ex1_3 ? ? ? P0). - -(* multiple existental quantifier (2, 1) *) - -inductive ex2_1 (A0:Type[0]) (P0,P1:A0→Prop) : Prop ≝ - | ex2_1_intro: ∀x0. P0 x0 → P1 x0 → ex2_1 ? ? ? -. - -interpretation "multiple existental quantifier (2, 1)" 'Ex P0 P1 = (ex2_1 ? P0 P1). - -(* multiple existental quantifier (2, 2) *) - -inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝ - | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ? -. - -interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1). - -(* multiple existental quantifier (2, 3) *) - -inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝ - | ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ? -. - -interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? P0 P1). - -(* multiple existental quantifier (3, 1) *) - -inductive ex3_1 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝ - | ex3_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3_1 ? ? ? ? -. - -interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3_1 ? P0 P1 P2). - -(* multiple existental quantifier (3, 2) *) - -inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝ - | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ? -. - -interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2). - -(* multiple existental quantifier (3, 3) *) - -inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝ - | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2). - -(* multiple existental quantifier (3, 4) *) - -inductive ex3_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2:A0→A1→A2→A3→Prop) : Prop ≝ - | ex3_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → ex3_4 ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (3, 4)" 'Ex P0 P1 P2 = (ex3_4 ? ? ? ? P0 P1 P2). - -(* multiple existental quantifier (4, 1) *) - -inductive ex4_1 (A0:Type[0]) (P0,P1,P2,P3:A0→Prop) : Prop ≝ - | ex4_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → P3 x0 → ex4_1 ? ? ? ? ? -. - -interpretation "multiple existental quantifier (4, 1)" 'Ex P0 P1 P2 P3 = (ex4_1 ? P0 P1 P2 P3). - -(* multiple existental quantifier (4, 2) *) - -inductive ex4_2 (A0,A1:Type[0]) (P0,P1,P2,P3:A0→A1→Prop) : Prop ≝ - | ex4_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → ex4_2 ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3). - -(* multiple existental quantifier (4, 3) *) - -inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝ - | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3). - -(* multiple existental quantifier (4, 4) *) - -inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝ - | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3). - -(* multiple existental quantifier (4, 5) *) - -inductive ex4_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→A4→Prop) : Prop ≝ - | ex4_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → ex4_5 ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (4, 5)" 'Ex P0 P1 P2 P3 = (ex4_5 ? ? ? ? ? P0 P1 P2 P3). - -(* multiple existental quantifier (5, 2) *) - -inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝ - | ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4). - -(* multiple existental quantifier (5, 3) *) - -inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝ - | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4). - -(* multiple existental quantifier (5, 4) *) - -inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝ - | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4). - -(* multiple existental quantifier (5, 5) *) - -inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝ - | ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4). - -(* multiple existental quantifier (6, 4) *) - -inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝ - | ex6_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → ex6_4 ? ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5). - -(* multiple existental quantifier (6, 5) *) - -inductive ex6_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→Prop) : Prop ≝ - | ex6_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → ex6_5 ? ? ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5). - -(* multiple existental quantifier (6, 6) *) - -inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝ - | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5). - -(* multiple existental quantifier (6, 7) *) - -inductive ex6_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝ - | ex6_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → ex6_7 ? ? ? ? ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5). - -(* multiple existental quantifier (7, 7) *) - -inductive ex7_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝ - | ex7_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → P6 x0 x1 x2 x3 x4 x5 x6 → ex7_7 ? ? ? ? ? ? ? ? ? ? ? ? ? ? -. - -interpretation "multiple existental quantifier (7, 7)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6). - -(* multiple disjunction connective (3) *) - -inductive or3 (P0,P1,P2:Prop) : Prop ≝ - | or3_intro0: P0 → or3 ? ? ? - | or3_intro1: P1 → or3 ? ? ? - | or3_intro2: P2 → or3 ? ? ? -. - -interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2). - -(* multiple disjunction connective (4) *) - -inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝ - | or4_intro0: P0 → or4 ? ? ? ? - | or4_intro1: P1 → or4 ? ? ? ? - | or4_intro2: P2 → or4 ? ? ? ? - | or4_intro3: P3 → or4 ? ? ? ? -. - -interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3). - -(* multiple conjunction connective (3) *) - -inductive and3 (P0,P1,P2:Prop) : Prop ≝ - | and3_intro: P0 → P1 → P2 → and3 ? ? ? -. - -interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2). - -(* multiple conjunction connective (4) *) - -inductive and4 (P0,P1,P2,P3:Prop) : Prop ≝ - | and4_intro: P0 → P1 → P2 → P3 → and4 ? ? ? ? -. - -interpretation "multiple conjunction connective (4)" 'And P0 P1 P2 P3 = (and4 P0 P1 P2 P3). - diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma b/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma deleted file mode 100644 index 6f614f2e5..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma +++ /dev/null @@ -1,270 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was generated by xoa.native: do not edit *********************) - -(* multiple existental quantifier (1, 2) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }. - -(* multiple existental quantifier (1, 3) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) }. - -(* multiple existental quantifier (2, 1) *) - -notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) }. - -notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }. - -(* multiple existental quantifier (2, 2) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }. - -(* multiple existental quantifier (2, 3) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) }. - -(* multiple existental quantifier (3, 1) *) - -notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }. - -notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }. - -(* multiple existental quantifier (3, 2) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }. - -(* multiple existental quantifier (3, 3) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }. - -(* multiple existental quantifier (3, 4) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) }. - -(* multiple existental quantifier (4, 1) *) - -notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) (λ${ident x0}.$P3) }. - -notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) (λ${ident x0}:$T0.$P3) }. - -(* multiple existental quantifier (4, 2) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) }. - -(* multiple existental quantifier (4, 3) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }. - -(* multiple existental quantifier (4, 4) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }. - -(* multiple existental quantifier (4, 5) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) }. - -(* multiple existental quantifier (5, 2) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) (λ${ident x0}.λ${ident x1}.$P4) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }. - -(* multiple existental quantifier (5, 3) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }. - -(* multiple existental quantifier (5, 4) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }. - -(* multiple existental quantifier (5, 5) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }. - -(* multiple existental quantifier (6, 4) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }. - -(* multiple existental quantifier (6, 5) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) }. - -(* multiple existental quantifier (6, 6) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }. - -(* multiple existental quantifier (6, 7) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) }. - -(* multiple existental quantifier (7, 7) *) - -notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P6) }. - -notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" - non associative with precedence 20 - for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P6) }. - -(* multiple disjunction connective (3) *) - -notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)" - non associative with precedence 30 - for @{ 'Or $P0 $P1 $P2 }. - -(* multiple disjunction connective (4) *) - -notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)" - non associative with precedence 30 - for @{ 'Or $P0 $P1 $P2 $P3 }. - -(* multiple conjunction connective (3) *) - -notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)" - non associative with precedence 35 - for @{ 'And $P0 $P1 $P2 }. - -(* multiple conjunction connective (4) *) - -notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2 break & term 34 P3)" - non associative with precedence 35 - for @{ 'And $P0 $P1 $P2 $P3 }. - diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa_props.ma b/matita/matita/contribs/lambda_delta/ground_2/xoa_props.ma deleted file mode 100644 index 71216d1c4..000000000 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa_props.ma +++ /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 "basics/logic.ma". -include "ground_2/xoa_notation.ma". -include "ground_2/xoa.ma". - -interpretation "logical false" 'false = False. - -interpretation "logical true" 'true = True. - -lemma ex2_1_comm: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0. -#A0 #P0 #P1 * /2 width=3/ -qed. diff --git a/matita/matita/contribs/lambda_delta/ma2etc.sh b/matita/matita/contribs/lambda_delta/ma2etc.sh deleted file mode 100644 index e546af776..000000000 --- a/matita/matita/contribs/lambda_delta/ma2etc.sh +++ /dev/null @@ -1 +0,0 @@ -for FILE in `find $1 -name "*.ma"`; do svn mv $FILE ${FILE/%.ma/.etc} ; done diff --git a/matita/matita/contribs/lambda_delta/orig.sh b/matita/matita/contribs/lambda_delta/orig.sh deleted file mode 100644 index 83b1fa183..000000000 --- a/matita/matita/contribs/lambda_delta/orig.sh +++ /dev/null @@ -1,4 +0,0 @@ -F=`find $1 -name "*.ma" -or -name "*.txt"` -while read A A A; do - if grep -q "$A" $F; then true; else echo $A; fi -done diff --git a/matita/matita/contribs/lambda_delta/replace.sh b/matita/matita/contribs/lambda_delta/replace.sh deleted file mode 100644 index 5e281b251..000000000 --- a/matita/matita/contribs/lambda_delta/replace.sh +++ /dev/null @@ -1,10 +0,0 @@ -#!/bin/sh -for MA in `find -name "*.ma"`; do - echo ${MA}; sed "s!$1!$2!g" ${MA} > ${MA}.new - if diff ${MA} ${MA}.new > /dev/null; - then rm -f ${MA}.new; - else mv -f ${MA} ${MA}.old; mv -f ${MA}.new ${MA}; - fi -done - -unset MA diff --git a/matita/matita/contribs/lambda_delta/root b/matita/matita/contribs/lambda_delta/root deleted file mode 100644 index c41bf7380..000000000 --- a/matita/matita/contribs/lambda_delta/root +++ /dev/null @@ -1 +0,0 @@ -baseuri=cic:/matita/lambda_delta/ diff --git a/matita/matita/contribs/lambdadelta/Makefile b/matita/matita/contribs/lambdadelta/Makefile new file mode 100644 index 000000000..7e267a2f3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/Makefile @@ -0,0 +1,130 @@ +H = @ +XOA_DIR = ../../../components/binaries/xoa +XOA = xoa.native +DEP_DIR = ../../../components/binaries/matitadep +DEP = matitadep.native +MAC_DIR = ../../../components/binaries/mac +MAC = mac.native + +XOA_CONF = ground_2/xoa.conf.xml +XOA_TARGETS = ground_2/xoa_notation.ma ground_2/xoa.ma + +ORIG = . ./orig.sh + +ORIGS = basic_2/basic_1.orig + +PACKAGES = ground_2 basic_2 apps_2 + +all: + +# xoa ######################################################################## + +xoa: $(XOA_TARGETS) + +$(XOA_TARGETS): $(XOA_CONF) + @echo " EXEC $(XOA) $(XOA_CONF)" + $(H)MATITA_RT_BASE_DIR=../.. $(XOA_DIR)/$(XOA) $(XOA_CONF) + +# orig ####################################################################### + +orig: $(ORIGS) + @echo " ORIG basic_2" + $(H)$(ORIG) basic_2 < $(ORIGS) + +# dep ######################################################################## + +deps: MAS = $(shell find $* -name "*.ma") + +deps: $(DEP_DIR)/$(DEP) + @echo " MATITADEP" + $(H)grep "include \"" $(MAS) | $< + +# stats ###################################################################### + +stats: $(PACKAGES:%=%.stats) + +%.stats: MAS = $(shell find $* -name "*.ma") + +%.stats: CHARS = $(shell $(MAC_DIR)/$(MAC) $(MAS)) + +%.stats: + @printf '\x1B[1;40;37m' + @printf '%-15s %-40s' 'Statistics for:' $* + @printf '\x1B[0m\n' + @printf '\x1B[1;40;35m' + @printf '%-8s %6i' Chars $(CHARS) + @printf ' %-8s %3i' Pages `echo $$(($(CHARS) / 5120))` + @printf ' %-23s' '' + @printf '\x1B[0m\n' + @printf '\x1B[1;40;36m' + @printf '%-8s %6i' Sources `ls $(MAS) | wc -l` + @printf ' %-38s' '' +# @printf ' %-8s %5i' Objs `ls *.vo | wc -l` +# @printf ' %-6s %3i' Files `ls *.v | wc -l` + @printf '\x1B[0m\n' + @printf '\x1B[1;40;32m' + @printf '%-8s %6i' Theorems `grep "theorem " $(MAS) | wc -l` + @printf ' %-8s %3i' Lemmas `grep "lemma " $(MAS) | wc -l` + @printf ' %-5s %3i' Facts `grep "fact " $(MAS) | wc -l` + @printf ' %-6s %4i' Proofs `grep qed $(MAS) | wc -l` + @printf '\x1B[0m\n' + @printf '\x1B[1;40;33m' + @printf '%-8s %6i' Declared `grep "inductive \|record " $(MAS) | wc -l` + @printf ' %-8s %3i' Defined `grep "definition \|let rec " $(MAS) | wc -l` + @printf ' %-23s' '' +# @printf ' %-8s %5i' Local `grep "Local" *.v | wc -l` + @printf '\x1B[0m\n' + @printf '\x1B[1;40;31m' + @printf '%-8s %6i' Axioms `grep axiom $(MAS) | wc -l` + @printf ' %-8s %3i' Comments `grep "(\*[^*:]*$$" $(MAS) | wc -l` + @printf ' %-5s %3i' Marks `grep "(\*\*)" $(MAS) | wc -l` + @printf ' %-11s' '' + @printf '\x1B[0m\n' + +# summary #################################################################### + +define SUMMARY_TEMPLATE + TBL_$(1) := $(1)/$(1)_sum.tbl + MAS_$(1) := $$(shell find $(1) -name "*.ma") + TBLS += $$(TBL_$(1)) + + $$(TBL_$(1)): V1 := $$(shell ls $$(MAS_$(1)) | wc -l) + $$(TBL_$(1)): V2 := $$(shell $$(MAC_DIR)/$$(MAC) $$(MAS_$(1))) + $$(TBL_$(1)): C1 := $$(shell grep "inductive \|record " $$(MAS_$(1)) | wc -l) + $$(TBL_$(1)): C2 := $$(shell grep "definition \|let rec " $$(MAS_$(1)) | wc -l) + $$(TBL_$(1)): C3 := $$(shell grep "inductive \|record \|definition \|let rec " $$(MAS_$(1)) | wc -l) + $$(TBL_$(1)): P1 := $$(shell grep "theorem " $$(MAS_$(1)) | wc -l) + $$(TBL_$(1)): P2 := $$(shell grep "lemma " $$(MAS_$(1)) | wc -l) + $$(TBL_$(1)): P3 := $$(shell grep "lemma \|theorem " $$(MAS_$(1)) | wc -l) + + $$(TBL_$(1)): $$(MAS_$(1)) + @printf ' SUMMARY $(1)\n' + @printf 'name "$$(basename $$(@F))"\n\n' > $$@ + @printf 'table {\n' >> $$@ + @printf ' class "grey" [ "category"\n' >> $$@ + @printf ' [ "objects" * ]\n' >> $$@ + @printf ' ]\n' >> $$@ + @printf ' class "cyan" [ "sizes"\n' >> $$@ + @printf ' [ "files" "$$(V1)" ]\n' >> $$@ + @printf ' [ "characters" "$$(V2)" ]\n' >> $$@ + @printf ' [ * ]\n' >> $$@ + @printf ' ]\n' >> $$@ + @printf ' class "green" [ "propositions"\n' >> $$@ + @printf ' [ "theorems" "$$(P1)" ]\n' >> $$@ + @printf ' [ "lemmas" "$$(P2)" ]\n' >> $$@ + @printf ' [ "total" "$$(P3)" ]\n' >> $$@ + @printf ' ]\n' >> $$@ + @printf ' class "yellow" [ "concepts"\n' >> $$@ + @printf ' [ "declared" "$$(C1)" ]\n' >> $$@ + @printf ' [ "defined" "$$(C2)" ]\n' >> $$@ + @printf ' [ "total" "$$(C3)" ]\n' >> $$@ + @printf ' ]\n' >> $$@ + @printf '}\n\n' >> $$@ + @printf 'class "component" { 0 }\n\n' >> $$@ + @printf 'class "plane" { 1 } { 3 } { 5 }\n\n' >> $$@ + @printf 'class "number" { 2 } { 4 } { 6 }\n\n' >> $$@ +endef + +$(foreach PKG, $(PACKAGES), $(eval $(call SUMMARY_TEMPLATE,$(PKG)))) + +tbls: $(TBLS) diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/dsubst.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/dsubst.ma new file mode 100644 index 000000000..f5847371c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/dsubst.ma @@ -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/unfold/delift_lift.ma". +include "apps_2/functional/lift.ma". + +(* FUNCTIONAL DELIFTING SUBSTITUTION ****************************************) + +let rec fdsubst W d U on U ≝ match U with +[ TAtom I ⇒ match I with + [ Sort _ ⇒ U + | LRef i ⇒ tri … i d (#i) (↑[0, i] W) (#(i-1)) + | GRef _ ⇒ U + ] +| TPair I V T ⇒ match I with + [ Bind2 a I ⇒ ⓑ{a,I} (fdsubst W d V). (fdsubst W (d+1) T) + | Flat2 I ⇒ ⓕ{I} (fdsubst W d V). (fdsubst W d T) + ] +]. + +interpretation + "functional delifting substitution" + 'DSubst V d T = (fdsubst V d T). + +(* Main properties **********************************************************) + +theorem fdsubst_delift: ∀K,V,T,L,d. + ⇩[0, d] L ≡ K. ⓓV → L ⊢ ▼*[d, 1] T ≡ [d ⬐ V] T. +#K #V #T elim T -T +[ * #i #L #d #HLK normalize in ⊢ (? ? ? ? ? %); /2 width=3/ + elim (lt_or_eq_or_gt i d) #Hid + [ -HLK >(tri_lt ?????? Hid) /3 width=3/ + | destruct >tri_eq /4 width=4 by tpss_strap2, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *) + | -HLK >(tri_gt ?????? Hid) /3 width=3/ + ] +| * /3 width=1/ /4 width=1/ +] +qed. + +(* Main inversion properties ************************************************) + +theorem fdsubst_inv_delift: ∀K,V,T1,L,T2,d. ⇩[0, d] L ≡ K. ⓓV → + L ⊢ ▼*[d, 1] T1 ≡ T2 → [d ⬐ V] T1 = T2. +#K #V #T1 elim T1 -T1 +[ * #i #L #T2 #d #HLK #H + [ -HLK >(delift_inv_sort1 … H) -H // + | elim (lt_or_eq_or_gt i d) #Hid normalize + [ -HLK >(delift_inv_lref1_lt … H) -H // /2 width=1/ + | destruct + elim (delift_inv_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0 + lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H >minus_plus (delift_inv_refl_O2 … HV2) -V >(flift_inv_lift … HVT2) -V2 // + | -HLK >(delift_inv_lref1_ge … H) -H // /2 width=1/ + ] + | -HLK >(delift_inv_gref1 … H) -H // + ] +| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H + [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // /2 width=1/ + | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/lift.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/lift.ma new file mode 100644 index 000000000..bf05ea36a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/lift.ma @@ -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/substitution/lift.ma". +include "apps_2/functional/notation.ma". + +(* FUNCTIONAL RELOCATION ****************************************************) + +let rec flift d e U on U ≝ match U with +[ TAtom I ⇒ match I with + [ Sort _ ⇒ U + | LRef i ⇒ #(tri … i d i (i + e) (i + e)) + | GRef _ ⇒ U + ] +| TPair I V T ⇒ match I with + [ Bind2 a I ⇒ ⓑ{a,I} (flift d e V). (flift (d+1) e T) + | Flat2 I ⇒ ⓕ{I} (flift d e V). (flift d e T) + ] +]. + +interpretation "functional relocation" 'Lift d e T = (flift d e T). + +(* Main properties **********************************************************) + +theorem flift_lift: ∀T,d,e. ⇧[d, e] T ≡ ↑[d, e] T. +#T elim T -T +[ * #i #d #e // + elim (lt_or_eq_or_gt i d) #Hid normalize + [ >(tri_lt ?????? Hid) /2 width=1/ + | /2 width=1/ + | >(tri_gt ?????? Hid) /3 width=2/ + ] +| * /2/ +] +qed. + +(* Main inversion properties ************************************************) + +theorem flift_inv_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ↑[d, e] T1 = T2. +#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize // +[ #i #d #e #Hid >(tri_lt ?????? Hid) // +| #i #d #e #Hid + elim (le_to_or_lt_eq … Hid) -Hid #Hid + [ >(tri_gt ?????? Hid) // + | destruct // + ] +] +qed-. + +(* Derived properties *******************************************************) + +lemma flift_join: ∀e1,e2,T. ⇧[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T. +#e1 #e2 #T +lapply (flift_lift T 0 (e1+e2)) #H +elim (lift_split … H e1 e1 ? ? ?) -H // #U #H +>(flift_inv_lift … H) -H // +qed. diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/notation.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/notation.ma new file mode 100644 index 000000000..1c60d6c18 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/notation.ma @@ -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 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE "functional" COMPONENT ********************************) + +notation "hvbox( ↑ [ term 46 d , break term 46 e ] break term 46 T )" + non associative with precedence 46 + for @{ 'Lift $d $e $T }. + +notation "hvbox( [ term 46 d ⬐ break term 46 V ] break term 46 T )" + non associative with precedence 46 + for @{ 'DSubst $V $d $T }. + +notation "hvbox( T1 ⇨ break term 46 T2 )" + non associative with precedence 45 + for @{ 'SRed $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/rtm.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/rtm.ma new file mode 100644 index 000000000..c7acff72e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/rtm.ma @@ -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/grammar/term_vector.ma". +include "basic_2/grammar/genv.ma". + +(* REDUCTION AND TYPE MACHINE ***********************************************) + +(* machine local environment *) +inductive xenv: Type[0] ≝ +| XAtom: xenv (* empty *) +| XQuad: xenv → bind2 → nat → xenv → term → xenv (* entry *) +. + +interpretation "atom (ext. local environment)" + 'Star = XAtom. + +interpretation "environment construction (quad)" + 'DxItem4 L I u K V = (XQuad L I u K V). + +(* machine stack *) +definition stack: Type[0] ≝ list2 xenv term. + +(* machine status *) +record rtm: Type[0] ≝ +{ rg: genv; (* global environment *) + ru: nat; (* current de Bruijn's level *) + re: xenv; (* extended local environment *) + rs: stack; (* application stack *) + rt: term (* code *) +}. + +(* initial state *) +definition rtm_i: genv → term → rtm ≝ + λG,T. mk_rtm G 0 (⋆) (⟠) T. + +(* update code *) +definition rtm_t: rtm → term → rtm ≝ + λM,T. match M with + [ mk_rtm G u E _ _ ⇒ mk_rtm G u E (⟠) T + ]. + +(* update closure *) +definition rtm_u: rtm → xenv → term → rtm ≝ + λM,E,T. match M with + [ mk_rtm G u _ _ _ ⇒ mk_rtm G u E (⟠) T + ]. + +(* get global environment *) +definition rtm_g: rtm → genv ≝ + λM. match M with + [ mk_rtm G _ _ _ _ ⇒ G + ]. + +(* get local reference level *) +definition rtm_l: rtm → nat ≝ + λM. match M with + [ mk_rtm _ u E _ _ ⇒ match E with + [ XAtom ⇒ u + | XQuad _ _ u _ _ ⇒ u + ] + ]. + +(* get stack *) +definition rtm_s: rtm → stack ≝ + λM. match M with + [ mk_rtm _ _ _ S _ ⇒ S + ]. + +(* get code *) +definition rtm_c: rtm → term ≝ + λM. match M with + [ mk_rtm _ _ _ _ T ⇒ T + ]. diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/rtm_step.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/rtm_step.ma new file mode 100644 index 000000000..ed16d5091 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/rtm_step.ma @@ -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 "apps_2/functional/rtm.ma". + +(* REDUCTION AND TYPE MACHINE ***********************************************) + +(* transitions *) +inductive rtm_step: relation rtm ≝ +| rtm_ldrop : ∀G,u,E,I,t,F,V,S,i. + rtm_step (mk_rtm G u (E. ④{I} {t, F, V}) S (#(i + 1))) + (mk_rtm G u E S (#i)) +| rtm_ldelta: ∀G,u,E,t,F,V,S. + rtm_step (mk_rtm G u (E. ④{Abbr} {t, F, V}) S (#0)) + (mk_rtm G u F S V) +| rtm_ltype : ∀G,u,E,t,F,V,S. + rtm_step (mk_rtm G u (E. ④{Abst} {t, F, V}) S (#0)) + (mk_rtm G u F S V) +| rtm_gdrop : ∀G,I,V,u,E,S,p. p < |G| → + rtm_step (mk_rtm (G. ⓑ{I} V) u E S (§p)) + (mk_rtm G u E S (§p)) +| rtm_gdelta: ∀G,V,u,E,S,p. p = |G| → + rtm_step (mk_rtm (G. ⓓV) u E S (§p)) + (mk_rtm G u E S V) +| rtm_gtype : ∀G,V,u,E,S,p. p = |G| → + rtm_step (mk_rtm (G. ⓛV) u E S (§p)) + (mk_rtm G u E S V) +| rtm_tau : ∀G,u,E,S,W,T. + rtm_step (mk_rtm G u E S (ⓝW. T)) + (mk_rtm G u E S T) +| rtm_appl : ∀G,u,E,S,V,T. + rtm_step (mk_rtm G u E S (ⓐV. T)) + (mk_rtm G u E ({E, V} @ S) T) +| rtm_beta : ∀G,u,E,F,V,S,W,T. + rtm_step (mk_rtm G u E ({F, V} @ S) (+ⓛW. T)) + (mk_rtm G u (E. ④{Abbr} {u, F, V}) S T) +| rtm_push : ∀G,u,E,W,T. + rtm_step (mk_rtm G u E ⟠ (+ⓛW. T)) + (mk_rtm G (u + 1) (E. ④{Abst} {u, E, W}) ⟠ T) +| rtm_theta : ∀G,u,E,S,V,T. + rtm_step (mk_rtm G u E S (+ⓓV. T)) + (mk_rtm G u (E. ④{Abbr} {u, E, V}) S T) +. + +interpretation "sequential reduction (RTM)" + 'SRed O1 O2 = (rtm_step O1 O2). diff --git a/matita/matita/contribs/lambdadelta/basic_2/basic_1.orig b/matita/matita/contribs/lambdadelta/basic_2/basic_1.orig new file mode 100644 index 000000000..0b48e942f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/basic_1.orig @@ -0,0 +1,712 @@ +aplus/props / aplus_ahead_simpl +aplus/props / aplus_asort_le_simpl +aplus/props / aplus_asort_O_simpl +aplus/props / aplus_asort_simpl +aplus/props / aplus_assoc +aplus/props / aplus_asucc +aplus/props / aplus_asucc_false +aplus/props / aplus_inj +aplus/props / aplus_reg_r +aplus/props / aplus_sort_O_S_simpl +aplus/props / aplus_sort_S_S_simpl +aprem/fwd / aprem_gen_head_O +aprem/fwd / aprem_gen_head_S +aprem/fwd / aprem_gen_sort +aprem/props / aprem_asucc +aprem/props / aprem_repl +arity/aprem / arity_aprem +arity/cimp / arity_cimp_conf +arity/fwd / arity_gen_abst +arity/fwd / arity_gen_appl +arity/fwd / arity_gen_appls +arity/fwd / arity_gen_bind +arity/fwd / arity_gen_cast +arity/fwd / arity_gen_lift +arity/fwd / arity_gen_lref +arity/fwd / arity_gen_sort +arity/lift1 / arity_lift1 +arity/pr3 / arity_sred_pr2 +arity/pr3 / arity_sred_pr3 +arity/pr3 / arity_sred_wcpr0_pr0 +arity/pr3 / arity_sred_wcpr0_pr1 +arity/props / arity_appls_abbr +arity/props / arity_appls_bind +arity/props / arity_appls_cast +arity/props / arity_lift +arity/props / arity_mono +arity/props / arity_repellent +arity/props / node_inh +arity/subst0 / arity_fsubst0 +arity/subst0 / arity_gen_cvoid +arity/subst0 / arity_gen_cvoid_subst0 +arity/subst0 / arity_subst0 +asucc/fwd / asucc_gen_head +asucc/fwd / asucc_gen_sort +cimp/props / cimp_bind +cimp/props / cimp_flat_dx +cimp/props / cimp_flat_sx +cimp/props / cimp_getl_conf +clear/drop / drop_clear +clear/drop / drop_clear_O +clear/drop / drop_clear_S +clear/fwd / clear_gen_all +clear/fwd / clear_gen_bind +clear/fwd / clear_gen_flat +clear/fwd / clear_gen_flat_r +clear/fwd / clear_gen_sort +clear/props / clear_cle +clear/props / clear_clear +clear/props / clear_ctail +clear/props / clear_mono +clear/props / clear_trans +clen/getl / getl_ctail_clen +clen/getl / getl_gen_tail +cnt/props / cnt_lift +C/props / chead_ctail +C/props / clt_cong +C/props / clt_head +C/props / clt_thead +C/props / clt_wf_ind +C/props clt_wf q_ind +C/props / c_tail_ind +csuba/arity / arity_appls_appl +csuba/arity / csuba_arity +csuba/arity / csuba_arity_rev +csuba/clear / csuba_clear_conf +csuba/clear / csuba_clear_trans +csuba/drop / csuba_drop_abbr +csuba/drop / csuba_drop_abbr_rev +csuba/drop / csuba_drop_abst +csuba/drop / csuba_drop_abst_rev +csuba/fwd / csuba_gen_abbr +csuba/fwd / csuba_gen_abbr_rev +csuba/fwd / csuba_gen_abst +csuba/fwd / csuba_gen_abst_rev +csuba/fwd / csuba_gen_bind +csuba/fwd / csuba_gen_bind_rev +csuba/fwd / csuba_gen_flat +csuba/fwd / csuba_gen_flat_rev +csuba/fwd / csuba_gen_void +csuba/fwd / csuba_gen_void_rev +csuba/getl / csuba_getl_abbr +csuba/getl / csuba_getl_abbr_rev +csuba/getl / csuba_getl_abst +csuba/getl / csuba_getl_abst_rev +csuba/props / csuba_refl +csubc/arity / csubc_arity_conf +csubc/arity / csubc_arity_trans +csubc/clear / csubc_clear_conf +csubc/csuba / csubc_csuba +csubc/drop1 / csubc_drop1_conf_rev +csubc/drop1 / drop1_csubc_trans +csubc/drop / csubc_drop_conf_O +csubc/drop / csubc_drop_conf_rev +csubc/drop / drop_csubc_trans +csubc/fwd / csubc_gen_head_l +csubc/fwd / csubc_gen_head_r +csubc/fwd / csubc_gen_sort_l +csubc/fwd / csubc_gen_sort_r +csubc/getl / csubc_getl_conf +csubc/props / csubc_refl +csubst0/clear / csubst0_clear_O +csubst0/clear / csubst0_clear_O_back +csubst0/clear / csubst0_clear_S +csubst0/clear / csubst0_clear_trans +csubst0/drop / csubst0_drop_eq +csubst0/drop / csubst0_drop_eq_back +csubst0/drop / csubst0_drop_gt +csubst0/drop / csubst0_drop_gt_back +csubst0/drop / csubst0_drop_lt +csubst0/drop / csubst0_drop_lt_back +csubst0/fwd / csubst0_gen_head +csubst0/fwd / csubst0_gen_S_bind_2 +csubst0/fwd / csubst0_gen_sort +csubst0/getl / csubst0_getl_ge +csubst0/getl / csubst0_getl_ge_back +csubst0/getl / csubst0_getl_lt +csubst0/getl / csubst0_getl_lt_back +csubst0/props / csubst0_both_bind +csubst0/props / csubst0_fst_bind +csubst0/props / csubst0_snd_bind +csubst1/fwd / csubst1_gen_head +csubst1/getl / csubst1_getl_ge +csubst1/getl / csubst1_getl_ge_back +csubst1/getl / csubst1_getl_lt +csubst1/getl / getl_csubst1 +csubst1/props / csubst1_bind +csubst1/props / csubst1_flat +csubst1/props / csubst1_head +csubt/clear / csubt_clear_conf +csubt/csuba / csubt_csuba +csubt/drop / csubt_drop_abbr +csubt/drop / csubt_drop_abst +csubt/drop / csubt_drop_flat +csubt/fwd / csubt_gen_abbr +csubt/fwd / csubt_gen_abst +csubt/fwd / csubt_gen_bind +csubt/fwd / csubt_gen_flat +csubt/getl / csubt_getl_abbr +csubt/getl / csubt_getl_abst +csubt/pc3 / csubt_pc3 +csubt/pc3 / csubt_pr2 +csubt/props / csubt_refl +csubt/ty3 / csubt_ty3 +csubt/ty3 / csubt_ty3_ld +csubv/clear / csubv_clear_conf +csubv/clear / csubv_clear_conf_void +csubv/drop / csubv_drop_conf +csubv/getl / csubv_getl_conf +csubv/getl / csubv_getl_conf_void +csubv/props / csubv_bind_same +csubv/props / csubv_refl +drop1/fwd / drop1_gen_pcons +drop1/fwd / drop1_gen_pnil +drop1/getl / drop1_getl_trans +drop1/props / drop1_cons_tail +drop1/props / drop1_skip_bind +drop1/props / drop1_trans +drop/fwd / drop_gen_drop +drop/fwd / drop_gen_refl +drop/fwd / drop_gen_skip_l +drop/fwd / drop_gen_skip_r +drop/fwd / drop_gen_sort +drop/props / drop_conf_ge +drop/props / drop_conf_lt +drop/props / drop_conf_rev +drop/props / drop_ctail +drop/props / drop_mono +drop/props / drop_S +drop/props / drop_skip_bind +drop/props / drop_skip_flat +drop/props / drop_trans_ge +drop/props / drop_trans_le +ex0/props / aplus_gz_ge +ex0/props / aplus_gz_le +ex0/props / leq_leqz +ex0/props / leqz_leq +ex0/props / next_plus_gz +ex1/props / ex1_arity +ex1/props ex1 leq_sort_SS +ex1/props / ex1_ty3 +ex2/props / ex2_arity +ex2/props / ex2_nf2 +flt/props / flt_arith0 +flt/props / flt_arith1 +flt/props / flt_arith2 +flt/props / flt_shift +flt/props / flt_thead_dx +flt/props / flt_thead_sx +flt/props / flt_trans +flt/props / flt_wf_ind +flt/props flt_wf q_ind +fsubst0/fwd / fsubst0_gen_base +getl/clear / clear_getl_trans +getl/clear / getl_clear_bind +getl/clear / getl_clear_conf +getl/clear / getl_clear_trans +getl/dec / getl_dec +getl/drop / drop_getl_trans_ge +getl/drop / drop_getl_trans_le +getl/drop / drop_getl_trans_lt +getl/drop / getl_conf_ge_drop +getl/drop / getl_drop +getl/drop / getl_drop_conf_ge +getl/drop / getl_drop_conf_lt +getl/drop / getl_drop_conf_rev +getl/drop / getl_drop_trans +getl/flt / getl_flt +getl/fwd / getl_gen_2 +getl/fwd / getl_gen_all +getl/fwd / getl_gen_bind +getl/fwd / getl_gen_flat +getl/fwd / getl_gen_O +getl/fwd / getl_gen_S +getl/fwd / getl_gen_sort +getl/getl / getl_conf_le +getl/getl / getl_trans +getl/props / getl_ctail +getl/props / getl_flat +getl/props / getl_head +getl/props / getl_mono +getl/props / getl_refl +iso/fwd / iso_flats_flat_bind_false +iso/fwd / iso_flats_lref_bind_false +iso/fwd / iso_gen_head +iso/fwd / iso_gen_lref +iso/fwd / iso_gen_sort +iso/props / iso_refl +iso/props / iso_trans +leq/asucc / asucc_inj +leq/asucc / asucc_repl +leq/asucc / leq_ahead_asucc_false +leq/asucc / leq_asucc +leq/asucc / leq_asucc_false +leq/fwd / leq_gen_head1 +leq/fwd / leq_gen_head2 +leq/fwd / leq_gen_sort1 +leq/fwd / leq_gen_sort2 +leq/props / ahead_inj_snd +leq/props / leq_ahead_false_1 +leq/props / leq_ahead_false_2 +leq/props / leq_eq +leq/props / leq_refl +leq/props / leq_sym +leq/props / leq_trans +lift1/fwd / lift1_bind +lift1/fwd / lift1_cons_tail +lift1/fwd / lift1_flat +lift1/fwd / lift1_lref +lift1/fwd / lift1_sort +lift1/fwd / lifts1_cons +lift1/fwd / lifts1_flat +lift1/fwd / lifts1_nil +lift1/props / lift1_free +lift1/props / lift1_lift1 +lift1/props / lift1_xhg +lift1/props / lifts1_xhg +lift/fwd / lift_bind +lift/fwd / lift_flat +lift/fwd / lift_gen_bind +lift/fwd / lift_gen_flat +lift/fwd / lift_gen_head +lift/fwd / lift_gen_lref +lift/fwd / lift_gen_lref_false +lift/fwd / lift_gen_lref_ge +lift/fwd / lift_gen_lref_lt +lift/fwd / lift_gen_sort +lift/fwd / lift_head +lift/fwd / lift_lref_ge +lift/fwd / lift_lref_lt +lift/fwd / lift_sort +lift/props / lift_d +lift/props / lift_free +lift/props / lift_gen_lift +lift/props / lift_inj +lift/props / lift_lref_gt +lift/props / lift_r +lift/props / lifts_inj +lift/props / lifts_tapp +lift/props / thead_x_lift_y_y +lift/tlt / lift_tlt_dx +lift/tlt / lift_weight +lift/tlt / lift_weight_add +lift/tlt / lift_weight_add_O +lift/tlt / lift_weight_map +llt/props / llt_head_dx +llt/props / llt_head_sx +llt/props / llt_repl +llt/props / llt_trans +llt/props / llt_wf_ind +llt/props llt_wf q_ind +llt/props / lweight_repl +next_plus/props / next_plus_assoc +next_plus/props / next_plus_lt +next_plus/props / next_plus_next +nf2/arity / arity_nf2_inv_all +nf2/dec / nf2_dec +nf2/fwd / nf2_gen_abbr +nf2/fwd / nf2_gen_abst +nf2/fwd / nf2_gen_beta +nf2/fwd / nf2_gen_cast +nf2/fwd / nf2_gen_flat +nf2/fwd / nf2_gen_lref +nf2/fwd nf2_gen nf2_gen_aux +nf2/fwd / nf2_gen_void +nf2/iso / nf2_iso_appls_lref +nf2/lift1 / nf2_lift1 +nf2/pr3 / nf2_pr3_confluence +nf2/pr3 / nf2_pr3_unfold +nf2/props / nf2_abst +nf2/props / nf2_abst_shift +nf2/props / nf2_appl_lref +nf2/props / nf2_appls_lref +nf2/props / nf2_csort_lref +nf2/props / nf2_lift +nf2/props / nf2_lref_abst +nf2/props / nf2_sort +nf2/props / nfs2_tapp +pc1/props / pc1_head +pc1/props / pc1_head_1 +pc1/props / pc1_head_2 +pc1/props / pc1_pr0_r +pc1/props / pc1_pr0_u +pc1/props / pc1_pr0_u2 +pc1/props / pc1_pr0_x +pc1/props / pc1_refl +pc1/props / pc1_s +pc1/props / pc1_t +pc3/dec / pc3_abst_dec +pc3/dec / pc3_dec +pc3/fsubst0 / pc3_fsubst0 +pc3/fsubst0 / pc3_pr2_fsubst0 +pc3/fsubst0 / pc3_pr2_fsubst0_back +pc3/fwd / pc3_gen_abst +pc3/fwd / pc3_gen_abst_shift +pc3/fwd / pc3_gen_lift +pc3/fwd / pc3_gen_lift_abst +pc3/fwd / pc3_gen_not_abst +pc3/fwd / pc3_gen_sort +pc3/fwd / pc3_gen_sort_abst +pc3/left / pc3_ind_left +pc3/left pc3_ind_left pc3_left_pc3 +pc3/left pc3_ind_left pc3_left_pr3 +pc3/left pc3_ind_left pc3_left_sym +pc3/left pc3_ind_left pc3_left_trans +pc3/left pc3_ind_left pc3_pc3_left +pc3/nf2 / pc3_nf2 +pc3/nf2 / pc3_nf2_unfold +pc3/pc1 / pc3_pc1 +pc3/props / clear_pc3_trans +pc3/props / pc3_eta +pc3/props / pc3_head_1 +pc3/props / pc3_head_12 +pc3/props / pc3_head_2 +pc3/props / pc3_head_21 +pc3/props / pc3_lift +pc3/props / pc3_pr0_pr2_t +pc3/props / pc3_pr2_pr2_t +pc3/props / pc3_pr2_pr3_t +pc3/props / pc3_pr2_r +pc3/props / pc3_pr2_u +pc3/props / pc3_pr2_u2 +pc3/props / pc3_pr2_x +pc3/props / pc3_pr3_conf +pc3/props / pc3_pr3_pc3_t +pc3/props / pc3_pr3_r +pc3/props / pc3_pr3_t +pc3/props / pc3_pr3_x +pc3/props / pc3_refl +pc3/props / pc3_s +pc3/props / pc3_t +pc3/props / pc3_thin_dx +pc3/subst1 / pc3_gen_cabbr +pc3/wcpr0 / pc3_wcpr0 +pc3/wcpr0 pc3_wcpr0 pc3_wcpr0_t_aux +pc3/wcpr0 / pc3_wcpr0_t +pr0/dec / nf0_dec +pr0/fwd / pr0_gen_abbr +pr0/fwd / pr0_gen_abst +pr0/fwd / pr0_gen_appl +pr0/fwd / pr0_gen_cast +pr0/fwd / pr0_gen_lift +pr0/fwd / pr0_gen_lref +pr0/fwd / pr0_gen_sort +pr0/fwd / pr0_gen_void +pr0/pr0 / pr0_confluence +pr0/pr0 pr0_confluence pr0_cong_delta +pr0/pr0 pr0_confluence pr0_cong_upsilon_cong +pr0/pr0 pr0_confluence pr0_cong_upsilon_delta +pr0/pr0 pr0_confluence pr0_cong_upsilon_refl +pr0/pr0 pr0_confluence pr0_cong_upsilon_zeta +pr0/pr0 pr0_confluence pr0_delta_delta +pr0/pr0 pr0_confluence pr0_delta_tau +pr0/pr0 pr0_confluence pr0_upsilon_upsilon +pr0/props / pr0_lift +pr0/props / pr0_subst0 +pr0/props / pr0_subst0_back +pr0/props / pr0_subst0_fwd +pr0/subst1 / pr0_delta1 +pr0/subst1 / pr0_subst1 +pr0/subst1 / pr0_subst1_back +pr0/subst1 / pr0_subst1_fwd +pr1/pr1 / pr1_confluence +pr1/pr1 / pr1_strip +pr1/props / pr1_comp +pr1/props / pr1_eta +pr1/props / pr1_head_1 +pr1/props / pr1_head_2 +pr1/props / pr1_pr0 +pr1/props / pr1_t +pr2/clen / pr2_gen_cbind +pr2/clen / pr2_gen_cflat +pr2/clen / pr2_gen_ctail +pr2/fwd / pr2_gen_abbr +pr2/fwd / pr2_gen_abst +pr2/fwd / pr2_gen_appl +pr2/fwd / pr2_gen_cast +pr2/fwd / pr2_gen_csort +pr2/fwd / pr2_gen_lift +pr2/fwd / pr2_gen_lref +pr2/fwd / pr2_gen_sort +pr2/fwd / pr2_gen_void +pr2/pr2 / pr2_confluence +pr2/pr2 pr2_confluence pr2_delta_delta +pr2/pr2 pr2_confluence pr2_free_delta +pr2/pr2 pr2_confluence pr2_free_free +pr2/props / clear_pr2_trans +pr2/props / pr2_cflat +pr2/props / pr2_change +pr2/props / pr2_ctail +pr2/props / pr2_head_1 +pr2/props / pr2_head_2 +pr2/props / pr2_lift +pr2/props / pr2_thin_dx +pr2/subst1 / pr2_delta1 +pr2/subst1 / pr2_gen_cabbr +pr2/subst1 / pr2_subst1 +pr3/fwd / pr3_gen_abbr +pr3/fwd / pr3_gen_abst +pr3/fwd / pr3_gen_appl +pr3/fwd / pr3_gen_bind +pr3/fwd / pr3_gen_cast +pr3/fwd / pr3_gen_lift +pr3/fwd / pr3_gen_lref +pr3/fwd / pr3_gen_sort +pr3/fwd / pr3_gen_void +pr3/iso / pr3_iso_appl_bind +pr3/iso / pr3_iso_appls_abbr +pr3/iso / pr3_iso_appls_appl_bind +pr3/iso / pr3_iso_appls_beta +pr3/iso / pr3_iso_appls_bind +pr3/iso / pr3_iso_appls_cast +pr3/iso / pr3_iso_beta +pr3/pr1 / pr3_pr1 +pr3/pr3 / pr3_confluence +pr3/pr3 / pr3_strip +pr3/props / clear_pr3_trans +pr3/props / pr3_cflat +pr3/props / pr3_eta +pr3/props / pr3_flat +pr3/props / pr3_head_1 +pr3/props / pr3_head_12 +pr3/props / pr3_head_2 +pr3/props / pr3_head_21 +pr3/props / pr3_lift +pr3/props / pr3_pr0_pr2_t +pr3/props / pr3_pr2 +pr3/props / pr3_pr2_pr2_t +pr3/props / pr3_pr2_pr3_t +pr3/props / pr3_pr3_pr3_t +pr3/props / pr3_t +pr3/props / pr3_thin_dx +pr3/subst1 / pr3_gen_cabbr +pr3/subst1 / pr3_subst1 +pr3/wcpr0 / pr3_wcpr0_t +r/props / r_arith0 +r/props / r_arith1 +r/props / r_dis +r/props / r_minus +r/props / r_plus +r/props / r_plus_sym +r/props / r_S +r/props / s_r +sc3/arity / sc3_arity +sc3/arity / sc3_arity_csubc +sc3/props / sc3_abbr +sc3/props / sc3_abst +sc3/props / sc3_appl +sc3/props / sc3_arity_gen +sc3/props / sc3_bind +sc3/props / sc3_cast +sc3/props / sc3_lift +sc3/props / sc3_lift1 +sc3/props sc3_props sc3_sn3_abst +sc3/props / sc3_repl +sc3/props / sc3_sn3 +sn3/fwd / sn3_gen_bind +sn3/fwd / sn3_gen_cflat +sn3/fwd / sn3_gen_flat +sn3/fwd / sn3_gen_head +sn3/fwd / sn3_gen_lift +sn3/lift1 / sns3_lifts1 +sn3/nf2 / nf2_sn3 +sn3/nf2 / sn3_nf2 +sn3/props / sn3_abbr +sn3/props / sn3_appl_abbr +sn3/props / sn3_appl_appl +sn3/props / sn3_appl_appls +sn3/props / sn3_appl_beta +sn3/props / sn3_appl_bind +sn3/props / sn3_appl_cast +sn3/props / sn3_appl_lref +sn3/props / sn3_appls_abbr +sn3/props / sn3_appls_beta +sn3/props / sn3_appls_bind +sn3/props / sn3_appls_cast +sn3/props / sn3_appls_lref +sn3/props / sn3_beta +sn3/props / sn3_bind +sn3/props / sn3_cast +sn3/props / sn3_cdelta +sn3/props / sn3_cflat +sn3/props / sn3_change +sn3/props / sn3_cpr3_trans +sn3/props / sn3_gen_def +sn3/props / sn3_lift +sn3/props / sn3_pr2_intro +sn3/props / sn3_pr3_trans +sn3/props / sn3_shift +sn3/props / sns3_lifts +s/props / minus_s_s +s/props / s_arith0 +s/props / s_arith1 +s/props / s_inc +s/props / s_inj +s/props / s_le +s/props / s_lt +s/props / s_minus +s/props / s_plus +s/props / s_plus_sym +s/props / s_S +sty0/fwd / sty0_gen_appl +sty0/fwd / sty0_gen_bind +sty0/fwd / sty0_gen_cast +sty0/fwd / sty0_gen_lref +sty0/fwd / sty0_gen_sort +sty0/props / sty0_correct +sty0/props / sty0_lift +sty1/cnt / sty1_cnt +sty1/props / sty1_abbr +sty1/props / sty1_appl +sty1/props / sty1_bind +sty1/props / sty1_cast2 +sty1/props / sty1_correct +sty1/props / sty1_lift +sty1/props / sty1_trans +subst0/dec / dnf_dec +subst0/dec / dnf_dec2 +subst0/fwd / subst0_gen_head +subst0/fwd / subst0_gen_lift_false +subst0/fwd / subst0_gen_lift_ge +subst0/fwd / subst0_gen_lift_lt +subst0/fwd / subst0_gen_lref +subst0/fwd / subst0_gen_sort +subst0/props / subst0_lift_ge +subst0/props / subst0_lift_ge_s +subst0/props / subst0_lift_ge_S +subst0/props / subst0_lift_lt +subst0/props / subst0_refl +subst0/subst0 / subst0_confluence_eq +subst0/subst0 / subst0_confluence_lift +subst0/subst0 / subst0_confluence_neq +subst0/subst0 / subst0_subst0 +subst0/subst0 / subst0_subst0_back +subst0/subst0 / subst0_trans +subst0/tlt / subst0_tlt +subst0/tlt / subst0_tlt_head +subst0/tlt / subst0_weight_le +subst0/tlt / subst0_weight_lt +subst1/fwd / subst1_gen_head +subst1/fwd / subst1_gen_lift_eq +subst1/fwd / subst1_gen_lift_ge +subst1/fwd / subst1_gen_lift_lt +subst1/fwd / subst1_gen_lref +subst1/fwd / subst1_gen_sort +subst1/props / subst1_ex +subst1/props / subst1_head +subst1/props / subst1_lift_ge +subst1/props / subst1_lift_lt +subst1/props / subst1_lift_S +subst1/subst1 / subst1_confluence_eq +subst1/subst1 / subst1_confluence_lift +subst1/subst1 / subst1_confluence_neq +subst1/subst1 / subst1_subst1 +subst1/subst1 / subst1_subst1_back +subst1/subst1 / subst1_trans +subst/fwd / subst_head +subst/fwd / subst_lref_eq +subst/fwd / subst_lref_gt +subst/fwd / subst_lref_lt +subst/fwd / subst_sort +subst/props / subst_lift_SO +subst/props / subst_subst0 +T/dec / abst_dec +T/dec / bind_dec_not +T/dec / binder_dec +T/dec / term_dec +T/dec terms_props bind_dec +T/dec terms_props flat_dec +T/dec terms_props kind_dec +tlist/props / tcons_tapp_ex +tlist/props / theads_tapp +tlist/props / tlist_ind_rev +tlist/props / tslt_wf_ind +tlist/props tslt_wf q_ind +tlt/props / tlt_head_dx +tlt/props / tlt_head_sx +tlt/props / tlt_trans +tlt/props / tlt_wf_ind +tlt/props tlt_wf q_ind +tlt/props / wadd_le +tlt/props / wadd_lt +tlt/props / wadd_O +tlt/props / weight_add_O +tlt/props / weight_add_S +tlt/props / weight_eq +tlt/props / weight_le +T/props / not_abbr_abst +T/props / not_abbr_void +T/props / not_abst_void +T/props / not_void_abst +T/props / thead_x_y_y +T/props / tweight_lt +ty3/arity / ty3_arity +ty3/arity_props / ty3_acyclic +ty3/arity_props / ty3_predicative +ty3/arity_props / ty3_repellent +ty3/arity_props / ty3_sn3 +ty3/dec / ty3_inference +ty3/fsubst0 / ty3_csubst0 +ty3/fsubst0 / ty3_fsubst0 +ty3/fsubst0 / ty3_subst0 +ty3/fwd / ty3_gen_appl +ty3/fwd / ty3_gen_bind +ty3/fwd / ty3_gen_cast +ty3/fwd / ty3_gen_lref +ty3/fwd / ty3_gen_sort +ty3/fwd / tys3_gen_cons +ty3/fwd / tys3_gen_nil +ty3/fwd_nf2 / ty3_gen_appl_nf2 +ty3/fwd_nf2 / ty3_inv_appls_lref_nf2 +ty3/fwd_nf2 / ty3_inv_lref_lref_nf2 +ty3/fwd_nf2 / ty3_inv_lref_nf2 +ty3/fwd_nf2 / ty3_inv_lref_nf2_pc3 +ty3/nf2 ty3_nf2_gen ty3_nf2_inv_abst_aux +ty3/nf2 / ty3_nf2_inv_abst +ty3/nf2 / ty3_nf2_inv_abst_premise_csort +ty3/nf2 / ty3_nf2_inv_all +ty3/nf2 / ty3_nf2_inv_sort +ty3/pr3 / ty3_sred_pr0 +ty3/pr3 / ty3_sred_pr1 +ty3/pr3 / ty3_sred_pr2 +ty3/pr3 / ty3_sred_pr3 +ty3/pr3 / ty3_sred_wcpr0_pr0 +ty3/pr3_props / ty3_cred_pr2 +ty3/pr3_props / ty3_cred_pr3 +ty3/pr3_props / ty3_gen_lift +ty3/pr3_props / ty3_sconv +ty3/pr3_props / ty3_sconv_pc3 +ty3/pr3_props / ty3_sred_back +ty3/pr3_props / ty3_tred +ty3/props / ty3_correct +ty3/props / ty3_gen_abst_abst +ty3/props / ty3_getl_subst0 +ty3/props / ty3_lift +ty3/props / ty3_typecheck +ty3/props / ty3_unique +ty3/sty0 / ty3_sty0 +ty3/subst1 / ty3_gen_cabbr +ty3/subst1 / ty3_gen_cvoid +wcpr0/fwd / wcpr0_gen_head +wcpr0/fwd / wcpr0_gen_sort +wcpr0/getl / wcpr0_drop +wcpr0/getl / wcpr0_drop_back +wcpr0/getl / wcpr0_getl +wcpr0/getl / wcpr0_getl_back +wf3/clear / clear_wf3_trans +wf3/clear / wf3_clear_conf +wf3/fwd / wf3_gen_bind1 +wf3/fwd / wf3_gen_flat1 +wf3/fwd / wf3_gen_head2 +wf3/fwd / wf3_gen_sort1 +wf3/getl / getl_wf3_trans +wf3/getl / wf3_getl_conf +wf3/props / ty3_shift1 +wf3/props / wf3_idem +wf3/props / wf3_mono +wf3/props / wf3_total +wf3/props / wf3_ty3 +wf3/ty3 / wf3_pc3_conf +wf3/ty3 / wf3_pr2_conf +wf3/ty3 / wf3_pr3_conf +wf3/ty3 / wf3_ty3_conf diff --git a/matita/matita/contribs/lambdadelta/basic_2/basic_1.txt b/matita/matita/contribs/lambdadelta/basic_2/basic_1.txt new file mode 100644 index 000000000..d64855d0d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/basic_1.txt @@ -0,0 +1,244 @@ +# waiting #################################################################### + +aplus/props aplus_reg_r +aplus/props aplus_assoc +aplus/props aplus_asucc +aplus/props aplus_sort_O_S_simpl +aplus/props aplus_sort_S_S_simpl +aplus/props aplus_asort_O_simpl +aplus/props aplus_asort_le_simpl +aplus/props aplus_asort_simpl +aplus/props aplus_ahead_simpl +aplus/props aplus_asucc_false +aplus/props aplus_inj +aprem/fwd aprem_gen_sort +aprem/fwd aprem_gen_head_O +aprem/fwd aprem_gen_head_S +aprem/props aprem_repl +aprem/props aprem_asucc +arity/aprem arity_aprem +arity/cimp arity_cimp_conf +arity/fwd arity_gen_sort +arity/fwd arity_gen_lref +arity/fwd arity_gen_bind +arity/fwd arity_gen_abst +arity/fwd arity_gen_appl +arity/fwd arity_gen_cast +arity/fwd arity_gen_appls +arity/fwd arity_gen_lift +arity/lift1 arity_lift1 +arity/pr3 arity_sred_wcpr0_pr0 +arity/pr3 arity_sred_wcpr0_pr1 +arity/pr3 arity_sred_pr2 +arity/pr3 arity_sred_pr3 +arity/props node_inh +arity/props arity_lift +arity/props arity_mono +arity/props arity_repellent +arity/props arity_appls_cast +arity/props arity_appls_abbr +arity/props arity_appls_bind +arity/subst0 arity_gen_cvoid_subst0 +arity/subst0 arity_gen_cvoid +arity/subst0 arity_fsubst0 +arity/subst0 arity_subst0 +asucc/fwd asucc_gen_sort +asucc/fwd asucc_gen_head +cnt/props cnt_lift +C/props clt_wf__q_ind +C/props clt_wf_ind + +csuba/arity csuba_arity +csuba/arity csuba_arity_rev +csuba/arity arity_appls_appl +csuba/clear csuba_clear_conf +csuba/clear csuba_clear_trans +csuba/drop csuba_drop_abbr +csuba/drop csuba_drop_abst +csuba/drop csuba_drop_abst_rev +csuba/drop csuba_drop_abbr_rev +csuba/fwd csuba_gen_abbr +csuba/fwd csuba_gen_void +csuba/fwd csuba_gen_abst +csuba/fwd csuba_gen_flat +csuba/fwd csuba_gen_bind +csuba/fwd csuba_gen_abst_rev +csuba/fwd csuba_gen_void_rev +csuba/fwd csuba_gen_abbr_rev +csuba/fwd csuba_gen_flat_rev +csuba/fwd csuba_gen_bind_rev +csuba/getl csuba_getl_abbr +csuba/getl csuba_getl_abst +csuba/getl csuba_getl_abst_rev +csuba/getl csuba_getl_abbr_rev +csuba/props csuba_refl + +csubc/arity csubc_arity_conf +csubc/arity csubc_arity_trans +csubc/drop1 drop1_csubc_trans +csubc/drop drop_csubc_trans + +csubt/csuba csubt_csuba +csubt/fwd csubt_gen_abbr +csubt/fwd csubt_gen_abst + +csubv/clear csubv_clear_conf +csubv/clear csubv_clear_conf_void +csubv/drop csubv_drop_conf +csubv/getl csubv_getl_conf +csubv/getl csubv_getl_conf_void +csubv/props csubv_bind_same +csubv/props csubv_refl +drop1/props drop1_cons_tail +ex0/props aplus_gz_le +ex0/props aplus_gz_ge +ex0/props next_plus_gz +ex0/props leqz_leq +ex0/props leq_leqz +ex1/props ex1__leq_sort_SS +ex1/props ex1_arity +ex1/props ex1_ty3 +ex2/props ex2_nf2 +ex2/props ex2_arity +leq/asucc asucc_repl +leq/asucc asucc_inj +leq/asucc leq_asucc +leq/asucc leq_ahead_asucc_false +leq/asucc leq_asucc_false +leq/fwd leq_gen_sort1 +leq/fwd leq_gen_head1 +leq/fwd leq_gen_sort2 +leq/fwd leq_gen_head2 +leq/props ahead_inj_snd +leq/props leq_refl +leq/props leq_eq +leq/props leq_sym +leq/props leq_trans +leq/props leq_ahead_false_1 +leq/props leq_ahead_false_2 +lift1/fwd lift1_cons_tail +lift1/fwd lifts1_nil +lift1/fwd lifts1_cons +lift/props thead_x_lift_y_y +lift/props lifts_tapp +lift/props lifts_inj +llt/props lweight_repl +llt/props llt_repl +llt/props llt_trans +llt/props llt_head_sx +llt/props llt_head_dx +llt/props llt_wf__q_ind +llt/props llt_wf_ind +next_plus/props next_plus_assoc +next_plus/props next_plus_next +next_plus/props next_plus_lt +nf2/arity arity_nf2_inv_all +nf2/fwd nf2_gen_lref +nf2/fwd nf2_gen_abst +nf2/fwd nf2_gen_cast +nf2/fwd nf2_gen_beta +nf2/fwd nf2_gen_flat +nf2/fwd nf2_gen__nf2_gen_aux +nf2/fwd nf2_gen_abbr +nf2/fwd nf2_gen_void +nf2/props nfs2_tapp +nf2/props nf2_appls_lref +pc1/props pc1_pr0_r +pc1/props pc1_pr0_x +pc1/props pc1_refl +pc1/props pc1_pr0_u +pc1/props pc1_s +pc1/props pc1_head_1 +pc1/props pc1_head_2 +pc1/props pc1_t +pc1/props pc1_pr0_u2 +pc1/props pc1_head + +pc3/dec pc3_dec +pc3/dec pc3_abst_dec +pc3/fwd pc3_gen_not_abst +pc3/fwd pc3_gen_lift_abst +pc3/nf2 pc3_nf2 +pc3/nf2 pc3_nf2_unfold +pc3/pc1 pc3_pc1 +pc3/props pc3_pr2_pr2_t +pc3/props pc3_pr2_pr3_t +pc3/props pc3_pr3_pc3_t +pc3/props pc3_eta + +pr0/fwd pr0_gen_void +pr0/dec nf0_dec + +pr1/props pr1_eta + +pr2/fwd pr2_gen_void +pr3/fwd pr3_gen_void +pr3/props pr3_eta +sn3/props sns3_lifts +sty1/cnt sty1_cnt +subst/fwd subst_sort +subst/fwd subst_lref_lt +subst/fwd subst_lref_eq +subst/fwd subst_lref_gt +subst/fwd subst_head +subst/props subst_lift_SO +subst/props subst_subst0 +T/dec binder_dec +T/dec abst_dec +tlist/props tslt_wf__q_ind +tlist/props tslt_wf_ind +tlist/props theads_tapp +tlist/props tcons_tapp_ex +tlist/props tlist_ind_rev +ty3/arity ty3_arity +ty3/arity_props ty3_predicative +ty3/arity_props ty3_repellent +ty3/arity_props ty3_acyclic +ty3/dec ty3_inference +ty3/fwd tys3_gen_nil +ty3/fwd tys3_gen_cons +ty3/fwd_nf2 ty3_gen_appl_nf2 +ty3/fwd_nf2 ty3_inv_lref_nf2_pc3 +ty3/fwd_nf2 ty3_inv_lref_nf2 +ty3/fwd_nf2 ty3_inv_appls_lref_nf2 +ty3/fwd_nf2 ty3_inv_lref_lref_nf2 +ty3/nf2 ty3_nf2_inv_abst_premise_csort +ty3/nf2 ty3_nf2_inv_all +ty3/nf2 ty3_nf2_inv_sort +ty3/nf2 ty3_nf2_gen__ty3_nf2_inv_abst_aux +ty3/nf2 ty3_nf2_inv_abst +ty3/pr3 ty3_sred_wcpr0_pr0 +ty3/pr3 ty3_sred_pr0 +ty3/pr3 ty3_sred_pr1 +ty3/pr3 ty3_sred_pr2 +ty3/pr3 ty3_sred_pr3 +ty3/pr3_props ty3_cred_pr2 +ty3/pr3_props ty3_cred_pr3 +ty3/pr3_props ty3_gen_lift +ty3/pr3_props ty3_tred +ty3/pr3_props ty3_sconv_pc3 +ty3/pr3_props ty3_sred_back +ty3/pr3_props ty3_sconv +ty3/props ty3_gen_abst_abst +ty3/sty0 ty3_sty0 +ty3/subst1 ty3_gen_cvoid + +wf3/clear wf3_clear_conf +wf3/clear clear_wf3_trans +wf3/fwd wf3_gen_sort1 +wf3/fwd wf3_gen_bind1 +wf3/fwd wf3_gen_flat1 +wf3/fwd wf3_gen_head2 +wf3/getl wf3_getl_conf +wf3/getl getl_wf3_trans +wf3/props wf3_mono +wf3/props wf3_total +wf3/props ty3_shift1 +wf3/props wf3_idem +wf3/props wf3_ty3 +wf3/ty3 wf3_pr2_conf +wf3/ty3 wf3_pr3_conf +wf3/ty3 wf3_pc3_conf +wf3/ty3 wf3_ty3_conf + +# check ###################################################################### diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/acp.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/acp.ma new file mode 100644 index 000000000..dc046b094 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/acp.ma @@ -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/unfold/ldrops.ma". + +(* ABSTRACT COMPUTATION PROPERTIES ******************************************) + +definition CP1 ≝ λRR:lenv→relation term. λRS:relation term. + ∀L,k. NF … (RR L) RS (⋆k). + +definition CP2 ≝ λRR:lenv→relation term. λRS:relation term. + ∀L,K,W,i. ⇩[0,i] L ≡ K. ⓛW → NF … (RR L) RS (#i). + +definition CP3 ≝ λRR:lenv→relation term. λRP:lenv→predicate term. + ∀L,V,k. RP L (ⓐ⋆k.V) → RP L V. + +definition CP4 ≝ λRR:lenv→relation term. λRS:relation term. + ∀L0,L,T,T0,d,e. NF … (RR L) RS T → + ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → NF … (RR L0) RS T0. + +definition CP4s ≝ λRR:lenv→relation term. λRS:relation term. + ∀L0,L,des. ⇩*[des] L0 ≡ L → + ∀T,T0. ⇧*[des] T ≡ T0 → + NF … (RR L) RS T → NF … (RR L0) RS T0. + +(* requirements for abstract computation properties *) +record acp (RR:lenv->relation term) (RS:relation term) (RP:lenv→predicate term) : Prop ≝ +{ cp1: CP1 RR RS; + cp2: CP2 RR RS; + cp3: CP3 RR RP; + cp4: CP4 RR RS +}. + +(* Basic properties *********************************************************) + +(* Basic_1: was: nf2_lift1 *) +lemma acp_lifts: ∀RR,RS. CP4 RR RS → CP4s RR RS. +#RR #RS #HRR #L1 #L2 #des #H elim H -L1 -L2 -des +[ #L #T1 #T2 #H #HT1 + <(lifts_inv_nil … H) -H // +| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2 + elim (lifts_inv_cons … H) -H /3 width=9/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma new file mode 100644 index 000000000..f4da11310 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma @@ -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/lifts_lifts.ma". +include "basic_2/unfold/ldrops_ldrops.ma". +include "basic_2/static/aaa_lifts.ma". +include "basic_2/static/aaa_aaa.ma". +include "basic_2/computation/lsubc_ldrops.ma". + +(* ABSTRACT COMPUTATION PROPERTIES ******************************************) + +(* Main propertis ***********************************************************) + +(* Basic_1: was: sc3_arity_csubc *) +theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP. + acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L0,des. ⇩*[des] L0 ≡ L1 → + ∀T0. ⇧*[des] T ≡ T0 → ∀L2. L2 ⊑[RP] L0 → + ⦃L2, T0⦄ ϵ[RP] 〚A〛. +#RR #RS #RP #H1RP #H2RP #L1 #T #A #H elim H -L1 -T -A +[ #L #k #L0 #des #HL0 #X #H #L2 #HL20 + >(lifts_inv_sort1 … H) -H + lapply (aacr_acr … H1RP H2RP ⓪) #HAtom + @(s2 … HAtom … ◊) // /2 width=2/ +| #I #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20 + lapply (aacr_acr … H1RP H2RP B) #HB + elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK1) #HK1b + elim (ldrops_ldrop_trans … HL01 … HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1 + >(at_mono … Hi1 … Hi0) -i1 + elim (ldrops_inv_skip2 … Hdes1 … H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct + elim (lsubc_ldrop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H + elim (lsubc_inv_pair2 … H) -H * + [ #K2 #HK20 #H destruct + generalize in match HLK2; generalize in match I; -HLK2 -I * #HLK2 + [ elim (lift_total V0 0 (i0 +1)) #V #HV0 + elim (lifts_lift_trans … Hi0 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2 + @(s4 … HB … ◊ … HV0 HLK2) /3 width=7/ (* uses IHB HL20 V2 HV0 *) + | @(s2 … HB … ◊) // /2 width=3/ + ] + | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0 + #K2 #V2 #A2 #HKV2A #HKV0A #_ #H1 #H2 destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK2b + lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B + >(aaa_mono … HKV0A … HKV0B) in HKV2A; -HKV0A -HKV0B #HKV2B + elim (lift_total V2 0 (i0 +1)) #V #HV2 + @(s4 … HB … ◊ … HV2 HLK2) + @(s7 … HB … HKV2B) // + ] +| #a #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20 + elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct + lapply (aacr_acr … H1RP H2RP A) #HA + lapply (aacr_acr … H1RP H2RP B) #HB + lapply (s1 … HB) -HB #HB + @(s5 … HA … ◊ ◊) // /3 width=5/ +| #a #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02 + elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct + @(aacr_abst … H1RP H2RP) + [ lapply (aacr_acr … H1RP H2RP B) #HB + @(s1 … HB) /2 width=5/ + | -IHB + #L3 #V3 #T3 #des3 #HL32 #HT03 #HB + elim (lifts_total des3 W0) #W2 #HW02 + elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20 + lapply (aaa_lifts … L2 W2 … (des @@ des3) … HLWB) -HLWB /2 width=3/ #HLW2B + @(IHA (L2. ⓛW2) … (des + 1 @@ des3 + 1)) -IHA + /2 width=3/ /3 width=5/ + ] +| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20 + elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct + /3 width=10/ +| #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20 + elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct + lapply (aacr_acr … H1RP H2RP A) #HA + lapply (s1 … HA) #H + @(s6 … HA … ◊) /2 width=5/ /3 width=5/ +] +qed. + +(* Basic_1: was: sc3_arity *) +lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀L,T,A. L ⊢ T ⁝ A → ⦃L, T⦄ ϵ[RP] 〚A〛. +/2 width=8/ qed. + +lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀L,T,A. L ⊢ T ⁝ A → RP L T. +#RR #RS #RP #H1RP #H2RP #L #T #A #HT +lapply (aacr_acr … H1RP H2RP A) #HA +@(s1 … HA) /2 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma new file mode 100644 index 000000000..b0b15e665 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma @@ -0,0 +1,174 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/aarity.ma". +include "basic_2/unfold/gr2_gr2.ma". +include "basic_2/unfold/lifts_lift_vector.ma". +include "basic_2/unfold/ldrops_ldrop.ma". +include "basic_2/computation/acp.ma". + +(* ABSTRACT COMPUTATION PROPERTIES ******************************************) + +(* Note: this is Girard's CR1 *) +definition S1 ≝ λRP,C:lenv→predicate term. + ∀L,T. C L T → RP L T. + +(* Note: this is Tait's iii, or Girard's CR4 *) +definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term. + ∀L,Vs. all … (RP L) Vs → + ∀T. 𝐒⦃T⦄ → NF … (RR L) RS T → C L (ⒶVs.T). + +(* Note: this is Tait's ii *) +definition S3 ≝ λRP,C:lenv→predicate term. + ∀a,L,Vs,V,T,W. C L (ⒶVs. ⓓ{a}V. T) → RP L W → C L (ⒶVs. ⓐV. ⓛ{a}W. T). + +definition S4 ≝ λRP,C:lenv→predicate term. ∀L,K,Vs,V1,V2,i. + C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 → + ⇩[0, i] L ≡ K. ⓓV1 → C L (Ⓐ Vs. #i). + +definition S5 ≝ λRP,C:lenv→predicate term. + ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s → + ∀a,V,T. C (L. ⓓV) (ⒶV2s. T) → RP L V → C L (ⒶV1s. ⓓ{a}V. T). + +definition S6 ≝ λRP,C:lenv→predicate term. + ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓝW. T). + +definition S7 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e. + C L1 T1 → ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2. + +definition S7s ≝ λC:lenv→predicate term. + ∀L1,L2,des. ⇩*[des] L2 ≡ L1 → + ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2. + +(* properties of the abstract candidate of reducibility *) +record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝ +{ s1: S1 RP C; + s2: S2 RR RS RP C; + s3: S3 RP C; + s4: S4 RP C; + s5: S5 RP C; + s6: S6 RP C; + s7: S7 C +}. + +(* the abstract candidate of reducibility associated to an atomic arity *) +let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝ +λT. match A with +[ AAtom ⇒ RP L T +| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 → + aacr RP A L0 (ⓐV0. T0) +]. + +interpretation + "candidate of reducibility of an atomic arity (abstract)" + 'InEInt RP L T A = (aacr RP A L T). + +(* Basic properties *********************************************************) + +(* Basic_1: was: sc3_lift1 *) +lemma acr_lifts: ∀C. S7 C → S7s C. +#C #HC #L1 #L2 #des #H elim H -L1 -L2 -des +[ #L #T1 #T2 #H #HT1 + <(lifts_inv_nil … H) -H // +| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2 + elim (lifts_inv_cons … H) -H /3 width=9/ +] +qed. + +lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) → + ∀des,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[des] V ≡ V0 → + RP L V → RP L0 V0. +#RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV +@acr_lifts /width=6/ +@(s7 … HRP) +qed. + +(* Basic_1: was only: sns3_lifts1 *) +lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) → + ∀des,L0,L,Vs,V0s. ⇧*[des] Vs ≡ V0s → ⇩*[des] L0 ≡ L → + all … (RP L) Vs → all … (RP L0) V0s. +#RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize // +#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s +@conj /2 width=1/ /2 width=6 by rp_lifts/ +qed. + +(* Basic_1: was: + sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift +*) +lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀A. acr RR RS RP (aacr RP A). +#RR #RS #RP #H1RP #H2RP #A elim A -A normalize // +#B #A #IHB #IHA @mk_acr normalize +[ #L #T #H + lapply (H ? (⋆0) ? ⟠ ? ? ?) -H + [1,3: // |2,4: skip + | @(s2 … IHB … ◊) // /2 width=2/ + | #H @(cp3 … H1RP … 0) @(s1 … IHA) // + ] +| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H + elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct + lapply (s1 … IHB … HB) #HV0 + @(s2 … IHA … (V0 @ V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/ +| #a #L #Vs #U #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H + elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct + elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct + elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct + @(s3 … IHA … (V0 @ V0s)) /2 width=6 by rp_lifts/ /4 width=5/ +| #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H + elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct + elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct + elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0 + >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02 + elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct + elim (lift_total W1 0 (i0 + 1)) #W2 #HW12 + elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2 + >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2 + @(s4 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=4/ +| #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H + elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct + elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct + elim (lift_total V10 0 1) #V20 #HV120 + elim (liftv_total 0 1 V10s) #V20s #HV120s + @(s5 … IHA … (V10 @ V10s) (V20 @ V20s)) /2 width=1/ /2 width=6 by rp_lifts/ + @(HA … (des + 1)) /2 width=1/ + [ @(s7 … IHB … HB … HV120) /2 width=1/ + | @lifts_applv // + elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s + >(liftv_mono … HV12s … HV10s) -V1s // + ] +| #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H + elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct + elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct + @(s6 … IHA … (V0 @ V0s)) /2 width=6 by rp_lifts/ /3 width=4/ +| /3 width=7/ +] +qed. + +lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀a,L,W,T,A,B. RP L W → ( + ∀L0,V0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des + 1] T ≡ T0 → + ⦃L0, V0⦄ ϵ[RP] 〚B〛 → ⦃L0. ⓓV0, T0⦄ ϵ[RP] 〚A〛 + ) → + ⦃L, ⓛ{a}W. T⦄ ϵ[RP] 〚②B. A〛. +#RR #RS #RP #H1RP #H2RP #a #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H +lapply (aacr_acr … H1RP H2RP A) #HCA +lapply (aacr_acr … H1RP H2RP B) #HCB +elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct +lapply (s1 … HCB) -HCB #HCB +@(s3 … HCA … ◊) /2 width=6 by rp_lifts/ +@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/ +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/computation/cpe.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpe.ma new file mode 100644 index 000000000..285e6e4fc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpe.ma @@ -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/computation/cprs.ma". +include "basic_2/computation/csn.ma". + +(* CONTEXT-SENSITIVE PARALLEL EVALUATION ON TERMS **************************) + +definition cpe: lenv → relation term ≝ + λL,T1,T2. L ⊢ T1 ➡* T2 ∧ L ⊢ 𝐍⦃T2⦄. + +interpretation "context-sensitive parallel evaluation (term)" + 'PEval L T1 T2 = (cpe L T1 T2). + +(* Basic_properties *********************************************************) + +(* Basic_1: was: nf2_sn3 *) +lemma cpe_csn: ∀L,T1. L ⊢ ⬊* T1 → ∃T2. L ⊢ T1 ➡* 𝐍⦃T2⦄. +#L #T1 #H @(csn_ind … H) -T1 +#T1 #_ #IHT1 +elim (cnf_dec L T1) /3 width=3/ +* #T #H1T1 #H2T1 +elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1 #T2 * /4 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpe_cpe.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpe_cpe.ma new file mode 100644 index 000000000..ec770787b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpe_cpe.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 *) +(* *) +(**************************************************************************) + +include "basic_2/computation/cprs_cprs.ma". +include "basic_2/computation/cpe.ma". + +(* CONTEXT-SENSITIVE PARALLEL EVALUATION ON TERMS **************************) + +(* Main properties *********************************************************) + +(* Basic_1: was: nf2_pr3_confluence *) +theorem cpe_mono: ∀L,T,T1. L ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. L ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2. +#L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2 +elim (cprs_conf … H1T1 … H1T2) -T #T #HT1 +>(cprs_inv_cnf1 … HT1 H2T1) -T1 #HT2 +>(cprs_inv_cnf1 … HT2 H2T2) -T2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma new file mode 100644 index 000000000..ae0c1ae62 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma @@ -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 *) +(* *) +(**************************************************************************) + +include "basic_2/reducibility/cnf.ma". +include "basic_2/computation/tprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Basic_1: includes: pr3_pr2 *) +definition cprs: lenv → relation term ≝ + λL. TC … (cpr L). + +interpretation "context-sensitive parallel computation (term)" + 'PRedStar L T1 T2 = (cprs L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 → + (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) → + ∀T2. L ⊢ T1 ➡* T2 → R T2. +#L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 → + (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) → + ∀T1. L ⊢ T1 ➡* T2 → R T1. +#L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: pr3_refl *) +lemma cprs_refl: ∀L,T. L ⊢ T ➡* T. +/2 width=1/ qed. + +lemma cprs_strap1: ∀L,T1,T,T2. + L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2. +/2 width=3/ qed. + +(* Basic_1: was: pr3_step *) +lemma cprs_strap2: ∀L,T1,T,T2. + L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2. +/2 width=3/ qed. + +(* Note: it does not hold replacing |L1| with |L2| *) +lemma cprs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 → + ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ➡* T2. +/3 width=3/ +qed. + +(* Basic_1: was only: pr3_thin_dx *) +lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 → + L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. +#I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/ +#T #T2 #_ #HT2 #IHT2 +@(cprs_strap1 … IHT2) -IHT2 /2 width=1/ +qed. + +(* Basic_1: was: pr3_pr1 *) +lemma tprs_cprs: ∀T1,T2. T1 ➡* T2 → ∀L. L ⊢ T1 ➡* T2. +#T1 #T2 #H @(tprs_ind … H) -T2 /2 width=1/ /3 width=3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Basic_1: was: pr3_gen_sort *) +lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k. +#L #U2 #k #H @(cprs_ind … H) -U2 // +#U2 #U #_ #HU2 #IHU2 destruct +>(cpr_inv_sort1 … HU2) -HU2 // +qed-. + +(* Basic_1: was: pr3_gen_cast *) +lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓝW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨ + ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓝW2.T2. +#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ +#U2 #U #_ #HU2 * /3 width=3/ * +#W #T #HW1 #HT1 #H destruct +elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ * +#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/ +qed-. + +(* Basic_1: was: nf2_pr3_unfold *) +lemma cprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U. +#L #T #U #H @(cprs_ind_dx … H) -T // +#T0 #T #H1T0 #_ #IHT #H2T0 +lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/ +qed-. + +lemma tprs_inv_cnf1: ∀T,U. T ➡* U → ⋆ ⊢ 𝐍⦃T⦄ → T = U. +/3 width=3 by tprs_cprs, cprs_inv_cnf1/ qed-. + +(* Basic_1: removed theorems 10: + clear_pr3_trans pr3_cflat pr3_gen_bind + pr3_head_1 pr3_head_2 pr3_head_21 pr3_head_12 + pr3_iso_appl_bind pr3_iso_appls_appl_bind pr3_iso_appls_bind +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_aaa.ma new file mode 100644 index 000000000..e04e3c784 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_aaa.ma @@ -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/reducibility/cpr_aaa.ma". +include "basic_2/computation/cprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Properties about atomic arity assignment on terms ************************) + +lemma aaa_cprs_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ T2 ⁝ A. +#L #T1 #A #HT1 #T2 #HT12 +@(TC_Conf3 … HT1 ? HT12) /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma new file mode 100644 index 000000000..8f94b0fa7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma @@ -0,0 +1,150 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_cpr.ma". +include "basic_2/reducibility/lfpr_cpr.ma". +include "basic_2/computation/cprs_lfpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Advanced properties ******************************************************) + +lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. + L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2. +#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HT12) -T2 +[ /3 width=2/ +| /3 width=6 by cprs_strap1, cpr_abst/ (**) (* /3 width=6/ is too slow *) +] +qed. + +lemma cprs_abbr1_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → + ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 +[ /3 width=5/ +| #T1 #T #HT1 #_ #IHT1 + @(cprs_strap2 … IHT1) -IHT1 /2 width=1/ +] +qed. + +lemma cpr_abbr1: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 → + ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. +/3 width=1/ qed. + +lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 → + ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 +lapply (lfpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/ +qed. + +(* Basic_1: was: pr3_strip *) +lemma cprs_strip: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡ T2 → + ∃∃T0. L ⊢ T1 ➡ T0 & L ⊢ T2 ➡* T0. +/3 width=3/ qed. + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was pr3_gen_appl *) +lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 → + ∨∨ ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 & + U2 = ⓐV2. T2 + | ∃∃a,V2,W,T. L ⊢ V1 ➡* V2 & + L ⊢ T1 ➡* ⓛ{a}W. T & L ⊢ ⓓ{a}V2. T ➡* U2 + | ∃∃a,V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 & + L ⊢ T1 ➡* ⓓ{a}V. T & L ⊢ ⓓ{a}V. ⓐV2. T ➡* U2. +#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ +#U #U2 #_ #HU2 * * +[ #V0 #T0 #HV10 #HT10 #H destruct + elim (cpr_inv_appl1 … HU2) -HU2 * + [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ + | #a #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/ + | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct + @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *) + ] +| /4 width=9/ +| /4 width=11/ +] +qed-. + +(* Main propertis ***********************************************************) + +(* Basic_1: was: pr3_confluence *) +theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 → + ∃∃T0. L ⊢ T1 ➡* T0 & L ⊢ T2 ➡* T0. +/3 width=3/ qed. + +(* Basic_1: was: pr3_t *) +theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2. +/2 width=3/ qed. + +(* Basic_1: was: pr3_flat *) +lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 → + L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. +#I #L #T1 #T2 #HT12 #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ +#V #V2 #_ #HV2 #IHV1 +@(cprs_trans … IHV1) -IHV1 /2 width=1/ +qed. + +lemma cprs_abst: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2. + L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2. +#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HV12) -V2 +[ lapply (cprs_lsubs_trans … HT12 (L.ⓛV1) ?) -HT12 /2 width=2/ +| #V0 #V2 #_ #HV02 #IHV01 + @(cprs_trans … IHV01) -V1 /2 width=2/ +] +qed. + +lemma cprs_abbr1: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 → + ∀a.L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. +#L #V1 #T1 #T2 #HT12 #V2 #HV12 #a @(cprs_ind … HV12) -V2 /2 width=1/ +#V #V2 #_ #HV2 #IHV1 +@(cprs_trans … IHV1) -IHV1 /2 width=1/ +qed. + +lemma cprs_abbr2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 → + ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 +[ /2 width=1/ +| #T1 #T #HT1 #_ #IHT1 + lapply (lfpr_cpr_trans (L. ⓓV1) … HT1) -HT1 /2 width=1/ #HT1 + @(cprs_trans … IHT1) -IHT1 /2 width=1/ +] +qed. + +lemma cprs_abbr2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 → + ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. +#L #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ +#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 #a +lapply (IHV1 T1 T1 ? a) -IHV1 // #HV1 +@(cprs_trans … HV1) -HV1 /2 width=1/ +qed. + +lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2. + L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 → + ∀a.L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2. +#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #a @(cprs_ind … HT12) -T2 +[ /3 width=1/ +| -HV12 #T #T2 #_ #HT2 #IHT1 + lapply (cpr_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 + @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/ +] +qed. + +(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *) +lemma lcpr_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → + ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. +#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT2 +@(cprs_trans … IHT2) /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_delift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_delift.ma new file mode 100644 index 000000000..6b7892611 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_delift.ma @@ -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: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 → L ⊢ +ⓓV.T1 ➡* T2. +#L #V #T1 #T2 * #T #HT1 #HT2 +@(cprs_strap2 … (+ⓓV.T)) [ /3 width=3/ | @inj /3 width=3/ ] (**) (* explicit constructor, /5 width=3/ is too slow *) +qed. + +(* Basic_1: was only: pr3_gen_cabbr *) +lemma thin_cprs_delift_conf: ∀L,U1,U2. L ⊢ U1 ➡* U2 → + ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 → + ∃∃T2. K ⊢ T1 ➡* T2 & L ⊢ ▼*[d, e] U2 ≡ T2. +#L #U1 #U2 #H @(cprs_ind … H) -U2 /2 width=3/ +#U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1 +elim (IHU1 … HLK … HTU1) -U1 #T #HT1 #HUT +elim (thin_cpr_delift_conf … HU2 … HLK … HUT) -U -HLK /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lfpr.ma new file mode 100644 index 000000000..a06643577 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lfpr.ma @@ -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/reducibility/ltpr_tps.ma". +include "basic_2/reducibility/cpr_ltpss.ma". +include "basic_2/reducibility/lfpr.ma". +include "basic_2/computation/cprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Properties concerning focalized parallel reduction on local environments *) + +lemma ltpr_tpss_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2,d,e. L2 ⊢ T1 ▶* [d, e] T2 → + ∃∃T. L1 ⊢ T1 ▶* [d, e] T & L1 ⊢ T ➡* T2. +#L1 #L2 #HL12 #T1 #T2 #d #e #H @(tpss_ind … H) -T2 +[ /2 width=3/ +| #T #T2 #_ #HT2 * #T0 #HT10 #HT0 + elim (ltpr_tps_trans … HT2 … HL12) -L2 #T3 #HT3 #HT32 + @(ex2_1_intro … HT10) -T1 (**) (* explicit constructors *) + @(cprs_strap1 … T3 …) /2 width=1/ -HT32 + @(cprs_strap1 … HT0) -HT0 /3 width=3/ +] +qed. + +(* Basic_1: was just: pr3_pr0_pr2_t *) +lemma ltpr_cpr_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2. +#L1 #L2 #HL12 #T1 #T2 * #T #HT1 +<(ltpr_fwd_length … HL12) #HT2 +elim (ltpr_tpss_trans … HL12 … HT2) -L2 /3 width=3/ +qed. + +(* Basic_1: was just: pr3_pr2_pr2_t *) +lemma lfpr_cpr_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2. +#L1 #L2 * /3 width=7/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lfprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lfprs.ma new file mode 100644 index 000000000..33620d01c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lfprs.ma @@ -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/computation/cprs_cprs.ma". +include "basic_2/computation/lfprs_lfprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Properties on focalized computation for local environments ***************) + +(* Basic_1: was just: pr3_pr3_pr3_t *) +lemma lfprs_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → + ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. +#L1 #L2 #HL12 @(lfprs_ind … HL12) -L2 // /3 width=3/ +qed. + +lemma lfprs_cpr_trans: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → + ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2. +/3 width=3 by lfprs_cprs_trans, inj/ qed. + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was pr3_gen_abbr *) +lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡* U2 → + (∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 & + U2 = ⓓ{a}V2. T2 + ) ∨ + ∃∃T2. L. ⓓV1 ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true. +#a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ +#U0 #U2 #_ #HU02 * * +[ #V0 #T0 #HV10 #HT10 #H destruct + elim (cpr_inv_abbr1 … HU02) -HU02 * + [ #V #V2 #T2 #HV0 #HV2 #HT02 #H destruct + lapply (cpr_intro … HV0 … HV2) -HV2 #HV02 + lapply (ltpr_cpr_trans (L.ⓓV0) … HT02) /2 width=1/ -V #HT02 + lapply (lfprs_cprs_trans (L. ⓓV1) … HT02) -HT02 /2 width=1/ /4 width=5/ + | #T2 #HT02 #HUT2 + lapply (lfprs_cpr_trans (L.ⓓV1) … HT02) -HT02 /2 width=1/ -V0 #HT02 + lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3/ + ] +| #U1 #HTU1 #HU01 + elim (lift_total U2 0 1) #U #HU2 + lapply (cpr_lift (L.ⓓV1) … HU01 … HU2 HU02) -U0 /2 width=1/ /4 width=3/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma new file mode 100644 index 000000000..36ce0ef83 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma @@ -0,0 +1,78 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/computation/cprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was: pr3_gen_lref *) +lemma cprs_inv_lref1: ∀L,T2,i. L ⊢ #i ➡* T2 → + T2 = #i ∨ + ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 & + K ⊢ V1 ➡* T1 & + ⇧[0, i + 1] T1 ≡ T2 & + i < |L|. +#L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1/ +#T #T2 #_ #HT2 * +[ #H destruct + elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1/ + * #K #V1 #T1 #HLK #HVT1 #HT12 #Hi + @or_intror @(ex4_3_intro … HLK … HT12) // /3 width=3/ (**) (* explicit constructors *) +| * #K #V1 #T1 #HLK #HVT1 #HT1 #Hi + lapply (ldrop_fwd_ldrop2 … HLK) #H0LK + elim (cpr_inv_lift1 … H0LK … HT1 … HT2) -H0LK -T /4 width=6/ +] +qed-. + +(* Basic_1: was: pr3_gen_abst *) +lemma cprs_inv_abst1: ∀I,W,a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡* U2 → + ∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓑ{I} W ⊢ T1 ➡* T2 & + U2 = ⓛ{a}V2. T2. +#I #W #a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/ +#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct +elim (cpr_inv_abst1 … HU2 I W) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ +qed-. + +lemma cprs_inv_abst: ∀a,L,V1,V2,T1,T2. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 → ∀I,W. + L ⊢ V1 ➡* V2 ∧ L. ⓑ{I} W ⊢ T1 ➡* T2. +#a #L #V1 #V2 #T1 #T2 #H #I #W +elim (cprs_inv_abst1 I W … H) -H #V #T #HV1 #HT1 #H destruct /2 width=1/ +qed-. + +(* Relocation properties ****************************************************) + +(* Basic_1: was: pr3_lift *) +lemma cprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 → + ∀T2. K ⊢ T1 ➡* T2 → ∀U2. ⇧[d, e] T2 ≡ U2 → + L ⊢ U1 ➡* U2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #HT12 @(cprs_ind … HT12) -T2 +[ -HLK #T2 #HT12 + <(lift_mono … HTU1 … HT12) -T1 // +| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (cpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/ +] +qed. + +(* Basic_1: was: pr3_gen_lift *) +lemma cprs_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K → + ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡* U2 → + ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡* T2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 #HU12 @(cprs_ind … HU12) -U2 /2 width=3/ +-HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 +elim (cpr_inv_lift1 … HLK … HTU … HU2) -U -HLK /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_ltpr.ma new file mode 100644 index 000000000..2682a7609 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_ltpr.ma @@ -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: ∀L1,T1,U1,d,e. L1 ⊢ T1 ▶* [d, e] U1 → + ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → + ∃∃U2. L2 ⊢ U1 ➡* U2 & L2 ⊢ T2 ▶* [d, e] U2. +#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2 +[ #T2 #HT12 + elim (cpr_tpss_ltpr … HL12 … HT12 … HTU1) -L1 -T1 /3 width=3/ +| #T #T2 #_ #HT2 * #U #HU1 #HTU + elim (cpr_tpss_ltpr … HT2 … HTU) -L1 -T // /3 width=3/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_tstc.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_tstc.ma new file mode 100644 index 000000000..f7afb8df7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_tstc.ma @@ -0,0 +1,92 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/tstc.ma". +include "basic_2/computation/cprs_lift.ma". +include "basic_2/computation/cprs_lfprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Forward lemmas involving same top term constructor ***********************) + +lemma cprs_fwd_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → ∀U. L ⊢ T ➡* U → T ≃ U. +#L #T #HT #U #H +>(cprs_inv_cnf1 … H HT) -L -T // +qed-. + +(* Basic_1: was: pr3_iso_beta *) +lemma cprs_fwd_beta: ∀a,L,V,W,T,U. L ⊢ ⓐV. ⓛ{a}W. T ➡* U → + ⓐV. ⓛ{a}W. T ≃ U ∨ L ⊢ ⓓ{a}V. T ➡* U. +#a #L #V #W #T #U #H +elim (cprs_inv_appl1 … H) -H * +[ #V0 #T0 #_ #_ #H destruct /2 width=1/ +| #b #V0 #W0 #T0 #HV0 #HT0 #HU + elim (cprs_inv_abst1 Abbr V … HT0) -HT0 #W1 #T1 #_ #HT1 #H destruct -W1 + @or_intror -W + @(cprs_trans … HU) -U /2 width=1/ (**) (* explicit constructor *) +| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_ + elim (cprs_inv_abst1 Abbr V … HT1) -HT1 #W2 #T2 #_ #_ #H destruct +] +qed-. + +(* Note: probably this is an inversion lemma *) +lemma cprs_fwd_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 → + ∀V2. ⇧[0, i + 1] V1 ≡ V2 → + ∀U. L ⊢ #i ➡* U → + #i ≃ U ∨ L ⊢ V2 ➡* U. +#L #K #V1 #i #HLK #V2 #HV12 #U #H +elim (cprs_inv_lref1 … H) -H /2 width=1/ +* #K0 #V0 #U0 #HLK0 #HVU0 #HU0 #_ +lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct +lapply (ldrop_fwd_ldrop2 … HLK) -HLK /3 width=9/ +qed-. + +lemma cprs_fwd_theta: ∀a,L,V1,V,T,U. L ⊢ ⓐV1. ⓓ{a}V. T ➡* U → + ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1. ⓓ{a}V. T ≃ U ∨ + L ⊢ ⓓ{a}V. ⓐV2. T ➡* U. +#a #L #V1 #V #T #U #H #V2 #HV12 +elim (cprs_inv_appl1 … H) -H * +[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1/ +| #b #V0 #W #T0 #HV10 #HT0 #HU + elim (cprs_inv_abbr1 … HT0) -HT0 * + [ #V3 #T3 #_ #_ #H destruct + | #X #HT2 #H #H0 destruct + elim (lift_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct + @or_intror @(cprs_trans … HU) -U (**) (* explicit constructor *) + @(cprs_trans … (+ⓓV.ⓐV2.ⓛ{b}W2.T2)) [ /3 width=1/ ] -T + @(cprs_strap2 … (ⓐV1.ⓛ{b}W.T0)) [ /5 width=7/ ] -V -V2 -W2 -T2 + @(cprs_strap2 … (ⓓ{b}V1.T0)) [ /3 width=1/ ] -W /2 width=1/ + ] +| #b #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU + @or_intror @(cprs_trans … HU) -U (**) (* explicit constructor *) + elim (cprs_inv_abbr1 … HT0) -HT0 * + [ #V5 #T5 #HV5 #HT5 #H destruct + lapply (cprs_lift (L.ⓓV) … HV12 … HV13 … HV34) -V1 -V3 /2 width=1/ + /3 width=1/ + | #X #HT1 #H #H0 destruct + elim (lift_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct + lapply (cprs_lift (L.ⓓV0) … HV12 … HV13 … HV34) -V3 /2 width=1/ #HV24 + @(cprs_trans … (+ⓓV.ⓐV2.ⓓ{b}V5.T5)) [ /3 width=1/ ] -T + @(cprs_strap2 … (ⓐV1.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V5 -T5 + @(cprs_strap2 … (ⓓ{b}V0.ⓐV2.T0)) [ /3 width=3/ ] -V1 /3 width=1/ + ] +] +qed-. + +lemma cprs_fwd_tau: ∀L,W,T,U. L ⊢ ⓝW. T ➡* U → + ⓝW. T ≃ U ∨ L ⊢ T ➡* U. +#L #W #T #U #H +elim (cprs_inv_cast1 … H) -H /2 width=1/ * +#W0 #T0 #_ #_ #H destruct /2 width=1/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_tstc_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_tstc_vector.ma new file mode 100644 index 000000000..fd3eb585e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_tstc_vector.ma @@ -0,0 +1,157 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/tstc_vector.ma". +include "basic_2/substitution/lift_vector.ma". +include "basic_2/computation/cprs_tstc.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Vector form of forward lemmas involving same top term constructor ********) + +(* Basic_1: was just: nf2_iso_appls_lref *) +lemma cprs_fwd_cnf_vector: ∀L,T. 𝐒⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ∀Vs,U. L ⊢ ⒶVs.T ➡* U → ⒶVs.T ≃ U. +#L #T #H1T #H2T #Vs elim Vs -Vs [ @(cprs_fwd_cnf … H2T) ] (**) (* /2 width=3 by cprs_fwd_cnf/ does not work *) +#V #Vs #IHVs #U #H +elim (cprs_inv_appl1 … H) -H * +[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/ +| #a #V0 #W0 #T0 #HV0 #HT0 #HU + lapply (IHVs … HT0) -IHVs -HT0 #HT0 + elim (tstc_inv_bind_appls_simple … HT0 ?) // +| #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU + lapply (IHVs … HT0) -IHVs -HT0 #HT0 + elim (tstc_inv_bind_appls_simple … HT0 ?) // +] +qed-. + +(* Basic_1: was: pr3_iso_appls_beta *) +lemma cprs_fwd_beta_vector: ∀a,L,Vs,V,W,T,U. L ⊢ ⒶVs. ⓐV. ⓛ{a}W. T ➡* U → + ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ L ⊢ ⒶVs. ⓓ{a}V. T ➡* U. +#a #L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_beta/ +#V0 #Vs #IHVs #V #W #T #U #H +elim (cprs_inv_appl1 … H) -H * +[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/ +| #b #V1 #W1 #T1 #HV01 #HT1 #HU + elim (IHVs … HT1) -IHVs -HT1 #HT1 + [ elim (tstc_inv_bind_appls_simple … HT1 ?) // + | @or_intror -W (**) (* explicit constructor *) + @(cprs_trans … HU) -U + @(cprs_strap1 … (ⓐV1.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=1/ + ] +| #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU + elim (IHVs … HT1) -IHVs -HT1 #HT1 + [ elim (tstc_inv_bind_appls_simple … HT1 ?) // + | @or_intror -W (**) (* explicit constructor *) + @(cprs_trans … HU) -U + @(cprs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=3/ + ] +] +qed-. + +lemma cprs_fwd_delta_vector: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 → + ∀V2. ⇧[0, i + 1] V1 ≡ V2 → + ∀Vs,U. L ⊢ ⒶVs.#i ➡* U → + ⒶVs.#i ≃ U ∨ L ⊢ ⒶVs.V2 ➡* U. +#L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=4 by cprs_fwd_delta/ +#V #Vs #IHVs #U #H -K -V1 +elim (cprs_inv_appl1 … H) -H * +[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/ +| #b #V0 #W0 #T0 #HV0 #HT0 #HU + elim (IHVs … HT0) -IHVs -HT0 #HT0 + [ elim (tstc_inv_bind_appls_simple … HT0 ?) // + | @or_intror -i (**) (* explicit constructor *) + @(cprs_trans … HU) -U + @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=1/ + ] +| #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU + elim (IHVs … HT0) -IHVs -HT0 #HT0 + [ elim (tstc_inv_bind_appls_simple … HT0 ?) // + | @or_intror -i (**) (* explicit constructor *) + @(cprs_trans … HU) -U + @(cprs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=3/ + ] +] +qed-. + +(* Basic_1: was: pr3_iso_appls_abbr *) +lemma cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s → + ∀a,V,T,U. L ⊢ ⒶV1s. ⓓ{a}V. T ➡* U → + ⒶV1s. ⓓ{a}V. T ≃ U ∨ L ⊢ ⓓ{a}V. ⒶV2s. T ➡* U. +#L #V1s #V2s * -V1s -V2s /3 width=1/ +#V1s #V2s #V1a #V2a #HV12a #HV12s #a +generalize in match HV12a; -HV12a +generalize in match V2a; -V2a +generalize in match V1a; -V1a +elim HV12s -V1s -V2s /2 width=1 by cprs_fwd_theta/ +#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H +elim (cprs_inv_appl1 … H) -H * +[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1/ +| #b #V0a #W0 #T0 #HV10a #HT0 #HU + elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0 + [ -HV12a -HV12b -HV10a -HU + elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct + | @or_intror -V1s (**) (* explicit constructor *) + @(cprs_trans … HU) -U + elim (cprs_inv_abbr1 … HT0) -HT0 * + [ -HV12a -HV12b -HV10a #V1 #T1 #_ #_ #H destruct + | -V1b #X #HT1 #H #H0 destruct + elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct + @(cprs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1/ ] -T -V2b -V2s + @(cprs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [ /5 width=7/ ] -V -V2a -W1 -T1 + @(cprs_strap2 … (ⓓ{b}V1a.T0)) [ /3 width=1/ ] -W0 /2 width=1/ + ] + ] +| #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU + elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0 + [ -HV12a -HV10a -HV0a -HU + elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct + | @or_intror -V1s -V1b (**) (* explicit constructor *) + @(cprs_trans … HU) -U + elim (cprs_inv_abbr1 … HT0) -HT0 * + [ #V1 #T1 #HV1 #HT1 #H destruct + lapply (cprs_lift (L.ⓓV) … HV12a … HV10a … HV0a) -V1a -V0a [ /2 width=1/ ] #HV2a + @(cprs_trans … (ⓓ{a}V.ⓐV2a.T1)) [ /3 width=1/ ] -T -V2b -V2s /3 width=1/ + | #X #HT1 #H #H0 destruct + elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct + lapply (cprs_lift (L.ⓓV0) … HV12a … HV10a … HV0a) -V0a [ /2 width=1/ ] #HV2a + @(cprs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1/ ] -T -V2b -V2s + @(cprs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V1 -T1 + @(cprs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) [ /3 width=3/ ] -V1a /3 width=1/ + ] + ] +] +qed-. + +(* Basic_1: was: pr3_iso_appls_cast *) +lemma cprs_fwd_tau_vector: ∀L,Vs,W,T,U. L ⊢ ⒶVs. ⓝW. T ➡* U → + ⒶVs. ⓝW. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U. +#L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_tau/ +#V #Vs #IHVs #W #T #U #H +elim (cprs_inv_appl1 … H) -H * +[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/ +| #b #V0 #W0 #T0 #HV0 #HT0 #HU + elim (IHVs … HT0) -IHVs -HT0 #HT0 + [ elim (tstc_inv_bind_appls_simple … HT0 ?) // + | @or_intror -W (**) (* explicit constructor *) + @(cprs_trans … HU) -U + @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=1/ + ] +| #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU + elim (IHVs … HT0) -IHVs -HT0 #HT0 + [ elim (tstc_inv_bind_appls_simple … HT0 ?) // + | @or_intror -W (**) (* explicit constructor *) + @(cprs_trans … HU) -U + @(cprs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=3/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma new file mode 100644 index 000000000..3ed310164 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma @@ -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/reducibility/cnf.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) + +definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …). + +interpretation + "context-sensitive strong normalization (term)" + 'SN L T = (csn L T). + +(* Basic eliminators ********************************************************) + +lemma csn_ind: ∀L. ∀R:predicate term. + (∀T1. L ⊢ ⬊* T1 → + (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → R T2) → + R T1 + ) → + ∀T. L ⊢ ⬊* T → R T. +#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1 +@H0 -H0 /3 width=1/ -IHT1 /4 width=1/ +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: sn3_pr2_intro *) +lemma csn_intro: ∀L,T1. + (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊* T2) → L ⊢ ⬊* T1. +/4 width=1/ qed. + +(* Basic_1: was: sn3_nf2 *) +lemma csn_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ ⬊* T. +/2 width=1/ qed. + +lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬊* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬊* T2. +#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12 +@csn_intro #T #HLT2 #HT2 +elim (term_eq_dec T1 T2) #HT12 +[ -IHT1 -HLT12 destruct /3 width=1/ +| -HT1 -HT2 /3 width=4/ +qed. + +(* Basic_1: was: sn3_cast *) +lemma csn_cast: ∀L,W. L ⊢ ⬊* W → ∀T. L ⊢ ⬊* T → L ⊢ ⬊* ⓝW. T. +#L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT +@csn_intro #X #H1 #H2 +elim (cpr_inv_cast1 … H1) -H1 +[ * #W0 #T0 #HLW0 #HLT0 #H destruct + elim (eq_false_inv_tpair_sn … H2) -H2 + [ /3 width=3/ + | -HLW0 * #H destruct /3 width=1/ + ] +| /3 width=3/ +] +qed. + +(* Basic forward lemmas *****************************************************) + +fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬊* T. +#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct +@csn_intro #T2 #HLT2 #HT2 +@(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/ +qed. + +(* Basic_1: was: sn3_gen_flat *) +lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬊* ⓕ{I} V. T → L ⊢ ⬊* T. +/2 width=5/ qed-. + +(* Basic_1: removed theorems 14: + sn3_cdelta + sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change + sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr + sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma new file mode 100644 index 000000000..67744a098 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma @@ -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/acp_aaa.ma". +include "basic_2/computation/csn_tstc_vector.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) + +(* Properties concerning atomic arity assignment ****************************) + +lemma csn_aaa: ∀L,T,A. L ⊢ T ⁝ A → L ⊢ ⬊* T. +#L #T #A #H +@(acp_aaa … csn_acp csn_acr … H) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma new file mode 100644 index 000000000..eeba707dc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma @@ -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/computation/cprs.ma". +include "basic_2/computation/csn.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) + +(* alternative definition of csn *) +definition csna: lenv → predicate term ≝ λL. SN … (cprs L) (eq …). + +interpretation + "context-sensitive strong normalization (term) alternative" + 'SNAlt L T = (csna L T). + +(* Basic eliminators ********************************************************) + +lemma csna_ind: ∀L. ∀R:predicate term. + (∀T1. L ⊢ ⬊⬊* T1 → + (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → R T2) → R T1 + ) → + ∀T. L ⊢ ⬊⬊* T → R T. +#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1 +@H0 -H0 /3 width=1/ -IHT1 /4 width=1/ +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: sn3_intro *) +lemma csna_intro: ∀L,T1. + (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → L ⊢ ⬊⬊* T2) → L ⊢ ⬊⬊* T1. +/4 width=1/ qed. + +fact csna_intro_aux: ∀L,T1. + (∀T,T2. L ⊢ T ➡* T2 → T1 = T → (T1 = T2 → ⊥) → L ⊢ ⬊⬊* T2) → L ⊢ ⬊⬊* T1. +/4 width=3/ qed-. + +(* Basic_1: was: sn3_pr3_trans (old version) *) +lemma csna_cprs_trans: ∀L,T1. L ⊢ ⬊⬊* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬊⬊* T2. +#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12 +@csna_intro #T #HLT2 #HT2 +elim (term_eq_dec T1 T2) #HT12 +[ -IHT1 -HLT12 destruct /3 width=1/ +| -HT1 -HT2 /3 width=4/ +qed. + +(* Basic_1: was: sn3_pr2_intro (old version) *) +lemma csna_intro_cpr: ∀L,T1. + (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊⬊* T2) → + L ⊢ ⬊⬊* T1. +#L #T1 #H +@csna_intro_aux #T #T2 #H @(cprs_ind_dx … H) -T +[ -H #H destruct #H + elim (H ?) // +| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct + elim (term_eq_dec T0 T) #HT0 + [ -HLT1 -HLT2 -H /3 width=1/ + | -IHT -HT12 /4 width=3/ + ] +] +qed. + +(* Main properties **********************************************************) + +theorem csn_csna: ∀L,T. L ⊢ ⬊* T → L ⊢ ⬊⬊* T. +#L #T #H @(csn_ind … H) -T /4 width=1/ +qed. + +theorem csna_csn: ∀L,T. L ⊢ ⬊⬊* T → L ⊢ ⬊* T. +#L #T #H @(csna_ind … H) -T /4 width=1/ +qed. + +(* Basic_1: was: sn3_pr3_trans *) +lemma csn_cprs_trans: ∀L,T1. L ⊢ ⬊* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬊* T2. +/4 width=3/ qed. + +(* Main eliminators *********************************************************) + +lemma csn_ind_alt: ∀L. ∀R:predicate term. + (∀T1. L ⊢ ⬊* T1 → + (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → R T2) → R T1 + ) → + ∀T. L ⊢ ⬊* T → R T. +#L #R #H0 #T1 #H @(csna_ind … (csn_csna … H)) -T1 #T1 #HT1 #IHT1 +@H0 -H0 /2 width=1/ -HT1 /3 width=1/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_cpr.ma new file mode 100644 index 000000000..ccfd6015b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_cpr.ma @@ -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/cpr_cpr.ma". +include "basic_2/computation/csn.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) + +(* Advanced forvard lemmas **************************************************) + +fact csn_fwd_pair_sn_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ②{I} V. T → L ⊢ ⬊* V. +#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct +@csn_intro #V2 #HLV2 #HV2 +@(IH (②{I} V2. T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/ +qed. + +(* Basic_1: was: sn3_gen_head *) +lemma csn_fwd_pair_sn: ∀I,L,V,T. L ⊢ ⬊* ②{I} V. T → L ⊢ ⬊* V. +/2 width=5/ qed. + +fact csn_fwd_bind_dx_aux: ∀L,U. L ⊢ ⬊* U → + ∀a,I,V,T. U = ⓑ{a,I} V. T → L. ⓑ{I} V ⊢ ⬊* T. +#L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct +@csn_intro #T2 #HLT2 #HT2 +@(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/ +qed. + +(* Basic_1: was: sn3_gen_bind *) +lemma csn_fwd_bind_dx: ∀a,I,L,V,T. L ⊢ ⬊* ⓑ{a,I} V. T → L. ⓑ{I} V ⊢ ⬊* T. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_cpr_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_cpr_vector.ma new file mode 100644 index 000000000..70c00eb11 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_cpr_vector.ma @@ -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/csn_cpr.ma". +include "basic_2/computation/csn_vector.ma". + +(* Advanced forward lemmas **************************************************) + +lemma csn_fwd_applv: ∀L,T,Vs. L ⊢ ⬊* Ⓐ Vs. T → L ⊢ ⬊* Vs ∧ L ⊢ ⬊* T. +#L #T #Vs elim Vs -Vs /2 width=1/ +#V #Vs #IHVs #HVs +lapply (csn_fwd_pair_sn … HVs) #HV +lapply (csn_fwd_flat_dx … HVs) -HVs #HVs +elim (IHVs HVs) -IHVs -HVs /3 width=1/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lfpr.ma new file mode 100644 index 000000000..444dcf8f3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lfpr.ma @@ -0,0 +1,147 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/tstc_tstc.ma". +include "basic_2/computation/cprs_cprs.ma". +include "basic_2/computation/csn_lift.ma". +include "basic_2/computation/csn_cpr.ma". +include "basic_2/computation/csn_alt.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) + +(* Advanced properties ******************************************************) + +lemma csn_lfpr_conf: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. L1 ⊢ ⬊* T → L2 ⊢ ⬊* T. +#L1 #L2 #HL12 #T #H @(csn_ind_alt … H) -T #T #_ #IHT +@csn_intro #T0 #HLT0 #HT0 +@IHT /2 width=2/ -IHT -HT0 /2 width=3/ +qed. + +lemma csn_abbr: ∀a,L,V. L ⊢ ⬊* V → ∀T. L. ⓓV ⊢ ⬊* T → L ⊢ ⬊* ⓓ{a}V. T. +#a #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_alt … HT) -T #T #HT #IHT +@csn_intro #X #H1 #H2 +elim (cpr_inv_abbr1 … H1) -H1 * +[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct + lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1 + lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1 + elim (eq_false_inv_tpair_sn … H2) -H2 + [ #HV1 @IHV // /2 width=1/ -HV1 + @(csn_lfpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/ + | -IHV -HLV1 * #H destruct /3 width=1/ + ] +| -IHV -IHT -H2 #T0 #HLT0 #HT0 + lapply (csn_cpr_trans … HT … HLT0) -T #HLT0 + lapply (csn_inv_lift … HLT0 … HT0) -T0 /2 width=3/ +] +qed. + +fact csn_appl_beta_aux: ∀a,L,W. L ⊢ ⬊* W → ∀U. L ⊢ ⬊* U → + ∀V,T. U = ⓓ{a}V. T → L ⊢ ⬊* ⓐV. ⓛ{a}W. T. +#a #L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V #T #H destruct +lapply (csn_fwd_pair_sn … HVT) #HV +lapply (csn_fwd_bind_dx … HVT) #HT -HVT +@csn_intro #X #H #H2 +elim (cpr_inv_appl1 … H) -H * +[ #V0 #Y #HLV0 #H #H0 destruct + elim (cpr_inv_abst1 … H Abbr V) -H #W0 #T0 #HLW0 #HLT0 #H destruct + elim (eq_false_inv_beta … H2) -H2 + [ -IHVT #HW0 @IHW -IHW [1,5: // |3: skip ] -HLW0 /2 width=1/ -HW0 + @csn_abbr /2 width=3/ -HV + @(csn_lfpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/ + | -IHW -HLW0 -HV -HT * #H #HVT0 destruct + @(IHVT … HVT0) -IHVT -HVT0 // /2 width=1/ + ] +| -IHW -IHVT -H2 #b #V0 #W0 #T0 #T1 #HLV0 #HLT01 #H1 #H2 destruct + lapply (lfpr_cpr_trans (L. ⓓV) … HLT01) -HLT01 /2 width=1/ #HLT01 + @csn_abbr /2 width=3/ -HV + @(csn_lfpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/ +| -IHW -IHVT -HV -HT -H2 #b #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct +] +qed. + +(* Basic_1: was: sn3_beta *) +lemma csn_appl_beta: ∀a,L,W. L ⊢ ⬊* W → ∀V,T. L ⊢ ⬊* ⓓ{a}V. T → + L ⊢ ⬊* ⓐV. ⓛ{a}W. T. +/2 width=3/ qed. + +fact csn_appl_theta_aux: ∀a,L,U. L ⊢ ⬊* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 → + ∀V,T. U = ⓓ{a}V. ⓐV2. T → L ⊢ ⬊* ⓐV1. ⓓ{a}V. T. +#a #L #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct +lapply (csn_fwd_pair_sn … HVT) #HV +lapply (csn_fwd_bind_dx … HVT) -HVT #HVT +@csn_intro #X #HL #H +elim (cpr_inv_appl1 … HL) -HL * +[ -HV #V0 #Y #HLV10 #HL #H0 destruct + elim (cpr_inv_abbr1 … HL) -HL * + [ #V3 #V4 #T3 #HV3 #HLV34 #HLT3 #H0 destruct + lapply (cpr_intro … HV3 HLV34) -HLV34 #HLV34 + elim (lift_total V0 0 1) #V5 #HV05 + elim (term_eq_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V4.ⓐV5.T3)) + [ -IHVT #H0 destruct + elim (eq_false_inv_tpair_sn … H) -H + [ -HLV10 -HLV34 -HV3 -HLT3 -HVT + >(lift_inj … HV12 … HV05) -V5 + #H elim (H ?) // + | * #_ #H elim (H ?) // + ] + | -H -HVT #H + lapply (cpr_lift (L. ⓓV) … HV12 … HV05 HLV10) -HLV10 -HV12 /2 width=1/ #HV25 + lapply (ltpr_cpr_trans (L. ⓓV) … HLT3) /2 width=1/ -HLT3 #HLT3 + @(IHVT … H … HV05) -IHVT // -H -HV05 /3 width=1/ + ] + | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct + lapply (csn_cpr_trans … HVT (ⓐV2.T0) ?) /2 width=1/ -T #HVT0 + lapply (csn_inv_lift L … 1 HVT0 ? ? ?) -HVT0 [ /2 width=4/ |2,3: skip | /2 width=1/ ] -V2 -T0 #HVY + @(csn_cpr_trans … HVY) /2 width=1/ + ] +| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #T0 #T1 #_ #_ #H destruct +| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HLW01 #HLT01 #HV03 #H1 #H2 destruct + lapply (cpr_lift (L. ⓓW0) … HV12 … HV03 HLV10) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23 + lapply (lfpr_cpr_trans (L. ⓓW0) … HLT01) -HLT01 /2 width=1/ #HLT01 + @csn_abbr /2 width=3/ -HV + @(csn_lfpr_conf (L. ⓓW0)) /2 width=1/ -W1 + @(csn_cprs_trans … HVT) -HVT /2 width=1/ +] +qed. + +lemma csn_appl_theta: ∀a,V1,V2. ⇧[0, 1] V1 ≡ V2 → + ∀L,V,T. L ⊢ ⬊* ⓓ{a}V. ⓐV2. T → L ⊢ ⬊* ⓐV1. ⓓ{a}V. T. +/2 width=5/ qed. + +(* Basic_1: was only: sn3_appl_appl *) +lemma csn_appl_simple_tstc: ∀L,V. L ⊢ ⬊* V → ∀T1. + L ⊢ ⬊* T1 → + (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → ⊥) → L ⊢ ⬊* ⓐV. T2) → + 𝐒⦃T1⦄ → L ⊢ ⬊* ⓐV. T1. +#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1 +@csn_intro #X #HL #H +elim (cpr_inv_appl1_simple … HL ?) -HL // +#V0 #T0 #HLV0 #HLT10 #H0 destruct +elim (eq_false_inv_tpair_sn … H) -H +[ -IHT1 #HV0 + @(csn_cpr_trans … (ⓐV0.T1)) /2 width=1/ -HLT10 + @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0 + #T2 #HLT12 #HT12 + @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0 + @H2T1 -H2T1 // -HLT12 /2 width=1/ +| -IHV -H1T1 -HLV0 * #H #H1T10 destruct + elim (tstc_dec T1 T0) #H2T10 + [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1 + #T2 #HLT02 #HT02 + @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/ + | -IHT1 -H3T1 -H1T10 + @H2T1 -H2T1 /2 width=1/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma new file mode 100644 index 000000000..4fd784d75 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma @@ -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/reducibility/cnf_lift.ma". +include "basic_2/computation/acp.ma". +include "basic_2/computation/csn.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************) + +(* Relocation properties ****************************************************) + +(* Basic_1: was: sn3_lift *) +lemma csn_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬊* T1 → + ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → L2 ⊢ ⬊* T2. +#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12 +@csn_intro #T #HLT2 #HT2 +elim (cpr_inv_lift1 … HL21 … HT12 … HLT2) -HLT2 #T0 #HT0 #HLT10 +@(IHT1 … HLT10) // -L1 -L2 #H destruct +>(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1/ +qed. + +(* Basic_1: was: sn3_gen_lift *) +lemma csn_inv_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬊* T1 → + ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → L2 ⊢ ⬊* T2. +#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21 +@csn_intro #T #HLT2 #HT2 +elim (lift_total T d e) #T0 #HT0 +lapply (cpr_lift … HL12 … HT21 … HT0 HLT2) -HLT2 #HLT10 +@(IHT1 … HLT10) // -L1 -L2 #H destruct +>(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1/ +qed. + +(* Advanced properties ******************************************************) + +(* Basic_1: was: sn3_abbr *) +lemma csn_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → K ⊢ ⬊* V → L ⊢ ⬊* #i. +#L #K #V #i #HLK #HV +@csn_intro #X #H #Hi +elim (cpr_inv_lref1 … H) -H +[ #H destruct elim (Hi ?) // +| -Hi * #K0 #V0 #V1 #HLK0 #HV01 #HV1 #_ + lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct + lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK + @(csn_lift … HLK HV1) -HLK -HV1 + @(csn_cpr_trans … HV) -HV + @(cpr_intro … HV01) -HV01 // +] +qed. + +lemma csn_abst: ∀a,L,W. L ⊢ ⬊* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬊* T → L ⊢ ⬊* ⓛ{a}W. T. +#a #L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT +@csn_intro #X #H1 #H2 +elim (cpr_inv_abst1 … H1 I V) -H1 +#W0 #T0 #HLW0 #HLT0 #H destruct +elim (eq_false_inv_tpair_sn … H2) -H2 +[ /3 width=5/ +| -HLW0 * #H destruct /3 width=1/ +] +qed. + +lemma csn_appl_simple: ∀L,V. L ⊢ ⬊* V → ∀T1. + (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊* ⓐV. T2) → + 𝐒⦃T1⦄ → L ⊢ ⬊* ⓐV. T1. +#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1 +@csn_intro #X #H1 #H2 +elim (cpr_inv_appl1_simple … H1 ?) // -H1 +#V0 #T0 #HLV0 #HLT10 #H destruct +elim (eq_false_inv_tpair_dx … H2) -H2 +[ -IHV -HT1 #HT10 + @(csn_cpr_trans … (ⓐV.T0)) /2 width=1/ -HLV0 + @IHT1 -IHT1 // /2 width=1/ +| -HLT10 * #H #HV0 destruct + @IHV -IHV // -HT1 /2 width=1/ -HV0 + #T2 #HLT02 #HT02 + @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0 + @IHT1 -IHT1 // -HLT02 /2 width=1/ +] +qed. + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was: sn3_gen_def *) +lemma csn_inv_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → L ⊢ ⬊* #i → K ⊢ ⬊* V. +#L #K #V #i #HLK #Hi +elim (lift_total V 0 (i+1)) #V0 #HV0 +lapply (ldrop_fwd_ldrop2 … HLK) #H0LK +@(csn_inv_lift … H0LK … HV0) -H0LK +@(csn_cpr_trans … Hi) -Hi /2 width=6/ +qed-. + +(* Main properties **********************************************************) + +theorem csn_acp: acp cpr (eq …) (csn …). +@mk_acp +[ /2 width=1/ +| /2 width=3/ +| /2 width=5/ +| @cnf_lift +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma new file mode 100644 index 000000000..cfee668b9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma @@ -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_2/computation/acp_cr.ma". +include "basic_2/computation/cprs_tstc_vector.ma". +include "basic_2/computation/csn_lfpr.ma". +include "basic_2/computation/csn_vector.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************) + +(* Advanced properties ******************************************************) + +(* Basic_1: was only: sn3_appls_lref *) +lemma csn_applv_cnf: ∀L,T. 𝐒⦃T⦄ → L ⊢ 𝐍⦃T⦄ → + ∀Vs. L ⊢ ⬊* Vs → L ⊢ ⬊* ⒶVs.T. +#L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(csn_cnf … H2T) ] (**) (* /2 width=1/ does not work *) +#V #Vs #IHV #H +elim (csnv_inv_cons … H) -H #HV #HVs +@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -HVs +#X #H #H0 +lapply (cprs_fwd_cnf_vector … H) -H // -H1T -H2T #H +elim (H0 ?) -H0 // +qed. + +(* Basic_1: was: sn3_appls_beta *) +lemma csn_applv_beta: ∀a,L,W. L ⊢ ⬊* W → + ∀Vs,V,T. L ⊢ ⬊* ⒶVs.ⓓ{a}V.T → + L ⊢ ⬊* ⒶVs. ⓐV.ⓛ{a}W. T. +#a #L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW +#V0 #Vs #IHV #V #T #H1T +lapply (csn_fwd_pair_sn … H1T) #HV0 +lapply (csn_fwd_flat_dx … H1T) #H2T +@csn_appl_simple_tstc // -HV0 /2 width=1/ -IHV -H2T +#X #H #H0 +elim (cprs_fwd_beta_vector … H) -H #H +[ -H1T elim (H0 ?) -H0 // +| -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/ +] +qed. + +lemma csn_applv_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 → + ∀V2. ⇧[0, i + 1] V1 ≡ V2 → + ∀Vs.L ⊢ ⬊* (ⒶVs. V2) → L ⊢ ⬊* (ⒶVs. #i). +#L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs +[ #H + lapply (ldrop_fwd_ldrop2 … HLK) #HLK0 + lapply (csn_inv_lift … H … HLK0 HV12) -V2 -HLK0 /2 width=4/ +| #V #Vs #IHV #H1T + lapply (csn_fwd_pair_sn … H1T) #HV + lapply (csn_fwd_flat_dx … H1T) #H2T + @csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T + #X #H #H0 + elim (cprs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H + [ -H1T elim (H0 ?) -H0 // + | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/ + ] +] +qed. + +(* Basic_1: was: sn3_appls_abbr *) +lemma csn_applv_theta: ∀a,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s → + ∀V,T. L ⊢ ⬊* ⓓ{a}V. ⒶV2s. T → L ⊢ ⬊* V → + L ⊢ ⬊* ⒶV1s. ⓓ{a}V. T. +#a #L #V1s #V2s * -V1s -V2s /2 width=1/ +#V1s #V2s #V1 #V2 #HV12 #H +generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1 +elim H -V1s -V2s /2 width=3/ +#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H #HV +lapply (csn_appl_theta … HW12 … H) -H -HW12 #H +lapply (csn_fwd_pair_sn … H) #HW1 +lapply (csn_fwd_flat_dx … H) #H1 +@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -HV -H1 #X #H1 #H2 +elim (cprs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12 +[ -H #H elim (H2 ?) -H2 // +| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/ +] +qed. + +(* Basic_1: was: sn3_appls_cast *) +lemma csn_applv_tau: ∀L,W. L ⊢ ⬊* W → + ∀Vs,T. L ⊢ ⬊* ⒶVs. T → + L ⊢ ⬊* ⒶVs. ⓝW. T. +#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW +#V #Vs #IHV #T #H1T +lapply (csn_fwd_pair_sn … H1T) #HV +lapply (csn_fwd_flat_dx … H1T) #H2T +@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T +#X #H #H0 +elim (cprs_fwd_tau_vector … H) -H #H +[ -H1T elim (H0 ?) -H0 // +| -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/ +] +qed. + +theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬊* T). +@mk_acr // +[ /3 width=1/ +| /2 width=1/ +| /2 width=6/ +| #L #V1 #V2 #HV12 #a #V #T #H #HVT + @(csn_applv_theta … HV12) -HV12 // + @(csn_abbr) // +| /2 width=1/ +| @csn_lift +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma new file mode 100644 index 000000000..7c26ef429 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma @@ -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/grammar/term_vector.ma". +include "basic_2/computation/csn.ma". + +(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************) + +definition csnv: lenv → predicate (list term) ≝ + λL. all … (csn L). + +interpretation + "context-sensitive strong normalization (term vector)" + 'SN L Ts = (csnv L Ts). + +(* Basic inversion lemmas ***************************************************) + +lemma csnv_inv_cons: ∀L,T,Ts. L ⊢ ⬊* T @ Ts → L ⊢ ⬊* T ∧ L ⊢ ⬊* Ts. +normalize // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma new file mode 100644 index 000000000..61c720754 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma @@ -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/reducibility/fpr.ma". + +(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************) + +definition fprs: bi_relation lenv term ≝ bi_TC … fpr. + +interpretation "context-free parallel computation (closure)" + 'FocalizedPRedStar L1 T1 L2 T2 = (fprs L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma fprs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 → + (∀L,L2,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → R L T → R L2 T2) → + ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L2 T2. +/3 width=7 by bi_TC_star_ind/ qed-. + +lemma fprs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 → + (∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → R L T → R L1 T1) → + ∀L1,T1. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → R L1 T1. +/3 width=7 by bi_TC_star_ind_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma fprs_refl: bi_reflexive … fprs. +/2 width=1/ qed. + +lemma fprs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → + ⦃L1, T1⦄ ➡* ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma fprs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → + ⦃L1, T1⦄ ➡* ⦃L2, T2⦄. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_aaa.ma new file mode 100644 index 000000000..b76637ff7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_aaa.ma @@ -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: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → + ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A. +#L1 #T1 #A #HT1 #L2 #T2 #HLT12 +@(bi_TC_Conf3 … HT1 ?? HLT12) /2 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_cprs.ma new file mode 100644 index 000000000..4e7a633c3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_cprs.ma @@ -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/reducibility/fpr_cpr.ma". +include "basic_2/computation/cprs.ma". +include "basic_2/computation/fprs.ma". + +(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************) + +(* Properties on context-sensitive parallel computation for terms ***********) + +lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄. +#L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/ +qed. +(* +(* Advanced propertis *******************************************************) + +lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 → + ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄. +#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I +elim (fpr_inv_all … H) /3 width=4/ +qed. + +(* Advanced forward lemmas **************************************************) + +lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2. + ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ → + ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2. +* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H +elim (fpr_inv_all … H) -H #L #HL1 #H #HL2 +[ elim (cpr_inv_abbr1 … H) -H * + [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/ + | #T #_ #_ #H destruct + ] +| elim (cpr_inv_abst1 … H Abst V2) -H + #V #T #HV1 #_ #H destruct /3 width=4/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +lamma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ → + ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & + ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ & + L2 = K2.ⓑ{I}V2. +#I1 #K1 #X #V1 #T1 #T2 #H +elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct +elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/ +qed-. + +lamma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ → + ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & + ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ & + L1 = K1.ⓑ{I}V1. +#I2 #X #K2 #V2 #T1 #T2 #H +elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct +elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/ +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_fprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_fprs.ma new file mode 100644 index 000000000..e0c1b3058 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs_fprs.ma @@ -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: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ➡ ⦃L1, T1⦄ → + ∀L2,T2. ⦃L0, T0⦄ ➡* ⦃L2, T2⦄ → + ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄. +#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 +/2 width=4/ qed. + +(* Main propertis ***********************************************************) + +theorem fprs_conf: bi_confluent … fprs. +/2 width=4/ qed. + +theorem fprs_trans: bi_transitive … fprs. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma new file mode 100644 index 000000000..a193f3c0a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma @@ -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/lfpr.ma". + +(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) + +definition lfprs: relation lenv ≝ TC … lfpr. + +interpretation + "focalized parallel computation (environment)" + 'FocalizedPRedStar L1 L2 = (lfprs L1 L2). + +(* Basic eliminators ********************************************************) + +lemma lfprs_ind: ∀L1. ∀R:predicate lenv. R L1 → + (∀L,L2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L⦄ ➡ ⦃L2⦄ → R L → R L2) → + ∀L2. ⦃L1⦄ ➡* ⦃L2⦄ → R L2. +#L1 #R #HL1 #IHL1 #L2 #HL12 +@(TC_star_ind … HL1 IHL1 … HL12) // +qed-. + +lemma lfprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 → + (∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ⦃L⦄ ➡* ⦃L2⦄ → R L → R L1) → + ∀L1. ⦃L1⦄ ➡* ⦃L2⦄ → R L1. +#L2 #R #HL2 #IHL2 #L1 #HL12 +@(TC_star_ind_dx … HL2 IHL2 … HL12) // +qed-. + +(* Basic properties *********************************************************) + +lemma lfprs_refl: ∀L. ⦃L⦄ ➡* ⦃L⦄. +/2 width=1/ qed. + +lemma lfprs_strap1: ∀L1,L,L2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄. +/2 width=3/ qed. + +lemma lfprs_strap2: ∀L1,L,L2. ⦃L1⦄ ➡ ⦃L⦄ → ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄. +/2 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_aaa.ma new file mode 100644 index 000000000..5c6cd31cb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_aaa.ma @@ -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/reducibility/lfpr_aaa.ma". +include "basic_2/computation/lfprs.ma". + +(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) + +(* Properties about atomic arity assignment on terms ************************) + +lemma aaa_lfprs_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡* ⦃L2⦄ → L2 ⊢ T ⁝ A. +#L1 #T #A #HT #L2 #HL12 +@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_cprs.ma new file mode 100644 index 000000000..b0f7c4a94 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_cprs.ma @@ -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/lfpr_cpr.ma". +include "basic_2/computation/cprs.ma". +include "basic_2/computation/lfprs.ma". + +(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) + +(* Advanced properties ******************************************************) + +lemma lfprs_pair_dx: ∀I,L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡* V2 → + ⦃L1. ⓑ{I} V1⦄ ➡* ⦃L2. ⓑ{I} V2⦄. +#I #L1 #L2 #HL12 #V1 #V2 #H @(cprs_ind … H) -V2 +/3 width=1/ /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_lfprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_lfprs.ma new file mode 100644 index 000000000..e3866fd2e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs_lfprs.ma @@ -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/reducibility/lfpr_lfpr.ma". +include "basic_2/computation/lfprs_cprs.ma". + +(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************) + +(* Advanced properties ******************************************************) + +lemma lfprs_strip: ∀L,L1. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → + ∃∃L0. ⦃L1⦄ ➡ ⦃L0⦄ & ⦃L2⦄ ➡* ⦃L0⦄. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem lfprs_conf: ∀L,L1. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → + ∃∃L0. ⦃L1⦄ ➡* ⦃L0⦄ & ⦃L2⦄ ➡* ⦃L0⦄. +/3 width=3/ qed. + +theorem lfprs_trans: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄. +/2 width=3/ qed. + +lemma lfprs_pair: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡* V2 → + ∀I. ⦃L1. ⓑ{I} V1⦄ ➡* ⦃L2. ⓑ{I} V2⦄. +#L1 #L2 #H @(lfprs_ind … H) -L2 /2 width=1/ +#L #L2 #_ #HL2 #IHL1 #V1 #V2 #HV12 #I +@(lfprs_trans … (L.ⓑ{I}V1)) /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc.ma new file mode 100644 index 000000000..bcf6c7714 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc.ma @@ -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/static/aaa.ma". +include "basic_2/computation/acp_cr.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) + +inductive lsubc (RP:lenv→predicate term): relation lenv ≝ +| lsubc_atom: lsubc RP (⋆) (⋆) +| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V) +| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ ϵ[RP] 〚A〛 → L2 ⊢ W ⁝ A → + lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW) +. + +interpretation + "local environment refinement (abstract candidates of reducibility)" + 'CrSubEq L1 RP L2 = (lsubc RP L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L1 = ⋆ → L2 = ⋆. +#RP #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #A #_ #_ #_ #H destruct +] +qed. + +(* Basic_1: was: csubc_gen_sort_r *) +lemma lsubc_inv_atom1: ∀RP,L2. ⋆ ⊑[RP] L2 → L2 = ⋆. +/2 width=4/ qed-. + +fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V → + (∃∃K2. K1 ⊑[RP] K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & + K1 ⊑[RP] K2 & + L2 = K2. ⓛW & I = Abbr. +#RP #L1 #L2 * -L1 -L2 +[ #I #K1 #V #H destruct +| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/ +| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/ +] +qed. + +(* Basic_1: was: csubc_gen_head_r *) +lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V ⊑[RP] L2 → + (∃∃K2. K1 ⊑[RP] K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & + K1 ⊑[RP] K2 & + L2 = K2. ⓛW & I = Abbr. +/2 width=3/ qed-. + +fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L2 = ⋆ → L1 = ⋆. +#RP #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #A #_ #_ #_ #H destruct +] +qed. + +(* Basic_1: was: csubc_gen_sort_l *) +lemma lsubc_inv_atom2: ∀RP,L1. L1 ⊑[RP] ⋆ → L1 = ⋆. +/2 width=4/ qed-. + +fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W → + (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & + K1 ⊑[RP] K2 & + L1 = K1. ⓓV & I = Abst. +#RP #L1 #L2 * -L1 -L2 +[ #I #K2 #W #H destruct +| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/ +| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/ +] +qed. + +(* Basic_1: was: csubc_gen_head_l *) +lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 ⊑[RP] K2. ⓑ{I} W → + (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A & + K1 ⊑[RP] K2 & + L1 = K1. ⓓV & I = Abst. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: csubc_refl *) +lemma lsubc_refl: ∀RP,L. L ⊑[RP] L. +#RP #L elim L -L // /2 width=1/ +qed. + +(* Basic_1: removed theorems 3: + csubc_clear_conf csubc_getl_conf csubc_csuba +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_ldrop.ma new file mode 100644 index 000000000..a7c7c7a99 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_ldrop.ma @@ -0,0 +1,67 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/computation/lsubc.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) + +(* Properties concerning basic local environment slicing ********************) + +(* Basic_1: was: csubc_drop_conf_O *) +(* Note: the constant 0 can not be generalized *) +lemma lsubc_ldrop_O1_trans: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ⊑[RP] K2. +#RP #L1 #L2 #H elim H -L1 -L2 +[ #X #e #H + >(ldrop_inv_atom1 … H) -H /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #X #e #H + elim (ldrop_inv_O1 … H) -H * #He #H destruct + [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=3/ + | elim (IHL12 … H) -L2 /3 width=3/ + ] +| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #X #e #H + elim (ldrop_inv_O1 … H) -H * #He #H destruct + [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=7/ + | elim (IHL12 … H) -L2 /3 width=3/ + ] +qed-. + +(* Basic_1: was: csubc_drop_conf_rev *) +lemma ldrop_lsubc_trans: ∀RR,RS,RP. + acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 ⊑[RP] K2 → + ∃∃L2. L1 ⊑[RP] L2 & ⇩[d, e] L2 ≡ K2. +#RR #RS #RP #Hacp #Hacr #L1 #K1 #d #e #H elim H -L1 -K1 -d -e +[ #d #e #X #H + >(lsubc_inv_atom1 … H) -H /2 width=3/ +| #L1 #I #V1 #X #H + elim (lsubc_inv_pair1 … H) -H * + [ #K1 #HLK1 #H destruct /3 width=3/ + | #K1 #W1 #A #HV1 #HW1 #HLK1 #H1 #H2 destruct /3 width=3/ + ] +| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 + elim (IHLK1 … HK12) -K1 /3 width=5/ +| #L1 #K1 #I #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H + elim (lsubc_inv_pair1 … H) -H * + [ #K2 #HK12 #H destruct + elim (IHLK1 … HK12) -K1 /3 width=5/ + | #K2 #W2 #A #HV2 #HW2 #HK12 #H1 #H2 destruct + elim (IHLK1 … HK12) #K #HL1K #HK2 + lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA + lapply (s7 … HA … HV2 … HLK1 HV21) -HV2 + elim (lift_total W2 d e) /4 width=9/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_ldrops.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_ldrops.ma new file mode 100644 index 000000000..4e26322a5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_ldrops.ma @@ -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/computation/lsubc_ldrop.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) + +(* Properties concerning generic local environment slicing ******************) + +(* Basic_1: was: csubc_drop1_conf_rev *) +lemma ldrops_lsubc_trans: ∀RR,RS,RP. + acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀L1,K1,des. ⇩*[des] L1 ≡ K1 → ∀K2. K1 ⊑[RP] K2 → + ∃∃L2. L1 ⊑[RP] L2 & ⇩*[des] L2 ≡ K2. +#RR #RS #RP #Hacp #Hacr #L1 #K1 #des #H elim H -L1 -K1 -des +[ /2 width=3/ +| #L1 #L #K1 #des #d #e #_ #HLK1 #IHL #K2 #HK12 + elim (ldrop_lsubc_trans … Hacp Hacr … HLK1 … HK12) -Hacp -Hacr -K1 #K #HLK #HK2 + elim (IHL … HLK) -IHL -HLK /3 width=5/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_lsuba.ma new file mode 100644 index 000000000..aad454f62 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_lsuba.ma @@ -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/static/lsuba.ma". +include "basic_2/computation/acp_aaa.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****) + +(* properties concerning lenv refinement for atomic arity assignment ********) + +lemma lsubc_lsuba: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) → + ∀L1,L2. L1 ⁝⊑ L2 → L1 ⊑[RP] L2. +#RR #RS #RP #H1RP #H2RP #L1 #L2 #H elim H -L1 -L2 +// /2 width=1/ /3 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma new file mode 100644 index 000000000..b7b0e1094 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma @@ -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/reducibility/ltpr.ma". +include "basic_2/computation/tprs.ma". + +(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************) + +definition ltprs: relation lenv ≝ TC … ltpr. + +interpretation + "context-free parallel computation (environment)" + 'PRedStar L1 L2 = (ltprs L1 L2). + +(* Basic eliminators ********************************************************) + +lemma ltprs_ind: ∀L1. ∀R:predicate lenv. R L1 → + (∀L,L2. L1 ➡* L → L ➡ L2 → R L → R L2) → + ∀L2. L1 ➡* L2 → R L2. +#L1 #R #HL1 #IHL1 #L2 #HL12 +@(TC_star_ind … HL1 IHL1 … HL12) // +qed-. + +lemma ltprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 → + (∀L1,L. L1 ➡ L → L ➡* L2 → R L → R L1) → + ∀L1. L1 ➡* L2 → R L1. +#L2 #R #HL2 #IHL2 #L1 #HL12 +@(TC_star_ind_dx … HL2 IHL2 … HL12) // +qed-. + +(* Basic properties *********************************************************) + +lemma ltprs_refl: reflexive … ltprs. +/2 width=1/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma ltprs_inv_atom1: ∀L2. ⋆ ➡* L2 → L2 = ⋆. +#L2 #H @(ltprs_ind … H) -L2 // +#L #L2 #_ #HL2 #IHL1 destruct +>(ltpr_inv_atom1 … HL2) -L2 // +qed-. + +lemma ltprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ➡* L2 → + ∃∃K2,V2. K1 ➡* K2 & V1 ➡* V2 & L2 = K2. ⓑ{I} V2. +#I #K1 #L2 #V1 #H @(ltprs_ind … H) -L2 /2 width=5/ +#L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct +elim (ltpr_inv_pair1 … HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/ +qed-. + +lemma ltprs_inv_atom2: ∀L1. L1 ➡* ⋆ → L1 = ⋆. +#L1 #H @(ltprs_ind_dx … H) -L1 // +#L1 #L #HL1 #_ #IHL2 destruct +>(ltpr_inv_atom2 … HL1) -L1 // +qed-. + +lemma ltprs_inv_pair2: ∀I,L1,K2,V2. L1 ➡* K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ➡* K2 & V1 ➡* V2 & L1 = K1. ⓑ{I} V1. +#I #L1 #K2 #V2 #H @(ltprs_ind_dx … H) -L1 /2 width=5/ +#L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct +elim (ltpr_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma ltprs_fwd_length: ∀L1,L2. L1 ➡* L2 → |L1| = |L2|. +#L1 #L2 #H @(ltprs_ind … H) -L2 // +#L #L2 #_ #HL2 #IHL1 +>IHL1 -L1 >(ltpr_fwd_length … HL2) -HL2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_alt.ma new file mode 100644 index 000000000..7d532c973 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_alt.ma @@ -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/computation/ltprs.ma". + +(* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************) + +(* alternative definition of ltprs *) +definition ltprsa: relation lenv ≝ lpx tprs. + +interpretation + "context-free parallel computation (environment) alternative" + 'PRedStarAlt L1 L2 = (ltprsa L1 L2). + +(* Basic properties *********************************************************) + +lemma ltprs_ltprsa: ∀L1,L2. L1 ➡* L2 → L1 ➡➡* L2. +/2 width=1/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma ltprsa_ltprs: ∀L1,L2. L1 ➡➡* L2 → L1 ➡* L2. +/2 width=1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_ldrop.ma new file mode 100644 index 000000000..a7c320089 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_ldrop.ma @@ -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_2/computation/ltprs_ltprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_ltprs.ma new file mode 100644 index 000000000..e529ee31a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs_ltprs.ma @@ -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: ∀L1. ∀L:term. L ➡* L1 → ∀L2. L ➡ L2 → + ∃∃L0. L1 ➡ L0 & L2 ➡* L0. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem ltprs_conf: Confluent … ltprs. +/3 width=3/ qed. + +theorem ltprs_trans: Transitive … ltprs. +/2 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma new file mode 100644 index 000000000..b094e66c1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma @@ -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/reducibility/tpr.ma". + +(* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************) + +(* Basic_1: includes: pr1_pr0 *) +definition tprs: relation term ≝ TC … tpr. + +interpretation "context-free parallel computation (term)" + 'PRedStar T1 T2 = (tprs T1 T2). + +(* Basic eliminators ********************************************************) + +lemma tprs_ind: ∀T1. ∀R:predicate term. R T1 → + (∀T,T2. T1 ➡* T → T ➡ T2 → R T → R T2) → + ∀T2. T1 ➡* T2 → R T2. +#T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma tprs_ind_dx: ∀T2. ∀R:predicate term. R T2 → + (∀T1,T. T1 ➡ T → T ➡* T2 → R T → R T1) → + ∀T1. T1 ➡* T2 → R T1. +#T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma tprs_refl: reflexive … tprs. +/2 width=1/ qed. + +lemma tprs_strap1: ∀T1,T,T2. T1 ➡* T → T ➡ T2 → T1 ➡* T2. +/2 width=3/ qed. + +lemma tprs_strap2: ∀T1,T,T2. T1 ➡ T → T ➡* T2 → T1 ➡* T2. +/2 width=3/ qed. + +(* Basic_1: was only: pr1_head_1 *) +lemma tprs_pair_sn: ∀I,T1,T2. T1 ➡ T2 → ∀V1,V2. V1 ➡* V2 → + ②{I} V1. T1 ➡* ②{I} V2. T2. +* [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind … H) -V2 +[1,3: /3 width=1/ +|2,4: #V #V2 #_ #HV2 #IHV1 + @(tprs_strap1 … IHV1) -IHV1 /2 width=1/ +] +qed. + +(* Basic_1: was only: pr1_head_2 *) +lemma tprs_pair_dx: ∀I,V1,V2. V1 ➡ V2 → ∀T1,T2. T1 ➡* T2 → + ②{I} V1. T1 ➡* ②{I} V2. T2. +* [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind … H) -T2 +[1,3: /3 width=1/ +|2,4: #T #T2 #_ #HT2 #IHT1 + @(tprs_strap1 … IHT1) -IHT1 /2 width=1/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma tprs_inv_atom1: ∀U2,k. ⋆k ➡* U2 → U2 = ⋆k. +#U2 #k #H @(tprs_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU1 destruct +>(tpr_inv_atom1 … HU2) -HU2 // +qed-. + +lemma tprs_inv_cast1: ∀W1,T1,U2. ⓝW1.T1 ➡* U2 → T1 ➡* U2 ∨ + ∃∃W2,T2. W1 ➡* W2 & T1 ➡* T2 & U2 = ⓝW2.T2. +#W1 #T1 #U2 #H @(tprs_ind … H) -U2 /3 width=5/ +#U #U2 #_ #HU2 * /3 width=3/ * +#W #T #HW1 #HT1 #H destruct +elim (tpr_inv_cast1 … HU2) -HU2 /3 width=3/ * +#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/tprs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs_lift.ma new file mode 100644 index 000000000..d0d173470 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs_lift.ma @@ -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: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡* U2 → + ∃∃V2,T2. V1 ➡* V2 & T1 ➡* T2 & U2 = ⓛ{a}V2. T2. +#a #V1 #T1 #U2 #H @(tprs_ind … H) -U2 /2 width=5/ +#U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct +elim (tpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/ +qed-. + +lemma tprs_inv_abst: ∀a,V1,V2,T1,T2. ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 → + V1 ➡* V2 ∧ T1 ➡* T2. +#a #V1 #V2 #T1 #T2 #H +elim (tprs_inv_abst1 … H) -H #V #T #HV1 #HT1 #H destruct /2 width=1/ +qed-. + +(* Relocation properties ****************************************************) + +(* Note: this was missing in basic_1 *) +lemma tprs_lift: t_liftable tprs. +/3 width=7/ qed. + +(* Note: this was missing in basic_1 *) +lemma tprs_inv_lift1: t_deliftable_sn tprs. +/3 width=3 by tpr_inv_lift1, t_deliftable_sn_TC/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/tprs_tprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs_tprs.ma new file mode 100644 index 000000000..232244510 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs_tprs.ma @@ -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: ∀T1,T. T ➡* T1 → ∀T2. T ➡ T2 → + ∃∃T0. T1 ➡ T0 & T2 ➡* T0. +/3 width=3/ qed. + +(* Main propertis ***********************************************************) + +(* Basic_1: was: pr1_confluence *) +theorem tprs_conf: Confluent … tprs. +/3 width=3/ qed. + +(* Basic_1: was: pr1_t *) +theorem tprs_trans: Transitive … tprs. +/2 width=3/ qed. + +(* Basic_1: was: pr1_comp *) +lemma tprs_pair: ∀I,V1,V2. V1 ➡* V2 → ∀T1,T2. T1 ➡* T2 → + ②{I} V1. T1 ➡* ②{I} V2. T2. +#I #V1 #V2 #H @(tprs_ind … H) -V2 /2 width=1/ +#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 +@(tprs_trans … (②{I}V.T2)) /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/xprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs.ma new file mode 100644 index 000000000..854c5da27 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs.ma @@ -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/static/lsubss.ma". +include "basic_2/reducibility/xpr.ma". +(* +include "basic_2/reducibility/cnf.ma". +*) +(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************) + +definition xprs: ∀h. sd h → lenv → relation term ≝ + λh,g,L. TC … (xpr h g L). + +interpretation "extended parallel computation (term)" + 'XPRedStar h g L T1 T2 = (xprs h g L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma xprs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 → + (∀T,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → R T → R T2) → + ∀T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T2. +#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma xprs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 → + (∀T1,T. ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → R T → R T1) → + ∀T1. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → R T1. +#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma xprs_refl: ∀h,g,L. reflexive … (xprs h g L). +/2 width=1/ qed. + +lemma xprs_strap1: ∀h,g,L,T1,T,T2. + ⦃h, L⦄ ⊢ T1 •➡*[g] T → ⦃h, L⦄ ⊢ T •➡[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2. +/2 width=3/ qed. + +lemma xprs_strap2: ∀h,g,L,T1,T,T2. + ⦃h, L⦄ ⊢ T1 •➡[g] T → ⦃h, L⦄ ⊢ T •➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2. +/2 width=3/ qed. + +(* Basic inversion lemmas ***************************************************) +(* +axiom xprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U. +#L #T #U #H @(xprs_ind_dx … H) -T // +#T0 #T #H1T0 #_ #IHT #H2T0 +lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/ +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_aaa.ma new file mode 100644 index 000000000..5beb8fe19 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_aaa.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_2/reducibility/xpr_aaa.ma". +include "basic_2/computation/xprs.ma". + +(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************) + +(* Properties on atomic arity assignment for terms **************************) + +lemma xprs_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •➡*[g] U → L ⊢ U ⁝ A. +#h #g #L #T #A #HT #U #H @(xprs_ind … H) -U // /2 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_cprs.ma new file mode 100644 index 000000000..13a4f8889 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_cprs.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_2/computation/cprs.ma". +include "basic_2/computation/xprs.ma". + +(* EXTENDED PARALLEL COMPUTATION ON TERMS ***********************************) + +(* properties on context sensitive parallel computation for terms ***********) + +lemma cprs_xprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 •➡*[g] T2. +#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_lift.ma new file mode 100644 index 000000000..cb151a194 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_lift.ma @@ -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: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •➡*[g] U2 → + ∃∃V2,T2. L ⊢ V1 ➡* V2 & U2 = ⓛ{a}V2. T2. +#h #g #a #L #V1 #T1 #U2 #H @(xprs_ind … H) -U2 /2 width=4/ +#U #U2 #_ #HU2 * #V #T #HV1 #H destruct +elim (xpr_inv_abst1 … HU2) -HU2 #V2 #T2 #HV2 #_ #H destruct /3 width=4/ +qed-. + +(* Relocation properties ****************************************************) + +lemma xprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 → + ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •➡*[g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 → + ⦃h, L⦄ ⊢ U1 •➡*[g] U2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 #HT12 @(xprs_ind … HT12) -T2 +[ -HLK #T2 #HT12 + <(lift_mono … HTU1 … HT12) -T1 // +| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (xpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/ +] +qed. + +lemma xprs_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K → + ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •➡*[g] U2 → + ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •➡*[g] T2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 #HU12 @(xprs_ind … HU12) -U2 /2 width=3/ +-HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 +elim (xpr_inv_lift1 … HLK … HTU … HU2) -U -HLK /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_lsubss.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_lsubss.ma new file mode 100644 index 000000000..c883c14f3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_lsubss.ma @@ -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: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → + ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2. +#h #g #L1 #L2 #HL12 #T1 #T2 #H @(xprs_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (lsubss_xpr_trans … HL12 … HT2) -L2 /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_xprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_xprs.ma new file mode 100644 index 000000000..9593f0550 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/xprs_xprs.ma @@ -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: ∀h,g,L. transitive … (xprs h g L). +/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma new file mode 100644 index 000000000..5fb614a8c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma @@ -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.ma". + +(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************) + +definition cpc: lenv → relation term ≝ + λL,T1,T2. L ⊢ T1 ➡ T2 ∨ L ⊢ T2 ➡ T1. + +interpretation + "context-sensitive parallel conversion (term)" + 'PConv L T1 T2 = (cpc L T1 T2). + +(* Basic properties *********************************************************) + +lemma cpc_refl: ∀L. reflexive … (cpc L). +/2 width=1/ qed. + +lemma cpc_sym: ∀L. symmetric … (cpc L). +#L #T1 #T2 * /2 width=1/ +qed. + +(* Basic forward lemmas *****************************************************) + +lemma cpc_fwd_cpr: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T. +#L #T1 #T2 * /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma new file mode 100644 index 000000000..dcea07a8b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma @@ -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/cpc.ma". + +(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************) + +(* Main properties **********************************************************) + +theorem cpc_conf: ∀L,T0,T1,T2. L ⊢ T0 ⬌ T1 → L ⊢ T0 ⬌ T2 → + ∃∃T. L ⊢ T1 ⬌ T & L ⊢ T2 ⬌ T. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/fpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/fpc.ma new file mode 100644 index 000000000..f552d5818 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/fpc.ma @@ -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/reducibility/fpr.ma". + +(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************) + +definition fpc: bi_relation lenv term ≝ + λL1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ ∨ ⦃L2, T2⦄ ➡ ⦃L1, T1⦄. + +interpretation + "context-free parallel conversion (closure)" + 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2). + +(* Basic properties *********************************************************) + +lemma fpc_refl: bi_reflexive … fpc. +/2 width=1/ qed. + +lemma fpc_sym: bi_symmetric … fpc. +#L1 #L2 #T1 #T2 * /2 width=1/ +qed. + +(* Basic forward lemmas *****************************************************) + +lemma fpc_fwd_fpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ → + ∃∃L,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄. +#L1 #L2 #T1 #T2 * /2 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/fpc_fpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/fpc_fpc.ma new file mode 100644 index 000000000..22fc16f37 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/fpc_fpc.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_2/conversion/fpc.ma". + +(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************) + +(* Main properties **********************************************************) + +theorem fpc_conf: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ → + ∀L2,T2. ⦃L0, T0⦄ ⬌ ⦃L2, T2⦄ → + ∃∃L,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄. +/3 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/lfpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/lfpc.ma new file mode 100644 index 000000000..273873abd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/lfpc.ma @@ -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/lfpr.ma". + +(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************) + +definition lfpc: relation lenv ≝ + λL1,L2. ⦃L1⦄ ➡ ⦃L2⦄ ∨ ⦃L2⦄ ➡ ⦃L1⦄. + +interpretation + "focalized parallel conversion (local environment)" + 'FocalizedPConv L1 L2 = (lfpc L1 L2). + +(* Basic properties *********************************************************) + +lemma lfpc_refl: ∀L. ⦃L⦄ ⬌ ⦃L⦄. +/2 width=1/ qed. + +lemma lfpc_sym: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L2⦄ ⬌ ⦃L1⦄. +#L1 #L2 * /2 width=1/ +qed. + +lemma lfpc_lfpr: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄. +#L1 #L2 * /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/lfpc_lfpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/lfpc_lfpc.ma new file mode 100644 index 000000000..69e444adb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/lfpc_lfpc.ma @@ -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: ∀L0,L1,L2. ⦃L0⦄ ⬌ ⦃L1⦄ → ⦃L0⦄ ⬌ ⦃L2⦄ → + ∃∃L. ⦃L1⦄ ⬌ ⦃L⦄ & ⦃L2⦄ ⬌ ⦃L⦄. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma new file mode 100644 index 000000000..2be571525 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma @@ -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/computation/cprs.ma". +include "basic_2/computation/xprs.ma". +include "basic_2/equivalence/cpcs.ma". + +(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) + +inductive snv (h:sh) (g:sd h): lenv → predicate term ≝ +| snv_sort: ∀L,k. snv h g L (⋆k) +| snv_lref: ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → snv h g K V → snv h g L (#i) +| snv_bind: ∀a,I,L,V,T. snv h g L V → snv h g (L.ⓑ{I}V) T → snv h g L (ⓑ{a,I}V.T) +| snv_appl: ∀a,L,V,W,W0,T,U,l. snv h g L V → snv h g L T → + ⦃h, L⦄ ⊢ V •[g, l + 1] W → L ⊢ W ➡* W0 → + ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U → snv h g L (ⓐV.T) +| snv_cast: ∀L,W,T,U,l. snv h g L W → snv h g L T → + ⦃h, L⦄ ⊢ T •[g, l + 1] U → L ⊢ U ⬌* W → snv h g L (ⓝW.T) +. + +interpretation "stratified native validity (term)" + 'NativeValid h g L T = (snv h g L T). + +(* Basic inversion lemmas ***************************************************) + +fact snv_inv_lref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀i. X = #i → + ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊩ V :[g]. +#h #g #L #X * -L -X +[ #L #k #i #H destruct +| #I #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/ +| #a #I #L #V #T #_ #_ #i #H destruct +| #a #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #i #H destruct +| #L #W #T #U #l #_ #_ #_ #_ #i #H destruct +] +qed. + +lemma snv_inv_lref: ∀h,g,L,i. ⦃h, L⦄ ⊩ #i :[g] → + ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊩ V :[g]. +/2 width=3/ qed-. + +fact snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀a,I,V,T. X = ⓑ{a,I}V.T → + ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g]. +#h #g #L #X * -L -X +[ #L #k #a #I #V #T #H destruct +| #I0 #L #K #V0 #i #_ #_ #a #I #V #T #H destruct +| #b #I0 #L #V0 #T0 #HV0 #HT0 #a #I #V #T #H destruct /2 width=1/ +| #b #L #V0 #W0 #W00 #T0 #U0 #l #_ #_ #_ #_ #_ #a #I #V #T #H destruct +| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #a #I #V #T #H destruct +] +qed. + +lemma snv_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊩ ⓑ{a,I}V.T :[g] → + ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g]. +/2 width=4/ qed-. + +fact snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀V,T. X = ⓐV.T → + ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] & + ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 & + ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U. +#h #g #L #X * -L -X +[ #L #k #V #T #H destruct +| #I #L #K #V0 #i #_ #_ #V #T #H destruct +| #a #I #L #V0 #T0 #_ #_ #V #T #H destruct +| #a #L #V0 #W0 #W00 #T0 #U0 #l #HV0 #HT0 #HVW0 #HW00 #HTU0 #V #T #H destruct /2 width=8/ +| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #V #T #H destruct +] +qed. + +lemma snv_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊩ ⓐV.T :[g] → + ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] & + ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 & + ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U. +/2 width=3/ qed-. + +fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀W,T. X = ⓝW.T → + ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] & + ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W. +#h #g #L #X * -L -X +[ #L #k #W #T #H destruct +| #I #L #K #V #i #_ #_ #W #T #H destruct +| #a #I #L #V #T0 #_ #_ #W #T #H destruct +| #a #L #V #W0 #W00 #T0 #U #l #_ #_ #_ #_ #_ #W #T #H destruct +| #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HUW0 #W #T #H destruct /2 width=4/ +] +qed. + +lemma snv_inv_cast: ∀h,g,L,W,T. ⦃h, L⦄ ⊩ ⓝW.T :[g] → + ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] & + ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W. +/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma new file mode 100644 index 000000000..3d4761da8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma @@ -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/computation/csn_aaa.ma". +include "basic_2/computation/xprs_aaa.ma". +include "basic_2/computation/xprs_cprs.ma". +include "basic_2/equivalence/cpcs_aaa.ma". +include "basic_2/dynamic/snv.ma". + +(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) + +(* Properties on atomic arity assignment for terms **************************) + +lemma snv_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃A. L ⊢ T ⁝ A. +#h #g #L #T #H elim H -L -T +[ /2 width=2/ +| #I #L #K #V #i #HLK #_ * /3 width=6/ +| #a * #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2/ +| #a #L #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU * #B #HV * #X #HT + lapply (xprs_aaa h g … HV W0 ?) [ /3 width=3/ ] -W #HW0 + lapply (xprs_aaa … HT … HTU) -HTU #H + elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct + lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4/ +| #L #W #T #U #l #_ #_ #HTU #HUW * #B #HW * #A #HT + lapply (aaa_cpcs_mono … HUW A … HW) -HUW /2 width=7/ -HTU #H destruct /3 width=3/ +] +qed-. + +lemma snv_csn: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → L ⊢ ⬊* T. +#h #g #L #T #H elim (snv_aaa … H) -H /2 width=2/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma new file mode 100644 index 000000000..6d79ef571 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma @@ -0,0 +1,79 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_lift.ma". +include "basic_2/equivalence/cpcs_cpcs.ma". +include "basic_2/dynamic/snv.ma". + +(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) + +(* Relocation properties ****************************************************) + +lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊩ T :[g] → ∀L,d,e. ⇩[d, e] L ≡ K → + ∀U. ⇧[d, e] T ≡ U → ⦃h, L⦄ ⊩ U :[g]. +#h #g #K #T #H elim H -K -T +[ #K #k #L #d #e #_ #X #H + >(lift_inv_sort1 … H) -X -K -d -e // +| #I #K #K0 #V #i #HK0 #_ #IHV #L #d #e #HLK #X #H + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (ldrop_trans_le … HLK … HK0 ?) -K /2 width=2/ #X #HL0 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #L0 #W #HLK0 #HVW #H destruct + /3 width=8/ + | lapply (ldrop_trans_ge … HLK … HK0 ?) -K // -Hid /3 width=8/ + ] +| #a #I #K #V #T #_ #_ #IHV #IHT #L #d #e #HLK #X #H + elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct + /4 width=4/ +| #a #K #V #V0 #V1 #T #T1 #l #_ #_ #HV0 #HV01 #HT1 #IHV #IHT #L #d #e #HLK #X #H + elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct + elim (lift_total V0 d e) #W0 #HVW0 + elim (lift_total V1 d e) #W1 #HVW1 + elim (lift_total T1 (d+1) e) #U1 #HTU1 + @(snv_appl … a … W0 … W1 … U1 l) + [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ] + @(xprs_lift … HLK … HTU … HT1) /2 width=1/ +| #K #V0 #T #V #l #_ #_ #HTV #HV0 #IHV0 #IHT #L #d #e #HLK #X #H + elim (lift_inv_flat1 … H) -H #W0 #U #HVW0 #HTU #H destruct + elim (lift_total V d e) #W #HVW + @(snv_cast … W l) [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ] +] +qed. + +lemma snv_inv_lift: ∀h,g,L,U. ⦃h, L⦄ ⊩ U :[g] → ∀K,d,e. ⇩[d, e] L ≡ K → + ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊩ T :[g]. +#h #g #L #U #H elim H -L -U +[ #L #k #K #d #e #_ #X #H + >(lift_inv_sort2 … H) -X -L -d -e // +| #I #L #L0 #W #i #HL0 #_ #IHW #K #d #e #HLK #X #H + elim (lift_inv_lref2 … H) * #Hid #H destruct + [ elim (ldrop_conf_le … HLK … HL0 ?) -L /2 width=2/ #X #HK0 #H + elim (ldrop_inv_skip1 … H ?) -H /2 width=1/ -Hid #K0 #V #HLK0 #HVW #H destruct + /3 width=8/ + | lapply (ldrop_conf_ge … HLK … HL0 ?) -L // -Hid /3 width=8/ + ] +| #a #I #L #W #U #_ #_ #IHW #IHU #K #d #e #HLK #X #H + elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=4/ +| #a #L #W #W0 #W1 #U #U1 #l #_ #_ #HW0 #HW01 #HU1 #IHW #IHU #K #d #e #HLK #X #H + elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct + elim (ssta_inv_lift1 … HW0 … HLK … HVW) -HW0 #V0 #HV0 #HVW0 + elim (cprs_inv_lift1 … HLK … HVW0 … HW01) -W0 #V1 #HVW1 #HV01 + elim (xprs_inv_lift1 … HLK … HTU … HU1) -HU1 #X #H #HTU + elim (lift_inv_bind2 … H) -H #Y #T1 #HY #HTU1 #H destruct + lapply (lift_inj … HY … HVW1) -HY #H destruct /3 width=8/ +| #L #W0 #U #W #l #_ #_ #HUW #HW0 #IHW0 #IHU #K #d #e #HLK #X #H + elim (lift_inv_flat2 … H) -H #V0 #T #HVW0 #HTU #H destruct + elim (ssta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HTV #HVW + lapply (cpcs_inv_lift … HLK … HVW … HVW0 ?) // -W /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma new file mode 100644 index 000000000..d96994e6c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma @@ -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/dynamic/snv.ma". + +(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) + +(* Properties on stratified static type assignment for terms ****************) + +lemma snv_ssta: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃∃U,l. ⦃h, L⦄ ⊢ T •[g, l] U. +#h #g #L #T #H elim H -L -T +[ #L #k elim (deg_total h g k) /3 width=3/ +| * #L #K #V #i #HLK #_ * #W #l0 #HVW + [ elim (lift_total W 0 (i+1)) /3 width=8/ + | elim (lift_total V 0 (i+1)) /3 width=8/ + ] +| #a #I #L #V #T #_ #_ #_ * /3 width=3/ +| #a #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/ +| #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *) +] +qed-. + +fact snv_ssta_conf_aux: ∀h,g,L,T. ( + ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] → + ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 → + #{L0, T0} < #{L, T} → ⦃h, L0⦄ ⊩ U0 :[g] + ) → + ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] → + ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 → + L0 = L → T0 = T → ⦃h, L0⦄ ⊩ U0 :[g]. +#h #g #L #T #IH1 #L0 #T0 * -L0 -T0 +[ +| +| +| #a #L0 #V #W #W0 #T0 #V0 #l0 #HV #HT0 #HVW #HW0 #HTV0 #X #l #H #H1 #H2 destruct + elim (ssta_inv_appl1 … H) -H #U0 #HTU0 #H destruct + lapply (IH1 … HT0 … HTU0 ?) // #HU0 + @(snv_appl … HV HU0 HVW HW0) -HV -HU0 -HVW -HW0 +| #L0 #W #T0 #W0 #l0 #_ #HT0 #_ #_ #U0 #l #H #H1 #H2 destruct -W0 + lapply (ssta_inv_cast1 … H) -H /2 width=5/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma new file mode 100644 index 000000000..9d9ceb942 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma @@ -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/conversion/cpc.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) + +definition cpcs: lenv → relation term ≝ + λL. TC … (cpc L). + +interpretation "context-sensitive parallel equivalence (term)" + 'PConvStar L T1 T2 = (cpcs L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 → + (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) → + ∀T2. L ⊢ T1 ⬌* T2 → R T2. +#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma cpcs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 → + (∀T1,T. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → R T → R T1) → + ∀T1. L ⊢ T1 ⬌* T2 → R T1. +#L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: pc3_refl *) +lemma cpcs_refl: ∀L. reflexive … (cpcs L). +/2 width=1/ qed. + +(* Basic_1: was: pc3_s *) +lemma cpcs_sym: ∀L. symmetric … (cpcs L). +/3 width=1/ qed. + +lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2. +/2 width=3/ qed. + +lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +/2 width=3/ qed. + +(* Basic_1: was: pc3_pr2_r *) +lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2. +/3 width=1/ qed. + +lemma cpcs_tpr_dx: ∀L,T1,T2. T1 ➡ T2 → L ⊢ T1 ⬌* T2. +/3 width=1/ qed. + +(* Basic_1: was: pc3_pr2_x *) +lemma cpcs_cpr_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2. +/3 width=1/ qed. + +lemma cpcs_tpr_sn: ∀L,T1,T2. T2 ➡ T1 → L ⊢ T1 ⬌* T2. +/3 width=1/ qed. + +lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2. +/3 width=3/ qed. + +(* Basic_1: was: pc3_pr2_u *) +lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +/3 width=3/ qed. + +lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2. +/3 width=3/ qed. + +lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2. +/3 width=3/ qed-. + +(* Basic_1: was: pc3_pr2_u2 *) +lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +/3 width=3/ qed. + +(* Basic_1: removed theorems 9: + clear_pc3_trans pc3_ind_left + pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21 + pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0 + Basic_1: removed local theorems 6: + pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left + pc3_wcpr0_t_aux +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma new file mode 100644 index 000000000..363b8acd3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma @@ -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/cprs_aaa.ma". +include "basic_2/equivalence/cpcs_cpcs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) + +(* Main properties about atomic arity assignment on terms *******************) + +theorem aaa_cpcs_mono: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → + ∀A1. L ⊢ T1 ⁝ A1 → ∀A2. L ⊢ T2 ⁝ A2 → + A1 = A2. +#L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2 +elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 +lapply (aaa_cprs_conf … HA1 … HT1) -T1 #HA1 +lapply (aaa_cprs_conf … HA2 … HT2) -T2 #HA2 +lapply (aaa_mono … HA1 … HA2) -L -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma new file mode 100644 index 000000000..ac9de9300 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma @@ -0,0 +1,204 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_lift.ma". +include "basic_2/computation/cprs_cprs.ma". +include "basic_2/conversion/cpc_cpc.ma". +include "basic_2/equivalence/cpcs_cprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) + +(* Advanced inversion lemmas ************************************************) + +lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → + ∃∃T. L ⊢ T1 ➡* T & L ⊢ T2 ➡* T. +#L #T1 #T2 #H @(cpcs_ind … H) -T2 +[ /3 width=3/ +| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 + [ elim (cprs_strip … HT0 … HT2) -T #T #HT0 #HT2 + lapply (cprs_strap1 … HT10 … HT0) -T0 /2 width=3/ + | lapply (cprs_strap2 … HT2 … HT0) -T /2 width=3/ + ] +] +qed-. + +(* Basic_1: was: pc3_gen_sort *) +lemma cpcs_inv_sort: ∀L,k1,k2. L ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2. +#L #k1 #k2 #H +elim (cpcs_inv_cprs … H) -H #T #H1 +>(cprs_inv_sort1 … H1) -T #H2 +lapply (cprs_inv_sort1 … H2) -L #H destruct // +qed-. + +(* Basic_1: was: pc3_gen_sort_abst *) +lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. L ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥. +#a #L #W #T #k #H +elim (cpcs_inv_cprs … H) -H #X #H1 +>(cprs_inv_sort1 … H1) -X #H2 +elim (cprs_inv_abst1 Abst W … H2) -H2 #W0 #T0 #_ #_ #H destruct +qed-. + +(* Basic_1: was: pc3_gen_abst *) +lemma cpcs_inv_abst: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → ∀I,V. + ∧∧ L ⊢ W1 ⬌* W2 & L. ②{I}V ⊢ T1 ⬌* T2 & a1 = a2. +#a1 #a2 #L #W1 #W2 #T1 #T2 #H #I #V +elim (cpcs_inv_cprs … H) -H #T #H1 #H2 +elim (cprs_inv_abst1 I V … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct +elim (cprs_inv_abst1 I V … H2) -H2 #W #T #HW2 #HT2 #H destruct /3 width=3/ +qed-. + +(* Basic_1: was: pc3_gen_abst_shift *) +lemma cpcs_inv_abst_shift: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → ∀W. + ∧∧ L ⊢ W1 ⬌* W2 & L. ⓛW ⊢ T1 ⬌* T2 & a1 = a2. +#a1 #a2 #L #W1 #W2 #T1 #T2 #H #W +lapply (cpcs_inv_abst … H Abst W) -H // +qed. + +lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. L ⊢ ⓛ{a}W1.T1 ⬌* T → + ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2. +#a #L #W1 #T1 #T #H +elim (cpcs_inv_cprs … H) -H #X #H1 #H2 +elim (cprs_inv_abst1 Abst W1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct +@(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *) +qed-. + +lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. L ⊢ T ⬌* ⓛ{a}W1.T1 → + ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2. +/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-. + +(* Basic_1: was: pc3_gen_lift *) +lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → + ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 → + L ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12 +elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2 +elim (cprs_inv_lift1 … HLK … HTU1 … HU1) -U1 #T #HTU #HT1 +elim (cprs_inv_lift1 … HLK … HTU2 … HU2) -L -U2 #X #HXU +>(lift_inj … HXU … HTU) -X -U -d -e /2 width=3/ +qed-. + +(* Advanced properties ******************************************************) + +lemma cpr_cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2. +#L #T #T1 #T2 #HT1 #HT2 +elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/ +qed-. + +lemma cprs_cpr_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1. +#L #T #T1 #T2 #HT1 #HT2 +elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/ +qed-. + +lemma cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2. +#L #T #T1 #T2 #HT1 #HT2 +elim (cprs_conf … HT1 … HT2) /2 width=3/ +qed-. + +(* Basic_1: was only: pc3_thin_dx *) +lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 → + ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 #I +elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2 +elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *) +qed. + +lemma cpcs_flat_dx_tpr_rev: ∀L,V1,V2. V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 → + ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2. +/3 width=1/ qed. + +lemma cpcs_abst: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → + ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛ{a}V1. T1 ⬌* ⓛ{a}V2. T2. +#a #L #V1 #V2 #HV12 #V #T1 #T2 #HT12 +elim (cpcs_inv_cprs … HV12) -HV12 +elim (cpcs_inv_cprs … HT12) -HT12 +/3 width=6 by cprs_div, cprs_abst/ (**) (* /3 width=6/ is a bit slow *) +qed. + +lemma cpcs_abbr_dx: ∀a,L,V,T1,T2. L.ⓓV ⊢ T1 ⬌* T2 → L ⊢ ⓓ{a}V. T1 ⬌* ⓓ{a}V. T2. +#a #L #V #T1 #T2 #HT12 +elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *) +qed. + +lemma cpcs_bind_dx: ∀a,I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 → + L ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2. +#a * /2 width=1/ /2 width=2/ qed. + +lemma cpcs_abbr_sn: ∀a,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓓ{a}V1. T ⬌* ⓓ{a}V2. T. +#a #L #V1 #V2 #T #HV12 +elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *) +qed. + +lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T. +#a * /2 width=1/ /2 width=2/ qed. + +lemma cpcs_beta_dx: ∀a,L,V1,V2,W,T1,T2. + L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ⬌* T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ⬌* ⓓ{a}V2.T2. +#a #L #V1 #V2 #W #T1 #T2 #HV12 #HT12 +elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 +lapply (cprs_beta_dx … HV12 HT1 a) -HV12 -HT1 #HT1 +lapply (cprs_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 +@(cprs_div … HT1) /2 width=1/ +qed. + +lemma cpcs_beta_dx_tpr_rev: ∀a,L,V1,V2,W,T1,T2. + V1 ➡ V2 → L.ⓛW ⊢ T2 ⬌* T1 → + L ⊢ ⓓ{a}V2.T2 ⬌* ⓐV1.ⓛ{a}W.T1. +/4 width=1/ qed. + +(* Note: it does not hold replacing |L1| with |L2| *) +lemma cpcs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 → + ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ⬌* T2. +#L1 #T1 #T2 #HT12 +elim (cpcs_inv_cprs … HT12) -HT12 +/3 width=5 by cprs_div, cprs_lsubs_trans/ (**) (* /3 width=5/ is a bit slow *) +qed. + +(* Basic_1: was: pc3_lift *) +lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → + ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 → + K ⊢ T1 ⬌* T2 → L ⊢ U1 ⬌* U2. +#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12 +elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 +elim (lift_total T d e) #U #HTU +lapply (cprs_lift … HLK … HTU1 … HT1 … HTU) -T1 #HU1 +lapply (cprs_lift … HLK … HTU2 … HT2 … HTU) -K -T2 -T -d -e /2 width=3/ +qed. + +lemma cpcs_strip: ∀L,T1,T. L ⊢ T ⬌* T1 → ∀T2. L ⊢ T ⬌ T2 → + ∃∃T0. L ⊢ T1 ⬌ T0 & L ⊢ T2 ⬌* T0. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +(* Basic_1: was pc3_t *) +theorem cpcs_trans: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +/2 width=3/ qed. + +theorem cpcs_canc_sn: ∀L,T,T1,T2. L ⊢ T ⬌* T1 → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +/3 width=3 by cpcs_trans, cpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) + +theorem cpcs_canc_dx: ∀L,T,T1,T2. L ⊢ T1 ⬌* T → L ⊢ T2 ⬌* T → L ⊢ T1 ⬌* T2. +/3 width=3 by cpcs_trans, cpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) + +lemma cpcs_abbr1: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV1 ⊢ T1 ⬌* T2 → + L ⊢ ⓓ{a}V1. T1 ⬌* ⓓ{a}V2. T2. +#a #L #V1 #V2 #HV12 #T1 #T2 #HT12 +@(cpcs_trans … (ⓓ{a}V1.T2)) /2 width=1/ +qed. + +lemma cpcs_abbr2: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV2 ⊢ T1 ⬌* T2 → + L ⊢ ⓓ{a}V1. T1 ⬌* ⓓ{a}V2. T2. +#a #L #V1 #V2 #HV12 #T1 #T2 #HT12 +@(cpcs_trans … (ⓓ{a}V2.T1)) /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma new file mode 100644 index 000000000..cb1f5d76a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma @@ -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/computation/cprs.ma". +include "basic_2/equivalence/cpcs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) + +(* Properties about context sensitive computation on terms ******************) + +(* Basic_1: was: pc3_pr3_r *) +lemma cpcs_cprs_dx: ∀L,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ T1 ⬌* T2. +#L #T1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /3 width=3/ +qed. + +(* Basic_1: was: pc3_pr3_x *) +lemma cpcs_cprs_sn: ∀L,T1,T2. L ⊢ T2 ➡* T1 → L ⊢ T1 ⬌* T2. +#L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /3 width=3/ +qed. + +lemma cpcs_cprs_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2. +#L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /2 width=3/ +qed. + +lemma cpcs_cprs_strap2: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +#L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /width=1/ /2 width=3/ +qed. + +lemma cpcs_cprs_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2. +#L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /2 width=3/ +qed. + +(* Basic_1: was: pc3_pr3_conf *) +lemma cpcs_cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. +#L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /width=1/ /2 width=3/ +qed. + +(* Basic_1: was: pc3_pr3_t *) +(* Basic_1: note: pc3_pr3_t should be renamed *) +lemma cprs_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2. +#L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=1/ /2 width=3/ +qed. + +lemma cprs_cpr_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2. +/3 width=5 by step, cprs_div/ qed-. + +lemma cpr_cprs_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2. +/3 width=3 by step, cprs_div/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_delift.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_delift.ma new file mode 100644 index 000000000..7012ec11e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_delift.ma @@ -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: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 → + L ⊢ T2 ⬌* +ⓓV.T1. +/3 width=1/ qed. + +(* Basic_1: was only: pc3_gen_cabbr *) +lemma thin_cpcs_delift_mono: ∀L,U1,U2. L ⊢ U1 ⬌* U2 → + ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 → + ∀T2. L ⊢ ▼*[d, e] U2 ≡ T2 → K ⊢ T1 ⬌* T2. +#L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2 +elim (cpcs_inv_cprs … H) -H #U #HU1 #HU2 +elim (thin_cprs_delift_conf … HU1 … HLK … HTU1) -U1 #T #HT1 #HUT +elim (thin_cprs_delift_conf … HU2 … HLK … HTU2) -U2 -HLK #X #HT2 #H +lapply (delift_mono … H … HUT) -L #H destruct /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_ltpr.ma new file mode 100644 index 000000000..13713e51e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_ltpr.ma @@ -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/cpr_ltpr.ma". +include "basic_2/equivalence/cpcs_cpcs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) + +(* Properties about context-free parallel reduction on local environments ***) + +(* Basic_1: was only: pc3_pr0_pr2_t *) +(* Basic_1: note: pc3_pr0_pr2_t should be renamed *) +lemma ltpr_cpr_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2. +#L1 #L2 #HL12 #T1 #T2 #HT12 +elim (cpr_ltpr_conf_eq … HT12 … HL12) -L1 #T #HT1 #HT2 +@(cprs_div … T) /2 width=1/ /3 width=1/ (**) (* /4 width=3/ is too long *) +qed. + +(* Basic_1: was: pc3_wcpr0_t *) +(* Basic_1: note: pc3_wcpr0_t should be renamed *) +lemma ltpr_cprs_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2. +#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 +@(cpcs_trans … IHT1) -T1 /2 width=3/ +qed. + +(* Basic_1: was: pc3_wcpr0 *) +lemma ltpr_cpcs_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2. +#L1 #L2 #HL12 #T1 #T2 #H +elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2 +@(cpcs_canc_dx … T) /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_ltpss.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_ltpss.ma new file mode 100644 index 000000000..43385668b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_ltpss.ma @@ -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/equivalence/cpcs_cpcs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) + +(* Properties concerning partial unfold on local environments ***************) + +lemma ltpss_dx_cpr_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → + ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2. +#L1 #L2 #d #e #HL12 #T1 #T2 * +lapply (ltpss_dx_weak_all … HL12) +>(ltpss_dx_fwd_length … HL12) -HL12 #HL12 #T #HT1 #HT2 +elim (ltpss_dx_tpss_conf … HT2 … HL12) -L1 #T0 #HT0 #HT20 +@(cprs_div … T0) /3 width=3/ (**) (* /4/ is too slow *) +qed. + +lemma ltpss_dx_cprs_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → + ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2. +#L1 #L2 #d #e #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 +@(cpcs_trans … IHT1) -T1 /2 width=5/ +qed. + +lemma ltpss_dx_cpcs_conf: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → + ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2. +#L1 #L2 #d #e #HL12 #T1 #T2 #H +elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2 +@(cpcs_canc_dx … T) /2 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma new file mode 100644 index 000000000..c0e02359b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma @@ -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/conversion/fpc.ma". + +(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************) + +definition fpcs: bi_relation lenv term ≝ bi_TC … fpc. + +interpretation "context-free parallel equivalence (closure)" + 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma fpcs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 → + (∀L,L2,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → R L T → R L2 T2) → + ∀L2,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L2 T2. +/3 width=7 by bi_TC_star_ind/ qed-. + +lemma fpcs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 → + (∀L1,L,T1,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → R L T → R L1 T1) → + ∀L1,T1. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L1 T1. +/3 width=7 by bi_TC_star_ind_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma fpcs_refl: bi_reflexive … fpcs. +/2 width=1/ qed. + +lemma fpcs_sym: bi_symmetric … fpcs. +/3 width=1/ qed. + +lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma fpcs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma fpcs_fpr_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=1/ qed. + +lemma fpcs_fpr_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡ ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=1/ qed. + +lemma fpcs_fpr_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → + ∀L2,T2. ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4/ qed. + +lemma fpcs_fpr_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → + ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4/ qed. + +lemma fpcs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → + ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4/ qed. + +lemma fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4/ qed-. + +lemma fpcs_fpr_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡ ⦃L1, T1⦄ → + ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_aaa.ma new file mode 100644 index 000000000..9f4327bff --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_aaa.ma @@ -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: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → + ∀A1. L1 ⊢ T1 ⁝ A1 → ∀A2. L2 ⊢ T2 ⁝ A2 → + A1 = A2. +#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2 +elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2 +lapply (aaa_fprs_conf … HT1 … H1) -L1 -T1 #HT1 +lapply (aaa_fprs_conf … HT2 … H2) -L2 -T2 #HT2 +lapply (aaa_mono … HT1 … HT2) -L -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_cpcs.ma new file mode 100644 index 000000000..4b51a7084 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_cpcs.ma @@ -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/fprs_cprs.ma". +include "basic_2/equivalence/cpcs_cpcs.ma". +include "basic_2/equivalence/fpcs_fprs.ma". + +(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************) + +(* Properties on context-sensitive parallel equivalence for terms ***********) + +lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄. +#L #T1 #T2 #H +elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_fpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_fpcs.ma new file mode 100644 index 000000000..270e8dc40 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_fpcs.ma @@ -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: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → + ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡* ⦃L, T⦄. +#L1 #L2 #T1 #T2 #H @(fpcs_ind … H) -L2 -T2 +[ /3 width=4/ +| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 + [ elim (fprs_strip … HT2 … HT0) -L -T #L #T #HT2 #HT0 + lapply (fprs_strap1 … HT10 … HT0) -L0 -T0 /2 width=4/ + | lapply (fprs_strap2 … HT2 … HT0) -L -T /2 width=4/ + ] +] +qed-. + +(* Advanced properties ******************************************************) + +lemma fpr_fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 +elim (fprs_strip … HT2 … HT1) /2 width=4 by fpr_fprs_div/ +qed-. + +lemma fprs_fpr_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L2, T2⦄ ⬌* ⦃L1, T1⦄. +#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 +elim (fprs_strip … HT2 … HT1) /2 width=4 by fprs_fpr_div/ +qed-. + +lemma fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L #L1 #L2 #T #T1 #T2 #HT1 #HT2 +elim (fprs_conf … HT1 … HT2) /2 width=4/ +qed-. + +lemma fpcs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ → + ∀L2,T2. ⦃L0, T0⦄ ⬌* ⦃L2, T2⦄ → + ∃∃L,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄. +/3 width=4/ qed. + +(* Main properties **********************************************************) + +theorem fpcs_trans: bi_transitive … fpcs. +/2 width=4/ qed. + +theorem fpcs_canc_sn: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ⬌* ⦃L1, T1⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) + +theorem fpcs_canc_dx: ∀L1,L2,L,T1,T2,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L2, T2⦄ ⬌* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_fprs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_fprs.ma new file mode 100644 index 000000000..1d3f71f9e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs_fprs.ma @@ -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: lemma 1000 *) +lemma fpcs_fprs_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /3 width=4/ +qed. + +lemma fpcs_fprs_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡* ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L2 #T1 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /3 width=4/ +qed. + +lemma fpcs_fprs_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /2 width=4/ +qed. + +lemma fpcs_fprs_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx … H) -L1 -T1 /width=1/ /2 width=4/ +qed. + +lemma fpcs_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /2 width=4/ +qed. + +lemma fpcs_fprs_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind … H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *) +qed. + +lemma fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx … H) -L2 -T2 /2 width=1/ /2 width=4/ +qed. + +lemma fprs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=7 by bi_step, fprs_div/ qed-. + +lemma fpr_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄. +/3 width=4 by bi_step, fprs_div/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma new file mode 100644 index 000000000..fded17bd1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma @@ -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/conversion/lfpc.ma". + +(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************) + +definition lfpcs: relation lenv ≝ TC … lfpc. + +interpretation "focalized parallel equivalence (local environment)" + 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2). + +(* Basic eliminators ********************************************************) + +lemma lfpcs_ind: ∀L1. ∀R:predicate lenv. R L1 → + (∀L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → R L → R L2) → + ∀L2. ⦃L1⦄ ⬌* ⦃L2⦄ → R L2. +#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) // +qed-. + +lemma lfpcs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 → + (∀L1,L. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → R L → R L1) → + ∀L1. ⦃L1⦄ ⬌* ⦃L2⦄ → R L1. +#L2 #R #HL2 #IHL2 #L1 #HL12 +@(TC_star_ind_dx … HL2 IHL2 … HL12) // +qed-. + +(* Basic properties *********************************************************) + +lemma lfpcs_refl: reflexive … lfpcs. +/2 width=1/ qed. + +lemma lfprs_sym: symmetric … lfpcs. +/3 width=1/ qed. + +lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/2 width=3/ qed. + +lemma lfpcs_strap2: ∀L1,L,L2. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/2 width=3/ qed. + +lemma lfpcs_lfpr_dx: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=1/ qed. + +lemma lfpcs_lfpr_sn: ∀L1,L2. ⦃L2⦄ ➡ ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=1/ qed. + +lemma lfpcs_lfpr_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=3/ qed. + +lemma lfpcs_lfpr_strap2: ∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=3/ qed. + +lemma lfpcs_lfpr_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡ ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=3/ qed. + +lemma lfpcs_lfpr_conf: ∀L1,L. ⦃L⦄ ➡ ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_aaa.ma new file mode 100644 index 000000000..b7cea0b7a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_aaa.ma @@ -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: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ → + ∀T,A1. L1 ⊢ T ⁝ A1 → ∀A2. L2 ⊢ T ⁝ A2 → + A1 = A2. +#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2 +elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2 +lapply (aaa_lfprs_conf … HT1 … HL1) -L1 #HT1 +lapply (aaa_lfprs_conf … HT2 … HL2) -L2 #HT2 +lapply (aaa_mono … HT1 … HT2) -L -T // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma new file mode 100644 index 000000000..434068e4f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma @@ -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: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ → + ∃∃L. ⦃L1⦄ ➡* ⦃L⦄ & ⦃L2⦄ ➡* ⦃L⦄. +#L1 #L2 #H @(lfpcs_ind … H) -L2 +[ /3 width=3/ +| #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0 + [ elim (lfprs_strip … HL0 … HL2) -L #L #HL0 #HL2 + lapply (lfprs_strap1 … HL10 … HL0) -L0 /2 width=3/ + | lapply (lfprs_strap2 … HL2 … HL0) -L /2 width=3/ + ] +] +qed-. + +(* Advanced properties ******************************************************) + +lemma lfpcs_strip: ∀L,L1. ⦃L⦄ ⬌* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌ ⦃L2⦄ → + ∃∃L0. ⦃L1⦄ ⬌ ⦃L0⦄ & ⦃L2⦄ ⬌* ⦃L0⦄. +/3 width=3/ qed. + +(* Main properties **********************************************************) + +theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/2 width=3/ qed. + +theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=3 by lfpcs_trans, lfprs_sym/ qed. + +theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +/3 width=3 by lfpcs_trans, lfprs_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfprs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfprs.ma new file mode 100644 index 000000000..baf2caf27 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfprs.ma @@ -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: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /3 width=3/ +qed. + +lemma lfpcs_lfprs_sn: ∀L1,L2. ⦃L2⦄ ➡* ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /3 width=3/ +qed. + +lemma lfpcs_lfprs_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L #HL1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /2 width=3/ +qed. + +lemma lfpcs_lfprs_strap2: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L #H #L2 #HL2 @(lfprs_ind_dx … H) -L1 /width=1/ /2 width=3/ +qed. + +lemma lfpcs_lfprs_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /2 width=3/ +qed. + +lemma lfpcs_lfprs_conf: ∀L1,L. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L #H #L2 #HL2 @(lfprs_ind … H) -L1 /width=1/ /2 width=3/ +qed. + +lemma lfprs_div: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄. +#L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /2 width=1/ /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse.ma new file mode 100644 index 000000000..6719c7afe --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse.ma @@ -0,0 +1,114 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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.ma". +include "basic_2/computation/cprs.ma". +include "basic_2/equivalence/cpcs.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) + +(* Note: this is not transitive *) +inductive lsubse (h:sh) (g:sd h): relation lenv ≝ +| lsubse_atom: lsubse h g (⋆) (⋆) +| lsubse_pair: ∀I,L1,L2,V. lsubse h g L1 L2 → + lsubse h g (L1. ⓑ{I} V) (L2. ⓑ{I} V) +| lsubse_abbr: ∀L1,L2,V1,V2,W1,W2,l. L1 ⊢ W1 ⬌* W2 → + ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 → + lsubse h g L1 L2 → lsubse h g (L1. ⓓV1) (L2. ⓛW2) +. + +interpretation + "local environment refinement (context-sensitive parallel equivalence)" + 'CrSubEqSE h g L1 L2 = (lsubse h g L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubse_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 = ⋆ → L2 = ⋆. +#h #g #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct +] +qed-. + +lemma lsubse_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊢•⊑[g] L2 → L2 = ⋆. +/2 width=5 by lsubse_inv_atom1_aux/ qed-. + +fact lsubse_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → + ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + (∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ + ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. +#h #g #L1 #L2 * -L1 -L2 +[ #J #K1 #U1 #H destruct +| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/ +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/ +] +qed-. + +lemma lsubse_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊢•⊑[g] L2 → + (∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ + ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. +/2 width=3 by lsubse_inv_pair1_aux/ qed-. + +fact lsubse_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L2 = ⋆ → L1 = ⋆. +#h #g #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct +] +qed-. + +lemma lsubse_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊢•⊑[g] ⋆ → L1 = ⋆. +/2 width=5 by lsubse_inv_atom2_aux/ qed-. + +fact lsubse_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → + ∀I,K2,W2. L2 = K2. ⓑ{I} W2 → + (∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ + ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. +#h #g #L1 #L2 * -L1 -L2 +[ #J #K2 #U2 #H destruct +| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/ +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=10/ +] +qed-. + +lemma lsubse_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊢•⊑[g] K2. ⓑ{I} W2 → + (∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ + ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. +/2 width=3 by lsubse_inv_pair2_aux/ qed-. + +(* Basic_forward lemmas *****************************************************) + +lemma lsubse_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 ≼[0, |L1|] L2. +#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +lemma lsubse_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 ≼[0, |L2|] L2. +#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +lemma lsubse_refl: ∀h,g,L. h ⊢ L ⊢•⊑[g] L. +#h #g #L elim L -L // /2 width=1/ +qed. + +lemma lsubse_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → + ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. +/3 width=5 by lsubse_fwd_lsubs2, cprs_lsubs_trans/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_cpcs.ma new file mode 100644 index 000000000..2cc3ce076 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_cpcs.ma @@ -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/equivalence/cpcs_cpcs.ma". +include "basic_2/equivalence/lsubse.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) + +(* Properties on context-sensitive parallel equivalence for terms ***********) + +lemma lsubse_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → + ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2. +/3 width=5 by lsubse_fwd_lsubs2, cpcs_lsubs_trans/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_ldrop.ma new file mode 100644 index 000000000..729f4b61f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_ldrop.ma @@ -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/equivalence/lsubse.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) + +(* Properties concerning basic local environment slicing ********************) + +(* Note: the constant 0 cannot be generalized *) +lemma lsubse_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → + ∀K1,e. ⇩[0, e] L1 ≡ K1 → + ∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & ⇩[0, e] L2 ≡ K2. +#h #g #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +] +qed-. + +(* Note: the constant 0 cannot be generalized *) +lemma lsubse_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → + ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & ⇩[0, e] L1 ≡ K1. +#h #g #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_ssta.ma new file mode 100644 index 000000000..b5dc3f5ac --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lsubse_ssta.ma @@ -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/xprs_lsubss.ma". +*) +include "basic_2/equivalence/lsubse.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **) + +(* Properties on stratified native type assignment **************************) + +axiom lsubse_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g,l] U2 → + ∀L1. h ⊢ L1 ⊢•⊑[g] L2 → + ∃∃U1. ⦃h, L1⦄ ⊢ T •[g,l] U1 & L1 ⊢ U1 ⬌* U2. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csup.etc new file mode 100644 index 000000000..dcfe086e9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csup.etc @@ -0,0 +1,157 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The 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 T1 ⦄ > break ⦃ L2 , break T2 ⦄ )" + non associative with precedence 45 + for @{ 'SupTerm $L1 $T1 $L2 $T2 }. + +include "basic_2/substitution/ldrop.ma". + +(* SUPCLOSURE ***************************************************************) + +inductive csup: bi_relation lenv term ≝ +| csup_lref : ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → csup L (#i) K V +| csup_bind_sn: ∀a,I,L,V,T. csup L (ⓑ{a,I}V.T) L V +| csup_bind_dx: ∀a,I,L,V,T. csup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T +| csup_flat_sn: ∀I,L,V,T. csup L (ⓕ{I}V.T) L V +| csup_flat_dx: ∀I,L,V,T. csup L (ⓕ{I}V.T) L T +. + +interpretation + "structural predecessor (closure)" + 'SupTerm L1 T1 L2 T2 = (csup L1 T1 L2 T2). + +(* Basic inversion lemmas ***************************************************) + +fact csup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ∀J. T1 = ⓪{J} → + ∃∃I,i. ⇩[0, i] L1 ≡ L2.ⓑ{I}T2 & J = LRef i. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #I #L #K #V #i #HLK #J #H destruct /2 width=4/ +| #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 csup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ > ⦃L2, T2⦄ → + ∃∃I,i. ⇩[0, i] L1 ≡ L2.ⓑ{I}T2 & J = LRef i. +/2 width=3 by csup_inv_atom1_aux/ qed-. + +fact csup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → + ∀b,J,W,U. T1 = ⓑ{b,J}W.U → + (L2 = L1 ∧ T2 = W) ∨ + (L2 = L1.ⓑ{J}W ∧ T2 = U). +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #I #L #K #V #i #_ #b #J #W #U #H destruct +| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ +| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ +| #I #L #V #T #b #J #W #U #H destruct +| #I #L #V #T #b #J #W #U #H destruct +] +qed-. + +lemma csup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ > ⦃L2, T2⦄ → + (L2 = L1 ∧ T2 = W) ∨ + (L2 = L1.ⓑ{J}W ∧ T2 = U). +/2 width=4 by csup_inv_bind1_aux/ qed-. + +fact csup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → + ∀J,W,U. T1 = ⓕ{J}W.U → + L2 = L1 ∧ (T2 = W ∨ T2 = U). +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #I #L #K #V #i #_ #J #W #U #H destruct +| #a #I #L #V #T #J #W #U #H destruct +| #a #I #L #V #T #J #W #U #H destruct +| #I #L #V #T #J #W #U #H destruct /3 width=1/ +| #I #L #V #T #J #W #U #H destruct /3 width=1/ +] +qed-. + +lemma csup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ > ⦃L2, T2⦄ → + L2 = L1 ∧ (T2 = W ∨ T2 = U). +/2 width=4 by csup_inv_flat1_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma csup_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=4 by ldrop_pair2_fwd_cw/ +qed-. + +lemma csup_fwd_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → + ∃i. ⇩[0, i] L1 ≡ L2 ∨ ⇩[0, i] L2 ≡ L1. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=2/ /4 width=2/ +#I #L1 #K1 #V1 #i #HLK1 +lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /3 width=2/ +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lift_csup_trans_eq: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → + ∀L,U2. ⦃L, U1⦄ > ⦃L, U2⦄ → + ∃T2. ⇧[d, e] T2 ≡ U2. +#T1 #U1 #d #e * -T1 -U1 -d -e +[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #_ #L #X #H + elim (csup_inv_bind1 … H) -H * + [ #_ #H destruct /2 width=2/ + | #H elim (discr_lpair_x_xy … H) + ] +|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H + elim (csup_inv_flat1 … H) -H #_ * #H destruct /2 width=2/ +] +#i #d #e [2,3: #_ ] #L #X #H +elim (csup_inv_atom1 … H) -H #I #j #HL #H destruct +lapply (ldrop_pair2_fwd_cw … HL X) -HL #H +elim (lt_refl_false … H) +qed-. +(* +lemma lift_csup_trans_gt: ∀L1,L2,U1,U2. ⦃L1, U1⦄ > ⦃L2, U2⦄ → + ⇩[0, 1] L2 ≡ L1 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → + ∃T2. ⇧[d + 1, e] T2 ≡ U2. +#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2 +[ #I #L1 #K1 #V #i #HLK1 #HKL1 + lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1 + lapply (ldrop_fwd_lw … HKL1) -HKL1 #HKL1 + lapply (transitive_le … HLK1 HKL1) -L1 normalize #H + + +| #a +| #a +] +#I #L1 #W1 #U1 #HL1 + + + + #X #d #e #H + lapply (ldrop_inv_refl … HL1) -HL1 +| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H + lapply (ldrop_inv_ldrop1 … HL1) + + elim (lift_inv_bind2 … H) -H #W2 #U2 #HW21 #HU21 #H destruct + + + /3 width=2/ /4 width=2/ + +*) + + + +(* Advanced inversion lemmas ************************************************) + +lemma csup_inv_lref2_be: ∀L,U,i. ⦃L, U⦄ > ⦃L, #i⦄ → + ∀T,d,e. ⇧[d, e] T ≡ U → d ≤ i → i < d + e → ⊥. +#L #U #i #H #T #d #e #HTU #Hdi #Hide +elim (lift_csup_trans_eq … HTU … H) -H -T #T #H +elim (lift_inv_lref2_be … H ? ?) // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csup_csup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csup_csup.etc new file mode 100644 index 000000000..813cb969d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csup_csup.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/ldrop_ldrop.ma". +include "basic_2/substitution/csup.ma". + +(* SUPCLOSURE ***************************************************************) + +(* Advanced inversion lemmas ************************************************) + +lemma csup_inv_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → + ∀J,W,j. ⇩[0, j] L1 ≡ L2.ⓑ{J}W → T1 = #j ∧ T2 = W. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #I #L #K #V #i #HLKV #J #W #j #HLKW + elim (ldrop_conf_div … HLKV … HLKW) -L /2 width=1/ +| #a +| #a +] +#I #L #V #T #J #W #j #H +lapply (ldrop_pair2_fwd_cw … H W) -H #H +[2: lapply (transitive_lt (#{L,W}) … H) /2 width=1/ -H #H ] +elim (lt_refl_false … H) +qed-. + +(* Main forward lemmas ******************************************************) + +theorem csup_trans_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ → + ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ → + L = L0 ∨ ⦃L, T1⦄ > ⦃L, T3⦄. +#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ +[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H + lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1 + lapply (csup_fwd_cw … H) -H #H2 + lapply (transitive_lt … H1 H2) -H1 -H2 #H + elim (lt_refl_false … H) +| #a #I #L0 #V2 #T2 #T3 #HT23 + elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ + qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csupp.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csupp.etc new file mode 100644 index 000000000..c28eaea73 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csupp.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 *) +(* *) +(**************************************************************************) + +notation "hvbox( ⦃ L1, break T1 ⦄ > + break ⦃ L2 , break T2 ⦄ )" + non associative with precedence 45 + for @{ 'SupTermPlus $L1 $T1 $L2 $T2 }. + +include "basic_2/substitution/csup.ma". + +(* PLUS-ITERATED SUPCLOSURE *************************************************) + +definition csupp: bi_relation lenv term ≝ bi_TC … csup. + +interpretation "plus-iterated structural predecessor (closure)" + 'SupTermPlus L1 T1 L2 T2 = (csupp L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma csupp_ind: ∀L1,T1. ∀R:relation2 lenv term. + (∀L2,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → R L2 T2) → + (∀L,T,L2,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → R L T → R L2 T2) → + ∀L2,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → R L2 T2. +#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H +@(bi_TC_ind … IH1 IH2 ? ? H) +qed-. + +lemma csupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. + (∀L1,T1. ⦃L1, T1⦄ > ⦃L2, T2⦄ → R L1 T1) → + (∀L1,L,T1,T. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >+ ⦃L2, T2⦄ → R L T → R L1 T1) → + ∀L1,T1. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → R L1 T1. +#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H +@(bi_TC_ind_dx … IH1 IH2 ? ? H) +qed-. + +(* Basic properties *********************************************************) + +lemma csup_csupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄. +/2 width=1/ qed. + +lemma csupp_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → + ⦃L1, T1⦄ >+ ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma csupp_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >+ ⦃L2, T2⦄ → + ⦃L1, T1⦄ >+ ⦃L2, T2⦄. +/2 width=4/ qed. + +(* Basic forward lemmas *****************************************************) + +lemma csupp_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}. +#L1 #L2 #T1 #T2 #H @(csupp_ind … H) -L2 -T2 +/3 width=3 by csup_fwd_cw, transitive_lt/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csupp_csupp.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csupp_csupp.etc new file mode 100644 index 000000000..5afdb68d4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csupp_csupp.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/csupp.ma". + +(* PLUS-ITERATED SUPCLOSURE *************************************************) + +(* Main propertis ***********************************************************) + +theorem csupp_trans: bi_transitive … csupp. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csups.etc new file mode 100644 index 000000000..7f5879426 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csups.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 *) +(* *) +(**************************************************************************) + +notation "hvbox( ⦃ L1, break T1 ⦄ > * break ⦃ L2 , break T2 ⦄ )" + non associative with precedence 45 + for @{ 'SupTermStar $L1 $T1 $L2 $T2 }. + +include "basic_2/substitution/csup.ma". +include "basic_2/unfold/csupp.ma". + +(* STAR-ITERATED SUPCLOSURE *************************************************) + +definition csups: bi_relation lenv term ≝ bi_star … csup. + +interpretation "star-iterated structural predecessor (closure)" + 'SupTermStar L1 T1 L2 T2 = (csups L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma csups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 → + (∀L,L2,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → R L T → R L2 T2) → + ∀L2,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → R L2 T2. +#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H +@(bi_star_ind … IH1 IH2 ? ? H) +qed-. + +lemma csups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 → + (∀L1,L,T1,T. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >* ⦃L2, T2⦄ → R L T → R L1 T1) → + ∀L1,T1. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → R L1 T1. +#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H +@(bi_star_ind_dx … IH1 IH2 ? ? H) +qed-. + +(* Basic properties *********************************************************) + +lemma csups_refl: bi_reflexive … csups. +/2 width=1/ qed. + +lemma csupp_csups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → ⦃L1, T1⦄ >* ⦃L2, T2⦄. +/2 width=1/ qed. + +lemma csup_csups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ⦃L1, T1⦄ >* ⦃L2, T2⦄. +/2 width=1/ qed. + +lemma csups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → + ⦃L1, T1⦄ >* ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma csups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >* ⦃L2, T2⦄ → + ⦃L1, T1⦄ >* ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma csups_csupp_csupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → + ⦃L, T⦄ >+ ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma csupp_csups_csupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ → + ⦃L, T⦄ >* ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄. +/2 width=4/ qed. + +(* Basic forward lemmas *****************************************************) + +lemma csups_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}. +#L1 #L2 #T1 #T2 #H @(csups_ind … H) -L2 -T2 // +/4 width=3 by csup_fwd_cw, lt_to_le_to_lt, lt_to_le/ (**) (* slow even with trace *) +qed-. + +(* Advanced inversion lemmas for csupp **************************************) + +lemma csupp_inv_atom1_csups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ >+ ⦃L2, T2⦄ → + ∃∃I,K,V,i. ⇩[0, i] L1 ≡ K.ⓑ{I}V & + ⦃K, V⦄ >* ⦃L2, T2⦄ & J = LRef i. +#J #L1 #L2 #T2 #H @(csupp_ind … H) -L2 -T2 +[ #L2 #T2 #H + elim (csup_inv_atom1 … H) -H * #i #HL12 #H destruct /2 width=7/ +| #L #T #L2 #T2 #_ #HT2 * #I #K #V #i #HLK #HVT #H destruct /3 width=8/ +] +qed-. + +lemma csupp_inv_bind1_csups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ >+ ⦃L2, T2⦄ → + ⦃L1, W⦄ >* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ >* ⦃L2, T2⦄. +#b #J #L1 #L2 #W #U #T2 #H @(csupp_ind … H) -L2 -T2 +[ #L2 #T2 #H + elim (csup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/ +| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ +] +qed-. + +lemma csupp_inv_flat1_csups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ >+ ⦃L2, T2⦄ → + ⦃L1, W⦄ >* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ >* ⦃L2, T2⦄. +#J #L1 #L2 #W #U #T2 #H @(csupp_ind … H) -L2 -T2 +[ #L2 #T2 #H + elim (csup_inv_flat1 … 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/csup/csups_csups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csups_csups.etc new file mode 100644 index 000000000..aa54d9bef --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/csups_csups.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/csup_csup.ma". +include "basic_2/unfold/csups.ma". + +(* STAR-ITERATED SUPCLOSURE *************************************************) + +(* Advanced forward lemmas **************************************************) + +(* +lemma csupp_strap2_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ > ⦃L0, T2⦄ → + ∀T3. ⦃L0, T2⦄ >+ ⦃L, T3⦄ → + L = L0 ∨ ⦃L, T1⦄ >+ ⦃L, T3⦄. +#L #L0 #T1 #T2 * -L -L0 -T1 -T2 /2 width=1/ +[ #I #L0 #K0 #V0 #i #HLK0 #T3 #H + lapply (ldrop_pair2_fwd_cw … HLK0 T3) -HLK0 #H1 + lapply (csupp_fwd_cw … H) -H #H2 + lapply (transitive_lt … H1 H2) -H1 -H2 #H + elim (lt_refl_false … H) +| #a #I #L0 #V2 #T2 #T3 #HT23 + /3 width=5/ + + elim (csup_inv_ldrop … HT23 I V2 0 ?) -HT23 // #H1 #H2 destruct /2 width=1/ + qed-. + + + + + + + + +lemma csups_strap1_fwd_refl: ∀L,L0,T1,T2. ⦃L, T1⦄ >* ⦃L0, T2⦄ → + ∀T3. ⦃L0, T2⦄ > ⦃L, T3⦄ → L = L0. +#L #L0 #T1 #T2 #H @(csups_ind_dx … H) -L -T1 // +#L1 #L #T1 #T #HL1 #_ #IHL0 #T3 #HL0 +lapply (csup_trans_fwd_refl … HL10) … HL0) -T2 +*) +lemma lift_csups_trans_aux: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → + ∀L1,L2,U2. ⦃L1, U1⦄ >* ⦃L2, U2⦄ → L1 = L2 → + ∃T2. ⇧[d, e] T2 ≡ U2. +#T1 #U1 #d #e #HTU1 #L1 #L2 #U2 #H @(csups_ind … H) -L2 -U2 /2 width=2/ -T1 +#L #L2 #U #U2 #HL1 #HL2 #IHL1 #H destruct + +* -T1 -U1 -d -e + +(* Main propertis ***********************************************************) + +theorem csups_trans: bi_transitive … csups. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ypr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ypr.etc new file mode 100644 index 000000000..f1510ab7e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ypr.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( h ⊢ break ⦃ L1, break T1 ⦄ • ⥸ break [ g ] break ⦃ L2 , break T2 ⦄ )" + non associative with precedence 45 + for @{ 'YPRed $h $g $L1 $T1 $L2 $T2 }. + +include "basic_2/substitution/csup.ma". +include "basic_2/reducibility/xpr.ma". + +(* HYPER PARALLEL REDUCTION ON CLOSURES *************************************) + +inductive ypr (h) (g) (L1) (T1): relation2 lenv term ≝ +| ypr_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → ypr h g L1 T1 L1 T2 +| ypr_ssta: ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] T2 → ypr h g L1 T1 L1 T2 +| ypr_csup: ∀L2,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ypr h g L1 T1 L2 T2 +. + +interpretation + "hyper parallel reduction (closure)" + 'YPRed h g L1 T1 L2 T2 = (ypr h g L1 T1 L2 T2). + +(* Basic properties *********************************************************) + +lemma ypr_refl: ∀h,g. bi_reflexive … (ypr h g). +/2 width=1/ qed. + +lemma xpr_ypr: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •➡[g] T2 → h ⊢ ⦃L, T1⦄ •⥸[g] ⦃L, T2⦄. +#h #g #L #T1 #T2 * /2 width=1/ /2 width=2/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs.etc new file mode 100644 index 000000000..86dc0c135 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs.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 *) +(* *) +(**************************************************************************) + +notation "hvbox( h ⊢ break ⦃ L1, break T1 ⦄ • ⥸ * break [ g ] break ⦃ L2 , break T2 ⦄ )" + non associative with precedence 45 + for @{ 'YPRedStar $h $g $L1 $T1 $L2 $T2 }. + +include "basic_2/reducibility/ypr.ma". + +(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************) + +definition yprs: ∀h. sd h → bi_relation lenv term ≝ + λh,g. bi_TC … (ypr h g). + +interpretation "hyper parallel computation (closure)" + 'YPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 → + (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ •⥸[g] ⦃L2, T2⦄ → R L T → R L2 T2) → + ∀L2,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → R L2 T2. +/3 width=7 by bi_TC_star_ind/ qed-. + +lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 → + (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ •⥸[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ •⥸*[g] ⦃L2, T2⦄ → R L T → R L1 T1) → + ∀L1,T1. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → R L1 T1. +/3 width=7 by bi_TC_star_ind_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g). +/2 width=1/ qed. + +lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L, T⦄ → + h ⊢ ⦃L, T⦄ •⥸[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄. +/2 width=4/ qed. + +lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ •⥸[g] ⦃L, T⦄ → + h ⊢ ⦃L, T⦄ •⥸*[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_csups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_csups.etc new file mode 100644 index 000000000..08c939d8d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_csups.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/unfold/csups.ma". +include "basic_2/computation/yprs.ma". + +(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************) + +(* Properties on iterated supclosure ****************************************) + +lemma csups_yprs: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → + h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄. +#h #g #L1 #L2 #T1 #T2 #H @(csups_ind … H) -L2 -T2 /3 width=1/ /3 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_xprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_xprs.etc new file mode 100644 index 000000000..2feb88a2f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_xprs.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/computation/xprs_cprs.ma". +include "basic_2/computation/yprs.ma". + +(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************) + +(* Properties on extended parallel computation for terms ********************) + +lemma xprs_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 → + h ⊢ ⦃L, T1⦄ •⥸*[g] ⦃L, T2⦄. +#h #g #L #T1 #T2 #H @(xprs_ind … H) -T2 // /3 width=4/ +qed. + +lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ •⥸*[g] ⦃L, T2⦄. +/3 width=1/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_yprs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_yprs.etc new file mode 100644 index 000000000..d737dd817 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/yprs_yprs.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/yprs.ma". + +(* HYPER PARALLEL COMPUTATION ON TERMS **************************************) + +theorem yprs_trans: ∀h,g. bi_transitive … (yprs h g). +/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ysteps.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ysteps.etc new file mode 100644 index 000000000..149e7895b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ysteps.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 *) +(* *) +(**************************************************************************) + +notation "hvbox( h ⊢ break ⦃ L1, break T1 ⦄ • ⭃ * break [ g ] break ⦃ L2 , break T2 ⦄ )" + non associative with precedence 45 + for @{ 'YPRedStepStar $h $g $L1 $T1 $L2 $T2 }. + +include "basic_2/substitution/csup.ma". +include "basic_2/computation/yprs.ma". + +(* ITERATED STEP OF HYPER PARALLEL COMPUTATION ON CLOSURES ******************) + +inductive ysteps (h) (g) (L1) (T1) (L2) (T2): Prop ≝ +| ysteps_intro: h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → (L1 = L2 → T1 = T2 → ⊥) → + ysteps h g L1 T1 L2 T2 +. + +interpretation "iterated step of hyper parallel computation (closure)" + 'YPRedStepStar h g L1 T1 L2 T2 = (ysteps h g L1 T1 L2 T2). + +(* Basic properties *********************************************************) + +lemma ssta_ysteps: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → + h ⊢ ⦃L, T⦄ •⭃*[g] ⦃L, U⦄. +#h #g #L #T #U #l #HTU +@ysteps_intro /3 width=2/ #_ #H destruct +elim (ssta_inv_refl … HTU) +qed. + +lemma csup_ysteps: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → + h ⊢ ⦃L1, T1⦄ •⭃*[g] ⦃L2, T2⦄. +#h #g #L1 #L2 #T1 #T2 #H +lapply (csup_fwd_cw … H) #H1 +@ysteps_intro /3 width=1/ -H #H2 #H3 destruct +elim (lt_refl_false … H1) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ysteps_csups.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ysteps_csups.etc new file mode 100644 index 000000000..2e48f396d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/csup/ysteps_csups.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/computation/yprs_csups.ma". +include "basic_2/computation/ysteps.ma". + +(* ITERATED STEP OF HYPER PARALLEL COMPUTATION ON CLOSURES ******************) + +(* Properties on iterated supclosure ****************************************) + +lemma csups_ysteps: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → + h ⊢ ⦃L1, T1⦄ •⭃*[g] ⦃L2, T2⦄. +#h #g #L1 #L2 #T1 #T2 #H +lapply (csups_fwd_cw … H) #H1 +@ysteps_intro /2 width=1/ -H #H2 #H3 destruct +elim (lt_refl_false … H1) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/hod/ntas.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/hod/ntas.etc new file mode 100644 index 000000000..8cfaa343b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/hod/ntas.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 *) +(* *) +(**************************************************************************) + +notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 : : * break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeStarAlt $h $L $T1 $T2 }. + +include "basic_2/dynamic/nta.ma". + +(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************) + +definition ntas: sh → lenv → relation term ≝ + λh,L. star … (nta h L). + +interpretation "higher order native type assignment (term)" + 'NativeTypeStar h L T U = (ntas h L T U). + +(* Basic eliminators ********************************************************) +(* +lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 → + (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) → + ∀T2. L ⊢ T1 ➡* T2 → R T2. +#L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. +*) +axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 → + (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) → + ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1. +(* +#h #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(star_ind_dx … HT2 IHT2 … HT12) // +qed-. +*) +(* Basic properties *********************************************************) + +lemma ntas_refl: ∀h,L,T. ⦃h, L⦄ ⊢ T :* T. +// qed. + +lemma ntas_strap1: ∀h,L,T1,T,T2. + ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2. +/2 width=3/ qed. + +lemma ntas_strap2: ∀h,L,T1,T,T2. + ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2. +/2 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/hod/ntas_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/hod/ntas_lift.etc new file mode 100644 index 000000000..1adced79d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/hod/ntas_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/dynamic/nta_lift.ma". +include "basic_2/hod/ntas.ma". + +(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************) + +(* Advanced properties on native type assignment for terms ******************) + +lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U → + ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U. +#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/ +qed. + +axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U → + ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z. +(* REQUIRES SUBJECT CONVERSION +#h #L #T #W #Y #H @(ntas_ind_dx … H) -T +[ #U #HYU + elim (nta_fwd_correct … HYU) #U0 #HU0 + elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U +*) + +(* Advanced inversion lemmas on native type assignment for terms ************) + +fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X → + ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & + L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. +#h #L #Z #U #H elim H -L -Z -U +[ #L #k #X #Y #H destruct +| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct +| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct +| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct +| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/ +| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct + elim (IHUW U Y ?) -IHUW // /3 width=9/ +| #L #Z #U #_ #_ #X #Y #H destruct +| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct + elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT + lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/ +] +qed. + +(* Basic_1: was only: ty3_gen_appl *) +lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U → + ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & + L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. +/2 width=3/ qed-. + +axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U → + ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W & + L ⊢ ⓐZ.ⓛY.W ⬌* U. +(* REQUIRES SUBJECT REDUCTION +#h #L #Z #Y #X #U #H +elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT +elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V +lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U +@(ex3_1_intro … HX0U) /2 width=2/ +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lcpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lcpcs.etc new file mode 100644 index 000000000..d815739fb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lcpcs.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( L1 ⊢ ⬌* break term 46 L2 )" + non associative with precedence 45 + for @{ 'CPConvStar $L1 $L2 }. + +include "basic_2/grammar/lenv_px_sn.ma". +include "basic_2/equivalence/cpcs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************) + +definition lcpcs: relation lenv ≝ lpx_sn … cpcs. + +interpretation "context-sensitive parallel equivalence (local environment)" + 'CPConvStar L1 L2 = (lcpcs L1 L2). + +(* Basic inversion lemmas ***************************************************) + +lemma lcpcs_inv_atom1: ∀L2. ⋆ ⊢ ⬌* L2 → L2 = ⋆. +/2 width=2 by lpx_sn_inv_atom1/ qed-. + +lemma lcpcs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ⬌* L2 → + ∃∃K2,V2. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L2 = K2. ⓑ{I} V2. +/2 width=1 by lpx_sn_inv_pair1/ qed-. + +lemma lcpcs_inv_atom2: ∀L1. L1 ⊢ ⬌* ⋆ → L1 = ⋆. +/2 width=2 by lpx_sn_inv_atom2/ qed-. + +lemma lcpcs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ⬌* K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L1 = K1. ⓑ{I} V1. +/2 width=1 by lpx_sn_inv_pair2/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lcpcs_fwd_length: ∀L1,L2. L1 ⊢ ⬌* L2 → |L1| = |L2|. +/2 width=2 by lpx_sn_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lcpcs_ltpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lcpcs_ltpr.etc new file mode 100644 index 000000000..ecc6be867 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lcpcs_ltpr.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/ltpr.ma". +include "basic_2/equivalence/lcpcs.ma". + +(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************) + +(* Properties on context-free parallel reduction for local environments *****) + +lemma ltpr_lcpcs: ∀L1,L2. L1 ➡ L2 → L1 ⊢ ⬌* L2. +#L1 #L2 #H elim H -L1 -L2 // /4 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lenv_px_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lenv_px_sn.etc new file mode 100644 index 000000000..fddab0332 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lenv_px/lenv_px_sn.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/grammar/lenv_length.ma". + +(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **********) + +inductive lpx_sn (R:lenv→relation term): relation lenv ≝ +| lpx_sn_stom: lpx_sn R (⋆) (⋆) +| lpx_sn_pair: ∀I,K1,K2,V1,V2. + lpx_sn R K1 K2 → R K1 V1 V2 → lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) +. + +(* Basic inversion lemmas ***************************************************) + +fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆. +/2 width=4 by lpx_sn_inv_atom1_aux/ qed-. + +fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2. +#R #L1 #L2 * -L1 -L2 +[ #J #K1 #V1 #H destruct +| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/ +] +qed-. + +lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 → + ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2. +/2 width=3 by lpx_sn_inv_pair1_aux/ qed-. + +fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆. +/2 width=4 by lpx_sn_inv_atom2_aux/ qed-. + +fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1. +#R #L1 #L2 * -L1 -L2 +[ #J #K2 #V2 #H destruct +| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/ +] +qed-. + +lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) → + ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1. +/2 width=3 by lpx_sn_inv_pair2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|. +#R #L1 #L2 #H elim H -L1 -L2 normalize // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubn/lsubn_lsubn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubn/lsubn_lsubn.etc new file mode 100644 index 000000000..9ef3dda88 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/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: ∀h,L1,L. h ⊢ L1 :⊑ L → ∀L2. h ⊢ L :⊑ L2 → h ⊢ L1 :⊑ L2. +#h #L1 #L #H elim H -L1 -L +[ #X #H >(lsubn_inv_atom1 … H) -H // +| #I #L1 #L #V #HL1 #H1W #IHL1 #X #H + elim (lsubn_inv_pair1 … H) -H * #L2 + [ #HL2 #H #H2W destruct /4 width=1/ + | #W #H1VW #H2VW #HL2 #H1 #H2 destruct /3 width=3/ + ] +| #L1 #L #V1 #W1 #H1VW1 #H2VW1 #HL1 #IHL1 #X #H + elim (lsubn_inv_pair1 … H) -H * #L2 + [ #HL2 #H #HW destruct /3 width=1/ + | #V #_ #_ #_ #_ #H destruct + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv.etc new file mode 100644 index 000000000..25122262a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv.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 *) +(* *) +(**************************************************************************) + +include "basic_2/dynamic/snv.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) + +(* Note: this is not transitive *) +inductive lsubsv (h:sh) (g:sd h): relation lenv ≝ +| lsubsv_atom: lsubsv h g (⋆) (⋆) +| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 → + lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V) +| lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → L1 ⊢ W2 ⬌* W1 → + ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 → + lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2) +. + +interpretation + "local environment refinement (stratified native validity)" + 'CrSubEqV h g L1 L2 = (lsubsv h g L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ → L2 = ⋆. +#h #g #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct +] +qed-. + +lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆. +/2 width=5 by lsubsv_inv_atom1_aux/ qed-. + +fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ + ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. +#h #g #L1 #L2 * -L1 -L2 +[ #J #K1 #U1 #H destruct +| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/ +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/ +] +qed-. + +lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 → + (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨ + ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr. +/2 width=3 by lsubsv_inv_pair1_aux/ qed-. + +fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆. +#h #g #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct +] +qed-. + +lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆. +/2 width=5 by lsubsv_inv_atom2_aux/ qed-. + +fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀I,K2,W2. L2 = K2. ⓑ{I} W2 → + (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ + ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. +#h #g #L1 #L2 * -L1 -L2 +[ #J #K2 #U2 #H destruct +| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/ +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/ +] +qed-. + +lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 → + (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨ + ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 & + K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst. +/2 width=3 by lsubsv_inv_pair2_aux/ qed-. + +(* Basic_forward lemmas *****************************************************) + +lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2. +#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2. +#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +lemma lsubsv_refl: ∀h,g,L. h ⊢ L ⊩:⊑[g] L. +#h #g #L elim L -L // /2 width=1/ +qed. + +lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. +/3 width=5 by lsubsv_fwd_lsubs2, cprs_lsubs_trans/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_cpcs.etc new file mode 100644 index 000000000..87a72e0fd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_cpcs.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/equivalence/cpcs_cpcs.ma". +include "basic_2/dynamic/lsubsv.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) + +(* Properties on context-sensitive parallel equivalence for terms ***********) + +lemma lsubsv_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2. +/3 width=5 by lsubsv_fwd_lsubs2, cpcs_lsubs_trans/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_ldrop.etc new file mode 100644 index 000000000..2c3381f86 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_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/lsubsv.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) + +(* Properties concerning basic local environment slicing ********************) + +(* Note: the constant 0 cannot be generalized *) +lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀K1,e. ⇩[0, e] L1 ≡ K1 → + ∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & ⇩[0, e] L2 ≡ K2. +#h #g #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +] +qed. + +(* Note: the constant 0 cannot be generalized *) +lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & ⇩[0, e] L1 ≡ K1. +#h #g #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=6/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_snv.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_snv.etc new file mode 100644 index 000000000..e5bd9951e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_snv.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/dynamic/lsubsv_ldrop.ma". +include "basic_2/dynamic/lsubsv_ssta.ma". +include "basic_2/dynamic/lsubsv_cpcs.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) + +(* Properties concerning stratified native validity *************************) + +axiom lsubsv_xprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → + ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2. +(* +/3 width=3 by lsubsv_fwd_lsubss, lsubss_xprs_trans/ qed-. +*) +axiom lsubsv_snv_trans: ∀h,g,L2,T. ⦃h, L2⦄ ⊩ T :[g] → + ∀L1. h ⊢ L1 ⊩:⊑[g] L2 → ⦃h, L1⦄ ⊩ T :[g]. +(* +#h #g #L2 #T #H elim H -L2 -T // +[ #I2 #L2 #K2 #V2 #i #HLK2 #_ #IHV2 #L1 #HL12 + elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 + elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -IHV2 ] + [ #HK12 #H destruct /3 width=5/ + | #V1 #l #HV1 #_ #_ #_ #H destruct /2 width=5/ + ] +| #a #I #L2 #V #T #_ #_ #IHV #IHT #L1 #HL12 /4 width=1/ +| #a #L2 #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU #IHV #IHT #L1 #HL12 + lapply (IHV … HL12) -IHV #HV + lapply (IHT … HL12) -IHT #HT + lapply (lsubsv_ssta_trans … HVW … HL12) -HVW #HVW + lapply (lsubsv_cprs_trans … HL12 … HW0) -HW0 #HW0 + lapply (lsubsv_xprs_trans … HL12 … HTU) -HL12 -HTU /2 width=8/ +| #L2 #W #T #U #l #_ #_ #HTU #HWU #IHW #IHT #L1 #HL12 + lapply (IHW … HL12) -IHW #HW + lapply (IHT … HL12) -IHT #HT + lapply (lsubsv_ssta_trans … HTU … HL12) -HTU #HTU + lapply (lsubsv_cpcs_trans … HL12 … HWU) -HL12 -HWU /2 width=4/ +] +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_ssta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_ssta.etc new file mode 100644 index 000000000..1e5b5fddd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubsv/lsubsv_ssta.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/xprs_lsubss.ma". +include "basic_2/dynamic/lsubsv.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************) + +(* Properties on stratified native type assignment **************************) + +axiom lsubsv_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g,l] U2 → + ∀L1. h ⊢ L1 ⊩:⊑[g] L2 → + ∃∃U1. L1 ⊢ U2 ⬌* U1 & ⦃h, L1⦄ ⊢ T •[g,l] U1. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn.etc new file mode 100644 index 000000000..c4359c35f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/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 ⊢ break term 46 L1 : ⊑ break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqN $h $L1 $L2 }. + +notation "hvbox( h ⊢ break term 46 L1 : : ⊑ break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqNAlt $h $L1 $L2 }. + +include "basic_2/dynamic/nta.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************) + +(* Note: may not be transitive *) +inductive lsubn (h:sh): relation lenv ≝ +| lsubn_atom: lsubn h (⋆) (⋆) +| lsubn_pair: ∀I,L1,L2,W. lsubn h L1 L2 → lsubn h (L1. ⓑ{I} W) (L2. ⓑ{I} W) +| lsubn_abbr: ∀L1,L2,V,W. ⦃h, L1⦄ ⊢ V : W → ⦃h, L2⦄ ⊢ V : W → + lsubn h L1 L2 → lsubn h (L1. ⓓV) (L2. ⓛW) +. + +interpretation + "local environment refinement (native type assigment)" + 'CrSubEqN h L1 L2 = (lsubn h L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 = ⋆ → L2 = ⋆. +#h #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #_ #_ #_ #H destruct +] +qed. + +lemma lsubn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑ L2 → L2 = ⋆. +/2 width=4/ qed-. + +fact lsubn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V → + (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & + h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr. +#h #L1 #L2 * -L1 -L2 +[ #I #K1 #V #H destruct +| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/ +| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/ +] +qed. + +lemma lsubn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑ L2 → + (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & + h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr. +/2 width=3/ qed-. + +fact lsubn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L2 = ⋆ → L1 = ⋆. +#h #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #_ #_ #_ #H destruct +] +qed. + +lemma lsubc_inv_atom2: ∀h,L1. h ⊢ L1 :⊑ ⋆ → L1 = ⋆. +/2 width=4/ qed-. + +fact lsubn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W → + (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & + h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst. +#h #L1 #L2 * -L1 -L2 +[ #I #K2 #W #H destruct +| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/ +| #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/ +] +qed. + +(* Basic_1: was: csubt_gen_bind *) +lemma lsubn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑ K2. ⓑ{I} W → + (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W & + h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst. +/2 width=3/ qed-. + +(* Basic_forward lemmas *****************************************************) + +lemma lsubn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L1|] L2. +#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +lemma lsubn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L2|] L2. +#h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: csubt_refl *) +lemma lsubn_refl: ∀h,L. h ⊢ L :⊑ L. +#h #L elim L -L // /2 width=1/ +qed. + +(* Basic_1: removed theorems 6: + csubt_gen_flat csubt_drop_flat csubt_clear_conf + csubt_getl_abbr csubt_getl_abst csubt_ty3_ld +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn_cpcs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn_cpcs.etc new file mode 100644 index 000000000..5f610bc96 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/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: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → + ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2. +/3 width=4 by lsubn_fwd_lsubs2, cpr_lsubs_trans/ qed. + +lemma cprs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → + ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. +/3 width=4 by lsubn_fwd_lsubs2, cprs_lsubs_trans/ qed. + +(* Basic_1: was: csubt_pc3 *) +lemma cpcs_lsubn_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → + ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2. +/3 width=4 by lsubn_fwd_lsubs2, cpcs_lsubs_trans/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn_ldrop.etc new file mode 100644 index 000000000..a16fff618 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/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: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 → + ∃∃K2. h ⊢ K1 :⊑ K2 & ⇩[0, e] L2 ≡ K2. +#h #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +] +qed. + +(* Note: the constant 0 cannot be generalized *) +(* Basic_1: was only: csubt_drop_abbr csubt_drop_abst *) +lemma lsubn_ldrop_O1_trans: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. h ⊢ K1 :⊑ K2 & ⇩[0, e] L1 ≡ K1. +#h #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V #W #H1VW #H2VW #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn_nta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/lsubn_nta.etc new file mode 100644 index 000000000..5832b00b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/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: ∀h,L2,T,U. ⦃h, L2⦄ ⊢ T : U → ∀L1. h ⊢ L1 :⊑ L2 → + ⦃h, L1⦄ ⊢ T : U. +#h #L2 #T #U #H elim H -L2 -T -U +[ // +| #L2 #K2 #V2 #W2 #U2 #i #HLK2 #_ #WU2 #IHVW2 #L1 #HL12 + elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 + elim (lsubn_inv_pair2 … H) -H * #K1 + [ #HK12 #H destruct /3 width=6/ + | #V1 #_ #_ #_ #_ #H destruct + ] +| #L2 #K2 #W2 #V2 #U2 #i #HLK2 #_ #HWU2 #IHWV2 #L1 #HL12 + elim (lsubn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 + elim (lsubn_inv_pair2 … H) -H * #K1 [ | -IHWV2 ] + [ #HK12 #H destruct /3 width=6/ + | #V1 #H1V1W2 #_ #_ #H #_ destruct /2 width=6/ + ] +| /4 width=2/ +| /3 width=1/ +| /3 width=2/ +| /3 width=1/ +| /4 width=6/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta.etc new file mode 100644 index 000000000..fa4a8ed8f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta.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/static/sh.ma". +include "basic_2/equivalence/cpcs.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +inductive nta (h:sh): lenv → relation term ≝ +| nta_sort: ∀L,k. nta h L (⋆k) (⋆(next h k)) +| nta_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → nta h K V W → + ⇧[0, i + 1] W ≡ U → nta h L (#i) U +| nta_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → nta h K W V → + ⇧[0, i + 1] W ≡ U → nta h L (#i) U +| nta_bind: ∀I,L,V,W,T,U. nta h L V W → nta h (L. ⓑ{I} V) T U → + nta h L (ⓑ{I}V.T) (ⓑ{I}V.U) +| nta_appl: ∀L,V,W,T,U. nta h L V W → nta h L (ⓛW.T) (ⓛW.U) → + nta h L (ⓐV.ⓛW.T) (ⓐV.ⓛW.U) +| nta_pure: ∀L,V,W,T,U. nta h L T U → nta h L (ⓐV.U) W → + nta h L (ⓐV.T) (ⓐV.U) +| nta_cast: ∀L,T,U. nta h L T U → nta h L (ⓝU. T) U +| nta_conv: ∀L,T,U1,U2,V2. nta h L T U1 → L ⊢ U1 ⬌* U2 → nta h L U2 V2 → + nta h L T U2 +. + +interpretation "native type assignment (term)" + 'NativeType h L T U = (nta h L T U). + +(* Basic properties *********************************************************) + +(* Basic_1: was: ty3_cast *) +lemma nta_cast_old: ∀h,L,W,T,U. + ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ U : W → ⦃h, L⦄ ⊢ ⓝU.T : ⓝW.U. +/4 width=3/ qed. + +(* Basic_1: was: ty3_typecheck *) +lemma nta_typecheck: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃T0. ⦃h, L⦄ ⊢ ⓝU.T : T0. +/3 width=2/ qed. + +(* Basic_1: removed theorems 4: + ty3_getl_subst0 ty3_fsubst0 ty3_csubst0 ty3_subst0 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_aaa.etc new file mode 100644 index 000000000..962856983 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_aaa.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/computation/csn_aaa.ma". +include "basic_2/equivalence/lcpcs_aaa.ma". +include "basic_2/dynamic/nta.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Forward lemmas on atomic arity assignment for terms **********************) + +lemma nta_fwd_aaa: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃∃A. L ⊢ T ⁝ A & L ⊢ U ⁝ A. +#h #L #T #U #H elim H -L -T -U +[ /2 width=3/ +| #L #K #V #W #U #i #HLK #_ #HWU * #B #HV #HW + lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/ +| #L #K #W #V #U #i #HLK #_ #HWU * #B #HW #_ -V + lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/ +| * #L #V #W #T #U #_ #_ * #B #HV #HW * #A #HT #HU + [ /3 width=3/ | /3 width=5/ ] +| #L #V #W #T #U #_ #_ * #B #HV #HW * #X #H1 #H2 + elim (aaa_inv_abst … H1) -H1 #B1 #A1 #HW1 #HT #H destruct + elim (aaa_inv_abst … H2) -H2 #B2 #A #_ #HU #H destruct + lapply (aaa_mono … HW1 … HW) -HW1 #H destruct /4 width=5/ +| #L #V #W #T #U #_ #_ * #X #HT #HUX * #A #H #_ -W + elim (aaa_inv_appl … H) -H #B #HV #HUA + lapply (aaa_mono … HUA … HUX) -HUX #H destruct /3 width=5/ +| #L #T #U #_ * #A #HT #HU /3 width=3/ +| #L #T #U1 #U2 #V2 #_ #HU12 #_ * #X #HT #HU1 * #A #HU2 #_ + lapply (aaa_cpcs_mono … HU12 … HU1 … HU2) -U1 #H destruct /2 width=3/ +] +qed-. + +(* Note: this is the stong normalization property *) +(* Basic_1: was only: ty3_sn3 *) +theorem nta_fwd_csn: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → L ⊢ ⬇* T ∧ L ⊢ ⬇* U. +#h #L #T #U #H elim (nta_fwd_aaa … H) -H /3 width=2/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_alt.etc new file mode 100644 index 000000000..8cbd59518 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_alt.etc @@ -0,0 +1,190 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/nta.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* alternative definition of nta *) +inductive ntaa (h:sh): lenv → relation term ≝ +| ntaa_sort: ∀L,k. ntaa h L (⋆k) (⋆(next h k)) +| ntaa_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → ntaa h K V W → + ⇧[0, i + 1] W ≡ U → ntaa h L (#i) U +| ntaa_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → ntaa h K W V → + ⇧[0, i + 1] W ≡ U → ntaa h L (#i) U +| ntaa_bind: ∀I,L,V,W,T,U. ntaa h L V W → ntaa h (L. ⓑ{I} V) T U → + ntaa h L (ⓑ{I}V.T) (ⓑ{I}V.U) +| ntaa_appl: ∀L,V,W,T,U. ntaa h L V W → ntaa h L (ⓛW.T) (ⓛW.U) → + ntaa h L (ⓐV.ⓛW.T) (ⓐV.ⓛW.U) +| ntaa_pure: ∀L,V,W,T,U. ntaa h L T U → ntaa h L (ⓐV.U) W → + ntaa h L (ⓐV.T) (ⓐV.U) +| ntaa_cast: ∀L,T,U,W. ntaa h L T U → ntaa h L U W → ntaa h L (ⓝU. T) U +| ntaa_conv: ∀L,T,U1,U2,V2. ntaa h L T U1 → L ⊢ U1 ⬌* U2 → ntaa h L U2 V2 → + ntaa h L T U2 +. + +interpretation "native type assignment (term) alternative" + 'NativeTypeAlt h L T U = (ntaa h L T U). + +(* Advanced inversion lemmas ************************************************) + +fact ntaa_inv_bind1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T :: U → ∀J,X,Y. T = ⓑ{J}Y.X → + ∃∃Z1,Z2. ⦃h, L⦄ ⊢ Y :: Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :: Z2 & + L ⊢ ⓑ{J}Y.Z2 ⬌* U. +#h #L #T #U #H elim H -L -T -U +[ #L #k #J #X #Y #H destruct +| #L #K #V #W #U #i #_ #_ #_ #_ #J #X #Y #H destruct +| #L #K #W #V #U #i #_ #_ #_ #_ #J #X #Y #H destruct +| #I #L #V #W #T #U #HVW #HTU #_ #_ #J #X #Y #H destruct /2 width=3/ +| #L #V #W #T #U #_ #_ #_ #_ #J #X #Y #H destruct +| #L #V #W #T #U #_ #_ #_ #_ #J #X #Y #H destruct +| #L #T #U #W #_ #_ #_ #_ #J #X #Y #H destruct +| #L #T #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #J #X #Y #H destruct + elim (IHTU1 ????) -IHTU1 [5: // |2,3,4: skip ] #Z1 #Z2 #HZ1 #HZ2 #HU1 + lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/ +] +qed. + +lemma ntaa_inv_bind1: ∀h,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X :: U → + ∃∃Z1,Z2. ⦃h, L⦄ ⊢ Y :: Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :: Z2 & + L ⊢ ⓑ{J}Y.Z2 ⬌* U. +/2 width=3/ qed-. + +lemma ntaa_nta: ∀h,L,T,U. ⦃h, L⦄ ⊢ T :: U → ⦃h, L⦄ ⊢ T : U. +#h #L #T #U #H elim H -L -T -U +// /2 width=1/ /2 width=2/ /2 width=3/ /2 width=6/ +qed-. + +(* Properties on relocation *************************************************) + +lemma ntaa_lift: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 :: U1 → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → + ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 :: U2. +#h #L1 #T1 #U1 #H elim H -L1 -T1 -U1 +[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2 + >(lift_inv_sort1 … H1) -X1 + >(lift_inv_sort1 … H2) -X2 // +| #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 + elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct + /3 width=8/ + | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 + lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ + ] +| #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (cprs_inv_sort1 … H) -H // +| +| +| +| +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #T2 #H + elim (cprs_inv_appl1 … H) -H * + [ #V2 #T0 #HV12 #HT10 #H destruct + elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U + @(nta_conv … (ⓐV2.U1)) (* /2 width=1/*) [ /4 width=2/] (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #HV12 #HT10 #HT02 + lapply (IHTU1 … HL12 (ⓛW2.T0) ?) -IHTU1 /2 width=1/ -HT10 #H + elim (nta_inv_bind1 … H) -H #W #U0 #HW2 #HTU0 #HU01 + elim (cpcs_inv_abst1 … HU01) -HU01 #W #U #HU1 #HU0 + lapply (IHUW1 … HL12 (ⓐV2.ⓛW.U) ?) -IHUW1 -HL12 /2 width=1/ -HV12 #H + + + + elim (nta_fwd_pure1 … H) -H #W0 #U2 #HVU2 #H #HW01 + elim (nta_inv_bind1 … H) -H #W3 #U3 #HW3 #HU3 #H + elim (cpcs_inv_abst1 … H) -H #W4 #U4 +*) +(* +axiom nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → + ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. +#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U +[ #L1 #k #L2 #_ #T2 #H + >(tpr_inv_atom1 … H) -H // +| #L1 #K1 #V1 #W #U #i #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H + >(tpr_inv_atom1 … H) -T2 + elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/ +| #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #HL12 #T2 #H + >(tpr_inv_atom1 … H) -T2 + elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK + elim (lift_total V1 0 (i+1)) #W #HW + lapply (nta_lift h … HLK … HWU1 … HW) /2 width=1/ -HLK -HW + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpr_lift … HW12 … HWU1 … HWU2) -HWU1 #HU12 + @(nta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /3 width=6/ is too slow *) +| #I #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H + elim (tpr_inv_bind1 … H) -H * + [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct + lapply (IHVW1 … HL12 … HV12) #HV2W1 + lapply (IHVW1 L2 … V1 ?) // -IHVW1 #HWV1 + lapply (IHTU1 (L2.ⓑ{I}V2) … HT10) -HT10 /2 width=1/ #HT0U1 + lapply (IHTU1 (L2.ⓑ{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H + lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02 + lapply (nta_tps_conf … HT0U1 … HT02) -T0 #HT2U1 + elim (nta_fwd_correct … H) -H #U2 #HU12 + @(nta_conv … (ⓑ{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | #T #HT1 #HTX #H destruct + lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HVW1 + elim (lift_total X 0 1) #Y #HXY + lapply (tpr_lift … HTX … HT1 … HXY) -T #H + lapply (IHTU1 (L2.ⓓV1) … H) -T1 /2 width=1/ -L1 #H + elim (nta_fwd_correct … H) #T1 #HUT1 + elim (nta_thin_conf … H L2 0 (0+1) ? ?) -H /2 width=1/ /3 width=1/ #T #U #HTU #H + normalize in ⊢ (??%??? → ?); #HU1 + lapply (delift_inv_lift1_eq … H L2 … HXY) -Y /2 width=1/ #H destruct + @(nta_conv … U) // /2 width=2/ + ] +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H + elim (tpr_inv_appl1 … H) -H * + [ #V2 #Y #HV12 #HY #H destruct + elim (tpr_inv_abst1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct + lapply (IHTU1 L2 ? (ⓛW1.T1) ?) // #H + elim (nta_fwd_correct … H) -H #X #H + elim (nta_inv_bind1 … H) -H #W #U #HW #HU #_ + @(nta_conv … (ⓐV2.ⓛW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + lapply (IHVW1 … HL12 … HV12) #HVW2 + lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HV1W2 + lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1 + elim (nta_fwd_correct … H1) #T #H2 + elim (nta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H + elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21 + elim (nta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0 + lapply (lsubn_nta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2 + @(nta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *) + | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct + ] +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #X #H + elim (tpr_inv_appl1 … H) -H * + [ #V2 #T2 #HV12 #HT12 #H destruct + elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U + @(nta_conv … (ⓐV2.U1)) /2 width=1/ /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 /2 width=1/ -T0 #H + elim (nta_inv_bind1 … H) -H #W #U2 #HW2 #HTU2 #HU21 + lapply (IHUW1 … HL12 (ⓐV2.U1) ?) -IHUW1 -HL12 /2 width=1/ #H + elim (nta_inv_pure1 … H) -H #V0 #U0 #U #HV20 #HU10 #HU0W1 #HU0 + @(nta_conv … (ⓓV2.U2)) + [2: @nta_bind // + @(lsubn_nta_trans … HTU2) @lsubn_abbr // +(* + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB + lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 + lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ +*) +*) +(* +axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y → + ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W. + +*) +(* SEGMENT 2 +| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct + lapply (cpr_tpss … HU12) /4 width=4/ +| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 + @(nta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) +] +qed. +*) + +(* SEGMENT 3 +fact nta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 → + ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. + + + | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct + elim (nta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 + lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 + lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0 + lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2 + @(nta_abbr … HW2) -HW2 + @(nta_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) + ] +| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct + elim (tpr_inv_cast1 … H) -H + [ * #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2 + lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ + | -HV1 #HT1X + lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/ + ] +] +qed. + +/2 width=9/ qed. + +axiom nta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A. +/2 width=5/ qed. + +axiom nta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A. +/2 width=5/ qed. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_ltpss.etc new file mode 100644 index 000000000..828fd82e0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_ltpss.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/equivalence/cpcs_ltpss.ma". +include "basic_2/dynamic/nta_nta.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Properties about parallel unfold *****************************************) + +lemma nta_ltpss_tpss_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 : U. +#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U +[ #L1 #k #L2 #d #e #_ #T2 #H + >(tpss_inv_sort1 … H) -H // +| #L1 #K1 #V1 #W #U #i #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H + [ #H destruct + elim (lt_or_ge i d) #Hdi + [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct + /3 width=7/ + | elim (lt_or_ge i (d + e)) #Hide [ | -Hdi ] + [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct + /3 width=7/ + | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=7/ + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 + elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct + lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2 + lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=9/ + ] +| #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ] + [ #H destruct + elim (lift_total V1 0 (i+1)) #W #HW + elim (lt_or_ge i d) #Hdi [ -HWV1 ] + [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK + lapply (nta_lift h … HLK … HWU1 … HW) /2 width=4/ -HW + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 + lapply (cpr_tpss … HU12) -HU12 #HU12 + @(nta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -HW -Hdi ] + [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK + lapply (nta_lift h … HLK … HWU1 … HW) /2 width=4/ -HW + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 + lapply (cpr_tpss … HU12) -HU12 #HU12 + @(nta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=6/ + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_ + elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct + lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct + ] +| #I #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpss … HV12) #HV + lapply (IHTU1 (L2.ⓑ{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H + elim (nta_fwd_correct … H) -H #U2 #HU12 + @(nta_conv … (ⓑ{I}V2.U1)) /3 width=2/ /3 width=4/ /4 width=4/ (**) (* explicit constructor, /5 width=6/ is too slow *) +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct + elim (tpss_inv_bind1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct + lapply (cpr_tpss … HV12) #HV + lapply (IHTU1 L2 d e ? (ⓛW1.T1) ?) // #H + elim (nta_fwd_correct … H) -H #X #H + elim (nta_inv_bind1 … H) -H #W #U #HW #HU #_ + @(nta_conv … (ⓐV2.ⓛW1.U1)) /3 width=2/ /3 width=4/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpss … HV12) #HV + elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=4/ ] #U #HU + @(nta_conv … (ⓐV2.U1)) // /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) +| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct + lapply (cpr_tpss … HU12) /4 width=4/ +| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 + @(nta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) +] +qed. + +lemma nta_ltpss_tps_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → ⦃h, L2⦄ ⊢ T2 : U. +/3 width=7/ qed. + +lemma nta_ltpss_conf: ∀h,L1,T,U. ⦃h, L1⦄ ⊢ T : U → + ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T : U. +/2 width=7/ qed. + +lemma nta_tpss_conf: ∀h,L,T1,U. ⦃h, L⦄ ⊢ T1 : U → + ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 : U. +/2 width=7/ qed. + +lemma nta_tps_conf: ∀h,L,T1,U. ⦃h, L⦄ ⊢ T1 : U → + ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 : U. +/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_nta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_nta.etc new file mode 100644 index 000000000..05eb6c55d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_nta.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/dynamic/nta_lift.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Main properties **********************************************************) + +(* Basic_1: was: ty3_unique *) +theorem nta_mono: ∀h,L,T,U1. ⦃h, L⦄ ⊢ T : U1 → ∀U2. ⦃h, L⦄ ⊢ T : U2 → + L ⊢ U1 ⬌* U2. +#h #L #T #U1 #H elim H -L -T -U1 +[ #L #k #X #H + lapply (nta_inv_sort1 … H) -H // +| #L #K #V #W11 #W12 #i #HLK #_ #HW112 #IHVW11 #X #H + elim (nta_inv_lref1 … H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX + lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK + @(cpcs_trans … HX) -X /3 width=9 by cpcs_lift/ (**) (* to slow without trace *) +| #L #K #W #V1 #V #i #HLK #_ #HWV #_ #X #H + elim (nta_inv_lref1 … H) -H * #K0 #W0 #V2 #V0 #HLK0 #_ #HWV0 #HX + lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H destruct + lapply (lift_mono … HWV0 … HWV) -HWV0 -HWV #H destruct // +| #I #L #V #W1 #T #U1 #_ #_ #_ #IHTU1 #X #H + elim (nta_inv_bind1 … H) -H #W2 #U2 #_ #HTU2 #H + @(cpcs_trans … H) -X /3 width=1/ +| #L #V #W1 #T #U1 #_ #_ #_ #IHTU1 #X #H + elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H + @(cpcs_trans … H) -X /3 width=1/ +| #L #V #W1 #T #U1 #_ #_ #IHTU1 #_ #X #H + elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H + @(cpcs_trans … H) -X /3 width=1/ +| #L #T #U1 #_ #_ #X #H + elim (nta_inv_cast1 … H) -H // +| #L #T #U11 #U12 #V12 #_ #HU112 #_ #IHTU11 #_ #U2 #HTU2 + @(cpcs_canc_sn … HU112) -U12 /2 width=1/ +] +qed-. + +(* Advanced properties ******************************************************) + +lemma nta_cast_alt: ∀h,L,T,W,U. ⦃h, L⦄ ⊢ T : W → ⦃h, L⦄ ⊢ T : U → + ⦃h, L⦄ ⊢ ⓝW.T : U. +#h #L #T #W #U #HTW #HTU +lapply (nta_mono … HTW … HTU) #HWU +elim (nta_fwd_correct … HTU) -HTU /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_sta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_sta.etc new file mode 100644 index 000000000..6268b98b1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_sta.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/static/sta.ma". +include "basic_2/equivalence/cpcs_cpcs.ma". +include "basic_2/dynamic/nta.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Properties on static type assignment *************************************) + +lemma nta_fwd_sta: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → + ∃∃U0. ⦃h, L⦄ ⊢ T • U0 & L ⊢ U0 ⬌* U. +#h #L #T #U #H elim H -L -T -U +[ /2 width=3/ +| #L #K #V #W1 #V1 #i #HLK #_ #HWV1 * #W0 #HVW0 #HW01 + elim (lift_total W0 0 (i+1)) #V0 #HWV0 + lapply (ldrop_fwd_ldrop2 … HLK) #HLK0 + lapply (cpcs_lift … HLK0 … HWV0 … HWV1 HW01) -HLK0 -HWV1 -HW01 /3 width=6/ +| #L #K #W #V1 #W1 #i #HLK #HWV1 #HW1 * /3 width=6/ +| #I #L #V #W #T #U #_ #_ * #W0 #_ #_ * /3 width=3/ +| #L #V #W #T #U #_ #_ * #W0 #_ #HW0 * #X #H #HX + elim (sta_inv_bind1 … H) -H #U0 #HTU0 #H destruct + @(ex2_1_intro … (ⓐV.ⓛW.U0)) /2 width=1/ /3 width=1/ +| #L #V #W #T #U #_ #_ * #U0 #HTU0 #HU0 #_ -W + @(ex2_1_intro … (ⓐV.U0)) /2 width=1/ +| #L #T #U #HTU * #U0 #HTU0 #HU0 /3 width=3/ +| #L #T #U1 #U2 #V2 #_ #HU12 #_ * #U0 #HTU0 #HU01 #_ + lapply (cpcs_trans … HU01 … HU12) -U1 /2 width=3/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_thin.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_thin.etc new file mode 100644 index 000000000..f927f841a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta/nta_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/nta_lift.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Properties on basic local environment thinning ***************************) + +(* Note: this is known as the substitution lemma *) +(* Basic_1: was only: ty3_gen_cabbr *) +lemma nta_thin_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 : U1 → + ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 → + ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 : U2 & + L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2. +#h #L1 #T1 #U1 #H elim H -L1 -T1 -U1 +[ /2 width=5/ +| #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12 + elim (lt_or_ge i d) #Hdi [ -HVW1 ] + [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H + lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1 + elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2 + elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12 + lapply (delift_mono … H … HV12) -H -HV12 #H destruct + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1 + lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 // + >minus_plus minus_plus #HU1 + lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2 + lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/ + | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei + lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1 + elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W + commutative_plus minus_plus commutative_plus 0 & L ⊢ U0 ⬌* U + ). +#h #L #T #U #l #H elim H -L -T -U -l +[ #L #k #j #H destruct +| #L #K #V #W #U #i #l #HLK #HVW #HWU #_ #j #H destruct /3 width=8/ +| #L #K #W #V #U #i #l #HLK #HWV #HWU #_ #j #H destruct /3 width=8/ +| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct +| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #j #H destruct +| #L #V #T #U #W #l #_ #_ #_ #_ #j #H destruct +| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #j #H destruct +| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #j #H destruct + elim (IHTU1 ??) -IHTU1 [4: // |2: skip ] * #K #V #W #U0 #HLK #HVW #HWU0 [2: #H ] #HU01 + lapply (cpcs_trans … HU01 … HU12) -U1 /3 width=8/ +] +qed. + +lemma snta_inv_lref1: ∀h,L,U,i,l. ⦃h, L⦄ ⊢ #i :[l] U → + (∃∃K,V,W,U0. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V :[l] W & + ⇧[0, i + 1] W ≡ U0 & L ⊢ U0 ⬌* U + ) ∨ + (∃∃K,W,V,U0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W :[l-1] V & + ⇧[0, i + 1] W ≡ U0 & l > 0 & L ⊢ U0 ⬌* U + ). +/2 width=3/ qed-. + +fact snta_inv_bind1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀J,X,Y. T = ⓑ{J}Y.X → + ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 & + ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 & + L ⊢ ⓑ{J}Y.Z2 ⬌* U. +#h #L #T #U #l #H elim H -L -T -U -l +[ #L #k #J #X #Y #H destruct +| #L #K #V #W #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct +| #L #K #W #V #U #i #l #_ #_ #_ #_ #J #X #Y #H destruct +| #I #L #V #W #T #U #l1 #l2 #HVW #HTU #_ #_ #J #X #Y #H destruct /2 width=3/ +| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct +| #L #V #T #U #W #l #_ #_ #_ #_ #J #X #Y #H destruct +| #L #T #U #W #l1 #l2 #_ #_ #_ #_ #J #X #Y #H destruct +| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #J #X #Y #H destruct + elim (IHTU1 ????) -IHTU1 [5: // |2,3,4: skip ] #Z1 #Z2 #l0 #HZ1 #HZ2 #HU1 + lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=3/ +] +qed. + +lemma snta_inv_bind1: ∀h,J,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{J}Y.X :[l] U → + ∃∃Z1,Z2,l0. ⦃h, L⦄ ⊢ Y :[l0] Z1 & ⦃h, L.ⓑ{J}Y⦄ ⊢ X :[l] Z2 & + L ⊢ ⓑ{J}Y.Z2 ⬌* U. +/2 width=3/ qed-. + +fact snta_inv_cast1_aux: ∀h,L,T,U,l. ⦃h, L⦄ ⊢ T :[l] U → ∀X,Y. T = ⓝY.X → + ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U. +#h #L #T #U #l #H elim H -L -T -U -l +[ #L #k #X #Y #H destruct +| #L #K #V #W #U #i #l #_ #_ #_ #_ #X #Y #H destruct +| #L #K #W #V #U #i #l #_ #_ #_ #_ #X #Y #H destruct +| #I #L #V #W #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct +| #L #V #W1 #W2 #T #U #l1 #l2 #_ #_ #_ #_ #X #Y #H destruct +| #L #V #T #U #W #l #_ #_ #_ #_ #X #Y #H destruct +| #L #T #U #W #l1 #l2 #HTU #_ #_ #_ #X #Y #H destruct /2 width=1/ +| #L #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct + elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #HXY #HU1 + lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=1/ +] +qed. + +lemma snta_inv_cast1: ∀h,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X :[l] U → + ⦃h, L⦄ ⊢ X :[l] Y ∧ L ⊢ Y ⬌* U. +/2 width=3/ qed-. + +(* Properties on relocation *************************************************) + +lemma snta_lift: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 → + ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → + ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 → + ⦃h, L2⦄ ⊢ T2 :[l] U2. +#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l +[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2 + >(lift_inv_sort1 … H1) -X1 + >(lift_inv_sort1 … H2) -X2 // +| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 + elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct + /3 width=8/ + | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 + lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ + ] +| #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (tpr_inv_atom1 … H) -H // +| #L1 #K1 #V1 #W #U #i1 #l #HLK1 #_ #HWU #IHV1 #L2 #HL12 #T2 #H #Hl -IH + >(tpr_inv_atom1 … H) -T2 + elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/ +| #L1 #K1 #W1 #V1 #U1 #i1 #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #HL12 #T2 #H #Hl -IH +(* + >(tpr_inv_atom1 … H) -T2 + elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK + elim (lift_total V1 0 (i+1)) #W #HW + lapply (snta_lift h … HLK … HWU1 … HW) /2 width=1/ -HLK -HW + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpr_lift … HW12 … HWU1 … HWU2) -HWU1 #HU12 + @(snta_conv … U2) /2 width=1/ /3 width=6/ (**) (* explicit constructor, /3 width=6/ is too slow *) +*) +| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH +(* + elim (tpr_inv_bind1 … H) -H * + [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct + lapply (IHVW1 … HL12 … HV12) #HV2W1 + lapply (IHVW1 L2 … V1 ?) // -IHVW1 #HWV1 + lapply (IHTU1 (L2.ⓑ{I}V2) … HT1) -HT1 /2 width=1/ #HTU1 + lapply (IHTU1 (L2.ⓑ{I}V1) ? T1 ?) -IHTU1 // /2 width=1/ -HL12 #H + lapply (tps_lsubs_trans … HT2 (L2.ⓑ{I}V2) ?) -HT2 /2 width=1/ #HT2 + lapply (snta_tps_conf … HTU1 … HT2) -T #HT2U1 + elim (snta_fwd_correct … H) -H #U2 #HU12 + @(snta_conv … (ⓑ{I}V2.U1)) /2 width=2/ /3 width=1/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | #T #HT1 #HTX #H destruct + lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HVW1 + lapply (IHTU1 (L2.ⓓV1) … HT1) -T1 /2 width=1/ -L1 #H + elim (snta_fwd_correct … H) #T1 #HUT1 + elim (snta_ldrop_conf … H L2 0 1 ? ?) -H // /2 width=1/ #T0 #U0 #HTU0 #H #HU10 + lapply (delift_inv_lift1_eq … H L2 … HTX) -H -HTX /2 width=1/ #H destruct + @(snta_conv … HTU0) /2 width=2/ + ] +*) +| #L1 #V1 #W11 #W2 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #HL12 #X #H #Hl -IH +(* + elim (tpr_inv_appl1 … H) -H * + [ #V2 #Y #HV12 #HY #H destruct + elim (tpr_inv_abst1 … HY) -HY #W2 #T2 #HW12 #HT12 #H destruct + lapply (IHTU1 L2 ? (ⓛW1.T1) ?) // #H + elim (snta_fwd_correct … H) -H #X #H + elim (snta_inv_bind1 … H) -H #W #U #HW #HU #_ + @(snta_conv … (ⓐV2.ⓛW1.U1)) /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + lapply (IHVW1 … HL12 … HV12) #HVW2 + lapply (IHVW1 … HL12 V1 ?) -IHVW1 // #HV1W2 + lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 -HL12 /2 width=1/ -HT02 #H1 + elim (snta_fwd_correct … H1) #T #H2 + elim (snta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H + elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21 + elim (snta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0 + lapply (lsubsn_snta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2 + @(snta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *) + | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct + ] +*) +| #L1 #V1 #T1 #U1 #W1 #l #_ #HUW1 #IHTU1 #_ #L2 #HL12 #X #H #Hl + elim (tpr_inv_appl1 … H) -H * + [ #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpr … HV12 L2) #HV + elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=6/ ] #U + @(snta_conv … (ⓐV2.U1)) /2 width=1/ -HV12 /4 width=8 by snta_pure, cprs_flat_dx/ (**) (* explicit constructor, /4 width=8/ is too slow without trace *) + | #V2 #W0 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + lapply (IHTU1 … HL12 (ⓛW0.T2) ? ?) -IHTU1 // /2 width=1/ -T0 #H1 + lapply (IH … (ⓐV2.U1) … HUW1 HL12 ?) // /3 width=1/ #H2 + lapply (snta_pure … H1 H2) -H2 #H + elim (snta_inv_bind1 … H1) -H1 #V0 #U2 #l1 #HWV0 #HTU2 #HU21 + @(snta_conv … (ⓓV2.U2)) (**) (* explicit constructor *) + [2: +(* + @snta_bind /3 width=2/ /3 width=6/ (**) (* /4 width=6/ is a bit slow *) +*) + |3: @(cpcs_cpr_conf … (ⓐV1.ⓛW0.U2)) /2 width=1/ + |4: /2 width=5/ + | skip + ] +(* + elim (snta_fwd_pure1 … H) -H #T1 #W2 #HVW2 #HUT1 #HTW1 + + elim (cpcs_inv_abst1 … HU21) #W3 #U3 #HU13 #H + elim (cprs_inv_abst … H Abst W0) -H #HW03 #_ + elim (pippo … IH … HUW1 ? V2 W3 U3 HL12 ? ?) -IH -HUW1 -HL12 // /3 width=1/ -HU13 #l2 #HV2W3 + lapply (snta_conv h L2 V2 W3 W0 V0 (l1+1) ? ? ?) /2 width=1/ -HV2W3 -HW03 -HWV0 #HV2W0 +*) +(* SEGMENT 1.5 + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB + lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 + lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ + +axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y → + ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W. + +*) +(* SEGMENT 2 +| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct + lapply (cpr_tpss … HU12) /4 width=4/ +| #L1 #T1 #U11 #U12 #U #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 + @(snta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) +] +qed. +*) + +(* SEGMENT 3 +fact snta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 → + ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. + + + | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct + elim (snta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 + lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 + lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0 + lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2 + @(snta_abbr … HW2) -HW2 + @(snta_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) + ] +| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct + elim (tpr_inv_cast1 … H) -H + [ * #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2 + lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ + | -HV1 #HT1X + lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/ + ] +] +qed. + +lemma snta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → + ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. + +/2 width=9/ qed. + +axiom snta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A. +/2 width=5/ qed. + +axiom snta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A. +/2 width=5/ qed. +*) +*)*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/snta/snta_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/snta/snta_ltpss.etc new file mode 100644 index 000000000..0e5f3930e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/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: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U. +#h #L1 #T1 #U #l #H elim H -L1 -T1 -U -l +[ #L1 #k #L2 #d #e #_ #T2 #H + >(tpss_inv_sort1 … H) -H // +| #L1 #K1 #V1 #W #U #i #l #HLK1 #_ #HWU #IHV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H + [ #H destruct + elim (lt_or_ge i d) #Hdi + [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct + /3 width=7/ + | elim (lt_or_ge i (d + e)) #Hide [ | -Hdi ] + [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct + /3 width=7/ + | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=7/ + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 + elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct + lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2 + lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=9/ + ] +| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ] + [ #H destruct + elim (lift_total V1 0 (i+1)) #W #HW + elim (lt_or_ge i d) #Hdi [ -HWV1 ] + [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK + lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 + lapply (cpr_tpss … HU12) -HU12 #HU12 + @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -HW -Hdi ] + [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #W2 #HK12 #HW12 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) #HLK + lapply (snta_lift h … HLK … HWU1 … HW) [ /2 width=4/ | skip ] -HW #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) -HLK -HWU1 // #HU12 + lapply (cpr_tpss … HU12) -HU12 #HU12 + @(snta_conv … U2) // /2 width=1/ /3 width=6/ (**) (* explicit constructor, /4 width=6/ is too slow *) + | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /2 width=6/ + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_ + elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct + lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct + ] +| #I #L1 #V1 #W1 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpss … HV12) #HV + lapply (IHTU1 (L2.ⓑ{I}V1) (d+1) e ? T1 ?) // /2 width=1/ #H + elim (snta_fwd_correct … H) -H #U2 #HU12 + @(snta_conv … (ⓑ{I}V2.U1)) /3 width=2/ /3 width=4/ /4 width=4/ (**) (* explicit constructor, /5 width=6/ is too slow *) +| #L1 #V1 #W11 #W12 #T1 #U1 #l1 #l2 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct + elim (tpss_inv_bind1 … HY) -HY #W21 #T2 #HW121 #HT12 #H destruct + lapply (cpr_tpss … HV12) #HVV12 + lapply (IHTU1 L2 d e ? (ⓛW21.T2) ?) -IHTU1 // /2 width=1/ -HW121 -HT12 #H0 + elim (snta_fwd_correct … H0) #X #H + elim (snta_inv_bind1 … H) -H #W #U #l0 #HW #HU #_ + @(snta_conv … (ⓐV2.ⓛW12.U1)) /3 width=2/ /3 width=4/ /3 width=5/ (**) (* explicit constructor, /4 width=5/ is too slow *) +| #L1 #V1 #T1 #U1 #W1 #l #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (cpr_tpss … HV12) #HV + elim (snta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) (l+1) ?) [2: /3 width=4/ ] #U + @(snta_conv … (ⓐV2.U1)) /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) +| #L1 #T1 #U1 #W1 #l1 #l2 #HTU1 #HUW1 #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H + elim (snta_fwd_correct … HTU1) -HTU1 #U #H + elim (snta_mono … HUW1 … H) -HUW1 -H #H #_ -U destruct + elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct + lapply (cpr_tpss … HU12) #HU /4 width=4/ +| #L1 #T1 #U11 #U12 #U #l #_ #HU112 #_ #IHTU11 #IHU12 #L2 #d #e #HL12 #T2 #HT12 + @(snta_conv … U11) /2 width=5/ (**) (* explicit constructor, /3 width=7/ is too slow *) +] +qed. + +lemma snta_ltpss_tps_conf: ∀h,L1,T1,U,l. ⦃h, L1⦄ ⊢ T1 :[l] U → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → ⦃h, L2⦄ ⊢ T2 :[l] U. +/3 width=7/ qed. + +lemma snta_ltpss_conf: ∀h,L1,T,U,l. ⦃h, L1⦄ ⊢ T :[l] U → + ∀L2,d,e. L1 ▶* [d, e] L2 → ⦃h, L2⦄ ⊢ T :[l] U. +/2 width=7/ qed. + +lemma snta_tpss_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U → + ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U. +/2 width=7/ qed. + +lemma snta_tps_conf: ∀h,L,T1,U,l. ⦃h, L⦄ ⊢ T1 :[l] U → + ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ⦃h, L⦄ ⊢ T2 :[l] U. +/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/snta/snta_snta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/snta/snta_snta.etc new file mode 100644 index 000000000..db71e1192 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/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: ∀h,L,T,U1,l1. ⦃h, L⦄ ⊢ T :[l1] U1 → + ∀U2,l2. ⦃h, L⦄ ⊢ T :[l2] U2 → l1 = l2 ∧ L ⊢ U1 ⬌* U2. +#h #L #T #U1 #l1 #H elim H -L -T -U1 -l1 +[ #L #k #X #l2 #H + lapply (snta_inv_sort1 … H) -H * /2 width=1/ +| #L #K #V #W11 #W12 #i #l1 #HLK #_ #HW112 #IHVW11 #X #l2 #H + elim (snta_inv_lref1 … H) -H * #K0 #V0 #W21 #W22 #HLK0 #HVW21 #HW212 #HX + lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK + elim (IHVW11 … HVW21) -IHVW11 -HVW21 #Hl12 #HW121 + lapply (cpcs_lift … HLK … HW112 … HW212 ?) // -K -W11 -W21 /3 width=3/ +| #L #K #W #V1 #V #i #l1 #HLK #_ #HWV #IHWV1 #X #l2 #H + elim (snta_inv_lref1 … H) -H * #K0 #W0 #V2 #V0 #HLK0 #HW0V2 #HWV0 [2: #HL2 ] #HX + lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H destruct + lapply (lift_mono … HWV0 … HWV) -HWV0 -HWV #H destruct + elim (IHWV1 … HW0V2) -IHWV1 -HW0V2 /3 width=1/ +| #I #L #V #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H + elim (snta_inv_bind1 … H) -H #W2 #U2 #l20 #_ #HTU2 #H + elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12 + lapply (cpcs_trans … (ⓑ{I}V.U1) … H) -H /2 width=1/ +| #L #V #W #W1 #T #U1 #l10 #l1 #_ #_ #_ #IHTU1 #X #l2 #H + elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H + elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12 + lapply (cpcs_trans … (ⓐV.ⓛW1.U1) … H) -H /2 width=1/ +| #L #V #T #U1 #W1 #l1 #_ #_ #IHTU1 #_ #X #l2 #H + elim (snta_fwd_pure1 … H) -H #U2 #W2 #l20 #_ #HTU2 #H + elim (IHTU1 … HTU2) -IHTU1 -HTU2 #Hl12 #HU12 + lapply (cpcs_trans … (ⓐV.U1) … H) -H /2 width=1/ +| #L #T #U1 #W1 #l10 #l1 #_ #_ #IHTU1 #_ #X #l2 #H + elim (snta_inv_cast1 … H) -H #HTU1 + elim (IHTU1 … HTU1) -IHTU1 -HTU1 /2 width=1/ +| #L #T #U11 #U12 #V12 #l1 #_ #HU112 #_ #IHTU11 #_ #U2 #l2 #HTU2 + elim (IHTU11 … HTU2) -IHTU11 -HTU2 #Hl12 #H + lapply (cpcs_canc_sn … HU112 … H) -U11 /2 width=1/ +] +qed-. + +(* Advanced properties ******************************************************) + +lemma snta_cast_alt: ∀h,L,T,W,U,l. ⦃h, L⦄ ⊢ T :[l] W → ⦃h, L⦄ ⊢ T :[l] U → + ⦃h, L⦄ ⊢ ⓝW.T :[l] U. +#h #L #T #W #U #l #HTW #HTU +elim (snta_mono … HTW … HTU) #_ #HWU +elim (snta_fwd_correct … HTU) -HTU /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/snta/snta_thin.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/snta/snta_thin.etc new file mode 100644 index 000000000..ceb5375bf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/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: ∀h,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 :[l] U1 → + ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 → + ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 :[l] U2 & + L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2. +#h #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l +[ /2 width=5/ +| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12 + elim (lt_or_ge i d) #Hdi [ -HVW1 ] + [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H + lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1 + elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2 + elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12 + lapply (delift_mono … H … HV12) -H -HV12 #H destruct + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1 + lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 // + >minus_plus minus_plus #HU1 + lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2 + lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/ + | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei + lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1 + elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W + commutative_plus minus_plus commutative_plus (deg_mono … Hkl HkO) -g -l // +| #T0 #U0 #l0 #HTU0 #_ #IHU0 #k #H #l #Hkl destruct + elim (ssta_inv_sort1 … HTU0) -L #HkS #H destruct + lapply (deg_mono … Hkl HkS) -Hkl #H destruct + >(IHU0 (next h k) ? l0) -IHU0 // /2 width=1/ >iter_SO >iter_n_Sm // +] +qed. + +lemma sstas_inv_sort1: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k •*[g] U → ∀l. deg h g k l → + U = ⋆((next h)^l k). +/2 width=6/ qed-. + +fact sstas_inv_bind1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → + ∀J,X,Y. T = ⓑ{J}Y.X → + ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z. +#h #g #L #T #U #H @(sstas_ind_alt … H) -T +[ #U0 #HU0 #J #X #Y #H destruct + elim (ssta_inv_bind1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/ +| #T0 #U0 #l #HTU0 #_ #IHU0 #J #X #Y #H destruct + elim (ssta_inv_bind1 … HTU0) -HTU0 #X0 #HX0 #H destruct + elim (IHU0 J X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/ +] +qed. + +lemma sstas_inv_bind1: ∀h,g,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X •*[g] U → + ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z. +/2 width=3/ qed-. + +fact sstas_inv_appl1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀X,Y. T = ⓐY.X → + ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z. +#h #g #L #T #U #H @(sstas_ind_alt … H) -T +[ #U0 #HU0 #X #Y #H destruct + elim (ssta_inv_appl1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/ +| #T0 #U0 #l #HTU0 #_ #IHU0 #X #Y #H destruct + elim (ssta_inv_appl1 … HTU0) -HTU0 #X0 #HX0 #H destruct + elim (IHU0 X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/ +] +qed. + +lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U → + ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z. +/2 width=3/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma sstas_fwd_correct: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → + ∃∃W. ⦃h, L⦄ ⊢ U •[g, 0] W & ⦃h, L⦄ ⊢ U •*[g] U. +#h #g #L #T #U #H @(sstas_ind_alt … H) -T /2 width=1/ /3 width=2/ +qed-. + +(* Basic_1: removed theorems 7: + sty1_bind sty1_abbr sty1_appl sty1_cast2 + sty1_lift sty1_correct sty1_trans +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_lift.etc new file mode 100644 index 000000000..838c7b6d4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_lift.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/static/ssta_lift.ma". +include "basic_2/unwind/sstas.ma". + +(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************) + +(* Advanced properties ******************************************************) + +lemma sstas_total_S: ∀h,g,L,l,T,U. ⦃h, L⦄ ⊢ T•[g, l + 1]U → + ∃∃W. ⦃h, L⦄ ⊢ T •*[g] W & ⦃h, L⦄ ⊢ U •*[g] W. +#h #g #L #l @(nat_ind_plus … l) -l +[ #T #U #HTU + elim (ssta_fwd_correct … HTU) /4 width=4/ +| #l #IHl #T #U #HTU + elim (ssta_fwd_correct … HTU) (lift_mono … HX … HU12) -X + elim (lift_total T1 d e) /3 width=10/ +| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL21 #T2 #HT02 #U2 #HU12 + elim (lift_total U0 d e) /3 width=10/ +] +qed. + +lemma sstas_inv_lift1: ∀h,g,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 → + ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 → + ∃∃U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 & ⇧[d, e] U1 ≡ U2. +#h #g #L2 #T2 #U2 #H @(sstas_ind_alt … H) -T2 +[ #T2 #HUT2 #L1 #d #e #HL21 #U1 #HU12 + elim (ssta_inv_lift1 … HUT2 … HL21 … HU12) -HUT2 -HL21 /3 width=3/ +| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L1 #d #e #HL21 #U1 #HU12 + elim (ssta_inv_lift1 … HTU0 … HL21 … HU12) -HTU0 -HU12 #U #HU1 #HU0 + elim (IHU01 … HL21 … HU0) -IHU01 -HL21 -U0 /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_ltpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_ltpss.etc new file mode 100644 index 000000000..e0aa94207 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_ltpss.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/static/ssta_ltpss.ma". +include "basic_2/unwind/sstas.ma". + +(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************) + +(* Properties about parallel unfold *****************************************) + +lemma sstas_ltpss_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → + ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & + L2 ⊢ U1 ▶* [d, e] U2. +#h #g #L1 #T1 #U1 #H @(sstas_ind_alt … H) -T1 +[ #T1 #HUT1 #L2 #d #e #HL12 #U2 #HU12 + elim (ssta_ltpss_tpss_conf … HUT1 … HL12 … HU12) -HUT1 -HL12 /3 width=3/ +| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0 + elim (ssta_ltpss_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0 + elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/ +] +qed. + +lemma sstas_ltpss_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → + ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2. +/3 width=5/ qed. + +lemma sstas_ltpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2. +/2 width=5/ qed. + +lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 → + ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → + ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2. +/2 width=5/ qed. + +lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 → + ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → + ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2. +/2 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_sstas.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_sstas.etc new file mode 100644 index 000000000..2f7e261b4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sstas/sstas_sstas.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/unfold/delift_lift.ma". +include "basic_2/static/ssta_ssta.ma". +include "basic_2/unwind/sstas_lift.ma". + +(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************) + +(* Advanced inversion lemmas ************************************************) + +lemma sstas_inv_O: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → + ∀T0. ⦃h, L⦄ ⊢ T •[g , 0] T0 → U = T. +#h #g #L #T #U #H @(sstas_ind_alt … H) -T // +#T0 #U0 #l0 #HTU0 #_ #_ #T1 #HT01 +elim (ssta_mono … HTU0 … HT01) (sstas_inv_O … HU12 … HUT1) -h -L -T1 -U2 // +| #T0 #U0 #l0 #HTU0 #_ #IHU01 #U2 #HU12 + lapply (sstas_inv_S … HU12 … HTU0) -T0 -l0 /2 width=1/ +] +qed-. + +(* More advancd inversion lemmas ********************************************) + +fact sstas_inv_lref1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀j. T = #j → + ∃∃I,K,V,W. ⇩[0, j] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V •*[g] W & + L ⊢ ▼*[0, j + 1] U ≡ W. +#h #g #L #T #U #H @(sstas_ind_alt … H) -T +[ #T #HUT #j #H destruct + elim (ssta_inv_lref1 … HUT) -HUT * #K #V #W [2: #l] #HLK #HVW #HVT + [ (sstas_mono … HWX … HWT) -X -W /3 width=7/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta.etc new file mode 100644 index 000000000..20302c623 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta.etc @@ -0,0 +1,128 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/ldrop.ma". +include "basic_2/static/sh.ma". + +(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************) + +inductive sta (h:sh): lenv → relation term ≝ +| sta_sort: ∀L,k. sta h L (⋆k) (⋆(next h k)) +| sta_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → sta h K V W → + ⇧[0, i + 1] W ≡ U → sta h L (#i) U +| sta_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → sta h K W V → + ⇧[0, i + 1] W ≡ U → sta h L (#i) U +| sta_bind: ∀I,L,V,T,U. sta h (L. ⓑ{I} V) T U → + sta h L (ⓑ{I}V.T) (ⓑ{I}V.U) +| sta_appl: ∀L,V,T,U. sta h L T U → + sta h L (ⓐV.T) (ⓐV.U) +| sta_cast: ∀L,W,T,U. sta h L T U → sta h L (ⓝW. T) U +. + +interpretation "static type assignment (term)" + 'StaticType h L T U = (sta h L T U). + +(* Basic inversion lemmas ************************************************) + +fact sta_inv_sort1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀k0. T = ⋆k0 → + U = ⋆(next h k0). +#h #L #T #U * -L -T -U +[ #L #k #k0 #H destruct // +| #L #K #V #W #U #i #_ #_ #_ #k0 #H destruct +| #L #K #W #V #U #i #_ #_ #_ #k0 #H destruct +| #I #L #V #T #U #_ #k0 #H destruct +| #L #V #T #U #_ #k0 #H destruct +| #L #W #T #U #_ #k0 #H destruct +qed. + +(* Basic_1: was: sty0_gen_sort *) +lemma sta_inv_sort1: ∀h,L,U,k. ⦃h, L⦄ ⊢ ⋆k • U → U = ⋆(next h k). +/2 width=4/ qed-. + +fact sta_inv_lref1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀j. T = #j → + (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V • W & + ⇧[0, j + 1] W ≡ U + ) ∨ + (∃∃K,W,V. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W • V & + ⇧[0, j + 1] W ≡ U + ). +#h #L #T #U * -L -T -U +[ #L #k #j #H destruct +| #L #K #V #W #U #i #HLK #HVW #HWU #j #H destruct /3 width=6/ +| #L #K #W #V #U #i #HLK #HWV #HWU #j #H destruct /3 width=6/ +| #I #L #V #T #U #_ #j #H destruct +| #L #V #T #U #_ #j #H destruct +| #L #W #T #U #_ #j #H destruct +] +qed. + +(* Basic_1: was sty0_gen_lref *) +lemma sta_inv_lref1: ∀h,L,U,i. ⦃h, L⦄ ⊢ #i • U → + (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V • W & + ⇧[0, i + 1] W ≡ U + ) ∨ + (∃∃K,W,V. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W • V & + ⇧[0, i + 1] W ≡ U + ). +/2 width=3/ qed-. + +fact sta_inv_bind1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀J,X,Y. T = ⓑ{J}Y.X → + ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X • Z & U = ⓑ{J}Y.Z. +#h #L #T #U * -L -T -U +[ #L #k #J #X #Y #H destruct +| #L #K #V #W #U #i #_ #_ #_ #J #X #Y #H destruct +| #L #K #W #V #U #i #_ #_ #_ #J #X #Y #H destruct +| #I #L #V #T #U #HTU #J #X #Y #H destruct /2 width=3/ +| #L #V #T #U #_ #J #X #Y #H destruct +| #L #W #T #U #_ #J #X #Y #H destruct +] +qed. + +(* Basic_1: was: sty0_gen_bind *) +lemma sta_inv_bind1: ∀h,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X • U → + ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X • Z & U = ⓑ{J}Y.Z. +/2 width=3/ qed-. + +fact sta_inv_appl1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀X,Y. T = ⓐY.X → + ∃∃Z. ⦃h, L⦄ ⊢ X • Z & U = ⓐY.Z. +#h #L #T #U * -L -T -U +[ #L #k #X #Y #H destruct +| #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct +| #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct +| #I #L #V #T #U #_ #X #Y #H destruct +| #L #V #T #U #HTU #X #Y #H destruct /2 width=3/ +| #L #W #T #U #_ #X #Y #H destruct +] +qed. + +(* Basic_1: was: sty0_gen_appl *) +lemma sta_inv_appl1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X • U → + ∃∃Z. ⦃h, L⦄ ⊢ X • Z & U = ⓐY.Z. +/2 width=3/ qed-. + +fact sta_inv_cast1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀X,Y. T = ⓝY.X → + ⦃h, L⦄ ⊢ X • U. +#h #L #T #U * -L -T -U +[ #L #k #X #Y #H destruct +| #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct +| #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct +| #I #L #V #T #U #_ #X #Y #H destruct +| #L #V #T #U #_ #X #Y #H destruct +| #L #W #T #U #HTU #X #Y #H destruct // +] +qed. + +(* Basic_1: was: sty0_gen_cast *) +lemma sta_inv_cast1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓝY.X • U → ⦃h, L⦄ ⊢ X • U. +/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_lift.etc new file mode 100644 index 000000000..c9fbda014 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_lift.etc @@ -0,0 +1,120 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/static/sta.ma". + +(* STATIC TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Properties on relocation *************************************************) + +(* Basic_1: was: sty0_lift *) +lemma sta_lift: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → + ∀T2. ⇧[d, e] T1 ≡ T2 → ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 • U2. +#h #L1 #T1 #U1 #H elim H -L1 -T1 -U1 +[ #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2 + >(lift_inv_sort1 … H1) -X1 + >(lift_inv_sort1 … H2) -X2 // +| #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 + elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct + /3 width=8/ + | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 + lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ + ] +| #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (lift_inv_sort2 … H) -X /2 width=3/ +| #L2 #K2 #V2 #W2 #W #i #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ] + [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 + elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HVW1 #HW12 + elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ + | minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ + | (tpss_inv_sort1 … H) -H /2 width=3/ +| #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H [ | -HVW1 ] + [ #H destruct + elim (lt_or_ge i d) #Hdi [ -HVW1 | ] + [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 + lapply (ldrop_fwd_ldrop2 … HLK2) #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 + >minus_plus minus_plus #H + lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H + lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H + lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus (sta_inv_sort1 … H) -X // +| #L #K #V #W #U1 #i #HLK #_ #HWU1 #IHVW #U2 #H + elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HW0U2 + lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct + lapply (IHVW … HVW0) -IHVW -HVW0 #H destruct + >(lift_mono … HWU1 … HW0U2) -W0 -U1 // +| #L #K #W #V #U1 #i #HLK #_ #HWU1 #IHWV #U2 #H + elim (sta_inv_lref1 … H) -H * #K0 #W0 #V0 #HLK0 #HWV0 #HV0U2 + lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct + lapply (IHWV … HWV0) -IHWV -HWV0 #H destruct + >(lift_mono … HWU1 … HV0U2) -W -U1 // +| #I #L #V #T #U1 #_ #IHTU1 #X #H + elim (sta_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1/ +| #L #V #T #U1 #_ #IHTU1 #X #H + elim (sta_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1/ +| #L #W #T #U1 #_ #IHTU1 #U2 #H + lapply (sta_inv_cast1 … H) -H /2 width=1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/top/lenv_top.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/top/lenv_top.etc new file mode 100644 index 000000000..ab90cebe7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/top/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 𝟙 break term 46 T2 )" + non associative with precedence 45 + for @{ 'RTop $T1 $T2 }. + +include "basic_2/grammar/lenv_px.ma". + +(* POINTWISE EXTENSION OF TOP RELATION FOR TERMS ****************************) + +definition ttop: relation term ≝ λT1,T2. True. + +definition ltop: relation lenv ≝ lpx ttop. + +interpretation + "top reduction (environment)" + 'RTop L1 L2 = (ltop L1 L2). + +(* Basic properties *********************************************************) + +lemma ltop_refl: reflexive … ltop. +/2 width=1/ qed. + +lemma ltop_sym: symmetric … ltop. +/2 width=1/ qed. + +lemma ltop_trans: transitive … ltop. +/2 width=3/ qed. + +lemma ltop_append: ∀K1,K2. K1 𝟙 K2 → ∀L1,L2. L1 𝟙 L2 → L1 @@ K1 𝟙 L2 @@ K2. +/2 width=1/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma ltop_inv_atom1: ∀L2. ⋆ 𝟙 L2 → L2 = ⋆. +/2 width=2 by lpx_inv_atom1/ qed-. + +lemma ltop_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 𝟙 L2 → + ∃∃K2,V2. K1 𝟙 K2 & L2 = K2. ⓑ{I} V2. +#K1 #I #V1 #L2 #H +elim (lpx_inv_pair1 … H) -H /2 width=4/ +qed-. + +lemma ltop_inv_atom2: ∀L1. L1 𝟙 ⋆ → L1 = ⋆. +/2 width=2 by lpx_inv_atom2/ qed-. + +lemma ltop_inv_pair2: ∀L1,K2,I,V2. L1 𝟙 K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 𝟙 K2 & L1 = K1. ⓑ{I} V1. +#L1 #K2 #I #V2 #H +elim (lpx_inv_pair2 … H) -H /2 width=4/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma ltop_fwd_length: ∀L1,L2. L1 𝟙 L2 → |L1| = |L2|. +/2 width=2 by lpx_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma new file mode 100644 index 000000000..7489da188 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma @@ -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 *) +(* *) +(**************************************************************************) + +(* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES + * Suggested invocation to start formal specifications with: + * - Patience on me to gain peace and perfection! - + *) + +include "ground_2/star.ma". +include "basic_2/notation.ma". + +(* ATOMIC ARITY *************************************************************) + +inductive aarity: Type[0] ≝ + | AAtom: aarity (* atomic aarity construction *) + | APair: aarity → aarity → aarity (* binary aarity construction *) +. + +interpretation "aarity construction (atomic)" + 'Item0 = AAtom. + +interpretation "aarity construction (binary)" + 'SnItem2 A1 A2 = (APair A1 A2). + +(* Basic inversion lemmas ***************************************************) + +lemma discr_apair_xy_x: ∀A,B. ②B. A = B → ⊥. +#A #B elim B -B +[ #H destruct +| #Y #X #IHY #_ #H destruct + -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *) + /2 width=1/ +] +qed-. + +lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → ⊥. +#B #A elim A -A +[ #H destruct +| #Y #X #_ #IHX #H destruct + -H (**) (* destruct: the destucted equality is not erased *) + /2 width=1/ +] +qed-. + +(* Basic properties *********************************************************) + +lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2). +#A1 elim A1 -A1 +[ #A2 elim A2 -A2 /2 width=1/ + #B2 #A2 #_ #_ @or_intror #H destruct +| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2 + [ -IHB1 -IHA1 @or_intror #H destruct + | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1 + [ #H destruct elim (IHA1 A2) -IHA1 + [ #H destruct /2 width=1/ + | #HA12 @or_intror #H destruct /2 width=1/ + ] + | -IHA1 #HB12 @or_intror #H destruct /2 width=1/ + ] + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_shift.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_shift.ma new file mode 100644 index 000000000..bbdc8e7d0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_shift.ma @@ -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/lenv_append.ma". + +(* SHIFT OF A CLOSURE *******************************************************) + +let rec shift L T on L ≝ match L with +[ LAtom ⇒ T +| LPair L I V ⇒ shift L (-ⓑ{I} V. T) +]. + +interpretation "shift (closure)" 'Append L T = (shift L T). + +(* Basic properties *********************************************************) + +lemma shift_append_assoc: ∀L,K. ∀T:term. (L @@ K) @@ T = L @@ K @@ T. +#L #K elim K -K // normalize // +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma shift_inj: ∀L1,L2. ∀T1,T2:term. L1 @@ T1 = L2 @@ T2 → |L1| = |L2| → + L1 = L2 ∧ T1 = T2. +#L1 elim L1 -L1 +[ * normalize /2 width=1/ + #L2 #I2 #V2 #T1 #T2 #_ append_length in H2; #H + elim (plus_xySz_x_false … H) +| #K1 #I1 #V1 #IH * normalize + [ #L1 #L2 #H1 #H2 destruct + normalize in H2; >append_length in H2; #H + elim (plus_xySz_x_false … (sym_eq … H)) + | #K2 #I2 #V2 #L1 #L2 #H1 #H2 + elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *) + elim (IH … H1 ?) -IH -H1 // -H2 /2 width=1/ + ] +] +qed-. + +lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆. +#L #K #H +elim (append_inj_dx … (⋆) … H ?) // +qed-. + +lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V. +#I #L #K #V #H +elim (append_inj_dx … (⋆.ⓑ{I}V) … H ?) // +qed-. + +lemma length_inv_pos_dx_append: ∀d,L. |L| = d + 1 → + ∃∃I,K,V. |K| = d & L = ⋆.ⓑ{I}V @@ K. +#d @(nat_ind_plus … d) -d +[ #L #H + elim (length_inv_pos_dx … H) -H #I #K #V #H + >(length_inv_zero_dx … H) -H #H destruct + @ex2_3_intro [4: /2 width=2/ |5: // |1,2,3: skip ] (**) (* /3/ does not work *) +| #d #IHd #L #H + elim (length_inv_pos_dx … H) -H #I #K #V #H + elim (IHd … H) -IHd -H #I0 #K0 #V0 #H1 #H2 #H3 destruct + @(ex2_3_intro … (K0.ⓑ{I}V)) // +] +qed-. + +(* Basic_eliminators ********************************************************) + +fact lenv_ind_dx_aux: ∀R:predicate lenv. R ⋆ → + (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) → + ∀d,L. |L| = d → R L. +#R #Hatom #Hpair #d @(nat_ind_plus … d) -d +[ #L #H >(length_inv_zero_dx … H) -H // +| #d #IH #L #H + elim (length_inv_pos_dx_append … H) -H #I #K #V #H1 #H2 destruct /3 width=1/ +] +qed-. + +lemma lenv_ind_dx: ∀R:predicate lenv. R ⋆ → + (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) → + ∀L. R L. +/3 width=2 by lenv_ind_dx_aux/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma length_inv_pos_sn_append: ∀d,L. 1 + d = |L| → + ∃∃I,K,V. d = |K| & L = ⋆. ⓑ{I}V @@ K. +#d >commutative_plus @(nat_ind_plus … d) -d +[ #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct + >(length_inv_zero_sn … H1) -K + @(ex2_3_intro … (⋆)) // (**) (* explicit constructor *) +| #d #IHd #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct + >H1 in IHd; -H1 #IHd + elim (IHd K ?) -IHd // #J #L #W #H1 #H2 destruct + @(ex2_3_intro … (L.ⓑ{I}V)) // (**) (* explicit constructor *) + >append_length /2 width=1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma new file mode 100644 index 000000000..faf6de02d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma @@ -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/grammar/lenv.ma". + +(* LENGTH OF A LOCAL ENVIRONMENT ********************************************) + +let rec length L ≝ match L with +[ LAtom ⇒ 0 +| LPair L _ _ ⇒ length L + 1 +]. + +interpretation "length (local environment)" 'card L = (length L). + +(* Basic inversion lemmas ***************************************************) + +lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆. +* // #L #I #V normalize (lpx_inv_atom1 … H) -X /2 width=3/ +| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H + elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct + elim (IHK01 … HK02) -K0 #K #HK1 #HK2 + elim (HR … HV01 … HV02) -HR -V0 /3 width=5/ +] +qed. + +lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2. +#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/ +qed. + +lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L → + ∀L2. lpx R L L2 → lpx (TC … R) L1 L2. +#R #L1 #L #H elim H -L /2 width=1/ +#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H +elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ +qed. + +lemma TC_lpx_pair_dx: ∀R. reflexive ? R → + ∀I,K,V1,V2. TC … R V1 V2 → + TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2). +#R #HR #I #K #V1 #V2 #H elim H -V2 +/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *) +qed. + +lemma TC_lpx_pair_sn: ∀R. reflexive ? R → + ∀I,V,K1,K2. TC … (lpx R) K1 K2 → + TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V). +#R #HR #I #V #K1 #K2 #H elim H -K2 +/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *) +qed. + +lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2. +#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/ +qed. + +lemma lpx_inv_TC: ∀R. reflexive ? R → + ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2. +#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/ +qed. + +lemma lpx_append: ∀R,K1,K2. lpx R K1 K2 → ∀L1,L2. lpx R L1 L2 → + lpx R (L1 @@ K1) (L2 @@ K2). +#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_px_bi.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_px_bi.ma new file mode 100644 index 000000000..931d075ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_px_bi.ma @@ -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 ≝ +| lpx_bi_stom: lpx_bi R (⋆) (⋆) +| lpx_bi_pair: ∀I,K1,K2,V1,V2. + lpx_bi R K1 K2 → R K1 V1 K2 V2 → + lpx_bi R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) +. + +(* Basic inversion lemmas ***************************************************) + +fact lpx_bi_inv_atom1_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L1 = ⋆ → L2 = ⋆. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_bi_inv_atom1: ∀R,L2. lpx_bi R (⋆) L2 → L2 = ⋆. +/2 width=4 by lpx_bi_inv_atom1_aux/ qed-. + +fact lpx_bi_inv_pair1_aux: ∀R,L1,L2. lpx_bi R L1 L2 → + ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. lpx_bi R K1 K2 & + R K1 V1 K2 V2 & L2 = K2. ⓑ{I} V2. +#R #L1 #L2 * -L1 -L2 +[ #J #K1 #V1 #H destruct +| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/ +] +qed-. + +lemma lpx_bi_inv_pair1: ∀R,I,K1,V1,L2. lpx_bi R (K1. ⓑ{I} V1) L2 → + ∃∃K2,V2. lpx_bi R K1 K2 & R K1 V1 K2 V2 & + L2 = K2. ⓑ{I} V2. +/2 width=3 by lpx_bi_inv_pair1_aux/ qed-. + +fact lpx_bi_inv_atom2_aux: ∀R,L1,L2. lpx_bi R L1 L2 → L2 = ⋆ → L1 = ⋆. +#R #L1 #L2 * -L1 -L2 +[ // +| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct +] +qed-. + +lemma lpx_bi_inv_atom2: ∀R,L1. lpx_bi R L1 (⋆) → L1 = ⋆. +/2 width=4 by lpx_bi_inv_atom2_aux/ qed-. + +fact lpx_bi_inv_pair2_aux: ∀R,L1,L2. lpx_bi R L1 L2 → + ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & + L1 = K1. ⓑ{I} V1. +#R #L1 #L2 * -L1 -L2 +[ #J #K2 #V2 #H destruct +| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/ +] +qed-. + +lemma lpx_bi_inv_pair2: ∀R,I,L1,K2,V2. lpx_bi R L1 (K2. ⓑ{I} V2) → + ∃∃K1,V1. lpx_bi R K1 K2 & R K1 V1 K2 V2 & + L1 = K1. ⓑ{I} V1. +/2 width=3 by lpx_bi_inv_pair2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lpx_bi_fwd_length: ∀R,L1,L2. lpx_bi R L1 L2 → |L1| = |L2|. +#R #L1 #L2 #H elim H -L1 -L2 normalize // +qed-. + +(* Basic properties *********************************************************) + +lemma lpx_bi_refl: ∀R. bi_reflexive ? ? R → reflexive … (lpx_bi R). +#R #HR #L elim L -L // /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_weight.ma new file mode 100644 index 000000000..59e2e6172 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_weight.ma @@ -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/grammar/term_weight.ma". +include "basic_2/grammar/lenv.ma". + +(* WEIGHT OF A LOCAL ENVIRONMENT ********************************************) + +let rec lw L ≝ match L with +[ LAtom ⇒ 0 +| LPair L _ V ⇒ lw L + #{V} +]. + +interpretation "weight (local environment)" 'Weight L = (lw L). + +(* Basic properties *********************************************************) + +lemma lw_pair: ∀I,L,V. #{L} < #{(L.ⓑ{I}V)}. +/3 width=1/ qed. + +(* Basic eliminators ********************************************************) + +axiom lw_ind: ∀R:predicate lenv. + (∀L2. (∀L1. #{L1} < #{L2} → R L1) → R L2) → + ∀L. R L. + +(* Basic_1: removed theorems 2: clt_cong clt_head clt_thead *) +(* Basic_1: note: clt_thead should be renamed clt_ctail *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma new file mode 100644 index 000000000..5ccb5e4ac --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma @@ -0,0 +1,135 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/item.ma". + +(* TERMS ********************************************************************) + +(* terms *) +inductive term: Type[0] ≝ + | TAtom: item0 → term (* atomic item construction *) + | TPair: item2 → term → term → term (* binary item construction *) +. + +interpretation "term construction (atomic)" + 'Item0 I = (TAtom I). + +interpretation "term construction (binary)" + 'SnItem2 I T1 T2 = (TPair I T1 T2). + +interpretation "term binding construction (binary)" + 'SnBind2 a I T1 T2 = (TPair (Bind2 a I) T1 T2). + +interpretation "term positive binding construction (binary)" + 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2). + +interpretation "term negative binding construction (binary)" + 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2). + +interpretation "term flat construction (binary)" + 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2). + +interpretation "sort (term)" + 'Star k = (TAtom (Sort k)). + +interpretation "local reference (term)" + 'LRef i = (TAtom (LRef i)). + +interpretation "global reference (term)" + 'GRef p = (TAtom (GRef p)). + +interpretation "abbreviation (term)" + 'SnAbbr a T1 T2 = (TPair (Bind2 a Abbr) T1 T2). + +interpretation "positive abbreviation (term)" + 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2). + +interpretation "negative abbreviation (term)" + 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2). + +interpretation "abstraction (term)" + 'SnAbst a T1 T2 = (TPair (Bind2 a Abst) T1 T2). + +interpretation "positive abstraction (term)" + 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2). + +interpretation "negative abstraction (term)" + 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2). + +interpretation "application (term)" + 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2). + +interpretation "native type annotation (term)" + 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2). + +(* Basic properties *********************************************************) + +(* Basic_1: was: term_dec *) +axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2). + +(* Basic inversion lemmas ***************************************************) + +lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → ⊥. +#I #T #V elim V -V +[ #J #H destruct +| #J #W #U #IHW #_ #H destruct + -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *) + /2 width=1/ +] +qed-. + +(* Basic_1: was: thead_x_y_y *) +lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → ⊥. +#I #V #T elim T -T +[ #J #H destruct +| #J #W #U #_ #IHU #H destruct + -H (**) (* destruct: the destucted equality is not erased *) + /2 width=1/ +] +qed-. + +lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2. + (②{I} V1. T1 = ②{I} V2. T2 → ⊥) → + (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)). +#I #V1 #T1 #V2 #T2 #H +elim (term_eq_dec V1 V2) /3 width=1/ #HV12 destruct +@or_intror @conj // #HT12 destruct /2 width=1/ +qed-. + +lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2. + (②{I} V1. T1 = ②{I} V2. T2 → ⊥) → + (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)). +#I #V1 #T1 #V2 #T2 #H +elim (term_eq_dec T1 T2) /3 width=1/ #HT12 destruct +@or_intror @conj // #HT12 destruct /2 width=1/ +qed-. + +lemma eq_false_inv_beta: ∀a,V1,V2,W1,W2,T1,T2. + (ⓐV1. ⓛ{a}W1. T1 = ⓐV2. ⓛ{a}W2 .T2 → ⊥) → + (W1 = W2 → ⊥) ∨ + (W1 = W2 ∧ (ⓓ{a}V1. T1 = ⓓ{a}V2. T2 → ⊥)). +#a #V1 #V2 #W1 #W2 #T1 #T2 #H +elim (eq_false_inv_tpair_sn … H) -H +[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/ + #H destruct @or_intror @conj // #H destruct /2 width=1/ +| * #H1 #H2 destruct + elim (eq_false_inv_tpair_sn … H2) -H2 /3 width=1/ + * #H #HT12 destruct + @or_intror @conj // #H destruct /2 width=1/ +] +qed. + +(* Basic_1: removed theorems 3: + not_void_abst not_abbr_void not_abst_void +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/term_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/term_simple.ma new file mode 100644 index 000000000..328dc55a6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/term_simple.ma @@ -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/grammar/term.ma". + +(* SIMPLE (NEUTRAL) TERMS ***************************************************) + +inductive simple: predicate term ≝ + | simple_atom: ∀I. simple (⓪{I}) + | simple_flat: ∀I,V,T. simple (ⓕ{I} V. T) +. + +interpretation "simple (term)" 'Simple T = (simple T). + +(* Basic inversion lemmas ***************************************************) +(* +lemma mt: ∀R1,R2:Prop. (R1 → R2) → (R2 → ⊥) → R1 → ⊥. +/3 width=1/ qed. +*) +fact simple_inv_bind_aux: ∀T. 𝐒⦃T⦄ → ∀a,J,W,U. T = ⓑ{a,J} W. U → ⊥. +#T * -T +[ #I #a #J #W #U #H destruct +| #I #V #T #a #J #W #U #H destruct +] +qed. + +lemma simple_inv_bind: ∀a,I,V,T. 𝐒⦃ⓑ{a,I} V. T⦄ → ⊥. +/2 width=7/ qed-. (**) (* auto fails if mt is enabled *) + +lemma simple_inv_pair: ∀I,V,T. 𝐒⦃②{I}V.T⦄ → ∃J. I = Flat2 J. +* /2 width=2/ #a #I #V #T #H +elim (simple_inv_bind … H) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/term_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/term_vector.ma new file mode 100644 index 000000000..7169d1af4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/term_vector.ma @@ -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 "ground_2/list.ma". +include "basic_2/grammar/term_simple.ma". + +(* TERMS ********************************************************************) + +let rec applv Vs T on Vs ≝ + match Vs with + [ nil ⇒ T + | cons hd tl ⇒ ⓐhd. (applv tl T) + ]. + +interpretation "application o vevtor (term)" + 'SnApplV Vs T = (applv Vs T). + +(* properties concerning simple terms ***************************************) + +lemma applv_simple: ∀T,Vs. 𝐒⦃T⦄ → 𝐒⦃ⒶVs.T⦄. +#T * // +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/term_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/term_weight.ma new file mode 100644 index 000000000..d8f39a3a1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/term_weight.ma @@ -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/grammar/term.ma". + +(* WEIGHT OF A TERM *********************************************************) + +let rec tw T ≝ match T with +[ TAtom _ ⇒ 1 +| TPair _ V T ⇒ tw V + tw T + 1 +]. + +interpretation "weight (term)" 'Weight T = (tw T). + +(* Basic properties *********************************************************) + +(* Basic_1: was: tweight_lt *) +lemma tw_pos: ∀T. 1 ≤ #{T}. +#T elim T -T // +qed. + +(* Basic eliminators ********************************************************) + +axiom tw_ind: ∀R:predicate term. + (∀T2. (∀T1. #{T1} < #{T2} → R T1) → R T2) → + ∀T. R T. + +(* Basic_1: removed theorems 11: + wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O + weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind + removed local theorems 1: q_ind +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/tshf.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/tshf.ma new file mode 100644 index 000000000..a8873c18b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/tshf.ma @@ -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/grammar/term_simple.ma". + +(* SAME HEAD TERM FORMS *****************************************************) + +inductive tshf: relation term ≝ + | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I}) + | tshf_abbr: ∀V1,V2,T1,T2. tshf (-ⓓV1. T1) (-ⓓV2. T2) + | tshf_abst: ∀a,V1,V2,T1,T2. tshf (ⓛ{a}V1. T1) (ⓛ{a}V2. T2) + | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄ → + tshf (ⓐV1. T1) (ⓐV2. T2) +. + +interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2). + +(* Basic properties *********************************************************) + +lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1. +#T1 #T2 #H elim H -T1 -T2 /2 width=1/ +qed. + +lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2. +#T1 #T2 #H elim H -T1 -T2 // /2 width=1/ +qed. + +lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1. +/3 width=2/ qed. + +lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. +#T1 #T2 #H elim H -T1 -T2 // +[ #V1 #V2 #T1 #T2 #H + elim (simple_inv_bind … H) +| #a #V1 #V2 #T1 #T2 #H + elim (simple_inv_bind … H) +] +qed. (**) (* remove from index *) + +lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. +/3 width=3/ qed-. + +(* Basic inversion lemmas ***************************************************) + +fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀a,I,W1,U1. T1 = ⓑ{a,I}W1.U1 → + ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 & + (Bind2 a I = Bind2 false Abbr ∨ I = Abst). +#T1 #T2 * -T1 -T2 +[ #J #a #I #W1 #U1 #H destruct +| #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/ +| #b #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/ +| #V1 #V2 #T1 #T2 #_ #_ #_ #a #I #W1 #U1 #H destruct +] +qed. + +lemma tshf_inv_bind1: ∀a,I,W1,U1,T2. ⓑ{a,I}W1.U1 ≈ T2 → + ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 & + (Bind2 a I = Bind2 false Abbr ∨ I = Abst). +/2 width=5/ qed-. + +fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 → + ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ & + I = Appl & T2 = ⓐW2. U2. +#T1 #T2 * -T1 -T2 +[ #J #I #W1 #U1 #H destruct +| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct +| #a #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct +| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/ +] +qed. + +lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 → + ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ & + I = Appl & T2 = ⓐW2. U2. +/2 width=4/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc.ma new file mode 100644 index 000000000..78a9b4987 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc.ma @@ -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/grammar/term_simple.ma". + +(* SAME TOP TERM CONSTRUCTOR ************************************************) + +inductive tstc: relation term ≝ + | tstc_atom: ∀I. tstc (⓪{I}) (⓪{I}) + | tstc_pair: ∀I,V1,V2,T1,T2. tstc (②{I} V1. T1) (②{I} V2. T2) +. + +interpretation "same top constructor (term)" 'Iso T1 T2 = (tstc T1 T2). + +(* Basic inversion lemmas ***************************************************) + +fact tstc_inv_atom1_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T1 = ⓪{I} → T2 = ⓪{I}. +#T1 #T2 * -T1 -T2 // +#J #V1 #V2 #T1 #T2 #I #H destruct +qed. + +(* Basic_1: was: iso_gen_sort iso_gen_lref *) +lemma tstc_inv_atom1: ∀I,T2. ⓪{I} ≃ T2 → T2 = ⓪{I}. +/2 width=3/ qed-. + +fact tstc_inv_pair1_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W1,U1. T1 = ②{I}W1.U1 → + ∃∃W2,U2. T2 = ②{I}W2. U2. +#T1 #T2 * -T1 -T2 +[ #J #I #W1 #U1 #H destruct +| #J #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/ +] +qed. + +(* Basic_1: was: iso_gen_head *) +lemma tstc_inv_pair1: ∀I,W1,U1,T2. ②{I}W1.U1 ≃ T2 → + ∃∃W2,U2. T2 = ②{I}W2. U2. +/2 width=5/ qed-. + +fact tstc_inv_atom2_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T2 = ⓪{I} → T1 = ⓪{I}. +#T1 #T2 * -T1 -T2 // +#J #V1 #V2 #T1 #T2 #I #H destruct +qed. + +lemma tstc_inv_atom2: ∀I,T1. T1 ≃ ⓪{I} → T1 = ⓪{I}. +/2 width=3/ qed-. + +fact tstc_inv_pair2_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W2,U2. T2 = ②{I}W2.U2 → + ∃∃W1,U1. T1 = ②{I}W1. U1. +#T1 #T2 * -T1 -T2 +[ #J #I #W2 #U2 #H destruct +| #J #V1 #V2 #T1 #T2 #I #W2 #U2 #H destruct /2 width=3/ +] +qed. + +lemma tstc_inv_pair2: ∀I,T1,W2,U2. T1 ≃ ②{I}W2.U2 → + ∃∃W1,U1. T1 = ②{I}W1. U1. +/2 width=5/ qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: iso_refl *) +lemma tstc_refl: ∀T. T ≃ T. +#T elim T -T // +qed. + +lemma tstc_sym: ∀T1,T2. T1 ≃ T2 → T2 ≃ T1. +#T1 #T2 #H elim H -T1 -T2 // +qed. + +lemma tstc_dec: ∀T1,T2. Decidable (T1 ≃ T2). +* #I1 [2: #V1 #T1 ] * #I2 [2,4: #V2 #T2 ] +[ elim (item2_eq_dec I1 I2) #HI12 + [ destruct /2 width=1/ + | @or_intror #H + elim (tstc_inv_pair1 … H) -H #V #T #H destruct /2 width=1/ + ] +| @or_intror #H + lapply (tstc_inv_atom1 … H) -H #H destruct +| @or_intror #H + lapply (tstc_inv_atom2 … H) -H #H destruct +| elim (item0_eq_dec I1 I2) #HI12 + [ destruct /2 width=1/ + | @or_intror #H + lapply (tstc_inv_atom2 … H) -H #H destruct /2 width=1/ + ] +] +qed. + +lemma simple_tstc_repl_dx: ∀T1,T2. T1 ≃ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. +#T1 #T2 * -T1 -T2 // +#I #V1 #V2 #T1 #T2 #H +elim (simple_inv_pair … H) -H #J #H destruct // +qed. (**) (* remove from index *) + +lemma simple_tstc_repl_sn: ∀T1,T2. T1 ≃ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. +/3 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc_tstc.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc_tstc.ma new file mode 100644 index 000000000..df6fe3729 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc_tstc.ma @@ -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/grammar/tstc.ma". + +(* SAME TOP TERM CONSTRUCTOR ************************************************) + +(* Main properties **********************************************************) + +(* Basic_1: was: iso_trans *) +theorem tstc_trans: ∀T1,T. T1 ≃ T → ∀T2. T ≃ T2 → T1 ≃ T2. +#T1 #T * -T1 -T // +#I #V1 #V #T1 #T #X #H +elim (tstc_inv_pair1 … H) -H #V2 #T2 #H destruct // +qed. + +theorem tstc_canc_sn: ∀T,T1. T ≃ T1 → ∀T2. T ≃ T2 → T1 ≃ T2. +/3 width=3/ qed. + +theorem tstc_canc_dx: ∀T1,T. T1 ≃ T → ∀T2. T2 ≃ T → T1 ≃ T2. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc_vector.ma new file mode 100644 index 000000000..1e35292ef --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/tstc_vector.ma @@ -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/grammar/term_vector.ma". +include "basic_2/grammar/tstc.ma". + +(* SAME TOP TERM CONSTRUCTOR ************************************************) + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *) +lemma tstc_inv_bind_appls_simple: ∀a,I,Vs,V2,T1,T2. ⒶVs.T1 ≃ ⓑ{a,I} V2. T2 → + 𝐒⦃T1⦄ → ⊥. +#a #I #Vs #V2 #T1 #T2 #H +elim (tstc_inv_pair2 … H) -H #V0 #T0 +elim Vs -Vs normalize +[ #H destruct #H + @(simple_inv_bind … H) +| #V #Vs #_ #H destruct +] +qed. + diff --git a/matita/matita/contribs/lambdadelta/basic_2/names.txt b/matita/matita/contribs/lambdadelta/basic_2/names.txt new file mode 100644 index 000000000..6c34328b4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/names.txt @@ -0,0 +1,48 @@ +NAMING CONVENTIONS FOR METAVARIABLES + +A,B : arity +C,D : candidate of reducibility +E,F : RTM environment +G : global environment +H : reserved: transient premise +IH : reserved: inductive premise +I,J : item +K,L : local environment +M,N : reserved: future use +O,P,Q : +R : generic predicate (relation) +S : RTM stack +T,U,V,W: term +X,Y,Z : reserved: transient objet denoted by a capital letter + +a,b : binder polarity +c : reserved: future use (lambda_delta 3) +d : relocation depth +e : relocation height +f : +g : sort degree parameter +h : sort hierarchy parameter +i,j : local reference position index (de Bruijn's) +k : sort index +l : term degree +m,n : reserved: future use +o : +p,q : global reference position +r,s : +t,u : local reference position level (de Bruijn's) +v,w : +x,y,z : reserved: transient objet denoted by a small letter + +NAMING CONVENTIONS FOR CONSTRUCTORS + +0: atomic +2: binary + +A: application to vector + +a: application +b: binder +d: abbreviation +f: flat +l: abstraction +n: native type annotation diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation.ma b/matita/matita/contribs/lambdadelta/basic_2/notation.ma new file mode 100644 index 000000000..3ff8f21e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/notation.ma @@ -0,0 +1,475 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +(* Grammar ******************************************************************) + +notation "⓪" + non associative with precedence 90 + for @{ 'Item0 }. + +notation "hvbox( ⓪ { term 46 I } )" + non associative with precedence 90 + for @{ 'Item0 $I }. + +notation "⋆" + non associative with precedence 90 + for @{ 'Star }. + +notation "hvbox( ⋆ term 90 k )" + non associative with precedence 90 + for @{ 'Star $k }. + +notation "hvbox( # term 90 i )" + non associative with precedence 90 + for @{ 'LRef $i }. + +notation "hvbox( § term 90 p )" + non associative with precedence 90 + for @{ 'GRef $p }. + +notation "hvbox( ② term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnItem2 $T1 $T }. + +notation "hvbox( ② { term 46 I } break term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnItem2 $I $T1 $T }. + +notation "hvbox( ⓑ { term 46 a , term 46 I } break term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnBind2 $a $I $T1 $T }. + +notation "hvbox( + ⓑ { term 46 I } break term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnBind2Pos $I $T1 $T }. + +notation "hvbox( - ⓑ { term 46 I } break term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnBind2Neg $I $T1 $T }. + +notation "hvbox( ⓕ { term 46 I } break term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnFlat2 $I $T1 $T }. + +notation "hvbox( ⓓ { term 46 a } term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAbbr $a $T1 $T2 }. + +notation "hvbox( + ⓓ term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAbbrPos $T1 $T2 }. + +notation "hvbox( - ⓓ term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAbbrNeg $T1 $T2 }. + +notation "hvbox( ⓛ { term 46 a } term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAbst $a $T1 $T2 }. + +notation "hvbox( + ⓛ term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAbstPos $T1 $T2 }. + +notation "hvbox( - ⓛ term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAbstNeg $T1 $T2 }. + +notation "hvbox( ⓐ term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnAppl $T1 $T2 }. + +notation "hvbox( ⓝ term 55 T1 . break term 55 T2 )" + non associative with precedence 55 + for @{ 'SnCast $T1 $T2 }. + +notation "hvbox( Ⓐ term 55 T1 . break term 55 T )" + non associative with precedence 55 + for @{ 'SnApplV $T1 $T }. + +notation > "hvbox( T . break ②{ term 46 I } break term 47 T1 )" + non associative with precedence 46 + for @{ 'DxBind2 $T $I $T1 }. + +notation "hvbox( T . break ⓑ { term 46 I } break term 48 T1 )" + non associative with precedence 47 + for @{ 'DxBind2 $T $I $T1 }. + +notation "hvbox( T1 . break ⓓ T2 )" + left associative with precedence 48 + for @{ 'DxAbbr $T1 $T2 }. + +notation "hvbox( T1 . break ⓛ T2 )" + left associative with precedence 49 + for @{ 'DxAbst $T1 $T2 }. + +notation "hvbox( T . break ④ { term 46 I } break { term 46 T1 , break term 46 T2 , break term 46 T3 } )" + non associative with precedence 50 + for @{ 'DxItem4 $T $I $T1 $T2 $T3 }. + +notation "hvbox( # { term 46 x } )" + non associative with precedence 90 + for @{ 'Weight $x }. + +notation "hvbox( # { term 46 x , break term 46 y } )" + non associative with precedence 90 + for @{ 'Weight $x $y }. + +notation "hvbox( 𝐒 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'Simple $T }. + +notation "hvbox( L ⊢ break term 46 T1 ≈ break term 46 T2 )" + non associative with precedence 45 + for @{ 'Hom $L $T1 $T2 }. + +notation "hvbox( T1 ≃ break term 46 T2 )" + non associative with precedence 45 + for @{ 'Iso $T1 $T2 }. + +(* Substitution *************************************************************) + +notation "hvbox( ⇧ [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'RLift $d $e $T1 $T2 }. + +notation "hvbox( T1 break ≼ [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'SubEq $T1 $d $e $T2 }. + +notation "hvbox( ≽ [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'SubEqBottom $d $e $T2 }. + +notation "hvbox( ⇩ [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'RDrop $e $L1 $L2 }. + +notation "hvbox( ⇩ [ term 46 d , break term 46 e ] break term 46 L1 ≡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'RDrop $d $e $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'RestSupTerm $L1 $T1 $L2 $T2 }. + +notation "hvbox( L ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )" + non associative with precedence 45 + for @{ 'ICM $L $T $k }. + +notation "hvbox( L ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubst $L $T1 $d $e $T2 }. + +(* Unfold *******************************************************************) + +notation "hvbox( @ ⦃ term 46 T1 , break term 46 f ⦄ ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'RAt $T1 $f $T2 }. + +notation "hvbox( T1 ▭ break term 46 T2 ≡ break term 46 T )" + non associative with precedence 45 + for @{ 'RMinus $T1 $T2 $T }. + +notation "hvbox( ⇧ * [ term 46 e ] break term 46 T1 ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'RLiftStar $e $T1 $T2 }. + +notation "hvbox( ⇩ * [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'RDropStar $e $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'RestSupTermPlus $L1 $T1 $L2 $T2 }. + +notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'RestSupTermStar $L1 $T1 $L2 $T2 }. + +notation "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $T1 $d $e $T2 }. + +notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $L $T1 $d $e $T2 }. + +notation "hvbox( L ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStarAlt $L $T1 $d $e $T2 }. + +notation "hvbox( T1 break ⊢ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStarSn $T1 $d $e $T2 }. + +notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }. + +notation "hvbox( ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'TSubst $T1 $d $e $T2 }. + +notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'TSubst $L $T1 $d $e $T2 }. + +notation "hvbox( ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'TSubstAlt $T1 $d $e $T2 }. + +notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'TSubstAlt $L $T1 $d $e $T2 }. + +(* Static typing ************************************************************) + +notation "hvbox( L ⊢ break term 46 T ⁝ break term 46 A )" + non associative with precedence 45 + for @{ 'AtomicArity $L $T $A }. + +notation "hvbox( T1 ⁝ ⊑ break term 46 T2 )" + non associative with precedence 45 + for @{ 'CrSubEqA $T1 $T2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )" + non associative with precedence 45 + for @{ 'BinaryArity $h $L $T $A }. + +notation "hvbox( h ⊢ break term 46 L1 ÷ ⊑ break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqB $h $L1 $L2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • break [ term 46 g , break term 46 l ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'StaticType $h $g $l $L $T1 $T2 }. + +notation "hvbox( h ⊢ break term 46 L1 • ⊑ [ term 46 g ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqS $h $g $L1 $L2 }. + +(* Unwind *******************************************************************) + +notation "hvbox( L1 ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'Unwind $L1 $T $L2 }. + +(* Reducibility *************************************************************) + +notation "hvbox( L ⊢ break 𝐑 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'Reducible $L $T }. + +notation "hvbox( L ⊢ break 𝐈 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'NotReducible $L $T }. + +notation "hvbox( L ⊢ break 𝐍 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'Normal $L $T }. + +(* this might be removed *) +notation "hvbox( 𝐇𝐑 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'HdReducible $T }. + +(* this might be removed *) +notation "hvbox( L ⊢ break 𝐇𝐑 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'HdReducible $L $T }. + +(* this might be removed *) +notation "hvbox( 𝐇𝐈 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'NotHdReducible $T }. + +(* this might be removed *) +notation "hvbox( L ⊢ break 𝐇𝐈 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'NotHdReducible $L $T }. + +(* this might be removed *) +notation "hvbox( 𝐇𝐍 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'HdNormal $T }. + +(* this might be removed *) +notation "hvbox( L ⊢ break 𝐇𝐍 ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'HdNormal $L $T }. + +notation "hvbox( T1 ➡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'PRed $T1 $T2 }. + +notation "hvbox( L ⊢ break term 46 T1 ➡ break term 46 T2 )" + non associative with precedence 45 + for @{ 'PRed $L $T1 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ➡ break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRed $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }. + +notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRedAlt $L1 $L2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • ➡ break [ term 46 g ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'XPRed $h $g $L $T1 $T2 }. + +(* Computation **************************************************************) + +notation "hvbox( T1 ➡ * break term 46 T2 )" + non associative with precedence 45 + for @{ 'PRedStar $T1 $T2 }. + +notation "hvbox( L ⊢ break term 46 T1 ➡ * break term 46 T2 )" + non associative with precedence 45 + for @{ 'PRedStar $L $T1 $T2 }. + +notation "hvbox( T1 ➡ ➡ * break term 46 T2 )" + non associative with precedence 45 + for @{ 'PRedStarAlt $T1 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ➡ * break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRedStar $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ * break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRedStarAlt $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPRedStarAlt $L1 $T1 $L2 $T2 }. + +notation "hvbox( L ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ Tterm 46 2 ⦄ )" + non associative with precedence 45 + for @{ 'PEval $L $T1 $T2 }. + +notation "hvbox( ⬊ * term 46 T )" + non associative with precedence 45 + for @{ 'SN $T }. + +notation "hvbox( L ⊢ ⬊ * break term 46 T )" + non associative with precedence 45 + for @{ 'SN $L $T }. + +notation "hvbox( L ⊢ ⬊ ⬊ * break term 46 T )" + non associative with precedence 45 + for @{ 'SNAlt $L $T }. + +notation "hvbox( ⦃ term 46 L, break term 46 T ⦄ ϵ break [ term 46 R ] break 〚term 46 A 〛 )" + non associative with precedence 45 + for @{ 'InEInt $R $L $T $A }. + +notation "hvbox( T1 ⊑ break [ term 46 R ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'CrSubEq $T1 $R $T2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • ➡ * break [ term 46 g ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'XPRedStar $h $g $L $T1 $T2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • ⬊ * break [ term 46 g ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'XSN $h $g $L $T }. + +(* Conversion ***************************************************************) + +notation "hvbox( L ⊢ break term 46 T1 ⬌ break term 46 T2 )" + non associative with precedence 45 + for @{ 'PConv $L $T1 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConv $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConvAlt $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }. + +(* Equivalence **************************************************************) + +notation "hvbox( L ⊢ break term 46 T1 ⬌* break term 46 T2 )" + non associative with precedence 45 + for @{ 'PConvStar $L $T1 $T2 }. + +notation "hvbox( h ⊢ break term 46 L1 ⊢ • ⊑ [ term 46 g ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqSE $h $g $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ * break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConvStar $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }. + +notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConvStarAlt $L1 $L2 }. + +notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }. + +(* Dynamic typing ***********************************************************) + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊩ break term 46 T : break [ term 46 g ] )" + non associative with precedence 45 + for @{ 'NativeValid $h $g $L $T }. + +notation "hvbox( h ⊢ break term 46 L1 ⊩ : ⊑ [ term 46 g ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'CrSubEqV $h $g $L1 $L2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeType $h $L $T1 $T2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeAlt $h $L $T1 $T2 }. + +(* Higher order dynamic typing **********************************************) + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeStar $h $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr.ma new file mode 100644 index 000000000..95c01d19f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr.ma @@ -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/reducibility/cpr.ma". +include "basic_2/reducibility/fpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************) + +definition cfpr: lenv → bi_relation lenv term ≝ + λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2. + +interpretation + "context-sensitive parallel reduction (closure)" + 'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2). + +(* Basic properties *********************************************************) + +lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L). +/2 width=1/ qed. + +lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄. +#L1 #L2 #T1 #T2 * /3 width=1/ +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆. +#L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/ +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ → + ∃∃K2,V2. V1 ➡ V2 & + ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ & + L2 = ⋆.ⓑ{I}V2@@K2. +#I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H +elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct +>shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H +elim (tpr_inv_bind1 … H) -H * +[ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/ +| #T0 #_ #_ #H destruct +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_aaa.ma new file mode 100644 index 000000000..4c9b8ac97 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_aaa.ma @@ -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: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → + ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → L2 ⊢ T2 ⁝ A. +#L1 #T1 #A #HT1 #L2 #T2 #H +elim (fpr_inv_all … H) -H +/4 width=5 by aaa_cpr_conf, aaa_ltpr_conf, aaa_ltpss_sn_conf/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_cfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_cfpr.ma new file mode 100644 index 000000000..f442be28b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_cfpr.ma @@ -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: ∀L. bi_confluent … (cfpr L). +#L #L0 #L1 #T0 #T1 * #HL01 #HT01 #L2 #T2 * >HL01 #HL12 #HT02 +elim (cpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2 +elim (cpr_fwd_shift1 … H1) #L0 #T0 #HL10 #H destruct /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_cpr.ma new file mode 100644 index 000000000..a46c9776b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_cpr.ma @@ -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/unfold/ltpss_sn_alt.ma". +include "basic_2/reducibility/cpr_tpss.ma". +include "basic_2/reducibility/cpr_cpr.ma". +include "basic_2/reducibility/cfpr_ltpss.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************) + +(* Advanced properties ******************************************************) + +lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 → + L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄. +#L1 #L #H elim H -L1 -L +[ #L2 #T1 #T2 #HT12 #HL2 + lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct + lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/ +| #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H + elim (ltpss_sn_inv_tpss21 … H ?) -H // append_length >append_length #H + lapply (injective_plus_r … H) -H #H + @(ex3_1_intro … (⋆.ⓑ{I}V@@Y)) append_length H -H >commutative_plus /3 width=1/ +] +qed-. + +lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → + ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2. +#L1 #L2 #T1 #T2 #H +lapply (fpr_cfpr … H) -H #H +elim (cfpr_inv_all … H) -H /2 width=4/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_ltpss.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_ltpss.ma new file mode 100644 index 000000000..424911376 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cfpr_ltpss.ma @@ -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.ma". +include "basic_2/reducibility/cfpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************) + +(* Advanced inversion lemmas ************************************************) + +lemma cfpr_inv_pair1: ∀I,L,K1,L2,V1,T1,T2. L ⊢ ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ → + ∃∃K2,V,V2. V1 ➡ V & L ⊢ V ▶* [0, |L|] V2 & + L.ⓑ{I}V ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ & + L2 = ⋆.ⓑ{I}V2@@K2. +* #L #K1 #L2 #V1 #T1 #T2 * >append_length #H +elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct +>shift_append_assoc >shift_append_assoc normalize in ⊢ (??%%→?); #H +[ elim (cpr_inv_abbr1 … H) -H * + [ #V #V0 #T0 #HV1 #HV0 #HT10 #H destruct /3 width=7/ + | #T0 #_ #_ #H destruct + ] +| elim (cpr_inv_abst1 … H Abst V2) -H + #V #T * #V0 #HV10 #HV0 #HT1 #H destruct + lapply (ltpss_sn_cpr_trans (L.ⓛV0) … 0 (|L|+1) … HT1) -HT1 /2 width=1/ #HT12 + /3 width=7/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cif.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cif.ma new file mode 100644 index 000000000..0ea9d519b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cif.ma @@ -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/reducibility/crf.ma". + +(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************) + +definition cif: lenv → predicate term ≝ λL,T. L ⊢ 𝐑⦃T⦄ → ⊥. + +interpretation "context-sensitive irreducibility (term)" + 'NotReducible L T = (cif L T). + +(* Basic inversion lemmas ***************************************************) + +lemma cif_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐈⦃#i⦄ → ⊥. +/3 width=3/ qed-. + +lemma cif_inv_ri2: ∀I,L,V,T. ri2 I → L ⊢ 𝐈⦃②{I}V.T⦄ → ⊥. +/3 width=1/ qed-. + +lemma cif_inv_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ → + L ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄. +/4 width=1/ qed-. + +lemma cif_inv_bind: ∀a,I,L,V,T. L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ → + ∧∧ L ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I. +#a * [ elim a -a ] +[ #L #V #T #H elim (H ?) -H /3 width=1/ +|*: #L #V #T #H elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=1/ +] +qed-. + +lemma cif_inv_appl: ∀L,V,T. L ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄. +#L #V #T #HVT @and3_intro /3 width=1/ +generalize in match HVT; -HVT elim T -T // +* // #a * #U #T #_ #_ #H elim (H ?) -H /2 width=1/ +qed-. + +lemma cif_inv_flat: ∀I,L,V,T. L ⊢ 𝐈⦃ⓕ{I}V.T⦄ → + ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl. +* #L #V #T #H +[ elim (cif_inv_appl … H) -H /2 width=1/ +| elim (cif_inv_ri2 … H) -H /2 width=1/ +] +qed-. + +(* Basic properties *********************************************************) + +lemma tif_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄. +/2 width=2 by trf_inv_atom/ qed. + +lemma cif_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄. +#a #I #L #V #T #HI #HV #HT #H +elim (crf_inv_ib2 … HI H) -HI -H /2 width=1/ +qed. + +lemma cif_appl: ∀L,V,T. L ⊢ 𝐈⦃V⦄ → L ⊢ 𝐈⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐈⦃ⓐV.T⦄. +#L #V #T #HV #HT #H1 #H2 +elim (crf_inv_appl … H2) -H2 /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cif_append.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cif_append.ma new file mode 100644 index 000000000..45fd178cf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cif_append.ma @@ -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/crf_append.ma". +include "basic_2/reducibility/cif.ma". + +(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************) + +(* Advanved properties ******************************************************) + +lemma cif_labst_last: ∀L,T,W. L ⊢ 𝐈⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄. +/3 width=2 by crf_inv_labst_last/ qed. + +lemma cif_tif: ∀T,W. ⋆ ⊢ 𝐈⦃T⦄ → ⋆.ⓛW ⊢ 𝐈⦃T⦄. +/3 width=2 by crf_inv_trf/ qed. + +(* Advanced inversion lemmas ************************************************) + +lemma cif_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃T⦄. +/3 width=1/ qed-. + +lemma cif_inv_tif: ∀T,W. ⋆.ⓛW ⊢ 𝐈⦃T⦄ → ⋆ ⊢ 𝐈⦃T⦄. +/3 width=1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf.ma new file mode 100644 index 000000000..02bbcf87a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf.ma @@ -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/reducibility/cpr.ma". + +(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************) + +definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …). + +interpretation + "context-sensitive normality (term)" + 'Normal L T = (cnf L T). + +(* Basic inversion lemmas ***************************************************) + +lemma cnf_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄. +#L #V1 #T1 #HVT1 @and3_intro +[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct // +| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct // +| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H + [ elim (lift_total V1 0 1) #V2 #HV12 + lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3/ -HV12 #H destruct + | lapply (H (ⓓ{a}V1.U1) ?) -H /3 width=1/ #H destruct +] +qed-. + +lemma cnf_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥. +#L #V #T #H elim (is_lift_dec T 0 1) +[ * #U #HTU + lapply (H U ?) -H /3 width=3 by cpr_tpr, tpr_zeta/ #H destruct (**) (* auto too slow without trace *) + elim (lift_inv_pair_xy_y … HTU) +| #HT + elim (tps_full (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12 + lapply (H (+ⓓV.T2) ?) -H /3 width=3 by cpr_tpr, tpr_delta/ -HT2 #H destruct /3 width=2/ (**) (* auto too slow without trace *) +] +qed. + +lemma cnf_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥. +#L #V #T #H lapply (H T ?) -H /2 width=1/ #H +@discr_tpair_xy_y // +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: nf2_sort *) +lemma cnf_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄. +#L #k #X #H +>(cpr_inv_sort1 … H) // +qed. + +(* Basic_1: was: nf2_dec *) +axiom cnf_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨ + ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥). + +(* Basic_1: removed theorems 1: nf2_abst_shift *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf_cif.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf_cif.ma new file mode 100644 index 000000000..51ce95aa0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf_cif.ma @@ -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: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2. +#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e +[ // +| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e + elim (cif_inv_delta … HLK ?) // +| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H + elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct + lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H + elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/ +] +qed. + +lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 #HT1 +lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/ +qed. + +lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2. +#T1 #T2 #H elim H -T1 -T2 +[ // +| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H + [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_ + >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // + | elim (cif_inv_ri2 … H) /2 width=1/ + ] +| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H + elim (cif_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H + [ lapply (tps_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 + elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct + lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct + lapply (IHT1 … HT1) -IHT1 #H destruct + lapply (tps_cif_eq … HT2 ?) -HT2 // + | <(tps_inv_refl_SO2 … HT2 ?) -HT2 // + elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/ + ] +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H + elim (cif_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +| #V1 #T1 #T #T2 #_ #_ #_ #L #H + elim (cif_inv_ri2 … H) /2 width=1/ +| #V1 #T1 #T2 #_ #_ #L #H + elim (cif_inv_ri2 … H) /2 width=1/ +] +qed. + +lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2. +#L #T1 #T2 * #T0 #HT10 #HT02 #HT1 +lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/ +qed. + +theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄. +/3 width=3/ qed. + +(* Note: this property is unusual *) +lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥. +#L #T #H elim H -L -T +[ #L #K #V #i #HLK #H + elim (cnf_inv_delta … HLK H) +| #L #V #T #_ #IHV #H + elim (cnf_inv_appl … H) -H /2 width=1/ +| #L #V #T #_ #IHT #H + elim (cnf_inv_appl … H) -H /2 width=1/ +| #I #L #V #T * #H1 #H2 destruct + [ elim (cnf_inv_zeta … H2) + | elim (cnf_inv_tau … H2) + ] +|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct + [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/ + |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/ + ] +| #a #L #V #W #T #H + elim (cnf_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +| #a #L #V #W #T #H + elim (cnf_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +] +qed. + +theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf_lift.ma new file mode 100644 index 000000000..0e1a8551f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cnf_lift.ma @@ -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/reducibility/cpr_lift.ma". +include "basic_2/reducibility/cpr_cpr.ma". +include "basic_2/reducibility/cnf.ma". + +(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************) + +(* Advanced inversion lemmas ************************************************) + +lemma cnf_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐍⦃#i⦄ → ⊥. +#L #K #V #i #HLK #H +elim (lift_total V 0 (i+1)) #W #HVW +lapply (H W ?) -H [ /3 width=6/ ] -HLK #H destruct +elim (lift_inv_lref2_be … HVW ? ?) -HVW // +qed-. + +lemma cnf_inv_abst: ∀a,L,V,T. L ⊢ 𝐍⦃ⓛ{a}V.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄. +#a #L #V1 #T1 #HVT1 @conj +[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct // +| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct // +] +qed-. + +lemma cnf_inv_abbr: ∀L,V,T. L ⊢ 𝐍⦃-ⓓV.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄. +#L #V1 #T1 #HVT1 @conj +[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct // +| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct // +] +qed-. + +(* Advanced properties ******************************************************) + +(* Basic_1: was only: nf2_csort_lref *) +lemma cnf_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍⦃#i⦄. +#L #i #HLK #X #H +elim (cpr_inv_lref1 … H) // * +#K0 #V0 #V1 #HLK0 #_ #_ #_ +lapply (ldrop_mono … HLK … HLK0) -L #H destruct +qed. + +(* Basic_1: was: nf2_lref_abst *) +lemma cnf_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍⦃#i⦄. +#L #K #V #i #HLK #X #H +elim (cpr_inv_lref1 … H) // * +#K0 #V0 #V1 #HLK0 #_ #_ #_ +lapply (ldrop_mono … HLK … HLK0) -L #H destruct +qed. + +(* Basic_1: was: nf2_abst *) +lemma cnf_abst: ∀a,I,L,V,W,T. L ⊢ 𝐍⦃W⦄ → L. ⓑ{I} V ⊢ 𝐍⦃T⦄ → L ⊢ 𝐍⦃ⓛ{a}W.T⦄. +#a #I #L #V #W #T #HW #HT #X #H +elim (cpr_inv_abst1 … H I V) -H #W0 #T0 #HW0 #HT0 #H destruct +>(HW … HW0) -W0 >(HT … HT0) -T0 // +qed. + +(* Basic_1: was only: nf2_appl_lref *) +lemma cnf_appl_simple: ∀L,V,T. L ⊢ 𝐍⦃V⦄ → L ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐍⦃ⓐV.T⦄. +#L #V #T #HV #HT #HS #X #H +elim (cpr_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct +>(HV … HV0) -V0 >(HT … HT0) -T0 // +qed. + +(* Relocation properties ****************************************************) + +(* Basic_1: was: nf2_lift *) +lemma cnf_lift: ∀L0,L,T,T0,d,e. + L ⊢ 𝐍⦃T⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍⦃T0⦄. +#L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H +elim (cpr_inv_lift1 … HL0 … HT0 … H) -L0 #T1 #HT10 #HT1 +<(HLT … HT1) in HT0; -L #HT0 +>(lift_mono … HT10 … HT0) -T1 -X // +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr.ma new file mode 100644 index 000000000..22fbfc148 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr.ma @@ -0,0 +1,103 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/unfold/tpss.ma". +include "basic_2/reducibility/tpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) + +(* Basic_1: includes: pr2_delta1 *) +definition cpr: lenv → relation term ≝ + λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2. + +interpretation + "context-sensitive parallel reduction (term)" + 'PRed L T1 T2 = (cpr L T1 T2). + +(* Basic properties *********************************************************) + +lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2. +/4 width=3/ qed-. + +(* Basic_1: was by definition: pr2_free *) +lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2. +/2 width=3/ qed. + +lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2. +/3 width=5/ qed. + +lemma cpr_refl: ∀L,T. L ⊢ T ➡ T. +/2 width=1/ qed. + +(* Note: new property *) +(* Basic_1: was only: pr2_thin_dx *) +lemma cpr_flat: ∀I,L,V1,V2,T1,T2. + L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2. +#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/ +qed. + +lemma cpr_cast: ∀L,V,T1,T2. + L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2. +#L #V #T1 #T2 * /3 width=3/ +qed. + +(* Note: it does not hold replacing |L1| with |L2| *) +(* Basic_1: was only: pr2_change *) +lemma cpr_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → + ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2. +#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12 +lapply (tpss_lsubs_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Basic_1: was: pr2_gen_csort *) +lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2. +#T1 #T2 * #T #HT normalize #HT2 +<(tpss_inv_refl_O2 … HT2) -HT2 // +qed-. + +(* Basic_1: was: pr2_gen_sort *) +lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k. +#L #T2 #k * #X #H +>(tpr_inv_atom1 … H) -H #H +>(tpss_inv_sort1 … H) -H // +qed-. + +(* Basic_1: was: pr2_gen_cast *) +lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → ( + ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 & + U2 = ⓝV2. T2 + ) ∨ L ⊢ T1 ➡ U2. +#L #V1 #T1 #U2 * #X #H #HU2 +elim (tpr_inv_cast1 … H) -H /3 width=3/ +* #V #T #HV1 #HT1 #H destruct +elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T → + ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. +#L #L1 #T1 #T * #X #H1 #H2 +elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct +elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/ +qed-. + +(* Basic_1: removed theorems 6: + pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans + pr2_gen_ctail pr2_ctail + Basic_1: removed local theorems 3: + pr2_free_free pr2_free_delta pr2_delta_delta +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_aaa.ma new file mode 100644 index 000000000..7c1273334 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_aaa.ma @@ -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: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A. +#L #T1 #A #HT1 #T2 * /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_cpr.ma new file mode 100644 index 000000000..8d2eb1f1d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_cpr.ma @@ -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: ∀a,I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 → + L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2. +#a #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12 +@ex2_1_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *) +qed. + +(* Basic_1: was only: pr2_gen_cbind *) +lemma cpr_bind_dx: ∀a,I,L,V1,V2,T1,T2. V1 ➡ V2 → L. ⓑ{I} V2 ⊢ T1 ➡ T2 → + L ⊢ ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2. +#a #I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2 +elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 (tpr_inv_atom1 … H) -H #H +elim (tpss_inv_lref1 … H) -H /2 width=1/ +* /3 width=6/ +qed-. + +(* Basic_1: was pr2_gen_abbr *) +lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡ U2 → + (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 & + L. ⓓV ⊢ T1 ➡ T2 & + U2 = ⓓ{a}V2. T2 + ) ∨ + ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2 & a = true. +#a #L #V1 #T1 #Y * #X #H1 #H2 +elim (tpr_inv_abbr1 … H1) -H1 * +[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct + elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct + lapply (tps_lsubs_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0 + lapply (tps_weak_all … HT0) -HT0 #HT0 + lapply (tpss_lsubs_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02 + lapply (tpss_weak_all … HT02) -HT02 #HT02 + lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/ +| #T2 #HT12 #HXT2 #H destruct + elim (lift_total Y 0 1) #Z #HYZ + lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H + lapply (cpr_intro … HT12 … H) -T2 /3 width=3/ +] +qed-. + +(* Basic_1: was: pr2_gen_abst *) +lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡ U2 → ∀I,W. + ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2. T2. +#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W +elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct +elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct +lapply (tpss_lsubs_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/ +qed-. + +(* Basic_1: was pr2_gen_appl *) +lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 → + ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 & + U2 = ⓐV2. T2 + | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 & + U0 = ⓛ{a}W. T1 & + U2 = ⓓ{a}V2. T2 + | ∃∃a,V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 & + ⇧[0,1] V2 ≡ V & + U0 = ⓓ{a}W1. T1 & + U2 = ⓓ{a}W2. ⓐV. T2. +#L #V1 #U0 #Y * #X #H1 #H2 +elim (tpr_inv_appl1 … H1) -H1 * +[ #V #U #HV1 #HU0 #H destruct + elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/ +| #a #V #W #T0 #T #HV1 #HT0 #H #H1 destruct + elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct + lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=9/ +| #a #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct + elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct + elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct + elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V (ltpr_fwd_length … HL12) in HT2; #HT2 +elim (tpr_tpss_ltpr … HL12 … HT2) -L1 /3 width=3/ +qed. + +lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → + ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 → + ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2. +#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 +elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2 +elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2 +lapply (tpss_weak_all … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_ltpss.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_ltpss.ma new file mode 100644 index 000000000..b728d9dc2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_ltpss.ma @@ -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/unfold/ltpss_sn_ltpss_sn.ma". +include "basic_2/reducibility/cpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) + +(* Properties concerning partial unfold on local environments ***************) + +lemma ltpss_sn_cpr_trans: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → + ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2. +#L1 #L2 #d #e #HL12 #T1 #T2 * +lapply (ltpss_sn_weak_all … HL12) +<(ltpss_sn_fwd_length … HL12) -HL12 /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_tpss.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_tpss.ma new file mode 100644 index 000000000..e5afe8737 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_tpss.ma @@ -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/unfold/tpss_tpss.ma". +include "basic_2/reducibility/cpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) + +(* Properties on partial unfold for terms ***********************************) + +lemma cpr_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡ T → + ∀T2. L ⊢ T ▶* [O, |L|] T2 → L ⊢ T1 ➡ T2. +#L #T1 #T * #T0 #HT10 #HT0 #T2 #HT2 +lapply (tpss_trans_eq … HT0 HT2) -T /2 width=3/ +qed. + +lemma cpr_tps_trans: ∀L,T1,T. L ⊢ T1 ➡ T → + ∀T2. L ⊢ T ▶ [O, |L|] T2 → L ⊢ T1 ➡ T2. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/crf.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/crf.ma new file mode 100644 index 000000000..5015f033c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/crf.ma @@ -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/substitution/ldrop.ma". + +(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************) + +(* reducible binary items *) +definition ri2: item2 → Prop ≝ + λI. I = Bind2 true Abbr ∨ I = Flat2 Cast. + +(* irreducible binary binders *) +definition ib2: bool → bind2 → Prop ≝ + λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr. + +(* reducible terms *) +inductive crf: lenv → predicate term ≝ +| crf_delta : ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → crf L (#i) +| crf_appl_sn: ∀L,V,T. crf L V → crf L (ⓐV. T) +| crf_appl_dx: ∀L,V,T. crf L T → crf L (ⓐV. T) +| crf_ri2 : ∀I,L,V,T. ri2 I → crf L (②{I}V. T) +| crf_ib2_sn : ∀a,I,L,V,T. ib2 a I → crf L V → crf L (ⓑ{a,I}V. T) +| crf_ib2_dx : ∀a,I,L,V,T. ib2 a I → crf (L.ⓑ{I}V) T → crf L (ⓑ{a,I}V. T) +| crf_beta : ∀a,L,V,W,T. crf L (ⓐV. ⓛ{a}W. T) +| crf_theta : ∀a,L,V,W,T. crf L (ⓐV. ⓓ{a}W. T) +. + +interpretation + "context-sensitive reducibility (term)" + 'Reducible L T = (crf L T). + +(* Basic inversion lemmas ***************************************************) + +fact trf_inv_atom_aux: ∀I,L,T. L ⊢ 𝐑⦃T⦄ → L = ⋆ → T = ⓪{I} → ⊥. +#I #L #T * -L -T +[ #L #K #V #i #HLK #H1 #H2 destruct + lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct +| #L #V #T #_ #_ #H destruct +| #L #V #T #_ #_ #H destruct +| #J #L #V #T #_ #_ #H destruct +| #a #J #L #V #T #_ #_ #_ #H destruct +| #a #J #L #V #T #_ #_ #_ #H destruct +| #a #L #V #W #T #_ #H destruct +| #a #L #V #W #T #_ #H destruct +] +qed. + +lemma trf_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥. +/2 width=6/ qed-. + +fact trf_inv_lref_aux: ∀L,T. L ⊢ 𝐑⦃T⦄ → ∀i. T = #i → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV. +#L #T * -L -T +[ #L #K #V #j #HLK #i #H destruct /2 width=3/ +| #L #V #T #_ #i #H destruct +| #L #V #T #_ #i #H destruct +| #J #L #V #T #_ #i #H destruct +| #a #J #L #V #T #_ #_ #i #H destruct +| #a #J #L #V #T #_ #_ #i #H destruct +| #a #L #V #W #T #i #H destruct +| #a #L #V #W #T #i #H destruct +] +qed. + +lemma trf_inv_lref: ∀L,i. L ⊢ 𝐑⦃#i⦄ → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV. +/2 width=3/ qed-. + +fact crf_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → L ⊢ 𝐑⦃T⦄ → T = ⓑ{a,I}W. U → + L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃U⦄. +#a #I #L #W #U #T #HI * -T +[ #L #K #V #i #_ #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #J #L #V #T #H1 #H2 destruct + elim H1 -H1 #H destruct + elim HI -HI #H destruct +| #b #J #L #V #T #_ #HV #H destruct /2 width=1/ +| #b #J #L #V #T #_ #HT #H destruct /2 width=1/ +| #b #L #V #W #T #H destruct +| #b #L #V #W #T #H destruct +] +qed. + +lemma crf_inv_ib2: ∀a,I,L,W,T. ib2 a I → L ⊢ 𝐑⦃ⓑ{a,I}W.T⦄ → + L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃T⦄. +/2 width=5/ qed-. + +fact crf_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW. U → + ∨∨ L ⊢ 𝐑⦃W⦄ | L ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥). +#L #W #U #T * -T +[ #L #K #V #i #_ #H destruct +| #L #V #T #HV #H destruct /2 width=1/ +| #L #V #T #HT #H destruct /2 width=1/ +| #J #L #V #T #H1 #H2 destruct + elim H1 -H1 #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W0 #T #H destruct + @or3_intro2 #H elim (simple_inv_bind … H) +| #a #L #V #W0 #T #H destruct + @or3_intro2 #H elim (simple_inv_bind … H) +] +qed. + +lemma crf_inv_appl: ∀L,V,T. L ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ L ⊢ 𝐑⦃V⦄ | L ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥). +/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/crf_append.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/crf_append.ma new file mode 100644 index 000000000..f50b97e95 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/crf_append.ma @@ -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/substitution/ldrop_append.ma". +include "basic_2/reducibility/crf.ma". + +(* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************) + +(* Advanved properties ******************************************************) + +lemma crf_labst_last: ∀L,T,W. L ⊢ 𝐑⦃T⦄ → ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄. +#L #T #W #H elim H -L -T /2 width=1/ +#L #K #V #i #HLK +lapply (ldrop_fwd_ldrop2_length … HLK) #Hi +lapply (ldrop_O1_append_sn_le … HLK … (⋆.ⓛW)) -HLK /2 width=2/ -Hi /2 width=3/ +qed. + +lemma crf_trf: ∀T,W. ⋆ ⊢ 𝐑⦃T⦄ → ⋆.ⓛW ⊢ 𝐑⦃T⦄. +#T #W #H lapply (crf_labst_last … W H) // +qed. + +(* Advanced inversion lemmas ************************************************) + +fact crf_inv_labst_last_aux: ∀L1,T,W. L1 ⊢ 𝐑⦃T⦄ → + ∀L2. L1 = ⋆.ⓛW @@ L2 → L2 ⊢ 𝐑⦃T⦄. +#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_ldrop2_length … HLK1) + >append_length >commutative_plus normalize in ⊢ (??% → ?); #H + elim (le_to_or_lt_eq i (|L2|) ?) /2 width=1/ -H #Hi destruct + [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2 + lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi + normalize #H destruct /2 width=3/ + | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // HL01 #HL12 #HT02 +elim (tpr_conf … HT01 HT02) -L0 -T0 #X #H1 #H2 +elim (tpr_fwd_shift1 … H1) #L #T #HL1 #H destruct /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr.ma new file mode 100644 index 000000000..5e1ba7992 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr.ma @@ -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/unfold/ltpss_sn.ma". +include "basic_2/reducibility/ltpr.ma". + +(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************) + +definition lfpr: relation lenv ≝ + λL1,L2. ∃∃L. L1 ➡ L & L ⊢ ▶* [0, |L|] L2 +. + +interpretation + "focalized parallel reduction (environment)" + 'FocalizedPRed L1 L2 = (lfpr L1 L2). + +(* Basic properties *********************************************************) + +(* Note: lemma 250 *) +lemma lfpr_refl: ∀L. ⦃L⦄ ➡ ⦃L⦄. +/2 width=3/ qed. + +lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄. +/3 width=5/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆. +#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_sn_inv_atom1 … HL2) -HL2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_aaa.ma new file mode 100644 index 000000000..6f6c49df3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_aaa.ma @@ -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: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → L2 ⊢ T ⁝ A. +#L1 #T #A #HT #L2 * /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_alt.ma new file mode 100644 index 000000000..adff7ad07 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_alt.ma @@ -0,0 +1,79 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_bi.ma". +include "basic_2/reducibility/fpr_cpr.ma". +include "basic_2/reducibility/lfpr_fpr.ma". + +(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************) + +(* alternative definition *) +definition lfpra: relation lenv ≝ lpx_bi fpr. + +interpretation + "focalized parallel reduction (environment) alternative" + 'FocalizedPRedAlt L1 L2 = (lfpra L1 L2). + +(* Basic properties *********************************************************) + +lemma lfpra_refl: reflexive … lfpra. +/2 width=1/ qed. + +lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄. +#L1 elim L1 -L1 +[ #L2 #T1 #T2 #H + elim (fpr_inv_atom1 … H) -H #_ #H destruct // +| #L1 #I #V1 #IH #L2 #T1 #T2 #H + elim (fpr_inv_pair1 … H) -H #L #V #HV1 #HL1 #H destruct /3 width=3/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆. +/2 width=2 by lpx_bi_inv_atom1/ qed-. + +lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ → + ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & + L2 = K2. ⓑ{I} V2. +/2 width=1 by lpx_bi_inv_pair1/ qed-. + +lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆. +/2 width=2 by lpx_bi_inv_atom2/ qed-. + +lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ → + ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ & + L1 = K1. ⓑ{I} V1. +/2 width=1 by lpx_bi_inv_pair2/ qed-. + +lemma lfpra_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ∀T.⦃L1, T⦄ ➡ ⦃L2, T⦄. +#L1 #L2 * -L1 -L2 // /3 width=1/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|. +/2 width=2 by lpx_bi_fwd_length/ qed-. + +(* Main properties **********************************************************) + +theorem lfpr_lfpra: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄. +#L1 #L2 #H +lapply (lfpr_inv_fpr … H (⋆0)) -H /2 width=3/ +qed. + +theorem lfpra_lfpr: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → ⦃L1⦄ ➡ ⦃L2⦄. +#L1 #L2 #H +lapply (lfpra_inv_fpr … H (⋆0)) -H /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_cpr.ma new file mode 100644 index 000000000..2a40f58bd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_cpr.ma @@ -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/ltpss_sn_ltpss_sn.ma". +include "basic_2/reducibility/cpr.ma". +include "basic_2/reducibility/lfpr.ma". + +(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************) + +(* Advanced properties ****************************************************) + +lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 → + ∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄. +#L1 #L2 * #L #HL1 #HL2 #V1 #V2 * +<(ltpss_sn_fwd_length … HL2) #V #HV1 #HV2 #I +lapply (ltpss_sn_tpss_trans_eq … HV2 … HL2) -HV2 #V2 +@(ex2_1_intro … (L.ⓑ{I}V)) /2 width=1/ (**) (* explicit constructor *) +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_fpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_fpr.ma new file mode 100644 index 000000000..9a226be58 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_fpr.ma @@ -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 ***********************) + +(* Inversion lemmas on context-free parallel reduction for closures *********) + +lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄. +#L1 #L2 #T1 #T2 #H +elim (fpr_inv_all … H) -H /2 width=3/ +qed. + +(* Inversion lemmas on context-free parallel reduction for closures *********) + +lemma lfpr_inv_fpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡ ⦃L2, T⦄. +#L1 #L2 * /2 width=4/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_lfpr.ma new file mode 100644 index 000000000..7031e792b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/lfpr_lfpr.ma @@ -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: ∀L0,L1,L2. ⦃L0⦄ ➡ ⦃L1⦄ → ⦃L0⦄ ➡ ⦃L2⦄ → + ∃∃L. ⦃L1⦄ ➡ ⦃L⦄ & ⦃L2⦄ ➡ ⦃L⦄. +#K0 #L1 #L2 * #K1 #HK01 #HKL1 * #K2 #HK02 #HKL2 +lapply (ltpr_fwd_length … HK01) #H +>(ltpr_fwd_length … HK02) in H; #H +elim (ltpr_conf … HK01 … HK02) -K0 #K #HK1 #HK2 +lapply (ltpss_sn_fwd_length … HKL1) #H1 +lapply (ltpss_sn_fwd_length … HKL2) #H2 +>H1 in HKL1 H; -H1 #HKL1 +>H2 in HKL2; -H2 #HKL2 #H +elim (ltpr_ltpss_sn_conf … HKL1 … HK1) -K1 #K1 #HK1 #HLK1 +elim (ltpr_ltpss_sn_conf … HKL2 … HK2) -K2 #K2 #HK2 #HLK2 +elim (ltpss_sn_conf … HK1 … HK2) -K #K #HK1 #HK2 +lapply (ltpr_fwd_length … HLK1) #H1 +lapply (ltpr_fwd_length … HLK2) #H2 +/3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr.ma new file mode 100644 index 000000000..a910ea7df --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr.ma @@ -0,0 +1,67 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/grammar/lenv_px.ma". +include "basic_2/reducibility/tpr.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +definition ltpr: relation lenv ≝ lpx tpr. + +interpretation + "context-free parallel reduction (environment)" + 'PRed L1 L2 = (ltpr L1 L2). + +(* Basic properties *********************************************************) + +lemma ltpr_refl: reflexive … ltpr. +/2 width=1/ qed. + +lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2. +/2 width=1/ qed. + +(* Basic inversion lemmas ***************************************************) + +(* Basic_1: was: wcpr0_gen_sort *) +lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆. +/2 width=2 by lpx_inv_atom1/ qed-. + +(* Basic_1: was: wcpr0_gen_head *) +lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 → + ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2. +/2 width=1 by lpx_inv_pair1/ qed-. + +lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆. +/2 width=2 by lpx_inv_atom2/ qed-. + +lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1. +/2 width=1 by lpx_inv_pair2/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|. +/2 width=2 by lpx_fwd_length/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L → + ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2. +/2 width=1 by lpx_inv_append1/ qed-. + +lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 → + ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1. +/2 width=1 by lpx_inv_append2/ qed-. + +(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_aaa.ma new file mode 100644 index 000000000..b45dbb99e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_aaa.ma @@ -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/static/aaa_ltpss_dx.ma". +include "basic_2/static/lsuba_aaa.ma". +include "basic_2/reducibility/ltpr_ldrop.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +(* Properties about atomic arity assignment on terms ************************) + +fact aaa_ltpr_tpr_conf_aux: ∀L,T,L1,T1,A. L1 ⊢ T1 ⁝ A → L = L1 → T = T1 → + ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A. +#L #T @(fw_ind … L T) -L -T #L #T #IH +#L1 #T1 #A * -L1 -T1 -A +[ -IH #L1 #k #H1 #H2 #L2 #_ #T2 #H destruct + >(tpr_inv_atom1 … H) -H // +| #I #L1 #K1 #V1 #B #i #HLK1 #HK1 #H1 #H2 #L2 #HL12 #T2 #H destruct + >(tpr_inv_atom1 … H) -T2 + lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1 + elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2 + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct + lapply (IH … HKV1 … HK1 … HK12 … HV12) // -L1 -K1 -V1 /2 width=5/ +| #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct + elim (tpr_inv_abbr1 … H) -H * + [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct + lapply (tps_lsubs_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2 + lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB + lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/ + | -B #T #HT1 #HXT #H destruct + lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA + @(aaa_inv_lift … HA … HXT) /2 width=1/ + ] +| #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct + elim (tpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB + lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/ +| #L1 #V1 #T1 #B #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct + elim (tpr_inv_appl1 … H) -H * + [ #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB + lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ /2 width=3/ + | #a #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + elim (aaa_inv_abst … HT1) -HT1 #B0 #A0 #HB0 #HA0 #H destruct + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB + lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 + lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -T0 -L1 -V1 /4 width=7/ + | #a #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct + elim (aaa_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 + lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 + lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0 + lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -V1 -T0 -L1 -W0 #HT2 + @(aaa_abbr … HW2) -HW2 + @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *) + ] +| #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct + elim (tpr_inv_cast1 … H) -H + [ * #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2 + lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/ + | -HV1 #HT1X + lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/ + ] +] +qed. + +lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 → + ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A. +/2 width=9/ qed. + +lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A. +/2 width=5/ qed. + +lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A. +/2 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ldrop.ma new file mode 100644 index 000000000..945279795 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ldrop.ma @@ -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/substitution/ldrop_lpx.ma". +include "basic_2/reducibility/tpr_lift.ma". +include "basic_2/reducibility/ltpr.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +(* 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. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpr.ma new file mode 100644 index 000000000..4a27a6e70 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpr.ma @@ -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: ∀L0:lenv. ∀L1. L0 ➡ L1 → ∀L2. L0 ➡ L2 → + ∃∃L. L1 ➡ L & L2 ➡ L. +#L0 #L1 #H elim H -L0 -L1 /2 width=3/ +#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H +elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct +elim (IHK01 … HK02) -K0 #K #HK1 #HK2 +elim (tpr_conf … HV01 HV02) -V0 /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpss_dx.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpss_dx.ma new file mode 100644 index 000000000..cee1cb49e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpss_dx.ma @@ -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/tpr_tpss.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +(* Properties concerning dx parallel unfold on local environments ***********) + +lemma ltpr_ltpss_dx_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 → + ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2. +#L1 #K1 #d #e #H elim H -L1 -K1 -d -e +[ /2 width=3/ +| #L1 #I #V1 #X #H + elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H + elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12 + elim (tpr_tpss_ltpr … HK12 … HV12 … HVW1) -V1 /3 width=5/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H + elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12 + elim (tpr_tpss_ltpr … HK12 … HV12 … HVW1) -V1 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpss_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpss_sn.ma new file mode 100644 index 000000000..823762356 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_ltpss_sn.ma @@ -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: ∀L1,K1,d,e. L1 ⊢ ▶* [d, e] K1 → ∀L2. L1 ➡ L2 → + ∃∃K2. L2 ⊢ ▶* [d, e] K2 & K1 ➡ K2. +#L1 #K1 #d #e #H +lapply (ltpss_sn_ltpssa … H) -H #H +@(ltpssa_ind … H) -K1 /2 width=3/ +#K #K1 #_ #HK1 #IHK #L2 #HL12 +elim (IHK … HL12) -L1 #K2 #HLK2 #HK2 +elim (ltpr_ltpss_dx_conf … HK1 … HK2) -K /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_tps.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_tps.ma new file mode 100644 index 000000000..75792eef0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ltpr_tps.ma @@ -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/reducibility/ltpr_ldrop.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +(* Properties concerning parallel substitution on terms *********************) + +lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 → + ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2. +#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e +[ /2 width=3/ +| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12 + elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H + elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2 + elim (lift_total V1 0 (i+1)) #W1 #HVW1 + lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/ +| #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 + elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2 + elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/ +| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12 + elim (IHV12 … HL12) -IHV12 + elim (IHT12 … HL12) -L2 /3 width=5/ +] +qed. + +lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 → + ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T. +#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e +[ /2 width=3/ +| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12 + elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2 + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1 + elim (lift_total V2 0 (i+1)) #W2 #HVW2 + lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/ +| #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 + elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2 + elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/ +| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12 + elim (IHV12 … HL12) -IHV12 + elim (IHT12 … HL12) -L1 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/thnf.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/thnf.ma new file mode 100644 index 000000000..ab864268f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/thnf.ma @@ -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/tshf.ma". +include "basic_2/reducibility/tpr.ma". + +(* CONTEXT-FREE WEAK HEAD NORMAL TERMS **************************************) + +definition thnf: predicate term ≝ NF … tpr tshf. + +interpretation + "context-free head normality (term)" + 'HdNormal T = (thnf T). + +(* Basic inversion lemmas ***************************************************) + +lemma thnf_inv_tshf: ∀T. 𝐇𝐍⦃T⦄ → T ≈ T. +normalize /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +lemma tpr_tshf: ∀T1,T2. T1 ➡ T2 → T1 ≈ T1 → T1 ≈ T2. +#T1 #T2 #H elim H -T1 -T2 // +[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H + elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct + lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2 + lapply (simple_tshf_repl_dx … HUT2 HT1) /2 width=1/ +| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H + elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H + elim (simple_inv_bind … H) +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H + elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct // +| #a #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H + elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H + elim (simple_inv_bind … H) +| #V #T #T1 #T2 #_ #_ #_ #H + elim (tshf_inv_bind1 … H) -H #W2 #U2 #H1 * #H2 destruct +| #V #T1 #T2 #_ #_ #H + elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct +] +qed. + +lemma thnf_tshf: ∀T. T ≈ T → 𝐇𝐍⦃T⦄. +/3 width=1/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr.ma new file mode 100644 index 000000000..957b5f3a8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr.ma @@ -0,0 +1,229 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Basic_1: includes: pr0_delta1 *) +inductive tpr: relation term ≝ +| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I}) +| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 → + tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) +| tpr_beta : ∀a,V1,V2,W,T1,T2. + tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2) +| tpr_delta: ∀a,I,V1,V2,T1,T,T2. + tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 → + tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) +| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2. + tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 → + tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2) +| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2 +| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2 +. + +interpretation + "context-free parallel reduction (term)" + 'PRed T1 T2 = (tpr T1 T2). + +(* Basic properties *********************************************************) + +lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2. +/2 width=3/ qed. + +(* Basic_1: was by definition: pr0_refl *) +lemma tpr_refl: reflexive … tpr. +#T elim T -T // +#I elim I -I /2 width=1/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}. +#U1 #U2 * -U1 -U2 +[ // +| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct +| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct +| #V #T1 #T #T2 #_ #_ #k #H destruct +| #V #T1 #T2 #_ #k #H destruct +] +qed. + +(* Basic_1: was: pr0_gen_sort pr0_gen_lref *) +lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}. +/2 width=3/ qed-. + +fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → + (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T & + ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 & + U2 = ⓑ{a,I} V2. T2 + ) ∨ + ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr. +#U1 #U2 * -U1 -U2 +[ #J #a #I #V #T #H destruct +| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct +| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct +| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/ +| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct +| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/ +| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct +] +qed. + +lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 → + (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T & + ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 & + U2 = ⓑ{a,I} V2. T2 + ) ∨ + ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr. +/2 width=3/ qed-. + +(* Basic_1: was pr0_gen_abbr *) +lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 → + (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T & + ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 & + U2 = ⓓ{a}V2. T2 + ) ∨ + ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true. +#a #V1 #T1 #U2 #H +elim (tpr_inv_bind1 … H) -H * /3 width=7/ +qed-. + +fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 → + ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 & + U2 = ⓕ{I} V2. T2 + | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 & + U0 = ⓛ{a}W. T1 & + U2 = ⓓ{a}V2. T2 & I = Appl + | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 & + ⇧[0,1] V2 ≡ V & + U0 = ⓓ{a}W1. T1 & + U2 = ⓓ{a}W2. ⓐV. T2 & + I = Appl + | (U0 ➡ U2 ∧ I = Cast). +#U1 #U2 * -U1 -U2 +[ #I #J #V #T #H destruct +| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/ +| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/ +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/ +| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct +| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/ +] +qed. + +lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 → + ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 & + U2 = ⓕ{I} V2. T2 + | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 & + U0 = ⓛ{a}W. T1 & + U2 = ⓓ{a}V2. T2 & I = Appl + | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 & + ⇧[0,1] V2 ≡ V & + U0 = ⓓ{a}W1. T1 & + U2 = ⓓ{a}W2. ⓐV. T2 & + I = Appl + | (U0 ➡ U2 ∧ I = Cast). +/2 width=3/ qed-. + +(* Basic_1: was pr0_gen_appl *) +lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 → + ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 & + U2 = ⓐV2. T2 + | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 & + U0 = ⓛ{a}W. T1 & + U2 = ⓓ{a}V2. T2 + | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 & + ⇧[0,1] V2 ≡ V & + U0 = ⓓ{a}W1. T1 & + U2 = ⓓ{a}W2. ⓐV. T2. +#V1 #U0 #U2 #H +elim (tpr_inv_flat1 … H) -H * +/3 width=5/ /3 width=9/ /3 width=13/ +#_ #H destruct +qed-. + +(* Note: the main property of simple terms *) +lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ → + ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & + U = ⓐV2. T2. +#V1 #T1 #U #H #HT1 +elim (tpr_inv_appl1 … H) -H * +[ /2 width=5/ +| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct + elim (simple_inv_bind … HT1) +| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct + elim (simple_inv_bind … HT1) +] +qed-. + +(* Basic_1: was: pr0_gen_cast *) +lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 → + (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2) + ∨ T1 ➡ U2. +#V1 #T1 #U2 #H +elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2 +[ #_ #_ #_ #_ #H destruct +| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct +] +qed-. + +fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i → + ∨∨ T1 = #i + | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T + | ∃∃V,T. T ➡ #i & T1 = ⓝV. T. +#T1 #T2 * -T1 -T2 +[ #I #i #H destruct /2 width=1/ +| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct +| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct +| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct + lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/ +| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/ +] +qed. + +lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i → + ∨∨ T1 = #i + | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T + | ∃∃V,T. T ➡ #i & T1 = ⓝV. T. +/2 width=3/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T → + ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. +#L1 @(lenv_ind_dx … L1) -L1 normalize +[ #T1 #T #HT1 + @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *) +| #I #L1 #V1 #IH #T1 #X + >shift_append_assoc normalize #H + elim (tpr_inv_bind1 … H) -H * + [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct + elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct + elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct + >append_length >HL1 >HL2 -L1 -L + @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) + | #T #_ #_ #H destruct + ] +] +qed-. + +(* Basic_1: removed theorems 3: + pr0_subst0_back pr0_subst0_fwd pr0_subst0 + Basic_1: removed local theorems: 1: pr0_delta_tau +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_delift.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_delift.ma new file mode 100644 index 000000000..99e621d15 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_delift.ma @@ -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/unfold/delift.ma". +include "basic_2/reducibility/tpr_tpss.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Properties on inverse basic term relocation ******************************) + +lemma tpr_delift_conf: ∀U1,U2. U1 ➡ U2 → ∀L,T1,d,e. L ⊢ ▼*[d, e] U1 ≡ T1 → + ∃∃T2. T1 ➡ T2 & L ⊢ ▼*[d, e] U2 ≡ T2. +#U1 #U2 #HU12 #L #T1 #d #e * #W1 #HUW1 #HTW1 +elim (tpr_tpss_conf … HU12 … HUW1) -U1 #U1 #HWU1 #HU21 +elim (tpr_inv_lift1 … HWU1 … HTW1) -W1 /3 width=5/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_lift.ma new file mode 100644 index 000000000..b4d76066a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_lift.ma @@ -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/substitution/tps_lift.ma". +include "basic_2/reducibility/tpr.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Relocation properties ****************************************************) + +(* Basic_1: was: pr0_lift *) +lemma tpr_lift: t_liftable tpr. +#T1 #T2 #H elim H -T1 -T2 +[ * #i #U1 #d #e #HU1 #U2 #HU2 + lapply (lift_mono … HU1 … HU2) -HU1 #H destruct + [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct // + | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct // + | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct // + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 + elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct + elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/ +| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 + elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct + elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct + elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/ +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2 + elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct + elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct + elim (lift_total T (d + 1) e) #U #HTU + @tpr_delta + [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *) +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2 + elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct + elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct + elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct + elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct + elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/ +| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2 + elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V + elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/ +| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2 + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/ +] +qed. + +(* Basic_1: was: pr0_gen_lift *) +lemma tpr_inv_lift1: t_deliftable_sn tpr. +#T1 #T2 #H elim H -T1 -T2 +[ * #i #X #d #e #HX + [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/ + | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/ + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX + elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct + elim (IHV12 … HV01) -V1 + elim (IHT12 … HT01) -T1 /3 width=5/ +| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX + elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct + elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct + elim (IHV12 … HV01) -V1 + elim (IHT12 … HT01) -T1 /3 width=5/ +| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX + elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct + elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12 + elim (IHT1 … HUT1) -T1 #U #HUT #HU1 + elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2 + @ex2_1_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *) +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX + elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct + elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct + elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03 + elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03 + elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03 + elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2 + @ex2_1_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *) +| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX + elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct + elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31 + elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/ +| #V #T1 #T2 #_ #IHT12 #X #d #e #HX + elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct + elim (IHT12 … HT01) -T1 /3 width=3/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +fact tpr_inv_abst1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,V1,T1. U1 = ⓛ{a}V1. T1 → + ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2. +#U1 #U2 * -U1 -U2 +[ #I #a #V #T #H destruct +| #I #V1 #V2 #T1 #T2 #_ #_ #a #V #T #H destruct +| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #V #T #H destruct +| #b #I #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #V0 #T0 #H destruct + <(tps_inv_refl_SO2 … HT2 ? ? ?) -T2 /2 width=5/ +| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #V0 #T0 #H destruct +| #V #T1 #T #T2 #_ #_ #a #V0 #T0 #H destruct +| #V #T1 #T2 #_ #a #V0 #T0 #H destruct +] +qed. + +(* Basic_1: was pr0_gen_abst *) +lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 → + ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2. +/2 width=3/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tpr.ma new file mode 100644 index 000000000..1522d00c0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tpr.ma @@ -0,0 +1,283 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_tpss.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Confluence lemmas ********************************************************) + +fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X. +/2 width=3/ qed. + +fact tpr_conf_flat_flat: + ∀I,V0,V1,T0,T1,V2,T2. ( + ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 → + ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0. +#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 +elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2 +elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/ +qed. + +fact tpr_conf_flat_beta: + ∀a,V0,V1,T1,V2,W0,U0,T2. ( + ∀X0:term. #{X0} < #{V0} + (#{W0} + #{U0} + 1) + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → V0 ➡ V2 → + U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 → + ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X. +#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H +elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct +elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/ +qed. + +(* Basic-1: was: + pr0_cong_upsilon_refl pr0_cong_upsilon_zeta + pr0_cong_upsilon_cong pr0_cong_upsilon_delta +*) +fact tpr_conf_flat_theta: + ∀a,V0,V1,T1,V2,V,W0,W2,U0,U2. ( + ∀X0:term. #{X0} < #{V0} + (#{W0} + #{U0} + 1) + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V → + W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 → + ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X. +#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H +elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2 +elim (lift_total VV 0 1) #VVV #HVV +lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV +elim (tpr_inv_abbr1 … H) -H * +(* case 1: delta *) +[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct + elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2 + elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2 + elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1 + @ex2_1_intro + [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ] + |1:skip + |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/ + ] (**) (* /5 width=14/ is too slow *) +(* case 3: zeta *) +| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1 + elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2 + elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct + @(ex2_1_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *) +] +qed. + +fact tpr_conf_flat_cast: + ∀X2,V0,V1,T0,T1. ( + ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 → + ∃∃X. ⓝV1. T1 ➡ X & X2 ➡ X. +#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02 +elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/ +qed. + +fact tpr_conf_beta_beta: + ∀a. ∀W0:term. ∀V0,V1,T0,T1,V2,T2. ( + ∀X0:term. #{X0} < #{V0} + (#{W0} + #{T0} + 1) + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 → + ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X. +#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 +elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ +elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/ +qed. + +(* Basic_1: was: pr0_cong_delta pr0_delta_delta *) +fact tpr_conf_delta_delta: + ∀a,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. ( + ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 → + ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 → + ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 → + ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X. +#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2 +elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2 +elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2 +elim (tpr_tps_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1 +elim (tpr_tps_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2 +elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2 +@ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *) +qed. + +fact tpr_conf_delta_zeta: + ∀X2,V0,V1,T0,T1,TT1,T2. ( + ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 → + T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 → + ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X. +#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2 +elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2 +elim (tpr_tps_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT +elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2 +lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /3 width=3/ +qed. + +(* Basic_1: was: pr0_upsilon_upsilon *) +fact tpr_conf_theta_theta: + ∀a,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. ( + ∀X0:term. #{X0} < #{V0} + (#{W0} + #{T0} + 1) + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 → + ⇧[O, 1] V1 ≡ VV1 → ⇧[O, 1] V2 ≡ VV2 → + ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X. +#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2 +elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2 +elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2 +elim (lift_total V 0 1) #VV #HVV +lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1 +lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2 +@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *) +qed. + +fact tpr_conf_zeta_zeta: + ∀V0:term. ∀X2,TT0,T0,T1,TT2. ( + ∀X0:term. #{X0} < #{V0} + #{TT0} + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 → + TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 → + ∃∃X. T1 ➡ X & X2 ➡ X. +#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2 +elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2 +elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10 +elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2 +lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/ +qed. + +fact tpr_conf_tau_tau: + ∀V0,T0:term. ∀X2,T1. ( + ∀X0:term. #{X0} < #{V0} + #{T0} + 1 → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + T0 ➡ T1 → T0 ➡ X2 → + ∃∃X. T1 ➡ X & X2 ➡ X. +#V0 #T0 #X2 #T1 #IH #HT01 #HT02 +elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/ +qed. + +(* Confluence ***************************************************************) + +fact tpr_conf_aux: + ∀Y0:term. ( + ∀X0:term. #{X0} < #{Y0} → + ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 → + ∃∃X. X1 ➡ X & X2 ➡ X + ) → + ∀X0,X1,X2. X0 ➡ X1 → X0 ➡ X2 → X0 = Y0 → + ∃∃X. X1 ➡ X & X2 ➡ X. +#Y0 #IH #X0 #X1 #X2 * -X0 -X1 +[ #I1 #H1 #H2 destruct + lapply (tpr_inv_atom1 … H1) -H1 +(* case 1: atom, atom *) + #H1 destruct // +| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct + elim (tpr_inv_flat1 … H1) -H1 * +(* case 2: flat, flat *) + [ #V2 #T2 #HV02 #HT02 #H destruct + /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *) +(* case 3: flat, beta *) + | #b #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct + /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *) +(* case 4: flat, theta *) + | #b #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct + /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *) +(* case 5: flat, tau *) + | #HT02 #H destruct + /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *) + ] +| #a #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct + elim (tpr_inv_appl1 … H1) -H1 * +(* case 6: beta, flat (repeated) *) + [ #V2 #T2 #HV02 #HT02 #H destruct + @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/ +(* case 7: beta, beta *) + | #b #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct + /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *) +(* case 8, beta, theta (excluded) *) + | #b #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct + ] +| #a #I1 #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct + elim (tpr_inv_bind1 … H1) -H1 * +(* case 9: delta, delta *) + [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct + /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *) +(* case 10: delta, zeta *) + | #T2 #HT20 #HTX2 #H1 #H2 destruct + /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *) + ] +| #a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct + elim (tpr_inv_appl1 … H1) -H1 * +(* case 11: theta, flat (repeated) *) + [ #V2 #T2 #HV02 #HT02 #H destruct + @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/ +(* case 12: theta, beta (repeated) *) + | #b #V2 #WW0 #TT0 #T2 #_ #_ #H destruct +(* case 13: theta, theta *) + | #b #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct + /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *) + ] +| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct + elim (tpr_inv_abbr1 … H1) -H1 * +(* case 14: zeta, delta (repeated) *) + [ #V2 #TT2 #T2 #HV02 #HTT02 #HTT2 #H destruct + @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/ +(* case 15: zeta, zeta *) + | #TT2 #HTT02 #HXTT2 + /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *) + ] +| #V0 #T0 #T1 #HT01 #H1 #H2 destruct + elim (tpr_inv_cast1 … H1) -H1 +(* case 16: tau, flat (repeated) *) + [ * #V2 #T2 #HV02 #HT02 #H destruct + @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/ +(* case 17: tau, tau *) + | #HT02 + /3 width=5 by tpr_conf_tau_tau/ + ] +] +qed. + +(* Basic_1: was: pr0_confluence *) +theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 → + ∃∃T. T1 ➡ T & T2 ➡ T. +#T @(tw_ind … T) -T /3 width=6 by tpr_conf_aux/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tps.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tps.ma new file mode 100644 index 000000000..12cf13c0f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tps.ma @@ -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 TERMS *********************************) + +(* Properties on parallel substitution for terms ****************************) + +(* Basic_1: was: pr0_subst1_fwd *) +lemma ltpr_tpr_conf: ∀L1,T,U1,d,e. L1 ⊢ T ▶ [d, e] U1 → ∀L2. L1 ➡ L2 → + ∃∃U2. U1 ➡ U2 & L2 ⊢ T ▶ [d, e] U2. +#L1 #T #U1 #d #e #H elim H -L1 -T -U1 -d -e +[ /2 width=3/ +| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12 + elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2 + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1 + elim (lift_total V2 0 (i+1)) #W2 #HVW2 + lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=6/ +| #L1 #a #I #V #W1 #T #U1 #d #e #_ #_ #IHV #IHT #L2 #HL12 + elim (IHV … HL12) -IHV #W2 #HW12 + elim (IHT (L2.ⓑ{I}W2) ?) -IHT /2 width=1/ -L1 /3 width=5/ +| #L1 #I #V #W1 #T #U1 #d #e #_ #_ #IHV #IHT #L2 #HL12 + elim (IHV … HL12) -IHV + elim (IHT … HL12) -IHT -HL12 /3 width=5/ +] +qed. + +(* Basic_1: was: pr0_subst1_back *) +lemma ltpr_tps_trans: ∀L2,T,U2,d,e. L2 ⊢ T ▶ [d, e] U2 → ∀L1. L1 ➡ L2 → + ∃∃U1. U1 ➡ U2 & L1 ⊢ T ▶ [d, e] U1. +#L2 #T #U2 #d #e #H elim H -L2 -T -U2 -d -e +[ /2 width=3/ +| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12 + elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H + elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2 + elim (lift_total V1 0 (i+1)) #W1 #HVW1 + lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=6/ +| #L2 #a #I #V #W2 #T #U2 #d #e #_ #_ #IHV #IHT #L1 #HL12 + elim (IHV … HL12) -IHV #W1 #HW12 + elim (IHT (L1.ⓑ{I}W1) ?) -IHT /2 width=1/ -L2 /3 width=5/ +| #L2 #I #V #W2 #T #U2 #d #e #_ #_ #IHV #IHT #L1 #HL12 + elim (IHV … HL12) -IHV + elim (IHT … HL12) -IHT -HL12 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tpss.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tpss.ma new file mode 100644 index 000000000..e1ead4e44 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/tpr_tpss.ma @@ -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/tpr_tps.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Unfold properties ********************************************************) + +(* Basic_1: was: pr0_subst1 *) +lemma tpr_tps_ltpr: ∀T1,T2. T1 ➡ T2 → + ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 → + ∀L2. L1 ➡ L2 → + ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2. +#T1 #T2 #H elim H -T1 -T2 +[ #I #L1 #d #e #U1 #H #L2 #HL12 + elim (ltpr_tpr_conf … H … HL12) -L1 /3 width=3/ +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct + elim (IHV12 … HVW1 … HL12) -V1 + elim (IHT12 … HTU1 … HL12) -T1 -HL12 /3 width=5/ +| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct + elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct + elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2 + elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 + lapply (tpss_lsubs_trans … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/ +| #a #I #V1 #V2 #T1 #T #T2 #HV12 #_ #HT2 #IHV12 #IHT1 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct + elim (IHV12 … HVW1 … HL12) -V1 #W2 #HW12 #HVW2 + elim (IHT1 … HTU1 (L2. ⓑ{I} W2) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU + elim (tpss_strip_neq … HTU … HT2 ?) -T /2 width=1/ #U2 #HU2 #HTU2 + lapply (tps_lsubs_trans … HU2 (L2. ⓑ{I} V2) ?) -HU2 /2 width=1/ #HU2 + elim (ltpss_dx_tps_conf … HU2 (L2. ⓑ{I} W2) (d + 1) e ?) -HU2 /2 width=1/ #U3 #HU3 #HU23 + lapply (tps_lsubs_trans … HU3 (⋆. ⓑ{I} W2) ?) -HU3 /2 width=1/ #HU3 + lapply (tpss_lsubs_trans … HU23 (L2. ⓑ{I} W2) ?) -HU23 /2 width=1/ #HU23 + lapply (tpss_trans_eq … HTU2 … HU23) -U2 /3 width=5/ +| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct + elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct + elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2 + elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2 + elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2 + elim (lift_total VV2 0 1) #VV #H2VV + lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV + @ex2_1_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *) +| #V #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_bind1 … H) -H #W #U1 #_ #HTU1 #H destruct -V + elim (IHT1 … HTU1 (L2.ⓓW) ?) -T1 /2 width=1/ -HL12 #U #HU1 #HTU + elim (tpss_inv_lift1_ge … HTU L2 … HT2 ?) -T (aaa_inv_sort … H) -H // +| #I1 #L #K1 #V1 #B #i #HLK1 #_ #IHA1 #A2 #H + elim (aaa_inv_lref … H) -H #I2 #K2 #V2 #HLK2 #HA2 + lapply (ldrop_mono … HLK1 … HLK2) -L #H destruct /2 width=1/ +| #a #L #V #T #B1 #A1 #_ #_ #_ #IHA1 #A2 #H + elim (aaa_inv_abbr … H) -H /2 width=1/ +| #a #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H + elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1/ +| #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H + elim (aaa_inv_appl … H) -H #B2 #_ #HA2 + lapply (IHA1 … HA2) -L #H destruct // +| #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H + elim (aaa_inv_cast … H) -H /2 width=1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma new file mode 100644 index 000000000..b40f1a9b1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma @@ -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 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/ldrop_ldrop.ma". +include "basic_2/static/aaa.ma". + +(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) + +(* Properties concerning basic relocation ***********************************) + +lemma aaa_lift: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → + ∀T2. ⇧[d, e] T1 ≡ T2 → L2 ⊢ T2 ⁝ A. +#L1 #T1 #A #H elim H -L1 -T1 -A +[ #L1 #k #L2 #d #e #_ #T2 #H + >(lift_inv_sort1 … H) -H // +| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #d #e #HL21 #T2 #H + elim (lift_inv_lref1 … H) -H * #Hid #H destruct + [ elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct + /3 width=8/ + | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ + ] +| #a #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H + elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + /4 width=4/ +| #a #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H + elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + /4 width=4/ +| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H + elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + /3 width=4/ +| #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #d #e #HL21 #X #H + elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + /3 width=4/ +] +qed. + +lemma aaa_inv_lift: ∀L2,T2,A. L2 ⊢ T2 ⁝ A → ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → + ∀T1. ⇧[d, e] T1 ≡ T2 → L1 ⊢ T1 ⁝ A. +#L2 #T2 #A #H elim H -L2 -T2 -A +[ #L2 #k #L1 #d #e #_ #T1 #H + >(lift_inv_sort2 … H) -H // +| #I #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #d #e #HL21 #T1 #H + elim (lift_inv_lref2 … H) -H * #Hid #H destruct + [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // -Hid /3 width=8/ + | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // -Hid /3 width=8/ + ] +| #a #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H + elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + /4 width=4/ +| #a #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H + elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + /4 width=4/ +| #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H + elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + /3 width=4/ +| #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #d #e #HL21 #X #H + elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma new file mode 100644 index 000000000..7514f6dc5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma @@ -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/unfold/ldrops.ma". +include "basic_2/static/aaa_lift.ma". + +(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) + +(* Properties concerning generic relocation *********************************) + +lemma aaa_lifts: ∀L1,L2,T2,A,des. ⇩*[des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 → + L1 ⊢ T1 ⁝ A → L2 ⊢ T2 ⁝ A. +#L1 #L2 #T2 #A #des #H elim H -L1 -L2 -des +[ #L #T1 #H #HT1 + <(lifts_inv_nil … H) -H // +| #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1 + elim (lifts_inv_cons … H) -H /3 width=9/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_ltpss_dx.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_ltpss_dx.ma new file mode 100644 index 000000000..2f2d07360 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_ltpss_dx.ma @@ -0,0 +1,79 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/unfold/tpss_tpss.ma". +include "basic_2/unfold/ltpss_dx_ldrop.ma". +include "basic_2/static/aaa_lift.ma". + +(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) + +(* Properties about dx parallel unfold **************************************) + +(* Note: lemma 500 *) +lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A. +#L1 #T1 #A #H elim H -L1 -T1 -A +[ #L1 #k #L2 #d #e #_ #T2 #H + >(tpss_inv_sort1 … H) -H // +| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H + [ #H destruct + elim (lt_or_ge i d) #Hdi + [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct + /3 width=8 by aaa_lref/ (**) (* too slow without trace *) + | elim (lt_or_ge i (d + e)) #Hide + [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct + /3 width=8 by aaa_lref/ (**) (* too slow without trace *) + | -Hdi + lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide + /3 width=8 by aaa_lref/ (**) (* too slow without trace *) + ] + ] + | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2 + elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0 + elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct + lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2 + lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/ + ] +| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/ +| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/ +| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ +| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H + elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ +] +qed. + +lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → + ∀L2,d,e. L1 ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A. +/3 width=7/ qed. + +lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → + ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A. +/2 width=7/ qed. + +lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → + ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A. +/2 width=7/ qed. + +lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → + ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A. +/2 width=7/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_ltpss_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_ltpss_sn.ma new file mode 100644 index 000000000..4f2a44827 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_ltpss_sn.ma @@ -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: ∀L1,T,A. L1 ⊢ T ⁝ A → + ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A. +#L1 #T #A #HT #L2 #d #e #HL12 +lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12 +@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/ +qed. + +lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → + ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A. +/3 width=5/ qed. + +lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → + ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → + ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A. +/3 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma new file mode 100644 index 000000000..aa4800fd5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma @@ -0,0 +1,92 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) + +inductive lsuba: relation lenv ≝ +| lsuba_atom: lsuba (⋆) (⋆) +| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V) +| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ⁝ A → L2 ⊢ W ⁝ A → + lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW) +. + +interpretation + "local environment refinement (atomic arity assigment)" + 'CrSubEqA L1 L2 = (lsuba L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆. +#L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #A #_ #_ #_ #H destruct +] +qed. + +lemma lsuba_inv_atom1: ∀L2. ⋆ ⁝⊑ L2 → L2 = ⋆. +/2 width=3/ qed-. + +fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V → + (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & + L2 = K2. ⓛW & I = Abbr. +#L1 #L2 * -L1 -L2 +[ #I #K1 #V #H destruct +| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/ +| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/ +] +qed. + +lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ⁝⊑ L2 → + (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & + L2 = K2. ⓛW & I = Abbr. +/2 width=3/ qed-. + +fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆. +#L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #A #_ #_ #_ #H destruct +] +qed. + +lemma lsubc_inv_atom2: ∀L1. L1 ⁝⊑ ⋆ → L1 = ⋆. +/2 width=3/ qed-. + +fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W → + (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & + L1 = K1. ⓓV & I = Abst. +#L1 #L2 * -L1 -L2 +[ #I #K2 #W #H destruct +| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/ +| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/ +] +qed. + +lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ⁝⊑ K2. ⓑ{I} W → + (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 & + L1 = K1. ⓓV & I = Abst. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +lemma lsuba_refl: ∀L. L ⁝⊑ L. +#L elim L -L // /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_aaa.ma new file mode 100644 index 000000000..66e802aae --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_aaa.ma @@ -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/static/aaa_aaa.ma". +include "basic_2/static/lsuba_ldrop.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) + +(* Properties concerning atomic arity assignment ****************************) + +lemma lsuba_aaa_conf: ∀L1,V,A. L1 ⊢ V ⁝ A → ∀L2. L1 ⁝⊑ L2 → L2 ⊢ V ⁝ A. +#L1 #V #A #H elim H -L1 -V -A +[ // +| #I #L1 #K1 #V1 #B #i #HLK1 #HV1B #IHV1 #L2 #HL12 + elim (lsuba_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2 + elim (lsuba_inv_pair1 … H) -H * #K2 + [ #HK12 #H destruct /3 width=5/ + | #V2 #A1 #HV1A1 #HV2 #_ #H1 #H2 destruct + >(aaa_mono … HV1B … HV1A1) -B -HV1A1 /2 width=5/ + ] +| /4 width=2/ +| /4 width=1/ +| /3 width=3/ +| /3 width=1/ +] +qed. + +lemma lsuba_aaa_trans: ∀L2,V,A. L2 ⊢ V ⁝ A → ∀L1. L1 ⁝⊑ L2 → L1 ⊢ V ⁝ A. +#L2 #V #A #H elim H -L2 -V -A +[ // +| #I #L2 #K2 #V2 #B #i #HLK2 #HV2B #IHV2 #L1 #HL12 + elim (lsuba_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 + elim (lsuba_inv_pair2 … H) -H * #K1 + [ #HK12 #H destruct /3 width=5/ + | #V1 #A1 #HV1 #HV2A1 #_ #H1 #H2 destruct + >(aaa_mono … HV2B … HV2A1) -B -HV2A1 /2 width=5/ + ] +| /4 width=2/ +| /4 width=1/ +| /3 width=3/ +| /3 width=1/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma new file mode 100644 index 000000000..247a8b221 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma @@ -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/static/lsuba.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) + +(* Properties concerning basic local environment slicing ********************) + +(* Note: the constant 0 cannot be generalized *) +lemma lsuba_ldrop_O1_conf: ∀L1,L2. L1 ⁝⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 → + ∃∃K2. K1 ⁝⊑ K2 & ⇩[0, e] L2 ≡ K2. +#L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +] +qed-. + +(* Note: the constant 0 cannot be generalized *) +lemma lsuba_ldrop_O1_trans: ∀L1,L2. L1 ⁝⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. K1 ⁝⊑ K2 & ⇩[0, e] L1 ≡ K1. +#L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_lsuba.ma new file mode 100644 index 000000000..5d64516a5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_lsuba.ma @@ -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/static/lsuba_aaa.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************) + +(* Main properties **********************************************************) + +theorem lsuba_trans: ∀L1,L. L1 ⁝⊑ L → ∀L2. L ⁝⊑ L2 → L1 ⁝⊑ L2. +#L1 #L #H elim H -L1 -L +[ #X #H >(lsuba_inv_atom1 … H) -H // +| #I #L1 #L #V #HL1 #IHL1 #X #H + elim (lsuba_inv_pair1 … H) -H * #L2 + [ #HL2 #H destruct /3 width=1/ + | #V #A #HLV #HL2V #HL2 #H1 #H2 destruct /3 width=3/ + ] +| #L1 #L #V1 #W #A1 #HV1 #HW #HL1 #IHL1 #X #H + elim (lsuba_inv_pair1 … H) -H * #L2 + [ #HL2 #H destruct /3 width=5/ + | #V #A2 #_ #_ #_ #_ #H destruct + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubss.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss.ma new file mode 100644 index 000000000..356d7fe11 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss.ma @@ -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/static/ssta.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******) + +(* Note: may not be transitive *) +inductive lsubss (h:sh) (g:sd h): relation lenv ≝ +| lsubss_atom: lsubss h g (⋆) (⋆) +| lsubss_pair: ∀I,L1,L2,W. lsubss h g L1 L2 → + lsubss h g (L1. ⓑ{I} W) (L2. ⓑ{I} W) +| lsubss_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V •[g, l+1] W → ⦃h, L2⦄ ⊢ V •[g, l+1] W → + lsubss h g L1 L2 → lsubss h g (L1. ⓓV) (L2. ⓛW) +. + +interpretation + "local environment refinement (stratified static type assigment)" + 'CrSubEqS h g L1 L2 = (lsubss h g L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lsubss_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 = ⋆ → L2 = ⋆. +#h #g #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #l #_ #_ #_ #H destruct +] +qed. + +lemma lsubss_inv_atom1: ∀h,g,L2. h ⊢ ⋆ •⊑[g] L2 → L2 = ⋆. +/2 width=5/ qed-. + +fact lsubss_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → + ∀I,K1,V. L1 = K1. ⓑ{I} V → + (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & + h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr. +#h #g #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 lsubss_inv_pair1: ∀h,g,I,K1,L2,V. h ⊢ K1. ⓑ{I} V •⊑[g] L2 → + (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨ + ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & + h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr. +/2 width=3/ qed-. + +fact lsubss_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L2 = ⋆ → L1 = ⋆. +#h #g #L1 #L2 * -L1 -L2 +[ // +| #I #L1 #L2 #V #_ #H destruct +| #L1 #L2 #V #W #l #_ #_ #_ #H destruct +] +qed. + +lemma lsubss_inv_atom2: ∀h,g,L1. h ⊢ L1 •⊑[g] ⋆ → L1 = ⋆. +/2 width=5/ qed-. + +fact lsubss_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → + ∀I,K2,W. L2 = K2. ⓑ{I} W → + (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & + h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst. +#h #g #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 lsubss_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 •⊑[g] K2. ⓑ{I} W → + (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨ + ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W & + h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst. +/2 width=3/ qed-. + +(* Basic_forward lemmas *****************************************************) + +lemma lsubss_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L1|] L2. +#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +lemma lsubss_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L2|] L2. +#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ +qed-. + +(* Basic properties *********************************************************) + +lemma lsubss_refl: ∀h,g,L. h ⊢ L •⊑[g] L. +#h #g #L elim L -L // /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_ldrop.ma new file mode 100644 index 000000000..82ede6149 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_ldrop.ma @@ -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/static/lsubss.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******) + +(* Properties concerning basic local environment slicing ********************) + +(* Note: the constant 0 cannot be generalized *) +lemma lsubss_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → + ∀K1,e. ⇩[0, e] L1 ≡ K1 → + ∃∃K2. h ⊢ K1 •⊑[g] K2 & ⇩[0, e] L2 ≡ K2. +#h #g #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K1 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK1 + [ destruct + elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK1) -L1 /3 width=3/ + ] +] +qed. + +(* Note: the constant 0 cannot be generalized *) +lemma lsubss_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → + ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. h ⊢ K1 •⊑[g] K2 & ⇩[0, e] L1 ≡ K1. +#h #g #L1 #L2 #H elim H -L1 -L2 +[ /2 width=3/ +| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +| #L1 #L2 #V #W #l #H1VW #H2VW #_ #IHL12 #K2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #HLK2 + [ destruct + elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H + <(ldrop_inv_refl … H) in HL12; -H /3 width=3/ + | elim (IHL12 … HLK2) -L2 /3 width=3/ + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_lsubss.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_lsubss.ma new file mode 100644 index 000000000..d9f9496ba --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_lsubss.ma @@ -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/static/lsubss_ssta.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STATIC TYPE ASSIGNMENT ******************) + +(* Main properties **********************************************************) + +theorem lsubss_trans: ∀h,g,L1,L. h ⊢ L1 •⊑[g] L → ∀L2. h ⊢ L •⊑[g] L2 → + h ⊢ L1 •⊑[g] L2. +#h #g #L1 #L #H elim H -L1 -L +[ #X #H >(lsubss_inv_atom1 … H) -H // +| #I #L1 #L #W #HL1 #IHL1 #X #H + elim (lsubss_inv_pair1 … H) -H * #L2 + [ #HL2 #H destruct /3 width=1/ + | #V #l #H1WV #H2WV #HL2 #H1 #H2 destruct /3 width=3/ + ] +| #L1 #L #V1 #W1 #l #H1VW1 #H2VW1 #HL1 #IHL1 #X #H + elim (lsubss_inv_pair1 … H) -H * #L2 + [ #HL2 #H destruct /3 width=5/ + | #V #l0 #_ #_ #_ #_ #H destruct + ] +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_ssta.ma new file mode 100644 index 000000000..f9c628921 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubss_ssta.ma @@ -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/static/ssta_lift.ma". +include "basic_2/static/ssta_ssta.ma". +include "basic_2/static/lsubss_ldrop.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******) + +(* Properties concerning stratified native type assignment ******************) + +lemma lsubss_ssta_trans: ∀h,g,L2,T,U,l. ⦃h, L2⦄ ⊢ T •[g,l] U → + ∀L1. h ⊢ L1 •⊑[g] L2 → ⦃h, L1⦄ ⊢ T •[g,l] U. +#h #g #L2 #T #U #l #H elim H -L2 -T -U -l +[ /2 width=1/ +| #L2 #K2 #V2 #W2 #U2 #i #l #HLK2 #_ #HWU2 #IHVW2 #L1 #HL12 + elim (lsubss_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 + elim (lsubss_inv_pair2 … H) -H * #K1 [ | -HWU2 -IHVW2 -HLK1 ] + [ #HK12 #H destruct /3 width=6/ + | #V1 #l0 #_ #_ #_ #_ #H destruct + ] +| #L2 #K2 #W2 #V2 #U2 #i #l #HLK2 #HWV2 #HWU2 #IHWV2 #L1 #HL12 + elim (lsubss_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1 + elim (lsubss_inv_pair2 … H) -H * #K1 [ -HWV2 | -IHWV2 ] + [ #HK12 #H destruct /3 width=6/ + | #V1 #l0 #H1 #H2 #_ #H #_ destruct + elim (ssta_fwd_correct … H2) -H2 #V #H + elim (ssta_mono … HWV2 … H) -HWV2 -H /2 width=6/ + ] +| /4 width=1/ +| /3 width=1/ +| /3 width=1/ +] +qed. + +lemma lsubss_ssta_conf: ∀h,g,L1,T,U,l. ⦃h, L1⦄ ⊢ T •[g,l] U → + ∀L2. h ⊢ L1 •⊑[g] L2 → ⦃h, L2⦄ ⊢ T •[g,l] U. +#h #g #L1 #T #U #l #H elim H -L1 -T -U -l +[ /2 width=1/ +| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #HL12 + elim (lsubss_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2 + elim (lsubss_inv_pair1 … H) -H * #K2 [ -HVW1 | -IHVW1 ] + [ #HK12 #H destruct /3 width=6/ + | #V2 #l0 #H1 #H2 #_ #H #_ destruct + elim (ssta_mono … HVW1 … H1) -HVW1 -H1 #H1 #H2 destruct + elim (ssta_fwd_correct … H2) -H2 /2 width=6/ + ] +| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #_ #HWU1 #IHWV1 #L2 #HL12 + elim (lsubss_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2 + elim (lsubss_inv_pair1 … H) -H * #K2 [ | -HWU1 -IHWV1 -HLK2 ] + [ #HK12 #H destruct /3 width=6/ + | #V2 #l0 #_ #_ #_ #_ #H destruct + ] +| /4 width=1/ +| /3 width=1/ +| /3 width=1/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma b/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma new file mode 100644 index 000000000..63143b19b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma @@ -0,0 +1,109 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/sh.ma". + +(* SORT DEGREE **************************************************************) + +(* sort degree specification *) +record sd (h:sh): Type[0] ≝ { + deg : relation nat; (* degree of the sort *) + deg_total: ∀k. ∃l. deg k l; (* functional relation axioms *) + deg_mono : ∀k,l1,l2. deg k l1 → deg k l2 → l1 = l2; + deg_next : ∀k,l. deg k l → deg (next h k) (l - 1) (* compatibility condition *) +}. + +(* Notable specifications ***************************************************) + +definition deg_O: relation nat ≝ λk,l. l = 0. + +definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O …. +// /2 width=1/ /2 width=2/ qed. + +inductive deg_SO (h:sh) (k:nat) (k0:nat): predicate nat ≝ +| deg_SO_pos : ∀l0. (next h)^l0 k0 = k → deg_SO h k k0 (l0 + 1) +| deg_SO_zero: ((∃l0. (next h)^l0 k0 = k) → ⊥) → deg_SO h k k0 0 +. + +fact deg_SO_inv_pos_aux: ∀h,k,k0,l0. deg_SO h k k0 l0 → ∀l. l0 = l + 1 → + (next h)^l k0 = k. +#h #k #k0 #l0 * -l0 +[ #l0 #Hl0 #l #H + lapply (injective_plus_l … H) -H #H destruct // +| #_ #l0 H -H #H + lapply (transitive_lt … H HK12) -k1 #H1 + lapply (nexts_le h k2 l) #H2 + lapply (le_to_lt_to_lt … H2 H1) -h -l #H + elim (lt_refl_false … H) +qed. + +definition sd_SO: ∀h. nat → sd h ≝ λh,k. mk_sd h (deg_SO h k) …. +[ #k0 + lapply (nexts_dec h k0 k) * [ * /3 width=2/ | /4 width=2/ ] +| #K0 #l1 #l2 * [ #l01 ] #H1 * [1,3: #l02 ] #H2 // + [ < H2 in H1; -H2 #H + lapply (nexts_inj … H) -H #H destruct // + | elim (H1 ?) /2 width=2/ + | elim (H2 ?) /2 width=2/ + ] +| #k0 #l0 * + [ #l #H destruct elim l -l normalize /2 width=1/ + | #H1 @deg_SO_zero * #l #H2 destruct + @H1 -H1 @(ex_intro … (S l)) /2 width=1/ (**) (* explicit constructor *) + ] +] +qed. + +let rec sd_l (h:sh) (k:nat) (l:nat) on l : sd h ≝ + match l with + [ O ⇒ sd_O h + | S l ⇒ match l with + [ O ⇒ sd_SO h k + | _ ⇒ sd_l h (next h k) l + ] + ]. + +(* Basic properties *********************************************************) + +lemma deg_pred: ∀h,g,k,l. deg h g (next h k) (l + 1) → deg h g k (l + 2). +#h #g #k #l #H1 +elim (deg_total h g k) #l0 #H0 +lapply (deg_next … H0) #H2 +lapply (deg_mono … H1 H2) -H1 -H2 #H +<(associative_plus l 1 1) >H (lift_inv_sort1 … H1) -X1 + >(lift_inv_sort1 … H2) -X2 /2 width=1/ +| #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2 + elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct + /3 width=8/ + | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2 + lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/ + ] +| #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hid #H destruct + [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // (lift_inv_sort2 … H) -X /3 width=3/ +| #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ] + [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 + elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HVW1 #HW12 + elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ + | minus_plus minus_minus_m_m /2 width=1/ /3 width=6/ + | (tpss_inv_sort1 … H) -H /3 width=3/ +| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H + elim (tpss_inv_lref1 … H) -H [ | -HVW1 ] + [ #H destruct + elim (lt_or_ge i d) #Hdi [ -HVW1 | ] + [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2 + elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct + elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12 + lapply (ldrop_fwd_ldrop2 … HLK2) #H + elim (lift_total W2 0 (i+1)) #U2 #HWU2 + lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 + >minus_plus minus_plus #H + lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus #H + lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus minus_plus minus_plus #H + lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus (deg_mono … Hkl2 … Hkl) -g -L -l2 /2 width=1/ +| #L #K #V #W #U1 #i #l1 #HLK #_ #HWU1 #IHVW #U2 #l2 #H + elim (ssta_inv_lref1 … H) -H * #K0 #V0 #W0 [2: #l0] #HLK0 #HVW0 #HW0U2 + lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct + lapply (IHVW … HVW0) -IHVW -HVW0 * #H1 #H2 destruct + >(lift_mono … HWU1 … HW0U2) -W0 -U1 /2 width=1/ +| #L #K #W #V #U1 #i #l1 #HLK #_ #HWU1 #IHWV #U2 #l2 #H + elim (ssta_inv_lref1 … H) -H * #K0 #W0 #V0 [2: #l0 ] #HLK0 #HWV0 #HV0U2 + lapply (ldrop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct + lapply (IHWV … HWV0) -IHWV -HWV0 * #H1 #H2 destruct + >(lift_mono … HWU1 … HV0U2) -W -U1 /2 width=1/ +| #a #I #L #V #T #U1 #l1 #_ #IHTU1 #X #l2 #H + elim (ssta_inv_bind1 … H) -H #U2 #HTU2 #H destruct + elim (IHTU1 … HTU2) -T /3 width=1/ +| #L #V #T #U1 #l1 #_ #IHTU1 #X #l2 #H + elim (ssta_inv_appl1 … H) -H #U2 #HTU2 #H destruct + elim (IHTU1 … HTU2) -T /3 width=1/ +| #L #W1 #T #U1 #l1 #_ #IHTU1 #U2 #l2 #H + lapply (ssta_inv_cast1 … H) -H #HTU2 + elim (IHTU1 … HTU2) -T /2 width=1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/frsup.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/frsup.ma new file mode 100644 index 000000000..31d6c9fee --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/frsup.ma @@ -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/grammar/cl_weight.ma". +include "basic_2/substitution/lift.ma". + +(* RESTRICTED SUPCLOSURE ****************************************************) + +inductive frsup: bi_relation lenv term ≝ +| frsup_bind_sn: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) L V +| frsup_bind_dx: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T +| frsup_flat_sn: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L V +| frsup_flat_dx: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L T +. + +interpretation + "restricted structural predecessor (closure)" + 'RestSupTerm L1 T1 L2 T2 = (frsup L1 T1 L2 T2). + +(* Basic inversion lemmas ***************************************************) + +fact frsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → + ∀J. T1 = ⓪{J} → ⊥. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #a #I #L #V #T #J #H destruct +| #a #I #L #V #T #J #H destruct +| #I #L #V #T #J #H destruct +| #I #L #V #T #J #H destruct +] +qed-. + +lemma frsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁ ⦃L2, T2⦄ → ⊥. +/2 width=7 by frsup_inv_atom1_aux/ qed-. + +fact frsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → + ∀b,J,W,U. T1 = ⓑ{b,J}W.U → + (L2 = L1 ∧ T2 = W) ∨ + (L2 = L1.ⓑ{J}W ∧ T2 = U). +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ +| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/ +| #I #L #V #T #b #J #W #U #H destruct +| #I #L #V #T #b #J #W #U #H destruct +] +qed-. + +lemma frsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁ ⦃L2, T2⦄ → + (L2 = L1 ∧ T2 = W) ∨ + (L2 = L1.ⓑ{J}W ∧ T2 = U). +/2 width=4 by frsup_inv_bind1_aux/ qed-. + +fact frsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → + ∀J,W,U. T1 = ⓕ{J}W.U → + L2 = L1 ∧ (T2 = W ∨ T2 = U). +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #a #I #L #V #T #J #W #U #H destruct +| #a #I #L #V #T #J #W #U #H destruct +| #I #L #V #T #J #W #U #H destruct /3 width=1/ +| #I #L #V #T #J #W #U #H destruct /3 width=1/ +] +qed-. + +lemma frsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁ ⦃L2, T2⦄ → + L2 = L1 ∧ (T2 = W ∨ T2 = U). +/2 width=4 by frsup_inv_flat1_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma frsup_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ +qed-. + +lemma frsup_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → #{L1} ≤ #{L2}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ +qed-. + +lemma frsup_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → #{T2} < #{T1}. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/ +qed-. + +lemma frsup_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L. +#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 +[ #a +| #a #I #L #V #_ @(ex_intro … (⋆.ⓑ{I}V)) // +] +#I #L #V #T @(ex_intro … (⋆)) // +qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lift_frsup_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → + ∀L,K,U2. ⦃L, U1⦄ ⧁ ⦃L @@ K, U2⦄ → + ∃T2. ⇧[d + |K|, e] T2 ≡ U2. +#T1 #U1 #d #e * -T1 -U1 -d -e +[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HTU1 #L #K #X #H + elim (frsup_inv_bind1 … H) -H * + [ -HTU1 #H1 #H2 destruct + >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/ + | -HVW1 #H1 #H2 destruct + >(append_inv_pair_dx … H1) -L -K normalize /2 width=2/ + ] +|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #K #X #H + elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct + >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/ +] +#i #d #e [2,3: #_ ] #L #K #X #H +elim (frsup_inv_atom1 … H) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/gdrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/gdrop.ma new file mode 100644 index 000000000..218389e1c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/gdrop.ma @@ -0,0 +1,80 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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". + +(* GLOBAL ENVIRONMENT SLICING ***********************************************) + +inductive gdrop (e:nat): relation genv ≝ +| gdrop_gt: ∀G. |G| ≤ e → gdrop e G (⋆) +| gdrop_eq: ∀G. |G| = e + 1 → gdrop e G G +| gdrop_lt: ∀I,G1,G2,V. e < |G1| → gdrop e G1 G2 → gdrop e (G1. ⓑ{I} V) G2 +. + +interpretation "global slicing" + 'RDrop e G1 G2 = (gdrop e G1 G2). + +(* basic inversion lemmas ***************************************************) + +lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆. +#G1 #G2 #e * -G1 -G2 // +[ #G #H >H -H >commutative_plus #H + lapply (le_plus_to_le_r … 0 H) -H #H + lapply (le_n_O_to_eq … H) -H #H destruct +| #I #G1 #G2 #V #H1 #_ #H2 + lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H + lapply (lt_plus_to_lt_l … 0 H) -H #H + elim (lt_zero_false … H) +] +qed-. + +lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2. +#G1 #G2 #e * -G1 -G2 // +[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H + lapply (le_plus_to_le_r … 0 H) -H #H + lapply (le_n_O_to_eq … H) -H #H destruct +| #I #G1 #G2 #V #H1 #_ normalize #H2 + <(injective_plus_l … H2) in H1; -H2 #H + elim (lt_refl_false … H) +] +qed-. + +fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V → + e < |G1| → ⇩[e] G1 ≡ G2. +#I #G #G1 #G2 #V #e * -G -G2 +[ #G #H1 #H destruct #H2 + lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H + lapply (lt_plus_to_lt_l … 0 H) -H #H + elim (lt_zero_false … H) +| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H + elim (lt_refl_false … H) +| #J #G #G2 #W #_ #HG2 #H destruct // +] +qed. + +lemma gdrop_inv_lt: ∀I,G1,G2,V,e. + ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2. +/2 width=5/ qed-. + +(* Basic properties *********************************************************) + +lemma gdrop_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2. +#e #G1 elim G1 -G1 /3 width=2/ +#I #V #G1 * #G2 #HG12 +elim (lt_or_eq_or_gt e (|G1|)) #He +[ /3 width=2/ +| destruct /3 width=2/ +| @ex_intro [2: @gdrop_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *) +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/gdrop_gdrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/gdrop_gdrop.ma new file mode 100644 index 000000000..0bc1a40d5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/gdrop_gdrop.ma @@ -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/gdrop.ma". + +(* GLOBAL ENVIRONMENT SLICING ***********************************************) + +(* Main properties **********************************************************) + +theorem gdrop_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2. +#G #G1 #e #H elim H -G -G1 +[ #G #He #G2 #H + >(gdrop_inv_gt … H He) -H -He // +| #G #He #G2 #H + >(gdrop_inv_eq … H He) -H -He // +| #I #G #G1 #V #He #_ #IHG1 #G2 #H + lapply (gdrop_inv_lt … H He) -H -He /2 width=1/ +] +qed-. + +lemma gdrop_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2). +#G1 #G2 #e +elim (gdrop_total e G1) #G #HG1 +elim (genv_eq_dec G G2) #HG2 +[ destruct /2 width=1/ +| @or_intror #HG12 + lapply (gdrop_mono … HG1 … HG12) -HG1 -HG12 /2 width=1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop.ma new file mode 100644 index 000000000..9511648aa --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop.ma @@ -0,0 +1,313 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/cl_weight.ma". +include "basic_2/substitution/lift.ma". +include "basic_2/substitution/lsubs.ma". + +(* LOCAL ENVIRONMENT SLICING ************************************************) + +(* Basic_1: includes: drop_skip_bind *) +inductive ldrop: nat → nat → relation lenv ≝ +| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆) +| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) +| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2 +| ldrop_skip : ∀L1,L2,I,V1,V2,d,e. + ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 → + ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) +. + +interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2). + +definition l_liftable: (lenv → relation term) → Prop ≝ + λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K → + ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2. + +definition l_deliftable_sn: (lenv → relation term) → Prop ≝ + λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K → + ∀T1. ⇧[d, e] T1 ≡ U1 → + ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2. + +definition dropable_sn: relation lenv → Prop ≝ + λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 → + ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2. + +definition dedropable_sn: relation lenv → Prop ≝ + λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 → + ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2. + +definition dropable_dx: relation lenv → Prop ≝ + λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 → + ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2. + +(* Basic inversion lemmas ***************************************************) + +fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ // +| // +| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +(* Basic_1: was: drop_gen_refl *) +lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2. +/2 width=5/ qed-. + +fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ → + L2 = ⋆. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V #e #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +(* Basic_1: was: drop_gen_sort *) +lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆. +/2 width=5/ qed-. + +fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → + ∀K,I,V. L1 = K. ⓑ{I} V → + (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ + (0 < e ∧ ⇩[d, e - 1] K ≡ L2). +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #K #I #V #H destruct +| #L #I #V #_ #K #J #W #HX destruct /3 width=1/ +| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 → + (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ + (0 < e ∧ ⇩[0, e - 1] K ≡ L2). +/2 width=3/ qed-. + +lemma ldrop_inv_pair1: ∀K,I,V,L2. ⇩[0, 0] K. ⓑ{I} V ≡ L2 → L2 = K. ⓑ{I} V. +#K #I #V #L2 #H +elim (ldrop_inv_O1 … H) -H * // #H destruct +elim (lt_refl_false … H) +qed-. + +(* Basic_1: was: drop_gen_drop *) +lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2. + ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2. +#e #K #I #V #L2 #H #He +elim (ldrop_inv_O1 … H) -H * // #H destruct +elim (lt_refl_false … He) +qed-. + +fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d → + ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 & + ⇧[d - 1, e] V2 ≡ V1 & + L2 = K2. ⓑ{I} V2. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #I #K #V #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) +| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/ +] +qed. + +(* Basic_1: was: drop_gen_skip_l *) +lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d → + ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 & + ⇧[d - 1, e] V2 ≡ V1 & + L2 = K2. ⓑ{I} V2. +/2 width=3/ qed-. + +fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d → + ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & + ⇧[d - 1, e] V2 ≡ V1 & + L1 = K1. ⓑ{I} V1. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #I #K #V #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) +| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/ +] +qed. + +(* Basic_1: was: drop_gen_skip_r *) +lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d → + ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 & + L1 = K1. ⓑ{I} V1. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was by definition: drop_refl *) +lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L. +#L elim L -L // +qed. + +lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e. + ⇩[0, e - 1] L1 ≡ L2 → 0 < e → ⇩[0, e] L1. ⓑ{I} V ≡ L2. +#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ +qed. + +lemma ldrop_skip_lt: ∀L1,L2,I,V1,V2,d,e. + ⇩[d - 1, e] L1 ≡ L2 → ⇧[d - 1, e] V2 ≡ V1 → 0 < d → + ⇩[d, e] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/ +qed. + +lemma ldrop_O1_le: ∀i,L. i ≤ |L| → ∃K. ⇩[0, i] L ≡ K. +#i @(nat_ind_plus … i) -i /2 width=2/ +#i #IHi * +[ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct +| #L #I #V normalize #H + elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/ +] +qed. + +lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V. +#L elim L -L +[ #i #H elim (lt_zero_false … H) +| #L #I #V #IHL #i @(nat_ind_plus … i) -i /2 width=4/ + #i #_ normalize #H + elim (IHL i ? ) -IHL /2 width=1/ -H /3 width=4/ +] +qed. + +lemma ldrop_lsubs_ldrop2_abbr: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → + ∀K2,V,i. ⇩[0, i] L2 ≡ K2. ⓓV → + d ≤ i → i < d + e → + ∃∃K1. K1 ≼ [0, d + e - i - 1] K2 & + ⇩[0, i] L1 ≡ K1. ⓓV. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e +[ #d #e #K1 #V #i #H + lapply (ldrop_inv_atom1 … H) -H #H destruct +| #L1 #L2 #K1 #V #i #_ #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie + elim (ldrop_inv_O1 … H) -H * #Hi #HLK1 + [ -IHL12 -Hie destruct + minus_minus_comm >arith_b1 // /4 width=3/ + ] +| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie + elim (ldrop_inv_O1 … H) -H * #Hi #HLK1 + [ -IHL12 -Hie -Hi destruct + | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/ + ] +| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide + elim (le_inv_plus_l … Hdi) #Hdim #Hi + lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1 + elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/ +] +qed. + +lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R). +#R #HR #L1 #K1 #d #e #HLK1 #L2 #H elim H -L2 +[ #L2 #HL12 + elim (HR … HLK1 … HL12) -HR -L1 /3 width=3/ +| #L #L2 #_ #HL2 * #K #HK1 #HLK + elim (HR … HLK … HL2) -HR -L /3 width=3/ +] +qed. + +lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R). +#R #HR #L1 #K1 #d #e #HLK1 #K2 #H elim H -K2 +[ #K2 #HK12 + elim (HR … HLK1 … HK12) -HR -K1 /3 width=3/ +| #K #K2 #_ #HK2 * #L #HL1 #HLK + elim (HR … HLK … HK2) -HR -K /3 width=3/ +] +qed. + +lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R). +#R #HR #L1 #L2 #H elim H -L2 +[ #L2 #HL12 #K2 #e #HLK2 + elim (HR … HL12 … HLK2) -HR -L2 /3 width=3/ +| #L #L2 #_ #HL2 #IHL1 #K2 #e #HLK2 + elim (HR … HL2 … HLK2) -HR -L2 #K #HLK #HK2 + elim (IHL1 … HLK) -L /3 width=5/ +] +qed. + +(* Basic forvard lemmas *****************************************************) + +(* Basic_1: was: drop_S *) +lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 → + ⇩[O, e + 1] L1 ≡ K2. +#L1 elim L1 -L1 +[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct +| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #H + [ -IHL1 destruct /2 width=1/ + | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/ + ] +] +qed-. + +lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → #{L2} ≤ #{L1}. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize +[ /2 width=3/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 + >(tw_lift … HV21) -HV21 /2 width=1/ +] +qed-. + +lemma ldrop_pair2_fwd_fw: ∀I,L,K,V,d,e. ⇩[d, e] L ≡ K. ⓑ{I} V → + ∀T. #{K, V} < #{L, T}. +#I #L #K #V #d #e #H #T +lapply (ldrop_fwd_lw … H) -H #H +@(le_to_lt_to_lt … H) -H /3 width=1/ +qed-. + +lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e. + ⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|. +#L1 elim L1 -L1 +[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct +| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #H + [ -IHL1 destruct // + | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/ + ] +] +qed-. + +lemma ldrop_fwd_O1_length: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → |L2| = |L1| - e. +#L1 elim L1 -L1 +[ #L2 #e #H >(ldrop_inv_atom1 … H) -H // +| #K1 #I1 #V1 #IHL1 #L2 #e #H + elim (ldrop_inv_O1 … H) -H * #He #H + [ -IHL1 destruct // + | lapply (IHL1 … H) -IHL1 -H #H >H -H normalize + >minus_le_minus_minus_comm // + ] +] +qed-. + +(* Basic_1: removed theorems 50: + drop_ctail drop_skip_flat + cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf + drop_clear drop_clear_O drop_clear_S + clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r + clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle + getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans + getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt + getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev + drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge + getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O + getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le + getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_append.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_append.ma new file mode 100644 index 000000000..359d39c80 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_append.ma @@ -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/ldrop.ma". + +(* DROPPING *****************************************************************) + +(* Properties on append for local environments ******************************) + +fact ldrop_O1_append_sn_le_aux: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → + d = 0 → e ≤ |L1| → + ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize // /4 width=1/ +#d #e #_ #H #L -d +lapply (le_n_O_to_eq … H) -H // +qed-. + +lemma ldrop_O1_append_sn_le: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → e ≤ |L1| → + ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2. +/2 width=3 by ldrop_O1_append_sn_le_aux/ qed. + +(* Inversion lemmas on append for local environments ************************) + +lemma ldrop_O1_inv_append1_ge: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K → + |L2| ≤ e → ⇩[0, e - |L2|] L1 ≡ K. +#K #L1 #L2 elim L2 -L2 normalize // +#L2 #I #V #IHL2 #e #H #H1e +elim (ldrop_inv_O1 … H) -H * #H2e #HL12 destruct +[ lapply (le_n_O_to_eq … H1e) -H1e -IHL2 + >commutative_plus normalize #H destruct +| minus_minus_comm /3 width=1/ +] +qed-. + +lemma ldrop_O1_inv_append1_le: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K → e ≤ |L2| → + ∀K2. ⇩[0, e] L2 ≡ K2 → K = L1 @@ K2. +#K #L1 #L2 elim L2 -L2 normalize +[ #e #H1 #H2 #K2 #H3 + lapply (le_n_O_to_eq … H2) -H2 #H2 + lapply (ldrop_inv_atom1 … H3) -H3 #H3 destruct + >(ldrop_inv_refl … H1) -H1 // +| #L2 #I #V #IHL2 #e @(nat_ind_plus … e) -e [ -IHL2 ] + [ #H1 #_ #K2 #H2 + lapply (ldrop_inv_refl … H1) -H1 #H1 + lapply (ldrop_inv_refl … H2) -H2 #H2 destruct // + | #e #_ #H1 #H1e #K2 #H2 + lapply (ldrop_inv_ldrop1 … H1 ?) -H1 // + lapply (ldrop_inv_ldrop1 … H2 ?) -H2 // /3 width=4/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_ldrop.ma new file mode 100644 index 000000000..07d9c53e4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_ldrop.ma @@ -0,0 +1,176 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/lift_lift.ma". +include "basic_2/substitution/ldrop.ma". + +(* DROPPING *****************************************************************) + +(* Main properties **********************************************************) + +(* Basic_1: was: drop_mono *) +theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 → + ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2. +#d #e #L #L1 #H elim H -d -e -L -L1 +[ #d #e #L2 #H + >(ldrop_inv_atom1 … H) -L2 // +| #K #I #V #L2 #HL12 + <(ldrop_inv_refl … HL12) -L2 // +| #L #K #I #V #e #_ #IHLK #L2 #H + lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/ +| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H + elim (ldrop_inv_skip1 … H ?) -H // (lift_inj … HVT1 … HVT2) -HVT1 -HVT2 + >(IHLK1 … HLK2) -IHLK1 -HLK2 // +] +qed-. + +(* Basic_1: was: drop_conf_ge *) +theorem ldrop_conf_ge: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 → + ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 + e1 ≤ e2 → + ⇩[0, e2 - e1] L1 ≡ L2. +#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1 +[ #d #e #e2 #L2 #H + >(ldrop_inv_atom1 … H) -L2 // +| // +| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2 + minus_minus_comm /3 width=1/ +| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2 + lapply (transitive_le 1 … Hdee2) // #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2 + lapply (transitive_le (1 + e) … Hdee2) // #Hee2 + @ldrop_ldrop_lt >minus_minus_comm /3 width=1/ (**) (* explicit constructor *) +] +qed. + +(* Note: apparently this was missing in basic_1 *) +theorem ldrop_conf_be: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → + ∃∃L. ⇩[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L. +#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct + lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ +| normalize #L0 #K0 #I #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HL20 + [ -IHLK0 -He21 destruct plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + minus_le_minus_minus_comm // /3 width=3/ + ] +] +qed. + +(* Basic_1: was: drop_trans_ge *) +theorem ldrop_trans_ge: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L → + ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2. +#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L +[ #d #e #e2 #L2 #H + >(ldrop_inv_atom1 … H) -H -L2 // +| // +| /3 width=1/ +| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2 + lapply (lt_to_le_to_lt 0 … Hde2) // #He2 + lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HL2 + @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2 width=1/ (**) (* explicit constructor *) +] +qed. + +(* Basic_1: was: drop_trans_le *) +theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L → + ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 → + ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2. +#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L +[ #d #e #e2 #L2 #H + >(ldrop_inv_atom1 … H) -L2 /2 width=3/ +| #K #I #V #e2 #L2 #HL2 #H + lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/ +| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H + lapply (le_n_O_to_eq … H) -H #H destruct + elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0 + lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d + elim (ldrop_inv_O1 … H) -H * + [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/ + | -HL12 -HV12 #He2 #HL2 + elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ] + ] +] +qed. + +(* Basic_1: was: drop_conf_rev *) +axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L → + ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1. + +(* Basic_1: was: drop_conf_lt *) +lemma ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 → + ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 → + e2 < d1 → let d ≝ d1 - e2 - 1 in + ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 & + ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2. +#d1 #e1 #L #L1 #H1 #e2 #K2 #I #V2 #H2 #He2d1 +elim (ldrop_conf_le … H1 … H2 ?) -L [2: /2 width=2/] #K #HL1K #HK2 +elim (ldrop_inv_skip1 … HK2 ?) -HK2 [2: /2 width=1/] #K1 #V1 #HK21 #HV12 #H destruct /2 width=5/ +qed. + +lemma ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L. + ⇩[d1, e1] L1 ≡ L → ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → + ⇩[0, e2 + e1] L1 ≡ L2. +#e1 #e1 #e2 >commutative_plus /2 width=5/ +qed. + +lemma ldrop_conf_div: ∀I1,L,K,V1,e1. ⇩[0, e1] L ≡ K. ⓑ{I1} V1 → + ∀I2,V2,e2. ⇩[0, e2] L ≡ K. ⓑ{I2} V2 → + ∧∧ e1 = e2 & I1 = I2 & V1 = V2. +#I1 #L #K #V1 #e1 #HLK1 #I2 #V2 #e2 #HLK2 +elim (le_or_ge e1 e2) #He +[ lapply (ldrop_conf_ge … HLK1 … HLK2 ?) +| lapply (ldrop_conf_ge … HLK2 … HLK1 ?) +] -HLK1 -HLK2 // #HK +lapply (ldrop_fwd_O1_length … HK) #H +elim (discr_minus_x_xy … H) -H +[1,3: normalize H in HK; #HK +lapply (ldrop_inv_refl … HK) -HK #H destruct +lapply (inv_eq_minus_O … H) -H /3 width=1/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_lpx.ma new file mode 100644 index 000000000..2605b921c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_lpx.ma @@ -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/substitution/ldrop.ma". + +(* DROPPING *****************************************************************) + +(* Properties on pointwise extension ****************************************) + +lemma lpx_deliftable_dropable: ∀R. t_deliftable_sn R → dropable_sn (lpx R). +#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e +[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/ +| #K1 #I #V1 #X #H + elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H + elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (IHLK1 … HL12) -L1 /3 width=3/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H + elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct + elim (HR … HV12 … HWV1) -V1 + elim (IHLK1 … HL12) -L1 /3 width=5/ +] +qed. + +lemma lpx_liftable_dedropable: ∀R. reflexive ? R → + t_liftable R → dedropable_sn (lpx R). +#R #H1R #H2R #L1 #K1 #d #e #H elim H -L1 -K1 -d -e +[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/ +| #K1 #I #V1 #X #H + elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 + elim (IHLK1 … HK12) -K1 /3 width=5/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H + elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct + elim (lift_total W2 d e) #V2 #HWV2 + lapply (H2R … HW12 … HWV1 … HWV2) -W1 + elim (IHLK1 … HK12) -K1 /3 width=5/ +] +qed. + +fact lpx_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx R L1 L2 → + d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx R K1 K2. +#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e +[ #d #e #X #H >(lpx_inv_atom2 … H) -H /2 width=3/ +| #K2 #I #V2 #X #H + elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/ +| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_ + elim (lpx_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct + elim (IHLK2 … HL12 ?) -L2 // /3 width=3/ +| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_ + >commutative_plus normalize #H destruct +] +qed. + +lemma ltpr_dropable: ∀R. dropable_dx (lpx R). +/2 width=5/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_sfr.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_sfr.ma new file mode 100644 index 000000000..78a15d70d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/ldrop_sfr.ma @@ -0,0 +1,92 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/lsubs_sfr.ma". +include "basic_2/substitution/ldrop_ldrop.ma". + +(* DROPPING *****************************************************************) + +(* Inversion lemmas about local env. full refinement for substitution *******) + +(* Note: ldrop_ldrop not needed *) +lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≽ [d, e] L → + d ≤ i → i < d + e → I = Abbr. +#I #L elim L -L +[ #K #V #i #H + lapply (ldrop_inv_atom1 … H) -H #H destruct +| #L #J #W #IHL #K #V #i #H + elim (ldrop_inv_O1 … H) -H * + [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct + lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct + lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H + elim (lsubs_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct // + | #Hi #HLK #d @(nat_ind_plus … d) -d + [ #e #H #_ #Hide + elim (sfr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct + @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/ + | #d #_ #e #H #Hdi #Hide + lapply (sfr_inv_skip … H ?) -H // #HL + @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/ + ] + ] +] +qed-. + +(* Properties about local env. full refinement for substitution *************) + +(* Note: ldrop_ldrop not needed *) +lemma sfr_ldrop: ∀L,d,e. + (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) → + ≽ [d, e] L. +#L elim L -L // +#L #I #V #IHL #d @(nat_ind_plus … d) -d +[ #e @(nat_ind_plus … e) -e // + #e #_ #HH + >(HH I L V 0 ? ? ?) // /5 width=6/ +| /5 width=6/ +] +qed. + +lemma sfr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 → + dd + ee ≤ d → ≽ [dd, ee] L2. +#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee +@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2 +lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid +elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H +elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct +@(sfr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 // +qed. + +lemma sfr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → + ∀dd,ee. ≽ [dd, ee] L1 → + dd ≤ d + e → d + e ≤ dd + ee → + ≽ [d, dd + ee - d - e] L2. +#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee +@sfr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2 +lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie +>commutative_plus in Hiddee; >minus_minus_comm commutative_plus // -Hddie /2 width=1/ +qed. + +lemma sfr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 → + d + e ≤ dd → ≽ [dd - e, ee] L2. +#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee +@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2 +elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd +>plus_minus in Hiddee; // #Hiddee +lapply (transitive_le … Hdde Hddi) -Hdde #Hid +lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2 +@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma new file mode 100644 index 000000000..36c353ba9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma @@ -0,0 +1,402 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_weight.ma". +include "basic_2/grammar/term_simple.ma". + +(* BASIC TERM RELOCATION ****************************************************) + +(* Basic_1: includes: + lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat +*) +inductive lift: nat → nat → relation term ≝ +| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k) +| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i) +| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e)) +| lift_gref : ∀p,d,e. lift d e (§p) (§p) +| lift_bind : ∀a,I,V1,V2,T1,T2,d,e. + lift d e V1 V2 → lift (d + 1) e T1 T2 → + lift d e (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) +| lift_flat : ∀I,V1,V2,T1,T2,d,e. + lift d e V1 V2 → lift d e T1 T2 → + lift d e (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) +. + +interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2). + +definition t_liftable: relation term → Prop ≝ + λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 → + ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2. + +definition t_deliftable_sn: relation term → Prop ≝ + λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → + ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2. + +(* Basic inversion lemmas ***************************************************) + +fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2. +#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/ +qed. + +lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2. +/2 width=4/ qed-. + +fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k. +#d #e #T1 #T2 * -d -e -T1 -T2 // +[ #i #d #e #_ #k #H destruct +| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +] +qed. + +lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k. +/2 width=5/ qed-. + +fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i → + (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). +#d #e #T1 #T2 * -d -e -T1 -T2 +[ #k #d #e #i #H destruct +| #j #d #e #Hj #i #Hi destruct /3 width=1/ +| #j #d #e #Hj #i #Hi destruct /3 width=1/ +| #p #d #e #i #H destruct +| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct +] +qed. + +lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → + (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). +/2 width=3/ qed-. + +lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i. +#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // +#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd +elim (lt_refl_false … Hdd) +qed-. + +lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e). +#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // +#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd +elim (lt_refl_false … Hdd) +qed-. + +fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p. +#d #e #T1 #T2 * -d -e -T1 -T2 // +[ #i #d #e #_ #k #H destruct +| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +] +qed. + +lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p. +/2 width=5/ qed-. + +fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → + ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 → + ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 & + T2 = ⓑ{a,I} V2. U2. +#d #e #T1 #T2 * -d -e -T1 -T2 +[ #k #d #e #a #I #V1 #U1 #H destruct +| #i #d #e #_ #a #I #V1 #U1 #H destruct +| #i #d #e #_ #a #I #V1 #U1 #H destruct +| #p #d #e #a #I #V1 #U1 #H destruct +| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/ +| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct +] +qed. + +lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⇧[d,e] ⓑ{a,I} V1. U1 ≡ T2 → + ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 & + T2 = ⓑ{a,I} V2. U2. +/2 width=3/ qed-. + +fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → + ∀I,V1,U1. T1 = ⓕ{I} V1.U1 → + ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 & + T2 = ⓕ{I} V2. U2. +#d #e #T1 #T2 * -d -e -T1 -T2 +[ #k #d #e #I #V1 #U1 #H destruct +| #i #d #e #_ #I #V1 #U1 #H destruct +| #i #d #e #_ #I #V1 #U1 #H destruct +| #p #d #e #I #V1 #U1 #H destruct +| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct +| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/ +] +qed. + +lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] ⓕ{I} V1. U1 ≡ T2 → + ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 & + T2 = ⓕ{I} V2. U2. +/2 width=3/ qed-. + +fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k. +#d #e #T1 #T2 * -d -e -T1 -T2 // +[ #i #d #e #_ #k #H destruct +| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +] +qed. + +(* Basic_1: was: lift_gen_sort *) +lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k. +/2 width=5/ qed-. + +fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i → + (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). +#d #e #T1 #T2 * -d -e -T1 -T2 +[ #k #d #e #i #H destruct +| #j #d #e #Hj #i #Hi destruct /3 width=1/ +| #j #d #e #Hj #i #Hi destruct (plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/ +qed. + +lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i. +/2 width=1/ qed-. + +(* Basic_1: was: lift_r *) +lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T. +#T elim T -T +[ * #i // #d elim (lt_or_ge i d) /2 width=1/ +| * /2 width=1/ +] +qed. + +lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2. +#T1 elim T1 -T1 +[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/ +| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e + elim (IHV1 d e) -IHV1 #V2 #HV12 + [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/ + | elim (IHT1 d e) -IHT1 /3 width=2/ + ] +] +qed. + +(* Basic_1: was: lift_free (right to left) *) +lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 → + ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 → + ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2. +#d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2 +[ /3 width=3/ +| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_ + lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/ +| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12 + lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21 + >(plus_minus_m_m e2 e1 ?) // /3 width=3/ +| /3 width=3/ +| #a #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 + elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b + elim (IHT (d2+1) … ? ? He12) /2 width=1/ /3 width=5/ +| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 + elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b + elim (IHT d2 … ? ? He12) // /3 width=5/ +] +qed. + +(* Basic_1: was only: dnf_dec2 dnf_dec *) +lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2). +#T1 elim T1 -T1 +[ * [1,3: /3 width=2/ ] #i #d #e + elim (lt_dec i d) #Hid + [ /4 width=2/ + | lapply (false_lt_to_le … Hid) -Hid #Hid + elim (lt_dec i (d + e)) #Hide + [ @or_intror * #T1 #H + elim (lift_inv_lref2_be … H Hid Hide) + | lapply (false_lt_to_le … Hide) -Hide /4 width=2/ + ] + ] +| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e + [ elim (IHV2 d e) -IHV2 + [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2 + [ * #T1 #HT12 @or_introl /3 width=2/ + | -V1 #HT2 @or_intror * #X #H + elim (lift_inv_bind2 … H) -H /3 width=2/ + ] + | -IHT2 #HV2 @or_intror * #X #H + elim (lift_inv_bind2 … H) -H /3 width=2/ + ] + | elim (IHV2 d e) -IHV2 + [ * #V1 #HV12 elim (IHT2 d e) -IHT2 + [ * #T1 #HT12 /4 width=2/ + | -V1 #HT2 @or_intror * #X #H + elim (lift_inv_flat2 … H) -H /3 width=2/ + ] + | -IHT2 #HV2 @or_intror * #X #H + elim (lift_inv_flat2 … H) -H /3 width=2/ + ] + ] +] +qed. + +lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R). +#R #HR #T1 #T2 #H elim H -T2 +[ /3 width=7/ +| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2 + elim (lift_total T d e) /3 width=9/ +] +qed. + +lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R). +#R #HR #U1 #U2 #H elim H -U2 +[ #U2 #HU12 #T1 #d #e #HTU1 + elim (HR … HU12 … HTU1) -U1 /3 width=3/ +| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1 + elim (IHU1 … HTU1) -U1 #T #HTU #HT1 + elim (HR … HU2 … HTU) -U /3 width=5/ +] +qed-. + +(* Basic_1: removed theorems 7: + lift_head lift_gen_head + lift_weight_map lift_weight lift_weight_add lift_weight_add_O + lift_tlt_dx +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma new file mode 100644 index 000000000..3e18bff32 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma @@ -0,0 +1,217 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/lift.ma". + +(* BASIC TERM RELOCATION ****************************************************) + +(* Main properies ***********************************************************) + +(* Basic_1: was: lift_inj *) +theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2. +#d #e #T1 #U #H elim H -d -e -T1 -U +[ #k #d #e #X #HX + lapply (lift_inv_sort2 … HX) -HX // +| #i #d #e #Hid #X #HX + lapply (lift_inv_lref2_lt … HX ?) -HX // +| #i #d #e #Hdi #X #HX + lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/ +| #p #d #e #X #HX + lapply (lift_inv_gref2 … HX) -HX // +| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ +] +qed-. + +(* Basic_1: was: lift_gen_lift *) +theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T → + d1 ≤ d2 → + ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T +[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 + lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12 + lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 + lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12 + elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct + [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/ + | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H + elim (le_inv_plus_l … H) -H #Hide2 #He2i + lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12 + >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i + /4 width=3/ + ] +| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 + lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/ +| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 + lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct + elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1 + >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /3 width=5/ +| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 + lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct + elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1 + elim (IHU … HU2 ?) // /3 width=5/ +] +qed. + +(* Note: apparently this was missing in basic_1 *) +theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T → + e ≤ e1 → e1 ≤ e + e2 → + ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T +[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/ +| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2 + >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ] +| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2 + elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2 + [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/ + | >(lift_inv_lref2_ge … H ?) -H // + lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i + elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1 + @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12 + @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ] + ] +| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/ +| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2 + elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2 + elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/ +| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2 + elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + elim (IHV1 … HV2 ? ?) -V // + elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/ +] +qed. + +theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2. +#d #e #T #U1 #H elim H -d -e -T -U1 +[ #k #d #e #X #HX + lapply (lift_inv_sort1 … HX) -HX // +| #i #d #e #Hid #X #HX + lapply (lift_inv_lref1_lt … HX ?) -HX // +| #i #d #e #Hdi #X #HX + lapply (lift_inv_lref1_ge … HX ?) -HX // +| #p #d #e #X #HX + lapply (lift_inv_gref1 … HX) -HX // +| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ +] +qed-. + +(* Basic_1: was: lift_free (left to right) *) +theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → + d1 ≤ d2 → d2 ≤ d1 + e1 → ⇧[d1, e1 + e2] T1 ≡ T2. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T +[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ + >(lift_inv_sort1 … HT2) -HT2 // +| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_ + lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 + lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1/ +| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21 + lapply (lift_inv_lref1_ge … HT2 ?) -HT2 + [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/ + | -Hd21 /2 width=1/ + ] +| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ + >(lift_inv_gref1 … HT2) -HT2 // +| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10 + lapply (IHT12 … HT20 ? ?) /2 width=1/ +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10 + lapply (IHT12 … HT20 ? ?) // /2 width=1/ +] +qed. + +(* Basic_1: was: lift_d (right to left) *) +theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d2 ≤ d1 → + ∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T0 ≡ T2. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T +[ #k #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_sort1 … HX) -HX /2 width=3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ + lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2 + elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /3 width=3/ /4 width=3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21 + lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2 + lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct + >plus_plus_comm_23 /4 width=3/ +| #p #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_gref1 … HX) -HX /2 width=3/ +| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + elim (IHV12 … HV20 ?) -IHV12 -HV20 // + elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /3 width=5/ +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + elim (IHV12 … HV20 ?) -IHV12 -HV20 // + elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/ +] +qed. + +(* Basic_1: was: lift_d (left to right) *) +theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → + ∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[d1, e1] T0 ≡ T2. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T +[ #k #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_sort1 … HX) -HX /2 width=3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded + lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e + lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1/ #Hid2e + lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2 + lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ + elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/ +| #p #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_gref1 … HX) -HX /2 width=3/ +| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + elim (IHV12 … HV20 ?) -IHV12 -HV20 // + elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T + (lift_mono … H … HT1) -T // +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma new file mode 100644 index 000000000..cdc11129d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma @@ -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/substitution/lift_lift.ma". +include "basic_2/substitution/lift_vector.ma". + +(* BASIC TERM VECTOR RELOCATION *********************************************) + +(* Main properies ***********************************************************) + +theorem liftv_mono: ∀Ts,U1s,d,e. ⇧[d,e] Ts ≡ U1s → + ∀U2s:list term. ⇧[d,e] Ts ≡ U2s → U1s = U2s. +#Ts #U1s #d #e #H elim H -Ts -U1s +[ #U2s #H >(liftv_inv_nil1 … H) -H // +| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct + elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct + >(lift_mono … HTU1 … HTU2) -T /3 width=1/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma new file mode 100644 index 000000000..35ecb6535 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma @@ -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/grammar/term_vector.ma". +include "basic_2/substitution/lift.ma". + +(* BASIC TERM VECTOR RELOCATION *********************************************) + +inductive liftv (d,e:nat) : relation (list term) ≝ +| liftv_nil : liftv d e ◊ ◊ +| liftv_cons: ∀T1s,T2s,T1,T2. + ⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s → + liftv d e (T1 @ T1s) (T2 @ T2s) +. + +interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s). + +(* Basic inversion lemmas ***************************************************) + +fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊. +#T1s #T2s #d #e * -T1s -T2s // +#T1s #T2s #T1 #T2 #_ #_ #H destruct +qed. + +lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊. +/2 width=5/ qed-. + +fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → + ∀U1,U1s. T1s = U1 @ U1s → + ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s & + T2s = U2 @ U2s. +#T1s #T2s #d #e * -T1s -T2s +[ #U1 #U1s #H destruct +| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5/ +] +qed. + +lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 @ U1s ≡ T2s → + ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s & + T2s = U2 @ U2s. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s. +#d #e #T1s elim T1s -T1s +[ /2 width=2/ +| #T1 #T1s * #T2s #HT12s + elim (lift_total T1 d e) /3 width=2/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lsubs.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsubs.ma new file mode 100644 index 000000000..f27883b02 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsubs.ma @@ -0,0 +1,194 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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". + +(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************) + +inductive lsubs: nat → nat → relation lenv ≝ +| lsubs_sort: ∀d,e. lsubs d e (⋆) (⋆) +| lsubs_OO: ∀L1,L2. lsubs 0 0 L1 L2 +| lsubs_abbr: ∀L1,L2,V,e. lsubs 0 e L1 L2 → + lsubs 0 (e + 1) (L1. ⓓV) (L2.ⓓV) +| lsubs_abst: ∀L1,L2,I,V1,V2,e. lsubs 0 e L1 L2 → + lsubs 0 (e + 1) (L1. ⓑ{I}V1) (L2. ⓛV2) +| lsubs_skip: ∀L1,L2,I1,I2,V1,V2,d,e. + lsubs d e L1 L2 → lsubs (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2) +. + +interpretation + "local environment refinement (substitution)" + 'SubEq L1 d e L2 = (lsubs d e L1 L2). + +definition lsubs_trans: ∀S. (lenv → relation S) → Prop ≝ λS,R. + ∀L2,s1,s2. R L2 s1 s2 → + ∀L1,d,e. L1 ≼ [d, e] L2 → R L1 s1 s2. + +(* Basic properties *********************************************************) + +lemma lsubs_bind_eq: ∀L1,L2,e. L1 ≼ [0, e] L2 → ∀I,V. + L1. ⓑ{I} V ≼ [0, e + 1] L2.ⓑ{I} V. +#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/ +qed. + +lemma lsubs_abbr_lt: ∀L1,L2,V,e. L1 ≼ [0, e - 1] L2 → 0 < e → + L1. ⓓV ≼ [0, e] L2.ⓓV. +#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ +qed. + +lemma lsubs_abst_lt: ∀L1,L2,I,V1,V2,e. L1 ≼ [0, e - 1] L2 → 0 < e → + L1. ⓑ{I}V1 ≼ [0, e] L2. ⓛV2. +#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/ +qed. + +lemma lsubs_skip_lt: ∀L1,L2,d,e. L1 ≼ [d - 1, e] L2 → 0 < d → + ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 ≼ [d, e] L2. ⓑ{I2} V2. +#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/ +qed. + +lemma lsubs_bind_lt: ∀I,L1,L2,V,e. L1 ≼ [0, e - 1] L2 → 0 < e → + L1. ⓓV ≼ [0, e] L2. ⓑ{I}V. +* /2 width=1/ qed. + +lemma lsubs_refl: ∀d,e,L. L ≼ [d, e] L. +#d elim d -d +[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/ +| #d #IHd #e #L elim L -L // /2 width=1/ +] +qed. + +lemma TC_lsubs_trans: ∀S,R. lsubs_trans S R → lsubs_trans S (λL. (TC … (R L))). +#S #R #HR #L1 #s1 #s2 #H elim H -s2 +[ /3 width=5/ +| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12 + lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +fact lsubs_inv_atom1_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → L1 = ⋆ → + L2 = ⋆ ∨ (d = 0 ∧ e = 0). +#L1 #L2 #d #e * -L1 -L2 -d -e +[ /2 width=1/ +| /3 width=1/ +| #L1 #L2 #W #e #_ #H destruct +| #L1 #L2 #I #W1 #W2 #e #_ #H destruct +| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct +] +qed. + +lemma lsubs_inv_atom1: ∀L2,d,e. ⋆ ≼ [d, e] L2 → + L2 = ⋆ ∨ (d = 0 ∧ e = 0). +/2 width=3/ qed-. + +fact lsubs_inv_skip1_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → + ∀I1,K1,V1. L1 = K1.ⓑ{I1}V1 → 0 < d → + ∃∃I2,K2,V2. K1 ≼ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2. +#L1 #L2 #d #e * -L1 -L2 -d -e +[ #d #e #I1 #K1 #V1 #H destruct +| #L1 #L2 #I1 #K1 #V1 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ +] +qed. + +lemma lsubs_inv_skip1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ≼ [d, e] L2 → 0 < d → + ∃∃I2,K2,V2. K1 ≼ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2. +/2 width=5/ qed-. + +fact lsubs_inv_atom2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → L2 = ⋆ → + L1 = ⋆ ∨ (d = 0 ∧ e = 0). +#L1 #L2 #d #e * -L1 -L2 -d -e +[ /2 width=1/ +| /3 width=1/ +| #L1 #L2 #W #e #_ #H destruct +| #L1 #L2 #I #W1 #W2 #e #_ #H destruct +| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct +] +qed. + +lemma lsubs_inv_atom2: ∀L1,d,e. L1 ≼ [d, e] ⋆ → + L1 = ⋆ ∨ (d = 0 ∧ e = 0). +/2 width=3/ qed-. + +fact lsubs_inv_abbr2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → + ∀K2,V. L2 = K2.ⓓV → d = 0 → 0 < e → + ∃∃K1. K1 ≼ [0, e - 1] K2 & L1 = K1.ⓓV. +#L1 #L2 #d #e * -L1 -L2 -d -e +[ #d #e #K1 #V #H destruct +| #L1 #L2 #K1 #V #_ #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #W #e #HL12 #K1 #V #H #_ #_ destruct /2 width=3/ +| #L1 #L2 #I #W1 #W2 #e #_ #K1 #V #H destruct +| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #K1 #V #_ >commutative_plus normalize #H destruct +] +qed. + +lemma lsubs_inv_abbr2: ∀L1,K2,V,e. L1 ≼ [0, e] K2.ⓓV → 0 < e → + ∃∃K1. K1 ≼ [0, e - 1] K2 & L1 = K1.ⓓV. +/2 width=5/ qed-. + +fact lsubs_inv_skip2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → + ∀I2,K2,V2. L2 = K2.ⓑ{I2}V2 → 0 < d → + ∃∃I1,K1,V1. K1 ≼ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1. +#L1 #L2 #d #e * -L1 -L2 -d -e +[ #d #e #I1 #K1 #V1 #H destruct +| #L1 #L2 #I1 #K1 #V1 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/ +] +qed. + +lemma lsubs_inv_skip2: ∀I2,L1,K2,V2,d,e. L1 ≼ [d, e] K2.ⓑ{I2}V2 → 0 < d → + ∃∃I1,K1,V1. K1 ≼ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1. +/2 width=5/ qed-. + +(* Basic forward lemmas *****************************************************) + +fact lsubs_fwd_length_full1_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → + d = 0 → e = |L1| → |L1| ≤ |L2|. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize +[ // +| /2 width=1/ +| /3 width=1/ +| /3 width=1/ +| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma lsubs_fwd_length_full1: ∀L1,L2. L1 ≼ [0, |L1|] L2 → |L1| ≤ |L2|. +/2 width=5/ qed-. + +fact lsubs_fwd_length_full2_aux: ∀L1,L2,d,e. L1 ≼ [d, e] L2 → + d = 0 → e = |L2| → |L2| ≤ |L1|. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize +[ // +| /2 width=1/ +| /3 width=1/ +| /3 width=1/ +| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma lsubs_fwd_length_full2: ∀L1,L2. L1 ≼ [0, |L2|] L2 → |L2| ≤ |L1|. +/2 width=5/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lsubs_sfr.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsubs_sfr.ma new file mode 100644 index 000000000..b71f25e51 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsubs_sfr.ma @@ -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/substitution/lsubs.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************) + +(* bottom element of the refinement *) +definition sfr: nat → nat → predicate lenv ≝ + λd,e. NF_sn … (lsubs d e) (lsubs d e …). + +interpretation + "local environment full refinement (substitution)" + 'SubEqBottom d e L = (sfr d e L). + +(* Basic properties *********************************************************) + +lemma sfr_atom: ∀d,e. ≽ [d, e] ⋆. +#d #e #L #H +elim (lsubs_inv_atom2 … H) -H +[ #H destruct // +| * #H1 #H2 destruct // +] +qed. + +lemma sfr_OO: ∀L. ≽ [0, 0] L. +// qed. + +lemma sfr_abbr: ∀L,V,e. ≽ [0, e] L → ≽ [0, e + 1] L.ⓓV. +#L #V #e #HL #K #H +elim (lsubs_inv_abbr2 … H ?) -H // (plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/ + | -Hdi -Hdj #Hid + generalize in match Hide; -Hide (**) (* rewriting in the premises, rewrites in the goal too *) + >(plus_minus_m_m … Hjde) in ⊢ (% → ?); -Hjde /4 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_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide + elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 // + -Hdi -Hide /3 width=5/ +] +qed. + +lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → + ∀i. d ≤ i → i ≤ d + e → + ∃∃T. L ⊢ T1 ▶ [i, d + e - i] T & + L ⊢ T ▶ [d, i - d] T2. +#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e +[ /2 width=3/ +| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde + elim (lt_or_ge i j) + [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /4 width=4/ + | -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_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide + elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 // + -Hdi -Hide /3 width=5/ +] +qed. + +lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 → + ∀L. L @@ K ⊢ T1 ▶ [d, e] T2. +#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/ +#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L +lapply (ldrop_fwd_ldrop2_length … HK0) #H +@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *) +@(ldrop_O1_append_sn_le … HK0) /2 width=2/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} → + T2 = ⓪{I} ∨ + ∃∃K,V,i. d ≤ i & i < d + e & + ⇩[O, i] L ≡ K. ⓓV & + ⇧[O, i + 1] V ≡ T2 & + I = LRef i. +#L #T1 #T2 #d #e * -L -T1 -T2 -d -e +[ #L #I #d #e #J #H destruct /2 width=1/ +| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/ +| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct +] +qed. + +lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶ [d, e] T2 → + T2 = ⓪{I} ∨ + ∃∃K,V,i. d ≤ i & i < d + e & + ⇩[O, i] L ≡ K. ⓓV & + ⇧[O, i + 1] V ≡ T2 & + I = LRef i. +/2 width=3/ qed-. + + +(* Basic_1: was: subst1_gen_sort *) +lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶ [d, e] T2 → T2 = ⋆k. +#L #T2 #k #d #e #H +elim (tps_inv_atom1 … H) -H // +* #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +(* Basic_1: was: subst1_gen_lref *) +lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶ [d, e] T2 → + T2 = #i ∨ + ∃∃K,V. d ≤ i & i < d + e & + ⇩[O, i] L ≡ K. ⓓV & + ⇧[O, i + 1] V ≡ T2. +#L #T2 #i #d #e #H +elim (tps_inv_atom1 … H) -H /2 width=1/ +* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/ +qed-. + +lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶ [d, e] T2 → T2 = §p. +#L #T2 #p #d #e #H +elim (tps_inv_atom1 … H) -H // +* #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 → + ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → + ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & + L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 & + U2 = ⓑ{a,I} V2. T2. +#d #e #L #U1 #U2 * -d -e -L -U1 -U2 +[ #L #k #d #e #a #I #V1 #T1 #H destruct +| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct +| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/ +| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct +] +qed. + +lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶ [d, e] U2 → + ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & + L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 & + U2 = ⓑ{a,I} V2. T2. +/2 width=3/ qed-. + +fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 → + ∀I,V1,T1. U1 = ⓕ{I} V1. T1 → + ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 & + U2 = ⓕ{I} V2. T2. +#d #e #L #U1 #U2 * -d -e -L -U1 -U2 +[ #L #k #d #e #I #V1 #T1 #H destruct +| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct +| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/ +] +qed. + +lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶ [d, e] U2 → + ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 & + U2 = ⓕ{I} V2. T2. +/2 width=3/ qed-. + +fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → e = 0 → T1 = T2. +#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e +[ // +| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct + lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide shift_append_assoc normalize #H + elim (tps_inv_bind1 … H) -H + #V0 #T0 #_ #HT10 #H destruct + elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct + >append_length >HL12 -HL12 + @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) +] +qed-. + +(* Basic_1: removed theorems 25: + subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt + subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans + subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s + subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt + subst0_confluence_neq subst0_confluence_eq subst0_tlt_head + subst0_confluence_lift subst0_tlt + subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/tps_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/tps_lift.ma new file mode 100644 index 000000000..5ffc94922 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/tps_lift.ma @@ -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: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 → + ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2. +#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1 +[ // +| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct + elim (lt_or_ge i (d+1)) #HiSd + [ -Hide1 -HV0 + lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct + lapply (ldrop_mono … HLK0 … HLK) #H destruct + | -V -Hdi /2 width=4/ + ] +| /4 width=3/ +| /3 width=3/ +] +qed. + +lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 → + ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2. +/2 width=3/ qed-. + +lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 → + ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2. +#L #T1 #T2 #d #HT12 #K #V #HLK +lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12 +lapply (tps_inv_refl_O2 … HT12) -HT12 // +qed-. + +(* Relocation properties ****************************************************) + +(* Basic_1: was: subst1_lift_lt *) +lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 → + ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K → + ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 → + dt + et ≤ d → + L ⊢ U1 ▶ [dt, et] U2. +#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et +[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ + >(lift_mono … H1 … H2) -H1 -H2 // +| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd + lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid + lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct + elim (lift_trans_ge … HVW … HWU2 ?) -W // (lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/ +| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *) +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ +] +qed. + +lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 → + ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K → + ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 → + dt ≤ d → d ≤ dt + et → + L ⊢ U1 ▶ [dt, et + e] U2. +#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et +[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_ + >(lift_mono … H1 … H2) -H1 -H2 // +| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_ + elim (lift_inv_lref1 … H) -H * #Hid #H destruct + [ -Hdtd + lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete + elim (lift_trans_ge … HVW … HWU2 ?) -W // (lift_mono … HVY … HVW) -V #H destruct /2 width=4/ + | -Hdti + lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti + lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/ + ] +| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ] + ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *) +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/ +] +qed. + +(* Basic_1: was: subst1_lift_ge *) +lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 → + ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K → + ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 → + d ≤ dt → + L ⊢ U1 ▶ [dt + e, et] U2. +#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et +[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ + >(lift_mono … H1 … H2) -H1 -H2 // +| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt + lapply (transitive_le … Hddt … Hdti) -Hddt #Hid + lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct + lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2 + lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/ +| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *) +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/ +] +qed. + +(* Basic_1: was: subst1_gen_lift_lt *) +lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt + et ≤ d → + ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et +[ #L * #i #dt #et #K #d #e #_ #T1 #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/ + | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/ + ] +| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd + lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid + lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct + elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV + elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus 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 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct + elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2 + elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *) + /3 width=5/ +| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet + elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct + elim (IHV12 … HLK … HWV1 ? ?) -V1 // + elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/ +] +qed. + +(* Basic_1: was: subst1_gen_lift_ge *) +lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + d + e ≤ dt → + ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et +[ #L * #i #dt #et #K #d #e #_ #T1 #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/ + | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/ + | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/ + ] +| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt + lapply (transitive_le … Hdedt … Hdti) #Hdei + elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt + elim (le_inv_plus_l … Hdei) #Hdie #Hei + lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct + lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV + elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie + #V0 #HV10 >plus_minus // plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *) +| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd + elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct + elim (le_inv_plus_l … Hdetd) #_ #Hedt + elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2 + elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ] + IHV12 // >IHT12 // +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct + >IHV12 // >IHT12 // +] +qed. +(* + Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?) + (subst0 i v t1 (lift h d u2)) -> + (le (plus d h) i) -> + (EX u1 | (subst0 (minus i h) v u1 u2) & + t1 = (lift h d u1) + ). + + + Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?) + (subst0 i v t1 (lift h d u2)) -> + (le d i) -> (lt i (plus d h)) -> + (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)). +*) +lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → + ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet +elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 +lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 +lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct +elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU +lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus (lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/ + ] +| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02 + elim (IHV01 … HV02) -V0 #V #HV1 #HV2 + elim (IHT01 … HT02) -T0 #T #HT1 #HT2 + lapply (tps_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ + lapply (tps_lsubs_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/ +| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02) -V0 + elim (IHT01 … HT02) -T0 /3 width=5/ +] +qed. + +(* Basic_1: was: subst1_confluence_neq *) +theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 → + ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 → + (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) → + ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T. +#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1 +[ /2 width=3/ +| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2 + elim (tps_inv_lref1 … H1) -H1 + [ #H destruct /4 width=4/ + | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded + [ -Hd1 -Hde2 + lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H + lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H + elim (lt_refl_false … H) + | -Hd2 -Hde1 + lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H + lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H + elim (lt_refl_false … H) + ] + ] +| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2 + elim (IHT01 … HT02 ?) -T0 + [ -H #T #HT1 #HT2 + lapply (tps_lsubs_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/ + lapply (tps_lsubs_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/ + | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H + [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *) + ] +| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02 H) -V0 + elim (IHT01 … HT02 H) -T0 -H /3 width=5/ +] +qed. + +(* Note: the constant 1 comes from tps_subst *) +(* Basic_1: was: subst1_trans *) +theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 → + ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e → + L ⊢ T1 ▶ [d, e] T2. +#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e +[ #L #I #d #e #T2 #H #He + elim (tps_inv_atom1 … H) -H + [ #H destruct // + | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct + lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/ + ] +| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He + lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2 + <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/ +| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct + lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02 + lapply (IHT10 … HT02 He) -T0 #HT12 + lapply (tps_lsubs_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/ +| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/ +] +qed. + +theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 → + ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 → + ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2. +#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1 +[ /2 width=3/ +| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1 + lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1 + lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2 + <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4 width=4/ +| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02 + elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V + elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2 + lapply (tps_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ + lapply (tps_lsubs_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/ +| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV10 … HV02 ?) -V0 // + elim (IHT10 … HT02 ?) -T0 // /3 width=6/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/delift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift.ma new file mode 100644 index 000000000..e8ac23dae --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift.ma @@ -0,0 +1,108 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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". + +(* INVERSE BASIC TERM RELOCATION *******************************************) + +definition delift: nat → nat → lenv → relation term ≝ + λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T. + +interpretation "inverse basic relocation (term)" + 'TSubst L T1 d e T2 = (delift d e L T1 T2). + +(* Basic properties *********************************************************) + +lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 → + ∀L. L ⊢ ▼*[d, e] T2 ≡ T1. +/2 width=3/ qed. + +lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T. +/2 width=3/ qed. + +lemma delift_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 → + ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2. +#L1 #T1 #T2 #d #e * /3 width=3/ +qed. + +lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k. +/2 width=3/ qed. + +lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i. +/3 width=3/ qed. + +lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e). +/3 width=3/ qed. + +lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p. +/2 width=3/ qed. + +lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e. + L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 → + L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2. +#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2 +lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/ +qed. + +lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e. + L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 → + L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2. +#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/ +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k. +#L #U2 #d #e #k * #U #HU +>(tpss_inv_sort1 … HU) -HU #HU2 +>(lift_inv_sort2 … HU2) -HU2 // +qed-. + +lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p. +#L #U #d #e #p * #U #HU +>(tpss_inv_gref1 … HU) -HU #HU2 +>(lift_inv_gref2 … HU2) -HU2 // +qed-. + +lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 → + ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 & + L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 & + U2 = ⓑ{a,I} V2. T2. +#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 +elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct +elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2 +lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ +qed-. + +lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 → + ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 & + L ⊢ ▼*[d, e] T1 ≡ T2 & + U2 = ⓕ{I} V2. T2. +#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2 +elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct +elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/ +qed-. + +lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2. +#L #T1 #T2 #d * #T #HT1 +>(tpss_inv_refl_O2 … HT1) -HT1 #HT2 +>(lift_inv_refl_O2 … HT2) -HT2 // +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → #{T1} ≤ #{T2}. +#L #T1 #T2 #d #e * #T #HT1 #HT2 +>(tw_lift … HT2) -T2 /2 width=4 by tpss_fwd_tw / +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_alt.ma new file mode 100644 index 000000000..9a3eb1b7c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_alt.ma @@ -0,0 +1,100 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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". + +(* INVERSE BASIC TERM RELOCATION *******************************************) + +(* alternative definition of inverse basic term relocation *) +inductive delifta: nat → nat → lenv → relation term ≝ +| delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k) +| delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i) +| delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → + ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 → + ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2 +| delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e)) +| delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p) +| delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e. + delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 → + delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) +| delifta_flat : ∀L,I,V1,V2,T1,T2,d,e. + delifta d e L V1 V2 → delifta d e L T1 T2 → + delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) +. + +interpretation "inverse basic relocation (term) alternative" + 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2). + +(* Basic properties *********************************************************) + +lemma delifta_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 → + ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2. +#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/ +[ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 + elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ +| /4 width=1/ +| /3 width=1/ +] +qed. + +lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2. +#L #T1 @(fw_ind … L T1) -L -T1 #L #T1 elim T1 -T1 +[ * #i #IH #T2 #d #e #H + [ >(delift_inv_sort1 … H) -H // + | elim (delift_inv_lref1 … H) -H * /2 width=1/ + #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2 + lapply (ldrop_pair2_fwd_fw … HLK) #H + lapply (IH … HV12) // -H /2 width=6/ + | >(delift_inv_gref1 … H) -H // + ] +| * [ #a ] #I #V1 #T1 #_ #_ #IH #X #d #e #H + [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12 + lapply (IH … HV12) -HV12 // #HV12 + lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12 + lapply (delifta_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ + | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + lapply (IH … HV12) -HV12 // + lapply (IH … HT12) -IH -HT12 // /2 width=1/ + ] +] +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2. +#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/ +qed-. + +lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term. + (∀L,d,e,k. R d e L (⋆k) (⋆k)) → + (∀L,d,e,i. i < d → R d e L (#i) (#i)) → + (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → + ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 → + ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2 + ) → + (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) → + (∀L,d,e,p. R d e L (§p) (§p)) → + (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 → + L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 → + R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) + ) → + (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 → + L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 → + R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) + ) → + ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2. +#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e +// /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_delift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_delift.ma new file mode 100644 index 000000000..a5c563565 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_delift.ma @@ -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: ∀L,T,T1,T2,d,e. + L ⊢ ▼*[d, e] T ≡ T1 → L ⊢ ▼*[d, e] T ≡ T2 → T1 = T2. +#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2 +elim (tpss_conf_eq … H1TU1 … H1TU2) -T #U #HU1 #HU2 +lapply (tpss_inv_lift1_eq … HU1 … H2TU1) -HU1 #H destruct +lapply (tpss_inv_lift1_eq … HU2 … H2TU2) -HU2 #H destruct +lapply (lift_inj … H2TU1 … H2TU2) // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_lift.ma new file mode 100644 index 000000000..01ee6108e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/delift_lift.ma @@ -0,0 +1,167 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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_sfr.ma". +include "basic_2/unfold/tpss_lift.ma". +include "basic_2/unfold/delift.ma". + +(* INVERSE BASIC TERM RELOCATION *******************************************) + +(* Advanced properties ******************************************************) + +lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e → + ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 → + ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2. +#L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2 +elim (lift_total V 0 (i+1)) #U #HVU +lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus commutative_plus in ⊢ (??%??→?); 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 #d #e #Hde #HL #H destruct + elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12 + elim (IH (L.ⓑ{I}V1) T1 ? ? (d+1) e ? ? ?) -IH [3,6: // |2: skip |4,5: /2 width=1/ ] -Hde -HL #T2 #HT12 + lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/ +| #I #V1 #T1 #d #e #Hde #HL #H destruct + elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12 + elim (IH … T1 … Hde HL ?) -IH -Hde -HL [3,4: // |2: skip ] /3 width=2/ +] +qed. + +lemma sfr_delift: ∀L,T1,d,e. d + e ≤ |L| → ≽ [d, e] L → + ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2. +/2 width=2/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i. +#L #U2 #i #d #e * #U #HU #HU2 #Hid +elim (tpss_inv_lref1 … HU) -HU +[ #H destruct >(lift_inv_lref2_lt … HU2) // +| * #K #V1 #V2 #Hdi + lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi + elim (lt_refl_false … Hi) +] +qed-. + +lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 → + d ≤ i → i < d + e → + ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 & + K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 & + ⇧[0, d] V2 ≡ U2. +#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide +elim (tpss_inv_lref1 … HU) -HU +[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) // +| * #K #V1 #V #_ #_ #HLK #HV1 #HVU + elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/ +] +qed-. + +lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → + d + e ≤ i → U2 = #(i - e). +#L #U2 #i #d #e * #U #HU #HU2 #Hdei +elim (tpss_inv_lref1 … HU) -HU +[ #H destruct >(lift_inv_lref2_ge … HU2) // +| * #K #V1 #V2 #_ #Hide + lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi + elim (lt_refl_false … Hi) +] +qed-. + +lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → + ∨∨ (i < d ∧ U2 = #i) + | (∃∃K,V1,V2. d ≤ i & i < d + e & + ⇩[0, i] L ≡ K. ⓓV1 & + K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 & + ⇧[0, d] V2 ≡ U2 + ) + | (d + e ≤ i ∧ U2 = #(i - e)). +#L #U2 #i #d #e #H +elim (lt_or_ge i d) #Hdi +[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/ +| elim (lt_or_ge i (d+e)) #Hide + [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/ + | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/ + ] +] +qed-. + +(* Properties on basic term relocation **************************************) + +lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 → + ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K → + ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 → + L ⊢ ▼*[dt, et] U1 ≡ U2. +#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2 +elim (lift_total T d e) #U #HTU +lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1 +elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU +>(lift_mono … HTU2 … HT2) -T2 /2 width=3/ +qed. + +lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 → + ∀L,U1,d,e. dt ≤ d → d ≤ dt + et → + ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → + L ⊢ ▼*[dt, et + e] U1 ≡ T2. +#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 +elim (lift_total T d e) #U #HTU +lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1 +lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/ +qed. + +lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 → + ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K → + ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → + L ⊢ ▼*[dt + e, et] U1 ≡ U2. +#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2 +elim (lift_total T d e) #U #HTU +lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1 +elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU +>(lift_mono … HTU2 … HT2) -T2 /2 width=3/ +qed. + +lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 → + ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2. +#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1 +lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct +lapply (lift_inj … HTU1 … HTU2) -U2 // +qed-. + +lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T → + ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e → + L ⊢ ▼*[d, e] T1 ≡ T2. +#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide +lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10 +lapply (lift_trans_be … HT2 … HT0 ? ?) -T // +>commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?); +append_assoc #H +elim (append_inj_dx … H ?) -H // #_ #H destruct +(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/ +| /2 width=5 by lift_frsupp_trans/ +] +qed-. + +(* Advanced inversion lemmas for frsupp **************************************) + +lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥. +#J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 // +#L2 #T2 #H elim (frsup_inv_atom1 … H) +qed-. + +lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ → + ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄. +#b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2 +[ #L2 #T2 #H + elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/ +| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ +] +qed-. + +lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ → + ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄. +#J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2 +[ #L2 #T2 #H + elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/ +| #L #T #L2 #T2 #_ #HT2 * /3 width=4/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/frsups_frsups.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/frsups_frsups.ma new file mode 100644 index 000000000..e7b7de26e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/frsups_frsups.ma @@ -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 … frsups. +/2 width=4/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2.ma new file mode 100644 index 000000000..562b79530 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2.ma @@ -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/grammar/term_vector.ma". + +(* GENERIC RELOCATION WITH PAIRS ********************************************) + +inductive at: list2 nat nat → relation nat ≝ +| at_nil: ∀i. at ⟠ i i +| at_lt : ∀des,d,e,i1,i2. i1 < d → + at des i1 i2 → at ({d, e} @ des) i1 i2 +| at_ge : ∀des,d,e,i1,i2. d ≤ i1 → + at des (i1 + e) i2 → at ({d, e} @ des) i1 i2 +. + +interpretation "application (generic relocation with pairs)" + 'RAt i1 des i2 = (at des i1 i2). + +(* Basic inversion lemmas ***************************************************) + +fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ⟠ → i1 = i2. +#des #i1 #i2 * -des -i1 -i2 +[ // +| #des #d #e #i1 #i2 #_ #_ #H destruct +| #des #d #e #i1 #i2 #_ #_ #H destruct +] +qed. + +lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2. +/2 width=3/ qed-. + +fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → + ∀d,e,des0. des = {d, e} @ des0 → + i1 < d ∧ @⦃i1, des0⦄ ≡ i2 ∨ + d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2. +#des #i1 #i2 * -des -i1 -i2 +[ #i #d #e #des #H destruct +| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/ +| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/ +] +qed. + +lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 → + i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨ + d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2. +/2 width=3/ qed-. + +lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 → + i1 < d → @⦃i1, des⦄ ≡ i2. +#des #d #e #i1 #e2 #H +elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d +lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd +elim (lt_refl_false … Hd) +qed-. + +lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 → + d ≤ i1 → @⦃i1 + e, des⦄ ≡ i2. +#des #d #e #i1 #e2 #H +elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1 +lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd +elim (lt_refl_false … Hd) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_gr2.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_gr2.ma new file mode 100644 index 000000000..20ce856d6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_gr2.ma @@ -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/gr2.ma". + +(* GENERIC RELOCATION WITH PAIRS ********************************************) + +(* Main properties **********************************************************) + +theorem at_mono: ∀des,i,i1. @⦃i, des⦄ ≡ i1 → ∀i2. @⦃i, des⦄ ≡ i2 → i1 = i2. +#des #i #i1 #H elim H -des -i -i1 +[ #i #x #H <(at_inv_nil … H) -x // +| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H + lapply (at_inv_cons_lt … H Hid) -H -Hid /2 width=1/ +| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H + lapply (at_inv_cons_ge … H Hdi) -H -Hdi /2 width=1/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_minus.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_minus.ma new file mode 100644 index 000000000..6138548cd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_minus.ma @@ -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 *) +(* *) +(**************************************************************************) + +include "basic_2/unfold/gr2.ma". + +(* GENERIC RELOCATION WITH PAIRS ********************************************) + +inductive minuss: nat → relation (list2 nat nat) ≝ +| minuss_nil: ∀i. minuss i ⟠ ⟠ +| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 → + minuss i ({d, e} @ des1) ({d - i, e} @ des2) +| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 → + minuss i ({d, e} @ des1) des2 +. + +interpretation "minus (generic relocation with pairs)" + 'RMinus des1 i des2 = (minuss i des1 des2). + +(* Basic inversion lemmas ***************************************************) + +fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ → des2 = ⟠. +#des1 #des2 #i * -des1 -des2 -i +[ // +| #des1 #des2 #d #e #i #_ #_ #H destruct +| #des1 #des2 #d #e #i #_ #_ #H destruct +] +qed. + +lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠. +/2 width=4/ qed-. + +fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → + ∀d,e,des. des1 = {d, e} @ des → + d ≤ i ∧ des ▭ e + i ≡ des2 ∨ + ∃∃des0. i < d & des ▭ i ≡ des0 & + des2 = {d - i, e} @ des0. +#des1 #des2 #i * -des1 -des2 -i +[ #i #d #e #des #H destruct +| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3/ +| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1/ +] +qed. + +lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 → + d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨ + ∃∃des. i < d & des1 ▭ i ≡ des & + des2 = {d - i, e} @ des. +/2 width=3/ qed-. + +lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 → + d ≤ i → des1 ▭ e + i ≡ des2. +#des1 #des2 #d #e #i #H +elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi +lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi +elim (lt_refl_false … Hi) +qed-. + +lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 → + i < d → + ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} @ des. +#des1 #des2 #d #e #i #H +elim (minuss_inv_cons1 … H) -H * /2 width=3/ #Hdi #_ #Hid +lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi +elim (lt_refl_false … Hi) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_plus.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_plus.ma new file mode 100644 index 000000000..bd8d1a9be --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/gr2_plus.ma @@ -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/gr2.ma". + +(* GENERIC RELOCATION WITH PAIRS ********************************************) + +let rec pluss (des:list2 nat nat) (i:nat) on des ≝ match des with +[ nil2 ⇒ ⟠ +| cons2 d e des ⇒ {d + i, e} @ pluss des i +]. + +interpretation "plus (generic relocation with pairs)" + 'plus x y = (pluss x y). + +(* Basic inversion lemmas ***************************************************) + +lemma pluss_inv_nil2: ∀i,des. des + i = ⟠ → des = ⟠. +#i * // normalize +#d #e #des #H destruct +qed. + +lemma pluss_inv_cons2: ∀i,d,e,des2,des. des + i = {d, e} @ des2 → + ∃∃des1. des1 + i = des2 & des = {d - i, e} @ des1. +#i #d #e #des2 * normalize +[ #H destruct +| #d1 #e1 #des1 #H destruct /2 width=3/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma new file mode 100644 index 000000000..b899bd273 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma @@ -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/substitution/ldrop.ma". +include "basic_2/unfold/gr2_minus.ma". +include "basic_2/unfold/lifts.ma". + +(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************) + +inductive ldrops: list2 nat nat → relation lenv ≝ +| ldrops_nil : ∀L. ldrops ⟠ L L +| ldrops_cons: ∀L1,L,L2,des,d,e. + ldrops des L1 L → ⇩[d,e] L ≡ L2 → ldrops ({d, e} @ des) L1 L2 +. + +interpretation "generic local environment slicing" + 'RDropStar des T1 T2 = (ldrops des T1 T2). + +(* Basic inversion lemmas ***************************************************) + +fact ldrops_inv_nil_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → des = ⟠ → L1 = L2. +#L1 #L2 #des * -L1 -L2 -des // +#L1 #L #L2 #d #e #des #_ #_ #H destruct +qed. + +(* Basic_1: was: drop1_gen_pnil *) +lemma ldrops_inv_nil: ∀L1,L2. ⇩*[⟠] L1 ≡ L2 → L1 = L2. +/2 width=3/ qed-. + +fact ldrops_inv_cons_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → + ∀d,e,tl. des = {d, e} @ tl → + ∃∃L. ⇩*[tl] L1 ≡ L & ⇩[d, e] L ≡ L2. +#L1 #L2 #des * -L1 -L2 -des +[ #L #d #e #tl #H destruct +| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct + /2 width=3/ +qed. + +(* Basic_1: was: drop1_gen_pcons *) +lemma ldrops_inv_cons: ∀L1,L2,d,e,des. ⇩*[{d, e} @ des] L1 ≡ L2 → + ∃∃L. ⇩*[des] L1 ≡ L & ⇩[d, e] L ≡ L2. +/2 width=3/ qed-. + +lemma ldrops_inv_skip2: ∀I,des,i,des2. des ▭ i ≡ des2 → + ∀L1,K2,V2. ⇩*[des2] L1 ≡ K2. ⓑ{I} V2 → + ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 & + ⇩*[des1] K1 ≡ K2 & + ⇧*[des1] V2 ≡ V1 & + L1 = K1. ⓑ{I} V1. +#I #des #i #des2 #H elim H -des -i -des2 +[ #i #L1 #K2 #V2 #H + >(ldrops_inv_nil … H) -L1 /2 width=7/ +| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H + elim (ldrops_inv_cons … H) -H #L #HL1 #H + elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ #K #V >minus_plus #HK2 #HV2 #H destruct + elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct + @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7/ | skip ] + normalize >plus_minus // @minuss_lt // /2 width=1/ (**) (* explicit constructors, /3 width=1/ is a bit slow *) +| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H + elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct + /4 width=7/ +] +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: drop1_skip_bind *) +lemma ldrops_skip: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 → + ∀I. ⇩*[des + 1] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2. +#L1 #L2 #des #H elim H -L1 -L2 -des +[ #L #V1 #V2 #HV12 #I + >(lifts_inv_nil … HV12) -HV12 // +| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I + elim (lifts_inv_cons … H) -H /3 width=5/ +]. +qed. + +(* Basic_1: removed theorems 1: drop1_getl_trans +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops_ldrop.ma new file mode 100644 index 000000000..6ca2f73df --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops_ldrop.ma @@ -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/substitution/ldrop_ldrop.ma". +include "basic_2/unfold/ldrops.ma". + +(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************) + +(* Properties concerning basic local environment slicing ********************) + +lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[des] L1 ≡ L → ∀L2,i. ⇩[0, i] L ≡ L2 → + ∃∃L0,des0,i0. ⇩[0, i0] L1 ≡ L0 & ⇩*[des0] L0 ≡ L2 & + @⦃i, des⦄ ≡ i0 & des ▭ i ≡ des0. +#L1 #L #des #H elim H -L1 -L -des +[ /2 width=7/ +| #L1 #L3 #L #des3 #d #e #_ #HL3 #IHL13 #L2 #i #HL2 + elim (lt_or_ge i d) #Hid + [ elim (ldrop_trans_le … HL3 … HL2 ?) -L /2 width=2/ #L #HL3 #HL2 + elim (IHL13 … HL3) -L3 /3 width=7/ + | lapply (ldrop_trans_ge … HL3 … HL2 ?) -L // #HL32 + elim (IHL13 … HL32) -L3 /3 width=7/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops_ldrops.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops_ldrops.ma new file mode 100644 index 000000000..7709561a2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops_ldrops.ma @@ -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/unfold/ldrops_ldrop.ma". + +(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************) + +(* Main properties **********************************************************) + +(* Basic_1: was: drop1_trans *) +theorem ldrops_trans: ∀L,L2,des2. ⇩*[des2] L ≡ L2 → ∀L1,des1. ⇩*[des1] L1 ≡ L → + ⇩*[des2 @@ des1] L1 ≡ L2. +#L #L2 #des2 #H elim H -L -L2 -des2 // /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts.ma new file mode 100644 index 000000000..40158acbe --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts.ma @@ -0,0 +1,150 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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/lift.ma". +include "basic_2/unfold/gr2_plus.ma". + +(* GENERIC TERM RELOCATION **************************************************) + +inductive lifts: list2 nat nat → relation term ≝ +| lifts_nil : ∀T. lifts ⟠ T T +| lifts_cons: ∀T1,T,T2,des,d,e. + ⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} @ des) T1 T2 +. + +interpretation "generic relocation (term)" + 'RLiftStar des T1 T2 = (lifts des T1 T2). + +(* Basic inversion lemmas ***************************************************) + +fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2. +#T1 #T2 #des * -T1 -T2 -des // +#T1 #T #T2 #d #e #des #_ #_ #H destruct +qed. + +lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2. +/2 width=3/ qed-. + +fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → + ∀d,e,tl. des = {d, e} @ tl → + ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2. +#T1 #T2 #des * -T1 -T2 -des +[ #T #d #e #tl #H destruct +| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct + /2 width=3/ +qed. + +lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} @ des] T1 ≡ T2 → + ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2. +/2 width=3/ qed-. + +(* Basic_1: was: lift1_sort *) +lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k. +#T2 #k #des elim des -des +[ #H <(lifts_inv_nil … H) -H // +| #d #e #des #IH #H + elim (lifts_inv_cons … H) -H #X #H + >(lift_inv_sort1 … H) -H /2 width=1/ +] +qed-. + +(* Basic_1: was: lift1_lref *) +lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 → + ∃∃i2. @⦃i1, des⦄ ≡ i2 & T2 = #i2. +#T2 #des elim des -des +[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3/ +| #d #e #des #IH #i1 #H + elim (lifts_inv_cons … H) -H #X #H1 #H2 + elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct + elim (IH … H2) -IH -H2 /3 width=3/ +] +qed-. + +lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p. +#T2 #p #des elim des -des +[ #H <(lifts_inv_nil … H) -H // +| #d #e #des #IH #H + elim (lifts_inv_cons … H) -H #X #H + >(lift_inv_gref1 … H) -H /2 width=1/ +] +qed-. + +(* Basic_1: was: lift1_bind *) +lemma lifts_inv_bind1: ∀a,I,T2,des,V1,U1. ⇧*[des] ⓑ{a,I} V1. U1 ≡ T2 → + ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 & + T2 = ⓑ{a,I} V2. U2. +#a #I #T2 #des elim des -des +[ #V1 #U1 #H + <(lifts_inv_nil … H) -H /2 width=5/ +| #d #e #des #IHdes #V1 #U1 #H + elim (lifts_inv_cons … H) -H #X #H #HT2 + elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct + elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct + /3 width=5/ +] +qed-. + +(* Basic_1: was: lift1_flat *) +lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 → + ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 & + T2 = ⓕ{I} V2. U2. +#I #T2 #des elim des -des +[ #V1 #U1 #H + <(lifts_inv_nil … H) -H /2 width=5/ +| #d #e #des #IHdes #V1 #U1 #H + elim (lifts_inv_cons … H) -H #X #H #HT2 + elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct + elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct + /3 width=5/ +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. +#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_dx/ +qed-. + +lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. +#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_sn/ +qed-. + +(* Basic properties *********************************************************) + +lemma lifts_bind: ∀a,I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 → + ∀T1. ⇧*[des + 1] T1 ≡ T2 → + ⇧*[des] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2. +#a #I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des +[ #V #T1 #H >(lifts_inv_nil … H) -H // +| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H + elim (lifts_inv_cons … H) -H /3 width=3/ +] +qed. + +lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 → + ∀T1. ⇧*[des] T1 ≡ T2 → + ⇧*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2. +#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des +[ #V #T1 #H >(lifts_inv_nil … H) -H // +| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H + elim (lifts_inv_cons … H) -H /3 width=3/ +] +qed. + +lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2. +#des elim des -des /2 width=2/ +#d #e #des #IH #T1 +elim (lift_total T1 d e) #T #HT1 +elim (IH T) -IH /3 width=4/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_lift.ma new file mode 100644 index 000000000..6ad3ff015 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_lift.ma @@ -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/substitution/lift_lift.ma". +include "basic_2/unfold/gr2_minus.ma". +include "basic_2/unfold/lifts.ma". + +(* GENERIC TERM RELOCATION **************************************************) + +(* Properties concerning basic term relocation ******************************) + +(* Basic_1: was: lift1_xhg (right to left) *) +lemma lifts_lift_trans_le: ∀T1,T,des. ⇧*[des] T1 ≡ T → ∀T2. ⇧[0, 1] T ≡ T2 → + ∃∃T0. ⇧[0, 1] T1 ≡ T0 & ⇧*[des + 1] T0 ≡ T2. +#T1 #T #des #H elim H -T1 -T -des +[ /2 width=3/ +| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2 + elim (IHT13 … HT2) -T #T #HT3 #HT2 + elim (lift_trans_le … HT13 … HT3 ?) -T3 // /3 width=5/ +] +qed-. + +(* Basic_1: was: lift1_free (right to left) *) +lemma lifts_lift_trans: ∀des,i,i0. @⦃i, des⦄ ≡ i0 → + ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 → + ∀T1,T0. ⇧*[des0] T1 ≡ T0 → + ∀T2. ⇧[O, i0 + 1] T0 ≡ T2 → + ∃∃T. ⇧[0, i + 1] T1 ≡ T & ⇧*[des] T ≡ T2. +#des elim des -des normalize +[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2 + <(at_inv_nil … H1) -x #HT02 + lapply (minuss_inv_nil1 … H2) -H2 #H + >(pluss_inv_nil2 … H) in HT10; -des0 #H + >(lifts_inv_nil … H) -T1 /2 width=3/ +| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02 + elim (at_inv_cons … H1) -H1 * #Hid #Hi0 + [ elim (minuss_inv_cons1_lt … H2 ?) -H2 [2: /2 width=1/ ] #des1 #Hdes1 minus_plus #HT1 #HT0 + elim (IHdes … Hi0 … Hdes1 … HT0 … HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02 + elim (lift_trans_le … HT1 … HT0 ?) -T /2 width=1/ #T #HT1 commutative_plus in Hi0; #Hi0 + lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] (liftv_inv_nil1 … H) -T1s /2 width=3/ +| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H + elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct + elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s + elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_lifts.ma new file mode 100644 index 000000000..72948f04b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_lifts.ma @@ -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/unfold/lifts_lift.ma". + +(* GENERIC RELOCATION *******************************************************) + +(* Main properties **********************************************************) + +(* Basic_1: was: lift1_lift1 (left to right) *) +theorem lifts_trans: ∀T1,T,des1. ⇧*[des1] T1 ≡ T → ∀T2:term. ∀des2. ⇧*[des2] T ≡ T2 → + ⇧*[des1 @@ des2] T1 ≡ T2. +#T1 #T #des1 #H elim H -T1 -T -des1 // /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_vector.ma new file mode 100644 index 000000000..9ea173a56 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lifts_vector.ma @@ -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/substitution/lift_vector.ma". +include "basic_2/unfold/lifts.ma". + +(* GENERIC TERM VECTOR RELOCATION *******************************************) + +inductive liftsv (des:list2 nat nat) : relation (list term) ≝ +| liftsv_nil : liftsv des ◊ ◊ +| liftsv_cons: ∀T1s,T2s,T1,T2. + ⇧*[des] T1 ≡ T2 → liftsv des T1s T2s → + liftsv des (T1 @ T1s) (T2 @ T2s) +. + +interpretation "generic relocation (vector)" + 'RLiftStar des T1s T2s = (liftsv des T1s T2s). + +(* Basic inversion lemmas ***************************************************) + +(* Basic_1: was: lifts1_flat (left to right) *) +lemma lifts_inv_applv1: ∀V1s,U1,T2,des. ⇧*[des] Ⓐ V1s. U1 ≡ T2 → + ∃∃V2s,U2. ⇧*[des] V1s ≡ V2s & ⇧*[des] U1 ≡ U2 & + T2 = Ⓐ V2s. U2. +#V1s elim V1s -V1s normalize +[ #T1 #T2 #des #HT12 + @(ex3_2_intro) [3,4: // |1,2: skip | // ] (**) (* explicit constructor *) +| #V1 #V1s #IHV1s #T1 #X #des #H + elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct + elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct + @(ex3_2_intro) [4: // |3: /2 width=2/ |1,2: skip | // ] (**) (* explicit constructor *) +] +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: lifts1_flat (right to left) *) +lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s → + ∀T1,T2. ⇧*[des] T1 ≡ T2 → + ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2. +#V1s #V2s #des #H elim H -V1s -V2s // /3 width=1/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_dx.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_dx.ma new file mode 100644 index 000000000..6ba09b962 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_dx.ma @@ -0,0 +1,274 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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". + +(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************) + +(* Basic_1: includes: csubst1_bind *) +inductive ltpss_dx: nat → nat → relation lenv ≝ +| ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆) +| ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) +| ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e. + ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 → + ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) +| ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e. + ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 → + ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) +. + +interpretation "parallel unfold (local environment, dx variant)" + 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2. +#d #e #L1 #L2 #H elim H -d -e -L1 -L2 // +[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct + >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) // +] +qed. + +lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2. +/2 width=4/ qed-. + +fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2. + L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e → + ∀K1,I,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. K1 ▶* [0, e - 1] K2 & + K2 ⊢ V1 ▶* [0, e - 1] V2 & + L2 = K2. ⓑ{I} V2. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e → + ∃∃K2,V2. K1 ▶* [0, e - 1] K2 & + K2 ⊢ V1 ▶* [0, e - 1] V2 & + L2 = K2. ⓑ{I} V2. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d → + ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. K1 ▶* [d - 1, e] K2 & + K2 ⊢ V1 ▶* [d - 1, e] V2 & + L2 = K2. ⓑ{I} V2. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #I #K1 #V1 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d → + ∃∃K2,V2. K1 ▶* [d - 1, e] K2 & + K2 ⊢ V1 ▶* [d - 1, e] V2 & + L2 = K2. ⓑ{I} V2. +/2 width=3/ qed-. + +fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2. + L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e → + ∀K2,I,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ▶* [0, e - 1] K2 & + K2 ⊢ V1 ▶* [0, e - 1] V2 & + L1 = K1. ⓑ{I} V1. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e → + ∃∃K1,V1. K1 ▶* [0, e - 1] K2 & + K2 ⊢ V1 ▶* [0, e - 1] V2 & + L1 = K1. ⓑ{I} V1. +/2 width=5/ qed-. + +fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d → + ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ▶* [d - 1, e] K2 & + K2 ⊢ V1 ▶* [d - 1, e] V2 & + L1 = K1. ⓑ{I} V1. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #I #K2 #V2 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d → + ∃∃K1,V1. K1 ▶* [d - 1, e] K2 & + K2 ⊢ V1 ▶* [d - 1, e] V2 & + L1 = K1. ⓑ{I} V1. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e. + L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 → + L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e. + L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 → + L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e. + L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 → + 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2. +#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He +>(plus_minus_m_m e 1) /2 width=1/ +qed. + +lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 → + 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd +>(plus_minus_m_m d 1) /2 width=1/ +qed. + +lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e. + L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 → + 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 → + 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2. +/3 width=1/ qed. + +(* Basic_1: was by definition: csubst1_refl *) +lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L. +#L elim L -L // +#L #I #V #IHL * /2 width=1/ * /2 width=1/ +qed. + +lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2. +#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 // +[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2 + lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2; + lapply (lt_to_le_to_lt 0 … Hde2) // #He2 + lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/ +| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12 + >plus_plus_comm_23 in Hde12; #Hde12 + elim (le_to_or_lt_eq 0 d2 ?) // #H destruct + [ lapply (le_plus_to_minus_r … Hde12) -Hde12 plus_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 … H) -H // +| // +| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 + elim (le_inv_plus_l … He12) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ +] +qed. + +lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2. +#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H // +| // +| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 + elim (le_inv_plus_l … He12) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/ +| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ +] +qed. + +lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → + ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L. +#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct + lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + (ldrop_inv_atom1 … H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct + lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + (ldrop_inv_atom1 … H) -H /2 width=3/ +| /2 width=3/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct + lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 … H) -H /2 width=3/ +| /2 width=3/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct + lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ +| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK10 -He2d1 destruct minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 … HL01) -IHVW2 + elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/ +] +qed. + +lemma ltpss_dx_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 → + ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 → + L1 ⊢ T2 ▶* [d2, e2] U2. +#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltpss_dx_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/ +qed. + +(* Basic_1: was: subst1_subst1 *) +lemma ltpss_dx_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 → + ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → + ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T & + L0 ⊢ T ▶* [d1, e1] U2. +#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2 +[ /2 width=3/ +| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 + elim (lt_or_ge i2 d1) #Hi2d1 + [ elim (ltpss_dx_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1 + elim (ltpss_dx_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 … HL10) -IHVW2 + elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_sn.ma new file mode 100644 index 000000000..0d13a5a3f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_sn.ma @@ -0,0 +1,255 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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". + +(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************) + +inductive ltpss_sn: nat → nat → relation lenv ≝ +| ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆) +| ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) +| ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e. + ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 → + ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) +| ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e. + ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 → + ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) +. + +interpretation "parallel unfold (local environment, sn variant)" + 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2. +#d #e #L1 #L2 #H elim H -d -e -L1 -L2 // +[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct + >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) // +] +qed. + +lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2. +/2 width=4/ qed-. + +fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2. + L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e → + ∀K1,I,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 & + K1 ⊢ V1 ▶* [0, e - 1] V2 & + L2 = K2. ⓑ{I} V2. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e → + ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 & + K1 ⊢ V1 ▶* [0, e - 1] V2 & + L2 = K2. ⓑ{I} V2. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d → + ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → + ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 & + K1 ⊢ V1 ▶* [d - 1, e] V2 & + L2 = K2. ⓑ{I} V2. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #I #K1 #V1 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d → + ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 & + K1 ⊢ V1 ▶* [d - 1, e] V2 & + L2 = K2. ⓑ{I} V2. +/2 width=3/ qed-. + +fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2. + L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e → + ∀K2,I,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 & + K1 ⊢ V1 ▶* [0, e - 1] V2 & + L1 = K1. ⓑ{I} V1. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e → + ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 & + K1 ⊢ V1 ▶* [0, e - 1] V2 & + L1 = K1. ⓑ{I} V1. +/2 width=5/ qed-. + +fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d → + ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → + ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 & + K1 ⊢ V1 ▶* [d - 1, e] V2 & + L1 = K1. ⓑ{I} V1. +#d #e #L1 #L2 * -d -e -L1 -L2 +[ #d #e #_ #I #K2 #V2 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ +] +qed. + +lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d → + ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 & + K1 ⊢ V1 ▶* [d - 1, e] V2 & + L1 = K1. ⓑ{I} V1. +/2 width=3/ qed-. + +(* Basic properties *********************************************************) + +lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e. + L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 → + L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e. + L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 → + L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e. + L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 → + 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2. +#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He +>(plus_minus_m_m e 1) /2 width=1/ +qed. + +lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 → + 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd +>(plus_minus_m_m d 1) /2 width=1/ +qed. + +lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e. + L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 → + 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 → + 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L. +#L elim L -L // +#L #I #V #IHL * /2 width=1/ * /2 width=1/ +qed. + +lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2. +#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 // +[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2 + lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2; + lapply (lt_to_le_to_lt 0 … Hde2) // #He2 + lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/ +| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12 + >plus_plus_comm_23 in Hde12; #Hde12 + elim (le_to_or_lt_eq 0 d2 ?) // #H destruct + [ lapply (le_plus_to_minus_r … Hde12) -Hde12 plus_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 … H) -H // +| // +| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 + elim (le_inv_plus_l … He12) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/ +] +qed. + +lemma ltpss_sn_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2. +#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H // +| // +| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 + elim (le_inv_plus_l … He12) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/ +| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/ +] +qed. + +lemma ltpss_sn_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → + ∃∃L. L2 ⊢ ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L. +#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1 +[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct + lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK01 -He21 destruct plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + (ldrop_inv_atom1 … H) -H /2 width=3/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct + lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK10 -He21 destruct plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 + elim (le_inv_plus_l … Hd1e2) #_ #He2 + (ldrop_inv_atom1 … H) -H /2 width=3/ +| /2 width=3/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct + lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK01 -He2d1 destruct (ldrop_inv_atom1 … H) -H /2 width=3/ +| /2 width=3/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct + lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/ +| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1 + lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2 + [ -IHK10 -He2d1 destruct shift_append_assoc #H + elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct + elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct + append_length minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H + lapply (tpss_lsubs_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 … HL01) -IHVW2 + elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/ +] +qed. + +lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 → + ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 → + L1 ⊢ T2 ▶* [d2, e2] U2. +#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/ +qed. + +lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 → + ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → + ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T & + L1 ⊢ T ▶* [d1, e1] U2. +#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2 +[ /2 width=3/ +| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 + elim (lt_or_ge i2 d1) #Hi2d1 + [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1 + elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct + lapply (ldrop_fwd_ldrop2 … HLK1) #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus minus_plus >commutative_plus /2 width=1/ + | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/ + ] + ] +| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H + lapply (tpss_lsubs_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 … HL10) -IHVW2 + elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/thin.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/thin.ma new file mode 100644 index 000000000..65fb76fe0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/thin.ma @@ -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.ma". + +(* BASIC LOCAL ENVIRONMENT THINNING *****************************************) + +definition thin: nat → nat → relation lenv ≝ + λd,e,L1,L2. ∃∃L. L1 ⊢ ▶* [d, e] L & ⇩[d, e] L ≡ L2. + +interpretation "basic thinning (local environment)" + 'TSubst L1 d e L2 = (thin d e L1 L2). + +(* Basic properties *********************************************************) + +lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ▼*[d, e] L1 ≡ L2. +/2 width=3/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma thin_inv_thin1: ∀I,K1,V1,L2,e. ▼*[0, e] K1.ⓑ{I} V1 ≡ L2 → 0 < e → + ▼*[0, e - 1] K1 ≡ L2. +#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e +elim (ltpss_sn_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct +lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /2 width=3/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/thin_delift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/thin_delift.ma new file mode 100644 index 000000000..b5ffc5e4f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/thin_delift.ma @@ -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/unfold/delift_tpss.ma". +include "basic_2/unfold/delift_ltpss.ma". +include "basic_2/unfold/thin.ma". + +(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************) + +(* Inversion lemmas on inverse basic term relocation ************************) + +lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d → + ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 & + K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 & + L2 = K2. ⓑ{I} V2. +#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e +elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct +elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/ +qed-. + +(* Properties on inverse basic term relocation ******************************) + +lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 → + ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2. +#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I +elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0 +lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct +lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/ +qed. + +lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 → + ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → + ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd → + ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & + L ⊢ ▼*[dd, ee] U2 ≡ T2. +#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd +lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1 +elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2 +elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT +lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT +lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/ +qed. + +lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 → + ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → + ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd → + ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & + L ⊢ ▼*[dd, ee] U2 ≡ T2. +/3 width=3/ qed. + +lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 → + ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → + ∀K. ▼*[dd, ee] L ≡ K → + d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee → + ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 & + L ⊢ ▼*[dd, ee] U2 ≡ T2. +#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee +lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1 +elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2 +elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT +lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT +lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/ +qed. + +lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 → + ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → + ∀K. ▼*[dd, ee] L ≡ K → + d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee → + ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 & + L ⊢ ▼*[dd, ee] U2 ≡ T2. +/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *) + +lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 → + ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → + ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e → + ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 & + L ⊢ ▼*[dd, ee] U2 ≡ T2. +#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee +lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1 +elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2 +elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT +lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT +lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/ +qed. + +lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 → + ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 → + ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e → + ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 & + L ⊢ ▼*[dd, ee] U2 ≡ T2. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/thin_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/thin_ldrop.ma new file mode 100644 index 000000000..498660e1c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/thin_ldrop.ma @@ -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/substitution/ldrop_ldrop.ma". +include "basic_2/unfold/ltpss_sn_ldrop.ma". +include "basic_2/unfold/thin.ma". + +(* BASIC LOCAL ENVIRONMENT THINNING *****************************************) + +(* Properties on local environment slicing **********************************) + +lemma thin_ldrop_conf_ge: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ⇩[0, e2 - e1] L1 ≡ L2. +#L0 #L1 #d1 #e1 * /3 width=8 by ltpss_sn_ldrop_conf_ge, ldrop_conf_ge/ +qed. + +lemma thin_ldrop_conf_be: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → + ∃∃L. ▼*[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L. +#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #Hd1e2 #He2de1 +elim (ltpss_sn_ldrop_conf_be … HL0 … HL02 ? ?) -L0 // #L0 #HL20 #HL0 +elim (ldrop_conf_be … HL1 … HL0 ? ?) -L // -Hd1e2 -He2de1 /3 width=3/ +qed. + +lemma thin_ldrop_conf_le: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 → + ∃∃L. ▼*[d1 - e2, e1] L2 ≡ L & ⇩[0, e2] L1 ≡ L. +#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #He2d1 +elim (ltpss_sn_ldrop_conf_le … HL0 … HL02 ?) -L0 // #L0 #HL20 #HL0 +elim (ldrop_conf_le … HL1 … HL0 ?) -L // -He2d1 /3 width=3/ +qed. + +lemma thin_ldrop_trans_ge: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → + d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2. +#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #Hd1e2 +lapply (ldrop_trans_ge … HL0 … HL02 ?) -L0 // #HL2 +lapply (ltpss_sn_ldrop_trans_ge … HL1 … HL2 ?) -L // /2 width=1/ +qed. + +lemma thin_ldrop_trans_le: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 → + ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 → + ∃∃L. ▼*[d1 - e2, e1] L ≡ L2 & ⇩[0, e2] L1 ≡ L. +#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #He2d1 +elim (ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL0 #HL02 +elim (ltpss_sn_ldrop_trans_le … HL1 … HL0 He2d1) -L -He2d1 /3 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma new file mode 100644 index 000000000..93916208b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma @@ -0,0 +1,182 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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.ma". + +(* PARTIAL UNFOLD ON TERMS **************************************************) + +definition tpss: nat → nat → lenv → relation term ≝ + λd,e,L. TC … (tps d e L). + +interpretation "partial unfold (term)" + 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 → + (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) → + ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2. +#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 → + (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) → + ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1. +#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma tpss_strap1: ∀L,T1,T,T2,d,e. + L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. +/2 width=3/ qed. + +lemma tpss_strap2: ∀L,T1,T,T2,d,e. + L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. +/2 width=3/ qed. + +lemma tpss_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 → + ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2. +/3 width=3/ qed. + +lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T. +/2 width=1/ qed. + +lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 → + ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 → + L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2. +#L #V1 #V2 #d #e #HV12 elim HV12 -V2 +[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2 + [ /3 width=5/ + | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) + ] +| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12 + lapply (tpss_lsubs_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12 + lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e. + L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 → + L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2. +#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2 +[ #V2 #HV12 #HT12 elim HT12 -T2 + [ /3 width=1/ + | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) + ] +| #V #V2 #_ #HV12 #IHV #HT12 + lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → + L ⊢ T1 ▶* [d2, e2] T2. +#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/ +] +qed. + +lemma tpss_weak_top: ∀L,T1,T2,d,e. + L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak_top … HT12) -HT12 /2 width=3/ +] +qed. + +lemma tpss_weak_all: ∀L,T1,T2,d,e. + L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2. +#L #T1 #T2 #d #e #HT12 +lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tpss_weak_top … HT12) // +qed. + +lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 → + ∀L. L @@ K ⊢ T1 ▶* [d, e] T2. +#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k. +#L #T2 #k #d #e #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct + >(tps_inv_sort1 … HT2) -HT2 // +] +qed-. + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p. +#L #T2 #p #d #e #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct + >(tps_inv_gref1 … HT2) -HT2 // +] +qed-. + +lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 → + ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & + L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 & + U2 = ⓑ{a,I} V2. T2. +#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ +] +qed-. + +lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 → + ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 & + U2 = ⓕ{I} V2. T2. +#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/ +] +qed-. + +lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2. +#L #T1 #T2 #d #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 // +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #{T1} ≤ #{T2}. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (tps_fwd_tw … HT2) -HT2 #HT2 +@(transitive_le … IHT1) // +qed-. + +lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T → + ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. +#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T +[ /2 width=4/ +| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct + elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/ +] +qed-. + \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_alt.ma new file mode 100644 index 000000000..ae1dcf624 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_alt.ma @@ -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/tpss_lift.ma". + +(* PARALLEL UNFOLD ON TERMS *************************************************) + +(* alternative definition of tpss *) +inductive tpssa: nat → nat → lenv → relation term ≝ +| tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I}) +| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → + ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 → + ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2 +| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e. + tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 → + tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) +| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e. + tpssa d e L V1 V2 → tpssa d e L T1 T2 → + tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2) +. + +interpretation "parallel unfold (term) alternative" + 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2). + +(* Basic properties *********************************************************) + +lemma tpssa_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 → + ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2. +#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e +[ // +| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 + elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/ +| /4 width=1/ +| /3 width=1/ +] +qed. + +lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T. +#T elim T -T // +#I elim I -I /2 width=1/ +qed. + +lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T → + ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2. +#L #T1 #T #d #e #H elim H -L -T1 -T -d -e +[ #L #I #d #e #X #H + elim (tps_inv_atom1 … H) -H // * /2 width=6/ +| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H + lapply (ldrop_fwd_ldrop2 … HLK) #H0LK + lapply (tps_weak … H 0 (d+e) ? ?) -H // #H + elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/ +| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H + elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + lapply (tps_lsubs_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2 + lapply (IHV1 … HV2) -IHV1 -HV2 #HV12 + lapply (IHT1 … HT2) -IHT1 -HT2 #HT12 + lapply (tpssa_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ +| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H + elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/ +] +qed. + +lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. +#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/ +qed-. + +lemma tpss_ind_alt: ∀R:ℕ→ℕ→lenv→relation term. + (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) → + (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e → + ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 → + ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2 + ) → + (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 → + L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 → + R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) + ) → + (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 → + L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 → + R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) + ) → + ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2. +#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e +// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_lift.ma new file mode 100644 index 000000000..a68f86e32 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_lift.ma @@ -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: ∀L,K,V,U1,i,d,e. + d ≤ i → i < d + e → + ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 → + ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2. +#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1 +[ /3 width=4/ +| #U #U1 #_ #HU1 #IHU #U2 #HU12 + elim (lift_total U 0 (i+1)) #U0 #HU0 + lapply (IHU … HU0) -IHU #H + lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK + lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02 + lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ] +] +qed. + +(* Advanced inverion lemmas *************************************************) + +lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 → + T2 = ⓪{I} ∨ + ∃∃K,V1,V2,i. d ≤ i & i < d + e & + ⇩[O, i] L ≡ K. ⓓV1 & + K ⊢ V1 ▶* [0, d + e - i - 1] V2 & + ⇧[O, i + 1] V2 ≡ T2 & + I = LRef i. +#L #T2 #I #d #e #H @(tpss_ind … H) -T2 +[ /2 width=1/ +| #T #T2 #_ #HT2 * + [ #H destruct + elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ] + | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI + lapply (ldrop_fwd_ldrop2 … HLK) #H + elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 (lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ +] +qed. + +lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 → + ∀L,U1,d,e. dt ≤ d → d ≤ dt + et → + ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → + ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2. +#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ +] +qed. + +lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 → + ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K → + ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → + L ⊢ U1 ▶* [dt + e, et] U2. +#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/ +] +qed. + +lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt + et ≤ d → + ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt ≤ d → d + e ≤ dt + et → + ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + d + e ≤ dt → + ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e. + L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2. +#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU destruct +<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 // +qed. + +lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → + ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 & + ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt ≤ d → dt + et ≤ d + e → + ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. + +lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt ≤ d → d ≤ dt + et → dt + et ≤ d + e → + ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2 +[ /2 width=3/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_tpss.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_tpss.ma new file mode 100644 index 000000000..3f41b0083 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss_tpss.ma @@ -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: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2. +#L #T1 #T2 #d #H @(tpss_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (tps_trans_ge … IHT1 … HT2 ?) // +qed-. + +(* Advanced properties ******************************************************) + +lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 → + ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → + ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T. +/3 width=3/ qed. + +lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 → + ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 → + (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) → + ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T. +/3 width=3/ qed. + +lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 → + ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 → + ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2. +/3 width=3/ qed. + +lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 → + ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 → + ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2. +/3 width=3/ qed. + +lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → + ∀i. d ≤ i → i ≤ d + e → + ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2. +#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2 +[ /2 width=3/ +| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 + elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02 + elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ] + /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *) +] +qed. + +lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 → + ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → + ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 & + ⇧[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet +elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 +lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1 +lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct +elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // minus_minus_comm >minus_le_minus_minus_comm // +qed. + +lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b. +#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1/ +qed. + +lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b. +/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed. + +lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b → + a1 - c1 + a2 - (b - c1) = a1 + a2 - b. +#a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/ +qed. + +(* Inversion & forward lemmas ***********************************************) + +axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2). + +axiom lt_dec: ∀n1,n2. Decidable (n1 < n2). + +lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m. +#m #n elim (lt_or_ge m n) /2 width=1/ +#H elim H -m /2 width=1/ +#m #Hm * #H /2 width=1/ /3 width=1/ +qed-. + +lemma lt_refl_false: ∀n. n < n → ⊥. +#n #H elim (lt_to_not_eq … H) -H /2 width=1/ +qed-. + +lemma lt_zero_false: ∀n. n < 0 → ⊥. +#n #H elim (lt_to_not_le … H) -H /2 width=1/ +qed-. + +lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x. +#x #y #H elim (decidable_lt x y) /2 width=1/ +#Hxy elim (H Hxy) +qed-. + +lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥. +#y #z #x elim x -x +[ #H lapply (le_n_O_to_eq … H) -H + commutative_plus // +qed. + +lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b). +#B #f #b #l elim l -l normalize // +qed. + +(* Trichotomy operator ******************************************************) + +(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *) +let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝ + match n1 with + [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ] + | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ] + ]. + +lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1. +#A #a1 #a2 #a3 #n2 elim n2 -n2 +[ #n1 #H elim (lt_zero_false … H) +| #n2 #IH #n1 elim n1 -n1 // /3 width=1/ +] +qed. + +lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2. +#A #a1 #a2 #a3 #n elim n -n normalize // +qed. + +lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3. +#A #a1 #a2 #a3 #n1 elim n1 -n1 +[ #n2 #H elim (lt_zero_false … H) +| #n1 #IH #n2 elim n2 -n2 // /3 width=1/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2/list.ma b/matita/matita/contribs/lambdadelta/ground_2/list.ma new file mode 100644 index 000000000..9a5ac0aeb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/list.ma @@ -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 "ground_2/arith.ma". + +(* LISTS ********************************************************************) + +inductive list (A:Type[0]) : Type[0] := + | nil : list A + | cons: A → list A → list A. + +interpretation "nil (list)" 'Nil = (nil ?). + +interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl). + +let rec all A (R:predicate A) (l:list A) on l ≝ + match l with + [ nil ⇒ ⊤ + | cons hd tl ⇒ R hd ∧ all A R tl + ]. + +inductive list2 (A1,A2:Type[0]) : Type[0] := + | nil2 : list2 A1 A2 + | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2. + +interpretation "nil (list of pairs)" 'Nil2 = (nil2 ? ?). + +interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl). + +let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with +[ nil2 ⇒ l2 +| cons2 a1 a2 tl ⇒ {a1, a2} @ append2 A1 A2 tl l2 +]. + +interpretation "append (list of pairs)" + 'Append l1 l2 = (append2 ? ? l1 l2). + +let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with +[ nil2 ⇒ 0 +| cons2 _ _ l ⇒ length2 A1 A2 l + 1 +]. + +interpretation "length (list of pairs)" + 'card l = (length2 ? ? l). diff --git a/matita/matita/contribs/lambdadelta/ground_2/notation.ma b/matita/matita/contribs/lambdadelta/ground_2/notation.ma new file mode 100644 index 000000000..4ac2e6e64 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/notation.ma @@ -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 *) +(* *) +(**************************************************************************) + +(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) + +(* Logic ********************************************************************) + +notation "⊥" + non associative with precedence 90 + for @{'false}. + +notation "⊤" + non associative with precedence 90 + for @{'true}. + +(* Lists ********************************************************************) + +notation "◊" + non associative with precedence 90 + for @{'Nil}. + +notation "hvbox( hd @ break tl )" + right associative with precedence 47 + for @{'Cons $hd $tl}. + +notation "hvbox( l1 @@ break l2 )" + right associative with precedence 47 + for @{'Append $l1 $l2 }. + +notation "⟠" + non associative with precedence 90 + for @{'Nil2}. + +notation "hvbox( { hd1 , break hd2 } @ break tl )" + non associative with precedence 47 + for @{'Cons $hd1 $hd2 $tl}. diff --git a/matita/matita/contribs/lambdadelta/ground_2/star.ma b/matita/matita/contribs/lambdadelta/ground_2/star.ma new file mode 100644 index 000000000..1e46a48c6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/star.ma @@ -0,0 +1,158 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM 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/star.ma". +include "ground_2/xoa_props.ma". +include "ground_2/notation.ma". + +(* PROPERTIES OF RELATIONS **************************************************) + +definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥). + +definition Confluent: ∀A. ∀R: relation A. Prop ≝ λA,R. + ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → + ∃∃a. R a1 a & R a2 a. + +definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R. + ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2. + +definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. + ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & R1 a2 a. + +definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. + ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & R1 a a2. + +definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R. + ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 → + ∃∃a,b. R a1 b1 a b & R a2 b2 a b. + +lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 → + ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & TC … R1 a2 a. +#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 +[ #a1 #Ha01 #a2 #Ha02 + elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/ +| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 + elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 + elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/ +] +qed. + +lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 → + ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 → + ∃∃a. TC … R2 a1 a & R1 a2 a. +#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 +[ #a2 #Ha02 #a1 #Ha01 + elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/ +| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01 + elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0 + elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/ +] +qed. + +lemma TC_confluent2: ∀A,R1,R2. + confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2). +#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 +[ #a1 #Ha01 #a2 #Ha02 + elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/ +| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 + elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 + elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/ +] +qed. + +lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 → + ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & TC … R1 a a2. +#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 +[ #a0 #Ha10 #a2 #Ha02 + elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/ +| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 + elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02 + elim (IHa … Ha0) -a /4 width=3/ +] +qed. + +lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 → + ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 → + ∃∃a. TC … R2 a1 a & R1 a a2. +#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 +[ #a2 #Ha02 #a1 #Ha10 + elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/ +| #a #a2 #_ #Ha02 #IHa #a1 #Ha10 + elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0 + elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/ +] +qed. + +lemma TC_transitive2: ∀A,R1,R2. + transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2). +#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 +[ #a0 #Ha10 #a2 #Ha02 + elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3/ +| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 + elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02 + elim (IHa … Ha0) -a /4 width=3/ +] +qed. + +definition NF: ∀A. relation A → relation A → predicate A ≝ + λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1. + +inductive SN (A) (R,S:relation A): predicate A ≝ +| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1 +. + +lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a. +#A #R #S #a1 #Ha1 +@SN_intro #a2 #HRa12 #HSa12 +elim (HSa12 ?) -HSa12 /2 width=1/ +qed. + +definition NF_sn: ∀A. relation A → relation A → predicate A ≝ + λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1. + +inductive SN_sn (A) (R,S:relation A): predicate A ≝ +| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2 +. + +lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a. +#A #R #S #a2 #Ha2 +@SN_sn_intro #a1 #HRa12 #HSa12 +elim (HSa12 ?) -HSa12 /2 width=1/ +qed. + +lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R → + ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 → + ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b. +#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2 +[ #a2 #b2 #H02 + elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4/ +| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2 + elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4/ +] +qed. + +lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R → + bi_confluent A B (bi_TC … R). +#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1 +[ #a1 #b1 #H01 #a2 #b2 #H02 + elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4/ +| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02 + elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20 + elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml b/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml new file mode 100644 index 000000000..c6a00c160 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa.conf.xml @@ -0,0 +1,45 @@ + + +
+ $(MATITA_RT_BASE_DIR) + +
+
+ contribs/lambda_delta/ground_2/ + xoa + xoa_notation + basics/pts.ma + 1 2 + 1 3 + 2 1 + 2 2 + 2 3 + 3 1 + 3 2 + 3 3 + 3 4 + 4 1 + 4 2 + 4 3 + 4 4 + 4 5 + 5 2 + 5 3 + 5 4 + 5 5 + 6 4 + 6 5 + 6 6 + 6 7 + 7 7 + 3 + 4 + 3 + 4 +
+ diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa.ma new file mode 100644 index 000000000..ac4c8f9f7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa.ma @@ -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 *) +(* *) +(**************************************************************************) + +(* This file was generated by xoa.native: do not edit *********************) + +include "basics/pts.ma". + +(* multiple existental quantifier (1, 2) *) + +inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝ + | ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ? +. + +interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0). + +(* multiple existental quantifier (1, 3) *) + +inductive ex1_3 (A0,A1,A2:Type[0]) (P0:A0→A1→A2→Prop) : Prop ≝ + | ex1_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → ex1_3 ? ? ? ? +. + +interpretation "multiple existental quantifier (1, 3)" 'Ex P0 = (ex1_3 ? ? ? P0). + +(* multiple existental quantifier (2, 1) *) + +inductive ex2_1 (A0:Type[0]) (P0,P1:A0→Prop) : Prop ≝ + | ex2_1_intro: ∀x0. P0 x0 → P1 x0 → ex2_1 ? ? ? +. + +interpretation "multiple existental quantifier (2, 1)" 'Ex P0 P1 = (ex2_1 ? P0 P1). + +(* multiple existental quantifier (2, 2) *) + +inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝ + | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ? +. + +interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1). + +(* multiple existental quantifier (2, 3) *) + +inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝ + | ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ? +. + +interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? P0 P1). + +(* multiple existental quantifier (3, 1) *) + +inductive ex3_1 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝ + | ex3_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3_1 ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3_1 ? P0 P1 P2). + +(* multiple existental quantifier (3, 2) *) + +inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝ + | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2). + +(* multiple existental quantifier (3, 3) *) + +inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝ + | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2). + +(* multiple existental quantifier (3, 4) *) + +inductive ex3_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2:A0→A1→A2→A3→Prop) : Prop ≝ + | ex3_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → ex3_4 ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 4)" 'Ex P0 P1 P2 = (ex3_4 ? ? ? ? P0 P1 P2). + +(* multiple existental quantifier (4, 1) *) + +inductive ex4_1 (A0:Type[0]) (P0,P1,P2,P3:A0→Prop) : Prop ≝ + | ex4_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → P3 x0 → ex4_1 ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 1)" 'Ex P0 P1 P2 P3 = (ex4_1 ? P0 P1 P2 P3). + +(* multiple existental quantifier (4, 2) *) + +inductive ex4_2 (A0,A1:Type[0]) (P0,P1,P2,P3:A0→A1→Prop) : Prop ≝ + | ex4_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → ex4_2 ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (4, 3) *) + +inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝ + | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (4, 4) *) + +inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝ + | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (4, 5) *) + +inductive ex4_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→A4→Prop) : Prop ≝ + | ex4_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → ex4_5 ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 5)" 'Ex P0 P1 P2 P3 = (ex4_5 ? ? ? ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (5, 2) *) + +inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝ + | ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4). + +(* multiple existental quantifier (5, 3) *) + +inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝ + | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4). + +(* multiple existental quantifier (5, 4) *) + +inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝ + | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4). + +(* multiple existental quantifier (5, 5) *) + +inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝ + | ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4). + +(* multiple existental quantifier (6, 4) *) + +inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝ + | ex6_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → ex6_4 ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5). + +(* multiple existental quantifier (6, 5) *) + +inductive ex6_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→Prop) : Prop ≝ + | ex6_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → ex6_5 ? ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5). + +(* multiple existental quantifier (6, 6) *) + +inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝ + | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5). + +(* multiple existental quantifier (6, 7) *) + +inductive ex6_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝ + | ex6_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → ex6_7 ? ? ? ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5). + +(* multiple existental quantifier (7, 7) *) + +inductive ex7_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝ + | ex7_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → P6 x0 x1 x2 x3 x4 x5 x6 → ex7_7 ? ? ? ? ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (7, 7)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6). + +(* multiple disjunction connective (3) *) + +inductive or3 (P0,P1,P2:Prop) : Prop ≝ + | or3_intro0: P0 → or3 ? ? ? + | or3_intro1: P1 → or3 ? ? ? + | or3_intro2: P2 → or3 ? ? ? +. + +interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2). + +(* multiple disjunction connective (4) *) + +inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝ + | or4_intro0: P0 → or4 ? ? ? ? + | or4_intro1: P1 → or4 ? ? ? ? + | or4_intro2: P2 → or4 ? ? ? ? + | or4_intro3: P3 → or4 ? ? ? ? +. + +interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3). + +(* multiple conjunction connective (3) *) + +inductive and3 (P0,P1,P2:Prop) : Prop ≝ + | and3_intro: P0 → P1 → P2 → and3 ? ? ? +. + +interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2). + +(* multiple conjunction connective (4) *) + +inductive and4 (P0,P1,P2,P3:Prop) : Prop ≝ + | and4_intro: P0 → P1 → P2 → P3 → and4 ? ? ? ? +. + +interpretation "multiple conjunction connective (4)" 'And P0 P1 P2 P3 = (and4 P0 P1 P2 P3). + diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa_notation.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa_notation.ma new file mode 100644 index 000000000..6f614f2e5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa_notation.ma @@ -0,0 +1,270 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was generated by xoa.native: do not edit *********************) + +(* multiple existental quantifier (1, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }. + +(* multiple existental quantifier (1, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) }. + +(* multiple existental quantifier (2, 1) *) + +notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) }. + +notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }. + +(* multiple existental quantifier (2, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }. + +(* multiple existental quantifier (2, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) }. + +(* multiple existental quantifier (3, 1) *) + +notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }. + +notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }. + +(* multiple existental quantifier (3, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }. + +(* multiple existental quantifier (3, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }. + +(* multiple existental quantifier (3, 4) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) }. + +(* multiple existental quantifier (4, 1) *) + +notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) (λ${ident x0}.$P3) }. + +notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) (λ${ident x0}:$T0.$P3) }. + +(* multiple existental quantifier (4, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) }. + +(* multiple existental quantifier (4, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }. + +(* multiple existental quantifier (4, 4) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }. + +(* multiple existental quantifier (4, 5) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) }. + +(* multiple existental quantifier (5, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) (λ${ident x0}.λ${ident x1}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }. + +(* multiple existental quantifier (5, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }. + +(* multiple existental quantifier (5, 4) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }. + +(* multiple existental quantifier (5, 5) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }. + +(* multiple existental quantifier (6, 4) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }. + +(* multiple existental quantifier (6, 5) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) }. + +(* multiple existental quantifier (6, 6) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }. + +(* multiple existental quantifier (6, 7) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) }. + +(* multiple existental quantifier (7, 7) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P6) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P6) }. + +(* multiple disjunction connective (3) *) + +notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)" + non associative with precedence 30 + for @{ 'Or $P0 $P1 $P2 }. + +(* multiple disjunction connective (4) *) + +notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)" + non associative with precedence 30 + for @{ 'Or $P0 $P1 $P2 $P3 }. + +(* multiple conjunction connective (3) *) + +notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)" + non associative with precedence 35 + for @{ 'And $P0 $P1 $P2 }. + +(* multiple conjunction connective (4) *) + +notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2 break & term 34 P3)" + non associative with precedence 35 + for @{ 'And $P0 $P1 $P2 $P3 }. + diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa_props.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa_props.ma new file mode 100644 index 000000000..71216d1c4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa_props.ma @@ -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 "basics/logic.ma". +include "ground_2/xoa_notation.ma". +include "ground_2/xoa.ma". + +interpretation "logical false" 'false = False. + +interpretation "logical true" 'true = True. + +lemma ex2_1_comm: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0. +#A0 #P0 #P1 * /2 width=3/ +qed. diff --git a/matita/matita/contribs/lambdadelta/ma2etc.sh b/matita/matita/contribs/lambdadelta/ma2etc.sh new file mode 100644 index 000000000..e546af776 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ma2etc.sh @@ -0,0 +1 @@ +for FILE in `find $1 -name "*.ma"`; do svn mv $FILE ${FILE/%.ma/.etc} ; done diff --git a/matita/matita/contribs/lambdadelta/orig.sh b/matita/matita/contribs/lambdadelta/orig.sh new file mode 100644 index 000000000..83b1fa183 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/orig.sh @@ -0,0 +1,4 @@ +F=`find $1 -name "*.ma" -or -name "*.txt"` +while read A A A; do + if grep -q "$A" $F; then true; else echo $A; fi +done diff --git a/matita/matita/contribs/lambdadelta/replace.sh b/matita/matita/contribs/lambdadelta/replace.sh new file mode 100644 index 000000000..5e281b251 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/replace.sh @@ -0,0 +1,10 @@ +#!/bin/sh +for MA in `find -name "*.ma"`; do + echo ${MA}; sed "s!$1!$2!g" ${MA} > ${MA}.new + if diff ${MA} ${MA}.new > /dev/null; + then rm -f ${MA}.new; + else mv -f ${MA} ${MA}.old; mv -f ${MA}.new ${MA}; + fi +done + +unset MA diff --git a/matita/matita/contribs/lambdadelta/root b/matita/matita/contribs/lambdadelta/root new file mode 100644 index 000000000..c41bf7380 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/root @@ -0,0 +1 @@ +baseuri=cic:/matita/lambda_delta/ diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma index 5b430d7c6..e80380217 100644 --- a/matita/matita/lib/arithmetics/nat.ma +++ b/matita/matita/lib/arithmetics/nat.ma @@ -505,6 +505,19 @@ lemma f_ind: ∀A. ∀f:A→ℕ. ∀P:predicate A. (∀n. (∀a. f a < n → P a) → ∀a. f a = n → P a) → ∀a. P a. #A #f #P #H #a @(f_ind_aux … H) -H [2: // | skip ] +qed-. + +fact f2_ind_aux: ∀A1,A2. ∀f:A1→A2→ℕ. ∀P:relation2 A1 A2. + (∀n. (∀a1,a2. f a1 a2 < n → P a1 a2) → ∀a1,a2. f a1 a2 = n → P a1 a2) → + ∀n,a1,a2. f a1 a2 = n → P a1 a2. +#A1 #A2 #f #P #H #n @(nat_elim1 … n) -n #n /3 width=3/ (**) (* auto slow (34s) without #n *) +qed-. + +lemma f2_ind: ∀A1,A2. ∀f:A1→A2→ℕ. ∀P:relation2 A1 A2. + (∀n. (∀a1,a2. f a1 a2 < n → P a1 a2) → ∀a1,a2. f a1 a2 = n → P a1 a2) → + ∀a1,a2. P a1 a2. +#A1 #A2 #f #P #H #a1 #a2 +@(f2_ind_aux … H) -H [2: // | skip ] qed-. (* More negated equalities **************************************************)