From 46a8a345410219548128c2533ce32b1a8eca6c06 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Fri, 20 Dec 2013 12:50:43 +0000 Subject: [PATCH] theory of extended iterated substitution begins ... --- .../notation/relations/extpsubststar_6.ma | 19 ++ .../lambdadelta/basic_2/relocation/cpy.ma | 7 +- .../lambdadelta/basic_2/substitution/cpys.ma | 188 ++++++++++++++++++ 3 files changed, 211 insertions(+), 3 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/basic_2/notation/relations/extpsubststar_6.ma create mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/extpsubststar_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/extpsubststar_6.ma new file mode 100644 index 000000000..bd89b8d85 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/extpsubststar_6.ma @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * × [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'ExtPSubstStar $G $L $T1 $d $e $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma index e1b2afd75..d6959e253 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma @@ -20,17 +20,18 @@ include "basic_2/relocation/lsuby.ma". (* CONTEXT-SENSITIVE EXTENDED PARALLEL SUBSTITUTION FOR TERMS ***************) +(* Note: this substitution is ordinary *) (* activate genv *) inductive cpy: nat → nat → relation4 genv lenv term term ≝ | cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I}) | cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ i → i < d + e → ⇩[0, i] L ≡ K.ⓑ{I}V → ⇧[0, i + 1] V ≡ W → cpy d e G L (#i) W | cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e. - cpy d e G L V1 V2 → cpy (d + 1) e G (L.ⓑ{I} V2) T1 T2 → - cpy d e G L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2) + cpy d e G L V1 V2 → cpy (d + 1) e G (L.ⓑ{I}V2) T1 T2 → + cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) | cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e. cpy d e G L V1 V2 → cpy d e G L T1 T2 → - cpy d e G L (ⓕ{I}V1. T1) (ⓕ{I}V2. T2) + cpy d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) . interpretation "context-sensitive extended parallel substritution (term)" diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma new file mode 100644 index 000000000..433718803 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma @@ -0,0 +1,188 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $L $T1 $d $e $T2 }. + +include "basic_2/substitution/tps.ma". + +(* PARTIAL UNFOLD ON TERMS **************************************************) + +definition tpss: nat → nat → lenv → relation term ≝ + λd,e,L. TC … (tps d e L). + +interpretation "partial unfold (term)" + 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 → + (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) → + ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2. +#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 → + (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) → + ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1. +#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. +/2 width=1/ qed. + +lemma tpss_strap1: ∀L,T1,T,T2,d,e. + L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. +/2 width=3/ qed. + +lemma tpss_strap2: ∀L,T1,T,T2,d,e. + L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. +/2 width=3/ qed. + +lemma tpss_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 → + ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2. +/3 width=3/ qed. + +lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T. +/2 width=1/ qed. + +lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 → + ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 → + L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2. +#L #V1 #V2 #d #e #HV12 elim HV12 -V2 +[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2 + [ /3 width=5/ + | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) + ] +| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12 + lapply (tpss_lsubr_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12 + lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e. + L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 → + L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2. +#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2 +[ #V2 #HV12 #HT12 elim HT12 -T2 + [ /3 width=1/ + | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) + ] +| #V #V2 #_ #HV12 #IHV #HT12 + lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → + L ⊢ T1 ▶* [d2, e2] T2. +#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/ +] +qed. + +lemma tpss_weak_top: ∀L,T1,T2,d,e. + L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak_top … HT12) -HT12 /2 width=3/ +] +qed. + +lemma tpss_weak_full: ∀L,T1,T2,d,e. + L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2. +#L #T1 #T2 #d #e #HT12 +lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tpss_weak_top … HT12) // +qed. + +lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 → + ∀L. L @@ K ⊢ T1 ▶* [d, e] T2. +#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k. +#L #T2 #k #d #e #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct + >(tps_inv_sort1 … HT2) -HT2 // +] +qed-. + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p. +#L #T2 #p #d #e #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct + >(tps_inv_gref1 … HT2) -HT2 // +] +qed-. + +lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 → + ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & + L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 & + U2 = ⓑ{a,I} V2. T2. +#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/ +] +qed-. + +lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 → + ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 & + U2 = ⓕ{I} V2. T2. +#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/ +] +qed-. + +lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2. +#L #T1 #T2 #d #H @(tpss_ind … H) -T2 +[ // +| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 // +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ♯{T1} ≤ ♯{T2}. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (tps_fwd_tw … HT2) -HT2 #HT2 +@(transitive_le … IHT1) // +qed-. + +lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T → + ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. +#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T +[ /2 width=4/ +| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct + elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/ +] +qed-. -- 2.39.2