From b8a7401daf0637a4ce8f86e960b180cb5f22ecb3 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Fri, 30 Mar 2012 18:15:10 +0000 Subject: [PATCH 1/1] - more on subject reduction of atomic arity assignment - some results on context-sensitive equivalence of terms --- .../contribs/lambda_delta/basic_2/basic_1.txt | 29 +--- .../basic_2/computation/cprs_cprs.ma | 30 +++- .../basic_2/computation/cprs_lift.ma | 1 - .../basic_2/computation/lcprs_aaa.ma | 35 +++++ .../lambda_delta/basic_2/equivalence/cpcs.ma | 47 +++++- .../basic_2/equivalence/cpcs_cpcs.ma | 136 ++++++++++++++++++ .../basic_2/equivalence/cpcs_cprs.ma | 53 +++++++ .../basic_2/reducibility/lcpr_aaa.ma | 34 +++++ .../basic_2/reducibility/ltpr_aaa.ma | 2 +- 9 files changed, 327 insertions(+), 40 deletions(-) create mode 100644 matita/matita/contribs/lambda_delta/basic_2/computation/lcprs_aaa.ma create mode 100644 matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma create mode 100644 matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cprs.ma create mode 100644 matita/matita/contribs/lambda_delta/basic_2/reducibility/lcpr_aaa.ma diff --git a/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt b/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt index ca83c049a..caa897dec 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt +++ b/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt @@ -164,52 +164,27 @@ 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/fsubst0 pc3_pr2_fsubst0 pc3/fsubst0 pc3_pr2_fsubst0_back pc3/fsubst0 pc3_fsubst0 -pc3/fwd pc3_gen_sort -pc3/fwd pc3_gen_abst -pc3/fwd pc3_gen_abst_shift -pc3/fwd pc3_gen_lift pc3/fwd pc3_gen_not_abst pc3/fwd pc3_gen_lift_abst -pc3/fwd pc3_gen_sort_abst -pc3/left pc3_ind_left__pc3_left_pr3 -pc3/left pc3_ind_left__pc3_left_trans -pc3/left pc3_ind_left__pc3_left_sym -pc3/left pc3_ind_left__pc3_left_pc3 -pc3/left pc3_ind_left__pc3_pc3_left -pc3/left pc3_ind_left pc3/nf2 pc3_nf2 pc3/nf2 pc3_nf2_unfold pc3/pc1 pc3_pc1 -pc3/props pc3_pr2_r -pc3/props pc3_pr2_x -pc3/props pc3_pr3_r -pc3/props pc3_pr3_x -pc3/props pc3_pr3_t -pc3/props pc3_s -pc3/props pc3_thin_dx -pc3/props pc3_head_1 -pc3/props pc3_head_2 -pc3/props pc3_pr2_u -pc3/props pc3_t -pc3/props pc3_pr2_u2 -pc3/props pc3_pr3_conf -pc3/props pc3_head_12 -pc3/props pc3_head_21 pc3/props pc3_pr0_pr2_t pc3/props pc3_pr2_pr2_t pc3/props pc3_pr2_pr3_t pc3/props pc3_pr3_pc3_t -pc3/props pc3_lift pc3/props pc3_eta pc3/subst1 pc3_gen_cabbr pc3/wcpr0 pc3_wcpr0__pc3_wcpr0_t_aux pc3/wcpr0 pc3_wcpr0_t pc3/wcpr0 pc3_wcpr0 + pr0/fwd pr0_gen_void pr0/dec nf0_dec pr0/subst1 pr0_subst1_back 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 index 518c2bd07..0104d8b27 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma @@ -21,8 +21,8 @@ include "basic_2/computation/cprs_lcpr.ma". (* Advanced properties ******************************************************) -lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → - L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. +lemma cprs_abbr1_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → + L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1 [ /3 width=5/ | #T1 #T #HT1 #_ #IHT1 @@ -30,8 +30,8 @@ lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ] qed. -lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 → - L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. +lemma cpr_abbr1: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 → + L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. /3 width=1/ qed. lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 → @@ -88,13 +88,31 @@ lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 @(cprs_trans … IHV1) -IHV1 /2 width=1/ qed. -lemma cprs_abbr: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 → - L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. +lemma cprs_abbr1: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 → + L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. #L #V1 #T1 #T2 #HT12 #V2 #HV12 @(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 → + L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1 +[ /2 width=1/ +| #T1 #T #HT1 #_ #IHT1 + lapply (lcpr_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 → + L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. +#L #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ +#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 +lapply (IHV1 T1 T1 ?) -IHV1 // #HV1 +@(cprs_trans … HV1) -HV1 /2 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. 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 index 32dcb2ce9..11ce81ff8 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lift.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_lift.ma @@ -70,4 +70,3 @@ lemma cprs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → -HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1 elim (cpr_inv_lift … HLK … HTU … HU2) -U -HLK /3 width=5/ qed. - diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lcprs_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lcprs_aaa.ma new file mode 100644 index 000000000..9be9b17e4 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/computation/lcprs_aaa.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/reducibility/lcpr_aaa.ma". +include "basic_2/computation/cprs.ma". +include "basic_2/computation/lcprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************) + +(* Properties about atomic arity assignment on terms ************************) + +lemma aaa_lcprs_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. + +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. + +lemma aaa_lcprs_cprs_conf: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2. L1 ⊢ ➡* L2 → + ∀T2. L2 ⊢ T1 ➡* T2 → L2 ⊢ T2 ÷ A. +/3 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma index 58e53ed78..e38825fe1 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs.ma @@ -30,18 +30,55 @@ lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 → #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,T. L ⊢ T ⬌* T. /2 width=1/ qed. -lemma cpcs_strap1: ∀L,T1,T,T2. - L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2. +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. +lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2. /2 width=3/ qed. -(* Basic_1: removed theorems 1: clear_pc3_trans *) +(* Basic_1: was: pc3_pr2_r *) +lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ 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_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. + +(* 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: was: pc3_s *) +lemma cprs_comm: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → L ⊢ T2 ⬌* T1. +#L #T1 #T2 #H @(cpcs_ind … H) -T2 // /3 width=3/ +qed. + +(* Basic_1: removed theorems 6: + clear_pc3_trans pc3_ind_left + pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21 + Basic_1: removed local theorems 5: + pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left +*) 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 new file mode 100644 index 000000000..b80429770 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma @@ -0,0 +1,136 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| 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: ∀L,W,T,k. L ⊢ ⋆k ⬌* ⓛW.T → False. +#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: ∀L,W1,W2,T1,T2. L ⊢ ⓛW1.T1 ⬌* ⓛW2.T2 → ∀I,V. + L ⊢ W1 ⬌* W2 ∧ L. ②{I}V ⊢ T1 ⬌* T2. +#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: ∀L,W1,W2,T1,T2. L ⊢ ⓛW1.T1 ⬌* ⓛW2.T2 → ∀W. + L ⊢ W1 ⬌* W2 ∧ L. ⓛW ⊢ T1 ⬌* T2. +#L #W1 #W2 #T1 #T2 #H #W +lapply (cpcs_inv_abst … H Abst W) -H // +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_lift … HLK … HTU1 … HU1) -U1 #T #HTU #HT1 +elim (cprs_inv_lift … HLK … HTU2 … HU2) -L -U2 #X #HXU +>(lift_inj … HXU … HTU) -X -U -d -e /2 width=3/ +qed-. + +(* advanced properties ******************************************************) + +(* 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_abbr_dx: ∀L,V,T1,T2. L.ⓓV ⊢ T1 ⬌* T2 → L ⊢ ⓓV. T1 ⬌* ⓓV. T2. +#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_abbr_sn: ∀L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓓV1. T ⬌* ⓓV2. T. +#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. + +(* Basic_1: was: pc3_gen_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, cprs_comm/ 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, cprs_comm/ qed. (**) (* /3 width=3/ is too slow *) + +lemma cpcs_abbr1: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV1 ⊢ T1 ⬌* T2 → + L ⊢ ⓓV1. T1 ⬌* ⓓV2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 +@(cpcs_trans … (ⓓV1.T2)) /2 width=1/ +qed. + +lemma cpcs_abbr2: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV2 ⊢ T1 ⬌* T2 → + L ⊢ ⓓV1. T1 ⬌* ⓓV2. T2. +#L #V1 #V2 #HV12 #T1 #T2 #HT12 +@(cpcs_trans … (ⓓ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 new file mode 100644 index 000000000..1fb994c71 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cprs.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/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. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/lcpr_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lcpr_aaa.ma new file mode 100644 index 000000000..61ed86b5b --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/reducibility/lcpr_aaa.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/static/aaa_ltpss.ma". +include "basic_2/reducibility/ltpr_aaa.ma". +include "basic_2/reducibility/cpr.ma". +include "basic_2/reducibility/lcpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *************) + +(* Properties about atomic arity assignment on terms ************************) + +lemma aaa_lcpr_conf: ∀L1,T,A. L1 ⊢ T ÷ A → ∀L2. L1 ⊢ ➡ L2 → L2 ⊢ T ÷ A. +#L1 #T #A #HT #L2 * /3 width=5/ +qed. + +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. + +lemma aaa_lcpr_cpr_conf: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2. L1 ⊢ ➡ L2 → + ∀T2. L2 ⊢ T1 ➡ T2 → L2 ⊢ T2 ÷ A. +/3 width=3/ qed. 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 index b4f73c9fa..baf1abb09 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_aaa.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_aaa.ma @@ -12,9 +12,9 @@ (* *) (**************************************************************************) -include "basic_2/reducibility/ltpr_ldrop.ma". include "basic_2/static/aaa_ltps.ma". include "basic_2/static/lsuba_aaa.ma". +include "basic_2/reducibility/ltpr_ldrop.ma". (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) -- 2.39.2