X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_equivalence%2Fcpcs_cpcs.ma;h=692bd0b6136d15d930a9fad21408fdefbf986da6;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=7b9645afa39f8f845e6c8c52cb368ba9363e4a77;hpb=d02c188ee3d4bd9885490447e63453adb2cb4ea1;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma index 7b9645afa..692bd0b61 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma @@ -12,201 +12,40 @@ (* *) (**************************************************************************) -include "basic_2/computation/lprs_cprs.ma". -include "basic_2/conversion/cpc_cpc.ma". -include "basic_2/equivalence/cpcs_cprs.ma". +include "basic_2/rt_equivalence/cpcs_lprs.ma". -(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************) - -(* Advanced inversion lemmas ************************************************) - -lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T. -#G #L #T1 #T2 #H @(cpcs_ind … H) -T2 -[ /3 width=3 by ex2_intro/ -| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 - [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/ - | /3 width=5 by cprs_strap2, ex2_intro/ - ] -] -qed-. - -(* Basic_1: was: pc3_gen_sort *) -lemma cpcs_inv_sort: ∀G,L,s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌* ⋆s2 → s1 = s2. -#G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H -#T #H1 >(cprs_inv_sort1 … H1) -T #H2 -lapply (cprs_inv_sort1 … H2) -L #H destruct // -qed-. - -lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T → - ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2. -#a #G #L #W1 #T1 #T #H -elim (cpcs_inv_cprs … H) -H #X #H1 #H2 -elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct -/3 width=6 by cprs_bind, ex2_2_intro/ -qed-. - -lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 → - ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2. -/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-. - -(* Basic_1: was: pc3_gen_sort_abst *) -lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌* ⓛ{a}W.T → ⊥. -#a #G #L #W #T #s #H -elim (cpcs_inv_cprs … H) -H #X #H1 ->(cprs_inv_sort1 … H1) -X #H2 -elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct -qed-. - -(* Basic_1: was: pc3_gen_lift *) -lemma cpcs_inv_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≘ K → - ∀T1,U1. ⬆[l, k] T1 ≘ U1 → ∀T2,U2. ⬆[l, k] T2 ≘ U2 → - ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2. -#G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12 -elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2 -elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1 -elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU ->(lift_inj … HXU … HTU) -X -U -l -k /2 width=3 by cprs_div/ -qed-. - -(* Advanced properties ******************************************************) - -lemma lpr_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2. -#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H -/4 width=5 by cprs_div, lpr_cprs_trans/ -qed-. - -lemma lprs_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → - ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2. -#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H -/4 width=5 by cprs_div, lprs_cprs_trans/ -qed-. - -lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2. -#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2 -/2 width=3 by cpr_cprs_div/ -qed-. - -lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1. -#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2 -/2 width=3 by cprs_cpr_div/ -qed-. - -lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2. -#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2 -/2 width=3 by cprs_div/ -qed-. - -lemma lprs_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2. -#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1 -/2 width=3 by cprs_div/ -qed-. - -(* Basic_1: was: pc3_wcpr0_t *) -(* Basic_1: note: pc3_wcpr0_t should be renamed *) -(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *) -lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2. -#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1 -/2 width=3 by cprs_div/ -qed-. - -(* Basic_1: was only: pc3_pr0_pr2_t *) -(* Basic_1: note: pc3_pr0_pr2_t should be renamed *) -lemma lpr_cpr_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2. -/3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-. - -(* Basic_1: was only: pc3_thin_dx *) -lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2. -#G #L #V1 #V2 #HV12 #T1 #T2 #HT12 -elim (cpcs_inv_cprs … HV12) -HV12 -elim (cpcs_inv_cprs … HT12) -HT12 -/3 width=5 by cprs_flat, cprs_div/ -qed. - -lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2. -/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed. - -lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 → - ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2. -#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12 -/3 width=5 by cprs_div, cprs_bind/ -qed. - -lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T. -#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12 -/3 width=5 by cprs_div, cprs_bind/ -qed. - -lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → - ∀L2. L2 ⫃ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2. -#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12 -/3 width=5 by cprs_div, lsubr_cprs_trans/ -qed-. - -(* Basic_1: was: pc3_lift *) -lemma cpcs_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≘ K → - ∀T1,U1. ⬆[l, k] T1 ≘ U1 → ∀T2,U2. ⬆[l, k] T2 ≘ U2 → - ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2. -#G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12 -elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 -elim (lift_total T l k) /3 width=12 by cprs_div, cprs_lift/ -qed. - -lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 → - ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0. -#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-. - -(* More inversion lemmas ****************************************************) - -(* Note: there must be a proof suitable for llpr *) -lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → - ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2. -#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H -elim (cpcs_inv_cprs … H) -H #T #H1 #H2 -elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct -elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct -lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2 -lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2 -/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/ -qed-. - -lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → - ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2. -#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12 -#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/ -qed-. +(* CONTEXT-SENSITIVE PARALLEL R-EQUIVALENCE FOR TERMS ***********************) (* Main properties **********************************************************) (* Basic_1: was pc3_t *) -theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2. -#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-. +theorem cpcs_trans (h) (G) (L): Transitive … (cpcs h G L). +#h #G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) +qed-. -theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2. +theorem cpcs_canc_sn (h) (G) (L): left_cancellable … (cpcs h G L). /3 width=3 by cpcs_trans, cpcs_sym/ qed-. -theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2. +theorem cpcs_canc_dx (h) (G) (L): right_cancellable … (cpcs h G L). /3 width=3 by cpcs_trans, cpcs_sym/ qed-. -lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → - ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 → - ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2. +(* Advanced properties ******************************************************) + +lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ❪G,L❫ ⊢ V1 ⬌*[h] V2 → + ∀I,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬌*[h] T2 → + ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬌*[h] ⓑ[p,I]V2.T2. /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed. -lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → - ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 → - ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2. +lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ❪G,L❫ ⊢ V1 ⬌*[h] V2 → + ∀I,T1,T2. ❪G,L.ⓑ[I]V2❫ ⊢ T1 ⬌*[h] T2 → + ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬌*[h] ⓑ[p,I]V2.T2. /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed. +(* Advanced properties with r-transition for full local environments ********) + (* Basic_1: was: pc3_wcpr0 *) -lemma lpr_cpcs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2. -#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H +lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ❪G,L1❫ ⊢ ➡[h] L2 → + ∀T1,T2. ❪G,L1❫ ⊢ T1 ⬌*[h] T2 → ❪G,L2❫ ⊢ T1 ⬌*[h] T2. +#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/ qed-.