From b5d702735754632652b2659c425dd67d7f92f24b Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Thu, 19 Dec 2013 15:16:45 +0000 Subject: [PATCH] theory of cpy is complete! --- .../lambdadelta/basic_2/relocation/cpy_cpy.ma | 160 +++++++++--------- 1 file changed, 78 insertions(+), 82 deletions(-) diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma index 7e146c21e..dd4227b22 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma @@ -12,50 +12,48 @@ (* *) (**************************************************************************) -include "basic_2/substitution/tps_lift.ma". +include "basic_2/relocation/cpy_lift.ma". -(* PARALLEL SUBSTITUTION ON TERMS *******************************************) +(* CONTEXT-SENSITIVE EXTENDED PARALLEL SUBSTITUTION FOR TERMS ***************) (* Main properties **********************************************************) -(* Basic_1: was: subst1_confluence_eq *) -theorem tps_conf_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. -#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1 -[ /2 width=3/ -| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H - elim (tps_inv_lref1 … H) -H - [ #HX destruct /3 width=6/ - | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2 +theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶×[d1, e1] T1 → + ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶×[d2, e2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶×[d1, e1] T. +#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1 +[ /2 width=3 by ex2_intro/ +| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H + elim (cpy_inv_lref1 … H) -H + [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/ + | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2 lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct - >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/ + >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/ ] -| #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_lsubr_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02 +| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 elim (IHV01 … HV02) -V0 #V #HV1 #HV2 elim (IHT01 … HT02) -T0 #T #HT1 #HT2 - lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubr_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 + lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/ +| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct elim (IHV01 … HV02) -V0 - elim (IHT01 … HT02) -T0 /3 width=5/ + elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/ ] -qed. +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 → +theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶×[d1, e1] T1 → + ∀L2,T2,d2,e2. ⦃G, 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 /3 width=6/ - | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded + ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶×[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶×[d1, e1] T. +#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1 +[ /2 width=3 by ex2_intro/ +| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2 + elim (cpy_inv_lref1 … H1) -H1 + [ #H destruct /3 width=7 by cpy_subst, ex2_intro/ + | -HLK1 -HVT1 * #I2 #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 @@ -66,67 +64,65 @@ theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 → 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 +| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2 - elim (IHT01 … HT02 ?) -T0 + elim (IHT01 … HT02) -T0 [ -H #T #HT1 #HT2 - lapply (tps_lsubr_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubr_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 *) + lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/ + | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H + /3 width=1 by monotonic_le_plus_l, or_intror, or_introl/ ] -| #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 +| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct elim (IHV01 … HV02 H) -V0 - elim (IHT01 … HT02 H) -T0 -H /3 width=5/ + elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/ ] -qed. +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 +(* Note: the constant 1 comes from cpy_subst *) +theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T0 → + ∀T2. ⦃G, L⦄ ⊢ T0 ▶×[d, 1] T2 → 1 ≤ e → ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2. +#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e +[ #I #G #L #d #e #T2 #H #He + elim (cpy_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/ + | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct + lapply (lt_to_le_to_lt … (d+e) Hide2 ?) /2 width=5 by cpy_subst, monotonic_lt_plus_r/ ] -| #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_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02 +| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He + lapply (cpy_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1 by le_S_S/ #HVT2 + <(cpy_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=5 by cpy_subst/ +| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct + lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 lapply (IHT10 … HT02 He) -T0 #HT12 - lapply (tps_lsubr_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/ + lapply (lsuby_cpy_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /3 width=1 by cpy_bind, lsuby_succ/ +| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/ ] -qed. +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 +theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T0 → + ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶×[d2, e2] T2 → d2 + e2 ≤ d1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T & ⦃G, L⦄ ⊢ T ▶×[d1, e1] T2. +#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1 +[ /2 width=3 by ex2_intro/ +| #I #G #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 /3 width=8/ -| #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_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02 + lapply (cpy_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1 by le_S/ -Hde2i1 #HWT2 + <(cpy_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=9 by cpy_subst, ex2_intro/ +| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 + elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (lsuby_cpy_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V - elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2 - lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ - lapply (tps_lsubr_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/ + elim (IHT10 … HT02 ?) -T0 /2 width=1 by le_S_S/ #T #HT1 #HT2 + lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/ +| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 + elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV10 … HV02) -V0 // + elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/ ] -qed. +qed-. -- 2.39.2