X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Fltps.ma;h=c4b6c87f0c2831e38f632753e584d455278c3ea5;hb=b5db76fe31ab35bae0257cb6684c511bcc531e45;hp=43c13fb22ac69f4856727217ec2d0469fdd5bc8e;hpb=d38087520d6ce1d696b28da40f3811291fc8a311;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps.ma index 43c13fb22..c4b6c87f0 100644 --- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps.ma +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps.ma @@ -19,13 +19,13 @@ include "Basic_2/substitution/tps.ma". (* Basic_1: includes: csubst1_bind *) inductive ltps: nat → nat → relation lenv ≝ | ltps_atom: ∀d,e. ltps d e (⋆) (⋆) -| ltps_pair: ∀L,I,V. ltps 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V) +| ltps_pair: ∀L,I,V. ltps 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) | ltps_tps2: ∀L1,L2,I,V1,V2,e. - ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ≫ V2 → - ltps 0 (e + 1) (L1. 𝕓{I} V1) L2. 𝕓{I} V2 + ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶ V2 → + ltps 0 (e + 1) (L1. ⓑ{I} V1) L2. ⓑ{I} V2 | ltps_tps1: ∀L1,L2,I,V1,V2,d,e. - ltps d e L1 L2 → L2 ⊢ V1 [d, e] ≫ V2 → - ltps (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2) + ltps d e L1 L2 → L2 ⊢ V1 [d, e] ▶ V2 → + ltps (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) . interpretation "parallel substritution (local environment)" @@ -34,42 +34,53 @@ interpretation "parallel substritution (local environment)" (* Basic properties *********************************************************) lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e. - L1 [0, e - 1] ≫ L2 → L2 ⊢ V1 [0, e - 1] ≫ V2 → - 0 < e → L1. 𝕓{I} V1 [0, e] ≫ L2. 𝕓{I} V2. + 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/ +>(plus_minus_m_m e 1) /2 width=1/ qed. lemma ltps_tps1_lt: ∀L1,L2,I,V1,V2,d,e. - L1 [d - 1, e] ≫ L2 → L2 ⊢ V1 [d - 1, e] ≫ V2 → - 0 < d → L1. 𝕓{I} V1 [d, e] ≫ L2. 𝕓{I} V2. + 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/ +>(plus_minus_m_m d 1) /2 width=1/ qed. (* Basic_1: was by definition: csubst1_refl *) -lemma ltps_refl: ∀L,d,e. L [d, e] ≫ L. +lemma ltps_refl: ∀L,d,e. L [d, e] ▶ L. #L elim L -L // -#L #I #V #IHL * /2/ * /2/ +#L #I #V #IHL * /2 width=1/ * /2 width=1/ qed. +lemma ltps_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → L1 [0, |L2|] ▶ L2. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e +// /3 width=2/ /3 width=3/ +qed. + +(* Basic forward lemmas *****************************************************) + +lemma ltps_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → |L1| = |L2|. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e +normalize // +qed-. + (* Basic inversion lemmas ***************************************************) -fact ltps_inv_refl_O2_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → e = 0 → L1 = L2. -#d #e #L1 #L2 #H elim H -H d e L1 L2 // -[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ #H - elim (plus_S_eq_O_false … H) -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct -e +fact ltps_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 // >(tps_inv_refl_O2 … HV12) // ] qed. -lemma ltps_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ≫ L2 → L1 = L2. -/2/ qed-. +lemma ltps_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶ L2 → L1 = L2. +/2 width=4/ qed-. fact ltps_inv_atom1_aux: ∀d,e,L1,L2. - L1 [d, e] ≫ L2 → L1 = ⋆ → L2 = ⋆. -#d #e #L1 #L2 * -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 @@ -77,50 +88,49 @@ fact ltps_inv_atom1_aux: ∀d,e,L1,L2. ] qed. -lemma ltps_inv_atom1: ∀d,e,L2. ⋆ [d, e] ≫ L2 → L2 = ⋆. +lemma ltps_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶ L2 → L2 = ⋆. /2 width=5/ qed-. -fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e → - ∀K1,I,V1. L1 = K1. 𝕓{I} V1 → - ∃∃K2,V2. K1 [0, e - 1] ≫ K2 & - K2 ⊢ V1 [0, e - 1] ≫ V2 & - L2 = K2. 𝕓{I} V2. -#d #e #L1 #L2 * -d e L1 L2 +fact ltps_inv_tps21_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 -L1 I V1 /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_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 ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. 𝕓{I} V1 [0, e] ≫ L2 → 0 < e → - ∃∃K2,V2. K1 [0, e - 1] ≫ K2 & K2 ⊢ V1 [0, e - 1] ≫ V2 & - L2 = K2. 𝕓{I} V2. +lemma ltps_inv_tps21: ∀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 ltps_inv_tps11_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → 0 < d → - ∀I,K1,V1. L1 = K1. 𝕓{I} V1 → - ∃∃K2,V2. K1 [d - 1, e] ≫ K2 & - K2 ⊢ V1 [d - 1, e] ≫ V2 & - L2 = K2. 𝕓{I} V2. -#d #e #L1 #L2 * -d e L1 L2 +fact ltps_inv_tps11_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 -L1 I V1 - /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/ ] qed. -lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. 𝕓{I} V1 [d, e] ≫ L2 → 0 < d → - ∃∃K2,V2. K1 [d - 1, e] ≫ K2 & - K2 ⊢ V1 [d - 1, e] ≫ V2 & - L2 = K2. 𝕓{I} V2. -/2/ qed-. +lemma ltps_inv_tps11: ∀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 ltps_inv_atom2_aux: ∀d,e,L1,L2. - L1 [d, e] ≫ L2 → L2 = ⋆ → L1 = ⋆. -#d #e #L1 #L2 * -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 @@ -128,46 +138,45 @@ fact ltps_inv_atom2_aux: ∀d,e,L1,L2. ] qed. -lemma ltps_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆. +lemma ltps_inv_atom2: ∀d,e,L1. L1 [d, e] ▶ ⋆ → L1 = ⋆. /2 width=5/ qed-. -fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e → - ∀K2,I,V2. L2 = K2. 𝕓{I} V2 → - ∃∃K1,V1. K1 [0, e - 1] ≫ K2 & - K2 ⊢ V1 [0, e - 1] ≫ V2 & - L1 = K1. 𝕓{I} V1. -#d #e #L1 #L2 * -d e L1 L2 +fact ltps_inv_tps22_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 -L2 I V2 /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_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 ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ≫ K2. 𝕓{I} V2 → 0 < e → - ∃∃K1,V1. K1 [0, e - 1] ≫ K2 & K2 ⊢ V1 [0, e - 1] ≫ V2 & - L1 = K1. 𝕓{I} V1. +lemma ltps_inv_tps22: ∀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 ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → 0 < d → - ∀I,K2,V2. L2 = K2. 𝕓{I} V2 → - ∃∃K1,V1. K1 [d - 1, e] ≫ K2 & - K2 ⊢ V1 [d - 1, e] ≫ V2 & - L1 = K1. 𝕓{I} V1. -#d #e #L1 #L2 * -d e L1 L2 +fact ltps_inv_tps12_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 -L2 I V2 - /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/ ] qed. -lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ≫ K2. 𝕓{I} V2 → 0 < d → - ∃∃K1,V1. K1 [d - 1, e] ≫ K2 & - K2 ⊢ V1 [d - 1, e] ≫ V2 & - L1 = K1. 𝕓{I} V1. -/2/ qed-. +lemma ltps_inv_tps12: ∀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_1: removed theorems 27: csubst0_clear_O csubst0_ldrop_lt csubst0_ldrop_gt csubst0_ldrop_eq