X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2Fbasic_2%2Funfold%2Fltpss.ma;h=1baa893e4bbc93d27a4c7aaab9836a5432b430f1;hb=de64015de66a48373ade6cab7508d8f8e2c43af9;hp=c445f68900fe38b1edacbd089c5118abd9321373;hpb=a8c166f1e1baeeae04553058bd179420ada8bbe7;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss.ma index c445f6890..1baa893e4 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss.ma @@ -12,94 +12,179 @@ (* *) (**************************************************************************) -include "basic_2/substitution/ltps.ma". include "basic_2/unfold/tpss.ma". -(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************) - -definition ltpss: nat → nat → relation lenv ≝ - λd,e. TC … (ltps d e). - -interpretation "partial unfold (local environment)" +(* PARALLEL UNFOLD ON LOCAL ENVIRONMENTS ************************************) + +(* Basic_1: includes: csubst1_bind *) +inductive ltpss: nat → nat → relation lenv ≝ +| ltpss_atom : ∀d,e. ltpss d e (⋆) (⋆) +| ltpss_pair : ∀L,I,V. ltpss 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) +| ltpss_tpss2: ∀L1,L2,I,V1,V2,e. + ltpss 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶* V2 → + ltpss 0 (e + 1) (L1. ⓑ{I} V1) L2. ⓑ{I} V2 +| ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e. + ltpss d e L1 L2 → L2 ⊢ V1 [d, e] ▶* V2 → + ltpss (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2) +. + +interpretation "parallel unfold (local environment)" 'PSubstStar L1 d e L2 = (ltpss d e L1 L2). -(* Basic eliminators ********************************************************) +(* Basic properties *********************************************************) -lemma ltpss_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 → - (∀L,L2. L1 [d, e] ▶* L → L [d, e] ▶ L2 → R L → R L2) → - ∀L2. L1 [d, e] ▶* L2 → R L2. -#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) // -qed-. +lemma ltpss_tps2: ∀L1,L2,I,V1,V2,e. + L1 [0, e] ▶* L2 → L2 ⊢ V1 [0, e] ▶ V2 → + L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_tps1: ∀L1,L2,I,V1,V2,d,e. + L1 [d, e] ▶* L2 → L2 ⊢ V1 [d, e] ▶ V2 → + L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2. +/3 width=1/ qed. + +lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e. + 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 width=1/ +qed. -(* Basic properties *********************************************************) +lemma ltpss_tpss1_lt: ∀L1,L2,I,V1,V2,d,e. + 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 width=1/ +qed. -lemma ltpss_strap: ∀L1,L,L2,d,e. - L1 [d, e] ▶ L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2. -/2 width=3/ qed. +lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e. + L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 → + 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2. +/3 width=1/ qed. +lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 [d - 1, e] ▶* L2 → L2 ⊢ V1 [d - 1, e] ▶ V2 → + 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2. +/3 width=1/ qed. + +(* Basic_1: was by definition: csubst1_refl *) lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L. -/2 width=1/ qed. +#L elim L -L // +#L #I #V #IHL * /2 width=1/ * /2 width=1/ +qed. lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2. -#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 // -#L #L2 #_ #HL2 ->(ltps_fwd_length … HL2) /3 width=5/ +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e +// /3 width=2/ /3 width=3/ qed. (* Basic forward lemmas *****************************************************) lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|. -#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 // -#L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12 -/2 width=3 by ltps_fwd_length/ +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e +normalize // qed-. (* Basic inversion lemmas ***************************************************) +fact ltpss_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 // >(tpss_inv_refl_O2 … HV12) // +] +qed. + lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2. -#d #L1 #L2 #H @(ltpss_ind … H) -L2 // -#L #L2 #_ #HL2 #IHL <(ltps_inv_refl_O2 … HL2) -HL2 // -qed-. +/2 width=4/ qed-. + +fact ltpss_inv_atom1_aux: ∀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 +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆. -#d #e #L2 #H @(ltpss_ind … H) -L2 // -#L #L2 #_ #HL2 #IHL destruct ->(ltps_inv_atom1 … HL2) -HL2 // -qed-. +/2 width=5/ qed-. -fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆. -#d #e #L1 #L2 #H @(ltpss_ind … H) -L2 // -#L2 #L #_ #HL2 #IHL2 #H destruct -lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/ +fact ltpss_inv_tpss21_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 /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct +] +qed. + +lemma ltpss_inv_tpss21: ∀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 ltpss_inv_tpss11_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 /2 width=5/ +] +qed. + +lemma ltpss_inv_tpss11: ∀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 ltpss_inv_atom2_aux: ∀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 +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] qed. lemma ltpss_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. ⓑ{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 + +fact ltpss_inv_tpss22_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 /2 width=5/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H) +| #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. ⓑ{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. +lemma ltpss_inv_tpss22: ∀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. ⓑ{I} V2 → - ∃∃K1,V1. K1 [d - 1, e] ▶ K2 & - K2 ⊢ V1 [d - 1, e] ▶ V2 & +fact ltpss_inv_tpss12_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 #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) @@ -107,9 +192,20 @@ fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d → ] 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=1/ qed. +lemma ltpss_inv_tpss12: ∀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_drop_lt csubst0_drop_gt csubst0_drop_eq + csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans + csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back + csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt + csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back + csubst0_snd_bind csubst0_fst_bind csubst0_both_bind + csubst1_head csubst1_flat csubst1_gen_head + csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1 + *)