X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Fdrop.ma;fp=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Fdrop.ma;h=0000000000000000000000000000000000000000;hb=035e3f52f8da3cb3cdb493aa20568ad673cc2cf5;hp=f3163abb27fe699f44690e6426574b7292a6d1de;hpb=83aea9a1662de32505512d6296921ebfffcfc53d;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/drop.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/drop.ma deleted file mode 100644 index f3163abb2..000000000 --- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/drop.ma +++ /dev/null @@ -1,231 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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/grammar/lenv_weight.ma". -include "Basic_2/grammar/lsubs.ma". -include "Basic_2/substitution/lift.ma". - -(* DROPPING *****************************************************************) - -(* Basic_1: includes: drop_skip_bind *) -inductive drop: nat → nat → relation lenv ≝ -| drop_atom: ∀d,e. drop d e (⋆) (⋆) -| drop_pair: ∀L,I,V. drop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V) -| drop_drop: ∀L1,L2,I,V,e. drop 0 e L1 L2 → drop 0 (e + 1) (L1. 𝕓{I} V) L2 -| drop_skip: ∀L1,L2,I,V1,V2,d,e. - drop d e L1 L2 → ↑[d,e] V2 ≡ V1 → - drop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2) -. - -interpretation "dropping" 'RDrop d e L1 L2 = (drop d e L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2. -#d #e #L1 #L2 * -d e L1 L2 -[ // -| // -| #L1 #L2 #I #V #e #_ #_ #H - elim (plus_S_eq_O_false … H) -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H - elim (plus_S_eq_O_false … H) -] -qed. - -(* Basic_1: was: drop_gen_refl *) -lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2. -/2 width=5/ qed. - -fact drop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ → - L2 = ⋆. -#d #e #L1 #L2 * -d e L1 L2 -[ // -| #L #I #V #H destruct -| #L1 #L2 #I #V #e #_ #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct -] -qed. - -(* Basic_1: was: drop_gen_sort *) -lemma drop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆. -/2 width=5/ qed. - -fact drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → - ∀K,I,V. L1 = K. 𝕓{I} V → - (e = 0 ∧ L2 = K. 𝕓{I} V) ∨ - (0 < e ∧ ↓[d, e - 1] K ≡ L2). -#d #e #L1 #L2 * -d e L1 L2 -[ #d #e #_ #K #I #V #H destruct -| #L #I #V #_ #K #J #W #HX destruct -L I V /3/ -| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H) -] -qed. - -lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 → - (e = 0 ∧ L2 = K. 𝕓{I} V) ∨ - (0 < e ∧ ↓[0, e - 1] K ≡ L2). -/2/ qed. - -(* Basic_1: was: drop_gen_drop *) -lemma drop_inv_drop1: ∀e,K,I,V,L2. - ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2. -#e #K #I #V #L2 #H #He -elim (drop_inv_O1 … H) -H * // #H destruct -e; -elim (lt_refl_false … He) -qed. - -fact drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d → - ∀I,K1,V1. L1 = K1. 𝕓{I} V1 → - ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 & - ↑[d - 1, e] V2 ≡ V1 & - L2 = K2. 𝕓{I} V2. -#d #e #L1 #L2 * -d e L1 L2 -[ #d #e #_ #I #K #V #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) -| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z - /2 width=5/ -] -qed. - -(* Basic_1: was: drop_gen_skip_l *) -lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d → - ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 & - ↑[d - 1, e] V2 ≡ V1 & - L2 = K2. 𝕓{I} V2. -/2/ qed. - -fact drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d → - ∀I,K2,V2. L2 = K2. 𝕓{I} V2 → - ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & - ↑[d - 1, e] V2 ≡ V1 & - L1 = K1. 𝕓{I} V1. -#d #e #L1 #L2 * -d e L1 L2 -[ #d #e #_ #I #K #V #H destruct -| #L #I #V #H elim (lt_refl_false … H) -| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) -| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z - /2 width=5/ -] -qed. - -(* Basic_1: was: drop_gen_skip_r *) -lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d → - ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 & - L1 = K1. 𝕓{I} V1. -/2/ qed. - -(* Basic properties *********************************************************) - -(* Basic_1: was by definition: drop_refl *) -lemma drop_refl: ∀L. ↓[0, 0] L ≡ L. -#L elim L -L // -qed. - -lemma drop_drop_lt: ∀L1,L2,I,V,e. - ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2. -#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/ -qed. - -lemma drop_lsubs_drop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 → - ∀K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{Abbr} V → - d ≤ i → i < d + e → - ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 & - ↓[0, i] L2 ≡ K2. 𝕓{Abbr} V. -#L1 #L2 #d #e #H elim H -H L1 L2 d e -[ #d #e #K1 #V #i #H - lapply (drop_inv_atom1 … H) -H #H destruct -| #L1 #L2 #K1 #V #i #_ #_ #H - elim (lt_zero_false … H) -| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie - elim (drop_inv_O1 … H) -H * #Hi #HLK1 - [ -IHL12 Hie; destruct -i K1 W; - arith_g1 // /3/ - ] -| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie - elim (drop_inv_O1 … H) -H * #Hi #HLK1 - [ -IHL12 Hie Hi; destruct - | elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/ - ] -| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide - lapply (plus_S_le_to_pos … Hdi) #Hi - lapply (drop_inv_drop1 … H ?) -H // #HLK1 - elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/ -] -qed. - -(* Basic forvard lemmas *****************************************************) - -(* Basic_1: was: drop_S *) -lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 → - ↓[O, e + 1] L1 ≡ K2. -#L1 elim L1 -L1 -[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct -| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H - elim (drop_inv_O1 … H) -H * #He #H - [ -IHL1; destruct -e K2 I2 V2 /2/ - | @drop_drop >(plus_minus_m_m e 1) /2/ - ] -] -qed. - -lemma drop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1]. -#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize -[ /2/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 - >(tw_lift … HV21) -HV21 /2/ -] -qed. - -lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e. - ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|. -#L1 elim L1 -L1 -[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct -| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H - elim (drop_inv_O1 … H) -H * #He #H - [ -IHL1; destruct -e K2 I2 V2 // - | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/ - ] -] -qed. - -lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e. -#L1 elim L1 -L1 -[ #L2 #e #H >(drop_inv_atom1 … H) -H // -| #K1 #I1 #V1 #IHL1 #L2 #e #H - elim (drop_inv_O1 … H) -H * #He #H - [ -IHL1; destruct -e L2 // - | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize - >minus_le_minus_minus_comm // - ] -] -qed. - -(* Basic_1: removed theorems 49: - drop_skip_flat - cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf - drop_clear drop_clear_O drop_clear_S - clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r - clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle - getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans - getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt - getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev - drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge - getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O - getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le - getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono -*)