X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fapps_2%2Fetc%2Fmodels%2Fmodel_lower.etc;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fapps_2%2Fetc%2Fmodels%2Fmodel_lower.etc;h=0000000000000000000000000000000000000000;hb=2976c347e18717e691825ebdf73a5ce941c57d1b;hp=e99004e7b081eb64cf663c90fa0487a9646ec471;hpb=a77d0bd6a04e94f765d329d47b37d9e04d349b14;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/apps_2/etc/models/model_lower.etc b/matita/matita/contribs/lambdadelta/apps_2/etc/models/model_lower.etc deleted file mode 100644 index e99004e7b..000000000 --- a/matita/matita/contribs/lambdadelta/apps_2/etc/models/model_lower.etc +++ /dev/null @@ -1,181 +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 "apps_2/notation/models/fdrop_4.ma". -include "apps_2/models/model_veq.ma". -include "apps_2/models/model_raise.ma". - -(* MODEL ********************************************************************) - -definition lower: ∀M. nat → nat → evaluation M → evaluation M ≝ - λM,l,m,lv,i. tri … i l (lv i) (lv (i+m)) (lv (i+m)). - -interpretation "evaluation lower (models)" - 'FDrop M l m lv = (lower M l m lv). - -(* Basic properties *********************************************************) - -lemma lower_lt: ∀M,lv,l,m,i. i < l → (↓[l, m]⦋M⦌ lv) i = lv i. -/2 width=1 by tri_lt/ qed-. - -lemma lower_ge: ∀M,lv,l,m,i. l ≤ i → (↓[l, m]⦋M⦌ lv) i = lv (i+m). -#M #lv #l #m #i #Hli elim (le_to_or_lt_eq … Hli) -Hli #Hli destruct -[ /2 width=1 by tri_gt/ -| /2 width=1 by tri_eq/ -] -qed-. - -lemma lower_veq: ∀M,v1,v2. v1 ≐⦋M⦌ v2 → ∀l,m. ↓[l, m] v1 ≐ ↓[l, m] v2. -#m #v1 #v2 #Hv12 #l #m #i elim (lt_or_ge i l) #Hli -[ >lower_lt // >lower_lt // -| >lower_ge // >lower_ge // -] -qed. - -lemma lower_refl: ∀M,v,l. ↓[l, 0] v ≐⦋M⦌ v. -#M #v #l #i elim (lt_or_ge … i l) #Hil -[ >lower_lt // -| >lower_ge // -] -qed. - -(* Main properties **********************************************************) - -theorem lower_lower_le_sym: ∀M,v,l1,l2,m1,m2. l1 ≤ l2 → - ↓[l1, m1] ↓[l2+m1, m2] v ≐⦋M⦌ ↓[l2, m2] ↓[l1, m1] v. -#M #v #l1 #l2 #m1 #m2 #Hl12 #j elim (lt_or_ge … j l1) #Hjl1 -[ lapply (lt_to_le_to_lt … Hjl1 Hl12) -Hl12 #Hjl2 - >lower_lt // >lower_lt /2 width=3 by lt_to_le_to_lt/ - >lower_lt // >lower_lt // -| >lower_ge // elim (lt_or_ge … j l2) #Hjl2 -Hl12 - [ >lower_lt /2 width=1 by lt_minus_to_plus/ - >lower_lt // >lower_ge // - | >lower_ge /2 width=1 by monotonic_le_plus_l/ - >lower_ge // >lower_ge /2 width=1 by le_plus/ - ] -] -qed. - -lemma lower_lower_le: ∀M,v,l1,l2,m1,m2. l1 ≤ l2 → - ↓[l2, m2] ↓[l1, m1] v ≐⦋M⦌ ↓[l1, m1] ↓[l2+m1, m2] v. -/3 width=1 by lower_lower_le_sym, veq_sym/ qed-. - -(* Properties on raise ******************************************************) - -lemma lower_raise_lt: ∀M,v,d,l,m,i. i ≤ l → ↓[l+1, m] [i⬐d] v ≐⦋M⦌ [i⬐d] ↓[l, m] v. -#M #v #d #l #m #i #Hil #j elim (lt_or_eq_or_gt … j i) #Hij destruct -[ lapply (lt_to_le_to_lt … Hij Hil) -Hil #Hjl - >lower_lt /2 width=1 by le_S/ >raise_lt // >raise_lt // >lower_lt // -| >lower_lt /2 width=1 by le_S_S/ >raise_eq >raise_eq // -| lapply (ltn_to_ltO … Hij) #Hj - >raise_gt // elim (lt_or_ge … j (l+1)) #Hlj - [ -Hil >lower_lt // >lower_lt /2 width=2 by lt_plus_to_minus/ >raise_gt // - | >lower_ge // >lower_ge /2 width=1 by le_plus_to_minus_r/ - >raise_gt /2 width=1 by le_plus/ >plus_minus // - ] -] -qed. - -lemma raise_lower_lt: ∀M,v,d,l,m,i. i ≤ l → [i⬐d] ↓[l, m] v ≐⦋M⦌ ↓[l+1, m] [i⬐d] v. -/3 width=1 by lower_raise_lt, veq_sym/ qed. - -lemma lower_raise_be: ∀M,v,d,l,m,i. l ≤ i → i ≤ l + m → ↓[l, m+1] [i⬐d] v ≐⦋M⦌ ↓[l, m] v. -#M #v #d #l #m #i #Hli #Hilm #j elim (lt_or_ge … j l) #Hlj -[ lapply (lt_to_le_to_lt … Hlj Hli) -Hli -Hilm #Hij - >lower_lt // >lower_lt // >raise_lt // -| lapply (transitive_le … (j+m) Hilm ?) -Hli -Hilm /2 width=1 by monotonic_le_plus_l/ #Hijm - >lower_ge // >lower_ge // >raise_gt /2 width=1 by le_S_S/ -] -qed. - -lemma lower_raise_be_sym: ∀M,v,d,l,m,i. l ≤ i → i ≤ l + m → ↓[l, m] v ≐⦋M⦌ ↓[l, m+1] [i⬐d] v. -/3 width=1 by lower_raise_be, veq_sym/ qed. - -lemma lower_raise: ∀M,v,d,l. ↓[l, 1] [l⬐d] v ≐⦋M⦌ v. -/3 width=3 by lower_raise_be, veq_trans/ qed. - -lemma lower_raise_sym: ∀M,v,d,l. v ≐⦋M⦌ ↓[l, 1] [l⬐d] v. -/2 width=1 by veq_sym/ qed. - -lemma raise_lower: ∀M,v,i. [i⬐v i] ↓[i,1] v ≐⦋M⦌ v. -#M #V #i #j elim (lt_or_eq_or_gt j i) #Hij destruct -[ >raise_lt // >lower_lt // -| >raise_eq // -| >raise_gt // >lower_ge /2 width=1 by monotonic_pred/ - lower_lt // >raise_lt // >lower_lt // -| >lower_ge // >raise_eq // -| lapply (ltn_to_ltO … Hlj) #Hj - >lower_ge /2 width=1 by lt_to_le/ >raise_gt // - >lower_ge /4 width=1 by plus_minus, monotonic_pred, eq_f/ -] -qed. - -lemma raise_lower_be_sym: ∀M,v,l,m. [l⬐v (l+m)] ↓[l, m+1] v ≐⦋M⦌ ↓[l, m] v. -/3 width=1 by raise_lower_be, veq_sym/ qed. - -(* Forward lemmas on raise **************************************************) - -lemma lower_fwd_raise_be_S: ∀M,v1,v2,d,l,m. ↓[l, m] v1 ≐⦋M⦌ [l⬐d] v2 → - ↓[l, m+1] v1 ≐ v2. -#M #v1 #v2 #d #l #m #Hv12 #j elim (lt_or_ge j l) #Hlj -[ lapply (Hv12 j) -Hv12 - >lower_lt // >lower_lt // >raise_lt // -| lapply (Hv12 (j+1)) - >lower_ge /2 width=1 by le_S/ >lower_ge // >raise_gt /2 width=1 by le_S_S/ -] -qed-. - -lemma raise_fwd_lower_be_S: ∀M,v1,v2,d,l,m. [l⬐d] v2 ≐⦋M⦌ ↓[l, m] v1 → - v2 ≐ ↓[l, m+1] v1. -/3 width=2 by lower_fwd_raise_be_S, veq_sym/ qed-. - -lemma lower_fwd_raise_be_O: ∀M,v1,v2,d,l,m. ↓[l, m] v1 ≐⦋M⦌ [l⬐d] v2 → v1 (l+m) = d. -#M #v1 #v2 #d #l #m #Hv12 lapply (Hv12 l) ->lower_ge // >raise_eq // -qed-. - -lemma raise_fwd_lower_be_O: ∀M,v1,v2,d,l,m. [l⬐d] v2 ≐⦋M⦌ ↓[l, m] v1 → d = v1 (l+m). -/4 width=7 by lower_fwd_raise_be_O, veq_sym, sym_eq/ qed-. - -(* Inversion lemmas on raise ************************************************) - -lemma raise_inv_lower_lt: ∀M,v1,v2,d,l,m,i. i ≤ l → [i⬐d] v1 ≐⦋M⦌ ↓[l+1, m] v2 → - ∃∃v. v1 ≐ ↓[l, m] v & v2 ≐ [i⬐d] v. -#M #v1 #v2 #d #l #m #i #Hil #Hv12 -lapply (Hv12 i) >raise_eq >lower_lt /2 width=1 by le_S_S/ #H destruct -@(ex2_intro … (↓[i, 1] v2)) // -@(veq_trans … (↓[i, 1] ↓[l+1, m] v2)) -/3 width=3 by lower_lower_le_sym, lower_veq, veq_trans/ -qed-. - -lemma lower_inv_raise_lt: ∀M,v1,v2,d,l,m,i. i ≤ l → ↓[l+1, m] v2 ≐⦋M⦌ [i⬐d] v1 → - ∃∃v. v1 ≐ ↓[l, m] v & v2 ≐ [i⬐d] v. -/3 width=1 by raise_inv_lower_lt, veq_sym/ qed-. - -lemma lower_inv_raise_be: ∀M,v1,v2,d,l,m. ↓[l, m] v1 ≐⦋M⦌ [l⬐d] v2 → - v1 (l+m) = d ∧ ↓[l, m+1] v1 ≐ v2. -/3 width=2 by lower_fwd_raise_be_O, lower_fwd_raise_be_S, conj/ qed-. - -lemma raise_inv_lower_be: ∀M,v1,v2,d,l,m. [l⬐d] v2 ≐⦋M⦌ ↓[l, m] v1 → - d = v1 (l+m) ∧ v2 ≐ ↓[l, m+1] v1. -/3 width=2 by raise_fwd_lower_be_O, raise_fwd_lower_be_S, conj/ qed-.