From b40ccca541b378319b076b1bb36c435dfdf2a55f Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Fri, 1 Dec 2017 14:00:04 +0000 Subject: [PATCH] work in progress on a new definition of voids ... --- .../rvoidstar_4.ma} | 6 +-- .../lambdadelta/basic_2/static/fle.ma | 22 ++++++----- .../lambdadelta/basic_2/syntax/voids.ma | 38 ++++++++++++++++--- .../basic_2/syntax/voids_length.ma | 12 +++--- 4 files changed, 54 insertions(+), 24 deletions(-) rename matita/matita/contribs/lambdadelta/basic_2/notation/{functions/voidstar_2.ma => relations/rvoidstar_4.ma} (86%) diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/functions/voidstar_2.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rvoidstar_4.ma similarity index 86% rename from matita/matita/contribs/lambdadelta/basic_2/notation/functions/voidstar_2.ma rename to matita/matita/contribs/lambdadelta/basic_2/notation/relations/rvoidstar_4.ma index 4c95b4170..f9b03f68c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/functions/voidstar_2.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rvoidstar_4.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( ⓧ* [ term 46 n ] term 46 L )" - non associative with precedence 46 - for @{ 'VoidStar $n $L }. +notation "hvbox( ⓧ* [ term 46 n1 ] term 46 L1 ≋ ⓧ* [ term 46 n2 ] term 46 L2 )" + non associative with precedence 45 + for @{ 'RVoidStar $n1 $L1 $n2 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/fle.ma b/matita/matita/contribs/lambdadelta/basic_2/static/fle.ma index 1a0a83d6b..7340a8abc 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/fle.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/fle.ma @@ -19,13 +19,13 @@ include "basic_2/static/frees.ma". (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************) -inductive fle (T2) (L2) (T1): predicate lenv ≝ -| fle_intro: ∀f1,f2,L1,n. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 → - |L1| = |L2| → ⫱*[n]f1 ⊆ f2 → fle T2 L2 T1 (ⓧ*[n]L1) +inductive fle (T1) (T2): relation lenv ≝ +| fle_intro: ∀f1,f2,L1,L2,n1,n2. ⓧ*[n1]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → ⓧ*[n2]L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 → + |L1| = |L2| → ⫱*[n1]f1 ⊆ ⫱*[n2]f2 → fle T1 T2 (ⓧ*[n1]L1) (ⓧ*[n2]L2) . interpretation "free variables inclusion (restricted closure)" - 'SubSetEq L1 T1 L2 T2 = (fle T2 L2 T1 L1). + 'SubSetEq L1 T1 L2 T2 = (fle T1 T2 L1 L2). (* Basic properties *********************************************************) @@ -37,11 +37,15 @@ lemma fle_gref: ∀L1,L2. |L1| = |L2| → ∀l1,l2. ⦃L1, §l1⦄ ⊆ ⦃L2, § (* Basic inversion lemmas ***************************************************) -fact fle_inv_voids_sn_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → - ∀K1,n. |K1| = |L2| → L1 = ⓧ*[n]K1 → - ∃∃f1,f2. ⓧ*[n]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n]f1 ⊆ f2. -#L1 #L2 #T1 #T2 * -L1 #f1 #f2 #L1 #n #Hf1 #Hf2 #HL12 #Hf12 #Y #x #HY #H destruct -elim (voids_inj_length … H) // -H -HL12 -HY #H1 #H2 destruct +fact fle_inv_voids_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → + ∀K1,K2,n1,n2. |K1| = |K2| → L1 = ⓧ*[n1]K1 → L2 = ⓧ*[n2]K2 → + ∃∃f1,f2. ⓧ*[n1]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & ⓧ*[n2]K2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2. +#L1 #L2 #T1 #T2 * -L1 -L2 +#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #HL12 #Hf12 #Y1 #Y2 #x1 #x2 #HY12 #H1 #H2 destruct +>H1 in Hf1; >H2 in Hf2; #Hf2 #Hf1 +@(ex3_2_intro … Hf1 Hf2) -Hf1 -Hf2 + +elim (voids_inj_length … H1) // -H -HL12 -HY #H1 #H2 destruct /2 width=5 by ex3_2_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma index 766eefd5c..8c58fd478 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma @@ -12,19 +12,44 @@ (* *) (**************************************************************************) -include "basic_2/notation/functions/voidstar_2.ma". +include "basic_2/notation/relations/rvoidstar_4.ma". include "basic_2/syntax/lenv.ma". -(* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************) +(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************) -rec definition voids (L:lenv) (n:nat) on n: lenv ≝ match n with -[ O ⇒ L | S m ⇒ (voids L m).ⓧ ]. +inductive voids: bi_relation nat lenv ≝ +| voids_atom : voids 0 (⋆) 0 (⋆) +| voids_pair_sn: ∀I1,I2,K1,K2,V1,n. voids n K1 n K2 → + voids 0 (K1.ⓑ{I1}V1) 0 (K2.ⓘ{I2}) +| voids_pair_dx: ∀I1,I2,K1,K2,V2,n. voids n K1 n K2 → + voids 0 (K1.ⓘ{I1}) 0 (K2.ⓑ{I2}V2) +| voids_void_sn: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 → + voids (⫯n1) (K1.ⓧ) n2 K2 +| voids_void_dx: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 → + voids n1 K1 (⫯n2) (K2.ⓧ) +. -interpretation "extension with exclusion binders (local environment)" - 'VoidStar n L = (voids L n). +interpretation "equivalence up to exclusion binders (local environment)" + 'RVoidStar n1 L1 n2 L2 = (voids n1 L1 n2 L2). (* Basic properties *********************************************************) +lemma voids_refl: ∀L. ∃n. ⓧ*[n]L ≋ ⓧ*[n]L. +#L elim L -L /2 width=2 by ex_intro, voids_atom/ +#L #I * #n #IH cases I -I /3 width=2 by ex_intro, voids_pair_dx/ +* /4 width=2 by ex_intro, voids_void_sn, voids_void_dx/ +qed-. + +lemma voids_sym: bi_symmetric … voids. +#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2 +/2 width=2 by voids_atom, voids_pair_sn, voids_pair_dx, voids_void_sn, voids_void_dx/ +qed-. + +(* + + + + lemma voids_zero: ∀L. L = ⓧ*[0]L. // qed. @@ -46,3 +71,4 @@ theorem voids_inj: ∀n. injective … (λL. ⓧ*[n]L). elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *) /2 width=1 by/ qed-. +*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/voids_length.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/voids_length.ma index 510a05f68..30738ee24 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/syntax/voids_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/voids_length.ma @@ -15,14 +15,14 @@ include "basic_2/syntax/lenv_length.ma". include "basic_2/syntax/voids.ma". -(* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************) +(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************) -(* Properties with length for local environments ****************************) +(* Forward lemmas with length for local environments ************************) -lemma length_void: ∀L,n. n+|L| = |ⓧ*[n]L|. -#L #n elim n -n // -#n #IH length_bind