X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Ffrees_drops.ma;h=fe24eadf5d0496b0dce3a9145e2e441f55e4174b;hp=476a459c87c71bb2dd3d9547c088062fa7f5e02f;hb=222044da28742b24584549ba86b1805a87def070;hpb=98fbba1b68d457807c73ebf70eb2a48696381da4 diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma index 476a459c8..fe24eadf5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma @@ -20,8 +20,8 @@ include "basic_2/static/frees_fqup.ma". (* Advanced properties ******************************************************) -lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≡ ⋆ → - ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≡ ↑*[i]⫯f. +lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ → + ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f. #b #L elim L -L /2 width=1 by frees_atom/ #L #I #IH * [ #H lapply (drops_fwd_isid … H ?) -H // #H destruct @@ -29,16 +29,16 @@ lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≡ ⋆ → ] qed. -lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≡ f → - ∀i,I,L. ⬇*[i] L ≡ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≡ ↑*[i] ⫯f. +lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f → + ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f. #f #K #V #Hf #i elim i -i [ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/ | #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/ ] qed. -lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≡ K.ⓤ{I} → - L ⊢ 𝐅*⦃#i⦄ ≡ ↑*[i] ⫯f. +lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} → + L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f. #f #Hf #I #K #i elim i -i [ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/ | #i #IH #Y #H elim (drops_inv_succ … H) -H @@ -46,16 +46,16 @@ lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≡ K.ⓤ{I ] qed. (* -lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≡ f → - ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃⋆s⦄ ≡ ↑*[i] f. +lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f → + ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f. #f #K #s #Hf #i elim i -i [ #L #H lapply (drops_fwd_isid … H ?) -H // | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/ ] qed. *) -lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≡ f → - ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≡ ↑*[i] f. +lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f → + ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f. #f #K #j #Hf #i elim i -i [ #L #H lapply (drops_fwd_isid … H ?) -H // | #i #IH #L #H elim (drops_inv_succ … H) -H @@ -63,8 +63,8 @@ lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≡ f → ] qed. (* -lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≡ f → - ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃§l⦄ ≡ ↑*[i] f. +lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f → + ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f. #f #K #l #Hf #i elim i -i [ #L #H lapply (drops_fwd_isid … H ?) -H // | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/ @@ -73,33 +73,33 @@ qed. *) (* Advanced inversion lemmas ************************************************) -lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≡ f → - ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ & 𝐈⦃g⦄ & f = ↑*[i] ⫯g - | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g & - ⬇*[i] L ≡ K.ⓑ{I}V & f = ↑*[i] ⫯g - | ∃∃g,I,K. ⬇*[i] L ≡ K.ⓤ{I} & f = ↑*[i] ⫯g. +lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f → + ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g + | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g & + ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g + | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g. #L elim L -L [ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H [ elim (frees_inv_atom … H) -H #f #Hf #H destruct /3 width=3 by or3_intro0, ex3_intro/ | elim (frees_inv_unit … H) -H #f #Hf #H destruct - /4 width=3 by drops_refl, or3_intro2, ex2_3_intro/ + /4 width=3 by drops_refl, or3_intro2, ex3_3_intro/ | elim (frees_inv_pair … H) -H #f #Hf #H destruct /4 width=7 by drops_refl, or3_intro1, ex3_4_intro/ | elim (frees_inv_lref … H) -H #f #Hf #H destruct elim (IH … Hf) -IH -Hf * [ /4 width=3 by drops_drop, or3_intro0, ex3_intro/ | /4 width=7 by drops_drop, or3_intro1, ex3_4_intro/ - | /4 width=3 by drops_drop, or3_intro2, ex2_3_intro/ + | /4 width=3 by drops_drop, or3_intro2, ex3_3_intro/ ] ] qed-. (* Properties with generic slicing for local environments *******************) -lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≡ f1 → - ∀f,L. ⬇*[b, f] L ≡ K → ∀U. ⬆*[f] T ≡ U → - ∀f2. f ~⊚ f1 ≡ f2 → L ⊢ 𝐅*⦃U⦄ ≡ f2. +lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 → + ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U → + ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2. #b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T [ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3 lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2 @@ -154,41 +154,48 @@ lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≡ f1 → ] qed-. +lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U → + ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f. +#b #L #K #HLK #T #U #HTU #f #Hf +@(frees_lifts b … Hf … HTU) // (**) (* auto fails *) +qed. + (* Forward lemmas with generic slicing for local environments ***************) -lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 → - ∀f,K. ⬇*[b, f] L ≡ K → ∀T. ⬆*[f] T ≡ U → - ∀f1. K ⊢ 𝐅*⦃T⦄ ≡ f1 → f ~⊚ f1 ≡ f2. +lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 → + ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U → + ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2. /4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-. (* Inversion lemmas with generic slicing for local environments *************) -lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 → - ∀f,K. ⬇*[b, f] L ≡ K → ∀T. ⬆*[f] T ≡ U → - ∃∃f1. f ~⊚ f1 ≡ f2 & K ⊢ 𝐅*⦃T⦄ ≡ f1. +lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 → + ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U → + ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1. #b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T) /3 width=9 by frees_fwd_coafter, ex2_intro/ qed-. -lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f → - ∀K. ⬇*[b, 𝐔❴1❵] L ≡ K → ∀T. ⬆*[1] T ≡ U → - K ⊢ 𝐅*⦃T⦄ ≡ ⫱f. +lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f → + ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U → + K ⊢ 𝐅*⦃T⦄ ≘ ⫱f. #b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U #f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf /3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/ qed-. -lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 → - ∀f,K. ⬇*[b, f] L ≡ K → ∀T. ⬆*[f] T ≡ U → - ∀f1. f ~⊚ f1 ≡ f2 → K ⊢ 𝐅*⦃T⦄ ≡ f1. +lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 → + ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U → + ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1. #b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U /3 width=7 by frees_eq_repl_back, coafter_inj/ qed-. -lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → - ∀I2,L2,V2,n. ⬇*[n] L1 ≡ L2.ⓑ{I2}V2 → - ∀g1. ⫯g1 = ⫱*[n] f1 → - ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≡ g2 & g2 ⊆ g1. +(* Note: this is used by rex_conf and might be modified *) +lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 → + ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 → + ∀g1. ↑g1 = ⫱*[n] f1 → + ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1. #f1 #L1 #T1 #H elim H -f1 -L1 -T1 [ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s lapply (isid_tls n … Hf1) -Hf1