X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Ffrees_drops.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Ffrees_drops.ma;h=0000000000000000000000000000000000000000;hb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;hp=fe24eadf5d0496b0dce3a9145e2e441f55e4174b;hpb=222044da28742b24584549ba86b1805a87def070;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma deleted file mode 100644 index fe24eadf5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma +++ /dev/null @@ -1,245 +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 "ground_2/relocation/nstream_coafter.ma". -include "basic_2/relocation/drops_drops.ma". -include "basic_2/static/frees_fqup.ma". - -(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) - -(* Advanced properties ******************************************************) - -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 -| /4 width=3 by frees_lref, drops_inv_drop1/ -] -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. -#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. -#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 - #J #L #HLK #H destruct /3 width=1 by frees_lref/ -] -qed. -(* -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. -#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 - #I #Y #HYK #H destruct /3 width=1 by frees_lref/ -] -qed. -(* -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/ -] -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} & 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, 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, 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. -#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 - >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/ -| #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3 - elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct - elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct - lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2 - elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf - elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i - lapply (at_inv_uni … Hj) -Hj #H destruct - /3 width=8 by frees_atom_drops, drops_trans/ -| #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3 - lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1 - lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct - elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H - elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct - elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct - lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW // - /2 width=5 by frees_pair_drops/ -| #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3 - lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct - elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct - lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2 - elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H - lapply (liftsb_inv_unit_sn … H) -H #H destruct - /2 width=3 by frees_unit_drops/ -| #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3 - lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1 - lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct - elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct - elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_ - elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct - lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY - lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ] - >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *) -| #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3 - lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2 - >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/ -| #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3 - elim (sor_inv_isfin3 … H1f1) // #Hf1V #H - lapply (isfin_inv_tl … H) -H - elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct - elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H - elim (coafter_inv_tl1 … H) -H - /5 width=5 by frees_bind, drops_skip, ext2_pair/ -| #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3 - elim (sor_inv_isfin3 … H1f1) // - elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct - elim (coafter_sor … H3 … H1f1) - /3 width=5 by coafter_isfin2_fwd, frees_flat/ -] -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. -/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. -#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. -#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. -#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-. - -(* 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

(injective_next … Hgf1) -g1 - /2 width=3 by ex2_intro/ - | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12 - #HL12 #g1 tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2 - /3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/ - ] -| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1 - lapply (sor_tls … Hf1 n) -Hf1