X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fs_computation%2Ffqus.ma;h=1289e8487587df00115ccd5468e4778622cd40d6;hp=fb6a7ea7def256a305786849b19089e9ddc1e302;hb=f308429a0fde273605a2330efc63268b4ac36c99;hpb=87f57ddc367303c33e19c83cd8989cd561f3185b diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma index fb6a7ea7d..1289e8487 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma @@ -31,15 +31,15 @@ interpretation "star-iterated structural successor (closure)" (* Basic eliminators ********************************************************) lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2. + (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2. #b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H @(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) // qed-. lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1. + (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1. #b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H @(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) // qed-. @@ -49,56 +49,56 @@ qed-. lemma fqus_refl: ∀b. tri_reflexive … (fqus b). /2 width=1 by tri_inj/ qed. -lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. /2 width=1 by tri_inj/ qed. -lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. /2 width=5 by tri_step/ qed-. -lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. /2 width=5 by tri_TC_strap/ qed-. (* Basic inversion lemmas ***************************************************) -lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨ - ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄. + ∃∃G,L,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/ #G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/ * #HG #HL #HT #_ destruct // qed-. -lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ → (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨ - ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}. + ∃∃J,L. ⦃G1,L,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}. #b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/ qed-. -lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2 - | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0 - | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j. + | ∃∃J,L,V. ⦃G1,L,V⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓑ{J}V & i = 0 + | ∃∃J,L,j. ⦃G1,L,#j⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J} & i = ↑j. #b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/ #G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/ qed-. -lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ → (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨ - ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}. + ∃∃J,L. ⦃G1,L,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}. #b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/ qed-. -lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2 - | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ⦃G1, L1.ⓧ, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ ∧ b = Ⓕ - | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. + | ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ⦃G1,L1.ⓧ,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ ∧ b = Ⓕ + | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. #b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/ #G #L #T #H elim (fqu_inv_bind1 … H) -H * [4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct @@ -106,21 +106,21 @@ lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ qed-. -lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ → +lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2 - | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄ - | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄ - | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. + | ⦃G1,L1,V1⦄ ⊐* ⦃G2,L2,T2⦄ + | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⊐* ⦃G2,L2,T2⦄ + | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐* ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ] /3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/ #_ #H destruct qed-. -lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2 - | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}. + | ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}. #b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/ #G #L #T #H elim (fqu_inv_flat1 … H) -H * [3: #J ] #H1 #H2 #H3 #H destruct @@ -129,35 +129,35 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2. #b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/ #G #L #T #H elim (fqu_inv_atom1 … H) qed-. -lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1,L1.ⓘ{I},⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1,L1,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_sort1_bind … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ qed-. -lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1,L1.ⓑ{I}V1,#0⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_zero1_pair … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ qed-. -lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1,L1.ⓘ{I},#↑i⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1,L1,#i⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_lref1_bind … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ qed-. -lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1,L1.ⓘ{I},§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1,L1,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_gref1_bind … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/