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=2d86769599eb4cad76e18a647d567d300bb43bfa;hp=a4d01db0880f68558ec9749c6558140128a6e064;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hpb=3b7b8afcb429a60d716d5226a5b6ab0d003228b1 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 a4d01db08..2d8676959 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma @@ -34,15 +34,15 @@ 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-. @@ -52,56 +52,56 @@ 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=6 by ex3_3_intro, or3_intro1, or3_intro2/ 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⦄ → - ∨∨ ∧∧ 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⦄ & b = Ⓣ - | ∧∧ ⦃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}. +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❫ & b = Ⓣ + | ∧∧ ❪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 [3,4: #Hb ] #H destruct @@ -109,21 +109,21 @@ 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⦄ → - ∨∨ ∧∧ 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}. +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]. #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,3,4: * ] /3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3/ #_ #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⦄ → - ∨∨ ∧∧ 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}. +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]. #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 @@ -132,35 +132,35 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⬂*[b] ⦃G2,L2,T2⦄ → - ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = 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/