X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpg.ma;h=377d39ea6cfc129ddf1f8628bfbb16c2b5feacdc;hp=c33b21973f94da02b7745e130145ff79ee91282a;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hpb=3b7b8afcb429a60d716d5226a5b6ab0d003228b1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma index c33b21973..377d39ea6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma @@ -31,17 +31,17 @@ include "static_2/relocation/lifts.ma". (* avtivate genv *) inductive cpg (Rt:relation rtc) (h): rtc → relation4 genv lenv term term ≝ -| cpg_atom : ∀I,G,L. cpg Rt h (𝟘𝟘) G L (⓪{I}) (⓪{I}) +| cpg_atom : ∀I,G,L. cpg Rt h (𝟘𝟘) G L (⓪[I]) (⓪[I]) | cpg_ess : ∀G,L,s. cpg Rt h (𝟘𝟙) G L (⋆s) (⋆(⫯[h]s)) | cpg_delta: ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 → ⇧*[1] V2 ≘ W2 → cpg Rt h c G (L.ⓓV1) (#0) W2 | cpg_ell : ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 → ⇧*[1] V2 ≘ W2 → cpg Rt h (c+𝟘𝟙) G (L.ⓛV1) (#0) W2 | cpg_lref : ∀c,I,G,L,T,U,i. cpg Rt h c G L (#i) T → - ⇧*[1] T ≘ U → cpg Rt h c G (L.ⓘ{I}) (#↑i) U + ⇧*[1] T ≘ U → cpg Rt h c G (L.ⓘ[I]) (#↑i) U | cpg_bind : ∀cV,cT,p,I,G,L,V1,V2,T1,T2. - cpg Rt h cV G L V1 V2 → cpg Rt h cT G (L.ⓑ{I}V1) T1 T2 → - cpg Rt h ((↕*cV)∨cT) G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) + cpg Rt h cV G L V1 V2 → cpg Rt h cT G (L.ⓑ[I]V1) T1 T2 → + cpg Rt h ((↕*cV)∨cT) G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2) | cpg_appl : ∀cV,cT,G,L,V1,V2,T1,T2. cpg Rt h cV G L V1 V2 → cpg Rt h cT G L T1 T2 → cpg Rt h ((↕*cV)∨cT) G L (ⓐV1.T1) (ⓐV2.T2) @@ -54,11 +54,11 @@ inductive cpg (Rt:relation rtc) (h): rtc → relation4 genv lenv term term ≝ | cpg_ee : ∀c,G,L,V1,V2,T. cpg Rt h c G L V1 V2 → cpg Rt h (c+𝟘𝟙) G L (ⓝV1.T) V2 | cpg_beta : ∀cV,cW,cT,p,G,L,V1,V2,W1,W2,T1,T2. cpg Rt h cV G L V1 V2 → cpg Rt h cW G L W1 W2 → cpg Rt h cT G (L.ⓛW1) T1 T2 → - cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2) + cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2) | cpg_theta: ∀cV,cW,cT,p,G,L,V1,V,V2,W1,W2,T1,T2. cpg Rt h cV G L V1 V → ⇧*[1] V ≘ V2 → cpg Rt h cW G L W1 W2 → cpg Rt h cT G (L.ⓓW1) T1 T2 → - cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2) + cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2) . interpretation @@ -68,22 +68,22 @@ interpretation (* Basic properties *********************************************************) (* Note: this is "∀Rt. reflexive … Rt → ∀h,g,L. reflexive … (cpg Rt h (𝟘𝟘) L)" *) -lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G,L⦄ ⊢ T ⬈[Rt,𝟘𝟘,h] T. +lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ❪G,L❫ ⊢ T ⬈[Rt,𝟘𝟘,h] T. #Rt #HRt #h #G #T elim T -T // * /2 width=1 by cpg_bind/ * /2 width=1 by cpg_appl, cpg_cast/ qed. (* Basic inversion lemmas ***************************************************) -fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ∀J. T1 = ⓪{J} → - ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘 +fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rt,c,h] T2 → ∀J. T1 = ⓪[J] → + ∨∨ T2 = ⓪[J] ∧ c = 𝟘𝟘 | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙 - | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV - | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙 - | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & - L = K.ⓘ{I} & J = LRef (↑i). + | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & + L = K.ⓘ[I] & J = LRef (↑i). #Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2 [ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/ | #G #L #s #J #H destruct /3 width=3 by or5_intro1, ex3_intro/ @@ -101,18 +101,18 @@ fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ] qed-. -lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[Rt,c,h] T2 → - ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘 +lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ⬈[Rt,c,h] T2 → + ∨∨ T2 = ⓪[J] ∧ c = 𝟘𝟘 | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙 - | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV - | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙 - | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & - L = K.ⓘ{I} & J = LRef (↑i). + | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & + L = K.ⓘ[I] & J = LRef (↑i). /2 width=3 by cpg_inv_atom1_aux/ qed-. -lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[Rt,c,h] T2 → +lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ⬈[Rt,c,h] T2 → ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(⫯[h]s) ∧ c = 𝟘𝟙. #Rt #c #h #G #L #T2 #s #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ @@ -122,11 +122,11 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ ] qed-. -lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[Rt,c,h] T2 → +lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ❪G,L❫ ⊢ #0 ⬈[Rt,c,h] T2 → ∨∨ T2 = #0 ∧ c = 𝟘𝟘 - | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV - | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & L = K.ⓛV1 & c = cV+𝟘𝟙. #Rt #c #h #G #L #T2 #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/ @@ -136,9 +136,9 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/ ] qed-. -lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[Rt,c,h] T2 → +lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ⬈[Rt,c,h] T2 → ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘 - | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ{I}. + | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ[I]. #Rt #c #h #G #L #T2 #i #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ [ #s #H destruct @@ -147,7 +147,7 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ ] qed-. -lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘. +lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ❪G,L❫ ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘. #Rt #c #h #G #L #T2 #l #H elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/ [ #s #H destruct @@ -156,11 +156,11 @@ elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/ ] qed-. -fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → - ∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & - U2 = ⓑ{p,J}V2.T2 & c = ((↕*cV)∨cT) - | ∃∃cT,T. ⇧*[1] T ≘ U1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 & +fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ❪G,L❫ ⊢ U ⬈[Rt,c,h] U2 → + ∀p,J,V1,U1. U = ⓑ[p,J]V1.U1 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓑ[J]V1❫ ⊢ U1 ⬈[Rt,cT,h] T2 & + U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT) + | ∃∃cT,T. ⇧*[1] T ≘ U1 & ❪G,L❫ ⊢ T ⬈[Rt,cT,h] U2 & p = true & J = Abbr & c = cT+𝟙𝟘. #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #q #J #W #U1 #H destruct @@ -179,39 +179,39 @@ fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → ] qed-. -lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U2 = ⓑ{p,I}V2.T2 & c = ((↕*cV)∨cT) - | ∃∃cT,T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 & +lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT) + | ∃∃cT,T. ⇧*[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[Rt,cT,h] U2 & p = true & I = Abbr & c = cT+𝟙𝟘. /2 width=3 by cpg_inv_bind1_aux/ qed-. -lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt,c,h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U2 = ⓓ{p}V2.T2 & c = ((↕*cV)∨cT) - | ∃∃cT,T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 & +lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT) + | ∃∃cT,T. ⇧*[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[Rt,cT,h] U2 & p = true & c = cT+𝟙𝟘. #Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * /3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/ qed-. -lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt,c,h] U2 → - ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U2 = ⓛ{p}V2.T2 & c = ((↕*cV)∨cT). +lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈[Rt,c,h] U2 → + ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U2 = ⓛ[p]V2.T2 & c = ((↕*cV)∨cT). #Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * [ /3 width=8 by ex4_4_intro/ | #c #T #_ #_ #_ #H destruct ] qed-. -fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → +fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ❪G,L❫ ⊢ U ⬈[Rt,c,h] U2 → ∀V1,U1. U = ⓐV1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT) - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘 - | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⇧*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. + | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘 + | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V & ⇧*[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #W #U1 #H destruct | #G #L #s #W #U1 #H destruct @@ -229,21 +229,21 @@ fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → ] qed-. -lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & +lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT) - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘 - | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⇧*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & - U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. + | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘 + | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V & ⇧*[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[Rt,cT,h] T2 & + U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. /2 width=3 by cpg_inv_appl1_aux/ qed-. -fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → +fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ❪G,L❫ ⊢ U ⬈[Rt,c,h] U2 → ∀V1,U1. U = ⓝV1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 & Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT) - | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘 - | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙. + | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘 + | ∃∃cV. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙. #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #W #U1 #H destruct | #G #L #s #W #U1 #H destruct @@ -261,37 +261,37 @@ fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → ] qed-. -lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & +lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 & Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT) - | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘 - | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙. + | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘 + | ∃∃cV. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙. /2 width=3 by cpg_inv_cast1_aux/ qed-. (* Advanced inversion lemmas ************************************************) -lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt,c,h] T2 → +lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[Rt,c,h] T2 → ∨∨ T2 = #0 ∧ c = 𝟘𝟘 - | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & I = Abbr & c = cV - | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & + | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 & I = Abst & c = cV+𝟘𝟙. #Rt #c #h #I #G #K #V1 #T2 #H elim (cpg_inv_zero1 … H) -H /2 width=1 by or3_intro0/ * #z #Y #X1 #X2 #HX12 #HXT2 #H1 #H2 destruct /3 width=5 by or3_intro1, or3_intro2, ex4_2_intro/ qed-. -lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt,c,h] T2 → +lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[Rt,c,h] T2 → ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘 - | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2. + | ∃∃T. ❪G,K❫ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2. #Rt #c #h #I #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/ * #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/ qed-. (* Basic forward lemmas *****************************************************) -lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt,c,h] T → ∀p. - ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] ⓑ{p,I}V2.T2 & - T = -ⓑ{I}V2.T2. +lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈[Rt,c,h] T → ∀p. + ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rt,c,h] ⓑ[p,I]V2.T2 & + T = -ⓑ[I]V2.T2. #Rt #c #h #I #G #L #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H * [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/ | #c #T2 #_ #_ #H destruct