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=ad020929be506ce06bda80d136da981a207a2294;hp=36b69f4c594186b97010d1e0a55c6d9dc8894228;hb=3c7b4071a9ac096b02334c1d47468776b948e2de;hpb=2f6f2b7c01d47d23f61dd48d767bcb37aecdcfea 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 36b69f4c5..ad020929b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma @@ -23,7 +23,6 @@ include "ground/steps/rtc_shift.ma". include "ground/steps/rtc_plus.ma". include "ground/steps/rtc_max.ma". include "basic_2/notation/relations/predty_7.ma". -include "static_2/syntax/sh.ma". include "static_2/syntax/lenv.ma". include "static_2/syntax/genv.ma". include "static_2/relocation/lifts.ma". @@ -31,63 +30,62 @@ include "static_2/relocation/lifts.ma". (* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************) (* 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_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 +inductive cpg (Rs:relation nat) (Rk:relation rtc): rtc → relation4 genv lenv term term ≝ +| cpg_atom : ∀I,G,L. cpg Rs Rk (𝟘𝟘) G L (⓪[I]) (⓪[I]) +| cpg_ess : ∀G,L,s1,s2. Rs s1 s2 → cpg Rs Rk (𝟘𝟙) G L (⋆s1) (⋆s2) +| cpg_delta: ∀c,G,L,V1,V2,W2. cpg Rs Rk c G L V1 V2 → + ⇧[1] V2 ≘ W2 → cpg Rs Rk c G (L.ⓓV1) (#0) W2 +| cpg_ell : ∀c,G,L,V1,V2,W2. cpg Rs Rk c G L V1 V2 → + ⇧[1] V2 ≘ W2 → cpg Rs Rk (c+𝟘𝟙) G (L.ⓛV1) (#0) W2 +| cpg_lref : ∀c,I,G,L,T,U,i. cpg Rs Rk c G L (#i) T → + ⇧[1] T ≘ U → cpg Rs Rk 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 Rs Rk cV G L V1 V2 → cpg Rs Rk cT G (L.ⓑ[I]V1) T1 T2 → + cpg Rs Rk ((↕*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) -| cpg_cast : ∀cU,cT,G,L,U1,U2,T1,T2. Rt cU cT → - cpg Rt h cU G L U1 U2 → cpg Rt h cT G L T1 T2 → - cpg Rt h (cU∨cT) G L (ⓝU1.T1) (ⓝU2.T2) -| cpg_zeta : ∀c,G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → cpg Rt h c G L T T2 → - cpg Rt h (c+𝟙𝟘) G L (+ⓓV.T1) T2 -| cpg_eps : ∀c,G,L,V,T1,T2. cpg Rt h c G L T1 T2 → cpg Rt h (c+𝟙𝟘) G L (ⓝV.T1) T2 -| 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 Rs Rk cV G L V1 V2 → cpg Rs Rk cT G L T1 T2 → + cpg Rs Rk ((↕*cV)∨cT) G L (ⓐV1.T1) (ⓐV2.T2) +| cpg_cast : ∀cU,cT,G,L,U1,U2,T1,T2. Rk cU cT → + cpg Rs Rk cU G L U1 U2 → cpg Rs Rk cT G L T1 T2 → + cpg Rs Rk (cU∨cT) G L (ⓝU1.T1) (ⓝU2.T2) +| cpg_zeta : ∀c,G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → cpg Rs Rk c G L T T2 → + cpg Rs Rk (c+𝟙𝟘) G L (+ⓓV.T1) T2 +| cpg_eps : ∀c,G,L,V,T1,T2. cpg Rs Rk c G L T1 T2 → cpg Rs Rk (c+𝟙𝟘) G L (ⓝV.T1) T2 +| cpg_ee : ∀c,G,L,V1,V2,T. cpg Rs Rk c G L V1 V2 → cpg Rs Rk (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 Rs Rk cV G L V1 V2 → cpg Rs Rk cW G L W1 W2 → cpg Rs Rk cT G (L.ⓛW1) T1 T2 → + cpg Rs Rk (((↕*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 Rs Rk cV G L V1 V → ⇧[1] V ≘ V2 → cpg Rs Rk cW G L W1 W2 → + cpg Rs Rk cT G (L.ⓓW1) T1 T2 → + cpg Rs Rk (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2) . interpretation "bound context-sensitive parallel rt-transition (term)" - 'PRedTy Rt c h G L T1 T2 = (cpg Rt h c G L T1 T2). + 'PRedTy Rs Rk c G L T1 T2 = (cpg Rs Rk c G L T1 T2). (* 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. -#Rt #HRt #h #G #T elim T -T // * /2 width=1 by cpg_bind/ +(* Note: this is "∀Rs,Rk. reflexive … Rk → ∀G,L. reflexive … (cpg Rs Rk (𝟘𝟘) G L)" *) +lemma cpg_refl (Rs) (Rk): + reflexive … Rk → ∀G,T,L. ❪G,L❫ ⊢ T ⬈[Rs,Rk,𝟘𝟘] T. +#Rk #HRk #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 - | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙 - | ∃∃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 & - 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). -#Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2 +fact cpg_inv_atom1_aux (Rs) (Rk) (c) (G) (L): + ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rs,Rk,c] T2 → ∀J. T1 = ⓪[J] → + ∨∨ ∧∧ T2 = ⓪[J] & c = 𝟘𝟘 + | ∃∃s1,s2. Rs s1 s2 & J = Sort s1 & T2 = ⋆s2 & c = 𝟘𝟙 + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙 + | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i). +#Rs #Rk #c #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/ +| #G #L #s1 #s2 #HRs #J #H destruct /3 width=5 by or5_intro1, ex4_2_intro/ | #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro2, ex5_4_intro/ | #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro3, ex5_4_intro/ | #c #I #G #L #T #U #i #HT #HTU #J #H destruct /3 width=8 by or5_intro4, ex4_4_intro/ @@ -102,70 +100,70 @@ 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 - | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙 - | ∃∃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 & - 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). +lemma cpg_inv_atom1 (Rs) (Rk) (c) (G) (L): + ∀J,T2. ❪G,L❫ ⊢ ⓪[J] ⬈[Rs,Rk,c] T2 → + ∨∨ ∧∧ T2 = ⓪[J] & c = 𝟘𝟘 + | ∃∃s1,s2. Rs s1 s2 & J = Sort s1 & T2 = ⋆s2 & c = 𝟘𝟙 + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙 + | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] 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 → - ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(⫯[h]s) ∧ c = 𝟘𝟙. -#Rt #c #h #G #L #T2 #s #H +lemma cpg_inv_sort1 (Rs) (Rk) (c) (G) (L): + ∀T2,s1. ❪G,L❫ ⊢ ⋆s1 ⬈[Rs,Rk,c] T2 → + ∨∨ ∧∧ T2 = ⋆s1 & c = 𝟘𝟘 + | ∃∃s2. Rs s1 s2 & T2 = ⋆s2 & c = 𝟘𝟙. +#Rs #Rk #c #G #L #T2 #s #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ -[ #s0 #H destruct /3 width=1 by or_intror, conj/ +[ #s1 #s2 #HRs #H1 #H2 #H3 destruct /3 width=3 by ex3_intro, or_intror/ |2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct | #I #K #T #i #_ #_ #_ #H destruct ] qed-. -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 & - L = K.ⓓV1 & c = cV - | ∃∃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 +lemma cpg_inv_zero1 (Rs) (Rk) (c) (G) (L): + ∀T2. ❪G,L❫ ⊢ #0 ⬈[Rs,Rk,c] T2 → + ∨∨ ∧∧ T2 = #0 & c = 𝟘𝟘 + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV + | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & c = cV+𝟘𝟙. +#Rs #Rk #c #G #L #T2 #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/ -[ #s #H destruct +[ #s1 #s2 #_ #H destruct |2,3: #cV #K #V1 #V2 #HV12 #HVT2 #H1 #_ #H2 destruct /3 width=8 by or3_intro1, or3_intro2, ex4_4_intro/ | #I #K #T #i #_ #_ #_ #H destruct ] qed-. -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]. -#Rt #c #h #G #L #T2 #i #H +lemma cpg_inv_lref1 (Rs) (Rk) (c) (G) (L): + ∀T2,i. ❪G,L❫ ⊢ #↑i ⬈[Rs,Rk,c] T2 → + ∨∨ ∧∧ T2 = #(↑i) & c = 𝟘𝟘 + | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I]. +#Rs #Rk #c #G #L #T2 #i #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ -[ #s #H destruct +[ #s1 #s2 #_ #H destruct |2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct | #I #K #T #j #HT #HT2 #H1 #H2 destruct /3 width=6 by ex3_3_intro, or_intror/ ] qed-. -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 +lemma cpg_inv_gref1 (Rs) (Rk) (c) (G) (L): + ∀T2,l. ❪G,L❫ ⊢ §l ⬈[Rs,Rk,c] T2 → ∧∧ T2 = §l & c = 𝟘𝟘. +#Rs #Rk #c #G #L #T2 #l #H elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/ -[ #s #H destruct +[ #s1 #s2 #_ #H destruct |2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct | #I #K #T #i #_ #_ #_ #H destruct ] 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 & - p = true & J = Abbr & c = cT+𝟙𝟘. -#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 +fact cpg_inv_bind1_aux (Rs) (Rk) (c) (G) (L): + ∀U,U2. ❪G,L❫ ⊢ U ⬈[Rs,Rk,c] U2 → + ∀p,J,V1,U1. U = ⓑ[p,J]V1.U1 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓑ[J]V1❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT) + | ∃∃cT,T. ⇧[1] T ≘ U1 & ❪G,L❫ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & J = Abbr & c = cT+𝟙𝟘. +#Rs #Rk #c #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #q #J #W #U1 #H destruct -| #G #L #s #q #J #W #U1 #H destruct +| #G #L #s1 #s2 #_ #q #J #W #U1 #H destruct | #c #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct | #c #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct | #c #I #G #L #T #U #i #_ #_ #q #J #W #U1 #H destruct @@ -180,42 +178,38 @@ 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 & - p = true & I = Abbr & c = cT+𝟙𝟘. +lemma cpg_inv_bind1 (Rs) (Rk) (c) (G) (L): + ∀p,I,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT) + | ∃∃cT,T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[Rs,Rk,cT] 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 & - p = true & c = cT+𝟙𝟘. -#Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * +lemma cpg_inv_abbr1 (Rs) (Rk) (c) (G) (L): + ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈[Rs,Rk,c] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT) + | ∃∃cT,T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & c = cT+𝟙𝟘. +#Rs #Rk #c #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). -#Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * +lemma cpg_inv_abst1 (Rs) (Rk) (c) (G) (L): + ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈[Rs,Rk,c] U2 → + ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓛ[p]V2.T2 & c = ((↕*cV)∨cT). +#Rs #Rk #c #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 → - ∀V1,U1. U = ⓐV1.U1 → - ∨∨ ∃∃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. -#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 +fact cpg_inv_appl1_aux (Rs) (Rk) (c) (G) (L): + ∀U,U2. ❪G,L❫ ⊢ U ⬈[Rs,Rk,c] U2 → + ∀V1,U1. U = ⓐV1.U1 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT) + | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rs,Rk,cT] 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 ⬈[Rs,Rk,cV] V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. +#Rs #Rk #c #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #W #U1 #H destruct -| #G #L #s #W #U1 #H destruct +| #G #L #s1 #s2 #_ #W #U1 #H destruct | #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct | #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct | #c #I #G #L #T #U #i #_ #_ #W #U1 #H destruct @@ -230,30 +224,28 @@ 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 & - 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. +lemma cpg_inv_appl1 (Rs) (Rk) (c) (G) (L): + ∀V1,U1,U2. ❪G,L❫ ⊢ ⓐV1.U1 ⬈[Rs,Rk,c] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT) + | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rs,Rk,cT] 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 ⬈[Rs,Rk,cV] V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[Rs,Rk,cT] 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 → - ∀V1,U1. U = ⓝV1.U1 → - ∨∨ ∃∃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. -#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 +fact cpg_inv_cast1_aux (Rs) (Rk) (c) (G) (L): + ∀U,U2. ❪G,L❫ ⊢ U ⬈[Rs,Rk,c] U2 → + ∀V1,U1. U = ⓝV1.U1 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT) + | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘 + | ∃∃cV. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙. +#Rs #Rk #c #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #W #U1 #H destruct -| #G #L #s #W #U1 #H destruct +| #G #L #s1 #s2 #_ #W #U1 #H destruct | #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct | #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct | #c #I #G #L #T #U #i #_ #_ #W #U1 #H destruct | #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct | #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct -| #cV #cT #G #L #V1 #V2 #T1 #T2 #HRt #HV12 #HT12 #W #U1 #H destruct /3 width=9 by or3_intro0, ex5_4_intro/ +| #cV #cT #G #L #V1 #V2 #T1 #T2 #HRk #HV12 #HT12 #W #U1 #H destruct /3 width=9 by or3_intro0, ex5_4_intro/ | #c #G #L #V #T1 #T #T2 #_ #_ #W #U1 #H destruct | #c #G #L #V #T1 #T2 #HT12 #W #U1 #H destruct /3 width=3 by or3_intro1, ex2_intro/ | #c #G #L #V1 #V2 #T #HV12 #W #U1 #H destruct /3 width=3 by or3_intro2, ex2_intro/ @@ -262,38 +254,38 @@ 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 & - 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. +lemma cpg_inv_cast1 (Rs) (Rk) (c) (G) (L): + ∀V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈[Rs,Rk,c] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT) + | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘 + | ∃∃cV. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] 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 → - ∨∨ T2 = #0 ∧ c = 𝟘𝟘 - | ∃∃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 & - 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/ +lemma cpg_inv_zero1_pair (Rs) (Rk) (c) (G) (K): + ∀I,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[Rs,Rk,c] T2 → + ∨∨ ∧∧ T2 = #0 & c = 𝟘𝟘 + | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abbr & c = cV + | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abst & c = cV+𝟘𝟙. +#Rs #Rk #c #G #K #I #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 → - ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘 - | ∃∃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/ +lemma cpg_inv_lref1_bind (Rs) (Rk) (c) (G) (K): + ∀I,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[Rs,Rk,c] T2 → + ∨∨ ∧∧ T2 = #(↑i) & c = 𝟘𝟘 + | ∃∃T. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2. +#Rs #Rk #c #G #K #I #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. -#Rt #c #h #I #G #L #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H * +lemma cpg_fwd_bind1_minus (Rs) (Rk) (c) (G) (L): + ∀I,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈[Rs,Rk,c] T → ∀p. + ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2. +#Rs #Rk #c #G #L #I #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 ]