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=0ca16ee93c18ee2065c2a1035cb8248dcbd376ca;hp=d270bd2be039c3779158008d00254e6659f0da94;hb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;hpb=a373e008bbacd40002c529f3f14da0939af1c404 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 d270bd2be..0ca16ee93 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma @@ -12,80 +12,80 @@ (* *) (**************************************************************************) -include "ground_2/steps/rtc_shift.ma". +include "ground_2/steps/rtc_max.ma". include "ground_2/steps/rtc_plus.ma". -include "basic_2/notation/relations/pred_6.ma". -include "basic_2/grammar/lenv.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/relocation/lifts.ma". -include "basic_2/static/sh.ma". +include "basic_2/notation/relations/predty_7.ma". +include "static_2/syntax/item_sh.ma". +include "static_2/syntax/lenv.ma". +include "static_2/syntax/genv.ma". +include "static_2/relocation/lifts.ma". -(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************) +(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************) (* avtivate genv *) -inductive cpg (h): rtc → relation4 genv lenv term term ≝ -| cpg_atom : ∀I,G,L. cpg h (𝟘𝟘) G L (⓪{I}) (⓪{I}) -| cpg_ess : ∀G,L,s. cpg h (𝟘𝟙) G L (⋆s) (⋆(next h s)) -| cpg_delta: ∀c,G,L,V1,V2,W2. cpg h c G L V1 V2 → - ⬆*[1] V2 ≡ W2 → cpg h c G (L.ⓓV1) (#0) W2 -| cpg_ell : ∀c,G,L,V1,V2,W2. cpg h c G L V1 V2 → - ⬆*[1] V2 ≡ W2 → cpg h ((↓c)+𝟘𝟙) G (L.ⓛV1) (#0) W2 -| cpg_lref : ∀c,I,G,L,V,T,U,i. cpg h c G L (#i) T → - ⬆*[1] T ≡ U → cpg h c G (L.ⓑ{I}V) (#⫯i) U +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) (⋆(next 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 | cpg_bind : ∀cV,cT,p,I,G,L,V1,V2,T1,T2. - cpg h cV G L V1 V2 → cpg h cT G (L.ⓑ{I}V1) T1 T2 → - cpg h ((↓cV)+cT) G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) -| cpg_flat : ∀cV,cT,I,G,L,V1,V2,T1,T2. - cpg h cV G L V1 V2 → cpg h cT G L T1 T2 → - cpg h ((↓cV)+cT) G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) -| cpg_zeta : ∀c,G,L,V,T1,T,T2. cpg h c G (L.ⓓV) T1 T → - ⬆*[1] T2 ≡ T → cpg h ((↓c)+𝟙𝟘) G L (+ⓓV.T1) T2 -| cpg_eps : ∀c,G,L,V,T1,T2. cpg h c G L T1 T2 → cpg h ((↓c)+𝟙𝟘) G L (ⓝV.T1) T2 -| cpg_ee : ∀c,G,L,V1,V2,T. cpg h c G L V1 V2 → cpg h ((↓c)+𝟘𝟙) G L (ⓝV1.T) V2 + 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) +| 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. cpg Rt h c G (L.ⓓV) T1 T → + ⬆*[1] T2 ≘ T → 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_beta : ∀cV,cW,cT,p,G,L,V1,V2,W1,W2,T1,T2. - cpg h cV G L V1 V2 → cpg h cW G L W1 W2 → cpg h cT G (L.ⓛW1) T1 T2 → - cpg h ((↓cV)+(↓cW)+(↓cT)+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.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_theta: ∀cV,cW,cT,p,G,L,V1,V,V2,W1,W2,T1,T2. - cpg h cV G L V1 V → ⬆*[1] V ≡ V2 → cpg h cW G L W1 W2 → - cpg h cT G (L.ⓓW1) T1 T2 → - cpg h ((↓cV)+(↓cW)+(↓cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.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) . interpretation - "context-sensitive generic parallel rt-transition (term)" - 'PRed c h G L T1 T2 = (cpg h c G L T1 T2). + "bound context-sensitive parallel rt-transition (term)" + 'PRedTy Rt c h G L T1 T2 = (cpg Rt h c G L T1 T2). (* Basic properties *********************************************************) -(* Note: this is "∀h,g,L. reflexive … (cpg h (𝟘𝟘) L)" *) -lemma cpg_refl: ∀h,G,T,L. ⦃G, L⦄ ⊢ T ➡[𝟘𝟘, h] T. -#h #G #T elim T -T // * /2 width=1 by cpg_bind, cpg_flat/ -qed. - -lemma cpg_pair_sn: ∀c,h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[c, h] V2 → - ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[↓c, h] ②{I}V2.T. -#c #h * /2 width=1 by cpg_bind, cpg_flat/ +(* 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/ +* /2 width=1 by cpg_appl, cpg_cast/ qed. (* Basic inversion lemmas ***************************************************) -fact cpg_inv_atom1_aux: ∀c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[c, h] T2 → ∀J. T1 = ⓪{J} → +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 = ⋆(next h s) & c = 𝟘𝟙 - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[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 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 & - L = K.ⓛV1 & J = LRef 0 & c = (↓cV)+𝟘𝟙 - | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 & - L = K.ⓑ{I}V & J = LRef (⫯i). -#c #h #G #L #T1 #T2 * -c -G -L -T1 -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). +#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/ | #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 #V #T #U #i #HT #HTU #J #H destruct /3 width=9 by or5_intro4, ex4_5_intro/ +| #c #I #G #L #T #U #i #HT #HTU #J #H destruct /3 width=8 by or5_intro4, ex4_4_intro/ | #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct -| #cV #cT #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct +| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct +| #cU #cT #G #L #U1 #U2 #T1 #T2 #_ #_ #_ #J #H destruct | #c #G #L #V #T1 #T #T2 #_ #_ #J #H destruct | #c #G #L #V #T1 #T2 #_ #J #H destruct | #c #G #L #V1 #V2 #T #_ #J #H destruct @@ -94,182 +94,198 @@ fact cpg_inv_atom1_aux: ∀c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[c, h] T2 → ∀ ] qed-. -lemma cpg_inv_atom1: ∀c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[c, h] T2 → +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 = ⋆(next h s) & c = 𝟘𝟙 - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[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 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 & - L = K.ⓛV1 & J = LRef 0 & c = (↓cV)+𝟘𝟙 - | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 & - L = K.ⓑ{I}V & J = LRef (⫯i). + | ∃∃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). /2 width=3 by cpg_inv_atom1_aux/ qed-. -lemma cpg_inv_sort1: ∀c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[c, h] T2 → - (T2 = ⋆s ∧ c = 𝟘𝟘) ∨ (T2 = ⋆(next h s) ∧ c = 𝟘𝟙). -#c #h #G #L #T2 #s #H +lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[Rt, c, h] T2 → + ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(next 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/ [ #s0 #H destruct /3 width=1 by or_intror, conj/ |2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct -| #I #K #V1 #V2 #i #_ #_ #_ #H destruct +| #I #K #T #i #_ #_ #_ #H destruct ] qed-. -lemma cpg_inv_zero1: ∀c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[c, h] T2 → - ∨∨ (T2 = #0 ∧ c = 𝟘𝟘) - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ 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 & L = K.ⓓV1 & c = cV - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 & - L = K.ⓛV1 & c = (↓cV)+𝟘𝟙. -#c #h #G #L #T2 #H + | ∃∃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/ [ #s #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 #V1 #V2 #i #_ #_ #_ #H destruct +| #I #K #T #i #_ #_ #_ #H destruct ] qed-. -lemma cpg_inv_lref1: ∀c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[c, h] T2 → - (T2 = #(⫯i) ∧ c = 𝟘𝟘) ∨ - ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V. -#c #h #G #L #T2 #i #H +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 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ [ #s #H destruct |2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct -| #I #K #V1 #V2 #j #HV2 #HVT2 #H1 #H2 destruct /3 width=7 by ex3_4_intro, or_intror/ +| #I #K #T #j #HT #HT2 #H1 #H2 destruct /3 width=6 by ex3_3_intro, or_intror/ ] qed-. -lemma cpg_inv_gref1: ∀c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[c, h] T2 → T2 = §l ∧ c = 𝟘𝟘. -#c #h #G #L #T2 #l #H +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 |2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct -| #I #K #V1 #V2 #i #_ #_ #_ #H destruct +| #I #K #T #i #_ #_ #_ #H destruct ] qed-. -fact cpg_inv_bind1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[c, h] U2 → - ∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 → ( - ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ➡[cT, h] T2 & - U2 = ⓑ{p,J}V2.T2 & c = (↓cV)+cT - ) ∨ - ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ U1 ➡[cT, h] T & ⬆*[1] U2 ≡ T & - p = true & J = Abbr & c = (↓cT)+𝟙𝟘. -#c #h #G #L #U #U2 * -c -G -L -U -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. ⦃G, L.ⓓV1⦄ ⊢ U1 ⬈[Rt, cT, h] T & ⬆*[1] U2 ≘ T & + 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 | #G #L #s #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 #V #T #U #i #_ #_ #q #J #W #U1 #H destruct -| #rv #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W #U1 #H destruct /3 width=8 by ex4_4_intro, or_introl/ -| #rv #cT #I #G #L #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct +| #c #I #G #L #T #U #i #_ #_ #q #J #W #U1 #H destruct +| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W #U1 #H destruct /3 width=8 by ex4_4_intro, or_introl/ +| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct +| #cU #cT #G #L #U1 #U2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct | #c #G #L #V #T1 #T #T2 #HT1 #HT2 #q #J #W #U1 #H destruct /3 width=5 by ex5_2_intro, or_intror/ | #c #G #L #V #T1 #T2 #_ #q #J #W #U1 #H destruct | #c #G #L #V1 #V2 #T #_ #q #J #W #U1 #H destruct -| #rv #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct -| #rv #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #q #J #W #U1 #H destruct +| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct +| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #q #J #W #U1 #H destruct ] qed-. -lemma cpg_inv_bind1: ∀c,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[c, h] U2 → ( - ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[cT, h] T2 & - U2 = ⓑ{p,I}V2.T2 & c = (↓cV)+cT - ) ∨ - ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[cT, h] T & ⬆*[1] U2 ≡ T & - p = true & I = Abbr & c = (↓cT)+𝟙𝟘. +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. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[Rt, cT, h] T & ⬆*[1] U2 ≘ T & + p = true & I = Abbr & c = cT+𝟙𝟘. /2 width=3 by cpg_inv_bind1_aux/ qed-. -lemma cpg_inv_abbr1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[c, h] U2 → ( - ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[cT, h] T2 & - U2 = ⓓ{p}V2.T2 & c = (↓cV)+cT - ) ∨ - ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[cT, h] T & ⬆*[1] U2 ≡ T & - p = true & c = (↓cT)+𝟙𝟘. -#c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * +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. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[Rt, cT, h] T & ⬆*[1] U2 ≘ T & + 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: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[c, h] U2 → - ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[cT, h] T2 & - U2 = ⓛ{p}V2.T2 & c = (↓cV)+cT. -#c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * +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_flat1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[c, h] U2 → - ∀J,V1,U1. U = ⓕ{J}V1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 & - U2 = ⓕ{J}V2.T2 & c = (↓cV)+cT - | ∃∃cT. ⦃G, L⦄ ⊢ U1 ➡[cT, h] U2 & J = Cast & c = (↓cT)+𝟙𝟘 - | ∃∃cV. ⦃G, L⦄ ⊢ V1 ➡[cV, h] U2 & J = Cast & c = (↓cV)+𝟘𝟙 - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[cT, h] T2 & - J = Appl & 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 ➡[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[cT, h] T2 & - J = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘. -#c #h #G #L #U #U2 * -c -G -L -U -U2 -[ #I #G #L #J #W #U1 #H destruct -| #G #L #s #J #W #U1 #H destruct -| #c #G #L #V1 #V2 #W2 #_ #_ #J #W #U1 #H destruct -| #c #G #L #V1 #V2 #W2 #_ #_ #J #W #U1 #H destruct -| #c #I #G #L #V #T #U #i #_ #_ #J #W #U1 #H destruct -| #rv #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct -| #rv #cT #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=8 by or5_intro0, ex4_4_intro/ -| #c #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct -| #c #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=3 by or5_intro1, ex3_intro/ -| #c #G #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=3 by or5_intro2, ex3_intro/ -| #rv #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=15 by or5_intro3, ex7_9_intro/ -| #rv #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=17 by or5_intro4, ex8_10_intro/ +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 +[ #I #G #L #W #U1 #H destruct +| #G #L #s #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 #HV12 #HT12 #W #U1 #H destruct /3 width=8 by or3_intro0, ex4_4_intro/ +| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #_ #W #U1 #H destruct +| #c #G #L #V #T1 #T #T2 #_ #_ #W #U1 #H destruct +| #c #G #L #V #T1 #T2 #_ #W #U1 #H destruct +| #c #G #L #V1 #V2 #T #_ #W #U1 #H destruct +| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #W #U1 #H destruct /3 width=15 by or3_intro1, ex6_9_intro/ +| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #W #U1 #H destruct /3 width=17 by or3_intro2, ex7_10_intro/ ] qed-. -lemma cpg_inv_flat1: ∀c,h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 & - U2 = ⓕ{I}V2.T2 & c = (↓cV)+cT - | ∃∃cT. ⦃G, L⦄ ⊢ U1 ➡[cT, h] U2 & I = Cast & c = (↓cT)+𝟙𝟘 - | ∃∃cV. ⦃G, L⦄ ⊢ V1 ➡[cV, h] U2 & I = Cast & c = (↓cV)+𝟘𝟙 - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[cT, h] T2 & - I = Appl & 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 ➡[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[cT, h] T2 & - I = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘. -/2 width=3 by cpg_inv_flat1_aux/ 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. +/2 width=3 by cpg_inv_appl1_aux/ qed-. -lemma cpg_inv_appl1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡[c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 & - U2 = ⓐV2.T2 & c = (↓cV)+cT - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[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 ➡[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[cT, h] T2 & - U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘. -#c #h #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H * -[ /3 width=8 by or3_intro0, ex4_4_intro/ -|2,3: #c #_ #H destruct -| /3 width=15 by or3_intro1, ex6_9_intro/ -| /3 width=17 by or3_intro2, ex7_10_intro/ +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 +[ #I #G #L #W #U1 #H destruct +| #G #L #s #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/ +| #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/ +| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #W #U1 #H destruct +| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #W #U1 #H destruct ] qed-. -lemma cpg_inv_cast1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 & - U2 = ⓝV2.T2 & c = (↓cV)+cT - | ∃∃cT. ⦃G, L⦄ ⊢ U1 ➡[cT, h] U2 & c = (↓cT)+𝟙𝟘 - | ∃∃cV. ⦃G, L⦄ ⊢ V1 ➡[cV, h] U2 & c = (↓cV)+𝟘𝟙. -#c #h #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H * -[ /3 width=8 by or3_intro0, ex4_4_intro/ -|2,3: /3 width=3 by or3_intro1, or3_intro2, ex2_intro/ -| #rv #cW #cT #p #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #H destruct -| #rv #cW #cT #p #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #H destruct -] +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. +/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/ +* #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/ +* #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/ qed-. (* Basic forward lemmas *****************************************************) -lemma cpg_fwd_bind1_minus: ∀c,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[c, h] T → ∀b. - ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[c, h] ⓑ{b,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. -#c #h #I #G #L #V1 #T1 #T #H #b elim (cpg_inv_bind1 … H) -H * +#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 ]