include "ground_2/steps/rtc_max.ma".
include "ground_2/steps/rtc_plus.ma".
include "basic_2/notation/relations/predty_7.ma".
-include "basic_2/syntax/item_sh.ma".
-include "basic_2/syntax/lenv.ma".
-include "basic_2/syntax/genv.ma".
-include "basic_2/relocation/lifts.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".
-(* COUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* 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) (⋆(next h s))
+| 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 →
- â¬\86*[1] V2 â\89¡ W2 → cpg Rt h c G (L.ⓓV1) (#0) W2
+ â\87§*[1] V2 â\89\98 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 →
- â¬\86*[1] V2 â\89¡ W2 → cpg Rt h (c+𝟘𝟙) G (L.ⓛV1) (#0) W2
+ â\87§*[1] V2 â\89\98 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 →
- â¬\86*[1] T â\89¡ U â\86\92 cpg Rt h c G (L.â\93\98{I}) (#⫯i) U
+ â\87§*[1] T â\89\98 U â\86\92 cpg Rt h c G (L.â\93\98{I}) (#â\86\91i) 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 ((â\86\93cV)∨cT) G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+ cpg Rt h ((â\86\95*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 ((â\86\93cV)∨cT) G L (ⓐV1.T1) (ⓐV2.T2)
+ cpg Rt h ((â\86\95*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_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_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 (((â\86\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+ cpg Rt h (((â\86\95*cV)â\88¨(â\86\95*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 â\86\92 â¬\86*[1] V â\89¡ V2 → cpg Rt h cW G L W1 W2 →
+ cpg Rt h cV G L V1 V â\86\92 â\87§*[1] V â\89\98 V2 → cpg Rt h cW G L W1 W2 →
cpg Rt h cT G (L.ⓓW1) T1 T2 →
- cpg Rt h (((â\86\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ cpg Rt h (((â\86\95*cV)â\88¨(â\86\95*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
.
interpretation
- "counted context-sensitive parallel rt-transition (term)"
+ "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 "∀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} →
+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 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+ | ∃∃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 &
+ | ∃∃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.â\93\98{I} & J = LRef (⫯i).
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 &
+ L = K.â\93\98{I} & J = LRef (â\86\91i).
#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/
]
qed-.
-lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[Rt, 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 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+ | ∃∃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 &
+ | ∃∃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.â\93\98{I} & J = LRef (⫯i).
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 &
+ L = K.â\93\98{I} & J = LRef (â\86\91i).
/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 = ⋆(next h s) ∧ c = 𝟘𝟙.
+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/
[ #s0 #H destruct /3 width=1 by or_intror, 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/
]
qed-.
-lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ⬈[Rt, c, h] T2 →
- â\88¨â\88¨ T2 = #(⫯i) ∧ c = 𝟘𝟘
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓘ{I}.
+lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[Rt,c,h] T2 →
+ â\88¨â\88¨ T2 = #(â\86\91i) ∧ 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
]
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
]
qed-.
-fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, 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 = â\93\91{p,J}V2.T2 & c = ((â\86\93cV)∨cT)
- | â\88\83â\88\83cT,T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ U1 â¬\88[Rt, cT, h] T & â¬\86*[1] U2 â\89¡ T &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
+ U2 = â\93\91{p,J}V2.T2 & c = ((â\86\95*cV)∨cT)
+ | â\88\83â\88\83cT,T. â\87§*[1] T â\89\98 U1 & â¦\83G,Lâ¦\84 â\8a¢ T â¬\88[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
]
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 = â\93\91{p,I}V2.T2 & c = ((â\86\93cV)∨cT)
- | â\88\83â\88\83cT,T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[Rt, cT, h] T & â¬\86*[1] U2 â\89¡ T &
+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 = â\93\91{p,I}V2.T2 & c = ((â\86\95*cV)∨cT)
+ | â\88\83â\88\83cT,T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 â\8a¢ T â¬\88[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 = â\93\93{p}V2.T2 & c = ((â\86\93cV)∨cT)
- | â\88\83â\88\83cT,T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[Rt, cT, h] T & â¬\86*[1] U2 â\89¡ T &
+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 = â\93\93{p}V2.T2 & c = ((â\86\95*cV)∨cT)
+ | â\88\83â\88\83cT,T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 â\8a¢ T â¬\88[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 = â\93\9b{p}V2.T2 & c = ((â\86\93cV)∨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 = â\93\9b{p}V2.T2 & c = ((â\86\95*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 &
- U2 = â\93\90V2.T2 & c = ((â\86\93cV)∨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 = â\93\9b{p}W1.T1 & U2 = â\93\93{p}â\93\9dW2.V2.T2 & c = ((â\86\93cV)â\88¨(â\86\93cW)∨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 = â\93\93{p}W1.T1 & U2 = â\93\93{p}W2.â\93\90V2.T2 & c = ((â\86\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘.
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
+ U2 = â\93\90V2.T2 & c = ((â\86\95*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 = â\93\9b{p}W1.T1 & U2 = â\93\93{p}â\93\9dW2.V2.T2 & c = ((â\86\95*cV)â\88¨(â\86\95*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 = â\93\93{p}W1.T1 & U2 = â\93\93{p}W2.â\93\90V2.T2 & c = ((â\86\95*cV)â\88¨(â\86\95*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
]
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 = â\93\90V2.T2 & c = ((â\86\93cV)∨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 = â\93\9b{p}W1.T1 & U2 = â\93\93{p}â\93\9dW2.V2.T2 & c = ((â\86\93cV)â\88¨(â\86\93cW)∨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 = â\93\93{p}W1.T1 & U2 = â\93\93{p}W2.â\93\90V2.T2 & c = ((â\86\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘.
+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 = â\93\90V2.T2 & c = ((â\86\95*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 = â\93\9b{p}W1.T1 & U2 = â\93\93{p}â\93\9dW2.V2.T2 & c = ((â\86\95*cV)â\88¨(â\86\95*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 = â\93\93{p}W1.T1 & U2 = â\93\93{p}W2.â\93\90V2.T2 & c = ((â\86\95*cV)â\88¨(â\86\95*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
]
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 →
- â\88¨â\88¨ T2 = #(⫯i) ∧ c = 𝟘𝟘
- | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2.
+lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt,c,h] T2 →
+ â\88¨â\88¨ T2 = #(â\86\91i) ∧ 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: ∀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 &
+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/