| 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 →
- â¬\86*[1] T â\89\98 U â\86\92 cpg Rt h c G (L.â\93\98{I}) (#⫯i) U
+ â¬\86*[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_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 → ⬆*[1] V ≘ 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
| ∃∃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).
+ 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/
| ∃∃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).
+ 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 →
]
qed-.
-lemma cpg_inv_lref1: â\88\80Rt,c,h,G,L,T2,i. â¦\83G, Lâ¦\84 â\8a¢ #⫯i ⬈[Rt, c, h] T2 →
- â\88¨â\88¨ T2 = #(⫯i) ∧ c = 𝟘𝟘
+lemma cpg_inv_lref1: â\88\80Rt,c,h,G,L,T2,i. â¦\83G, Lâ¦\84 â\8a¢ #â\86\91i ⬈[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/
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)
+ U2 = â\93\91{p,J}V2.T2 & c = ((â\86\95*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
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)
+ U2 = â\93\91{p,I}V2.T2 & c = ((â\86\95*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: ∀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)
+ U2 = â\93\93{p}V2.T2 & c = ((â\86\95*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 *
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).
+ 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
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)
+ 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\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘
+ 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\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘.
+ 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
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)
+ 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\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘
+ 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\93cV)â\88¨(â\86\93cW)∨cT)+𝟙𝟘.
+ 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 →
* #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: â\88\80Rt,c,h,I,G,K,T2,i. â¦\83G, K.â\93\98{I}â¦\84 â\8a¢ #⫯i ⬈[Rt, c, h] T2 →
- â\88¨â\88¨ T2 = #(⫯i) ∧ c = 𝟘𝟘
+lemma cpg_inv_lref1_bind: â\88\80Rt,c,h,I,G,K,T2,i. â¦\83G, K.â\93\98{I}â¦\84 â\8a¢ #â\86\91i ⬈[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/