| 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 →
- â¬\86*[1] V2 â\89¡ W2 → cpg Rt h c G (L.ⓓV1) (#0) W2
+ â¬\86*[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¬\86*[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 → cpg Rt h c G (L.ⓘ{I}) (#⫯i) U
+ â¬\86*[1] T â\89\98 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 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 →
- â¬\86*[1] T2 â\89¡ T → cpg Rt h (c+𝟙𝟘) G L (+ⓓV.T1) T2
+ â¬\86*[1] T2 â\89\98 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 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 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¬\86*[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 (((↓cV)∨(↓cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
.
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
- | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
- | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | â\88\83â\88\83I,K,T,i. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[Rt, c, h] T & â¬\86*[1] T â\89¡ T2 &
+ | â\88\83â\88\83I,K,T,i. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[Rt, c, h] T & â¬\86*[1] T â\89\98 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/
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
- | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
- | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | â\88\83â\88\83I,K,T,i. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[Rt, c, h] T & â¬\86*[1] T â\89¡ T2 &
+ | â\88\83â\88\83I,K,T,i. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[Rt, c, h] T & â¬\86*[1] T â\89\98 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 = ⋆(next h s) ∧ c = 𝟘𝟙).
+ ∨∨ 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/
qed-.
lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[Rt, c, h] T2 →
- ∨∨ (T2 = #0 ∧ c = 𝟘𝟘)
- | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ ∨∨ T2 = #0 ∧ c = 𝟘𝟘
+ | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 T2 &
L = K.ⓓV1 & c = cV
- | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83cV,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 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 →
- (T2 = #(⫯i) ∧ c = 𝟘𝟘) ∨
- ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓘ{I}.
+ ∨∨ 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
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. ⦃G, L.ⓓV1⦄ ⊢ U1 ⬈[Rt, cT, h] T & ⬆*[1] U2 ≡ T &
- p = true & J = Abbr & c = cT+𝟙𝟘.
+ ∀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
]
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. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[Rt, 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: ∀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.
+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-.
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
- | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V & â¬\86*[1] V â\89¡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V & â¬\86*[1] V â\89\98 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
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
- | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V & â¬\86*[1] V â\89¡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V & â¬\86*[1] V â\89\98 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-.
(* 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)
- | â\88\83â\88\83cV,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ ∨∨ T2 = #0 ∧ c = 𝟘𝟘
+ | â\88\83â\88\83cV,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 T2 &
I = Abbr & c = cV
- | â\88\83â\88\83cV,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83cV,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[Rt, cV, h] V2 & â¬\86*[1] V2 â\89\98 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.
+ ∨∨ 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-.
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