/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)
+ ∨∨ 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 &
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-.
(* 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)
+ ∨∨ 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 &
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-.