| 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.
∀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)
- | 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\98 T &
+ | â\88\83â\88\83cT,T. â¬\86*[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
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)
- | 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\98 T &
+ | â\88\83â\88\83cT,T. â¬\86*[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 = ⓓ{p}V2.T2 & c = ((↕*cV)∨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\98 T &
+ | â\88\83â\88\83cT,T. â¬\86*[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/