- ∀I,K. ⇩*[i] L ≘ K.ⓘ{I} →
- ∨∨ ∧∧ ∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
- | ∃∃n,p,W,V1,V2,W2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U & ⦃G,K⦄ ⊢ V1 ⬌η[h] V2
- & ⇧*[↑i] V2 ≘ W2 & I = BPair Abst W & +ⓛW2.ⓐ#0.#(↑i) = X2.
+ ∀I,K,W. ⇩*[i] L ≘ K.ⓑ{I}W →
+ ∨∨ ∧∧ Abbr = I & #i = X2
+ | ∧∧ Abst = I & ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
+ | ∃∃n,p,W1,W2,V,V1,V2,U. Abst = I & ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
+ & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[↑i] W1 ≘ W2
+ & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[↑i] V1 ≘ V2
+ & ⓝW2.+ⓛV2.ⓐ#0.#(↑i) = X2.
+
+axiom cpce_inv_lref_sn_drops_ldef (h) (G) (i) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
+ ∀K,V. ⇩*[i] L ≘ K.ⓓV → #i = X2.
+
+axiom cpce_inv_lref_sn_drops_ldec (h) (G) (i) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
+ ∀K,W. ⇩*[i] L ≘ K.ⓛW →
+ ∨∨ ∧∧ ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
+ | ∃∃n,p,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
+ & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[↑i] W1 ≘ W2
+ & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[↑i] V1 ≘ V2
+ & ⓝW2.+ⓛV2.ⓐ#0.#(↑i) = X2.
+(*