inductive lsubc (RP) (G): relation lenv ≝
| lsubc_atom: lsubc RP G (⋆) (⋆)
| lsubc_bind: ∀I,L1,L2. lsubc RP G L1 L2 → lsubc RP G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G,L1,V⦄ ϵ[RP] 〚A〛 → ⦃G,L1,W⦄ ϵ[RP] 〚A〛 → ⦃G,L2⦄ ⊢ W ⁝ A →
lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
.
fact lsubc_inv_bind1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1. L1 = K1.ⓘ{I} →
(∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L2 = K2. ⓛW & I = BPair Abbr (ⓝW.V).
#RP #G #L1 #L2 * -L1 -L2
(* Basic_1: was: csubc_gen_head_r *)
lemma lsubc_inv_bind1: ∀RP,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃[RP] L2 →
(∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V).
/2 width=3 by lsubc_inv_bind1_aux/ qed-.
fact lsubc_inv_bind2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2. L2 = K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = BPair Abst W.
#RP #G #L1 #L2 * -L1 -L2
(* Basic_1: was just: csubc_gen_head_l *)
lemma lsubc_inv_bind2: ∀RP,I,G,L1,K2. G ⊢ L1 ⫃[RP] K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = BPair Abst W.
/2 width=3 by lsubc_inv_bind2_aux/ qed-.