inductive lsubc (RP) (G): relation lenv ≝
| lsubc_atom: lsubc RP G (⋆) (⋆)
-| lsubc_pair: ∀I,L1,L2,V. lsubc RP G L1 L2 → lsubc RP G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| 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 RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
.
fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
#RP #G #L1 #L2 * -L1 -L2
[ //
-| #I #L1 #L2 #V #_ #H destruct
+| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
]
qed-.
lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
/2 width=5 by lsubc_inv_atom1_aux/ qed-.
-fact lsubc_inv_pair1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (â\88\83â\88\83K2. G â\8a¢ K1 â«\83[RP] K2 & L2 = K2.â\93\91{I}X) ∨
+fact lsubc_inv_bind1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ (â\88\83â\88\83K2. G â\8a¢ K1 â«\83[RP] K2 & L2 = K2.â\93\98{I}) ∨
∃∃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 & X = ⓝW.V & I = Abbr.
+ L2 = K2. ⓛW & I = BPair Abbr (ⓝW.V).
#RP #G #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10 by ex7_4_intro, or_intror/
+[ #I #K1 #H destruct
+| #J #L1 #L2 #HL12 #I #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #H destruct
+ /3 width=10 by ex6_4_intro, or_intror/
]
qed-.
(* Basic_1: was: csubc_gen_head_r *)
-lemma lsubc_inv_pair1: ∀RP,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃[RP] L2 →
- (â\88\83â\88\83K2. G â\8a¢ K1 â«\83[RP] K2 & L2 = K2.â\93\91{I}X) ∨
+lemma lsubc_inv_bind1: ∀RP,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃[RP] L2 →
+ (â\88\83â\88\83K2. G â\8a¢ K1 â«\83[RP] K2 & L2 = K2.â\93\98{I}) ∨
∃∃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 & X = ⓝW.V & I = Abbr.
-/2 width=3 by lsubc_inv_pair1_aux/ qed-.
+ L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V).
+/2 width=3 by lsubc_inv_bind1_aux/ qed-.
fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
#RP #G #L1 #L2 * -L1 -L2
[ //
-| #I #L1 #L2 #V #_ #H destruct
+| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
]
qed-.
lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
/2 width=5 by lsubc_inv_atom2_aux/ qed-.
-fact lsubc_inv_pair2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
- (â\88\83â\88\83K1. G â\8a¢ K1 â«\83[RP] K2 & L1 = K1. â\93\91{I} W) ∨
- ∃∃K1,V,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 = Abst.
+fact lsubc_inv_bind2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ (â\88\83â\88\83K1. G â\8a¢ K1 â«\83[RP] K2 & L1 = K1. â\93\98{I}) ∨
+ ∃∃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
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8 by ex6_3_intro, or_intror/
+[ #I #K2 #H destruct
+| #J #L1 #L2 #HL12 #I #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #H destruct
+ /3 width=10 by ex6_4_intro, or_intror/
]
qed-.
(* Basic_1: was just: csubc_gen_head_l *)
-lemma lsubc_inv_pair2: ∀RP,I,G,L1,K2,W. G ⊢ L1 ⫃[RP] K2.ⓑ{I} W →
- (â\88\83â\88\83K1. G â\8a¢ K1 â«\83[RP] K2 & L1 = K1.â\93\91{I} W) ∨
- ∃∃K1,V,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 = Abst.
-/2 width=3 by lsubc_inv_pair2_aux/ qed-.
+lemma lsubc_inv_bind2: ∀RP,I,G,L1,K2. G ⊢ L1 ⫃[RP] K2.ⓘ{I} →
+ (â\88\83â\88\83K1. G â\8a¢ K1 â«\83[RP] K2 & L1 = K1.â\93\98{I}) ∨
+ ∃∃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-.
(* Basic properties *********************************************************)
(* Basic_1: was just: csubc_refl *)
lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
-#RP #G #L elim L -L /2 width=1 by lsubc_pair/
+#RP #G #L elim L -L /2 width=1 by lsubc_bind/
qed.
(* Basic_1: removed theorems 3: