inductive tshf: relation term ≝
| tshf_atom: ∀I. tshf (⓪{I}) (⓪{I})
- | tshf_abst: ∀V1,V2,T1,T2. tshf (ⓛV1. T1) (ⓛV2. T2)
- | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒[T1] → 𝐒[T2] →
+ | tshf_abbr: ∀V1,V2,T1,T2. tshf (-ⓓV1. T1) (-ⓓV2. T2)
+ | tshf_abst: ∀a,V1,V2,T1,T2. tshf (ⓛ{a}V1. T1) (ⓛ{a}V2. T2)
+ | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄ →
tshf (ⓐV1. T1) (ⓐV2. T2)
.
lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1.
/3 width=2/ qed.
-lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒[T1] → 𝐒[T2].
+lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
#T1 #T2 #H elim H -T1 -T2 //
-#V1 #V2 #T1 #T2 #H
-elim (simple_inv_bind … H)
+[ #V1 #V2 #T1 #T2 #H
+ elim (simple_inv_bind … H)
+| #a #V1 #V2 #T1 #T2 #H
+ elim (simple_inv_bind … H)
+]
qed. (**) (* remove from index *)
-lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒[T2] → 𝐒[T1].
+lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
/3 width=3/ qed-.
(* Basic inversion lemmas ***************************************************)
-fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓑ{I}W1.U1 →
- ∃∃W2,U2. I = Abst & T2 = ⓛW2. U2.
+fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀a,I,W1,U1. T1 = ⓑ{a,I}W1.U1 →
+ ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
+ (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
#T1 #T2 * -T1 -T2
-[ #J #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/
-| #V1 #V2 #T1 #T2 #H_ #_ #_ #I #W1 #U1 #H destruct
+[ #J #a #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
+| #b #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
+| #V1 #V2 #T1 #T2 #_ #_ #_ #a #I #W1 #U1 #H destruct
]
qed.
-lemma tshf_inv_bind1: ∀I,W1,U1,T2. ⓑ{I}W1.U1 ≈ T2 →
- ∃∃W2,U2. I = Abst & T2 = ⓛW2. U2.
+lemma tshf_inv_bind1: ∀a,I,W1,U1,T2. ⓑ{a,I}W1.U1 ≈ T2 →
+ ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
+ (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
/2 width=5/ qed-.
fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 →
- ∃∃W2,U2. U1 ≈ U2 & 𝐒[U1] & 𝐒[U2] &
+ ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
I = Appl & T2 = ⓐW2. U2.
#T1 #T2 * -T1 -T2
[ #J #I #W1 #U1 #H destruct
| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
+| #a #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/
]
qed.
lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 →
- ∃∃W2,U2. U1 ≈ U2 & 𝐒[U1] & 𝐒[U2] &
+ ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
I = Appl & T2 = ⓐW2. U2.
/2 width=4/ qed-.