(* Basic properties *********************************************************)
(* Basic_2A1: uses: lprs_pair_refl *)
-lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
+lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lprs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
+lemma lprs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
-lemma lprs_refl (h) (G): ∀L. ⦃G, L⦄ ⊢ ➡*[h] L.
+lemma lprs_refl (h) (G): ∀L. ⦃G,L⦄ ⊢ ➡*[h] L.
/2 width=1 by lex_refl/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: lprs_inv_atom1 *)
-lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
+lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
(* Basic_2A1: was: lprs_inv_pair1 *)
lemma lprs_inv_pair_sn (h) (G):
- ∀I,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2.
+ ∀I,K1,L2,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
(* Basic_2A1: uses: lprs_inv_atom2 *)
-lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
+lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
(* Basic_2A1: was: lprs_inv_pair2 *)
lemma lprs_inv_pair_dx (h) (G):
- ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1.
+ ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
(* Basic eliminators ********************************************************)
lemma lprs_ind (h) (G): ∀Q:relation lenv.
Q (⋆) (⋆) → (
∀I,K1,K2.
- ⦃G, K1⦄ ⊢ ➡*[h] K2 →
+ ⦃G,K1⦄ ⊢ ➡*[h] K2 →
Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
) → (
∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ➡*[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,K1⦄ ⊢ ➡*[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 →
Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1 L2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1 L2.
/3 width=4 by lex_ind/ qed-.