(* Basic_1: was just: sn3_intro *)
lemma csx_intro_cpxs (G) (L):
- â\88\80T1. (â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88* T2 â\86\92 (T1 â\89\85 T2 â\86\92 â\8a¥) â\86\92 â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T2) →
- â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T1.
+ â\88\80T1. (â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\88* T2 â\86\92 (T1 â\89\85 T2 â\86\92 â\8a¥) â\86\92 â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T2) →
+ â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T1.
/4 width=1 by cpx_cpxs, csx_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans *)
lemma csx_cpxs_trans (G) (L):
- â\88\80T1. â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T1 →
- â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88* T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T2.
+ â\88\80T1. â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T1 →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\88* T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T2.
#G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
/2 width=3 by csx_cpx_trans/
qed-.
fact csx_ind_cpxs_aux (G) (L):
∀Q:predicate term.
- (â\88\80T1. â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T1 →
- (â\88\80T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈* T2 → (T1 ≅ T2 → ⊥) → Q T2) → Q T1
+ (â\88\80T1. â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T1 →
+ (â\88\80T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬈* T2 → (T1 ≅ T2 → ⊥) → Q T2) → Q T1
) →
- â\88\80T1. â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T1 →
- â\88\80T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈* T2 → Q T2.
+ â\88\80T1. â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T1 →
+ â\88\80T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬈* T2 → Q T2.
#G #L #Q #IH #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IH1 #T2 #HT12
@IH -IH /2 width=3 by csx_cpxs_trans/ -HT1 #V2 #HTV2 #HnTV2
(* Basic_2A1: was: csx_ind_alt *)
lemma csx_ind_cpxs (G) (L) (Q:predicate …):
- (â\88\80T1. â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T1 →
- (â\88\80T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈* T2 → (T1 ≅ T2 → ⊥) → Q T2) → Q T1
+ (â\88\80T1. â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T1 →
+ (â\88\80T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬈* T2 → (T1 ≅ T2 → ⊥) → Q T2) → Q T1
) →
- â\88\80T. â\9dªG,Lâ\9d« ⊢ ⬈*𝐒 T → Q T.
+ â\88\80T. â\9d¨G,Lâ\9d© ⊢ ⬈*𝐒 T → Q T.
#G #L #Q #IH #T #HT
@(csx_ind_cpxs_aux … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.