(* Properties with unbound context-sensitive rt-computation for terms *******)
(* Basic_1: was just: sn3_intro *)
-lemma csx_intro_cpxs: ∀h,G,L,T1.
- (∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T2❫) →
- ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫.
+lemma csx_intro_cpxs (h) (G) (L):
+ ∀T1. (∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ❪G,L❫ ⊢ ⬈*𝐒[h] T2) →
+ ❪G,L❫ ⊢ ⬈*𝐒[h] T1.
/4 width=1 by cpx_cpxs, csx_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_cpxs_trans: ∀h,G,L,T1. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- ∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T2❫.
+lemma csx_cpxs_trans (h) (G) (L):
+ ∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
+ ∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → ❪G,L❫ ⊢ ⬈*𝐒[h] T2.
#h #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
/2 width=3 by csx_cpx_trans/
qed-.
(* Eliminators with unbound context-sensitive rt-computation for terms ******)
-lemma csx_ind_cpxs_teqx: ∀h,G,L. ∀Q:predicate term.
- (∀T1. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- (∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
- ) →
- ∀T1. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- ∀T0. ❪G,L❫ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2.
+lemma csx_ind_cpxs_teqx (h) (G) (L):
+ ∀Q:predicate term.
+ (∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
+ (∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ ) →
+ ∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
+ ∀T0. ❪G,L❫ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2.
#h #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
@IH -IH /3 width=3 by csx_cpxs_trans, csx_teqx_trans/ -HT1 #V2 #HTV2 #HnTV2
qed-.
(* Basic_2A1: was: csx_ind_alt *)
-lemma csx_ind_cpxs: ∀h,G,L. ∀Q:predicate term.
- (∀T1. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- (∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
- ) →
- ∀T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫ → Q T.
+lemma csx_ind_cpxs (h) (G) (L) (Q:predicate …):
+ (∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
+ (∀T2. ❪G,L❫ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
+ ) →
+ ∀T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q T.
#h #G #L #Q #IH #T #HT
@(csx_ind_cpxs_teqx … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.