definition IH_cnv_cpm_teqx_cpm_trans (h) (a): relation3 genv lenv term ≝
λG,L,T1. ❪G,L❫ ⊢ T1 ![h,a] →
- ∀n1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T → T1 ≛ T →
- ∀n2,T2. ❪G,L❫ ⊢ T ➡[n2,h] T2 →
- ∃∃T0. ❪G,L❫ ⊢ T1 ➡[n2,h] T0 & ❪G,L❫ ⊢ T0 ➡[n1,h] T2 & T0 ≛ T2.
+ ∀n1,T. ❪G,L❫ ⊢ T1 ➡[h,n1] T → T1 ≛ T →
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡[h,n2] T2 →
+ ∃∃T0. ❪G,L❫ ⊢ T1 ➡[h,n2] T0 & ❪G,L❫ ⊢ T0 ➡[h,n1] T2 & T0 ≛ T2.
(* Transitive properties restricted rt-transition for terms *****************)
fact cnv_cpm_teqx_cpm_trans_sub (h) (a) (G0) (L0) (T0):
- (∀G,L,T. ❪G0,L0,T0❫ >[h] ❪G,L,T❫ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ❪G0,L0,T0❫ > ❪G,L,T❫ → IH_cnv_cpm_trans_lpr h a G L T) →
(∀G,L,T. ❪G0,L0,T0❫ ⬂+ ❪G,L,T❫ → IH_cnv_cpm_teqx_cpm_trans h a G L T) →
∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_teqx_cpm_trans h a G L T1.
#h #a #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
qed-.
fact cnv_cpm_teqx_cpm_trans_aux (h) (a) (G0) (L0) (T0):
- (∀G,L,T. ❪G0,L0,T0❫ >[h] ❪G,L,T❫ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ❪G0,L0,T0❫ > ❪G,L,T❫ → IH_cnv_cpm_trans_lpr h a G L T) →
IH_cnv_cpm_teqx_cpm_trans h a G0 L0 T0.
#h #a #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #IH0