definition IH_cnv_cpm_trans_lpr (h) (a): relation3 genv lenv term ≝
λG,L1,T1. ❪G,L1❫ ⊢ T1 ![h,a] →
- ∀n,T2. ❪G,L1❫ ⊢ T1 ➡[n,h] T2 →
- ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T2 ![h,a].
+ ∀n,T2. ❪G,L1❫ ⊢ T1 ➡[h,n] T2 →
+ ∀L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → ❪G,L2❫ ⊢ T2 ![h,a].
definition IH_cnv_cpms_trans_lpr (h) (a): relation3 genv lenv term ≝
λG,L1,T1. ❪G,L1❫ ⊢ T1 ![h,a] →
- ∀n,T2. ❪G,L1❫ ⊢ T1 ➡*[n,h] T2 →
- ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T2 ![h,a].
+ ∀n,T2. ❪G,L1❫ ⊢ T1 ➡*[h,n] T2 →
+ ∀L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → ❪G,L2❫ ⊢ T2 ![h,a].
definition IH_cnv_cpm_conf_lpr (h) (a): relation3 genv lenv term ≝
λG,L0,T0. ❪G,L0❫ ⊢ T0 ![h,a] →
- ∀n1,T1. ❪G,L0❫ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ❪G,L0❫ ⊢ T0 ➡[n2,h] T2 →
- ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G,L0❫ ⊢ ➡[h] L2 →
- ∃∃T. ❪G,L1❫ ⊢ T1 ➡*[n2-n1,h] T & ❪G,L2❫ ⊢ T2 ➡*[n1-n2,h] T.
+ ∀n1,T1. ❪G,L0❫ ⊢ T0 ➡[h,n1] T1 → ∀n2,T2. ❪G,L0❫ ⊢ T0 ➡[h,n2] T2 →
+ ∀L1. ❪G,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G,L0❫ ⊢ ➡[h,0] L2 →
+ ∃∃T. ❪G,L1❫ ⊢ T1 ➡*[h,n2-n1] T & ❪G,L2❫ ⊢ T2 ➡*[h,n1-n2] T.
definition IH_cnv_cpms_strip_lpr (h) (a): relation3 genv lenv term ≝
λG,L0,T0. ❪G,L0❫ ⊢ T0 ![h,a] →
- ∀n1,T1. ❪G,L0❫ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ❪G,L0❫ ⊢ T0 ➡[n2,h] T2 →
- ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G,L0❫ ⊢ ➡[h] L2 →
- ∃∃T. ❪G,L1❫ ⊢ T1 ➡*[n2-n1,h] T & ❪G,L2❫ ⊢ T2 ➡*[n1-n2,h] T.
+ ∀n1,T1. ❪G,L0❫ ⊢ T0 ➡*[h,n1] T1 → ∀n2,T2. ❪G,L0❫ ⊢ T0 ➡[h,n2] T2 →
+ ∀L1. ❪G,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G,L0❫ ⊢ ➡[h,0] L2 →
+ ∃∃T. ❪G,L1❫ ⊢ T1 ➡*[h,n2-n1] T & ❪G,L2❫ ⊢ T2 ➡*[h,n1-n2] T.
definition IH_cnv_cpms_conf_lpr (h) (a): relation3 genv lenv term ≝
λG,L0,T0. ❪G,L0❫ ⊢ T0 ![h,a] →
- ∀n1,T1. ❪G,L0❫ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ❪G,L0❫ ⊢ T0 ➡*[n2,h] T2 →
- ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G,L0❫ ⊢ ➡[h] L2 →
- ∃∃T. ❪G,L1❫ ⊢ T1 ➡*[n2-n1,h] T & ❪G,L2❫ ⊢ T2 ➡*[n1-n2,h] T.
+ ∀n1,T1. ❪G,L0❫ ⊢ T0 ➡*[h,n1] T1 → ∀n2,T2. ❪G,L0❫ ⊢ T0 ➡*[h,n2] T2 →
+ ∀L1. ❪G,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G,L0❫ ⊢ ➡[h,0] L2 →
+ ∃∃T. ❪G,L1❫ ⊢ T1 ➡*[h,n2-n1] T & ❪G,L2❫ ⊢ T2 ➡*[h,n1-n2] T.
(* Auxiliary properties for preservation ************************************)
fact cnv_cpms_trans_lpr_sub (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ❪G0,L0,T0❫ >[h] ❪G1,L1,T1❫ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
- ∀G1,L1,T1. ❪G0,L0,T0❫ >[h] ❪G1,L1,T1❫ → IH_cnv_cpms_trans_lpr h a G1 L1 T1.
+ (∀G1,L1,T1. ❪G0,L0,T0❫ > ❪G1,L1,T1❫ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ❪G0,L0,T0❫ > ❪G1,L1,T1❫ → IH_cnv_cpms_trans_lpr h a G1 L1 T1.
#h #a #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2
/3 width=7 by fpbg_cpms_trans/
fact cnv_cpm_conf_lpr_sub (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ❪G0,L0,T0❫ >[h] ❪G1,L1,T1❫ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
- ∀G1,L1,T1. ❪G0,L0,T0❫ >[h] ❪G1,L1,T1❫ → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
+ (∀G1,L1,T1. ❪G0,L0,T0❫ > ❪G1,L1,T1❫ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ❪G0,L0,T0❫ > ❪G1,L1,T1❫ → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
fact cnv_cpms_strip_lpr_sub (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ❪G0,L0,T0❫ >[h] ❪G1,L1,T1❫ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
- ∀G1,L1,T1. ❪G0,L0,T0❫ >[h] ❪G1,L1,T1❫ → IH_cnv_cpms_strip_lpr h a G1 L1 T1.
+ (∀G1,L1,T1. ❪G0,L0,T0❫ > ❪G1,L1,T1❫ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ❪G0,L0,T0❫ > ❪G1,L1,T1❫ → IH_cnv_cpms_strip_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.