(* *)
(**************************************************************************)
+include "basic_2/rt_transition/lpr.ma".
include "basic_2/rt_computation/cpms_fpbg.ma".
include "basic_2/dynamic/cnv.ma".
(* Inductive premises for the preservation results **************************)
-definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡[n,h] T2 →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h].
+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 ➡[h,n] T2 →
+ ∀L2. ❨G,L1❩ ⊢ ➡[h,0] L2 → ❨G,L2❩ ⊢ T2 ![h,a].
-definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h].
+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 ➡*[h,n] T2 →
+ ∀L2. ❨G,L1❩ ⊢ ➡[h,0] L2 → ❨G,L2❩ ⊢ T2 ![h,a].
-definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
- ∀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.
+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 ➡[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 (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
- ∀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.
+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 ➡*[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 (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
- ∀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.
+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 ➡*[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 (a) (h):
+fact cnv_cpms_trans_lpr_sub (h) (a):
∀G0,L0,T0.
- (â\88\80G1,L1,T1. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G1,L1,T1â¦\84 â\86\92 IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- â\88\80G1,L1,T1. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G1,L1,T1â¦\84 â\86\92 IH_cnv_cpms_trans_lpr a h G1 L1 T1.
-#a #h #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
+ (â\88\80G1,L1,T1. â\9d¨G0,L0,T0â\9d© > â\9d¨G1,L1,T1â\9d© â\86\92 IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ â\88\80G1,L1,T1. â\9d¨G0,L0,T0â\9d© > â\9d¨G1,L1,T1â\9d© â\86\92 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/
qed-.
-fact cnv_cpm_conf_lpr_sub (a) (h):
+fact cnv_cpm_conf_lpr_sub (h) (a):
∀G0,L0,T0.
- (â\88\80G1,L1,T1. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G1,L1,T1â¦\84 â\86\92 IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- â\88\80G1,L1,T1. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G1,L1,T1â¦\84 â\86\92 IH_cnv_cpm_conf_lpr a h G1 L1 T1.
+ (â\88\80G1,L1,T1. â\9d¨G0,L0,T0â\9d© > â\9d¨G1,L1,T1â\9d© â\86\92 IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ â\88\80G1,L1,T1. â\9d¨G0,L0,T0â\9d© > â\9d¨G1,L1,T1â\9d© â\86\92 IH_cnv_cpm_conf_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
-fact cnv_cpms_strip_lpr_sub (a) (h):
+fact cnv_cpms_strip_lpr_sub (h) (a):
∀G0,L0,T0.
- (â\88\80G1,L1,T1. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G1,L1,T1â¦\84 â\86\92 IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- â\88\80G1,L1,T1. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G1,L1,T1â¦\84 â\86\92 IH_cnv_cpms_strip_lpr a h G1 L1 T1.
+ (â\88\80G1,L1,T1. â\9d¨G0,L0,T0â\9d© > â\9d¨G1,L1,T1â\9d© â\86\92 IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ â\88\80G1,L1,T1. â\9d¨G0,L0,T0â\9d© > â\9d¨G1,L1,T1â\9d© â\86\92 IH_cnv_cpms_strip_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.