(* Basic_2A1: was: cprs_ind_dx *)
lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …):
Q T2 →
- (â\88\80T1,T. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡[h] T â\86\92 â¦\83G,Lâ¦\84 ⊢ T ➡*[h] T2 → Q T → Q T1) →
- â\88\80T1. â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T2 → Q T1.
+ (â\88\80T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h] T â\86\92 â\9dªG,Lâ\9d« ⊢ T ➡*[h] T2 → Q T → Q T1) →
+ â\88\80T1. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2 → Q T1.
#h #G #L #T2 #Q #IH1 #IH2 #T1
@(insert_eq_0 … 0) #n #H
@(cpms_ind_sn … H) -n -T1 //
(* Basic_2A1: was: cprs_ind *)
lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …):
Q T1 →
- (â\88\80T,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[h] T â\86\92 â¦\83G,Lâ¦\84 ⊢ T ➡[h] T2 → Q T → Q T2) →
- â\88\80T2. â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T2 → Q T2.
+ (â\88\80T,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h] T â\86\92 â\9dªG,Lâ\9d« ⊢ T ➡[h] T2 → Q T → Q T2) →
+ â\88\80T2. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2 → Q T2.
#h #G #L #T1 #Q #IH1 #IH2 #T2
@(insert_eq_0 … 0) #n #H
@(cpms_ind_dx … H) -n -T2 //
(* Basic_1: was: pr3_step *)
(* Basic_2A1: was: cprs_strap2 *)
lemma cprs_step_sn (h) (G) (L):
- â\88\80T1,T. â¦\83G,Lâ¦\84 ⊢ T1 ➡[h] T →
- â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T2.
+ â\88\80T1,T. â\9dªG,Lâ\9d« ⊢ T1 ➡[h] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_sn/ qed-.
(* Basic_2A1: was: cprs_strap1 *)
lemma cprs_step_dx (h) (G) (L):
- â\88\80T1,T. â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T →
- â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡[h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T2.
+ â\88\80T1,T. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_dx/ qed-.
(* Basic_1: was only: pr3_thin_dx *)
lemma cprs_flat_dx (h) (I) (G) (L):
- â\88\80V1,V2. â¦\83G,Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡*[h] â\93\95{I}V2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\95[I]V1.T1 â\9e¡*[h] â\93\95[I]V2.T2.
#h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
lemma cprs_flat_sn (h) (I) (G) (L):
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡[h] T2 â\86\92 â\88\80V1,V2. â¦\83G,Lâ¦\84 ⊢ V1 ➡*[h] V2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\95{I} V1. T1 â\9e¡*[h] â\93\95{I} V2. T2.
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h] T2 â\86\92 â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡*[h] V2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\95[I] V1. T1 â\9e¡*[h] â\93\95[I] V2. T2.
#h #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_sn … H) -V1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1 (h) (G) (L): â\88\80X2,s. â¦\83G,Lâ¦\84 ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
+lemma cprs_inv_sort1 (h) (G) (L): â\88\80X2,s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
/2 width=4 by cpms_inv_sort1/ qed-.
(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1 (h) (G) (L): â\88\80W1,T1,X2. â¦\83G,Lâ¦\84 ⊢ ⓝW1.T1 ➡*[h] X2 →
- â\88¨â\88¨ â\88\83â\88\83W2,T2. â¦\83G,Lâ¦\84 â\8a¢ W1 â\9e¡*[h] W2 & â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
- | â¦\83G,Lâ¦\84 ⊢ T1 ➡*[h] X2.
+lemma cprs_inv_cast1 (h) (G) (L): â\88\80W1,T1,X2. â\9dªG,Lâ\9d« ⊢ ⓝW1.T1 ➡*[h] X2 →
+ â\88¨â\88¨ â\88\83â\88\83W2,T2. â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡*[h] W2 & â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
+ | â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] X2.
#h #G #L #W1 #T1 #X2 #H
elim (cpms_inv_cast1 … H) -H
[ /2 width=1 by or_introl/