include "basic_2/rt_transition/cpr.ma".
include "basic_2/rt_computation/cpms.ma".
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION FOR TERMS *************************)
+(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR TERMS ***********************)
(* Basic eliminators ********************************************************)
(* Basic_2A1: was: cprs_ind_dx *)
-lemma cprs_ind_sn (h) (G) (L) (T2) (R:predicate …):
- R T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ⦃G, L⦄ ⊢ T ➡*[h] T2 → R T → R T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → R T1.
-#h #G #L #T2 #R #IH1 #IH2 #T1
+lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …):
+ Q T2 →
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ⦃G,L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ 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 //
#n1 #n2 #T1 #T #HT1 #HT2 #IH #H
qed-.
(* Basic_2A1: was: cprs_ind *)
-lemma cprs_ind_dx (h) (G) (L) (T1) (R:predicate …):
- R T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ T ➡[h] T2 → R T → R T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → R T2.
-#h #G #L #T1 #R #IH1 #IH2 #T2
+lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …):
+ Q T1 →
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T → ⦃G,L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ 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 //
#n1 #n2 #T #T2 #HT1 #IH #HT2 #H
(* Basic_1: was: pr3_step *)
(* Basic_2A1: was: cprs_strap2 *)
lemma cprs_step_sn (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_sn/ qed-.
(* Basic_2A1: was: cprs_strap1 *)
lemma cprs_step_dx (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ 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):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{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):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{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): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
+lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆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): ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 →
- ∨∨ ⦃G, L⦄ ⊢ T1 ➡*[h] X2
- | ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2.
-#h #G #L #W1 #T1 #X2 #H @(cprs_ind_dx … H) -X2
-[ /3 width=5 by ex3_2_intro, or_intror/
-| #X #X2 #_ #HX2 * /3 width=3 by cprs_step_dx, or_introl/ *
- #W #T #HW1 #HT1 #H destruct
- elim (cpr_inv_cast1 … HX2) -HX2 /3 width=3 by cprs_step_dx, or_introl/ *
- #W2 #T2 #HW2 #HT2 #H destruct
- /4 width=5 by cprs_step_dx, ex3_2_intro, or_intror/
+lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ➡*[h] X2.
+#h #G #L #W1 #T1 #X2 #H
+elim (cpms_inv_cast1 … H) -H
+[ /2 width=1 by or_introl/
+| /2 width=1 by or_intror/
+| * #m #_ #H destruct
]
qed-.