include "ground_2/lib/ltc.ma".
include "basic_2/notation/relations/predstar_6.ma".
-include "basic_2/notation/relations/predstar_5.ma".
include "basic_2/rt_transition/cpm.ma".
(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
(* Basic_2A1: uses: scpds *)
definition cpms (h) (G) (L): relation3 nat term term ≝
- ltc … plus … (cpm h G L).
+ ltc … plus … (cpm h G L).
interpretation
"t-bound context-sensitive parallel rt-computarion (term)"
- 'PRedStar n h G L T1 T2 = (cpms h G L n T1 T2).
-
-interpretation
- "context-sensitive parallel r-computation (term)"
- 'PRedStar h G L T1 T2 = (cpms h G L O T1 T2).
+ 'PRedStar h n G L T1 T2 = (cpms h G L n T1 T2).
(* Basic eliminators ********************************************************)
lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
- Q 0 T2 →
- (∀n1,n2,T1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T → ❪G,L❫ ⊢ T ➡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 → Q n T1.
+ Q 0 T2 →
+ (∀n1,n2,T1,T. ❪G,L❫ ⊢ T1 ➡[h,n1] T → ❪G,L❫ ⊢ T ➡*[h,n2] T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 → Q n T1.
#h #G #L #T2 #Q @ltc_ind_sn_refl //
qed-.
lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
- Q 0 T1 →
- (∀n1,n2,T,T2. ❪G,L❫ ⊢ T1 ➡*[n1,h] T → Q n1 T → ❪G,L❫ ⊢ T ➡[n2,h] T2 → Q (n1+n2) T2) →
- ∀n,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 → Q n T2.
+ Q 0 T1 →
+ (∀n1,n2,T,T2. ❪G,L❫ ⊢ T1 ➡*[h,n1] T → Q n1 T → ❪G,L❫ ⊢ T ➡[h,n2] T2 → Q (n1+n2) T2) →
+ ∀n,T2. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 → Q n T2.
#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
(* Basic_1: includes: pr1_pr0 *)
(* Basic_1: uses: pr3_pr2 *)
(* Basic_2A1: includes: cpr_cprs *)
-lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n,h] T2.
+lemma cpm_cpms (h) (G) (L):
+ ∀n,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → ❪G,L❫ ⊢ T1 ➡*[h,n] T2.
/2 width=1 by ltc_rc/ qed.
-lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T →
- ∀n2,T2. ❪G,L❫ ⊢ T ➡*[n2,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2.
+lemma cpms_step_sn (h) (G) (L):
+ ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[h,n1] T →
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡*[h,n2] T2 → ❪G,L❫ ⊢ T1 ➡*[h,n1+n2] T2.
/2 width=3 by ltc_sn/ qed-.
-lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T →
- ∀n2,T2. ❪G,L❫ ⊢ T ➡[n2,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2.
+lemma cpms_step_dx (h) (G) (L):
+ ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[h,n1] T →
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡[h,n2] T2 → ❪G,L❫ ⊢ T1 ➡*[h,n1+n2] T2.
/2 width=3 by ltc_dx/ qed-.
(* Basic_2A1: uses: cprs_bind_dx *)
-lemma cpms_bind_dx (n) (h) (G) (L):
- ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡*[n,h] T2 →
- ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[n,h] ⓑ[p,I]V2.T2.
-#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
+lemma cpms_bind_dx (h) (n) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 →
+ ∀I,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡*[h,n] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[h,n] ⓑ[p,I]V2.T2.
+#h #n #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
-lemma cpms_appl_dx (n) (h) (G) (L):
- ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
- ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
- ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
-#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
+lemma cpms_appl_dx (h) (n) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 →
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 →
+ ❪G,L❫ ⊢ ⓐV1.T1 ➡*[h,n] ⓐV2.T2.
+#h #n #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
qed.
-lemma cpms_zeta (n) (h) (G) (L):
- ∀T1,T. ⇧[1] T ≘ T1 →
- ∀V,T2. ❪G,L❫ ⊢ T ➡*[n,h] T2 → ❪G,L❫ ⊢ +ⓓV.T1 ➡*[n,h] T2.
-#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
+lemma cpms_zeta (h) (n) (G) (L):
+ ∀T1,T. ⇧[1] T ≘ T1 →
+ ∀V,T2. ❪G,L❫ ⊢ T ➡*[h,n] T2 → ❪G,L❫ ⊢ +ⓓV.T1 ➡*[h,n] T2.
+#h #n #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_zeta *)
-lemma cpms_zeta_dx (n) (h) (G) (L):
- ∀T2,T. ⇧[1] T2 ≘ T →
- ∀V,T1. ❪G,L.ⓓV❫ ⊢ T1 ➡*[n,h] T → ❪G,L❫ ⊢ +ⓓV.T1 ➡*[n,h] T2.
-#n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
+lemma cpms_zeta_dx (h) (n) (G) (L):
+ ∀T2,T. ⇧[1] T2 ≘ T →
+ ∀V,T1. ❪G,L.ⓓV❫ ⊢ T1 ➡*[h,n] T → ❪G,L❫ ⊢ +ⓓV.T1 ➡*[h,n] T2.
+#h #n #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_eps *)
-lemma cpms_eps (n) (h) (G) (L):
- ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
- ∀V. ❪G,L❫ ⊢ ⓝV.T1 ➡*[n,h] T2.
-#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
+lemma cpms_eps (h) (n) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 →
+ ∀V. ❪G,L❫ ⊢ ⓝV.T1 ➡*[h,n] T2.
+#h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
qed.
-lemma cpms_ee (n) (h) (G) (L):
- ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[n,h] U2 →
- ∀T. ❪G,L❫ ⊢ ⓝU1.T ➡*[↑n,h] U2.
-#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
+lemma cpms_ee (h) (n) (G) (L):
+ ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[h,n] U2 →
+ ∀T. ❪G,L❫ ⊢ ⓝU1.T ➡*[h,↑n] U2.
+#h #n #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=1 by cpm_cpms, cpm_ee/
| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
/3 width=3 by cpms_step_sn, cpm_ee/
qed.
(* Basic_2A1: uses: cprs_beta_dx *)
-lemma cpms_beta_dx (n) (h) (G) (L):
- ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
- ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
- ∀T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
- ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
-#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+lemma cpms_beta_dx (h) (n) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 →
+ ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h,0] W2 →
+ ∀T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[h,n] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[h,n] ⓓ[p]ⓝW2.V2.T2.
+#h #n #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
qed.
(* Basic_2A1: uses: cprs_theta_dx *)
-lemma cpms_theta_dx (n) (h) (G) (L):
- ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h] V →
- ∀V2. ⇧[1] V ≘ V2 →
- ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
- ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
- ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
-#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+lemma cpms_theta_dx (h) (n) (G) (L):
+ ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h,0] V →
+ ∀V2. ⇧[1] V ≘ V2 →
+ ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h,0] W2 →
+ ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[h,n] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[h,n] ⓓ[p]W2.ⓐV2.T2.
+#h #n #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
qed.
(* Basic properties with r-transition ***************************************)
(* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
+lemma cprs_refl (h) (G) (L):
+ reflexive … (cpms h G L 0).
/2 width=1 by cpm_cpms/ qed.
(* Advanced properties ******************************************************)
-lemma cpms_sort (h) (G) (L) (n):
- ∀s. ❪G,L❫ ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
+lemma cpms_sort (h) (G) (L):
+ ∀n,s. ❪G,L❫ ⊢ ⋆s ➡*[h,n] ⋆((next h)^n s).
#h #G #L #n elim n -n [ // ]
#n #IH #s <plus_SO_dx
/3 width=3 by cpms_step_dx, cpm_sort/
(* Basic inversion lemmas ***************************************************)
-lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ❪G,L❫ ⊢ ⋆s ➡*[n,h] X2 → X2 = ⋆(((next h)^n) s).
-#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
+lemma cpms_inv_sort1 (h) (n) (G) (L):
+ ∀X2,s. ❪G,L❫ ⊢ ⋆s ➡*[h,n] X2 → X2 = ⋆(((next h)^n) s).
+#h #n #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
#n1 #n2 #X #X2 #_ #IH #HX2 destruct
elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
qed-.
-lemma cpms_inv_lref1_ctop (n) (h) (G):
- ∀X2,i. ❪G,⋆❫ ⊢ #i ➡*[n,h] X2 → ∧∧ X2 = #i & n = 0.
-#n #h #G #X2 #i #H @(cpms_ind_dx … H) -X2
+lemma cpms_inv_lref1_ctop (h) (n) (G):
+ ∀X2,i. ❪G,⋆❫ ⊢ #i ➡*[h,n] X2 → ∧∧ X2 = #i & n = 0.
+#h #n #G #X2 #i #H @(cpms_ind_dx … H) -X2
[ /2 width=1 by conj/
| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
elim (cpm_inv_lref1_ctop … HX2) -HX2 #H1 #H2 destruct
]
qed-.
-lemma cpms_inv_zero1_unit (n) (h) (I) (K) (G):
- ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡*[n,h] X2 → ∧∧ X2 = #0 & n = 0.
-#n #h #I #G #K #X2 #H @(cpms_ind_dx … H) -X2
+lemma cpms_inv_zero1_unit (h) (n) (I) (K) (G):
+ ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡*[h,n] X2 → ∧∧ X2 = #0 & n = 0.
+#h #n #I #G #K #X2 #H @(cpms_ind_dx … H) -X2
[ /2 width=1 by conj/
| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
elim (cpm_inv_zero1_unit … HX2) -HX2 #H1 #H2 destruct
]
qed-.
-lemma cpms_inv_gref1 (n) (h) (G) (L):
- ∀X2,l. ❪G,L❫ ⊢ §l ➡*[n,h] X2 → ∧∧ X2 = §l & n = 0.
-#n #h #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
+lemma cpms_inv_gref1 (h) (n) (G) (L):
+ ∀X2,l. ❪G,L❫ ⊢ §l ➡*[h,n] X2 → ∧∧ X2 = §l & n = 0.
+#h #n #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
[ /2 width=1 by conj/
| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
elim (cpm_inv_gref1 … HX2) -HX2 #H1 #H2 destruct
qed-.
lemma cpms_inv_cast1 (h) (n) (G) (L):
- ∀W1,T1,X2. ❪G,L❫ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
- ∨∨ ∃∃W2,T2. ❪G,L❫ ⊢ W1 ➡*[n,h] W2 & ❪G,L❫ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
- | ❪G,L❫ ⊢ T1 ➡*[n,h] X2
- | ∃∃m. ❪G,L❫ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+ ∀W1,T1,X2. ❪G,L❫ ⊢ ⓝW1.T1 ➡*[h,n] X2 →
+ ∨∨ ∃∃W2,T2. ❪G,L❫ ⊢ W1 ➡*[h,n] W2 & ❪G,L❫ ⊢ T1 ➡*[h,n] T2 & X2 = ⓝW2.T2
+ | ❪G,L❫ ⊢ T1 ➡*[h,n] X2
+ | ∃∃m. ❪G,L❫ ⊢ W1 ➡*[h,m] X2 & n = ↑m.
#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
[ /3 width=5 by or3_intro0, ex3_2_intro/
| #n1 #n2 #X #X2 #_ * [ * || * ]
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