include "ground_2/steps/rtc_isrt_plus.ma".
include "ground_2/steps/rtc_isrt_max_shift.ma".
include "basic_2/notation/relations/pred_6.ma".
-include "basic_2/notation/relations/pred_5.ma".
include "basic_2/rt_transition/cpg.ma".
(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
(* Basic_2A1: includes: cpr *)
definition cpm (h) (G) (L) (n): relation2 term term ≝
- λT1,T2. ∃∃c. 𝐑𝐓❪n,c❫ & ❪G,L❫ ⊢ T1 ⬈[eq_t,c,h] T2.
+ λT1,T2. ∃∃c. 𝐑𝐓❪n,c❫ & ❪G,L❫ ⊢ T1 ⬈[eq_t,c,h] T2.
interpretation
"t-bound context-sensitive parallel rt-transition (term)"
- 'PRed n h G L T1 T2 = (cpm h G L n T1 T2).
-
-interpretation
- "context-sensitive parallel r-transition (term)"
- 'PRed h G L T1 T2 = (cpm h G L O T1 T2).
+ 'PRed h n G L T1 T2 = (cpm h G L n T1 T2).
(* Basic properties *********************************************************)
-lemma cpm_ess: ∀h,G,L,s. ❪G,L❫ ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
+lemma cpm_ess: ∀h,G,L,s. ❪G,L❫ ⊢ ⋆s ➡[h,1] ⋆(⫯[h]s).
/2 width=3 by cpg_ess, ex2_intro/ qed.
-lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 →
- ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ➡[n,h] W2.
-#n #h #G #K #V1 #V2 #W2 *
+lemma cpm_delta: ∀h,n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 →
+ ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ➡[h,n] W2.
+#h #n #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_delta, ex2_intro/
qed.
-lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 →
- ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡[↑n,h] W2.
-#n #h #G #K #V1 #V2 #W2 *
+lemma cpm_ell: ∀h,n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 →
+ ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡[h,↑n] W2.
+#h #n #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_ell, ex2_intro, isrt_succ/
qed.
-lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[n,h] T →
- ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡[n,h] U.
-#n #h #I #G #K #T #U #i *
+lemma cpm_lref: ∀h,n,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h,n] T →
+ ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡[h,n] U.
+#h #n #I #G #K #T #U #i *
/3 width=5 by cpg_lref, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_bind *)
-lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
- ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[n,h] T2 →
- ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[n,h] ⓑ[p,I]V2.T2.
-#n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
+lemma cpm_bind: ∀h,n,p,I,G,L,V1,V2,T1,T2.
+ ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 →
+ ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] ⓑ[p,I]V2.T2.
+#h #n #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
/5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/
qed.
-lemma cpm_appl: ∀n,h,G,L,V1,V2,T1,T2.
- ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ T1 ➡[n,h] T2 →
- ❪G,L❫ ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2.
-#n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
+lemma cpm_appl: ∀h,n,G,L,V1,V2,T1,T2.
+ ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ ❪G,L❫ ⊢ ⓐV1.T1 ➡[h,n] ⓐV2.T2.
+#h #n #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
/5 width=5 by isrt_max_O1, isr_shift, cpg_appl, ex2_intro/
qed.
-lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2.
- ❪G,L❫ ⊢ U1 ➡[n,h] U2 → ❪G,L❫ ⊢ T1 ➡[n,h] T2 →
- ❪G,L❫ ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2.
-#n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
+lemma cpm_cast: ∀h,n,G,L,U1,U2,T1,T2.
+ ❪G,L❫ ⊢ U1 ➡[h,n] U2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ ❪G,L❫ ⊢ ⓝU1.T1 ➡[h,n] ⓝU2.T2.
+#h #n #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
/4 width=6 by cpg_cast, isrt_max_idem1, isrt_mono, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_zeta *)
-lemma cpm_zeta (n) (h) (G) (L):
- ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ➡[n,h] T2 →
- ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ➡[n,h] T2.
-#n #h #G #L #T1 #T #HT1 #T2 *
+lemma cpm_zeta (h) (n) (G) (L):
+ ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ➡[h,n] T2 →
+ ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ➡[h,n] T2.
+#h #n #G #L #T1 #T #HT1 #T2 *
/3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_eps *)
-lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → ❪G,L❫ ⊢ ⓝV.T1 ➡[n,h] T2.
-#n #h #G #L #V #T1 #T2 *
+lemma cpm_eps: ∀h,n,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → ❪G,L❫ ⊢ ⓝV.T1 ➡[h,n] T2.
+#h #n #G #L #V #T1 #T2 *
/3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
qed.
-lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[n,h] V2 → ❪G,L❫ ⊢ ⓝV1.T ➡[↑n,h] V2.
-#n #h #G #L #V1 #V2 #T *
+lemma cpm_ee: ∀h,n,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] V2 → ❪G,L❫ ⊢ ⓝV1.T ➡[h,↑n] V2.
+#h #n #G #L #V1 #V2 #T *
/3 width=3 by cpg_ee, isrt_succ, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_beta *)
-lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
- ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[n,h] T2 →
- ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡[n,h] ⓓ[p]ⓝW2.V2.T2.
-#n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
+lemma cpm_beta: ∀h,n,p,G,L,V1,V2,W1,W2,T1,T2.
+ ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 →
+ ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡[h,n] ⓓ[p]ⓝW2.V2.T2.
+#h #n #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
/6 width=7 by cpg_beta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_theta *)
-lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ❪G,L❫ ⊢ V1 ➡[h] V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ➡[h] W2 →
- ❪G,L.ⓓW1❫ ⊢ T1 ➡[n,h] T2 →
- ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡[n,h] ⓓ[p]W2.ⓐV2.T2.
-#n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
+lemma cpm_theta: ∀h,n,p,G,L,V1,V,V2,W1,W2,T1,T2.
+ ❪G,L❫ ⊢ V1 ➡[h,0] V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 →
+ ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 →
+ ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡[h,n] ⓓ[p]W2.ⓐV2.T2.
+#h #n #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
/6 width=9 by cpg_theta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
qed.
(* Advanced properties ******************************************************)
lemma cpm_sort (h) (G) (L):
- ∀n. n ≤ 1 → ∀s. ❪G,L❫ ⊢ ⋆s ➡[n,h] ⋆((next h)^n s).
+ ∀n. n ≤ 1 → ∀s. ❪G,L❫ ⊢ ⋆s ➡[h,n] ⋆((next h)^n s).
#h #G #L * //
#n #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ➡[n,h] T2 →
+lemma cpm_inv_atom1: ∀h,n,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h,n] T2 →
∨∨ T2 = ⓪[J] ∧ n = 0
| ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
- | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 & ⇧[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[m,h] V2 & ⇧[1] V2 ≘ T2 &
+ | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & n = ↑m
- | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[n,h] T & ⇧[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 &
L = K.ⓘ[I] & J = LRef (↑i).
-#n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
+#h #n #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
| #s #H1 #H2 #H3 destruct /4 width=3 by isrt_inv_01, or5_intro1, ex3_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
]
qed-.
-lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ➡[n,h] T2 →
+lemma cpm_inv_sort1: ∀h,n,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ➡[h,n] T2 →
∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1.
-#n #h #G #L #T2 #s * #c #Hc #H
+#h #n #G #L #T2 #s * #c #Hc #H
elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
[ lapply (isrt_inv_00 … Hc) | lapply (isrt_inv_01 … Hc) ] -Hc
#H destruct /2 width=1 by conj/
qed-.
-lemma cpm_inv_zero1: ∀n,h,G,L,T2. ❪G,L❫ ⊢ #0 ➡[n,h] T2 →
+lemma cpm_inv_zero1: ∀h,n,G,L,T2. ❪G,L❫ ⊢ #0 ➡[h,n] T2 →
∨∨ T2 = #0 ∧ n = 0
- | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 & ⇧[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 &
L = K.ⓓV1
- | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[m,h] V2 & ⇧[1] V2 ≘ T2 &
+ | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 &
L = K.ⓛV1 & n = ↑m.
-#n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
+#h #n #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
/4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
]
qed-.
-lemma cpm_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
+lemma cpm_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
elim (cpm_inv_zero1 … H) -H *
[ #H1 #H2 destruct /2 width=1 by conj/
| #Y #X1 #X2 #_ #_ #H destruct
]
qed.
-lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ➡[n,h] T2 →
+lemma cpm_inv_lref1: ∀h,n,G,L,T2,i. ❪G,L❫ ⊢ #↑i ➡[h,n] T2 →
∨∨ T2 = #(↑i) ∧ n = 0
- | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[n,h] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
-#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
+ | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+#h #n #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
| #I #K #V2 #HV2 #HVT2 #H destruct
/4 width=6 by ex3_3_intro, ex2_intro, or_intror/
]
qed-.
-lemma cpm_inv_lref1_ctop (n) (h) (G):
- ∀X2,i. ❪G,⋆❫ ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0.
-#n #h #G #X2 * [| #i ] #H
+lemma cpm_inv_lref1_ctop (h) (n) (G):
+ ∀X2,i. ❪G,⋆❫ ⊢ #i ➡[h,n] X2 → ∧∧ X2 = #i & n = 0.
+#h #n #G #X2 * [| #i ] #H
[ elim (cpm_inv_zero1 … H) -H *
[ #H1 #H2 destruct /2 width=1 by conj/
| #Y #X1 #X2 #_ #_ #H destruct
]
qed.
-lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ❪G,L❫ ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
-#n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
+lemma cpm_inv_gref1: ∀h,n,G,L,T2,l. ❪G,L❫ ⊢ §l ➡[h,n] T2 → T2 = §l ∧ n = 0.
+#h #n #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
qed-.
(* Basic_2A1: includes: cpr_inv_bind1 *)
-lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[n,h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[n,h] T2 &
+lemma cpm_inv_bind1: ∀h,n,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 &
U2 = ⓑ[p,I]V2.T2
- | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[n,h] U2 &
+ | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[h,n] U2 &
p = true & I = Abbr.
-#n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
+#h #n #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
(* Basic_2A1: includes: cpr_inv_abbr1 *)
-lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ➡[n,h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ➡[n,h] T2 &
+lemma cpm_inv_abbr1: ∀h,n,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ➡[h,n] U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ➡[h,n] T2 &
U2 = ⓓ[p]V2.T2
- | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[n,h] U2 & p = true.
-#n #h #p #G #L #V1 #T1 #U2 #H
+ | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[h,n] U2 & p = true.
+#h #n #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
[ /3 width=1 by or_introl/
| * /3 width=3 by ex3_intro, or_intror/
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
(* Basic_2A1: includes: cpr_inv_abst1 *)
-lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ➡[n,h] U2 →
- ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[n,h] T2 &
+lemma cpm_inv_abst1: ∀h,n,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ➡[h,n] U2 →
+ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[h,n] T2 &
U2 = ⓛ[p]V2.T2.
-#n #h #p #G #L #V1 #T1 #U2 #H
+#h #n #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
[ /3 width=1 by or_introl/
| * #T #_ #_ #_ #H destruct
]
qed-.
-lemma cpm_inv_abst_bi: ∀n,h,p1,p2,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ ⓛ[p1]V1.T1 ➡[n,h] ⓛ[p2]V2.T2 →
- ∧∧ ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[n,h] T2 & p1 = p2.
-#n #h #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H
+lemma cpm_inv_abst_bi: ∀h,n,p1,p2,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ ⓛ[p1]V1.T1 ➡[h,n] ⓛ[p2]V2.T2 →
+ ∧∧ ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[h,n] T2 & p1 = p2.
+#h #n #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H
elim (cpm_inv_abst1 … H) -H #XV #XT #HV #HT #H destruct
/2 width=1 by and3_intro/
qed-.
(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
(* Basic_2A1: includes: cpr_inv_appl1 *)
-lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ➡[n,h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L❫ ⊢ U1 ➡[n,h] T2 &
+lemma cpm_inv_appl1: ∀h,n,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ➡[h,n] U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ U1 ➡[h,n] T2 &
U2 = ⓐV2.T2
- | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L❫ ⊢ W1 ➡[h] W2 &
- ❪G,L.ⓛW1❫ ⊢ T1 ➡[n,h] T2 &
+ | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ W1 ➡[h,0] W2 &
+ ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 &
U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
- | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V & ⇧[1] V ≘ V2 &
- ❪G,L❫ ⊢ W1 ➡[h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ➡[n,h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V & ⇧[1] V ≘ V2 &
+ ❪G,L❫ ⊢ W1 ➡[h,0] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 &
U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
-#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
+#h #n #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
]
qed-.
-lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ➡[n,h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[n,h] V2 & ❪G,L❫ ⊢ U1 ➡[n,h] T2 &
+lemma cpm_inv_cast1: ∀h,n,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ➡[h,n] U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,n] V2 & ❪G,L❫ ⊢ U1 ➡[h,n] T2 &
U2 = ⓝV2.T2
- | ❪G,L❫ ⊢ U1 ➡[n,h] U2
- | ∃∃m. ❪G,L❫ ⊢ V1 ➡[m,h] U2 & n = ↑m.
-#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
+ | ❪G,L❫ ⊢ U1 ➡[h,n] U2
+ | ∃∃m. ❪G,L❫ ⊢ V1 ➡[h,m] U2 & n = ↑m.
+#h #n #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
lapply (isrt_eq_t_trans … HcV HcVT) -HcVT #H
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: cpr_fwd_bind1_minus *)
-lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ➡[n,h] T → ∀p.
- ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[n,h] ⓑ[p,I]V2.T2 &
+lemma cpm_fwd_bind1_minus: ∀h,n,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ➡[h,n] T → ∀p.
+ ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] ⓑ[p,I]V2.T2 &
T = -ⓑ[I]V2.T2.
-#n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
+#h #n #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex2_intro/
qed-.
lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
(∀I,G,L. Q 0 G L (⓪[I]) (⓪[I])) →
(∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
- (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
+ (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → Q n G K V1 V2 →
⇧[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
- ) → (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
+ ) → (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → Q n G K V1 V2 →
⇧[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
- ) → (∀n,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[n,h] T → Q n G K (#i) T →
+ ) → (∀n,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h,n] T → Q n G K (#i) T →
⇧[1] T ≘ U → Q n G (K.ⓘ[I]) (#↑i) (U)
- ) → (∀n,p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[n,h] T2 →
+ ) → (∀n,p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 →
Q 0 G L V1 V2 → Q n G (L.ⓑ[I]V1) T1 T2 → Q n G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ T1 ➡[n,h] T2 →
+ ) → (∀n,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[n,h] V2 → ❪G,L❫ ⊢ T1 ➡[n,h] T2 →
+ ) → (∀n,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,n] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
- ) → (∀n,G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → ❪G,L❫ ⊢ T ➡[n,h] T2 →
+ ) → (∀n,G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → ❪G,L❫ ⊢ T ➡[h,n] T2 →
Q n G L T T2 → Q n G L (+ⓓV.T1) T2
- ) → (∀n,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 →
+ ) → (∀n,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
- ) → (∀n,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[n,h] V2 →
+ ) → (∀n,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] V2 →
Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
- ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[n,h] T2 →
+ ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 →
Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
Q n G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
- ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[n,h] T2 →
+ ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 →
Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
⇧[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
) →
- ∀n,G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2.
+ ∀n,G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → Q n G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
* #c #HC #H generalize in match HC; -HC generalize in match n; -n
elim H -c -G -L -T1 -T2
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