(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_6_6.ma".
+include "ground_2/xoa/ex_7_7.ma".
+include "ground_2/xoa/or_4.ma".
+include "ground_2/insert_eq/insert_eq_0.ma".
include "basic_2/rt_transition/cpm.ma".
(* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
(* Note: cpr_flat: does not hold in basic_1 *)
(* Basic_1: includes: pr2_thin_dx *)
lemma cpr_flat: ∀h,I,G,L,V1,V2,T1,T2.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 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\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 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 * /2 width=1 by cpm_cast, cpm_appl/
-qed.
+qed.
(* Basic_1: was: pr2_head_1 *)
-lemma cpr_pair_sn: â\88\80h,I,G,L,V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\91¡{I}V1.T â\9e¡[h] â\91¡{I}V2.T.
+lemma cpr_pair_sn: â\88\80h,I,G,L,V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80T. â\9dªG,Lâ\9d« â\8a¢ â\91¡[I]V1.T â\9e¡[h] â\91¡[I]V2.T.
#h * /2 width=1 by cpm_bind, cpr_flat/
qed.
(* Basic inversion properties ***********************************************)
-lemma cpr_inv_atom1: â\88\80h,J,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{J} ➡[h] T2 →
- ∨∨ T2 = ⓪{J}
- | â\88\83â\88\83K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+lemma cpr_inv_atom1: â\88\80h,J,G,L,T2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[J] ➡[h] T2 →
+ ∨∨ T2 = ⓪[J]
+ | â\88\83â\88\83K,V1,V2. â\9dªG,Kâ\9d« â\8a¢ V1 â\9e¡[h] V2 & â\87§*[1] V2 â\89\98 T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 &
- L = K.ⓑ{I}V & J = LRef (⫯i).
+ | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[h] T & ⇧*[1] T ≘ T2 &
+ L = K.ⓘ[I] & J = LRef (↑i).
#h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
-/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_3_intro/
+[2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
[ #n #_ #_ #H destruct
| #n #K #V1 #V2 #_ #_ #_ #_ #H destruct
]
qed-.
(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: â\88\80h,G,L,T2,s. â¦\83G, Lâ¦\84 ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
-#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H * // #_ #H destruct
+lemma cpr_inv_sort1: â\88\80h,G,L,T2,s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
+#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H //
qed-.
-lemma cpr_inv_zero1: â\88\80h,G,L,T2. â¦\83G, Lâ¦\84 ⊢ #0 ➡[h] T2 →
- T2 = #0 ∨
- ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
- L = K.ⓓV1.
+lemma cpr_inv_zero1: â\88\80h,G,L,T2. â\9dªG,Lâ\9d« ⊢ #0 ➡[h] T2 →
+ ∨∨ T2 = #0
+ | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h] V2 & ⇧*[1] V2 ≘ T2 &
+ L = K.ⓓV1.
#h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
#n #K #V1 #V2 #_ #_ #_ #H destruct
qed-.
-lemma cpr_inv_lref1: â\88\80h,G,L,T2,i. â¦\83G, Lâ¦\84 â\8a¢ #⫯i ➡[h] T2 →
- T2 = #(⫯i) ∨
- ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+lemma cpr_inv_lref1: â\88\80h,G,L,T2,i. â\9dªG,Lâ\9d« â\8a¢ #â\86\91i ➡[h] T2 →
+ ∨∨ T2 = #(↑i)
+ | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ[I].
#h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
-/3 width=7 by ex3_4_intro, or_introl, or_intror/
+/3 width=6 by ex3_3_intro, or_introl, or_intror/
qed-.
-lemma cpr_inv_gref1: â\88\80h,G,L,T2,l. â¦\83G, Lâ¦\84 ⊢ §l ➡[h] T2 → T2 = §l.
+lemma cpr_inv_gref1: â\88\80h,G,L,T2,l. â\9dªG,Lâ\9d« ⊢ §l ➡[h] T2 → T2 = §l.
#h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H //
qed-.
(* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: â\88\80h,G,L,V1,U1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\9d V1. U1 â\9e¡[h] U2 â\86\92 (
- â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G, Lâ¦\84 ⊢ U1 ➡[h] T2 &
- U2 = ⓝV2.T2
- ) ∨ ⦃G, L⦄ ⊢ U1 ➡[h] U2.
+lemma cpr_inv_cast1: â\88\80h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\9d V1.U1 â\9e¡[h] U2 â\86\92
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ➡[h] T2 &
+ U2 = ⓝV2.T2
+ | ❪G,L❫ ⊢ U1 ➡[h] U2.
#h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
/2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
qed-.
-lemma cpr_inv_flat1: â\88\80h,I,G,L,V1,U1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.U1 ➡[h] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G, Lâ¦\84 ⊢ U1 ➡[h] T2 &
- U2 = ⓕ{I}V2.T2
- | (â¦\83G, Lâ¦\84 ⊢ U1 ➡[h] U2 ∧ I = Cast)
- | â\88\83â\88\83p,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G, Lâ¦\84 ⊢ W1 ➡[h] W2 &
- â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡[h] T2 & U1 = â\93\9b{p}W1.T1 &
- U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V & â¬\86*[1] V â\89¡ V2 &
- â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 & â¦\83G, L.â\93\93W1â¦\84 ⊢ T1 ➡[h] T2 &
- U1 = ⓓ{p}W1.T1 &
- U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
+lemma cpr_inv_flat1: â\88\80h,I,G,L,V1,U1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\95[I]V1.U1 ➡[h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ➡[h] T2 &
+ U2 = ⓕ[I]V2.T2
+ | (â\9dªG,Lâ\9d« ⊢ U1 ➡[h] U2 ∧ I = Cast)
+ | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 & â\9dªG,Lâ\9d« ⊢ W1 ➡[h] W2 &
+ â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â\9e¡[h] T2 & U1 = â\93\9b[p]W1.T1 &
+ U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V & â\87§*[1] V â\89\98 V2 &
+ â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ➡[h] T2 &
+ U1 = ⓓ[p]W1.T1 &
+ U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
#h * #G #L #V1 #U1 #U2 #H
[ elim (cpm_inv_appl1 … H) -H *
/3 width=13 by or4_intro0, or4_intro2, or4_intro3, ex7_7_intro, ex6_6_intro, ex3_2_intro/
]
qed-.
+(* Basic eliminators ********************************************************)
+
+lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
+ (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
+ (∀G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
+ ⇧*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
+ ) → (∀I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h] T → Q G K (#i) T →
+ ⇧*[1] T ≘ U → Q G (K.ⓘ[I]) (#↑i) (U)
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h] T2 →
+ Q G L V1 V2 → Q G (L.ⓑ[I]V1) T1 T2 → Q G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
+ ) → (∀I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ T1 ➡[h] T2 →
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
+ ) → (∀G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → ❪G,L❫ ⊢ T ➡[h] T2 →
+ Q G L T T2 → Q G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h] T2 →
+ Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
+ Q G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+ ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h] T2 →
+ Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
+ ⇧*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
+ ) →
+ ∀G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
+@(insert_eq_0 … 0) #n #H
+@(cpm_ind … H) -G -L -T1 -T2 -n [2,4,11:|*: /3 width=4 by/ ]
+[ #G #L #s #H destruct
+| #n #G #K #V1 #V2 #W2 #_ #_ #_ #H destruct
+| #n #G #L #U1 #U2 #T #_ #_ #H destruct
+]
+qed-.
+
(* Basic_1: removed theorems 12:
pr0_subst0_back pr0_subst0_fwd pr0_subst0
pr0_delta1
pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
pr2_gen_ctail pr2_ctail
*)
-(* Basic_1: removed local theorems 4:
- pr0_delta_eps pr0_cong_delta
- pr2_free_free pr2_free_delta
-*)
projects / helm.git / blobdiff