include "ground/xoa/ex_6_7.ma".
include "ground/xoa/ex_7_7.ma".
include "ground/xoa/or_4.ma".
-include "basic_2/notation/relations/predty_5.ma".
+include "basic_2/notation/relations/predty_4.ma".
include "basic_2/rt_transition/cpg.ma".
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
-definition cpx (h): relation4 genv lenv term term ≝
- λG,L,T1,T2. ∃c. ❪G,L❫ ⊢ T1 ⬈[eq_f,c,h] T2.
+definition sort_eq_f: relation nat ≝ λs1,s2. ⊤.
+
+definition cpx (G) (L): relation2 term term ≝
+ λT1,T2. ∃c. ❪G,L❫ ⊢ T1 ⬈[sort_eq_f,rtc_eq_f,c] T2.
interpretation
- "unbound context-sensitive parallel rt-transition (term)"
- 'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
+ "extended context-sensitive parallel rt-transition (term)"
+ 'PRedTy G L T1 T2 = (cpx G L T1 T2).
(* Basic properties *********************************************************)
-(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: ∀h,G,L,s. ❪G,L❫ ⊢ ⋆s ⬈[h] ⋆(⫯[h]s).
-/2 width=2 by cpg_ess, ex_intro/ qed.
+(* Basic_2A1: uses: cpx_st *)
+lemma cpx_qu (G) (L): ∀s1,s2. ❪G,L❫ ⊢ ⋆s1 ⬈ ⋆s2.
+/3 width=2 by cpg_ess, ex_intro/ qed.
-lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈[h] V2 →
- ⇧[1] V2 ≘ W2 → ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[h] W2.
-#h * #G #K #V1 #V2 #W2 *
+lemma cpx_delta (G) (K):
+ ∀I,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈ V2 →
+ ⇧[1] V2 ≘ W2 → ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈ W2.
+#G #K * #V1 #V2 #W2 *
/3 width=4 by cpg_delta, cpg_ell, ex_intro/
qed.
-lemma cpx_lref: ∀h,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ⬈[h] T →
- ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[h] U.
-#h #I #G #K #T #U #i *
+lemma cpx_lref (G) (K):
+ ∀I,T,U,i. ❪G,K❫ ⊢ #i ⬈ T →
+ ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈ U.
+#G #K #I #T #U #i *
/3 width=4 by cpg_lref, ex_intro/
qed.
-lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
- ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[h] T2 →
- ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[h] ⓑ[p,I]V2.T2.
-#h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
+lemma cpx_bind (G) (L):
+ ∀p,I,V1,V2,T1,T2.
+ ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ T2 →
+ ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2.
+#G #L #p #I #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_bind, ex_intro/
qed.
-lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
- ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L❫ ⊢ T1 ⬈[h] T2 →
- ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬈[h] ⓕ[I]V2.T2.
-#h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
+lemma cpx_flat (G) (L):
+ ∀I,V1,V2,T1,T2.
+ ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ T1 ⬈ T2 →
+ ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬈ ⓕ[I]V2.T2.
+#G #L * #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=5 by cpg_appl, cpg_cast, ex_intro/
qed.
-lemma cpx_zeta (h) (G) (L):
- ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ⬈[h] T2 →
- ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ⬈[h] T2.
-#h #G #L #T1 #T #HT1 #T2 *
+lemma cpx_zeta (G) (L):
+ ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ⬈ T2 →
+ ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ⬈ T2.
+#G #L #T1 #T #HT1 #T2 *
/3 width=4 by cpg_zeta, ex_intro/
qed.
-lemma cpx_eps: ∀h,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → ❪G,L❫ ⊢ ⓝV.T1 ⬈[h] T2.
-#h #G #L #V #T1 #T2 *
+lemma cpx_eps (G) (L):
+ ∀V,T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → ❪G,L❫ ⊢ ⓝV.T1 ⬈ T2.
+#G #L #V #T1 #T2 *
/3 width=2 by cpg_eps, ex_intro/
qed.
(* Basic_2A1: was: cpx_ct *)
-lemma cpx_ee: ∀h,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L❫ ⊢ ⓝV1.T ⬈[h] V2.
-#h #G #L #V1 #V2 #T *
+lemma cpx_ee (G) (L):
+ ∀V1,V2,T. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ ⓝV1.T ⬈ V2.
+#G #L #V1 #V2 #T *
/3 width=2 by cpg_ee, ex_intro/
qed.
-lemma cpx_beta: ∀h,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 →
- ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈[h] ⓓ[p]ⓝW2.V2.T2.
-#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
+lemma cpx_beta (G) (L):
+ ∀p,V1,V2,W1,W2,T1,T2.
+ ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 →
+ ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈ ⓓ[p]ⓝW2.V2.T2.
+#G #L #p #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
/3 width=2 by cpg_beta, ex_intro/
qed.
-lemma cpx_theta: ∀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 ⬈[h] T2 →
- ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈[h] ⓓ[p]W2.ⓐV2.T2.
-#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
+lemma cpx_theta (G) (L):
+ ∀p,V1,V,V2,W1,W2,T1,T2.
+ ❪G,L❫ ⊢ V1 ⬈ V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 →
+ ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈ ⓓ[p]W2.ⓐV2.T2.
+#G #L #p #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
/3 width=4 by cpg_theta, ex_intro/
qed.
(* Basic_2A1: includes: cpx_atom *)
-lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
+lemma cpx_refl (G) (L): reflexive … (cpx G L).
/3 width=2 by cpg_refl, ex_intro/ qed.
(* Advanced properties ******************************************************)
-lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ❪G,L❫ ⊢ V1 ⬈[h] V2 →
- ∀T. ❪G,L❫ ⊢ ②[I]V1.T ⬈[h] ②[I]V2.T.
-#h * /2 width=2 by cpx_flat, cpx_bind/
+lemma cpx_pair_sn (G) (L):
+ ∀I,V1,V2. ❪G,L❫ ⊢ V1 ⬈ V2 →
+ ∀T. ❪G,L❫ ⊢ ②[I]V1.T ⬈ ②[I]V2.T.
+#G #L * /2 width=2 by cpx_flat, cpx_bind/
qed.
-lemma cpg_cpx (h) (Rt) (c) (G) (L):
- ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rt,c,h] T2 → ❪G,L❫ ⊢ T1 ⬈[h] T2.
-#h #Rt #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
+lemma cpg_cpx (Rs) (Rk) (c) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rs,Rk,c] T2 → ❪G,L❫ ⊢ T1 ⬈ T2.
+#Rs #Rk #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
/2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpx_inv_atom1: ∀h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ⬈[h] T2 →
- ∨∨ T2 = ⓪[J]
- | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s
- | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[h] V2 & ⇧[1] V2 ≘ T2 &
- L = K.ⓑ[I]V1 & J = LRef 0
- | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[h] T & ⇧[1] T ≘ T2 &
- L = K.ⓘ[I] & J = LRef (↑i).
-#h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
-/4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/
+lemma cpx_inv_atom1 (G) (L):
+ ∀J,T2. ❪G,L❫ ⊢ ⓪[J] ⬈ T2 →
+ ∨∨ T2 = ⓪[J]
+ | ∃∃s1,s2. T2 = ⋆s2 & J = Sort s1
+ | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1 & J = LRef 0
+ | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+#G #L #J #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
+/4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_2_intro, ex_intro/
qed-.
-lemma cpx_inv_sort1: ∀h,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ⬈[h] T2 →
- ∨∨ T2 = ⋆s | T2 = ⋆(⫯[h]s).
-#h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
-/2 width=1 by or_introl, or_intror/
+lemma cpx_inv_sort1 (G) (L):
+ ∀T2,s1. ❪G,L❫ ⊢ ⋆s1 ⬈ T2 →
+ ∃s2. T2 = ⋆s2.
+#G #L #T2 #s1 * #c #H elim (cpg_inv_sort1 … H) -H *
+/2 width=2 by ex_intro/
qed-.
-lemma cpx_inv_zero1: ∀h,G,L,T2. ❪G,L❫ ⊢ #0 ⬈[h] T2 →
- ∨∨ T2 = #0
- | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[h] V2 & ⇧[1] V2 ≘ T2 &
- L = K.ⓑ[I]V1.
-#h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
+lemma cpx_inv_zero1 (G) (L):
+ ∀T2. ❪G,L❫ ⊢ #0 ⬈ T2 →
+ ∨∨ T2 = #0
+ | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1.
+#G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_lref1: ∀h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ⬈[h] T2 →
- ∨∨ T2 = #(↑i)
- | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈[h] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
-#h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
+lemma cpx_inv_lref1 (G) (L):
+ ∀T2,i. ❪G,L❫ ⊢ #↑i ⬈ T2 →
+ ∨∨ T2 = #(↑i)
+ | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+#G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
/4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_gref1: ∀h,G,L,T2,l. ❪G,L❫ ⊢ §l ⬈[h] T2 → T2 = §l.
-#h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
+lemma cpx_inv_gref1 (G) (L):
+ ∀T2,l. ❪G,L❫ ⊢ §l ⬈ T2 → T2 = §l.
+#G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
qed-.
-lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[h] T2 &
- U2 = ⓑ[p,I]V2.T2
- | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[h] U2 &
- p = true & I = Abbr.
-#h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
+lemma cpx_inv_bind1 (G) (L):
+ ∀p,I,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈ U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ T2 & U2 = ⓑ[p,I]V2.T2
+ | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈ U2 & p = true & I = Abbr.
+#G #L #p #I #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
/4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈[h] T2 &
- U2 = ⓓ[p]V2.T2
- | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[h] U2 & p = true.
-#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
+lemma cpx_inv_abbr1 (G) (L):
+ ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈ U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈ T2 & U2 = ⓓ[p]V2.T2
+ | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈ U2 & p = true.
+#G #L #p #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
/4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈[h] U2 →
- ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈[h] T2 &
- U2 = ⓛ[p]V2.T2.
-#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
+lemma cpx_inv_abst1 (G) (L):
+ ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈ U2 →
+ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈ T2 & U2 = ⓛ[p]V2.T2.
+#G #L #p #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
/3 width=5 by ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ U1 ⬈[h] T2 &
- U2 = ⓐV2.T2
- | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ W1 ⬈[h] W2 &
- ❪G,L.ⓛW1❫ ⊢ T1 ⬈[h] 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 ⬈[h] T2 &
- U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
-#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
+lemma cpx_inv_appl1 (G) (L):
+ ∀V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ⬈ U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ U1 ⬈ T2 & U2 = ⓐV2.T2
+ | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
+ | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
+#G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
/4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ U1 ⬈[h] T2 &
- U2 = ⓝV2.T2
- | ❪G,L❫ ⊢ U1 ⬈[h] U2
- | ❪G,L❫ ⊢ V1 ⬈[h] U2.
-#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
+lemma cpx_inv_cast1 (G) (L):
+ ∀V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈ U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ U1 ⬈ T2 & U2 = ⓝV2.T2
+ | ❪G,L❫ ⊢ U1 ⬈ U2
+ | ❪G,L❫ ⊢ V1 ⬈ U2.
+#G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
/4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[h] T2 →
- ∨∨ T2 = #0
- | ∃∃V2. ❪G,K❫ ⊢ V1 ⬈[h] V2 & ⇧[1] V2 ≘ T2.
-#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
+lemma cpx_inv_zero1_pair (G) (K):
+ ∀I,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈ T2 →
+ ∨∨ T2 = #0
+ | ∃∃V2. ❪G,K❫ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2.
+#G #K #I #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
/4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
qed-.
-lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[h] T2 →
- ∨∨ T2 = #(↑i)
- | ∃∃T. ❪G,K❫ ⊢ #i ⬈[h] T & ⇧[1] T ≘ T2.
-#h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
+lemma cpx_inv_lref1_bind (G) (K):
+ ∀I,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈ T2 →
+ ∨∨ T2 = #(↑i)
+ | ∃∃T. ❪G,K❫ ⊢ #i ⬈ T & ⇧[1] T ≘ T2.
+#G #K #I #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
/4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓕ[I]V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ U1 ⬈[h] T2 &
- U2 = ⓕ[I]V2.T2
- | (❪G,L❫ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
- | (❪G,L❫ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
- | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ W1 ⬈[h] W2 &
- ❪G,L.ⓛW1❫ ⊢ T1 ⬈[h] T2 &
- U1 = ⓛ[p]W1.T1 &
- U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
- | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V & ⇧[1] V ≘ V2 &
- ❪G,L❫ ⊢ W1 ⬈[h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[h] T2 &
- U1 = ⓓ[p]W1.T1 &
- U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
-#h * #G #L #V1 #U1 #U2 #H
+lemma cpx_inv_flat1 (G) (L):
+ ∀I,V1,U1,U2. ❪G,L❫ ⊢ ⓕ[I]V1.U1 ⬈ U2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ U1 ⬈ T2 & U2 = ⓕ[I]V2.T2
+ | (❪G,L❫ ⊢ U1 ⬈ U2 ∧ I = Cast)
+ | (❪G,L❫ ⊢ V1 ⬈ U2 ∧ I = Cast)
+ | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
+ | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
+#G #L * #V1 #U1 #U2 #H
[ elim (cpx_inv_appl1 … H) -H *
/3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
| elim (cpx_inv_cast1 … H) -H [ * ]
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈[h] T → ∀p.
- ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[h] ⓑ[p,I]V2.T2 &
- T = -ⓑ[I]V2.T2.
-#h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
+lemma cpx_fwd_bind1_minus (G) (L):
+ ∀I,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈ T → ∀p.
+ ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
+#G #L #I #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex_intro/
qed-.
(* Basic eliminators ********************************************************)
-lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
- (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
- (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
- (∀I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
- ⇧[1] V2 ≘ W2 → Q G (K.ⓑ[I]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
- ) → (∀G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
- Q G L (ⓝV1.T) V2
- ) → (∀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 #IH10 #IH11 #G #L #T1 #T2
+lemma cpx_ind (Q:relation4 …):
+ (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
+ (∀G,L,s1,s2. Q G L (⋆s1) (⋆s2)) →
+ (∀I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈ V2 → Q G K V1 V2 →
+ ⇧[1] V2 ≘ W2 → Q G (K.ⓑ[I]V1) (#0) W2
+ ) → (∀I,G,K,T,U,i. ❪G,K❫ ⊢ #i ⬈ 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 ⬈ V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ 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 ⬈ V2 → ❪G,L❫ ⊢ T1 ⬈ 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 ⬈ T2 → Q G L T T2 →
+ Q G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
+ ) → (∀G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈ V2 → Q G L V1 V2 →
+ Q G L (ⓝV1.T) V2
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈ 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 ⬈ V → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈ 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 ⬈ T2 → Q G L T1 T2.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
qed-.
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