(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
definition cpx (h): relation4 genv lenv term term ≝
- λG,L,T1,T2. â\88\83c. â¦\83G,Lâ¦\84 ⊢ T1 ⬈[eq_f,c,h] T2.
+ λG,L,T1,T2. â\88\83c. â\9dªG,Lâ\9d« ⊢ T1 ⬈[eq_f,c,h] T2.
interpretation
"unbound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: â\88\80h,G,L,s. â¦\83G,Lâ¦\84 ⊢ ⋆s ⬈[h] ⋆(⫯[h]s).
+lemma cpx_ess: â\88\80h,G,L,s. â\9dªG,Lâ\9d« ⊢ ⋆s ⬈[h] ⋆(⫯[h]s).
/2 width=2 by cpg_ess, ex_intro/ qed.
-lemma cpx_delta: â\88\80h,I,G,K,V1,V2,W2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[h] V2 →
- â\87§*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\91{I}V1â¦\84 ⊢ #0 ⬈[h] W2.
+lemma cpx_delta: â\88\80h,I,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[h] V2 →
+ â\87§*[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈[h] W2.
#h * #G #K #V1 #V2 #W2 *
/3 width=4 by cpg_delta, cpg_ell, ex_intro/
qed.
-lemma cpx_lref: â\88\80h,I,G,K,T,U,i. â¦\83G,Kâ¦\84 ⊢ #i ⬈[h] T →
- â\87§*[1] T â\89\98 U â\86\92 â¦\83G,K.â\93\98{I}â¦\84 ⊢ #↑i ⬈[h] U.
+lemma cpx_lref: â\88\80h,I,G,K,T,U,i. â\9dªG,Kâ\9d« ⊢ #i ⬈[h] T →
+ â\87§*[1] T â\89\98 U â\86\92 â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈[h] U.
#h #I #G #K #T #U #i *
/3 width=4 by cpg_lref, ex_intro/
qed.
lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ T1 ⬈[h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â¬\88[h] â\93\91{p,I}V2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈[h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â¬\88[h] â\93\91[p,I]V2.T2.
#h #p #I #G #L #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.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[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¬\88[h] â\93\95{I}V2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[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¬\88[h] â\93\95[I]V2.T2.
#h * #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):
- â\88\80T1,T. â\87§*[1] T â\89\98 T1 â\86\92 â\88\80T2. â¦\83G,Lâ¦\84 ⊢ T ⬈[h] T2 →
- â\88\80V. â¦\83G,Lâ¦\84 ⊢ +ⓓV.T1 ⬈[h] T2.
+ â\88\80T1,T. â\87§*[1] T â\89\98 T1 â\86\92 â\88\80T2. â\9dªG,Lâ\9d« ⊢ T ⬈[h] T2 →
+ â\88\80V. â\9dªG,Lâ\9d« ⊢ +ⓓV.T1 ⬈[h] T2.
#h #G #L #T1 #T #HT1 #T2 *
/3 width=4 by cpg_zeta, ex_intro/
qed.
-lemma cpx_eps: â\88\80h,G,L,V,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\88[h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓝV.T1 ⬈[h] T2.
+lemma cpx_eps: â\88\80h,G,L,V,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝV.T1 ⬈[h] T2.
#h #G #L #V #T1 #T2 *
/3 width=2 by cpg_eps, ex_intro/
qed.
(* Basic_2A1: was: cpx_ct *)
-lemma cpx_ee: â\88\80h,G,L,V1,V2,T. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓝV1.T ⬈[h] V2.
+lemma cpx_ee: â\88\80h,G,L,V1,V2,T. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝV1.T ⬈[h] V2.
#h #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.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 â\86\92 â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ⬈[h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{p}W1.T1 â¬\88[h] â\93\93{p}ⓝW2.V2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[h] W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\9b[p]W1.T1 â¬\88[h] â\93\93[p]ⓝW2.V2.T2.
#h #p #G #L #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.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V â\86\92 â\87§*[1] V â\89\98 V2 â\86\92 â¦\83G,Lâ¦\84 ⊢ W1 ⬈[h] W2 →
- â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ⬈[h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{p}W1.T1 â¬\88[h] â\93\93{p}W2.ⓐV2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V â\86\92 â\87§*[1] V â\89\98 V2 â\86\92 â\9dªG,Lâ\9d« ⊢ W1 ⬈[h] W2 →
+ â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈[h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\93[p]W1.T1 â¬\88[h] â\93\93[p]W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
/3 width=4 by cpg_theta, ex_intro/
qed.
(* Advanced properties ******************************************************)
-lemma cpx_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¬\88[h] â\91¡{I}V2.T.
+lemma cpx_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¬\88[h] â\91¡[I]V2.T.
#h * /2 width=2 by cpx_flat, cpx_bind/
qed.
lemma cpg_cpx (h) (Rt) (c) (G) (L):
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\88[Rt,c,h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ⬈[h] T2.
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[Rt,c,h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈[h] T2.
#h #Rt #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: â\88\80h,J,G,L,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93ª{J} ⬈[h] T2 →
- ∨∨ T2 = ⓪{J}
+lemma cpx_inv_atom1: â\88\80h,J,G,L,T2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[J] ⬈[h] T2 →
+ ∨∨ T2 = ⓪[J]
| ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s
- | â\88\83â\88\83I,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2 &
- L = K.ⓑ{I}V1 & J = LRef 0
- | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 ⊢ #i ⬈[h] T & ⇧*[1] T ≘ T2 &
- L = K.ⓘ{I} & J = LRef (↑i).
+ | â\88\83â\88\83I,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2 &
+ L = K.ⓑ[I]V1 & J = LRef 0
+ | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #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/
qed-.
-lemma cpx_inv_sort1: â\88\80h,G,L,T2,s. â¦\83G,Lâ¦\84 ⊢ ⋆s ⬈[h] T2 →
+lemma cpx_inv_sort1: â\88\80h,G,L,T2,s. â\9dªG,Lâ\9d« ⊢ ⋆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/
qed-.
-lemma cpx_inv_zero1: â\88\80h,G,L,T2. â¦\83G,Lâ¦\84 ⊢ #0 ⬈[h] T2 →
+lemma cpx_inv_zero1: â\88\80h,G,L,T2. â\9dªG,Lâ\9d« ⊢ #0 ⬈[h] T2 →
∨∨ T2 = #0
- | â\88\83â\88\83I,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2 &
- L = K.ⓑ{I}V1.
+ | â\88\83â\88\83I,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2 &
+ L = K.ⓑ[I]V1.
#h #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: â\88\80h,G,L,T2,i. â¦\83G,Lâ¦\84 ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1: â\88\80h,G,L,T2,i. â\9dªG,Lâ\9d« ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â¬\88[h] T & â\87§*[1] T â\89\98 T2 & L = K.â\93\98{I}.
+ | â\88\83â\88\83I,K,T. â\9dªG,Kâ\9d« â\8a¢ #i â¬\88[h] T & â\87§*[1] T â\89\98 T2 & L = K.â\93\98[I].
#h #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: â\88\80h,G,L,T2,l. â¦\83G,Lâ¦\84 ⊢ §l ⬈[h] T2 → T2 = §l.
+lemma cpx_inv_gref1: â\88\80h,G,L,T2,l. â\9dªG,Lâ\9d« ⊢ §l ⬈[h] T2 → T2 = §l.
#h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
qed-.
-lemma cpx_inv_bind1: â\88\80h,p,I,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 ⬈[h] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ T1 ⬈[h] T2 &
- U2 = ⓑ{p,I}V2.T2
- | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 ⊢ T ⬈[h] U2 &
+lemma cpx_inv_bind1: â\88\80h,p,I,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 ⬈[h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈[h] T2 &
+ U2 = ⓑ[p,I]V2.T2
+ | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ⬈[h] U2 &
p = true & I = Abbr.
#h #p #I #G #L #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: â\88\80h,p,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\93{p}V1.T1 ⬈[h] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,L.â\93\93V1â¦\84 ⊢ T1 ⬈[h] T2 &
- U2 = ⓓ{p}V2.T2
- | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 ⊢ T ⬈[h] U2 & p = true.
+lemma cpx_inv_abbr1: â\88\80h,p,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\93[p]V1.T1 ⬈[h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,L.â\93\93V1â\9d« ⊢ T1 ⬈[h] T2 &
+ U2 = ⓓ[p]V2.T2
+ | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ⬈[h] U2 & p = true.
#h #p #G #L #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: â\88\80h,p,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\9b{p}V1.T1 ⬈[h] U2 →
- â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,L.â\93\9bV1â¦\84 ⊢ T1 ⬈[h] T2 &
- U2 = ⓛ{p}V2.T2.
+lemma cpx_inv_abst1: â\88\80h,p,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\9b[p]V1.T1 ⬈[h] U2 →
+ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,L.â\93\9bV1â\9d« ⊢ T1 ⬈[h] T2 &
+ U2 = ⓛ[p]V2.T2.
#h #p #G #L #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: â\88\80h,G,L,V1,U1,U2. â¦\83G,Lâ¦\84 ⊢ ⓐ V1.U1 ⬈[h] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_appl1: â\88\80h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓐ V1.U1 ⬈[h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[h] T2 &
U2 = ⓐV2.T2
- | â\88\83â\88\83p,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,Lâ¦\84 ⊢ W1 ⬈[h] W2 &
- â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ⬈[h] T2 &
- U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
- | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 ⊢ V1 ⬈[h] V & ⇧*[1] V ≘ V2 &
- â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 & â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ⬈[h] T2 &
- U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
+ | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,Lâ\9d« ⊢ W1 ⬈[h] W2 &
+ â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[h] T2 &
+ U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« ⊢ V1 ⬈[h] V & ⇧*[1] V ≘ V2 &
+ â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[h] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ 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 *
/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: â\88\80h,G,L,V1,U1,U2. â¦\83G,Lâ¦\84 ⊢ ⓝV1.U1 ⬈[h] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_cast1: â\88\80h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓝV1.U1 ⬈[h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[h] T2 &
U2 = ⓝV2.T2
- | â¦\83G,Lâ¦\84 ⊢ U1 ⬈[h] U2
- | â¦\83G,Lâ¦\84 ⊢ V1 ⬈[h] U2.
+ | â\9dªG,Lâ\9d« ⊢ U1 ⬈[h] U2
+ | â\9dªG,Lâ\9d« ⊢ V1 ⬈[h] U2.
#h #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: â\88\80h,I,G,K,V1,T2. â¦\83G,K.â\93\91{I}V1â¦\84 ⊢ #0 ⬈[h] T2 →
+lemma cpx_inv_zero1_pair: â\88\80h,I,G,K,V1,T2. â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈[h] T2 →
∨∨ T2 = #0
- | â\88\83â\88\83V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2.
+ | â\88\83â\88\83V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2.
#h #I #G #L #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: â\88\80h,I,G,K,T2,i. â¦\83G,K.â\93\98{I}â¦\84 ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1_bind: â\88\80h,I,G,K,T2,i. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | â\88\83â\88\83T. â¦\83G,Kâ¦\84 ⊢ #i ⬈[h] T & ⇧*[1] T ≘ T2.
+ | â\88\83â\88\83T. â\9dªG,Kâ\9d« ⊢ #i ⬈[h] T & ⇧*[1] T ≘ T2.
#h #I #G #L #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: 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¬\88[h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[h] T2 &
- U2 = ⓕ{I}V2.T2
- | (â¦\83G,Lâ¦\84 ⊢ U1 ⬈[h] U2 ∧ I = Cast)
- | (â¦\83G,Lâ¦\84 ⊢ V1 ⬈[h] U2 ∧ I = Cast)
- | â\88\83â\88\83p,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G,Lâ¦\84 ⊢ W1 ⬈[h] W2 &
- â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ⬈[h] T2 &
- U1 = ⓛ{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 ⊢ V1 ⬈[h] V & ⇧*[1] V ≘ V2 &
- â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 & â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ⬈[h] T2 &
- U1 = ⓓ{p}W1.T1 &
- U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
+lemma cpx_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¬\88[h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[h] T2 &
+ U2 = ⓕ[I]V2.T2
+ | (â\9dªG,Lâ\9d« ⊢ U1 ⬈[h] U2 ∧ I = Cast)
+ | (â\9dªG,Lâ\9d« ⊢ V1 ⬈[h] U2 ∧ I = Cast)
+ | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 & â\9dªG,Lâ\9d« ⊢ W1 ⬈[h] W2 &
+ â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[h] T2 &
+ U1 = ⓛ[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« ⊢ V1 ⬈[h] V & ⇧*[1] V ≘ V2 &
+ â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[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 (cpx_inv_appl1 … H) -H *
/3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: â\88\80h,I,G,L,V1,T1,T. â¦\83G,Lâ¦\84 â\8a¢ -â\93\91{I}V1.T1 ⬈[h] T → ∀p.
- â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â¬\88[h] â\93\91{p,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
+lemma cpx_fwd_bind1_minus: â\88\80h,I,G,L,V1,T1,T. â\9dªG,Lâ\9d« â\8a¢ -â\93\91[I]V1.T1 ⬈[h] T → ∀p.
+ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â¬\88[h] â\93\91[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
/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})) →
+ (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
(∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
- (â\88\80I,G,K,V1,V2,W2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
- ⇧*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
- ) â\86\92 (â\88\80I,G,K,T,U,i. â¦\83G,Kâ¦\84 ⊢ #i ⬈[h] T → Q G K (#i) T →
- ⇧*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
- ) â\86\92 (â\88\80p,I,G,L,V1,V2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ 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)
- ) â\86\92 (â\88\80I,G,L,V1,V2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ⬈[h] T2 →
- Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) â\86\92 (â\88\80G,L,V,T1,T,T2. â\87§*[1] T â\89\98 T1 â\86\92 â¦\83G,Lâ¦\84 ⊢ T ⬈[h] T2 → Q G L T T2 →
+ (â\88\80I,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
+ ⇧*[1] V2 ≘ W2 → Q G (K.ⓑ[I]V1) (#0) W2
+ ) â\86\92 (â\88\80I,G,K,T,U,i. â\9dªG,Kâ\9d« ⊢ #i ⬈[h] T → Q G K (#i) T →
+ ⇧*[1] T ≘ U → Q G (K.ⓘ[I]) (#↑i) (U)
+ ) â\86\92 (â\88\80p,I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« ⊢ 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)
+ ) â\86\92 (â\88\80I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈[h] T2 →
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
+ ) â\86\92 (â\88\80G,L,V,T1,T,T2. â\87§*[1] T â\89\98 T1 â\86\92 â\9dªG,Lâ\9d« ⊢ T ⬈[h] T2 → Q G L T T2 →
Q G L (+ⓓV.T1) T2
- ) â\86\92 (â\88\80G,L,V,T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
+ ) â\86\92 (â\88\80G,L,V,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
Q G L (ⓝV.T1) T2
- ) â\86\92 (â\88\80G,L,V1,V2,T. â¦\83G,Lâ¦\84 ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
+ ) â\86\92 (â\88\80G,L,V1,V2,T. â\9dªG,Lâ\9d« ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
Q G L (ⓝV1.T) V2
- ) â\86\92 (â\88\80p,G,L,V1,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 â\86\92 â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ⬈[h] T2 →
+ ) â\86\92 (â\88\80p,G,L,V1,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[h] W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« ⊢ 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)
- ) â\86\92 (â\88\80p,G,L,V1,V,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[h] V â\86\92 â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 â\86\92 â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ⬈[h] T2 →
+ Q G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+ ) â\86\92 (â\88\80p,G,L,V1,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[h] W2 â\86\92 â\9dªG,L.â\93\93W1â\9d« ⊢ 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)
+ ⇧*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
) →
- â\88\80G,L,T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
+ â\88\80G,L,T1,T2. â\9dªG,Lâ\9d« ⊢ 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
* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
qed-.
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