(* *)
(**************************************************************************)
-include "basic_2/notation/relations/pred_5.ma".
+include "basic_2/notation/relations/predty_5.ma".
include "basic_2/rt_transition/cpg.ma".
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS *****************)
+(* UNCOUNTED 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 â\8a¢ T1 â\9e¡[c, h] T2.
+ λG,L,T1,T2. â\88\83c. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[c, h] T2.
interpretation
"uncounted context-sensitive parallel reduction (term)"
- 'PRed h G L T1 T2 = (cpx h G L T1 T2).
+ 'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
(* Basic properties *********************************************************)
-lemma cpx_atom: â\88\80h,I,G,L. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{I} â\9e¡[h] ⓪{I}.
+lemma cpx_atom: â\88\80h,I,G,L. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{I} â¬\88[h] ⓪{I}.
/2 width=2 by cpg_atom, ex_intro/ qed.
(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: â\88\80h,G,L,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â\9e¡[h] ⋆(next h s).
+lemma cpx_ess: â\88\80h,G,L,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â¬\88[h] ⋆(next 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 â\8a¢ V1 â\9e¡[h] V2 →
- â¬\86*[1] V2 â\89¡ W2 â\86\92 â¦\83G, K.â\93\91{I}V1â¦\84 â\8a¢ #0 â\9e¡[h] W2.
+lemma cpx_delta: â\88\80h,I,G,K,V1,V2,W2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[h] V2 →
+ â¬\86*[1] V2 â\89¡ W2 â\86\92 â¦\83G, K.â\93\91{I}V1â¦\84 â\8a¢ #0 â¬\88[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,V,T,U,i. â¦\83G, Kâ¦\84 â\8a¢ #i â\9e¡[h] T →
- â¬\86*[1] T â\89¡ U â\86\92 â¦\83G, K.â\93\91{I}Vâ¦\84 â\8a¢ #⫯i â\9e¡[h] U.
+lemma cpx_lref: â\88\80h,I,G,K,V,T,U,i. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[h] T →
+ â¬\86*[1] T â\89¡ U â\86\92 â¦\83G, K.â\93\91{I}Vâ¦\84 â\8a¢ #⫯i â¬\88[h] U.
#h #I #G #K #V #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\9e¡[h] V2 â\86\92 â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡[h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡[h] ⓑ{p,I}V2.T2.
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â¬\88[h] T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â¬\88[h] ⓑ{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\9e¡[h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡[h] ⓕ{I}V2.T2.
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h] T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â¬\88[h] ⓕ{I}V2.T2.
#h #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_flat, ex_intro/
qed.
-lemma cpx_zeta: â\88\80h,G,L,V,T1,T,T2. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â\9e¡[h] T →
- â¬\86*[1] T2 â\89¡ T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ +â\93\93V.T1 â\9e¡[h] T2.
+lemma cpx_zeta: â\88\80h,G,L,V,T1,T,T2. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â¬\88[h] T →
+ â¬\86*[1] T2 â\89¡ T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ +â\93\93V.T1 â¬\88[h] T2.
#h #G #L #V #T1 #T #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\9e¡[h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV.T1 â\9e¡[h] T2.
+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 â\8a¢ â\93\9dV.T1 â¬\88[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\9e¡[h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV1.T â\9e¡[h] V2.
+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 â\8a¢ â\93\9dV1.T â¬\88[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\9e¡[h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡[h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{p}W1.T1 â\9e¡[h] ⓓ{p}ⓝW2.V2.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 â\8a¢ T1 â¬\88[h] T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{p}W1.T1 â¬\88[h] ⓓ{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\9e¡[h] V â\86\92 â¬\86*[1] V â\89¡ V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 →
- â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡[h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{p}W1.T1 â\9e¡[h] ⓓ{p}W2.ⓐV2.T2.
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V â\86\92 â¬\86*[1] V â\89¡ V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 →
+ â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â¬\88[h] T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{p}W1.T1 â¬\88[h] ⓓ{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.
lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
/2 width=2 by ex_intro/ qed.
-lemma cpx_pair_sn: â\88\80h,I,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 →
- â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\91¡{I}V1.T â\9e¡[h] ②{I}V2.T.
+lemma cpx_pair_sn: â\88\80h,I,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 →
+ â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\91¡{I}V1.T â¬\88[h] ②{I}V2.T.
#h #I #G #L #V1 #V2 *
/3 width=2 by cpg_pair_sn, ex_intro/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpx_inv_atom1: â\88\80h,J,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{J} â\9e¡[h] T2 →
+lemma cpx_inv_atom1: â\88\80h,J,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{J} â¬\88[h] T2 →
∨∨ T2 = ⓪{J}
| ∃∃s. T2 = ⋆(next h s) & J = Sort s
- | â\88\83â\88\83I,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & ⬆*[1] V2 ≡ T2 &
+ | â\88\83â\88\83I,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓑ{I}V1 & J = LRef 0
- | â\88\83â\88\83I,K,V,T,i. â¦\83G, Kâ¦\84 â\8a¢ #i â\9e¡[h] T & ⬆*[1] T ≡ T2 &
+ | â\88\83â\88\83I,K,V,T,i. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[h] T & ⬆*[1] T ≡ T2 &
L = K.ⓑ{I}V & J = LRef (⫯i).
#h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
/4 width=9 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_5_intro, ex4_4_intro, ex2_intro, ex_intro/
qed-.
-lemma cpx_inv_sort1: â\88\80h,G,L,T2,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â\9e¡[h] T2 →
+lemma cpx_inv_sort1: â\88\80h,G,L,T2,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â¬\88[h] T2 →
T2 = ⋆s ∨ T2 = ⋆(next 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 â\8a¢ #0 â\9e¡[h] T2 →
+lemma cpx_inv_zero1: â\88\80h,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ #0 â¬\88[h] T2 →
T2 = #0 ∨
- â\88\83â\88\83I,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & ⬆*[1] V2 ≡ T2 &
+ â\88\83â\88\83I,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[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 â\8a¢ #⫯i â\9e¡[h] T2 →
+lemma cpx_inv_lref1: â\88\80h,G,L,T2,i. â¦\83G, Lâ¦\84 â\8a¢ #⫯i â¬\88[h] T2 →
T2 = #(⫯i) ∨
- â\88\83â\88\83I,K,V,T. â¦\83G, Kâ¦\84 â\8a¢ #i â\9e¡[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+ â\88\83â\88\83I,K,V,T. â¦\83G, Kâ¦\84 â\8a¢ #i â¬\88[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
#h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_gref1: â\88\80h,G,L,T2,l. â¦\83G, Lâ¦\84 â\8a¢ §l â\9e¡[h] T2 → T2 = §l.
+lemma cpx_inv_gref1: â\88\80h,G,L,T2,l. â¦\83G, Lâ¦\84 â\8a¢ §l â¬\88[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 â\9e¡[h] U2 → (
- â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡[h] T2 &
+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 â¬\88[h] U2 → (
+ 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 â\8a¢ T1 â¬\88[h] T2 &
U2 = ⓑ{p,I}V2.T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h] T & ⬆*[1] U2 ≡ T &
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[h] T & ⬆*[1] U2 ≡ T &
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 â\9e¡[h] U2 → (
- â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h] T2 &
+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 â¬\88[h] U2 → (
+ â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[h] T2 &
U2 = ⓓ{p}V2.T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h] T & ⬆*[1] U2 ≡ T & p = true.
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[h] T & ⬆*[1] U2 ≡ T & 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 â\9e¡[h] U2 →
- â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G, L.â\93\9bV1â¦\84 â\8a¢ T1 â\9e¡[h] T2 &
+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 â¬\88[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 â\8a¢ T1 â¬\88[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_flat1: â\88\80h,I,G,L,V1,U1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.U1 â\9e¡[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 â\8a¢ U1 â\9e¡[h] T2 &
+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 â¬\88[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 â\8a¢ U1 â¬\88[h] T2 &
U2 = ⓕ{I}V2.T2
- | (â¦\83G, Lâ¦\84 â\8a¢ U1 â\9e¡[h] U2 ∧ I = Cast)
- | (â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[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 â\8a¢ W1 â\9e¡[h] W2 &
- â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡[h] T2 &
+ | (â¦\83G, Lâ¦\84 â\8a¢ U1 â¬\88[h] U2 ∧ I = Cast)
+ | (â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[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 â\8a¢ W1 â¬\88[h] W2 &
+ â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â¬\88[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 â\8a¢ V1 â\9e¡[h] V & ⬆*[1] V ≡ V2 &
- â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 & â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡[h] T2 &
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V & ⬆*[1] V ≡ V2 &
+ â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 & â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â¬\88[h] T2 &
U1 = ⓓ{p}W1.T1 &
U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
#h #I #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_flat1 … H) -H *
/4 width=14 by or5_intro0, or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro, ex_intro, conj/
qed-.
-lemma cpx_inv_appl1: â\88\80h,G,L,V1,U1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\90 V1.U1 â\9e¡[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 â\8a¢ U1 â\9e¡[h] T2 &
+lemma cpx_inv_appl1: â\88\80h,G,L,V1,U1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\90 V1.U1 â¬\88[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 â\8a¢ U1 â¬\88[h] T2 &
U2 = ⓐV2.T2
- | 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 â\8a¢ W1 â\9e¡[h] W2 &
- â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡[h] 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 â\8a¢ W1 â¬\88[h] W2 &
+ â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â¬\88[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 â\8a¢ V1 â\9e¡[h] V & ⬆*[1] V ≡ V2 &
- â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 & â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡[h] T2 &
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V & ⬆*[1] V ≡ V2 &
+ â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 & â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â¬\88[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 â\8a¢ â\93\9dV1.U1 â\9e¡[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 â\8a¢ U1 â\9e¡[h] T2 &
+lemma cpx_inv_cast1: â\88\80h,G,L,V1,U1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV1.U1 â¬\88[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 â\8a¢ U1 â¬\88[h] T2 &
U2 = ⓝV2.T2
- | â¦\83G, Lâ¦\84 â\8a¢ U1 â\9e¡[h] U2
- | â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] U2.
+ | â¦\83G, Lâ¦\84 â\8a¢ U1 â¬\88[h] U2
+ | â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[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-.
(* 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 â\9e¡[h] T → ∀p.
- â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡[h] ⓑ{p,I}V2.T2 &
+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 â¬\88[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] ⓑ{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/
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