(* Basic_2A1: includes: cpr *)
definition cpm (h) (G) (L) (n): relation2 term term ≝
- λT1,T2. â\88\83â\88\83c. ð\9d\90\91ð\9d\90\93â¦\83n,câ¦\84 & â¦\83G,Lâ¦\84 ⊢ T1 ⬈[eq_t,c,h] T2.
+ λT1,T2. â\88\83â\88\83c. ð\9d\90\91ð\9d\90\93â\9dªn,câ\9d« & â\9dªG,Lâ\9d« ⊢ T1 ⬈[eq_t,c,h] T2.
interpretation
"t-bound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
-lemma cpm_ess: â\88\80h,G,L,s. â¦\83G,Lâ¦\84 ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
+lemma cpm_ess: â\88\80h,G,L,s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
/2 width=3 by cpg_ess, ex2_intro/ qed.
-lemma cpm_delta: â\88\80n,h,G,K,V1,V2,W2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[n,h] V2 →
- â\87§*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\93V1â¦\84 ⊢ #0 ➡[n,h] W2.
+lemma cpm_delta: â\88\80n,h,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ➡[n,h] V2 →
+ â\87§*[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\93V1â\9d« ⊢ #0 ➡[n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_delta, ex2_intro/
qed.
-lemma cpm_ell: â\88\80n,h,G,K,V1,V2,W2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[n,h] V2 →
- â\87§*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\9bV1â¦\84 ⊢ #0 ➡[↑n,h] W2.
+lemma cpm_ell: â\88\80n,h,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ➡[n,h] V2 →
+ â\87§*[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\9bV1â\9d« ⊢ #0 ➡[↑n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_ell, ex2_intro, isrt_succ/
qed.
-lemma cpm_lref: â\88\80n,h,I,G,K,T,U,i. â¦\83G,Kâ¦\84 ⊢ #i ➡[n,h] T →
- â\87§*[1] T â\89\98 U â\86\92 â¦\83G,K.â\93\98{I}â¦\84 ⊢ #↑i ➡[n,h] U.
+lemma cpm_lref: â\88\80n,h,I,G,K,T,U,i. â\9dªG,Kâ\9d« ⊢ #i ➡[n,h] T →
+ â\87§*[1] T â\89\98 U â\86\92 â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ➡[n,h] U.
#n #h #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.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ T1 ➡[n,h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡[n,h] â\93\91{p,I}V2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ➡[n,h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â\9e¡[n,h] â\93\91[p,I]V2.T2.
#n #h #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.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ➡[n,h] T2 →
- â¦\83G,Lâ¦\84 ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡[n,h] T2 →
+ â\9dªG,Lâ\9d« ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2.
#n #h #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.
- â¦\83G,Lâ¦\84 â\8a¢ U1 â\9e¡[n,h] U2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ➡[n,h] T2 →
- â¦\83G,Lâ¦\84 ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ U1 â\9e¡[n,h] U2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡[n,h] T2 →
+ â\9dªG,Lâ\9d« ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2.
#n #h #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):
- â\88\80T1,T. â\87§*[1] T â\89\98 T1 â\86\92 â\88\80T2. â¦\83G,Lâ¦\84 ⊢ T ➡[n,h] T2 →
- â\88\80V. â¦\83G,Lâ¦\84 ⊢ +ⓓV.T1 ➡[n,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 ➡[n,h] T2 →
+ â\88\80V. â\9dªG,Lâ\9d« ⊢ +ⓓV.T1 ➡[n,h] T2.
#n #h #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: â\88\80n,h,G,L,V,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓝV.T1 ➡[n,h] T2.
+lemma cpm_eps: â\88\80n,h,G,L,V,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝV.T1 ➡[n,h] T2.
#n #h #G #L #V #T1 #T2 *
/3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
qed.
-lemma cpm_ee: â\88\80n,h,G,L,V1,V2,T. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓝV1.T ➡[↑n,h] V2.
+lemma cpm_ee: â\88\80n,h,G,L,V1,V2,T. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝV1.T ➡[↑n,h] V2.
#n #h #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.
- 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 ⊢ T1 ➡[n,h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{p}W1.T1 â\9e¡[n,h] â\93\93{p}ⓝW2.V2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h] W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ➡[n,h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\9b[p]W1.T1 â\9e¡[n,h] â\93\93[p]ⓝW2.V2.T2.
#n #h #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.
- â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[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 ➡[n,h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{p}W1.T1 â\9e¡[n,h] â\93\93{p}W2.ⓐV2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[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 ➡[n,h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\93[p]W1.T1 â\9e¡[n,h] â\93\93[p]W2.ⓐV2.T2.
#n #h #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):
- â\88\80n. n â\89¤ 1 â\86\92 â\88\80s. â¦\83G,Lâ¦\84 ⊢ ⋆s ➡[n,h] ⋆((next h)^n s).
+ â\88\80n. n â\89¤ 1 â\86\92 â\88\80s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡[n,h] ⋆((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: â\88\80n,h,J,G,L,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93ª{J} ➡[n,h] T2 →
- ∨∨ T2 = ⓪{J} ∧ n = 0
+lemma cpm_inv_atom1: â\88\80n,h,J,G,L,T2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[J] ➡[n,h] T2 →
+ ∨∨ T2 = ⓪[J] ∧ n = 0
| ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
- | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
+ | â\88\83â\88\83K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
+ | â\88\83â\88\83m,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & n = ↑m
- | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 ⊢ #i ➡[n,h] T & ⇧*[1] T ≘ T2 &
- L = K.ⓘ{I} & J = LRef (↑i).
+ | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #i ➡[n,h] 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 *
[ #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/
]
qed-.
-lemma cpm_inv_sort1: â\88\80n,h,G,L,T2,s. â¦\83G,Lâ¦\84 ⊢ ⋆s ➡[n,h] T2 →
+lemma cpm_inv_sort1: â\88\80n,h,G,L,T2,s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡[n,h] T2 →
∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1.
#n #h #G #L #T2 #s * #c #Hc #H
elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
#H destruct /2 width=1 by conj/
qed-.
-lemma cpm_inv_zero1: â\88\80n,h,G,L,T2. â¦\83G,Lâ¦\84 ⊢ #0 ➡[n,h] T2 →
+lemma cpm_inv_zero1: â\88\80n,h,G,L,T2. â\9dªG,Lâ\9d« ⊢ #0 ➡[n,h] T2 →
∨∨ T2 = #0 ∧ n = 0
- | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
+ | â\88\83â\88\83K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
L = K.ⓓV1
- | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
+ | â\88\83â\88\83m,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
L = K.ⓛV1 & n = ↑m.
#n #h #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/
qed-.
lemma cpm_inv_zero1_unit (n) (h) (I) (K) (G):
- â\88\80X2. â¦\83G,K.â\93¤{I}â¦\84 ⊢ #0 ➡[n,h] X2 → ∧∧ X2 = #0 & n = 0.
+ â\88\80X2. â\9dªG,K.â\93¤[I]â\9d« ⊢ #0 ➡[n,h] X2 → ∧∧ X2 = #0 & n = 0.
#n #h #I #G #K #X2 #H
elim (cpm_inv_zero1 … H) -H *
[ #H1 #H2 destruct /2 width=1 by conj/
]
qed.
-lemma cpm_inv_lref1: â\88\80n,h,G,L,T2,i. â¦\83G,Lâ¦\84 ⊢ #↑i ➡[n,h] T2 →
+lemma cpm_inv_lref1: â\88\80n,h,G,L,T2,i. â\9dªG,Lâ\9d« ⊢ #↑i ➡[n,h] T2 →
∨∨ T2 = #(↑i) ∧ n = 0
- | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[n,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\9e¡[n,h] T & â\87§*[1] T â\89\98 T2 & L = K.â\93\98[I].
#n #h #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
qed-.
lemma cpm_inv_lref1_ctop (n) (h) (G):
- â\88\80X2,i. â¦\83G,â\8b\86â¦\84 ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0.
+ â\88\80X2,i. â\9dªG,â\8b\86â\9d« ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0.
#n #h #G #X2 * [| #i ] #H
[ elim (cpm_inv_zero1 … H) -H *
[ #H1 #H2 destruct /2 width=1 by conj/
]
qed.
-lemma cpm_inv_gref1: â\88\80n,h,G,L,T2,l. â¦\83G,Lâ¦\84 ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
+lemma cpm_inv_gref1: â\88\80n,h,G,L,T2,l. â\9dªG,Lâ\9d« ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
#n #h #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: â\88\80n,h,p,I,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 ➡[n,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,L.â\93\91{I}V1â¦\84 ⊢ T1 ➡[n,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 ➡[n,h] U2 &
+lemma cpm_inv_bind1: â\88\80n,h,p,I,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 ➡[n,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,L.â\93\91[I]V1â\9d« ⊢ T1 ➡[n,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 ➡[n,h] U2 &
p = true & I = Abbr.
#n #h #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
(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
(* Basic_2A1: includes: cpr_inv_abbr1 *)
-lemma cpm_inv_abbr1: â\88\80n,h,p,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\93{p}V1.T1 ➡[n,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,L.â\93\93V1â¦\84 ⊢ T1 ➡[n,h] T2 &
- U2 = ⓓ{p}V2.T2
- | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 ⊢ T ➡[n,h] U2 & p = true.
+lemma cpm_inv_abbr1: â\88\80n,h,p,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\93[p]V1.T1 ➡[n,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,L.â\93\93V1â\9d« ⊢ T1 ➡[n,h] T2 &
+ U2 = ⓓ[p]V2.T2
+ | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ➡[n,h] U2 & p = true.
#n #h #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
[ /3 width=1 by or_introl/
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
(* Basic_2A1: includes: cpr_inv_abst1 *)
-lemma cpm_inv_abst1: â\88\80n,h,p,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\9b{p}V1.T1 ➡[n,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 ⊢ T1 ➡[n,h] T2 &
- U2 = ⓛ{p}V2.T2.
+lemma cpm_inv_abst1: â\88\80n,h,p,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\9b[p]V1.T1 ➡[n,h] U2 →
+ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 & â\9dªG,L.â\93\9bV1â\9d« ⊢ T1 ➡[n,h] T2 &
+ U2 = ⓛ[p]V2.T2.
#n #h #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
[ /3 width=1 by or_introl/
]
qed-.
-lemma cpm_inv_abst_bi: â\88\80n,h,p1,p2,G,L,V1,V2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93\9b{p1}V1.T1 â\9e¡[n,h] â\93\9b{p2}V2.T2 →
- â\88§â\88§ â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¦\83G,L.â\93\9bV1â¦\84 ⊢ T1 ➡[n,h] T2 & p1 = p2.
+lemma cpm_inv_abst_bi: â\88\80n,h,p1,p2,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ â\93\9b[p1]V1.T1 â\9e¡[n,h] â\93\9b[p2]V2.T2 →
+ â\88§â\88§ â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 & â\9dªG,L.â\93\9bV1â\9d« ⊢ T1 ➡[n,h] T2 & p1 = p2.
#n #h #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/
(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
(* Basic_2A1: includes: cpr_inv_appl1 *)
-lemma cpm_inv_appl1: â\88\80n,h,G,L,V1,U1,U2. â¦\83G,Lâ¦\84 ⊢ ⓐ V1.U1 ➡[n,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 ➡[n,h] T2 &
+lemma cpm_inv_appl1: â\88\80n,h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓐ V1.U1 ➡[n,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 ➡[n,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 ⊢ W1 ➡[h] W2 &
- â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ➡[n,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\9e¡[h] W2 & â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ➡[n,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\9e¡[h] V2 & â\9dªG,Lâ\9d« ⊢ W1 ➡[h] W2 &
+ â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ➡[n,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\9e¡[h] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ➡[n,h] 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 *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
]
qed-.
-lemma cpm_inv_cast1: â\88\80n,h,G,L,V1,U1,U2. â¦\83G,Lâ¦\84 ⊢ ⓝV1.U1 ➡[n,h] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ➡[n,h] T2 &
+lemma cpm_inv_cast1: â\88\80n,h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓝV1.U1 ➡[n,h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ➡[n,h] T2 &
U2 = ⓝV2.T2
- | â¦\83G,Lâ¦\84 ⊢ U1 ➡[n,h] U2
- | â\88\83â\88\83m. â¦\83G,Lâ¦\84 ⊢ V1 ➡[m,h] U2 & n = ↑m.
+ | â\9dªG,Lâ\9d« ⊢ U1 ➡[n,h] U2
+ | â\88\83â\88\83m. â\9dªG,Lâ\9d« ⊢ V1 ➡[m,h] U2 & n = ↑m.
#n #h #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
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: cpr_fwd_bind1_minus *)
-lemma cpm_fwd_bind1_minus: â\88\80n,h,I,G,L,V1,T1,T. â¦\83G,Lâ¦\84 â\8a¢ -â\93\91{I}V1.T1 ➡[n,h] T → ∀p.
- â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡[n,h] â\93\91{p,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
+lemma cpm_fwd_bind1_minus: â\88\80n,h,I,G,L,V1,T1,T. â\9dªG,Lâ\9d« â\8a¢ -â\93\91[I]V1.T1 ➡[n,h] T → ∀p.
+ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â\9e¡[n,h] â\93\91[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
/3 width=4 by ex2_2_intro, ex2_intro/
qed-.
(* Basic eliminators ********************************************************)
lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
- (∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
+ (∀I,G,L. Q 0 G L (⓪[I]) (⓪[I])) →
(∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
- (â\88\80n,G,K,V1,V2,W2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
+ (â\88\80n,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
⇧*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
- ) â\86\92 (â\88\80n,G,K,V1,V2,W2. â¦\83G,Kâ¦\84 ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
+ ) â\86\92 (â\88\80n,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
⇧*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
- ) â\86\92 (â\88\80n,I,G,K,T,U,i. â¦\83G,Kâ¦\84 ⊢ #i ➡[n,h] T → Q n G K (#i) T →
- ⇧*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
- ) â\86\92 (â\88\80n,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 ⊢ T1 ➡[n,h] 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)
- ) â\86\92 (â\88\80n,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 ➡[n,h] T2 →
+ ) â\86\92 (â\88\80n,I,G,K,T,U,i. â\9dªG,Kâ\9d« ⊢ #i ➡[n,h] T → Q n G K (#i) T →
+ ⇧*[1] T ≘ U → Q n G (K.ⓘ[I]) (#↑i) (U)
+ ) â\86\92 (â\88\80n,p,I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ➡[n,h] 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)
+ ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
- ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ➡[n,h] T2 →
+ ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡[n,h] T2 →
Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
- ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â\87§*[1] T â\89\98 T1 â\86\92 â¦\83G,Lâ¦\84 ⊢ T ➡[n,h] T2 →
+ ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â\87§*[1] T â\89\98 T1 â\86\92 â\9dªG,Lâ\9d« ⊢ T ➡[n,h] T2 →
Q n G L T T2 → Q n G L (+ⓓV.T1) T2
- ) â\86\92 (â\88\80n,G,L,V,T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ➡[n,h] T2 →
+ ) â\86\92 (â\88\80n,G,L,V,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡[n,h] T2 →
Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
- ) â\86\92 (â\88\80n,G,L,V1,V2,T. â¦\83G,Lâ¦\84 ⊢ V1 ➡[n,h] V2 →
+ ) â\86\92 (â\88\80n,G,L,V1,V2,T. â\9dªG,Lâ\9d« ⊢ V1 ➡[n,h] V2 →
Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
- ) â\86\92 (â\88\80n,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 ⊢ T1 ➡[n,h] T2 →
+ ) â\86\92 (â\88\80n,p,G,L,V1,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h] W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ➡[n,h] 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)
- ) â\86\92 (â\88\80n,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¦\83G,Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 â\86\92 â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ➡[n,h] T2 →
+ Q n G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+ ) â\86\92 (â\88\80n,p,G,L,V1,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h] W2 â\86\92 â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ➡[n,h] 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)
+ ⇧*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
) →
- â\88\80n,G,L,T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2.
+ â\88\80n,G,L,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡[n,h] 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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