(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_4_1.ma".
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/xoa/ex_5_6.ma".
+include "ground_2/xoa/ex_6_7.ma".
include "basic_2/notation/relations/pred_6.ma".
include "basic_2/notation/relations/pred_5.ma".
include "basic_2/rt_transition/cpg.ma".
(* Basic_2A1: includes: cpr *)
definition cpm (h) (G) (L) (n): relation2 term term ≝
- λT1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[eq_t, c, h] T2.
+ λT1,T2. ∃∃c. 𝐑𝐓⦃n,c⦄ & ⦃G,L⦄ ⊢ T1 ⬈[eq_t,c,h] T2.
interpretation
"t-bound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
-lemma cpm_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[1, h] ⋆(next h s).
+lemma cpm_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
/2 width=3 by cpg_ess, ex2_intro/ qed.
-lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
- â¬\86*[1] V2 â\89\98 W2 â\86\92 â¦\83G, K.â\93\93V1â¦\84 â\8a¢ #0 â\9e¡[n, h] W2.
+lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
+ â\87§*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\93V1â¦\84 â\8a¢ #0 â\9e¡[n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_delta, ex2_intro/
qed.
-lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
- â¬\86*[1] V2 â\89\98 W2 â\86\92 â¦\83G, K.â\93\9bV1â¦\84 â\8a¢ #0 â\9e¡[â\86\91n, h] W2.
+lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
+ â\87§*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\9bV1â¦\84 â\8a¢ #0 â\9e¡[â\86\91n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_ell, ex2_intro, isrt_succ/
qed.
-lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T →
- â¬\86*[1] T â\89\98 U â\86\92 â¦\83G, K.â\93\98{I}â¦\84 â\8a¢ #â\86\91i â\9e¡[n, h] U.
+lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T →
+ â\87§*[1] T â\89\98 U â\86\92 â¦\83G,K.â\93\98{I}â¦\84 â\8a¢ #â\86\91i â\9e¡[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.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{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.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐ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.
- ⦃G, L⦄ ⊢ U1 ➡[n, h] U2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n, h] ⓝU2.T2.
+ ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓝ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,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[n, h] T →
- ⬆*[1] T2 ≘ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
-#n #h #G #L #V #T1 #T #T2 *
+lemma cpm_zeta (n) (h) (G) (L):
+ ∀T1,T. ⇧*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ +ⓓ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: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2.
+lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ ⓝ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: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[↑n, h] V2.
+lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ ⓝ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.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n,h] ⓓ{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.
- ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V → ⇧*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n,h] ⓓ{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.
lemma cpr_refl: ∀h,G,L. reflexive … (cpm h G L 0).
/3 width=3 by cpg_refl, ex2_intro/ qed.
+(* Advanced properties ******************************************************)
+
+lemma cpm_sort (h) (G) (L):
+ ∀n. n ≤ 1 → ∀s. ⦃G,L⦄ ⊢ ⋆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: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 →
+lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[n,h] T2 →
∨∨ T2 = ⓪{J} ∧ n = 0
- | ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≘ T2 &
- L = K.ⓛV1 & J = LRef 0 & n = ↑k
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
+ L = K.ⓛV1 & J = LRef 0 & n = ↑m
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #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/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
/4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
- elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+ elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
/4 width=9 by or5_intro3, ex5_4_intro, ex2_intro/
| #I #K #V2 #i #HV2 #HVT2 #H1 #H2 destruct
/4 width=8 by or5_intro4, ex4_4_intro, ex2_intro/
]
qed-.
-lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[n,h] T2 →
- â\88¨â\88¨ T2 = â\8b\86s â\88§ n = 0
- | T2 = ⋆(next h s) ∧ n = 1.
-#n #h #G #L #T2 #s * #c #Hc #H elim (cpg_inv_sort1 … H) -H *
-#H1 #H2 destruct
-/4 width=1 by isrt_inv_01, isrt_inv_00, or_introl, or_intror, conj/
+lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] T2 →
+ â\88§â\88§ T2 = â\8b\86(((next h)^n) s) & n â\89¤ 1.
+#n #h #G #L #T2 #s * #c #Hc #H
+elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
+[ lapply (isrt_inv_00 … Hc) | lapply (isrt_inv_01 … Hc) ] -Hc
+#H destruct /2 width=1 by conj/
qed-.
-lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 →
+lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 →
∨∨ T2 = #0 ∧ n = 0
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
L = K.ⓓV1
- | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≘ T2 &
- L = K.ⓛV1 & n = ↑k.
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ 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/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
/4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
- elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+ elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
/4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
]
qed-.
-lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[n, h] T2 →
+lemma cpm_inv_zero1_unit (n) (h) (I) (K) (G):
+ ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #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/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 →
∨∨ T2 = #(↑i) ∧ n = 0
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ{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_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = §l ∧ n = 0.
+lemma cpm_inv_lref1_ctop (n) (h) (G):
+ ∀X2,i. ⦃G,⋆⦄ ⊢ #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/
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpm_inv_lref1 … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G,L⦄ ⊢ §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/
+#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
qed-.
(* Basic_2A1: includes: cpr_inv_bind1 *)
-lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓑ{p,I}V2.T2
- | â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[n, h] T & â¬\86*[1] U2 â\89\98 T &
+ | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡[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
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
/4 width=5 by ex3_2_intro, ex2_intro, or_introl/
-| #cT #T2 #HT12 #HUT2 #H1 #H2 #H3 destruct
- /5 width=3 by isrt_inv_plus_O_dx, ex4_intro, ex2_intro, or_intror/
+| #cT #T2 #HT21 #HTU2 #H1 #H2 #H3 destruct
+ /5 width=5 by isrt_inv_plus_O_dx, ex4_intro, ex2_intro, or_intror/
]
qed-.
(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
(* Basic_2A1: includes: cpr_inv_abbr1 *)
-lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T & ⬆*[1] U2 ≘ T & p = true.
-#n #h #p #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_abbr1 … H) -H *
-[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
- elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
- elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
- /4 width=5 by ex3_2_intro, ex2_intro, or_introl/
-| #cT #T2 #HT12 #HUT2 #H1 #H2 destruct
- /5 width=3 by isrt_inv_plus_O_dx, ex3_intro, ex2_intro, or_intror/
+ | ∃∃T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ 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/
+| * /3 width=3 by ex3_intro, or_intror/
]
qed-.
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
(* Basic_2A1: includes: cpr_inv_abst1 *)
-lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡[n,h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓛ{p}V2.T2.
-#n #h #p #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_abst1 … H) -H
-#cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
-elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
-elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
-/3 width=5 by ex3_2_intro, ex2_intro/
+#n #h #p #G #L #V1 #T1 #U2 #H
+elim (cpm_inv_bind1 … H) -H
+[ /3 width=1 by or_introl/
+| * #T #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma cpm_inv_abst_bi: ∀n,h,p1,p2,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}V1.T1 ➡[n,h] ⓛ{p2}V2.T2 →
+ ∧∧ ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ 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/
qed-.
(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
(* Basic_2A1: includes: cpr_inv_appl1 *)
-lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
+lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,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 ➡[n, h] T2 &
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,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 ➡[n, h] 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 ➡[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
]
qed-.
-lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
+lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ➡[n, h] U2
- | ∃∃k. ⦃G, L⦄ ⊢ V1 ➡[k, h] U2 & n = ↑k.
+ | ⦃G,L⦄ ⊢ U1 ➡[n,h] U2
+ | ∃∃m. ⦃G,L⦄ ⊢ 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
| #cU #U12 #H destruct
/4 width=3 by isrt_inv_plus_O_dx, or3_intro1, ex2_intro/
| #cU #H12 #H destruct
- elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+ elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
/4 width=3 by or3_intro2, ex2_intro/
]
qed-.
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: cpr_fwd_bind1_minus *)
-lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 &
+lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n,h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{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/
(* Basic eliminators ********************************************************)
-lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n, ↕*c1 ∨ c2⦄ →
- ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃n, c2⦄.
-#n #c1 #c2 #H
-elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
-elim (isrt_inv_shift … Hc1) -Hc1 #Hc1 * -n1
-/2 width=1 by conj/
-qed-.
-
-lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → eq_t c1 c2 →
- ∧∧ 𝐑𝐓⦃n, c1⦄ & 𝐑𝐓⦃n, c2⦄.
-#n #c1 #c2 #H #Hc12
-elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
-lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H
-<(isrt_inj … H … Hc2) -Hc2
-<idempotent_max /2 width=1 by conj/
-qed-.
-
-lemma cpm_ind (h): ∀R:relation5 nat genv lenv term term.
- (∀I,G,L. R 0 G L (⓪{I}) (⓪{I})) →
- (∀G,L,s. R 1 G L (⋆s) (⋆(next h s))) →
- (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → R n G K V1 V2 →
- ⬆*[1] V2 ≘ W2 → R n G (K.ⓓV1) (#0) W2
- ) → (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → R n G K V1 V2 →
- ⬆*[1] V2 ≘ W2 → R (↑n) G (K.ⓛV1) (#0) W2
- ) → (∀n,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → R n G K (#i) T →
- ⬆*[1] T ≘ U → R n G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
- R 0 G L V1 V2 → R n G (L.ⓑ{I}V1) T1 T2 → R n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- R 0 G L V1 V2 → R n G L T1 T2 → R n G L (ⓐV1.T1) (ⓐV2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- R n G L V1 V2 → R n G L T1 T2 → R n G L (ⓝV1.T1) (ⓝV2.T2)
- ) → (∀n,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[n, h] T → R n G (L.ⓓV) T1 T →
- ⬆*[1] T2 ≘ T → R n G L (+ⓓV.T1) T2
- ) → (∀n,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- R n G L T1 T2 → R n G L (ⓝV.T1) T2
- ) → (∀n,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 →
- R n G L V1 V2 → R (↑n) G L (ⓝV1.T) V2
- ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
- R 0 G L V1 V2 → R 0 G L W1 W2 → R n G (L.ⓛW1) T1 T2 →
- R n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
- R 0 G L V1 V → R 0 G L W1 W2 → R n G (L.ⓓW1) T1 T2 →
- ⬆*[1] V ≘ V2 → R n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
+ (∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
+ (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
+ (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
+ ⇧*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
+ ) → (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
+ ⇧*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
+ ) → (∀n,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → Q n G K (#i) T →
+ ⇧*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
+ ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ 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)
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
+ ) → (∀n,G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
+ Q n G L T T2 → Q n G L (+ⓓV.T1) T2
+ ) → (∀n,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
+ ) → (∀n,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 →
+ Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
+ ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ 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)
+ ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ 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)
) →
- ∀n,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → R n G L T1 T2.
-#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
+ ∀n,G,L,T1,T2. ⦃G,L⦄ ⊢ 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
[ #I #G #L #n #H <(isrt_inv_00 … H) -H //
| #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #n #H
elim (isrt_inv_max_eq_t … H) -H // #HcV #HcT
/3 width=3 by ex2_intro/
-| #c #G #L #V #T1 #T2 #T #HT12 #HT2 #IH #n #H
+| #c #G #L #V #T1 #T #T2 #HT1 #HT2 #IH #n #H
lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
/3 width=4 by ex2_intro/
| #c #G #L #U #T1 #T2 #HT12 #IH #n #H
projects / helm.git / blobdiff