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.
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 →
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/
+ 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_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¬\86*[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-.
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-.
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 *)
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)
- ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â\9e¡[n, h] T â\86\92 Q n G (L.â\93\93V) T1 T →
- ⬆*[1] T2 ≘ T → Q n G L (+ⓓV.T1) T2
+ ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â¬\86*[1] T â\89\98 T1 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[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 →
| #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