(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_6_6.ma".
+include "ground_2/xoa/ex_7_7.ma".
+include "ground_2/xoa/or_4.ma".
include "ground_2/insert_eq/insert_eq_0.ma".
include "basic_2/rt_transition/cpm.ma".
⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 →
⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
#h * /2 width=1 by cpm_cast, cpm_appl/
-qed.
+qed.
(* Basic_1: was: pr2_head_1 *)
lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[h] T2 →
∨∨ T2 = ⓪{J}
- | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¬\86*[1] V2 ≘ T2 &
+ | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â\87§*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[h] T & â¬\86*[1] T ≘ T2 &
+ | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[h] T & â\87§*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
[2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
[ #n #_ #_ #H destruct
| #n #K #V1 #V2 #_ #_ #_ #_ #H destruct
-]
+]
qed-.
(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[h] T2 →
∨∨ T2 = #0
- | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â¬\86*[1] V2 ≘ T2 &
+ | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 & â\87§*[1] V2 ≘ T2 &
L = K.ⓓV1.
#h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[h] T2 →
∨∨ T2 = #(↑i)
- | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[h] T & â¬\86*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[h] T & â\87§*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
qed-.
| ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
⦃G,L.ⓛW1⦄ ⊢ 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 â\8a¢ V1 â\9e¡[h] V & â¬\86*[1] V ≘ V2 &
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V & â\87§*[1] V ≘ V2 &
⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
U1 = ⓓ{p}W1.T1 &
U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
(∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
(∀G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
- â¬\86*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
+ â\87§*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
- â¬\86*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
+ â\87§*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ 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)
) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ 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¬\86*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[h] T2 →
+ ) â\86\92 (â\88\80G,L,V,T1,T,T2. â\87§*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[h] T2 →
Q G L T T2 → Q G L (+ⓓV.T1) T2
) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
Q G L (ⓝV.T1) T2
Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
- â¬\86*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ â\87§*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2