1 include "basic_2/computation/cpxs_lift.ma".
2 include "basic_2/multiple/fqus_fqus.ma".
4 (* Advanced properties ******************************************************)
6 lemma lstas_cpxs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
7 ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
8 #h #o #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
9 [ /3 width=3 by cpxs_sort, da_inv_sort/
10 | #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
11 elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
12 lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpxs_delta/
13 | #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
14 elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
15 lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
16 #HV1 #H destruct lapply (le_plus_to_le_c … Hd21) -Hd21
17 /3 width=7 by cpxs_delta/
18 | /4 width=3 by cpxs_bind_dx, da_inv_bind/
19 | /4 width=3 by cpxs_flat_dx, da_inv_flat/
20 | /4 width=3 by cpxs_eps, da_inv_flat/
24 (* Properties on supclosure *************************************************)
26 lemma fqu_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
27 ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
28 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
29 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
30 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
31 #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
34 lemma fquq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
35 ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
36 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
37 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
38 [ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
39 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
43 lemma fquq_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
44 ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
45 ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
46 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
47 /3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
49 lemma fqup_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
50 ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
51 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
52 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
53 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
54 #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
57 lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
58 ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
59 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
60 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
61 [ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
62 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
66 lemma fqus_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
67 ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
68 ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
69 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
70 /3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.