(* Basic properties *********************************************************)
-lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed.
lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed.
-lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄.
+lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G,L,②{I}V.T⦄ ⊐+[b] ⦃G,L,V⦄.
/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
-lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
+lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G,L.ⓑ{I}V,T⦄.
/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
-lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄.
+lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G,L.ⓧ,T⦄.
/3 width=1 by fqu_clear, fqu_fqup/ qed.
-lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄.
+lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G,L,ⓕ{I}V.T⦄ ⊐+[b] ⦃G,L,T⦄.
/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
-lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄.
+lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G,L,V2⦄.
/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
-lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
+lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G,L,ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G,L.ⓑ{I1}V1,T⦄.
/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
-lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
+lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G,L.ⓑ{I2}V2,T⦄.
/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
(* Basic eliminators ********************************************************)
lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
#b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H
@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
qed-.
lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+ (∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
qed-.