+/3 width=5 by mq, mr/ qed-.
+
+lemma seq_canc_sn (M): is_model M → left_cancellable … (sq M).
+/3 width=3 by seq_trans, seq_sym/ qed-.
+
+lemma seq_canc_dx (M): is_model M → right_cancellable … (sq M).
+/3 width=3 by seq_trans, seq_sym/ qed-.
+
+lemma co_inv_dx (M): is_model M → is_injective M →
+ ∀p,d1,d3,d2,d4. d1⊕[p]d2 ≗{M} d3⊕[p]d4 → d2 ≗ d4.
+#M #H1M #H2M * #d1 #d3 #d2 #d4 #Hd
+/3 width=5 by mj, mz, mq/
+qed-.
+
+lemma ti_lref_vpush_eq (M): is_model M →
+ ∀gv,lv,d,i. ⟦#i⟧[gv,⫯[i←d]lv] ≗{M} d.
+#M #HM #gv #lv #d #i
+@(seq_trans … HM) [2: @ml // | skip ]
+>vpush_eq /2 width=1 by mr/
+qed.
+
+lemma ti_lref_vpush_gt (M): is_model M →
+ ∀gv,lv,d,i. ⟦#(↑i)⟧[gv,⫯[0←d]lv] ≗{M} ⟦#i⟧[gv,lv].
+#M #HM #gv #lv #d #i
+@(mq … HM) [4,5: @(seq_sym … HM) /2 width=2 by ml/ |1,2: skip ]
+>vpush_gt /2 width=1 by mr/
+qed.
+
+(* Basic Forward lemmas *****************************************************)
+
+lemma ti_fwd_mx_dx (M): is_model M → is_injective M →
+ ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2.
+ ⟦ⓛ[p]W1.T1⟧[gv1,lv1] ≗ ⟦ⓛ[p]W2.T2⟧[gv2,lv2] →
+ ∀d. ⟦T1⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←d]lv2].
+#M #H1M #H2M #p #gv1 #gv2 #lv1 #lv2 #W1 #W2 #T1 #T2 #H12 #d
+@(co_inv_dx … p d d)
+/4 width=5 by mb, mp, mq, mr/
+qed-.
+
+lemma ti_fwd_abbr_dx (M): is_model M → is_injective M →
+ ∀p,gv1,gv2,lv1,lv2,V1,V2,T1,T2.
+ ⟦ⓓ[p]V1.T1⟧[gv1,lv1] ≗ ⟦ⓓ[p]V2.T2⟧[gv2,lv2] →
+ ⟦T1⟧{M}[gv1,⫯[0←⟦V1⟧[gv1,lv1]]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←⟦V2⟧[gv2,lv2]]lv2].
+#M #H1M #H2M #p #gv1 #gv2 #lv1 #lv2 #V1 #V2 #T1 #T2 #H12
+@(co_inv_dx … p (⟦V1⟧[gv1,lv1]) (⟦V2⟧[gv2,lv2]))
+/3 width=5 by md, mq/
+qed-.