+lemma pippo (h) (gv) (lv) (T): ●[gv,lv]T = ⟦T⟧{TM h}[gv,lv].
+// qed.
+
+lemma tm_mi (h) (gv1) (gv2) (lv1) (lv2) (p) (W) (T):
+ ⟦W⟧[gv1,lv1] ≗{TM h} ⟦W⟧[gv2,lv2] →
+ (∀d. ⟦T⟧[gv1,⫯[0←d]lv1] ≗ ⟦T⟧[gv2,⫯[0←d]lv2]) →
+ ⟦ⓛ{p}W.T⟧[gv1,lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2,lv2].
+#h #gv1 #gv2 #lv1 #lv2 #p #W #T #HW #HT
+>tm_ti_bind >tm_ti_bind
+@(cpcs_bind1 … HW)
+
+
+
+<pippo in ⊢ (????%?); >(mf_comp … T) in ⊢ (????%?);
+[2: @@tm_vpush_vlift_join_O
+
+<pippo in ⊢ (????%?);
+
+lapply (HT (#0)) -HT #HT
+*)
+
+lemma tm_md (h) (p) (gv) (lv) (V) (T):
+ ⓓ{p}V.⟦T⟧{TM h}[⇡[0]gv,⇡[0←#0]lv] ≗{TM h} V⊕{TM h}[p]⟦T⟧{TM h}[gv,⫯{TM h}[0←V]lv].
+#h #p #gv #lv #V #T
+>tm_co_rw >(mf_lifts_basic_SO_dx T 0)
+>(mf_comp … T) in ⊢ (???%);
+[2: @tm_vpush_vlift_join_O |4: @exteq_refl |3,5: skip ]
+/6 width=4 by mf_delta_drops, cpcs_bind1, cpc_cpcs, drops_refl, or_introl/
+qed.
+
+lemma tm_mz (h) (V) (T): V ⊕[Ⓣ] T ≗{TM h} T.
+/4 width=3 by cpc_cpcs, cpm_zeta, or_introl/ qed.
+
+lemma tm_me (h) (gv) (lv) (U) (T):
+ ⟦ⓝU.T⟧[gv,lv] ≗{TM h} ⟦T⟧[gv,lv].
+/4 width=1 by cpc_cpcs, cpm_eps, or_introl/ qed.
+
+lemma tm_mb (h) (p) (gv) (lv) (d) (W) (T):
+ d@⟦ⓛ{p}W.T⟧[gv,lv] ≗{TM h} d⊕[p]⟦T⟧[gv,⫯[0←d]lv].
+#h #p #gv #lv #d #W #T
+@cpcs_repl [5: @tm_md |4: /4 width=2 by cpc_cpcs, cpm_beta, or_intror/ |1,2: skip ]
+/5 width=1 by cpcs_bind1, cpc_cpcs, cpm_eps, or_introl/
+qed.
+
+lemma ti_mh (h) (p) (d1) (d2) (d3):
+ d1@(d2⊕[p]d3) ≗{TM h} d2⊕[p](d1@d3).
+/4 width=3 by cpc_cpcs, cpm_theta, or_introl/ qed.