X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpx.ma;h=3da67484aaa0d1f427d6cb6decbcb85a7b363226;hp=c7dde663e8dfa0e6fc0ea7d379f44a6461995da9;hb=19a25bf176255055193372554437729a6fa1894c;hpb=632e2b3b8444d363c5f8acb87c3695fc781fe35b

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
index c7dde663e..3da67484a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
@@ -15,53 +15,51 @@
 include "basic_2/notation/relations/predty_5.ma".
 include "basic_2/rt_transition/cpg.ma".
 
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
 
 definition cpx (h): relation4 genv lenv term term ≝
-                    λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[c, h] T2.
+                    λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2.
 
 interpretation
-   "uncounted context-sensitive parallel rt-transition (term)"
+   "unbound context-sensitive parallel rt-transition (term)"
    'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma cpx_atom: ∀h,I,G,L. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] ⓪{I}.
-/2 width=2 by cpg_atom, ex_intro/ qed.
-
 (* Basic_2A1: was: cpx_st *)
 lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
 /2 width=2 by cpg_ess, ex_intro/ qed.
 
 lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
-                 ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
+                 ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
 #h * #G #K #V1 #V2 #W2 *
 /3 width=4 by cpg_delta, cpg_ell, ex_intro/
 qed.
 
-lemma cpx_lref: ∀h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
-                ⬆*[1] T ≡ U → ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[h] U.
-#h #I #G #K #V #T #U #i *
+lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
+                ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
+#h #I #G #K #T #U #i *
 /3 width=4 by cpg_lref, ex_intro/
 qed.
 
 lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
-                 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
-                 ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+                ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+                ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
 #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
 /3 width=2 by cpg_bind, ex_intro/
 qed.
 
 lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
-                 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
-                 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
-#h #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
-/3 width=2 by cpg_flat, ex_intro/
+                ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+                ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
+#h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
+/3 width=5 by cpg_appl, cpg_cast, ex_intro/
 qed.
 
-lemma cpx_zeta: ∀h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T →
-                ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
-#h #G #L #V #T1 #T #T2 *
+lemma cpx_zeta (h) (G) (L):
+               ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 →
+               ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
+#h #G #L #T1 #T #HT1 #T2 *
 /3 width=4 by cpg_zeta, ex_intro/
 qed.
 
@@ -84,20 +82,28 @@ lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
 qed.
 
 lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
-                 ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
+                 ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
                  ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
                  ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
 #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 * 
 /3 width=4 by cpg_theta, ex_intro/
 qed.
 
+(* Basic_2A1: includes: cpx_atom *)
 lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
-/2 width=2 by ex_intro/ qed.
+/3 width=2 by cpg_refl, ex_intro/ qed.
+
+(* Advanced properties ******************************************************)
 
 lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
-#h #I #G #L #V1 #V2 *
-/3 width=2 by cpg_pair_sn, ex_intro/
+#h * /2 width=2 by cpx_flat, cpx_bind/
+qed.
+
+lemma cpg_cpx (h) (Rt) (c) (G) (L):
+              ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2.
+#h #Rt #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
+/2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
@@ -105,54 +111,52 @@ qed.
 lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
                      ∨∨ T2 = ⓪{J}
                       | ∃∃s. T2 = ⋆(next h s) & J = Sort s
-                      | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
                                      L = K.ⓑ{I}V1 & J = LRef 0
-                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 &
-                                     L = K.ⓑ{I}V & J = LRef (⫯i).
+                      | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
+                                   L = K.ⓘ{I} & J = LRef (↑i).
 #h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
-/4 width=9 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_5_intro, ex4_4_intro, ex2_intro, ex_intro/
+/4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/
 qed-.
 
 lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 →
-                     T2 = ⋆s ∨ T2 = ⋆(next h s).
+                     ∨∨ T2 = ⋆s | T2 = ⋆(next h s).
 #h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
 /2 width=1 by or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 →
-                     T2 = #0 ∨
-                     ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
-                                  L = K.ⓑ{I}V1.
+                     ∨∨ T2 = #0
+                      | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+                                     L = K.ⓑ{I}V1.
 #h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ⬈[h] T2 →
-                     T2 = #(⫯i) ∨
-                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[h] T2 →
+                     ∨∨ T2 = #(↑i)
+                      | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
 #h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
-/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
+/4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
 #h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
 qed-.
 
-lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
-                              U2 = ⓑ{p,I}V2.T2
-                     ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T &
-                          p = true & I = Abbr.
+lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
+                                 U2 = ⓑ{p,I}V2.T2
+                      | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 & 
+                             p = true & I = Abbr.
 #h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
 /4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
-                              U2 = ⓓ{p}V2.T2
-                     ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T & p = true.
+lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
+                                 U2 = ⓓ{p}V2.T2
+                      | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 & p = true.
 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
 /4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
 qed-.
@@ -164,30 +168,13 @@ lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 
 /3 width=5 by ex3_2_intro, ex_intro/
 qed-.
 
-lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
-                                 U2 = ⓕ{I}V2.T2
-                      | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
-                      | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
-                      | ∃∃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
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[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.
-#h #I #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_flat1 … H) -H *
-/4 width=14 by or5_intro0, or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro, ex_intro, conj/
-qed-.
-
 lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
                                  U2 = ⓐV2.T2
                       | ∃∃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
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
                                               ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
                                               U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
@@ -203,6 +190,43 @@ lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
 /4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
 qed-.
 
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
+                          ∨∨ T2 = #0
+                           | ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2.
+#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
+/4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
+qed-.
+
+lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
+                          ∨∨ T2 = #(↑i)
+                           | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2.
+#h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
+/4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
+qed-.
+
+lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+                                 U2 = ⓕ{I}V2.T2
+                      | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
+                      | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
+                      | ∃∃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
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[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.
+#h * #G #L #V1 #U1 #U2 #H
+[ elim (cpx_inv_appl1 … H) -H *
+  /3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
+| elim (cpx_inv_cast1 … H) -H [ * ]
+  /3 width=14 by or5_intro0, or5_intro1, or5_intro2, ex3_2_intro, conj/
+] 
+qed-.
+
 (* Basic forward lemmas *****************************************************)
 
 lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
@@ -211,3 +235,34 @@ lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h
 #h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
 /3 width=4 by ex2_2_intro, ex_intro/
 qed-.
+
+(* Basic eliminators ********************************************************)
+
+lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
+               (∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
+               (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+               (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
+                 ⬆*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
+               ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
+                 ⬆*[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)
+               ) → (∀G,L,V,T1,T,T2. ⬆*[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
+               ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
+                  Q G L (ⓝV1.T) V2
+               ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+                  Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) 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 →
+                  ⬆*[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 #IH10 #IH11 #G #L #T1 #T2
+* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
+qed-.