X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpm.ma;h=341ecee8cfde5b23e00b615b04c4d3ef40673cc7;hp=7af9a5730843248e9d722c97b796c2f3acb698e8;hb=5c92c318030a05c766b3f6070dbd23589cbdee04;hpb=747b42f3b9aac5487047f57742f1fcf05b56b57d

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
index 7af9a5730..341ecee8c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
@@ -19,16 +19,16 @@ include "basic_2/rt_transition/cpg.ma".
 (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
 
 (* Basic_2A1: includes: cpr *)
-definition cpm (n) (h): relation4 genv lenv term term ≝
-                        λG,L,T1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[eq_t, c, h] T2.
+definition cpm (h) (G) (L) (n): relation2 term term ≝
+                                λT1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[eq_t, c, h] T2.
 
 interpretation
-   "semi-counted context-sensitive parallel rt-transition (term)"
-   'PRed n h G L T1 T2 = (cpm n h G L T1 T2).
+   "t-bound context-sensitive parallel rt-transition (term)"
+   'PRed n h G L T1 T2 = (cpm h G L n T1 T2).
 
 interpretation
    "context-sensitive parallel r-transition (term)"
-   'PRed h G L T1 T2 = (cpm O h G L T1 T2).
+   'PRed h G L T1 T2 = (cpm h G L O T1 T2).
 
 (* Basic properties *********************************************************)
 
@@ -36,19 +36,19 @@ lemma cpm_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[1, h] ⋆(next h s).
 /2 width=3 by cpg_ess, ex2_intro/ qed.
 
 lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
-                 ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡[n, h] W2.
+                 ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡[n, h] W2.
 #n #h #G #K #V1 #V2 #W2 *
 /3 width=5 by cpg_delta, ex2_intro/
 qed.
 
 lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
-               ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡[⫯n, h] W2.
+               ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡[↑n, h] W2.
 #n #h #G #K #V1 #V2 #W2 *
 /3 width=5 by cpg_ell, ex2_intro, isrt_succ/
 qed.
 
 lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T →
-                ⬆*[1] T ≡ U → ⦃G, K.ⓘ{I}⦄ ⊢ #⫯i ➡[n, h] U.
+                ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ➡[n, h] U.
 #n #h #I #G #K #T #U #i *
 /3 width=5 by cpg_lref, ex2_intro/
 qed.
@@ -76,9 +76,10 @@ lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2.
 qed.
 
 (* Basic_2A1: includes: cpr_zeta *)
-lemma cpm_zeta: ∀n,h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[n, h] T →
-                ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
-#n #h #G #L #V #T1 #T #T2 *
+lemma cpm_zeta (n) (h) (G) (L):
+               ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[n,h] T2 →
+               ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
+#n #h #G #L #T1 #T #HT1 #T2 *
 /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
 qed.
 
@@ -88,7 +89,7 @@ lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄
 /3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
 qed.
 
-lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[⫯n, h] V2.
+lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[↑n, h] V2.
 #n #h #G #L #V1 #V2 #T *
 /3 width=3 by cpg_ee, isrt_succ, ex2_intro/
 qed.
@@ -103,38 +104,47 @@ qed.
 
 (* Basic_2A1: includes: cpr_theta *)
 lemma cpm_theta: ∀n,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 ➡[n, h] T2 →
                  ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n, h] ⓓ{p}W2.ⓐV2.T2.
 #n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
 /6 width=9 by cpg_theta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
 qed.
 
-(* Basic properties on r-transition *****************************************)
+(* Basic properties with r-transition ***************************************)
 
+(* Note: this is needed by cpms_ind_sn and cpms_ind_dx *)
 (* Basic_1: includes by definition: pr0_refl *)
 (* Basic_2A1: includes: cpr_atom *)
-lemma cpr_refl: ∀h,G,L. reflexive … (cpm 0 h G L).
+lemma cpr_refl: ∀h,G,L. reflexive … (cpm h G L 0).
 /3 width=3 by cpg_refl, ex2_intro/ qed.
 
+(* Advanced properties ******************************************************)
+
+lemma cpm_sort (h) (G) (L):
+               ∀n. n ≤ 1 → ∀s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] ⋆((next h)^n s).
+#h #G #L * //
+#n #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/
+qed.
+
 (* Basic inversion lemmas ***************************************************)
 
 lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 →
                      ∨∨ T2 = ⓪{J} ∧ n = 0
                       | ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1
-                      | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
                                    L = K.ⓓV1 & J = LRef 0
-                      | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≡ T2 &
-                                     L = K.ⓛV1 & J = LRef 0 & n = ⫯k
-                      | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≡ T2 &
-                                   L = K.ⓘ{I} & J = LRef (⫯i).
+                      | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+                                     L = K.ⓛV1 & J = LRef 0 & n = ↑m
+                      | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 &
+                                   L = K.ⓘ{I} & J = LRef (↑i).
 #n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
 | #s #H1 #H2 #H3 destruct /4 width=3 by isrt_inv_01, or5_intro1, ex3_intro/
 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
   /4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
-  elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+  elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
   /4 width=9 by or5_intro3, ex5_4_intro, ex2_intro/
 | #I #K #V2 #i #HV2 #HVT2 #H1 #H2 destruct
   /4 width=8 by or5_intro4, ex4_4_intro, ex2_intro/
@@ -142,32 +152,32 @@ lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 →
 qed-.
 
 lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[n,h] T2 →
-                     ∨∨ T2 = ⋆s ∧ n = 0
-                      | T2 = ⋆(next h s) ∧ n = 1.
-#n #h #G #L #T2 #s * #c #Hc #H elim (cpg_inv_sort1 … H) -H *
-#H1 #H2 destruct
-/4 width=1 by isrt_inv_01, isrt_inv_00, or_introl, or_intror, conj/
+                     ∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1.
+#n #h #G #L #T2 #s * #c #Hc #H
+elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
+[ lapply (isrt_inv_00 … Hc) | lapply (isrt_inv_01 … Hc) ] -Hc
+#H destruct /2 width=1 by conj/
 qed-.
 
 lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 →
                      ∨∨ T2 = #0 ∧ n = 0
-                      | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
                                    L = K.ⓓV1
-                      | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≡ T2 &
-                                     L = K.ⓛV1 & n = ⫯k.
+                      | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+                                     L = K.ⓛV1 & n = ↑m.
 #n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
   /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
-  elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+  elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
   /4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
 ]
 qed-.
 
-lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[n, h] T2 →
-                     ∨∨ T2 = #(⫯i) ∧ n = 0
-                      | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≡ T2 & L = K.ⓘ{I}.
+lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[n, h] T2 →
+                     ∨∨ T2 = #(↑i) ∧ n = 0
+                      | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
 #n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
 | #I #K #V2 #HV2 #HVT2 #H destruct
@@ -184,15 +194,15 @@ qed-.
 lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 →
                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 &
                                  U2 = ⓑ{p,I}V2.T2
-                      | ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T & ⬆*[1] U2 ≡ T &
+                      | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 & 
                              p = true & I = Abbr.
 #n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
   elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
   /4 width=5 by ex3_2_intro, ex2_intro, or_introl/
-| #cT #T2 #HT12 #HUT2 #H1 #H2 #H3 destruct
-  /5 width=3 by isrt_inv_plus_O_dx, ex4_intro, ex2_intro, or_intror/
+| #cT #T2 #HT21 #HTU2 #H1 #H2 #H3 destruct
+  /5 width=5 by isrt_inv_plus_O_dx, ex4_intro, ex2_intro, or_intror/
 ]
 qed-.
 
@@ -201,14 +211,11 @@ qed-.
 lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 →
                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 &
                                  U2 = ⓓ{p}V2.T2
-                      | ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T & ⬆*[1] U2 ≡ T & p = true.
-#n #h #p #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_abbr1 … H) -H *
-[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
-  elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
-  elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
-  /4 width=5 by ex3_2_intro, ex2_intro, or_introl/
-| #cT #T2 #HT12 #HUT2 #H1 #H2 destruct
-  /5 width=3 by isrt_inv_plus_O_dx, ex3_intro, ex2_intro, or_intror/
+                      | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 & p = true.
+#n #h #p #G #L #V1 #T1 #U2 #H
+elim (cpm_inv_bind1 … H) -H
+[ /3 width=1 by or_introl/
+| * /3 width=3 by ex3_intro, or_intror/
 ]
 qed-.
 
@@ -217,11 +224,18 @@ qed-.
 lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 →
                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 &
                               U2 = ⓛ{p}V2.T2.
-#n #h #p #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_abst1 … H) -H
-#cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
-elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
-elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
-/3 width=5 by ex3_2_intro, ex2_intro/
+#n #h #p #G #L #V1 #T1 #U2 #H
+elim (cpm_inv_bind1 … H) -H
+[ /3 width=1 by or_introl/
+| * #T #_ #_ #_ #H destruct  
+]
+qed-.
+
+lemma cpm_inv_abst_bi: ∀n,h,p1,p2,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}V1.T1 ➡[n,h] ⓛ{p2}V2.T2 →
+                       ∧∧ ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 & p1 = p2.
+#n #h #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H
+elim (cpm_inv_abst1 … H) -H #XV #XT #HV #HT #H destruct
+/2 width=1 by and3_intro/  
 qed-.
 
 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
@@ -232,7 +246,7 @@ lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2
                       | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
                                             ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, 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 ➡[n, h] T2 &
                                               U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
@@ -261,7 +275,7 @@ lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 
                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
                                  U2 = ⓝV2.T2
                       | ⦃G, L⦄ ⊢ U1 ➡[n, h] U2
-                      | ∃∃k. ⦃G, L⦄ ⊢ V1 ➡[k, h] U2 & n = ⫯k.
+                      | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & n = ↑m.
 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
 [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
   elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
@@ -271,7 +285,7 @@ lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 
 | #cU #U12 #H destruct
   /4 width=3 by isrt_inv_plus_O_dx, or3_intro1, ex2_intro/
 | #cU #H12 #H destruct
-  elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+  elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
   /4 width=3 by or3_intro2, ex2_intro/
 ]
 qed-.
@@ -285,3 +299,75 @@ lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡
 #n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
 /3 width=4 by ex2_2_intro, ex2_intro/
 qed-.
+
+(* Basic eliminators ********************************************************)
+
+lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
+                   (∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
+                   (∀G,L,s. Q 1 G L (⋆s) (⋆(next h s))) →
+                   (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 →
+                     ⬆*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
+                   ) → (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 →
+                     ⬆*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
+                   ) → (∀n,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → Q n G K (#i) T →
+                     ⬆*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
+                   ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
+                     Q 0 G L V1 V2 → Q n G (L.ⓑ{I}V1) T1 T2 → Q n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+                   ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+                     Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
+                   ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+                     Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
+                   ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[n, h] T2 →
+                     Q n G L T T2 → Q n G L (+ⓓV.T1) T2
+                   ) → (∀n,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+                     Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
+                   ) → (∀n,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 →
+                     Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
+                   ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
+                     Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
+                     Q n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+                   ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
+                     Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
+                     ⬆*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+                   ) →
+                   ∀n,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → Q n G L T1 T2.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
+* #c #HC #H generalize in match HC; -HC generalize in match n; -n
+elim H -c -G -L -T1 -T2
+[ #I #G #L #n #H <(isrt_inv_00 … H) -H //
+| #G #L #s #n #H <(isrt_inv_01 … H) -H //
+| /3 width=4 by ex2_intro/
+| #c #G #L #V1 #V2 #W2 #HV12 #HVW2 #IH #x #H
+  elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
+  /3 width=4 by ex2_intro/
+| /3 width=4 by ex2_intro/
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
+  elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
+  /3 width=3 by ex2_intro/
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
+  elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
+  /3 width=3 by ex2_intro/
+| #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #n #H
+  elim (isrt_inv_max_eq_t … H) -H // #HcV #HcT
+  /3 width=3 by ex2_intro/
+| #c #G #L #V #T1 #T #T2 #HT1 #HT2 #IH #n #H
+  lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
+  /3 width=4 by ex2_intro/
+| #c #G #L #U #T1 #T2 #HT12 #IH #n #H
+  lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
+  /3 width=3 by ex2_intro/
+| #c #G #L #U1 #U2 #T #HU12 #IH #x #H
+  elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
+  /3 width=3 by ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #IHV #IHW #IHT #n #H
+  lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
+  elim (isrt_inv_max_shift_sn … H) -H #H #HcT
+  elim (isrt_O_inv_max … H) -H #HcV #HcW
+  /3 width=3 by ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #IHV #IHW #IHT #n #H
+  lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
+  elim (isrt_inv_max_shift_sn … H) -H #H #HcT
+  elim (isrt_O_inv_max … H) -H #HcV #HcW
+  /3 width=4 by ex2_intro/
+]
+qed-.