X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpm.ma;h=ac451b66c580f81de831319b96bbc455013c211c;hb=3c7b4071a9ac096b02334c1d47468776b948e2de;hp=61badb90a0e9ca3cc8e71c46446f8e3cb5aaf202;hpb=75f395f0febd02de8e0f881d918a8812b1425c8d;p=helm.git

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 61badb90a..ac451b66c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
@@ -12,230 +12,282 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground/xoa/ex_4_1.ma".
+include "ground/xoa/ex_4_3.ma".
+include "ground/xoa/ex_5_6.ma".
+include "ground/xoa/ex_6_7.ma".
+include "ground/steps/rtc_max_shift.ma".
+include "ground/steps/rtc_isrt_plus.ma".
+include "ground/steps/rtc_isrt_max_shift.ma".
+include "static_2/syntax/sh.ma".
 include "basic_2/notation/relations/pred_6.ma".
-include "basic_2/notation/relations/pred_5.ma".
 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 ⬈[sh_is_next h,rtc_eq_t,c] T2.
 
 interpretation
-   "semi-counted context-sensitive parallel rt-transition (term)"
-   'PRed n h G L T1 T2 = (cpm n h G L T1 T2).
-
-interpretation
-   "context-sensitive parallel r-transition (term)"
-   'PRed h G L T1 T2 = (cpm O h G L T1 T2).
+  "t-bound context-sensitive parallel rt-transition (term)"
+  'PRed h n G L T1 T2 = (cpm h G L n T1 T2).
 
 (* Basic properties *********************************************************)
 
-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_ess (h) (G) (L):
+      ∀s. ❪G,L❫ ⊢ ⋆s ➡[h,1] ⋆(⫯[h]s).
+/3 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.
-#n #h #G #K #V1 #V2 #W2 *
+lemma cpm_delta (h) (n) (G) (K):
+      ∀V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 →
+      ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ➡[h,n] W2.
+#h #n #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.
-#n #h #G #K #V1 #V2 #W2 *
+lemma cpm_ell (h) (n) (G) (K):
+      ∀V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 →
+      ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡[h,↑n] W2.
+#h #n #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.
-#n #h #I #G #K #T #U #i *
+lemma cpm_lref (h) (n) (G) (K):
+      ∀I,T,U,i. ❪G,K❫ ⊢ #i ➡[h,n] T →
+      ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡[h,n] U.
+#h #n #G #K #I #T #U #i *
 /3 width=5 by cpg_lref, ex2_intro/
 qed.
 
 (* Basic_2A1: includes: cpr_bind *)
-lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
-                ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
-                ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
-#n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
+lemma cpm_bind (h) (n) (G) (L):
+      ∀p,I,V1,V2,T1,T2.
+      ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 →
+      ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] ⓑ[p,I]V2.T2.
+#h #n #G #L #p #I #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
 /5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/
 qed.
 
-lemma cpm_appl: ∀n,h,G,L,V1,V2,T1,T2.
-                ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
-                ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] ⓐV2.T2.
-#n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
+lemma cpm_appl (h) (n) (G) (L):
+      ∀V1,V2,T1,T2.
+      ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+      ❪G,L❫ ⊢ ⓐV1.T1 ➡[h,n] ⓐV2.T2.
+#h #n #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
 /5 width=5 by isrt_max_O1, isr_shift, cpg_appl, ex2_intro/
 qed.
 
-lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2.
-                ⦃G, L⦄ ⊢ U1 ➡[n, h] U2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
-                ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n, h] ⓝU2.T2.
-#n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
+lemma cpm_cast (h) (n) (G) (L):
+      ∀U1,U2,T1,T2.
+      ❪G,L❫ ⊢ U1 ➡[h,n] U2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+      ❪G,L❫ ⊢ ⓝU1.T1 ➡[h,n] ⓝU2.T2.
+#h #n #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
 /4 width=6 by cpg_cast, isrt_max_idem1, isrt_mono, ex2_intro/
 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 (h) (n) (G) (L):
+      ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ➡[h,n] T2 →
+      ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ➡[h,n] T2.
+#h #n #G #L #T1 #T #HT1 #T2 *
 /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
 qed.
 
 (* Basic_2A1: includes: cpr_eps *)
-lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2.
-#n #h #G #L #V #T1 #T2 *
+lemma cpm_eps (h) (n) (G) (L):
+      ∀V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → ❪G,L❫ ⊢ ⓝV.T1 ➡[h,n] T2.
+#h #n #G #L #V #T1 #T2 *
 /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.
-#n #h #G #L #V1 #V2 #T *
+lemma cpm_ee (h) (n) (G) (L):
+      ∀V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] V2 → ❪G,L❫ ⊢ ⓝV1.T ➡[h,↑n] V2.
+#h #n #G #L #V1 #V2 #T *
 /3 width=3 by cpg_ee, isrt_succ, ex2_intro/
 qed.
 
 (* Basic_2A1: includes: cpr_beta *)
-lemma cpm_beta: ∀n,h,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 →
-                ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2.
-#n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
+lemma cpm_beta (h) (n) (G) (L):
+      ∀p,V1,V2,W1,W2,T1,T2.
+      ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 →
+      ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡[h,n] ⓓ[p]ⓝW2.V2.T2.
+#h #n #G #L #p #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
 /6 width=7 by cpg_beta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
 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.ⓓ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 *
+lemma cpm_theta (h) (n) (G) (L):
+      ∀p,V1,V,V2,W1,W2,T1,T2.
+      ❪G,L❫ ⊢ V1 ➡[h,0] V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 →
+      ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 → ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡[h,n] ⓓ[p]W2.ⓐV2.T2.
+#h #n #G #L #p #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) (n) (G) (L): n ≤ 1 →
+      ∀s. ❪G,L❫ ⊢ ⋆s ➡[h,n] ⋆((next h)^n s).
+#h * //
+#n #G #L #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 &
-                                   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).
-#n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
+lemma cpm_inv_atom1 (h) (n) (G) (L):
+      ∀J,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h,n] T2 →
+      ∨∨ ∧∧ T2 = ⓪[J] & n = 0
+       | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
+       | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0
+       | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & n = ↑m
+       | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+#h #n #G #L #J #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/
+| #s1 #s2 #H1 #H2 #H3 #H4 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/
 ]
 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/
+lemma cpm_inv_sort1 (h) (n) (G) (L):
+      ∀T2,s1. ❪G,L❫ ⊢ ⋆s1 ➡[h,n] T2 →
+      ∧∧ T2 = ⋆(((next h)^n) s1) & n ≤ 1.
+#h #n #G #L #T2 #s1 * #c #Hc #H
+elim (cpg_inv_sort1 … H) -H *
+[ #H1 #H2 destruct
+  lapply (isrt_inv_00 … Hc)
+| #s2 #H1 #H2 #H3 destruct
+  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 &
-                                   L = K.ⓓV1
-                      | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≘ T2 &
-                                     L = K.ⓛV1 & n = ⫯k.
-#n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
+lemma cpm_inv_zero1 (h) (n) (G) (L):
+      ∀T2. ❪G,L❫ ⊢ #0 ➡[h,n] T2 →
+      ∨∨ ∧∧ T2 = #0 & n = 0
+       | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1
+       | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & n = ↑m.
+#h #n #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}.
-#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
+lemma cpm_inv_zero1_unit (h) (n) (I) (K) (G):
+      ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡[h,n] X2 → ∧∧ X2 = #0 & n = 0.
+#h #n #I #G #K #X2 #H
+elim (cpm_inv_zero1 … H) -H *
+[ #H1 #H2 destruct /2 width=1 by conj/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cpm_inv_lref1 (h) (n) (G) (L):
+      ∀T2,i. ❪G,L❫ ⊢ #↑i ➡[h,n] T2 →
+      ∨∨ ∧∧ T2 = #(↑i) & n = 0
+       | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+#h #n #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
  /4 width=6 by ex3_3_intro, ex2_intro, or_intror/
 ]
 qed-.
 
-lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = §l ∧ n = 0.
-#n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
-#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/ 
+lemma cpm_inv_lref1_ctop (h) (n) (G):
+      ∀X2,i. ❪G,⋆❫ ⊢ #i ➡[h,n] X2 → ∧∧ X2 = #i & n = 0.
+#h #n #G #X2 * [| #i ] #H
+[ elim (cpm_inv_zero1 … H) -H *
+  [ #H1 #H2 destruct /2 width=1 by conj/
+  | #Y #X1 #X2 #_ #_ #H destruct
+  | #m #Y #X1 #X2 #_ #_ #H destruct
+  ]
+| elim (cpm_inv_lref1 … H) -H *
+  [ #H1 #H2 destruct /2 width=1 by conj/
+  | #Z #Y #X0 #_ #_ #H destruct
+  ]
+]
+qed.
+
+lemma cpm_inv_gref1 (h) (n) (G) (L):
+      ∀T2,l. ❪G,L❫ ⊢ §l ➡[h,n] T2 → ∧∧ T2 = §l & n = 0.
+#h #n #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
+#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
 qed-.
 
 (* Basic_2A1: includes: cpr_inv_bind1 *)
-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 &
-                             p = true & I = Abbr.
-#n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
+lemma cpm_inv_bind1 (h) (n) (G) (L):
+      ∀p,I,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] U2 →
+      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 & U2 = ⓑ[p,I]V2.T2
+       | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[h,n] U2 & p = true & I = Abbr.
+#h #n #G #L #p #I #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-.
 
 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
 (* Basic_2A1: includes: cpr_inv_abbr1 *)
-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/
+lemma cpm_inv_abbr1 (h) (n) (G) (L):
+      ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ➡[h,n] U2 →
+      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ➡[h,n] T2 & U2 = ⓓ[p]V2.T2
+       | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[h,n] U2 & p = true.
+#h #n #G #L #p #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-.
 
 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
 (* Basic_2A1: includes: cpr_inv_abst1 *)
-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/
+lemma cpm_inv_abst1 (h) (n) (G) (L):
+      ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ➡[h,n] U2 →
+      ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[h,n] T2 & U2 = ⓛ[p]V2.T2.
+#h #n #G #L #p #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 (h) (n) (G) (L):
+      ∀p1,p2,V1,V2,T1,T2. ❪G,L❫ ⊢ ⓛ[p1]V1.T1 ➡[h,n] ⓛ[p2]V2.T2 →
+      ∧∧ ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[h,n] T2 & p1 = p2.
+#h #n #G #L #p1 #p2 #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 *)
 (* Basic_2A1: includes: cpr_inv_appl1 *)
-lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, 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 ➡[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 &
-                                              ⦃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 *
+lemma cpm_inv_appl1 (h) (n) (G) (L):
+      ∀V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ➡[h,n] U2 →
+      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ U1 ➡[h,n] T2 & U2 = ⓐV2.T2
+       | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ W1 ➡[h,0] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
+       | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ➡[h,0] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
+#h #n #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … 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
@@ -257,12 +309,12 @@ lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2
 ]
 qed-.
 
-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.
-#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
+lemma cpm_inv_cast1 (h) (n) (G) (L):
+      ∀V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ➡[h,n] U2 →
+      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,n] V2 & ❪G,L❫ ⊢ U1 ➡[h,n] T2 & U2 = ⓝV2.T2
+       | ❪G,L❫ ⊢ U1 ➡[h,n] U2
+       | ∃∃m. ❪G,L❫ ⊢ V1 ➡[h,m] U2 & n = ↑m.
+#h #n #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
   lapply (isrt_eq_t_trans … HcV HcVT) -HcVT #H
@@ -271,7 +323,7 @@ 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-.
@@ -279,9 +331,81 @@ qed-.
 (* Basic forward lemmas *****************************************************)
 
 (* Basic_2A1: includes: cpr_fwd_bind1_minus *)
-lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p.
-                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 &
-                                    T = -ⓑ{I}V2.T2.
-#n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
+lemma cpm_fwd_bind1_minus (h) (n) (G) (L):
+      ∀I,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ➡[h,n] T → ∀p.
+      ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
+#h #n #G #L #I #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 …):
+      (∀I,G,L. Q 0 G L (⓪[I]) (⓪[I])) →
+      (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
+      (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] 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,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] T2 →
+        Q n G L T T2 → Q n G L (+ⓓV.T1) T2
+      ) → (∀n,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+        Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
+      ) → (∀n,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] 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,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] 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,0] V → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] 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 #s1 #s2 #HRs #n #H <(isrt_inv_01 … H) -H destruct //
+| /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-.