update in basic_2 and apps_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / cpms_cpms.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_cpms.ma
index e9c66826faee8fa611aa2e4b0a79e9346287b4ef..7c30116675486db53467efb50b449afbc20b4db1 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_cpms.ma
@@ -23,22 +23,22 @@ include "basic_2/rt_computation/cprs.ma".
 (* Main properties **********************************************************)
 
 (* Basic_2A1: includes: cprs_bind *)
-theorem cpms_bind (n) (h) (G) (L):
-                  ∀I,V1,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡*[n,h] T2 →
-                  ∀V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
-                  ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[n,h] ⓑ[p,I]V2.T2.
-#n #h #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
+theorem cpms_bind (h) (n) (G) (L):
+                  ∀I,V1,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡*[h,n] T2 →
+                  ∀V2. ❪G,L❫ ⊢ V1 ➡*[h,0] V2 →
+                  ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[h,n] ⓑ[p,I]V2.T2.
+#h #n #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
 [ /2 width=1 by cpms_bind_dx/
 | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
   /3 width=3 by cpr_pair_sn, cpms_step_dx/
 ]
 qed.
 
-theorem cpms_appl (n) (h) (G) (L):
-                  ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
-                  ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
-                  ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
-#n #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
+theorem cpms_appl (h) (n) (G) (L):
+                  ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 →
+                  ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h,0] V2 →
+                  ❪G,L❫ ⊢ ⓐV1.T1 ➡*[h,n] ⓐV2.T2.
+#h #n #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
 [ /2 width=1 by cpms_appl_dx/
 | #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12
   /3 width=3 by cpr_pair_sn, cpms_step_dx/
@@ -46,12 +46,12 @@ theorem cpms_appl (n) (h) (G) (L):
 qed.
 
 (* Basic_2A1: includes: cprs_beta_rc *)
-theorem cpms_beta_rc (n) (h) (G) (L):
-                     ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
-                     ∀W1,T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
-                     ∀W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
-                     ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
-#n #h #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
+theorem cpms_beta_rc (h) (n) (G) (L):
+                     ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 →
+                     ∀W1,T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[h,n] T2 →
+                     ∀W2. ❪G,L❫ ⊢ W1 ➡*[h,0] W2 →
+                     ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[h,n] ⓓ[p]ⓝW2.V2.T2.
+#h #n #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
 [ /2 width=1 by cpms_beta_dx/
 | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
   /4 width=3 by cpr_pair_sn, cpms_step_dx/
@@ -59,12 +59,12 @@ theorem cpms_beta_rc (n) (h) (G) (L):
 qed.
 
 (* Basic_2A1: includes: cprs_beta *)
-theorem cpms_beta (n) (h) (G) (L):
-                  ∀W1,T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
-                  ∀W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
-                  ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
-                  ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
-#n #h #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
+theorem cpms_beta (h) (n) (G) (L):
+                  ∀W1,T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[h,n] T2 →
+                  ∀W2. ❪G,L❫ ⊢ W1 ➡*[h,0] W2 →
+                  ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h,0] V2 →
+                  ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[h,n] ⓓ[p]ⓝW2.V2.T2.
+#h #n #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
 [ /2 width=1 by cpms_beta_rc/
 | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
   /4 width=5 by cpms_step_dx, cpr_pair_sn, cpm_cast/
@@ -72,12 +72,12 @@ theorem cpms_beta (n) (h) (G) (L):
 qed.
 
 (* Basic_2A1: includes: cprs_theta_rc *)
-theorem cpms_theta_rc (n) (h) (G) (L):
-                      ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h] V → ∀V2. ⇧[1] V ≘ V2 →
-                      ∀W1,T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
-                      ∀W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
-                      ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
-#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
+theorem cpms_theta_rc (h) (n) (G) (L):
+                      ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h,0] V → ∀V2. ⇧[1] V ≘ V2 →
+                      ∀W1,T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[h,n] T2 →
+                      ∀W2. ❪G,L❫ ⊢ W1 ➡*[h,0] W2 →
+                      ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[h,n] ⓓ[p]W2.ⓐV2.T2.
+#h #n #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
 [ /2 width=3 by cpms_theta_dx/
 | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
   /3 width=3 by cpr_pair_sn, cpms_step_dx/
@@ -85,12 +85,12 @@ theorem cpms_theta_rc (n) (h) (G) (L):
 qed.
 
 (* Basic_2A1: includes: cprs_theta *)
-theorem cpms_theta (n) (h) (G) (L):
-                   ∀V,V2. ⇧[1] V ≘ V2 → ∀W1,W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
-                   ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
-                   ∀V1. ❪G,L❫ ⊢ V1 ➡*[h] V →
-                   ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
-#n #h #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
+theorem cpms_theta (h) (n) (G) (L):
+                   ∀V,V2. ⇧[1] V ≘ V2 → ∀W1,W2. ❪G,L❫ ⊢ W1 ➡*[h,0] W2 →
+                   ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[h,n] T2 →
+                   ∀V1. ❪G,L❫ ⊢ V1 ➡*[h,0] V →
+                   ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[h,n] ⓓ[p]W2.ⓐV2.T2.
+#h #n #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
 [ /2 width=3 by cpms_theta_rc/
 | #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12
   /3 width=3 by cpr_pair_sn, cpms_step_sn/
@@ -99,32 +99,25 @@ qed.
 
 (* Basic_2A1: uses: lstas_scpds_trans scpds_strap2 *)
 theorem cpms_trans (h) (G) (L):
-                   ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T →
-                   ∀n2,T2. ❪G,L❫ ⊢ T ➡*[n2,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2.
+        ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[h,n1] T →
+        ∀n2,T2. ❪G,L❫ ⊢ T ➡*[h,n2] T2 → ❪G,L❫ ⊢ T1 ➡*[h,n1+n2] T2.
 /2 width=3 by ltc_trans/ qed-.
 
 (* Basic_2A1: uses: scpds_cprs_trans *)
-theorem cpms_cprs_trans (n) (h) (G) (L):
-                        ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[n,h] T →
-                        ∀T2. ❪G,L❫ ⊢ T ➡*[h] T2 → ❪G,L❫ ⊢ T1 ➡*[n,h] T2.
-#n #h #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
+theorem cpms_cprs_trans (h) (n) (G) (L):
+                        ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h,n] T →
+                        ∀T2. ❪G,L❫ ⊢ T ➡*[h,0] T2 → ❪G,L❫ ⊢ T1 ➡*[h,n] T2.
+#h #n #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
 /2 width=3 by cpms_trans/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpms_inv_appl_sn (n) (h) (G) (L):
-                       ∀V1,T1,X2. ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] X2 →
-                       ∨∨ ∃∃V2,T2.
-                            ❪G,L❫ ⊢ V1 ➡*[h] V2 & ❪G,L❫ ⊢ T1 ➡*[n,h] T2 &
-                            X2 = ⓐV2.T2
-                        | ∃∃n1,n2,p,W,T.
-                            ❪G,L❫ ⊢ T1 ➡*[n1,h] ⓛ[p]W.T & ❪G,L❫ ⊢ ⓓ[p]ⓝW.V1.T ➡*[n2,h] X2 &
-                            n1 + n2 = n
-                        | ∃∃n1,n2,p,V0,V2,V,T.
-                            ❪G,L❫ ⊢ V1 ➡*[h] V0 & ⇧[1] V0 ≘ V2 &
-                            ❪G,L❫ ⊢ T1 ➡*[n1,h] ⓓ[p]V.T & ❪G,L❫ ⊢ ⓓ[p]V.ⓐV2.T ➡*[n2,h] X2 &
-                            n1 + n2 = n.
-#n #h #G #L #V1 #T1 #U2 #H
+lemma cpms_inv_appl_sn (h) (n) (G) (L):
+      ∀V1,T1,X2. ❪G,L❫ ⊢ ⓐV1.T1 ➡*[h,n] X2 →
+      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡*[h,0] V2 & ❪G,L❫ ⊢ T1 ➡*[h,n] T2 & X2 = ⓐV2.T2
+       | ∃∃n1,n2,p,W,T. ❪G,L❫ ⊢ T1 ➡*[h,n1] ⓛ[p]W.T & ❪G,L❫ ⊢ ⓓ[p]ⓝW.V1.T ➡*[h,n2] X2 & n1 + n2 = n
+       | ∃∃n1,n2,p,V0,V2,V,T. ❪G,L❫ ⊢ V1 ➡*[h,0] V0 & ⇧[1] V0 ≘ V2 & ❪G,L❫ ⊢ T1 ➡*[h,n1] ⓓ[p]V.T & ❪G,L❫ ⊢ ⓓ[p]V.ⓐV2.T ➡*[h,n2] X2 & n1 + n2 = n.
+#h #n #G #L #V1 #T1 #U2 #H
 @(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
 #n1 #n2 #U #U2 #_ * *
 [ #V0 #T0 #HV10 #HT10 #H #HU2 destruct
@@ -146,8 +139,9 @@ lemma cpms_inv_appl_sn (n) (h) (G) (L):
 ]
 qed-.
 
-lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2 →
-                                 ∃∃T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T & ❪G,L❫ ⊢ T ➡*[n2,h] T2.
+lemma cpms_inv_plus (h) (G) (L):
+      ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n1+n2] T2 →
+      ∃∃T. ❪G,L❫ ⊢ T1 ➡*[h,n1] T & ❪G,L❫ ⊢ T ➡*[h,n2] T2.
 #h #G #L #n1 elim n1 -n1 /2 width=3 by ex2_intro/
 #n1 #IH #n2 #T1 #T2 <plus_S1 #H
 elim (cpms_inv_succ_sn … H) -H #T0 #HT10 #HT02
@@ -158,11 +152,11 @@ qed-.
 
 (* Advanced main properties *************************************************)
 
-theorem cpms_cast (n) (h) (G) (L):
-                  ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
-                  ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[n,h] U2 →
-                  ❪G,L❫ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
-#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
+theorem cpms_cast (h) (n) (G) (L):
+        ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 →
+        ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[h,n] U2 →
+        ❪G,L❫ ⊢ ⓝU1.T1 ➡*[h,n] ⓝU2.T2.
+#h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
 [ /3 width=3 by cpms_cast_sn/
 | #n1 #n2 #T1 #T #HT1 #_ #IH #U1 #U2 #H
   elim (cpms_inv_plus … H) -H #U #HU1 #HU2
@@ -171,8 +165,8 @@ theorem cpms_cast (n) (h) (G) (L):
 qed.
 
 theorem cpms_trans_swap (h) (G) (L) (T1):
-        ∀n1,T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T → ∀n2,T2. ❪G,L❫ ⊢ T ➡*[n2,h] T2 →
-        ∃∃T0. ❪G,L❫ ⊢ T1 ➡*[n2,h] T0 & ❪G,L❫ ⊢ T0 ➡*[n1,h] T2.
+        ∀n1,T. ❪G,L❫ ⊢ T1 ➡*[h,n1] T → ∀n2,T2. ❪G,L❫ ⊢ T ➡*[h,n2] T2 →
+        ∃∃T0. ❪G,L❫ ⊢ T1 ➡*[h,n2] T0 & ❪G,L❫ ⊢ T0 ➡*[h,n1] T2.
 #h #G #L #T1 #n1 #T #HT1 #n2 #T2 #HT2
 lapply (cpms_trans … HT1 … HT2) -T <commutative_plus #HT12
 /2 width=1 by cpms_inv_plus/