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[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 fcd97f3b905b5192dc3c7cc908bfebe0ef7bab2a..0e09b27f0835401c3ace850d4ded61fecde069e3 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
@@ -23,9 +23,9 @@ include "basic_2/rt_computation/cprs.ma".
 
 (* 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.
+                  ∀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
 [ /2 width=1 by cpms_bind_dx/
 | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
@@ -34,9 +34,9 @@ theorem cpms_bind (n) (h) (G) (L):
 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.
+                  ∀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
 [ /2 width=1 by cpms_appl_dx/
 | #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12
@@ -46,10 +46,10 @@ 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.
+                     ∀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
 [ /2 width=1 by cpms_beta_dx/
 | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
@@ -59,10 +59,10 @@ 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.
+                  ∀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
 [ /2 width=1 by cpms_beta_rc/
 | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
@@ -72,10 +72,10 @@ 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.
+                      ∀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
 [ /2 width=3 by cpms_theta_dx/
 | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
@@ -85,10 +85,10 @@ 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.
+                   ∀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
 [ /2 width=3 by cpms_theta_rc/
 | #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12
@@ -98,30 +98,30 @@ 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 ➡*[n1,h] T →
+                   ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] 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.
+                        ∀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)
 /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 →
+                       ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] X2 →
                        ∨∨ ∃∃V2,T2.
-                            ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L⦄ ⊢ T1 ➡*[n, h] 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 &
+                            ⦃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 &
+                            ⦃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
 @(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
@@ -145,8 +145,8 @@ 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 ➡*[n1+n2,h] T2 →
+                                 ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T ➡*[n2,h] 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
@@ -158,9 +158,9 @@ 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.
+                  ∀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
 [ /3 width=3 by cpms_cast_sn/
 | #n1 #n2 #T1 #T #HT1 #_ #IH #U1 #U2 #H