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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
index 454ee83127c6499287d18df4e8decbff5ce1ae03..d1817d6ff7479f56220a26507a47fc36024f41e8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
@@ -35,15 +35,15 @@ interpretation
 
 lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
                   Q 0 T2 →
-                  (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
-                  ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
+                  (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) →
+                  ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T1.
 #h #G #L #T2 #Q @ltc_ind_sn_refl //
 qed-.
 
 lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
                   Q 0 T1 →
-                  (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
-                  ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
+                  (∀n1,n2,T,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → Q n1 T → ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → Q (n1+n2) T2) →
+                  ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T2.
 #h #G #L #T1 #Q @ltc_ind_dx_refl //
 qed-.
 
@@ -52,36 +52,36 @@ qed-.
 (* Basic_1: includes: pr1_pr0 *)
 (* Basic_1: uses: pr3_pr2 *)
 (* Basic_2A1: includes: cpr_cprs *)
-lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
+lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
 /2 width=1 by ltc_rc/ qed.
 
-lemma cpms_step_sn (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.
+lemma cpms_step_sn (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.
 /2 width=3 by ltc_sn/ qed-.
 
-lemma cpms_step_dx (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.
+lemma cpms_step_dx (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.
 /2 width=3 by ltc_dx/ qed-.
 
 (* Basic_2A1: uses: cprs_bind_dx *)
 lemma cpms_bind_dx (n) (h) (G) (L):
-                   ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
-                   ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
-                   ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+                   ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+                   ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n,h] T2 →
+                   ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
 #n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
 
 lemma cpms_appl_dx (n) (h) (G) (L):
-                   ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
-                   ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
-                   ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2.
+                   ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+                   ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+                   ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
 #n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
 qed.
 
 lemma cpms_zeta (n) (h) (G) (L):
                 ∀T1,T. ⬆*[1] T ≘ T1 →
-                ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+                ∀V,T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2.
 #n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
 /3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
 qed.
@@ -89,22 +89,22 @@ qed.
 (* Basic_2A1: uses: cprs_zeta *)
 lemma cpms_zeta_dx (n) (h) (G) (L):
                    ∀T2,T. ⬆*[1] T2 ≘ T →
-                   ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+                   ∀V,T1. ⦃G,L.ⓓV⦄ ⊢ T1 ➡*[n,h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2.
 #n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
 qed.
 
 (* Basic_2A1: uses: cprs_eps *)
 lemma cpms_eps (n) (h) (G) (L):
-               ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
-               ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[n, h] T2.
+               ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+               ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ➡*[n,h] T2.
 #n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
 qed.
 
 lemma cpms_ee (n) (h) (G) (L):
-              ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
-              ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2.
+              ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+              ∀T. ⦃G,L⦄ ⊢ ⓝU1.T ➡*[↑n,h] U2.
 #n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
 [ /3 width=1 by cpm_cpms, cpm_ee/
 | #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
@@ -114,21 +114,21 @@ qed.
 
 (* Basic_2A1: uses: cprs_beta_dx *)
 lemma cpms_beta_dx (n) (h) (G) (L):
-                   ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
-                   ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
-                   ∀T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
-                   ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+                   ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+                   ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+                   ∀T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+                   ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
 #n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
 /4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
 qed.
 
 (* Basic_2A1: uses: cprs_theta_dx *)
 lemma cpms_theta_dx (n) (h) (G) (L):
-                    ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V →
+                    ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V →
                     ∀V2. ⬆*[1] V ≘ V2 →
-                    ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
-                    ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
-                    ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+                    ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+                    ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+                    ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
 #n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
 /4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
 qed.
@@ -150,17 +150,17 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
+lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] X2 → X2 = ⋆(((next h)^n) s).
 #n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
 #n1 #n2 #X #X2 #_ #IH #HX2 destruct
 elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
 qed-.
 
 lemma cpms_inv_cast1 (h) (n) (G) (L):
-      ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
-      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
-       | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2
-       | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+      ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+      ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+       | ⦃G,L⦄ ⊢ T1 ➡*[n,h] X2
+       | ∃∃m. ⦃G,L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
 #h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
 [ /3 width=5 by or3_intro0, ex3_2_intro/
 | #n1 #n2 #X #X2 #_ * [ * || * ]