partial commit: just the components before "static" ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / cpxs_tstc_vector.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma
index dffa50fba239043ca1e6766a35b7073f9530daea..a166abc439f3c49ea4568908578965d27d101be0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma
@@ -21,13 +21,13 @@ include "basic_2/computation/cpxs_tstc.ma".
 (* Vector form of forward lemmas involving same top term constructor ********)
 
 (* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,g,L,T.  𝐒⦃T⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ →
-                           ∀Vs,U. ⦃h, L⦄ ⊢ ⒶVs.T ➡*[g] U → ⒶVs.T ≃ U.
+lemma cpxs_fwd_cnx_vector: ∀h,g,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
+                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≃ U.
 #h #g #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
 #V #Vs #IHVs #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #a #V0 #W0 #T0 #HV0 #HT0 #HU
+| #a #W0 #T0 #HT0 #HU
   lapply (IHVs … HT0) -IHVs -HT0 #HT0
   elim (tstc_inv_bind_appls_simple … HT0) //
 | #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
@@ -36,27 +36,48 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
+lemma cpxs_fwd_sort_vector: ∀h,g,L,k,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆k ➡*[h, g] U →
+                            ⒶVs.⋆k ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
+#h #g #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
+#V #Vs #IHVs #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
+| #a #W1 #T1 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tstc_inv_bind_appls_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV.ⓛ{a}W1.T1)) [ /2 width=1/ ] -Vs /3 width=1/
+  ]
+| #a #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tstc_inv_bind_appls_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV1.ⓓ{a}V3.T1)) [ /2 width=1/ ] -Vs /3 width=3/
+  ]
+]
+qed-.
+
+
 (* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,g,a,L,Vs,V,W,T,U. ⦃h, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[g] U →
-                            ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨
-                            ∃∃T0.⦃h, L.ⓛW⦄ ⊢ T ➡*[g] T0 & ⦃h, L⦄ ⊢ ⒶVs.ⓓ{a}V.T0 ➡*[g] U.
+lemma cpxs_fwd_beta_vector: ∀h,g,a,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
+                            ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
 #h #g #a #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
 #V0 #Vs #IHVs #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
-| #b #V1 #W1 #T1 #HV01 #HT1 #HU
-  elim (IHVs … HT1) -IHVs -HT1
-  [ #HT1 elim (tstc_inv_bind_appls_simple … HT1) //
-  | * #T0 #HT0 #HT01
-    @or_intror @(ex2_intro … HT0) -W (**) (* explicit constructor *)
+| #b #W1 #T1 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tstc_inv_bind_appls_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
-    @(cpxs_strap1 … (ⓐV1.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=1/
+    @(cpxs_strap1 … (ⓐV0.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -Vs -T /3 width=1/
   ]
 | #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
-  elim (IHVs … HT1) -IHVs -HT1
-  [ #HT1 elim (tstc_inv_bind_appls_simple … HT1) //
-  | * #T0 #HT0 #HT01
-    @or_intror @(ex2_intro … HT0) -W (**) (* explicit constructor *)
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tstc_inv_bind_appls_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
     @(cpxs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=3/
   ]
@@ -65,18 +86,18 @@ qed-.
 
 lemma cpxs_fwd_delta_vector: ∀h,g,I,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
                              ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
-                             ∀Vs,U. ⦃h, L⦄ ⊢ ⒶVs.#i ➡*[g] U →
-                             ⒶVs.#i ≃ U ∨ ⦃h, L⦄ ⊢ ⒶVs.V2 ➡*[g] U.
+                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
+                             ⒶVs.#i ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
 #h #g #I #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
 #V #Vs #IHVs #U #H -K -V1
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #b #V0 #W0 #T0 #HV0 #HT0 #HU
+| #b #W0 #T0 #HT0 #HU
   elim (IHVs … HT0) -IHVs -HT0 #HT0
   [ elim (tstc_inv_bind_appls_simple … HT0) //
   | @or_intror -i (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
-    @(cpxs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=1/
+    @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V2 -Vs /3 width=1/
   ]
 | #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
   elim (IHVs … HT0) -IHVs -HT0 #HT0
@@ -90,8 +111,8 @@ qed-.
 
 (* Basic_1: was just: pr3_iso_appls_abbr *)
 lemma cpxs_fwd_theta_vector: ∀h,g,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
-                             ∀a,V,T,U. ⦃h, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[g] U →
-                             ⒶV1s. ⓓ{a}V. T ≃ U ∨ ⦃h, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[g] U.
+                             ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
+                             ⒶV1s. ⓓ{a}V. T ≃ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
 #h #g #L #V1s #V2s * -V1s -V2s /3 width=1/
 #V1s #V2s #V1a #V2a #HV12a #HV12s #a
 generalize in match HV12a; -HV12a
@@ -101,19 +122,19 @@ elim HV12s -V1s -V2s /2 width=1 by cpxs_fwd_theta/
 #V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #b #V0a #W0 #T0 #HV10a #HT0 #HU
+| #b #W0 #T0 #HT0 #HU
   elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
-  [ -HV12a -HV12b -HV10a -HU
+  [ -HV12a -HV12b -HU
     elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
   | @or_intror -V1s (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
     elim (cpxs_inv_abbr1 … HT0) -HT0 *
-    [ -HV12a -HV12b -HV10a #V1 #T1 #_ #_ #H destruct
+    [ -HV12a -HV12b #V1 #T1 #_ #_ #H destruct
     | -V1b #X #HT1 #H #H0 destruct
       elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
       @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1/ ] -T -V2b -V2s
-      @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [ /5 width=7/ ] -V -V2a -W1 -T1
-      @(cpxs_strap2 … (ⓓ{b}V1a.T0)) [ /3 width=1/ ] -W0 /2 width=1/
+      @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [2: /2 width=1/ ]
+      /4 width=7 by cpx_zeta, lift_bind, lift_flat/ (* auto too slow without trace *)
     ]
   ]
 | #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
@@ -130,7 +151,7 @@ elim (cpxs_inv_appl1 … H) -H *
       elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
       lapply (cpxs_lift … HV10a (L.ⓓV0) … HV12a … HV0a) -V0a [ /2 width=1/ ] #HV2a
       @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1/ ] -T -V2b -V2s
-      @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V1 -T1
+      @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1 (* auto too slow without trace *)
       @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) [ /3 width=3/ ] -V1a /3 width=1/
     ]
   ]
@@ -138,25 +159,33 @@ elim (cpxs_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_tau_vector: ∀h,g,L,Vs,W,T,U. ⦃h, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[g] U →
-                           ⒶVs. ⓝW. T ≃ U ∨ ⦃h, L⦄ ⊢ ⒶVs.T ➡*[g] U.
-#h #g #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_tau/
+lemma cpxs_fwd_cast_vector: ∀h,g,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
+                            ∨∨ ⒶVs. ⓝW. T ≃ U
+                             | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
+                             | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
+#h #g #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
 #V #Vs #IHVs #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #b #V0 #W0 #T0 #HV0 #HT0 #HU
+| #b #W0 #T0 #HT0 #HU
   elim (IHVs … HT0) -IHVs -HT0 #HT0
   [ elim (tstc_inv_bind_appls_simple … HT0) //
-  | @or_intror -W (**) (* explicit constructor *)
+  | @or3_intro1 -W (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1/
+  | @or3_intro2 -T (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
-    @(cpxs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=1/
+    @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1/
   ]
 | #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
   elim (IHVs … HT0) -IHVs -HT0 #HT0
   [ elim (tstc_inv_bind_appls_simple … HT0) //
-  | @or_intror -W (**) (* explicit constructor *)
+  | @or3_intro1 -W (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /2 width=3/
+  | @or3_intro2 -T (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
-    @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=3/
+    @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -W /2 width=3/  
   ]
 ]
 qed-.