- ng_refiner:

[helm.git] / matita / matita / contribs / lambda / paths / labeled_st_computation.ma

diff --git a/matita/matita/contribs/lambda/paths/labeled_st_computation.ma b/matita/matita/contribs/lambda/paths/labeled_st_computation.ma
index 35ece177513365417bd74fea4af0b636fb4480f2..29dd727691021f5226d52f0b5f118afdb0a22dce 100644 (file)
--- a/matita/matita/contribs/lambda/paths/labeled_st_computation.ma
+++ b/matita/matita/contribs/lambda/paths/labeled_st_computation.ma
@@ -90,7 +90,7 @@ lemma pl_sts_inv_rc_abst_dx: ∀b2,s,F1,T2. F1 Ⓡ↦*[s] {b2}𝛌.T2 → ∀r.
 | #p #s #F1 #F #HF1 #_ #IHF2 #r #H -b2
   elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
   elim (pl_st_inv_rc … HF1 … Hp) -HF1 -p #b1 #T1 #T #HT1 #HF1 #HF destruct
-  elim (IHF2 ??) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
+  elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
   #b0 #T0 #HT02 #H destruct
   lapply (pl_sts_step_sn … HT1 … HT02) -T /2 width=4/
 ]
@@ -103,7 +103,7 @@ lemma pl_sts_inv_sn_appl_dx: ∀b2,s,F1,V2,T2. F1 Ⓡ↦*[s] {b2}@V2.T2 → ∀r
 | #p #s #F1 #F #HF1 #_ #IHF2 #r #H -b2
   elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
   elim (pl_st_inv_sn … HF1 … Hp) -HF1 -p #b1 #V1 #V #T1 #HV1 #HF1 #HF destruct
-  elim (IHF2 ??) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
+  elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
   #b0 #V0 #T0 #HV02 #H destruct
   lapply (pl_sts_step_sn … HV1 … HV02) -V /2 width=5/
 ]
@@ -116,7 +116,7 @@ lemma pl_sts_inv_dx_appl_dx: ∀b,s,F1,V,T2. F1 Ⓡ↦*[s] {b}@V.T2 → ∀r. dx
 | #p #s #F1 #F #HF1 #_ #IHF2 #r #H
   elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
   elim (pl_st_inv_dx … HF1 … Hp) -HF1 -p #b0 #V0 #T1 #T #HT1 #HF1 #HF destruct
-  elim (IHF2 ??) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
+  elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
   #T0 #HT02 #H destruct
   lapply (pl_sts_step_sn … HT1 … HT02) -T /2 width=3/
 ]
@@ -150,17 +150,17 @@ lemma pl_sts_fwd_dx_sn_appl_dx: ∀b2,s,r,F1,V2,T2. F1 Ⓡ↦*[(dx:::s)@(sn:::r)
                                 ∃∃b1,V1,T1,T0. V1 Ⓡ↦*[r] V2 & T1 Ⓡ↦*[s] T0 & {b1}@V1.T1 = F1.
 #b2 #s #r #F1 #V2 #T2 #H
 elim (pl_sts_inv_trans … H) -H #F #HF1 #H
-elim (pl_sts_inv_sn_appl_dx … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
+elim (pl_sts_inv_sn_appl_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
 #b #V #T #HV2 #H destruct
-elim (pl_sts_inv_dx_appl_dx … HF1 ??) -HF1 [3: // |2: skip ] (**) (* simplify line *)
+elim (pl_sts_inv_dx_appl_dx … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
 #T1 #HT1 #H destruct /2 width=7/
 qed-.
 
 theorem pl_sts_fwd_is_standard: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → is_standard s.
 #s elim s -s // #p1 * //
 #p2 #s #IHs #F1 #F2 #H
-elim (pl_sts_inv_cons … H ???) -H [4: // |2,3: skip ] #F3 #HF13 #H (**) (* simplify line *)
-elim (pl_sts_inv_cons … H ???) [2: // |3,4: skip ] #F4 #HF34 #_ (**) (* simplify line *)
+elim (pl_sts_inv_cons … H …) -H [4: // |2,3: skip ] #F3 #HF13 #H (**) (* simplify line *)
+elim (pl_sts_inv_cons … H …) [2: // |3,4: skip ] #F4 #HF34 #_ (**) (* simplify line *)
 lapply (pl_st_fwd_sle … HF13 … HF34) -F1 -F4 /3 width=3/
 qed-.
 
@@ -172,7 +172,7 @@ lapply (pl_sts_fwd_is_standard … H)
 [ #_ @(ex2_2_intro … ◊ ◊) // (**) (* auto needs some help here *)
 | #p #s #F1 #F #HF1 #HF2 #IHF1 #Hs
   lapply (is_standard_fwd_cons … Hs) #H
-  elim (IHF1 ?) // -IHF1 -H #r1 #r2 #Hr1 #H destruct
+  elim (IHF1 …) // -IHF1 -H #r1 #r2 #Hr1 #H destruct
   elim (in_whd_or_in_inner p) #Hp
   [ -Hs -F1 -F -T2 -b2
     @(ex2_2_intro … (p::r1) r2) // /2 width=1/ (**) (* auto needs some help here *)
@@ -182,17 +182,17 @@ lapply (pl_sts_fwd_is_standard … H)
     elim (in_inner_inv … Hp) -Hp * #q [3: #IHq ] #Hp
 (* case 1: dx *)
     [ -IHq
-      elim (pl_sts_inv_rc_abst_dx … HF2 ??) -b2 [3: // |2: skip ] (**) (* simplify line *)
+      elim (pl_sts_inv_rc_abst_dx … HF2 …) -b2 [3: // |2: skip ] (**) (* simplify line *)
       #b #T #_ #HT -T2
-      elim (pl_st_inv_dx … HF1 ??) -HF1 [3: // |2: skip ] (**) (* simplify line *)
+      elim (pl_st_inv_dx … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
       #c #V #T1 #T0 #_ #_ #HT0 -q -T1 -F1 destruct
 (* case 2: rc *)
     | destruct -F1 -F -T2 -b2
       @(ex2_2_intro … ◊ (q::r2)) // (**) (* auto needs some help here *)
 (* case 3: sn *)
-    | elim (pl_sts_inv_rc_abst_dx … HF2 ??) -b2 [3: // |2: skip ] (**) (* simplify line *)
+    | elim (pl_sts_inv_rc_abst_dx … HF2 …) -b2 [3: // |2: skip ] (**) (* simplify line *)
       #b #T #_ #HT -T2
-      elim (pl_st_inv_sn … HF1 ??) -HF1 [3: // |2: skip ] (**) (* simplify line *)
+      elim (pl_st_inv_sn … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
       #c #V1 #V #T0 #_ #_ #HT0 -c -q -V1 -F1 destruct
     ]
   ]
@@ -208,7 +208,7 @@ lapply (pl_sts_fwd_is_standard … H)
 [ #_ @(ex3_3_intro … ◊ ◊ ◊) // (**) (* auto needs some help here *)
 | #p #s #F1 #F #HF1 #HF2 #IHF1 #Hs
   lapply (is_standard_fwd_cons … Hs) #H
-  elim (IHF1 ?) // -IHF1 -H #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
+  elim (IHF1 …) // -IHF1 -H #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
   elim (in_whd_or_in_inner p) #Hp
   [ -Hs -F1 -F -V2 -T2 -b2
     @(ex3_3_intro … (p::r1) r2 r3) // /2 width=1/ (**) (* auto needs some help here *)
@@ -234,11 +234,10 @@ lapply (pl_sts_fwd_is_standard … H)
 ]
 qed-.
 
-axiom pl_sred_is_standard_pl_st: ∀p,M,M2. M ↦[p] M2 → ∀F. ⇓F = M →
+lemma pl_sred_is_standard_pl_st: ∀p,M,M2. M ↦[p] M2 → ∀F. ⇓F = M →
                                  ∀s,M1.{⊤}⇑ M1 Ⓡ↦*[s] F →
                                  is_standard (s@(p::◊)) →
                                  ∃∃F2. F Ⓡ↦[p] F2 & ⇓F2 = M2.
-(*
 #p #M #M2 #H elim H -p -M -M2
 [ #B #A #F #HF #s #M1 #HM1 #Hs
   lapply (is_standard_fwd_is_whd … Hs) -Hs // #Hs
@@ -248,38 +247,74 @@ axiom pl_sred_is_standard_pl_st: ∀p,M,M2. M ↦[p] M2 → ∀F. ⇓F = M →
   elim (carrier_inv_abst … HF) -HF #b #T #HT #HF destruct
   elim (pl_sts_fwd_abst_dx … HM1) #r1 #r2 #Hr1 #H destruct
   elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
-  elim (pl_sts_inv_pl_sreds … HM1 ?) // #M0 #_ #H -M1 -Hr1 destruct
-  elim (pl_sts_inv_rc_abst_dx … HT ??) -HT [3: // |2: skip ] #b0 #T0 #HT02 #H (**) (* simplify line *)
-  elim (boolean_inv_abst … (sym_eq … H)) -H #A0 #_ #H #_ -b0 -M0 destruct
+  elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
   >associative_append in Hs; #Hs
   lapply (is_standard_fwd_append_dx … Hs) -r1
   <(map_cons_append … r2 (p::◊)) #H
-  lapply (is_standard_inv_compatible_rc … H) -H #H
-  elim (IHA12 … HT02 ?) // -r2 -A0 -IHA12 #F2 #HF2 #H
+  lapply (is_standard_inv_compatible_rc … H) -H #Hp
+  elim (pl_sts_inv_rc_abst_dx … HT …) -HT [3: // |2: skip ] #b0 #T0 #HT02 #H (**) (* simplify line *)
+  elim (boolean_inv_abst … (sym_eq … H)) -H #A0 #_ #H #_ -b0 -M0 destruct
+  elim (IHA12 … HT02 …) // -r2 -A0 -IHA12 #F2 #HF2 #H
   @(ex2_intro … ({⊥}𝛌.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
 | #p #B1 #B2 #A #_ #IHB12 #F #HF #s #M1 #HM1 #Hs
   elim (carrier_inv_appl … HF) -HF #b #V #T #HV #HT #HF destruct
   elim (pl_sts_fwd_appl_dx … HM1) #r1 #r2 #r3 #Hr1 #_ #H destruct
   elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
-  elim (pl_sts_inv_pl_sreds … HM1 ?) // #M0 #_ #H -M1 -Hr1 destruct
-  elim (pl_sts_fwd_dx_sn_appl_dx … HT) -HT #b0 #V0 #T0 #T1 #HV0 #_ #H -T1
-  elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #H #_ #_ -b0 -M0 -T0 destruct
-  >associative_append in Hs; #Hs
-  lapply (is_standard_fwd_append_dx … Hs) -r1 #Hs
+  elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
   >associative_append in Hs; #Hs
-  lapply (is_standard_fwd_append_dx … Hs) -r2
+  lapply (is_standard_fwd_append_dx … Hs) -r1
+  >associative_append #Hs
+  lapply (is_standard_fwd_append_dx … Hs) -Hs
   <(map_cons_append … r3 (p::◊)) #H
-  lapply (is_standard_inv_compatible_sn … H) -H #H
-  elim (IHB12 … HV0 ?) // -r3 -B0 -IHB12 #F2 #HF2 #H
-  @(ex2_intro … ({⊥}@F2.{⊥}⇕T)) normalize // /2 width=1/ (**) (* auto needs some help here *)
-*)
+  lapply (is_standard_inv_compatible_sn … H) -H #Hp
+  elim (pl_sts_fwd_dx_sn_appl_dx … HT) -HT #b0 #V0 #T0 #T1 #HV0 #_ #H -T1 -r2
+  elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #H #_ #_ -b0 -M0 -T0 destruct
+  elim (IHB12 … HV0 …) // -r3 -B0 -IHB12 #G2 #HG2 #H
+  @(ex2_intro … ({⊥}@G2.{⊥}⇕T)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+| #p #B #A1 #A2 #_ #IHA12 #F #HF #s #M1 #HM1 #Hs
+  elim (carrier_inv_appl … HF) -HF #b #V #T #HV #HT #HF destruct
+  elim (pl_sts_fwd_appl_dx … HM1) #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
+  elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
+  elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
+  >associative_append in Hs; #Hs
+  lapply (is_standard_fwd_append_dx … Hs) -r1
+  >associative_append #Hs
+  elim (list_inv … r3)
+  [ #H destruct
+    elim (in_whd_or_in_inner p) #Hp
+    [ lapply (is_standard_fwd_is_whd … Hs) -Hs /2 width=1/ -Hp #Hs
+      lapply (is_whd_inv_dx … Hs) -Hs #H
+      lapply (is_whd_is_inner_inv … Hr2) -Hr2 // -H #H destruct
+      lapply (pl_sts_inv_nil … HT ?) -HT // #H
+      elim (boolean_inv_appl … H) -H #B0 #A0 #_ #_ #H #_ -M0 -B0 destruct
+      elim (IHA12 … A0 …) -IHA12 [3,5,6: // |2,4: skip ] (* simplify line *)
+      #F2 #HF2 #H
+      @(ex2_intro … ({b}@V.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+    | <(map_cons_append … r2 (p::◊)) in Hs; #H
+      lapply (is_standard_inv_compatible_dx … H ?) -H /3 width=1/ -Hp #Hp
+      >append_nil in HT; #HT
+      elim (pl_sts_inv_dx_appl_dx … HT …) -HT [3: // |2: skip ] (* simplify line *) 
+      #T0 #HT0 #H
+      elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #_ #H #_ -M0 -B0 destruct
+      elim (IHA12 … HT0 …) // -r2 -A0 -IHA12 #F2 #HF2 #H
+      @(ex2_intro … ({b}@V.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+    ]
+  | -IHA12 -Hr2 -M0 * #q #r #H destruct
+    lapply (is_standard_fwd_append_dx … Hs) -r2 #Hs
+    lapply (is_standard_fwd_sle … Hs) -r #H
+    elim (sle_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
+    #q0 #_ #H destruct
+  ]
+]
+qed-.
+
 theorem pl_sreds_is_standard_pl_sts: ∀s,M1,M2. M1 ↦*[s] M2 → is_standard s →
                                      ∃∃F2. {⊤}⇑ M1 Ⓡ↦*[s] F2 & ⇓F2 = M2.
 #s #M1 #M2 #H @(lstar_ind_r … s M2 H) -s -M2 /2 width=3/
 #p #s #M #M2 #_ #HM2 #IHM1 #Hsp
 lapply (is_standard_fwd_append_sn … Hsp) #Hs
 elim (IHM1 Hs) -IHM1 -Hs #F #HM1 #H
-elim (pl_sred_is_standard_pl_st … HM2 … HM1 ?) -HM2 // -M -Hsp #F2 #HF2 #HFM2
+elim (pl_sred_is_standard_pl_st … HM2 … HM1 …) -HM2 // -M -Hsp #F2 #HF2 #HFM2
 lapply (pl_sts_step_dx … HM1 … HF2) -F
 #H @(ex2_intro … F2) // (**) (* auto needs some help here *)
 qed-.