X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Fcpx.ma;h=7b99bd8de0098a3536129dee3d27f92dd7d2b430;hb=2ba2dc23443ad764adab652e06d6f5ed10bd912d;hp=194871b5010e5977e2efbdc9c0bbc7ee370d2f63;hpb=fdb2c62b58006b82c015ba70b494d50c7860e28f;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
index 194871b50..7b99bd8de 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
@@ -23,8 +23,8 @@ inductive cpx (h) (g): relation4 genv lenv term term ≝
 | cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
 | cpx_sort : ∀G,L,k,l. deg h g k (l+1) → cpx h g G L (⋆k) (⋆(next h k))
 | cpx_delta: ∀I,G,L,K,V,V2,W2,i.
-             ⇩[0, i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
-             ⇧[0, i + 1] V2 ≡ W2 → cpx h g G L (#i) W2
+             ⇩[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
+             ⇧[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
 | cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
              cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
              cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
@@ -53,51 +53,54 @@ interpretation
 lemma lsubr_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) lsubr.
 #h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
 [ //
-| /2 width=2/
+| /2 width=2 by cpx_sort/
 | #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
   elim (lsubr_fwd_ldrop2_bind … HL12 … HLK1) -HL12 -HLK1 *
-  [ /3 width=7/ | /4 width=7/ ]
-|4,9: /4 width=1/
-|5,7,8: /3 width=1/
-|6,10: /4 width=3/
+  /4 width=7 by cpx_delta, cpx_ti/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lsubr_bind/
+|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsubr_bind/
 ]
 qed-.
 
 (* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
 lemma cpx_refl: ∀h,g,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, g] T.
-#h #g #G #T elim T -T // * /2 width=1/
+#h #g #G #T elim T -T // * /2 width=1 by cpx_bind, cpx_flat/
 qed.
 
 lemma cpr_cpx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=3/ /2 width=7/
+#h #g #G #L #T1 #T2 #H elim H -L -T1 -T2
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_beta, cpx_theta/
 qed.
 
 lemma cpx_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, g] ②{I}V2.T.
-#h #g * /2 width=1/ qed.
+#h #g * /2 width=1 by cpx_bind, cpx_flat/
+qed.
 
-lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
+lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓑ{I}V) →
                   ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⇧[d, 1] T ≡ T2.
 #h #g #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK /2 width=4/
-  elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
+[ * #i #L #d /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
+  elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
   destruct
   elim (lift_total V 0 (i+1)) #W #HVW
-  elim (lift_split … HVW i i) // /3 width=7/
+  elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/
 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
-  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+  [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /3 width=9 by cpx_bind, ldrop_drop, lift_bind, ex2_2_intro/
+  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, ex2_2_intro/
   ]
 ]
 qed-.
 
 lemma cpx_append: ∀h,g,G. l_appendable_sn … (cpx h g G).
-#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2 // /2 width=1/ /2 width=3/
+#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
+/2 width=3 by cpx_sort, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_ti, cpx_beta, cpx_theta/
 #I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
 lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpx_delta … I … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
+@(cpx_delta … I … (L@@K0) V1 … HVW2) // 
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
 qed.
 
 (* Basic inversion lemmas ***************************************************)
@@ -105,12 +108,12 @@ qed.
 fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
                         ∨∨ T2 = ⓪{J}
                          | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
-                         | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
-                                         ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
+                         | ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                         ⇧[O, i+1] V2 ≡ T2 & J = LRef i.
 #G #h #g #L #T1 #T2 * -L -T1 -T2
-[ #I #G #L #J #H destruct /2 width=1/
-| #G #L #k #l #Hkl #J #H destruct /3 width=5/
-| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9/
+[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
+| #G #L #k #l #Hkl #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
+| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by or3_intro2, ex4_5_intro/
 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
 | #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
@@ -124,30 +127,36 @@ qed-.
 lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
                      ∨∨ T2 = ⓪{J}
                       | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
-                      | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
-                                      ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
+                      | ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                      ⇧[O, i+1] V2 ≡ T2 & J = LRef i.
 /2 width=3 by cpx_inv_atom1_aux/ qed-.
 
 lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
                      ∃∃l. deg h g k (l+1) & T2 = ⋆(next h k).
 #h #g #G #L #T2 #k #H
-elim (cpx_inv_atom1 … H) -H /2 width=1/ *
-[ #k0 #l0 #Hkl0 #H1 #H2 destruct /3 width=4/
+elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+[ #k0 #l0 #Hkl0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
 | #I #K #V #V2 #i #_ #_ #_ #H destruct
 ]
 qed-.
 
 lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
                      T2 = #i ∨
-                     ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
-                                 ⇧[O, i + 1] V2 ≡ T2.
+                     ∃∃I,K,V,V2. ⇩[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                 ⇧[O, i+1] V2 ≡ T2.
 #h #g #G #L #T2 #i #H
-elim (cpx_inv_atom1 … H) -H /2 width=1/ *
+elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
 [ #k #l #_ #_ #H destruct
-| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7/
+| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
 ]
 qed-.
 
+lemma cpx_inv_lref1_ge: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 → |L| ≤ i → T2 = #i.
+#h #g #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
+#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (ldrop_fwd_length_lt2 … HLK) -K -I -V1
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed-.
+
 lemma cpx_inv_gref1: ∀h,g,G,L,T2,p.  ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
 #h #g #G #L #T2 #p #H
 elim (cpx_inv_atom1 … H) -H // *
@@ -167,9 +176,9 @@ fact cpx_inv_bind1_aux: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
 [ #I #G #L #b #J #W #U1 #H destruct
 | #G #L #k #l #_ #b #J #W #U1 #H destruct
 | #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/
+| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
-| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/
+| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
 | #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
 | #G #L #V1 #V2 #T #_ #b #J #W #U1 #H destruct
 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
@@ -191,7 +200,7 @@ lemma cpx_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, g]
                      ) ∨
                      ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T & a = true.
 #h #g #a #G #L #V1 #T1 #U2 #H
-elim (cpx_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
+elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_abst1: ∀h,g,a,G,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g] U2 →
@@ -199,7 +208,7 @@ lemma cpx_inv_abst1: ∀h,g,a,G,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g
                               U2 = ⓛ{a} V2. T2.
 #h #g #a #G #L #V1 #T1 #U2 #H
 elim (cpx_inv_bind1 … H) -H *
-[ /3 width=5/
+[ /3 width=5 by ex3_2_intro/
 | #T #_ #_ #_ #H destruct
 ]
 qed-.
@@ -223,12 +232,12 @@ fact cpx_inv_flat1_aux: ∀h,g,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
 | #G #L #k #l #_ #J #W #U1 #H destruct
 | #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/
+| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or5_intro0, ex3_2_intro/
 | #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
-| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/
-| #G #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=1/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/
+| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or5_intro1, conj/
+| #G #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=1 by or5_intro2, conj/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or5_intro3, ex6_6_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or5_intro4, ex7_7_intro/
 ]
 qed-.
 
@@ -257,10 +266,10 @@ lemma cpx_inv_appl1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2
                                               ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
                                               U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
 #h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
-[ /3 width=5/
+[ /3 width=5 by or3_intro0, ex3_2_intro/
 |2,3: #_ #H destruct
-| /3 width=11/
-| /3 width=13/
+| /3 width=11 by or3_intro1, ex5_6_intro/
+| /3 width=13 by or3_intro2, ex6_7_intro/
 ]
 qed-.
 
@@ -270,7 +279,7 @@ lemma cpx_inv_appl1_simple: ∀h,g,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, g
                                      U = ⓐV2.T2.
 #h #g #G #L #V1 #T1 #U #H #HT1
 elim (cpx_inv_appl1 … H) -H *
-[ /2 width=5/
+[ /2 width=5 by ex3_2_intro/
 | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
   elim (simple_inv_bind … HT1)
 | #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
@@ -284,8 +293,8 @@ lemma cpx_inv_cast1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, g] U2 
                       | ⦃G, L⦄ ⊢ U1 ➡[h, g] U2
                       | ⦃G, L⦄ ⊢ V1 ➡[h, g] U2.
 #h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
-[ /3 width=5/
-|2,3: /2 width=1/
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+|2,3: /2 width=1 by or3_intro1, or3_intro2/
 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
 ]
@@ -298,7 +307,7 @@ lemma cpx_fwd_bind1_minus: ∀h,g,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡
                                     T = -ⓑ{I}V2.T2.
 #h #g #I #G #L #V1 #T1 #T #H #b
 elim (cpx_inv_bind1 … H) -H *
-[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpx_bind, ex2_2_intro/
 | #T2 #_ #_ #H destruct
 ]
 qed-.
@@ -314,7 +323,7 @@ lemma cpx_fwd_shift1: ∀h,g,G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T 
   [ #V0 #T0 #_ #HT10 #H destruct
     elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
     >append_length >HL12 -HL12
-    @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+    @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *)
   | #T #_ #_ #H destruct
   ]
 ]