- test.ma on the disambiguation bug moved to ONAG (just out of the way)

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lift.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lift.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lift.ma
index 7e7961eabe9f4611e081169f8a39ccd3a7061049..ba117d867f12573db855baa9ca910921ae324bd8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lift.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/relations/rlift_4.ma".
 include "basic_2/grammar/term_weight.ma".
 include "basic_2/grammar/term_simple.ma".
 
@@ -20,7 +21,7 @@ include "basic_2/grammar/term_simple.ma".
 (* Basic_1: includes:
             lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
 *)
-inductive lift: nat → nat → relation term ≝
+inductive lift: relation4 nat nat term term ≝
 | lift_sort   : ∀k,d,e. lift d e (⋆k) (⋆k)
 | lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
 | lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
@@ -35,22 +36,14 @@ inductive lift: nat → nat → relation term ≝
 
 interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
 
-definition t_liftable: relation term → Prop ≝
-                       λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 →
-                       ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2.
-
-definition t_deliftable_sn: relation term → Prop ≝
-                            λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
-                            ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2.
-
 (* Basic inversion lemmas ***************************************************)
 
-fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+fact lift_inv_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
 #d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
-qed.
+qed-.
 
-lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
-/2 width=4/ qed-.
+lemma lift_inv_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
+/2 width=4 by lift_inv_O2_aux/ qed-.
 
 fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
 #d #e #T1 #T2 * -d -e -T1 -T2 //
@@ -58,10 +51,10 @@ fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k 
 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 ]
-qed.
+qed-.
 
 lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed-.
+/2 width=5 by lift_inv_sort1_aux/ qed-.
 
 fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i →
                          (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
@@ -73,11 +66,11 @@ fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i →
 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 ]
-qed.
+qed-.
 
 lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
                       (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_lref1_aux/ qed-.
 
 lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i.
 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
@@ -97,10 +90,10 @@ fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p →
 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 ]
-qed.
+qed-.
 
 lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
-/2 width=5/ qed-.
+/2 width=5 by lift_inv_gref1_aux/ qed-.
 
 fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
@@ -114,12 +107,12 @@ fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/
 | #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
 ]
-qed.
+qed-.
 
 lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⇧[d,e] ⓑ{a,I} V1. U1 ≡ T2 →
                       ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
                                T2 = ⓑ{a,I} V2. U2.
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_bind1_aux/ qed-.
 
 fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
@@ -133,12 +126,12 @@ fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
 ]
-qed.
+qed-.
 
 lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] ⓕ{I} V1. U1 ≡ T2 →
                       ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
                                T2 = ⓕ{I} V2. U2.
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_flat1_aux/ qed-.
 
 fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
 #d #e #T1 #T2 * -d -e -T1 -T2 //
@@ -146,11 +139,11 @@ fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k 
 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 ]
-qed.
+qed-.
 
 (* Basic_1: was: lift_gen_sort *)
 lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5/ qed-.
+/2 width=5 by lift_inv_sort2_aux/ qed-.
 
 fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i →
                          (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
@@ -162,12 +155,12 @@ fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i →
 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 ]
-qed.
+qed-.
 
 (* Basic_1: was: lift_gen_lref *)
 lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
                       (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_lref2_aux/ qed-.
 
 (* Basic_1: was: lift_gen_lref_lt *)
 lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
@@ -200,10 +193,10 @@ fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p →
 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 ]
-qed.
+qed-.
 
 lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
-/2 width=5/ qed-.
+/2 width=5 by lift_inv_gref2_aux/ qed-.
 
 fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
@@ -217,13 +210,13 @@ fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5/
 | #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
 ]
-qed.
+qed-.
 
 (* Basic_1: was: lift_gen_bind *)
 lemma lift_inv_bind2: ∀d,e,T1,a,I,V2,U2. ⇧[d,e] T1 ≡ ⓑ{a,I} V2. U2 →
                       ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
                                T1 = ⓑ{a,I} V1. U1.
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_bind2_aux/ qed-.
 
 fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
@@ -237,13 +230,13 @@ fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V2 #U2 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
 ]
-qed.
+qed-.
 
 (* Basic_1: was: lift_gen_flat *)
 lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡  ⓕ{I} V2. U2 →
                       ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
                                T1 = ⓕ{I} V1. U1.
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_flat2_aux/ qed-.
 
 lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → ⊥.
 #d #e #J #V elim V -V
@@ -276,6 +269,22 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
+lemma lift_fwd_pair1: ∀I,T2,V1,U1,d,e. ⇧[d,e] ②{I}V1.U1 ≡ T2 →
+                      ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & T2 = ②{I}V2.U2.
+* [ #a ] #I #T2 #V1 #U1 #d #e #H
+[ elim (lift_inv_bind1 … H) -H /2 width=4/
+|  elim (lift_inv_flat1 … H) -H /2 width=4/
+] 
+qed-.
+
+lemma lift_fwd_pair2: ∀I,T1,V2,U2,d,e. ⇧[d,e] T1 ≡ ②{I}V2.U2 →
+                      ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & T1 = ②{I}V1.U1.
+* [ #a ] #I #T1 #V2 #U2 #d #e #H
+[ elim (lift_inv_bind2 … H) -H /2 width=4/
+|  elim (lift_inv_flat2 … H) -H /2 width=4/
+] 
+qed-.
+
 lemma lift_fwd_tw: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ♯{T1} = ♯{T2}.
 #d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
 qed-.
@@ -378,24 +387,6 @@ lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
 ]
 qed.
 
-lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R).
-#R #HR #T1 #T2 #H elim H -T2
-[ /3 width=7/
-| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2
-  elim (lift_total T d e) /3 width=9/
-]
-qed.
-
-lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R).
-#R #HR #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #T1 #d #e #HTU1
-  elim (HR … HU12 … HTU1) -U1 /3 width=3/
-| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1
-  elim (IHU1 … HTU1) -U1 #T #HTU #HT1
-  elim (HR … HU2 … HTU) -U /3 width=5/
-]
-qed-.
-
 (* Basic_1: removed theorems 7:
             lift_head lift_gen_head
             lift_weight_map lift_weight lift_weight_add lift_weight_add_O