- xoa: more existential types

[helm.git] / matita / matita / lib / lambda-delta / substitution / lift_defs.ma

diff --git a/matita/matita/lib/lambda-delta/substitution/lift_defs.ma b/matita/matita/lib/lambda-delta/substitution/lift_defs.ma
index 86e1b751143fc32aecc392ad03cf4839ccb69ed3..1f9be7fe2f4aeafb984152b6cfe641e38b7cad4b 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/lift_defs.ma
+++ b/matita/matita/lib/lambda-delta/substitution/lift_defs.ma
@@ -29,12 +29,27 @@ interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
 
 (* The basic properties *****************************************************)
 
-lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d,e] #(i - e) ≡ #i.
+lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
 #d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
 qed.
 
+lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
+#T elim T -T
+[ //
+| #i #d elim (lt_or_ge i d) /2/
+| #I elim I -I /2/
+]
+qed.
+
 (* The basic inversion lemmas ***********************************************)
 
+lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
+qed.
+
+lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
+/2/ qed.
+
 lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
 [ #i #d #e #_ #k #H destruct
@@ -44,8 +59,7 @@ lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k 
 qed.
 
 lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-#d #e #T2 #k #H lapply (lift_inv_sort1_aux … H) /2/
-qed.
+/2 width=5/ qed.
 
 lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
                           (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
@@ -60,7 +74,18 @@ qed.
 
 lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
                       (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-#d #e #T2 #i #H lapply (lift_inv_lref1_aux … H) /2/
+/2/ 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 * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (lt_false … Hdd)
+qed.
+
+lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (lt_false … Hdd)
 qed.
 
 lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
@@ -79,8 +104,7 @@ qed.
 lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
                       ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
                                T2 = 𝕓{I} V2. U2.
-#d #e #T2 #I #V1 #U1 #H lapply (lift_inv_bind1_aux … H) /2/
-qed.
+/2/ qed.
 
 lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                           ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
@@ -98,8 +122,7 @@ qed.
 lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
                       ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
                                T2 = 𝕗{I} V2. U2.
-#d #e #T2 #I #V1 #U1 #H lapply (lift_inv_flat1_aux … H) /2/
-qed.
+/2/ qed.
 
 lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
@@ -110,8 +133,7 @@ lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k 
 qed.
 
 lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-#d #e #T1 #k #H lapply (lift_inv_sort2_aux … H) /2/
-qed.
+/2 width=5/ qed.
 
 lemma 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)).
@@ -126,7 +148,18 @@ qed.
 
 lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
                       (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-#d #e #T1 #i #H lapply (lift_inv_lref2_aux … H) /2/
+/2/ qed.
+
+lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (plus_lt_false … Hdd)
+qed.
+
+lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (plus_lt_false … Hdd)
 qed.
 
 lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
@@ -145,8 +178,7 @@ qed.
 lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡  𝕓{I} V2. U2 →
                       ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
                                T1 = 𝕓{I} V1. U1.
-#d #e #T1 #I #V2 #U2 #H lapply (lift_inv_bind2_aux … H) /2/
-qed.
+/2/ qed.
 
 lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                           ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
@@ -164,5 +196,4 @@ qed.
 lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡  𝕗{I} V2. U2 →
                       ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
                                T1 = 𝕗{I} V1. U1.
-#d #e #T1 #I #V2 #U2 #H lapply (lift_inv_flat2_aux … H) /2/
-qed.
+/2/ qed.