nat: we added a non-indexed theorem

[helm.git] / matita / matita / lib / lambda / levels / interpretations.ma

diff --git a/matita/matita/lib/lambda/levels/interpretations.ma b/matita/matita/lib/lambda/levels/interpretations.ma
index 0c06ab442dacaace5f6215b232b8d3308f8631f3..99643b17adfdf165067f374bf17d4bbed589b227 100644 (file)
--- a/matita/matita/lib/lambda/levels/interpretations.ma
+++ b/matita/matita/lib/lambda/levels/interpretations.ma
@@ -17,7 +17,7 @@ include "lambda/notation/functions/forward_3.ma".
 include "lambda/notation/functions/backward_1.ma".
 include "lambda/notation/functions/backward_3.ma".
 include "lambda/terms/iterated_abstraction.ma".
-include "lambda/levels/term.ma".
+include "lambda/levels/iterated_abstraction.ma".
 
 (* INTERPRETATIONS **********************************************************)
 
@@ -33,6 +33,10 @@ interpretation "forward interpretation (term by depth) general"
 interpretation "forward interpretation (term by depth)"
    'Forward M = (bylevel O O M).
 
+lemma bylevel_abst: ∀i,h,d,M. ⇑[d, h] 𝛌i. M = ⇑[i+d, i+h] M.
+#i elim i -i normalize //
+qed.
+
 let rec bydepth h d M on M ≝ match M with
 [ LVRef i e   ⇒ 𝛌i.#(tri … e (d+i-h) (d+i-h-e-1) e e)
 | LAppl i C A ⇒ 𝛌i.@(bydepth h (d+i) C).(bydepth h (d+i) A)
@@ -44,27 +48,43 @@ interpretation "backward interpretation (term by level) general"
 interpretation "backward interpretation (term by level)"
    'Backward M = (bydepth O O M).
 
-theorem by_depth_level_gen: ∀M,e,d,h. d ≤ e + h → ⇓[e, e+h-d] ⇑[d, h] M = 𝛌h.M.
+lemma by_depth_level_gen: ∀M,e,d,h. d ≤ e + h → ⇓[e, e+h-d] ⇑[d, h] M = 𝛌h.M.
 #M elim M -M normalize
 [ #i #e #d #h #Hdeh >(minus_minus_m_m … Hdeh)
   elim (lt_or_eq_or_gt i d) #Hid
   [ >(tri_lt ???? … Hid) >(tri_lt ???? d (d-i-1))
-    [ >minus_minus_associative /2 width=1 by monotonic_le_minus_r/
-      <minus_plus_m_m >minus_minus_associative /2 width=1 by lt_to_le/
-    | /2 width=1 by monotonic_lt_minus_l/
-    ]
+    /5 width=1 by minus_le_minus_minus_comm, monotonic_lt_minus_l, eq_f/ 
   | destruct >(tri_eq ???? …) >(tri_eq ???? …) //
   | >(tri_gt ???? … Hid) >(tri_gt ???? … Hid) //
   ]
 | #A #IHA #e #d #h #Hdeh lapply (IHA e (d+1) (h+1) ?) -IHA
   /2 width=1 by le_S_S, eq_f2/
 | #C #A #IHC #IHA #e #d #h #Hdeh
-  lapply (IHC (e+h) d 0 ?) -IHC
-  lapply (IHA (e+h) d 0 ?) -IHA
+  lapply (IHC (e+h) d 0 ?) -IHC // lapply (IHA (e+h) d 0 ?) -IHA //
   normalize /2 width=1 by/
 ]
-qed.
+qed-.
 
-lemma by_depth_level: ∀M. ⇓⇑M = M.
+theorem by_depth_level: ∀M. ⇓⇑M = M.
 #M lapply (by_depth_level_gen M 0 0 0 ?) normalize //
 qed.
+
+lemma by_level_depth_gen: ∀M,e,d,h. d ≤ e → ⇑[d, h] ⇓[e, e-d] M = 𝛌h.M.
+#M elim M -M
+[ #i #k #e #d #h #Hde >bylevel_abst normalize >(minus_plus_minus_l … Hde)
+  elim (lt_or_eq_or_gt k (i+d)) #Hkid
+  [ >(tri_lt ???? … Hkid) >(tri_lt ???? (i+d) (i+d-k-1))
+    /5 width=1 by minus_le_minus_minus_comm, monotonic_lt_minus_l, eq_f/ 
+  | destruct >(tri_eq ???? …) >(tri_eq ???? …) //
+  | >(tri_gt ???? … Hkid) >(tri_gt ???? … Hkid) //
+  ]
+| #i #C #A #IHC #IHA #e #d #h #Hdeh >bylevel_abst normalize
+  lapply (IHC (e+i) (i+d) 0 ?) -IHC /2 width=1 by monotonic_le_plus_r/
+  lapply (IHA (e+i) (i+d) 0 ?) -IHA /2 width=1 by monotonic_le_plus_r/
+  /3 width=1 by eq_f3, eq_f2/
+]
+qed-.
+
+theorem by_level_depth: ∀M. ⇑⇓M = M.
+#M lapply (by_level_depth_gen M 0 0 0 ?) //
+qed.