From b8e8c61042dd7d4d8bc00971e1ebcd6858064682 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Fri, 28 Dec 2018 16:25:21 +0100
Subject: [PATCH] reverse_complexity lib restored

the toolkit.ma file contains errors. I have
commented out a few places
---
 .../matita/lib/reverse_complexity/almost.ma   |   24 +
 .../matita/lib/reverse_complexity/basics.ma   |  106 ++
 .../reverse_complexity/big_O.ma               |    0
 .../lib/reverse_complexity/bigops_compl.ma    |  607 ++++++
 .../lib/reverse_complexity/complexity.ma      |  290 +++
 .../reverse_complexity/gap.ma                 |    0
 .../reverse_complexity/hierarchy.ma           |  210 +--
 .../lib/reverse_complexity/sigma_diseq.ma     |  153 ++
 .../reverse_complexity/speed_clean.ma         |    0
 .../reverse_complexity/speed_def.ma           |    0
 .../reverse_complexity/speed_new.ma           |    0
 .../reverse_complexity/speedup.ma}            |  324 +---
 .../matita/lib/reverse_complexity/toolkit.ma  | 1623 +++++++++++++++++
 13 files changed, 2908 insertions(+), 429 deletions(-)
 create mode 100644 matita/matita/lib/reverse_complexity/almost.ma
 create mode 100644 matita/matita/lib/reverse_complexity/basics.ma
 rename matita/matita/{broken_lib => lib}/reverse_complexity/big_O.ma (100%)
 create mode 100644 matita/matita/lib/reverse_complexity/bigops_compl.ma
 create mode 100644 matita/matita/lib/reverse_complexity/complexity.ma
 rename matita/matita/{broken_lib => lib}/reverse_complexity/gap.ma (100%)
 rename matita/matita/{broken_lib => lib}/reverse_complexity/hierarchy.ma (75%)
 create mode 100644 matita/matita/lib/reverse_complexity/sigma_diseq.ma
 rename matita/matita/{broken_lib => lib}/reverse_complexity/speed_clean.ma (100%)
 rename matita/matita/{broken_lib => lib}/reverse_complexity/speed_def.ma (100%)
 rename matita/matita/{broken_lib => lib}/reverse_complexity/speed_new.ma (100%)
 rename matita/matita/{broken_lib/reverse_complexity/toolkit.ma => lib/reverse_complexity/speedup.ma} (71%)
 create mode 100644 matita/matita/lib/reverse_complexity/toolkit.ma

diff --git a/matita/matita/lib/reverse_complexity/almost.ma b/matita/matita/lib/reverse_complexity/almost.ma
new file mode 100644
index 000000000..75578d0f9
--- /dev/null
+++ b/matita/matita/lib/reverse_complexity/almost.ma
@@ -0,0 +1,24 @@
+include "reverse_complexity/speedup.ma".
+
+definition aes : (nat → nat) → (nat → nat) → Prop ≝ λf,g.
+  ∃n.∀m. n ≤ m → f m = g m.
+
+lemma CF_almost: ∀f,g,s. CF s g → aes g f → CF s f.
+#f #g #s #CFg * #n lapply CFg -CFg lapply g -g
+elim n 
+  [#g #CFg #H @(ext_CF … g) [#m @H // |//]
+  |#i #Hind #g #CFg #H
+   @(Hind (λx. if eqb i x then f i else g x))
+    [@CF_if 
+      [@(O_to_CF … MSC) [@le_to_O @(MSC_in … CFg)] @CF_eqb //
+      |@(O_to_CF … MSC) [@le_to_O @(MSC_in … CFg)] @CF_const
+      |@CFg
+      ]
+    |#m #leim cases (le_to_or_lt_eq … leim)
+      [#ltim lapply (lt_to_not_eq … ltim) #noteqim 
+       >(not_eq_to_eqb_false … noteqim) @H @ltim
+      |#eqim >eqim >eqb_n_n //
+      ]
+    ]
+  ]
+qed.
\ No newline at end of file
diff --git a/matita/matita/lib/reverse_complexity/basics.ma b/matita/matita/lib/reverse_complexity/basics.ma
new file mode 100644
index 000000000..de383d1d0
--- /dev/null
+++ b/matita/matita/lib/reverse_complexity/basics.ma
@@ -0,0 +1,106 @@
+include "basics/types.ma".
+include "arithmetics/nat.ma".
+
+(********************************** pairing ***********************************)
+axiom pair: nat → nat → nat.
+axiom fst : nat → nat.
+axiom snd : nat → nat.
+
+interpretation "abstract pair" 'pair f g = (pair f g). 
+
+axiom fst_pair: ∀a,b. fst 〈a,b〉 = a.
+axiom snd_pair: ∀a,b. snd 〈a,b〉 = b.
+axiom surj_pair: ∀x. ∃a,b. x = 〈a,b〉. 
+
+axiom le_fst : ∀p. fst p ≤ p.
+axiom le_snd : ∀p. snd p ≤ p.
+axiom le_pair: ∀a,a1,b,b1. a ≤ a1 → b ≤ b1 → 〈a,b〉 ≤ 〈a1,b1〉.
+
+lemma expand_pair: ∀x. x = 〈fst x, snd x〉. 
+#x lapply (surj_pair x) * #a * #b #Hx >Hx >fst_pair >snd_pair //
+qed.
+
+(************************************* U **************************************)
+axiom U: nat → nat →nat → option nat. 
+
+axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
+  U i x n = Some ? y → U i x m = Some ? y.
+  
+lemma unique_U: ∀i,x,n,m,yn,ym.
+  U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
+#i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
+  [#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
+  |#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
+   >Hn #HS destruct (HS) //
+  ]
+qed.
+
+definition code_for ≝ λf,i.∀x.
+  ∃n.∀m. n ≤ m → U i x m = f x.
+
+definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y. 
+
+notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}. 
+
+lemma terminate_dec: ∀i,x,n. {i ⊙ x} ↓ n ∨ ¬ {i ⊙ x} ↓ n.
+#i #x #n normalize cases (U i x n)
+  [%2 % * #y #H destruct|#y %1 %{y} //]
+qed.
+  
+lemma monotonic_terminate: ∀i,x,n,m.
+  n ≤ m → {i ⊙ x} ↓ n → {i ⊙ x} ↓ m.
+#i #x #n #m #lenm * #z #H %{z} @(monotonic_U … H) //
+qed.
+
+definition termb ≝ λi,x,t. 
+  match U i x t with [None ⇒ false |Some y ⇒ true].
+
+lemma termb_true_to_term: ∀i,x,t. termb i x t = true → {i ⊙ x} ↓ t.
+#i #x #t normalize cases (U i x t) normalize [#H destruct | #y #_ %{y} //]
+qed.    
+
+lemma term_to_termb_true: ∀i,x,t. {i ⊙ x} ↓ t → termb i x t = true.
+#i #x #t * #y #H normalize >H // 
+qed.    
+
+definition out ≝ λi,x,r. 
+  match U i x r with [ None ⇒ 0 | Some z ⇒ z]. 
+
+definition bool_to_nat: bool → nat ≝ 
+  λb. match b with [true ⇒ 1 | false ⇒ 0]. 
+  
+coercion bool_to_nat. 
+
+definition pU : nat → nat → nat → nat ≝ λi,x,r.〈termb i x r,out i x r〉.
+
+lemma pU_vs_U_Some : ∀i,x,r,y. pU i x r = 〈1,y〉 ↔ U i x r = Some ? y.
+#i #x #r #y % normalize
+  [cases (U i x r) normalize 
+    [#H cut (0=1) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H @H] 
+     #H1 destruct
+    |#a #H cut (a=y) [lapply (eq_f … snd … H) >snd_pair >snd_pair #H1 @H1] 
+     #H1 //
+    ]
+  |#H >H //]
+qed.
+  
+lemma pU_vs_U_None : ∀i,x,r. pU i x r = 〈0,0〉 ↔ U i x r = None ?.
+#i #x #r % normalize
+  [cases (U i x r) normalize //
+   #a #H cut (1=0) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H1 @H1] 
+   #H1 destruct
+  |#H >H //]
+qed.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+qed.
+
+
+definition total ≝ λf.λx:nat. Some nat (f x).
+
+
diff --git a/matita/matita/broken_lib/reverse_complexity/big_O.ma b/matita/matita/lib/reverse_complexity/big_O.ma
similarity index 100%
rename from matita/matita/broken_lib/reverse_complexity/big_O.ma
rename to matita/matita/lib/reverse_complexity/big_O.ma
diff --git a/matita/matita/lib/reverse_complexity/bigops_compl.ma b/matita/matita/lib/reverse_complexity/bigops_compl.ma
new file mode 100644
index 000000000..032174a0c
--- /dev/null
+++ b/matita/matita/lib/reverse_complexity/bigops_compl.ma
@@ -0,0 +1,607 @@
+include "reverse_complexity/complexity.ma".
+include "reverse_complexity/sigma_diseq.ma".
+
+include alias "reverse_complexity/basics.ma".
+
+lemma bigop_prim_rec: ∀a,b,c,p,f,x.
+  bigop (b x-a x) (λi:ℕ.p 〈i+a x,x〉) ? (c 〈a x,x〉) plus (λi:ℕ.f 〈i+a x,x〉) = 
+    prim_rec c 
+    (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 
+        then plus (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma bigop_prim_rec_dec: ∀a,b,c,p,f,x.
+  bigop (b x-a x) (λi:ℕ.p 〈b x -i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x-i,x〉) = 
+    prim_rec c 
+    (λi.if p 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉 
+        then plus (f 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈b x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma bigop_prim_rec_dec1: ∀a,b,c,p,f,x.
+  bigop (S(b x)-a x) (λi:ℕ.p 〈b x - i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x- i,x〉) = 
+    prim_rec c 
+    (λi.if p 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉 
+        then plus (f 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (S(b x)-a x) 〈b x,x〉.
+#a #b #c #p #f #x elim (S(b x)-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma sum_prim_rec1: ∀a,b,p,f,x.
+  ∑_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 
+        then f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 + fst (snd i) 
+        else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed. 
+
+lemma bigop_plus_c: ∀k,p,f,c.
+  c k + bigop k (λi.p i) ? 0 plus (λi.f i) = 
+    bigop k (λi.p i) ? (c k) plus (λi.f i).
+#k #p #f elim k [normalize //]
+#i #Hind #c cases (true_or_false (p i)) #Hcase
+[>bigop_Strue // >bigop_Strue // <associative_plus >(commutative_plus ? (f i))
+ >associative_plus @eq_f @Hind
+|>bigop_Sfalse // >bigop_Sfalse // 
+]
+qed.
+
+
+(*********************************** maximum **********************************) 
+
+lemma max_gen: ∀a,b,p,f,x. a ≤b → 
+  max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = max_{i < b | leb a i ∧ p 〈i,x〉 }(f 〈i,x〉).
+#a #b #p #f #x @(bigop_I_gen ????? MaxA) 
+qed.
+
+lemma max_prim_rec_base: ∀a,b,p,f,x. a ≤b → 
+  max_{i < b| p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b 
+  [normalize //
+  |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma max_prim_rec: ∀a,b,p,f,x. a ≤b → 
+  max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if leb a (fst i) ∧ p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b 
+  [normalize //
+  |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+   cases (true_or_false (leb a i ∧ p 〈i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma max_prim_rec1: ∀a,b,p,f,x.
+  max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 
+        then max (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+(* the argument is 〈b-a,〈a,x〉〉 *)
+
+definition max_unary_pr ≝ λp,f.unary_pr (λx.0) 
+    (λi. 
+      let k ≝ fst i in
+      let r ≝ fst (snd i) in
+      let a ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      if p 〈k + a,x〉 then max (f 〈k+a,x〉) r else r).
+
+lemma max_unary_pr1: ∀a,b,p,f,x.
+  max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    ((max_unary_pr p f) ∘ (λx.〈b x - a x,〈a x,x〉〉)) x.
+#a #b #p #f #x normalize >fst_pair >snd_pair @max_prim_rec1
+qed.
+      
+definition aux_compl ≝ λhp,hf.λi.
+  let k ≝ fst i in 
+  let r ≝ fst (snd i) in 
+  let a ≝ fst (snd (snd i)) in 
+  let x ≝ snd (snd (snd i)) in 
+  hp 〈k+a,x〉 + hf 〈k+a,x〉 + (* was MSC r*) MSC i .
+  
+definition aux_compl1 ≝ λhp,hf.λi.
+  let k ≝ fst i in 
+  let r ≝ fst (snd i) in 
+  let a ≝ fst (snd (snd i)) in 
+  let x ≝ snd (snd (snd i)) in 
+  hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+
+lemma aux_compl1_def: ∀k,r,m,hp,hf. 
+  aux_compl1 hp hf 〈k,〈r,m〉〉 = 
+  let a ≝ fst m in 
+  let x ≝ snd m in 
+  hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #m #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma aux_compl1_def1: ∀k,r,a,x,hp,hf. 
+  aux_compl1 hp hf 〈k,〈r,〈a,x〉〉〉 = hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #a #x #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair 
+>fst_pair >snd_pair //
+qed.
+
+axiom Oaux_compl: ∀hp,hf. O (aux_compl1 hp hf) (aux_compl hp hf).
+
+lemma MSC_max: ∀f,h,x. CF h f → MSC (max_{i < x}(f i)) ≤ max_{i < x}(h i).
+#f #h #x #hf elim x // #i #Hind >bigop_Strue [|//] >bigop_Strue [|//]
+whd in match (max ??); 
+cases (true_or_false (leb (f i) (bigop i (λi0:ℕ.true) ? 0 max(λi0:ℕ.f i0))))
+#Hcase >Hcase 
+  [@(transitive_le … Hind) @le_maxr //
+  |@(transitive_le … (MSC_out … hf i)) @le_maxl //
+  ]
+qed.
+  
+lemma CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+  CF ha a → CF hb b → CF hp p → CF hf f → 
+  O s (λx.ha x + hb x + 
+       (∑_{i ∈[a x ,b x[ }
+         (hp 〈i,x〉 + hf 〈i,x〉 + max_{i ∈ [a x, b x [ }(hf 〈i,x〉)))) →
+  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #HO 
+@ext_CF1 [|#x @max_unary_pr1]
+@(CF_comp ??? (λx.ha x + hb x))
+  [|@CF_comp_pair
+    [@CF_minus [@(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+    |@CF_comp_pair 
+      [@(O_to_CF … CFa) @O_plus_l // 
+      | @(O_to_CF … CF_id) @O_plus_r @le_to_O @(MSC_in … CFb)
+      ]
+    ]
+  |@(CF_prim_rec … MSC … (aux_compl1 hp hf))
+     [@CF_const
+     |@(O_to_CF … (Oaux_compl … ))
+      @CF_if 
+       [@(CF_comp p … MSC … CFp) 
+         [@CF_comp_pair 
+           [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+           |@CF_comp_snd @CF_comp_snd @CF_snd]
+         |@le_to_O #x normalize >commutative_plus >associative_plus @le_plus //
+         ]
+       |@CF_max1 
+         [@(CF_comp f … MSC … CFf) 
+           [@CF_comp_pair 
+             [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+             |@CF_comp_snd @CF_comp_snd @CF_snd]
+           |@le_to_O #x normalize >commutative_plus // 
+           ]
+         |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+         ]
+       |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+       ]  
+     |@O_refl
+     ]
+  |@(O_trans … HO) -HO
+   @(O_trans ? (λx:ℕ
+   .ha x+hb x
+    +bigop (b x-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus
+     (λi:ℕ
+      .(λi0:ℕ
+        .hp 〈i0,x〉+hf 〈i0,x〉
+         +bigop (b x-a x) (λi1:ℕ.true) ? 0 max
+          (λi1:ℕ.(λi2:ℕ.hf 〈i2,x〉) (i1+a x))) (i+a x))))
+    [
+   @le_to_O #n @le_plus // whd in match (unary_pr ???);
+   >fst_pair >snd_pair >bigop_prim_rec elim (b n - a n)
+    [normalize //
+    |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >aux_compl1_def1
+     >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair 
+     >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+      [-Hind @le_plus // normalize >fst_pair >snd_pair 
+       @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+         (λi1:ℕ.hf 〈i1+a n,n〉)))
+        [elim x [normalize @MSC_le]
+         #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+         >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+          [cases (true_or_false (leb (f 〈x0+a n,n〉)
+            (prim_rec (λx00:ℕ.O)
+             (λi:ℕ
+            .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉 
+             then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+                           (fst (snd i)) 
+                      then fst (snd i) 
+                      else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉  
+             else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+           [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+           |@(transitive_le … (MSC_out … CFf …)) @(le_maxl … (le_n …))
+           ]
+         |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+         ]
+       |@(le_maxr … (le_n …))
+       ]
+     |@(transitive_le … Hind) -Hind 
+      generalize in match (bigop x (λi:ℕ.true) ? 0 max
+              (λi1:ℕ.(λi2:ℕ.hf 〈i2,n〉) (i1+a n))); #c
+      generalize in match (hf 〈x+a n,n〉); #c1
+      elim x [//] #x0 #Hind 
+      >prim_rec_S >prim_rec_S normalize >fst_pair >fst_pair >snd_pair 
+      >snd_pair >snd_pair >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair 
+      >fst_pair @le_plus 
+       [@le_plus // cases (true_or_false (leb c1 c)) #Hcase 
+        >Hcase normalize // @lt_to_le @not_le_to_lt @(leb_false_to_not_le ?? Hcase)
+       |@Hind
+       ]
+     ]
+   ]
+ |@O_plus [@O_plus_l //] @le_to_O #x 
+  <bigop_plus_c @le_plus // @(transitive_le … (MSC_pair …)) @le_plus 
+   [@MSC_out @CFa | @MSC_out @(O_to_CF MSC … (CF_const x)) @le_to_O @(MSC_in … CFb)]
+ ]
+qed.
+
+axiom daemon : ∀P:Prop.P.
+axiom O_extl: ∀s1,s2,f. (∀x.s1 x = s2 x) → O s1 f → O s2 f.
+
+lemma CF_max2: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+  CF ha a → CF hb b → CF hp p → CF hf f → 
+  O s (λx.ha x + hb x + 
+       (b x - a x)*max_{i ∈ [a x, b x [ }(hp 〈i,x〉 + hf 〈i,x〉)) →
+  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #Os
+@(O_to_CF … Os (CF_max … CFa CFb CFp CFf ?)) @O_plus 
+  [@O_plus_l @O_refl
+  |@O_plus_r @O_ext2 [|| #x @(bigop_op  … plusAC)]
+   @O_plus
+    [@le_to_O normalize #x @sigma_to_Max 
+    |@le_to_O #x @transitive_le [|@sigma_const] @le_times //  
+     @Max_le #i #lti #_ @(transitive_le ???? (le_MaxI … ))
+      [@le_plus_n | // | @lt_minus_to_plus_r // | //]
+    ]
+  ]
+qed.
+
+(*
+lemma CF_max_monotonic: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+  CF ha a → CF hb b → CF hp p → CF hf f → 
+  O s (λx.ha x + hb x + 
+       (b x - a x)*max_{i ∈ [a x, b x [ }(hp 〈i,x〉 + hf 〈i,x〉)) →
+  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #Os
+@(O_to_CF … Os (CF_max … CFa CFb CFp CFf ?)) @O_plus 
+  [@O_plus_l @O_refl
+  |@O_plus_r @O_ext2 [|| #x @(bigop_op  … plusAC)]
+   @O_plus
+    [@le_to_O normalize #x @sigma_to_Max 
+    |@le_to_O #x @transitive_le [|@sigma_const] @le_times //  
+     @Max_le #i #lti #_ @(transitive_le ???? (le_MaxI … ))
+      [@le_plus_n | // | @lt_minus_to_plus_r // | //]
+    ]
+  ]
+qed.
+*)
+
+(* old
+axiom CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+  CF ha a → CF hb b → CF hp p → CF hf f → 
+  O s (λx.ha x + hb x + ∑_{i ∈[a x ,b x[ }(hp 〈i,x〉 + hf 〈i,x〉)) →
+  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)). *)
+  
+(******************************** minimization ********************************) 
+
+let rec my_minim a f x k on k ≝
+  match k with
+  [O ⇒ a
+  |S p ⇒ if eqb (my_minim a f x p) (a+p) 
+         then if f 〈a+p,x〉 then a+p else S(a+p)
+         else (my_minim a f x p) ].
+         
+lemma my_minim_S : ∀a,f,x,k. 
+  my_minim a f x (S k) = 
+    if eqb (my_minim a f x k) (a+k) 
+    then if f 〈a+k,x〉 then a+k else S(a+k)
+    else (my_minim a f x k) .
+// qed.
+  
+lemma my_minim_fa : ∀a,f,x,k. f〈a,x〉 = true → my_minim a f x k = a.
+#a #f #x #k #H elim k // #i #Hind normalize >Hind
+cases (true_or_false (eqb a (a+i))) #Hcase >Hcase normalize //
+<(eqb_true_to_eq … Hcase) >H //
+qed.
+
+lemma my_minim_nfa : ∀a,f,x,k. f〈a,x〉 = false → 
+my_minim a f x (S k) = my_minim (S a) f x k.
+#a #f #x #k #H elim k  
+  [normalize <plus_n_O >H >eq_to_eqb_true // 
+  |#i #Hind >my_minim_S >Hind >my_minim_S <plus_n_Sm //
+  ]
+qed.
+
+lemma my_min_eq : ∀a,f,x,k.
+  min k a (λi.f 〈i,x〉) = my_minim a f x k.
+#a #f #x #k lapply a -a elim k // #i #Hind #a normalize in ⊢ (??%?);
+cases (true_or_false (f 〈a,x〉)) #Hcase >Hcase 
+  [>(my_minim_fa … Hcase) // | >Hind @sym_eq @(my_minim_nfa … Hcase) ]
+qed.
+
+(* cambiare qui: togliere S *)
+
+
+definition minim_unary_pr ≝ λf.unary_pr (λx.S(fst x))
+    (λi. 
+      let k ≝ fst i in
+      let r ≝ fst (snd i) in
+      let b ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      if f 〈b-k,x〉 then b-k else r).
+      
+definition compl_minim_unary ≝ λhf:nat → nat.λi. 
+      let k ≝ fst i in
+      let b ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+
+definition compl_minim_unary_aux ≝ λhf,i. 
+      let k ≝ fst i in
+      let r ≝ fst (snd i) in
+      let b ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      hf 〈b-k,x〉 + MSC i.
+
+lemma compl_minim_unary_aux_def : ∀hf,k,r,b,x.
+  compl_minim_unary_aux hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈r,〈b,x〉〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def : ∀hf,k,r,b,x.
+  compl_minim_unary hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_aux_def2 : ∀hf,k,r,x.
+  compl_minim_unary_aux hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈r,x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def2 : ∀hf,k,r,x.
+  compl_minim_unary hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈S(fst x),snd x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma min_aux: ∀a,f,x,k. min (S k) (a x) (λi:ℕ.f 〈i,x〉) =
+  ((minim_unary_pr f) ∘ (λx.〈S k,〈a x + k,x〉〉)) x.
+#a #f #x #k whd in ⊢ (???%); >fst_pair >snd_pair
+lapply a -a (* @max_prim_rec1 *)
+elim k
+  [normalize #a >fst_pair >snd_pair >fst_pair >snd_pair >snd_pair >fst_pair 
+   <plus_n_O <minus_n_O >fst_pair //
+  |#i #Hind #a normalize in ⊢ (??%?); >prim_rec_S >fst_pair >snd_pair
+   >fst_pair >snd_pair >snd_pair >fst_pair <plus_n_Sm <(Hind (λx.S (a x)))
+   whd in ⊢ (???%); >minus_S_S <(minus_plus_m_m (a x) i) //
+qed.
+
+lemma minim_unary_pr1: ∀a,b,f,x.
+  μ_{i ∈[a x,b x]}(f 〈i,x〉) = 
+    if leb (a x) (b x) then ((minim_unary_pr f) ∘ (λx.〈S (b x) - a x,〈b x,x〉〉)) x
+    else (a x).
+#a #b #f #x cases (true_or_false (leb (a x) (b x))) #Hcase >Hcase
+  [cut (b x = a x + (b x - a x))
+    [>commutative_plus <plus_minus_m_m // @leb_true_to_le // ]
+   #Hcut whd in ⊢ (???%); >minus_Sn_m [|@leb_true_to_le //]
+   >min_aux whd in ⊢ (??%?); <Hcut //
+  |>eq_minus_O // @not_le_to_lt @leb_false_to_not_le //
+  ]
+qed.
+
+axiom sum_inv: ∀a,b,f.∑_{i ∈ [a,S b[ }(f i) = ∑_{i ∈ [a,S b[ }(f (a + b - i)).
+
+(*
+#a #b #f @(bigop_iso … plusAC) whd %{(λi.b - a - i)} %{(λi.b - a -i)} %
+  [%[#i #lti #_ normalize @eq_f >commutative_plus <plus_minus_associative 
+    [2: @le_minus_to_plus_r //
+     [// @eq_f @@sym_eq @plus_to_minus 
+    |#i #Hi #_ % [% [@le_S_S 
+*)
+
+example sum_inv_check : ∑_{i ∈ [3,6[ }(i*i) = ∑_{i ∈ [3,6[ }((8-i)*(8-i)).
+normalize // qed.
+
+(* rovesciamo la somma *)
+
+lemma CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉 + MSC 〈b x - i,〈S(b x),x〉〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #ha #hb #hf #s #CFa #CFb #CFf #HO 
+@ext_CF1 [|#x @minim_unary_pr1]
+@CF_if 
+  [@CF_le @(O_to_CF … HO) 
+    [@(O_to_CF … CFa) @O_plus_l @O_plus_l  @O_refl
+    |@(O_to_CF … CFb) @O_plus_l @O_plus_r @O_refl
+    ]
+  |@(CF_comp ??? (λx.ha x + hb x))
+    [|@CF_comp_pair
+      [@CF_minus [@CF_compS @(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+      |@CF_comp_pair 
+        [@(O_to_CF … CFb) @O_plus_r //
+        |@(O_to_CF … CF_id) @O_plus_r @le_to_O @(MSC_in … CFb)
+        ]
+      ]
+    |@(CF_prim_rec_gen … MSC … (compl_minim_unary_aux hf))
+      [@((λx:ℕ.fst (snd x)
+          +compl_minim_unary hf
+          〈fst x,
+          〈unary_pr fst
+            (λi:ℕ
+             .(let (k:ℕ) ≝fst i in 
+               let (r:ℕ) ≝fst (snd i) in 
+               let (b:ℕ) ≝fst (snd (snd i)) in 
+               let (x:ℕ) ≝snd (snd (snd i)) in if f 〈b-S k,x〉 then b-S k else r ))
+          〈fst x,snd (snd x)〉,
+          snd (snd x)〉〉))
+      |@CF_compS @CF_fst
+      |@CF_if 
+        [@(CF_comp f … MSC … CFf) 
+          [@CF_comp_pair 
+            [@CF_minus [@CF_comp_fst @CF_comp_snd @CF_snd|@CF_fst]
+            |@CF_comp_snd @CF_comp_snd @CF_snd]
+          |@le_to_O #x normalize >commutative_plus //
+          ]
+        |@(O_to_CF … MSC)
+          [@le_to_O #x normalize //
+          |@CF_minus
+            [@CF_comp_fst @CF_comp_snd @CF_snd |@CF_fst]
+          ]
+        |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+        ]
+      |@O_plus    
+        [@O_plus_l @O_refl 
+        |@O_plus_r @O_ext2 [||#x >compl_minim_unary_aux_def2 @refl]
+         @O_trans [||@le_to_O #x >compl_minim_unary_def2 @le_n]
+         @O_plus [@O_plus_l //] 
+         @O_plus_r 
+         @O_trans [|@le_to_O #x @MSC_pair] @O_plus 
+           [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_fst …)) 
+            >fst_pair @le_n]
+         @O_trans [|@le_to_O #x @MSC_pair] @O_plus
+           [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …))
+            >snd_pair @(transitive_le ???? (le_fst …)) >fst_pair 
+            normalize >snd_pair >fst_pair lapply (surj_pair x)
+            * #x1 * #x2 #Hx >Hx >fst_pair >snd_pair elim x1 //
+            #i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+            cases (f ?) [@le_S // | //]]
+         @le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …)) >snd_pair
+         >(expand_pair (snd (snd x))) in ⊢ (?%?); @le_pair //
+        ]
+      ] 
+    |cut (O s (λx.ha x + hb x + 
+            ∑_{i ∈[a x ,S(b x)[ }(hf 〈a x+b x-i,x〉 + MSC 〈b x -(a x+b x-i),〈S(b x),x〉〉)))
+      [@(O_ext2 … HO) #x @eq_f @sum_inv] -HO #HO
+     @(O_trans … HO) -HO
+     @(O_trans ? (λx:ℕ.ha x+hb x
+       +bigop (S(b x)-a x) (λi:ℕ.true) ? 
+        (MSC 〈b x,x〉) plus (λi:ℕ.(λj.hf j + MSC 〈b x - fst j,〈S(b (snd j)),snd j〉〉) 〈b x- i,x〉)))
+      [@le_to_O #n @le_plus // whd in match (unary_pr ???); 
+       >fst_pair >snd_pair >(bigop_prim_rec_dec1 a b ? (λi.true)) 
+        (* it is crucial to recall that the index is bound by S(b x) *)
+        cut (S (b n) - a n ≤ S (b n)) [//]
+        elim (S(b n) - a n)
+        [normalize //  
+        |#x #Hind #lex >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair 
+         >compl_minim_unary_def >prim_rec_S >fst_pair >snd_pair >fst_pair 
+         >snd_pair >fst_pair >snd_pair >fst_pair whd in ⊢ (??%); >commutative_plus 
+         @le_plus [2:@Hind @le_S @le_S_S_to_le @lex] -Hind >snd_pair 
+         >minus_minus_associative // @le_S_S_to_le // 
+        ]
+      |@O_plus [@O_plus_l //] @O_ext2 [||#x @bigop_plus_c] 
+       @O_plus 
+        [@O_plus_l @O_trans [|@le_to_O #x @MSC_pair]
+         @O_plus 
+           [@O_plus_r @le_to_O @(MSC_out … CFb)
+           |@O_plus_r @le_to_O @(MSC_in … CFb)
+           ]
+        |@O_plus_r @(O_ext2 … (O_refl …)) #x @same_bigop
+          [//|#i #H #_ @eq_f2 [@eq_f @eq_f2 //|>fst_pair @eq_f @eq_f2  //]   
+        ]
+      ]
+    ]
+  ] 
+  |@(O_to_CF … CFa) @(O_trans … HO) @O_plus_l @O_plus_l @O_refl
+  ]
+  
+qed.
+
+lemma CF_mu2: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉 + MSC〈S(b x),x〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #sa #sb #sf #s #CFa #CFb #CFf #HO
+@(O_to_CF … HO (CF_mu … CFa CFb CFf ?)) @O_plus [@O_plus_l @O_refl] 
+@O_plus_r @O_ext2 [|| #x @(bigop_op  … plusAC)]
+@O_plus [@le_to_O #x @le_sigma //] 
+@(O_trans ? (λx.∑_{i ∈[a x ,S(b x)[ }(MSC(b x -i)+MSC 〈S(b x),x〉))) 
+  [@le_to_O #x @le_sigma //]
+@O_ext2 [|| #x @(bigop_op  … plusAC)] @O_plus  
+  [@le_to_O #x @le_sigma // #i #lti #_ @(transitive_le … (MSC 〈S (b x),x〉)) //
+   @monotonic_MSC @(transitive_le … (S(b x))) // @le_S //
+  |@le_to_O #x @le_sigma //
+  ]
+qed.
+
+(* uses MSC_S *)
+
+lemma CF_mu3: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s. (∀x.sf x > 0) →
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉 + MSC〈b x,x〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #sa #sb #sf #s #sfpos #CFa #CFb #CFf #HO
+@(O_to_CF … HO (CF_mu2 … CFa CFb CFf ?)) @O_plus [@O_plus_l @O_refl] 
+@O_plus_r @O_ext2 [|| #x @(bigop_op  … plusAC)]
+@O_plus [@le_to_O #x @le_sigma //] 
+@le_to_O #x @le_sigma // #x #lti #_ >MSC_pair_eq >MSC_pair_eq <associative_plus
+@le_plus // @(transitive_le … (MSC_sublinear … )) /2 by monotonic_lt_plus_l/
+qed.
+
+lemma CF_mu4: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s. (∀x.sf x > 0) →
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + (S(b x) - a x)*Max_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #sa #sb #sf #s #sfpos #CFa #CFb #CFf #HO
+@(O_to_CF … HO (CF_mu3 … sfpos CFa CFb CFf ?)) @O_plus [@O_plus_l @O_refl] 
+@O_ext2 [|| #x @(bigop_op  … plusAC)] @O_plus_r @O_plus
+  [@le_to_O #x @sigma_to_Max
+  |lapply (MSC_in …  CFf) #Hle
+   %{1} %{0} #n #_ @(transitive_le … (sigma_const …)) 
+   >(commutative_times 1) <times_n_1 
+   cases (decidable_le (S (b n)) (a n)) #H
+    [>(eq_minus_O … H) //
+    |lapply (le_S_S_to_le … (not_le_to_lt … H)) -H #H
+     @le_times // @(transitive_le … (Hle … ))
+     cut (b n = b n - a n + a n) [<plus_minus_m_m // ]
+     #Hcut >Hcut in ⊢ (?%?); @(le_Max (λi.sf 〈i+a n,n〉)) /2/
+    ]
+  ]
+qed.
+
+(*
+axiom CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)). *)
+  
+
+  
\ No newline at end of file
diff --git a/matita/matita/lib/reverse_complexity/complexity.ma b/matita/matita/lib/reverse_complexity/complexity.ma
new file mode 100644
index 000000000..ae7baceca
--- /dev/null
+++ b/matita/matita/lib/reverse_complexity/complexity.ma
@@ -0,0 +1,290 @@
+include "reverse_complexity/basics.ma".
+include "reverse_complexity/big_O.ma". 
+
+(********************************* complexity *********************************)
+
+(* We assume operations have a minimal structural complexity MSC. 
+For instance, for time complexity, MSC is equal to the size of input.
+For space complexity, MSC is typically 0, since we only measure the space 
+required in addition to dimension of the input. *)
+
+axiom MSC : nat → nat.
+axiom MSC_sublinear : ∀n. MSC (S n) ≤ S (MSC n). 
+(* axiom MSC_S: O MSC (λx.MSC (S x)) . *)
+axiom MSC_le: ∀n. MSC n ≤ n.
+
+axiom monotonic_MSC: monotonic ? le MSC.
+axiom MSC_pair_eq: ∀a,b. MSC 〈a,b〉 = MSC a + MSC b.
+
+
+lemma MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b. // qed.
+
+(* C s i means i is running in O(s) *)
+ 
+definition C ≝ λs,i.∃c.∃a.∀x.a ≤ x → ∃y. 
+  U i x (c*(s x)) = Some ? y.
+
+(* C f s means f ∈ O(s) where MSC ∈O(s) *)
+definition CF ≝ λs,f.∃i.code_for (total f) i ∧ C s i.
+
+axiom MSC_in: ∀f,h. CF h f → ∀x. MSC x ≤ h x.
+axiom MSC_out: ∀f,h. CF h f → ∀x. MSC (f x) ≤ h x.
+
+ 
+lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
+#f #g #s #Hext * #i * #Hcode #HC %{i} %
+  [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //] 
+qed.
+
+lemma ext_CF1 : ∀f,g,s. (∀n. f n = g n) → CF s g → CF s f.
+#f #g #s #Hext * #i * #Hcode #HC %{i} %
+  [#x cases (Hcode x) #a #H %{a} whd in match (total ??); >Hext @H | //] 
+qed.
+
+lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤  s2 x) → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #i * #Hcode #Hs1 
+%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 %{c} %{a} #n #lean 
+cases(Hs1 n lean) #y #Hy %{y} @(monotonic_U …Hy) @le_times // 
+qed.
+
+lemma O_to_CF: ∀s1,s2,f.O s2 s1 → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #i * #Hcode #Hs1 
+%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 
+cases H #c1 * #a1 #Ha1 %{(c*c1)} %{(a+a1)} #n #lean 
+cases(Hs1 n ?) [2:@(transitive_le … lean) //] #y #Hy %{y} @(monotonic_U …Hy)
+>associative_times @le_times // @Ha1 @(transitive_le … lean) //
+qed.
+
+lemma timesc_CF: ∀s,f,c.CF (λx.c*s x) f → CF s f.
+#s #f #c @O_to_CF @O_times_c 
+qed.
+
+(********************************* composition ********************************)
+axiom CF_comp: ∀f,g,sf,sg,sh. CF sg g → CF sf f → 
+  O sh (λx. sg x + sf (g x)) → CF sh (f ∘ g).
+  
+lemma CF_comp_ext: ∀f,g,h,sh,sf,sg. CF sg g → CF sf f → 
+  (∀x.f(g x) = h x) → O sh (λx. sg x + sf (g x)) → CF sh h.
+#f #g #h #sh #sf #sg #Hg #Hf #Heq #H @(ext_CF (f ∘ g))
+  [#n normalize @Heq | @(CF_comp … H) //]
+qed.
+
+(************************************* smn ************************************)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+(****************************** constructibility ******************************)
+ 
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+  (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //] 
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) → 
+  constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1)  #x >Hext //
+qed.
+
+
+(***************************** primitive recursion ****************************)
+
+(* no arguments *)
+
+let rec prim_rec0 (k:nat) (h:nat →nat) n on n ≝ 
+  match n with 
+  [ O ⇒ k
+  | S a ⇒ h 〈a, prim_rec0 k h a〉].
+  
+lemma prim_rec0_S: ∀k,h,n.
+  prim_rec0 k h (S n) = h 〈n, prim_rec0 k h n〉.
+// qed.
+
+let rec prim_rec0_compl (k,sk:nat) (h,sh:nat →nat) n on n ≝ 
+  match n with 
+  [ O ⇒ sk
+  | S a ⇒ prim_rec0_compl k sk h sh a + sh (prim_rec0 k h a)].
+
+axiom CF_prim_rec0_gen: ∀k,h,sk,sh,sh1,sf. CF sh h →
+  O sh1 (λx.snd x + sh 〈fst x,prim_rec0 k h (fst x)〉) → 
+  O sf (prim_rec0 sk sh1) →
+   CF sf (prim_rec0 k h).
+
+lemma CF_prim_rec0: ∀k,h,sk,sh,sf. CF sh h → 
+  O sf (prim_rec0 sk (λx. snd x + sh 〈fst x,prim_rec0 k h (fst x)〉))
+   → CF sf (prim_rec0 k h).
+#k #h #sk #sh #sf #Hh #HO @(CF_prim_rec0_gen … Hh … HO) @O_refl
+qed.
+
+(* with arguments. m is a vector of arguments *)
+let rec prim_rec (k,h:nat →nat) n m on n ≝ 
+  match n with 
+  [ O ⇒ k m
+  | S a ⇒ h 〈a,〈prim_rec k h a m, m〉〉].
+  
+lemma prim_rec_S: ∀k,h,n,m.
+  prim_rec k h (S n) m = h 〈n,〈prim_rec k h n m, m〉〉.
+// qed.
+
+lemma prim_rec_le: ∀k1,k2,h1,h2. (∀x.k1 x ≤ k2 x) → 
+(∀x,y.x ≤y → h1 x ≤ h2 y) →
+  ∀x,m. prim_rec k1 h1 x m ≤ prim_rec k2 h2 x m.
+#k1 #k2 #h1 #h2 #lek #leh #x #m elim x //
+#n #Hind normalize @leh @le_pair // @le_pair //
+qed.
+ 
+definition unary_pr ≝ λk,h,x. prim_rec k h (fst x) (snd x).
+
+lemma prim_rec_unary: ∀k,h,a,b. 
+  prim_rec k h a b = unary_pr k h 〈a,b〉.
+#k #h #a #b normalize >fst_pair >snd_pair //
+qed.
+  
+let rec prim_rec_compl (k,h,sk,sh:nat →nat) n m on n ≝ 
+  match n with 
+  [ O ⇒ sk m
+  | S a ⇒ prim_rec_compl k h sk sh a m + sh 〈a,〈prim_rec k h a m,m〉〉].
+
+axiom CF_prim_rec_gen: ∀k,h,sk,sh,sh1. CF sk k → CF sh h →  
+  O sh1 (λx. fst (snd x) + 
+             sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉) → 
+   CF (unary_pr sk sh1) (unary_pr k h).
+
+lemma CF_prim_rec: ∀k,h,sk,sh,sf. CF sk k → CF sh h → 
+  O sf (unary_pr sk 
+        (λx. fst (snd x) + 
+             sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉)) 
+   → CF sf (unary_pr k h).
+#k #h #sk #sh #sf #Hk #Hh #Osf @(O_to_CF … Osf) @(CF_prim_rec_gen … Hk Hh) 
+@O_refl
+qed.
+
+lemma constr_prim_rec: ∀s1,s2. constructible s1 → constructible s2 →
+  (∀n,r,m. 2 * r ≤ s2 〈n,〈r,m〉〉) → constructible (unary_pr s1 s2).
+#s1 #s2 #Hs1 #Hs2 #Hincr @(CF_prim_rec … Hs1 Hs2) whd %{2} %{0} 
+#x #_ lapply (surj_pair x) * #a * #b #eqx >eqx whd in match (unary_pr ???); 
+>fst_pair >snd_pair
+whd in match (unary_pr ???); >fst_pair >snd_pair elim a
+  [normalize //
+  |#n #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
+   >prim_rec_S @transitive_le [| @(monotonic_le_plus_l … Hind)]
+   @transitive_le [| @(monotonic_le_plus_l … (Hincr n ? b))] 
+   whd in match (unary_pr ???); >fst_pair >snd_pair //
+  ]
+qed.
+
+(**************************** primitive operations*****************************)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+axiom CF_const: ∀k.CF MSC (λx.k).
+axiom CF_comp_fst: ∀h,f. CF h f → CF h (fst ∘ f).
+axiom CF_comp_snd: ∀h,f. CF h f → CF h (snd ∘ f). 
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉). 
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed. 
+
+(**************************** arithmetic operations ***************************)
+
+axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
+axiom CF_le: ∀h,f,g. CF h f → CF h g → CF h (λx. leb (f x) (g x)). 
+axiom CF_eqb: ∀h,f,g. CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
+axiom CF_max1: ∀h,f,g. CF h f → CF h g → CF h (λx. max (f x) (g x)). 
+axiom CF_plus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x + g x). 
+axiom CF_minus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x - g x). 
+
+(******************************** if then else ********************************)
+
+lemma if_prim_rec: ∀b:nat → bool. ∀f,g:nat → nat.∀x:nat.
+  if b x then f x else g x = prim_rec g (f ∘ snd ∘ snd) (b x) x.
+#b #f #g #x cases (b x) normalize // 
+qed.
+  
+lemma CF_if: ∀b:nat → bool. ∀f,g,s. CF s b → CF s f → CF s g → 
+  CF s (λx. if b x then f x else g x).
+#b #f #g #s #CFb #CFf #CFg @(ext_CF (λx.unary_pr g (f ∘ snd ∘ snd) 〈b x,x〉))
+  [#n @sym_eq normalize >fst_pair >snd_pair @if_prim_rec
+  |@(CF_comp ??? s)
+    [|@CF_comp_pair // @(O_to_CF … CF_id) @le_to_O @MSC_in // 
+    |@(CF_prim_rec_gen ??? (λx.MSC x + s(snd (snd x))) … CFg) [3:@O_refl|] 
+     @(CF_comp … (λx.MSC x + s(snd x)) … CF_snd)
+      [@(CF_comp … s … CF_snd CFf) @O_refl 
+      |@O_plus 
+        [@O_plus_l @O_refl
+        |@O_plus 
+          [@O_plus_l @le_to_O #x @monotonic_MSC //
+          |@O_plus_r @O_refl
+          ]
+        ]
+      ]
+    |%{6} %{0} #n #_ normalize in ⊢ (??%); <plus_n_O cases (b n) normalize 
+      >snd_pair >fst_pair normalize 
+      [@le_plus [//] >fst_pair >fst_pair >snd_pair >fst_pair 
+       @le_plus [//] >snd_pair >snd_pair >snd_pair >snd_pair 
+       <associative_plus <associative_plus @le_plus [|//] 
+       @(transitive_le … (MSC_pair …)) >associative_plus @le_plus
+        [@(transitive_le ???? (MSC_in … CFf n)) @monotonic_MSC //
+        |@(transitive_le … (MSC_pair …)) @le_plus [|@(MSC_in … CFf)]
+         normalize @MSC_out @CFg
+        ]
+      |@le_plus // 
+      ]
+    ]
+  ]
+qed.
+
+(********************************* simulation *********************************)
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom sU : nat → nat.
+axiom CF_U : CF sU pU_unary. 
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+  sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+lemma monotonic_sU_aux : ∀x1,x2. fst x1 ≤ fst x2 → fst (snd x1) ≤ fst (snd x2) →
+snd (snd x1) ≤ snd (snd x2) → sU x1 ≤ sU x2.
+#x1 #x2 cases (surj_pair x1) #a1 * #y #eqx1 >eqx1 -eqx1 cases (surj_pair y) 
+#b1 * #c1 #eqy >eqy -eqy
+cases (surj_pair x2) #a2 * #y2 #eqx2 >eqx2 -eqx2 cases (surj_pair y2) 
+#b2 * #c2 #eqy2 >eqy2 -eqy2 >fst_pair >snd_pair >fst_pair >snd_pair 
+>fst_pair >snd_pair >fst_pair >snd_pair @monotonic_sU
+qed.
+   
+axiom sU_pos: ∀x. 0 < sU x.
+
+axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+
+lemma sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+#i #x #s @(transitive_le ???? (MSC_in … CF_U 〈i,〈x,s〉〉)) @monotonic_MSC //
+qed.
+
+lemma sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+#i #x #s @(transitive_le ???? (MSC_in … CF_U 〈i,〈x,s〉〉)) @monotonic_MSC 
+@(transitive_le … 〈x,s〉) //
+qed.
+
+
+
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary. 
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in  ⊢ (??(?%)?); >fst_pair % 
+qed.
+
+lemma CF_out: CF sU out_unary. 
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in  ⊢ (??(?%)?); >snd_pair % 
+qed.
+
diff --git a/matita/matita/broken_lib/reverse_complexity/gap.ma b/matita/matita/lib/reverse_complexity/gap.ma
similarity index 100%
rename from matita/matita/broken_lib/reverse_complexity/gap.ma
rename to matita/matita/lib/reverse_complexity/gap.ma
diff --git a/matita/matita/broken_lib/reverse_complexity/hierarchy.ma b/matita/matita/lib/reverse_complexity/hierarchy.ma
similarity index 75%
rename from matita/matita/broken_lib/reverse_complexity/hierarchy.ma
rename to matita/matita/lib/reverse_complexity/hierarchy.ma
index f288e98ca..d833a5164 100644
--- a/matita/matita/broken_lib/reverse_complexity/hierarchy.ma
+++ b/matita/matita/lib/reverse_complexity/hierarchy.ma
@@ -1,10 +1,9 @@
 
 include "arithmetics/nat.ma".
-include "arithmetics/log.ma".
-(* include "arithmetics/ord.ma". *)
+include "arithmetics/log.ma". 
 include "arithmetics/bigops.ma".
 include "arithmetics/bounded_quantifiers.ma".
-include "arithmetics/pidgeon_hole.ma".
+include "arithmetics/pidgeon_hole.ma". 
 include "basics/sets.ma".
 include "basics/types.ma".
 
@@ -118,21 +117,12 @@ lemma O_absorbr: ∀f,g,s. O s f → O f g → O s (f+g).
 #f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
 qed.
 
-(* 
-lemma O_ff: ∀f,s. O s f → O s (f+f).
-#f #s #Osf /2/ 
-qed. *)
-
 lemma O_ext2: ∀f,g,s. O s f → (∀x.f x = g x) → O s g.
 #f #g #s * #c * #a #Osf #eqfg %{c} %{a} #n #lean <eqfg @Osf //
 qed.    
 
-
 definition not_O ≝ λf,g.∀c,n0.∃n. n0 ≤ n ∧ c* (f n) < g n .
 
-(* this is the only classical result *)
-axiom not_O_def: ∀f,g. ¬ O f g → not_O f g.
-
 (******************************* small O notation *****************************)
 
 (*  o f g means g ∈ o(f) *)
@@ -164,56 +154,10 @@ interpretation "abstract pair" 'pair f g = (pair f g).
 
 (************************ basic complexity notions ****************************)
 
-(* u is the deterministic configuration relation of the universal machine (one 
-   step) 
-
-axiom u: nat → option nat.
-
-let rec U c n on n ≝ 
-  match n with  
-  [ O ⇒ None ?
-  | S m ⇒ match u c with 
-    [ None ⇒ Some ? c (* halting case *)
-    | Some c1 ⇒ U c1 m
-    ]
-  ].
- 
-lemma halt_U: ∀i,n,y. u i = None ? → U i n = Some ? y → y = i.
-#i #n #y #H cases n
-  [normalize #H1 destruct |#m normalize >H normalize #H1 destruct //]
-qed. 
-
-lemma Some_to_halt: ∀n,i,y. U i n = Some ? y → u y = None ? .
-#n elim n
-  [#i #y normalize #H destruct (H)
-  |#m #Hind #i #y normalize 
-   cut (u i = None ? ∨ ∃c. u i = Some ? c) 
-    [cases (u i) [/2/ | #c %2 /2/ ]] 
-   *[#H >H normalize #H1 destruct (H1) // |* #c #H >H normalize @Hind ]
-  ]
-qed. *)
-
 axiom U: nat → nat → nat → option nat. 
-(*
-lemma monotonici_U: ∀y,n,m,i.
-  U i m = Some ? y → U i (n+m) = Some ? y.
-#y #n #m elim m 
-  [#i normalize #H destruct 
-  |#p #Hind #i <plus_n_Sm normalize
-    cut (u i = None ? ∨ ∃c. u i = Some ? c) 
-    [cases (u i) [/2/ | #c %2 /2/ ]] 
-   *[#H1 >H1 normalize // |* #c #H >H normalize #H1 @Hind //]
-  ]
-qed. *)
 
 axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
   U i x n = Some ? y → U i x m = Some ? y.
-(* #i #n #m #y #lenm #H >(plus_minus_m_m m n) // @monotonici_U //
-qed. *)
-
-(* axiom U: nat → nat → option nat. *)
-(* axiom monotonic_U: ∀i,n,m,y.n ≤m →
-   U i n = Some ? y → U i m = Some ? y. *)
   
 lemma unique_U: ∀i,x,n,m,yn,ym.
   U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
@@ -228,7 +172,7 @@ definition code_for ≝ λf,i.∀x.
   ∃n.∀m. n ≤ m → U i x m = f x.
 
 definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y.
-notation "[i,x] ↓ r" with precedence 60 for @{terminate $i $x $r}.
+notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}. 
 
 definition lang ≝ λi,x.∃r,y. U i x r = Some ? y ∧ 0  < y. 
 
@@ -262,11 +206,6 @@ lemma size_f_def: ∀f,n. size_f f n =
   Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
 // qed.
 
-(*
-definition Max_const : ∀f,p,n,a. a < n → p a →
-  ∀n. f n = g n →
-  Max_{i < n | p n}(f n) = *)
-
 lemma size_f_size : ∀f,n. size_f (f ∘ size) n = |(f n)|.
 #f #n @le_to_le_to_eq
   [@Max_le #a #lta #Ha normalize >(eqb_true_to_eq  … Ha) //
@@ -289,12 +228,9 @@ lemma size_f_fst : ∀n. size_f fst n ≤ n.
 #n @Max_le #a #lta #Ha <(eqb_true_to_eq  … Ha) //
 qed.
 
-(* definition def ≝ λf:nat → option nat.λx.∃y. f x = Some ? y.*)
-
 (* C s i means that the complexity of i is O(s) *)
 
-definition C ≝ λs,i.∃c.∃a.∀x.a ≤ |x| → ∃y. 
-  U i x (c*(s(|x|))) = Some ? y.
+definition C ≝ λs,i.∃c.∃a.∀x.a ≤ |x| → {i ⊙ x} ↓ (c*(s(|x|))).
 
 definition CF ≝ λs,f.∃i.code_for f i ∧ C s i.
 
@@ -310,6 +246,18 @@ cases HCs1 #c2 * #b #H2 %{(c2*c1)} %{(max a b)}
 %{y} @(monotonic_U …Hy) >associative_times @le_times // @H @(le_maxl … Hmax)
 qed. 
 
+(*********************** The hierachy theorem (left) **************************)
+
+theorem hierarchy_theorem_left: ∀s1,s2:nat→nat.
+   O(s1) ⊆ O(s2) → CF s1 ⊆ CF s2.
+#s1 #s2 #HO #f * #i * #Hcode * #c * #a #Hs1_i %{i} % //
+cases (sub_O_to_O … HO) -HO #c1 * #b #Hs1s2 
+%{(c*c1)} %{(max a b)} #x #lemax 
+cases (Hs1_i x ?) [2: @(le_maxl …lemax)]
+#y #Hy %{y} @(monotonic_U … Hy) >associative_times
+@le_times // @Hs1s2 @(le_maxr … lemax)
+qed.
+
 (************************** The diagonal language *****************************)
 
 (* the diagonal language used for the hierarchy theorem *)
@@ -318,7 +266,7 @@ definition diag ≝ λs,i.
   U (fst i) i (s (|i|)) = Some ? 0. 
 
 lemma equiv_diag: ∀s,i. 
-  diag s i ↔ [fst i,i] ↓ s (|i|) ∧ ¬lang (fst i) i.
+  diag s i ↔ {fst i ⊙ i} ↓ s(|i|) ∧ ¬lang (fst i) i.
 #s #i %
   [whd in ⊢ (%→?); #H % [%{0} //] % * #x * #y *
    #H1 #Hy cut (0 = y) [@(unique_U … H H1)] #eqy /2/
@@ -360,7 +308,39 @@ lemma absurd1: ∀P. iff P (¬ P) →False.
 #H3 @(absurd ?? H3) @H2 @H3 
 qed.
 
-(* axiom weak_pad : ∀a,∃a0.∀n. a0 < n → ∃b. |〈a,b〉| = n. *)
+let rec f_img (f:nat →nat) n on n ≝
+  match n with
+  [O ⇒ [ ]
+  |S m ⇒ f m::f_img f m
+  ].
+
+(* a few lemma to prove injective_to_exists. This is a general result; a nice
+example of the pidgeon hole pricniple *)
+
+lemma f_img_to_exists: 
+  ∀f.∀n,a. a ∈ f_img f n → ∃b. b < n ∧ f b = a.
+#f #n #a elim n normalize [@False_ind]
+#m #Hind *
+  [#Ha %{m} /2/ |#H cases(Hind H) #b * #Hb #Ha %{b} % // @le_S //]
+qed. 
+
+lemma length_f_img: ∀f,n. |f_img f n| = n.
+#f #n elim n // normalize //
+qed. 
+
+lemma unique_f_img: ∀f,n. injective … f → unique ? (f_img f n).
+#f #n #Hinj elim n normalize //
+#m #Hind % // % #H lapply(f_img_to_exists …H) * #b * #ltbm
+#eqbm @(absurd … ltbm) @le_to_not_lt >(Hinj… eqbm) //
+qed.
+
+lemma injective_to_exists: ∀f. injective nat nat f → 
+  ∀n.(∀i.i < n → f i < n) → ∀a. a < n → ∃b. b<n ∧ f b = a.
+#f #finj #n #H1 #a #ltan @(f_img_to_exists f n a)
+@(eq_length_to_mem_all …  (length_f_img …) (unique_f_img …finj …) …ltan)
+#x #Hx cases(f_img_to_exists … Hx) #b * #ltbn #eqx <eqx @H1 //
+qed.
+
 lemma weak_pad1 :∀n,a.∃b. n ≤ 〈a,b〉. 
 #n #a 
 cut (∀i.decidable (〈a,i〉 < n))
@@ -374,7 +354,7 @@ cut (∀i.decidable (〈a,i〉 < n))
   |#H lapply(not_forall_to_exists … Hdec H) 
    * #b * #H1 #H2 %{b} @not_lt_to_le @H2
   ]
-qed. 
+qed.
 
 lemma pad : ∀n,a. ∃b. n ≤ |〈a,b〉|.
 #n #a cases (weak_pad1 (of_size n) a) #b #Hb 
@@ -382,7 +362,7 @@ lemma pad : ∀n,a. ∃b. n ≤ |〈a,b〉|.
 qed.
 
 lemma o_to_ex: ∀s1,s2. o s1 s2 → ∀i. C s2 i →
-  ∃b.[i, 〈i,b〉] ↓ s1 (|〈i,b〉|).
+  ∃b.{i ⊙ 〈i,b〉} ↓ s1 (|〈i,b〉|).
 #s1 #s2  #H #i * #c * #x0 #H1 
 cases (H c) #n0 #H2 cases (pad (max x0 n0) i) #b #Hmax
 %{b} cases (H1 〈i,b〉 ?)
@@ -415,14 +395,9 @@ definition opt_comp ≝ λf,g:nat → option nat. λx.
   [ None ⇒ None ?
   | Some y ⇒ f y ].   
 
-(* axiom CFU: ∀h,g,s. CF s (to_Some h)  → CF s (to_Some (of_size ∘ g)) → 
-  CF (Slow s) (λx.U (h x) (g x)). *)
-  
 axiom sU2: nat → nat → nat.
 axiom sU: nat → nat → nat → nat.
 
-(* axiom CFU: CF sU (λx.U (fst x) (snd x)). *)
-
 axiom CFU_new: ∀h,g,f,s. 
   CF s (to_Some h)  → CF s (to_Some g) → CF s (to_Some f) → 
   O s (λx. sU (size_f h x) (size_f g x) (size_f f x)) → 
@@ -445,18 +420,9 @@ axiom monotonic_sU: ∀a1,a2,b1,b2,c1,c2. a1 ≤ a2 → b1 ≤ b2 → c1 ≤c2 
 
 axiom superlinear_sU: ∀i,x,r. r ≤ sU i x r.
 
+(* not used *)
 definition sU_space ≝ λi,x,r.i+x+r.
-definition sU_time ≝ λi,x,r.i+x+(i^2)*r*(log 2 r).
-
-(*
-axiom CF_comp: ∀f,g,s1, s2. CF s1 f → CF s2 g → 
-  CF (λx.s2 x + s1 (size (deopt (g x)))) (opt_comp f g).
-
-(* axiom CF_comp: ∀f,g,s1, s2. CF s1 f → CF s2 g → 
-  CF (s1 ∘ (λx. size (deopt (g x)))) (opt_comp f g). *)
-  
-axiom CF_comp_strong: ∀f,g,s1,s2. CF s1 f → CF s2 g → 
-  CF (s1 ∘ s2) (opt_comp f g). *)
+definition sU_time ≝ λi,x,r.i+x+(i^2)*r*(log 2 r). 
 
 definition IF ≝ λb,f,g:nat →option nat. λx.
   match b x with 
@@ -485,16 +451,11 @@ axiom CF_pair: ∀f,g,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (g x)) →
 
 axiom CF_fst: ∀f,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (fst (f x))).
 
-definition minimal ≝ λs. CF s (λn. Some ? n) ∧ ∀c. CF s (λn. Some ? c).
-
-
-(*
-axiom le_snd: ∀n. |snd n| ≤ |n|.
-axiom daemon: ∀P:Prop.P. *)
+definition sufficiently_large ≝ λs. CF s (λn. Some ? n) ∧ ∀c. CF s (λn. Some ? c).
 
 definition constructible ≝ λs. CF s (λx.Some ? (of_size (s (|x|)))).
 
-lemma diag_s: ∀s. minimal s → constructible s → 
+lemma diag_s: ∀s. sufficiently_large s → constructible s → 
   CF (λx.sU x x (s x)) (diag_cf s).
 #s * #Hs_id #Hs_c #Hs_constr 
 cut (O (λx:ℕ.sU x x (s x)) s) [%{1} %{0} #n //]
@@ -507,61 +468,4 @@ cut (O (λx:ℕ.sU x x (s x)) s) [%{1} %{0} #n //]
   |%{1} %{0} #n #_ >commutative_times <times_n_1
    @monotonic_sU // >size_f_size >size_of_size //
   ]
-qed. 
-
-(*
-lemma diag_s: ∀s. minimal s → constructible s → 
-  CF (λx.s x + sU x x (s x)) (diag_cf s).
-#s * #Hs_id #Hs_c #Hs_constr 
-@ext_CF [2: #n @sym_eq @diag_cf_def | skip]
-@IF_CF_new [2,3:@(monotonic_CF … (Hs_c ?)) @O_plus_l //]
-@CFU_new
-  [@CF_fst @(monotonic_CF … Hs_id) @O_plus_l //
-  |@(monotonic_CF … Hs_id) @O_plus_l //
-  |@(monotonic_CF … Hs_constr) @O_plus_l //
-  |@O_plus_r %{1} %{0} #n #_ >commutative_times <times_n_1
-   @monotonic_sU // >size_f_size >size_of_size //
-  ]
-qed. *)
-
-(* proof with old axioms
-lemma diag_s: ∀s. minimal s → constructible s → 
-  CF (λx.s x + sU x x (s x)) (diag_cf s).
-#s * #Hs_id #Hs_c #Hs_constr 
-@ext_CF [2: #n @sym_eq @diag_cf_def | skip]
-@(monotonic_CF ???? (IF_CF (λi:ℕ.U (pair (fst i) i) (|of_size (s (|i|))|))
-   … (λi.s i + s i + s i + (sU (size_f fst i) (size_f (λi.i) i) (size_f (λi.of_size (s (|i|))) i))) … (Hs_c 1) (Hs_c 0) … ))
-  [2: @CFU [@CF_fst // | // |@Hs_constr]
-  |@(O_ext2 (λn:ℕ.s n+sU (size_f fst n) n (s n) + (s n+s n+s n+s n))) 
-    [2: #i >size_f_size >size_of_size >size_f_id //] 
-   @O_absorbr 
-    [%{1} %{0} #n #_ >commutative_times <times_n_1 @le_plus //
-     @monotonic_sU // 
-    |@O_plus_l @(O_plus … (O_refl s)) @(O_plus … (O_refl s)) 
-     @(O_plus … (O_refl s)) //
-  ]
-qed.
-*)
-
-(*************************** The hierachy theorem *****************************)
-
-(*
-theorem hierarchy_theorem_right: ∀s1,s2:nat→nat. 
-  O s1 idN → constructible s1 →
-    not_O s2 s1 → ¬ CF s1 ⊆ CF s2.
-#s1 #s2 #Hs1 #monos1 #H % #H1 
-@(absurd … (CF s2 (diag_cf s1)))
-  [@H1 @diag_s // |@(diag1_not_s1 … H)]
-qed.
-*)
-
-theorem hierarchy_theorem_left: ∀s1,s2:nat→nat.
-   O(s1) ⊆ O(s2) → CF s1 ⊆ CF s2.
-#s1 #s2 #HO #f * #i * #Hcode * #c * #a #Hs1_i %{i} % //
-cases (sub_O_to_O … HO) -HO #c1 * #b #Hs1s2 
-%{(c*c1)} %{(max a b)} #x #lemax 
-cases (Hs1_i x ?) [2: @(le_maxl …lemax)]
-#y #Hy %{y} @(monotonic_U … Hy) >associative_times
-@le_times // @Hs1s2 @(le_maxr … lemax)
-qed.
-
+qed.
\ No newline at end of file
diff --git a/matita/matita/lib/reverse_complexity/sigma_diseq.ma b/matita/matita/lib/reverse_complexity/sigma_diseq.ma
new file mode 100644
index 000000000..4356cd24e
--- /dev/null
+++ b/matita/matita/lib/reverse_complexity/sigma_diseq.ma
@@ -0,0 +1,153 @@
+include "arithmetics/sigma_pi.ma".
+
+(************************* notation for minimization **************************)
+notation "μ_{ ident i < n } p" 
+  with precedence 80 for @{min $n 0 (λ${ident i}.$p)}.
+
+notation "μ_{ ident i ≤ n } p" 
+  with precedence 80 for @{min (S $n) 0 (λ${ident i}.$p)}.
+  
+notation "μ_{ ident i ∈ [a,b[ } p"
+  with precedence 80 for @{min ($b-$a) $a (λ${ident i}.$p)}.
+  
+notation "μ_{ ident i ∈ [a,b] } p"
+  with precedence 80 for @{min (S $b-$a) $a (λ${ident i}.$p)}.
+  
+(************************************ MAX *************************************)
+notation "Max_{ ident i < n | p } f"
+  with precedence 80
+for @{'bigop $n max 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
+
+notation "Max_{ ident i < n } f"
+  with precedence 80
+for @{'bigop $n max 0 (λ${ident i}.true) (λ${ident i}. $f)}.
+
+notation "Max_{ ident j ∈ [a,b[ } f"
+  with precedence 80
+for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
+  (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+  
+notation "Max_{ ident j ∈ [a,b[ | p } f"
+  with precedence 80
+for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
+  (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+lemma Max_assoc: ∀a,b,c. max (max a b) c = max a (max b c).
+#a #b #c normalize cases (true_or_false (leb a b)) #leab >leab normalize
+  [cases (true_or_false (leb b c )) #lebc >lebc normalize
+    [>(le_to_leb_true a c) // @(transitive_le ? b) @leb_true_to_le //
+    |>leab //
+    ]
+  |cases (true_or_false (leb b c )) #lebc >lebc normalize //
+   >leab normalize >(not_le_to_leb_false a c) // @lt_to_not_le 
+   @(transitive_lt ? b) @not_le_to_lt @leb_false_to_not_le //
+  ]
+qed.
+
+lemma Max0 : ∀n. max 0 n = n.
+// qed.
+
+lemma Max0r : ∀n. max n 0 = n.
+#n >commutative_max //
+qed.
+
+definition MaxA ≝ 
+  mk_Aop nat 0 max Max0 Max0r (λa,b,c.sym_eq … (Max_assoc a b c)). 
+
+definition MaxAC ≝ mk_ACop nat 0 MaxA commutative_max.
+
+lemma le_Max: ∀f,p,n,a. a < n → p a = true →
+  f a ≤  Max_{i < n | p i}(f i).
+#f #p #n #a #ltan #pa 
+>(bigop_diff p ? 0 MaxAC f a n) // @(le_maxl … (le_n ?))
+qed.
+
+lemma le_MaxI: ∀f,p,n,m,a. m ≤ a → a < n → p a = true →
+  f a ≤  Max_{i ∈ [m,n[ | p i}(f i).
+#f #p #n #m #a #lema #ltan #pa 
+>(bigop_diff ? ? 0 MaxAC (λi.f (i+m)) (a-m) (n-m)) 
+  [<plus_minus_m_m // @(le_maxl … (le_n ?))
+  |<plus_minus_m_m //
+  |/2 by monotonic_lt_minus_l/
+  ]
+qed.
+
+lemma Max_le: ∀f,p,n,b. 
+  (∀a.a < n → p a = true → f a ≤ b) → Max_{i < n | p i}(f i) ≤ b.
+#f #p #n elim n #b #H // 
+#b0 #H1 cases (true_or_false (p b)) #Hb
+  [>bigop_Strue [2:@Hb] @to_max [@H1 // | @H #a #ltab #pa @H1 // @le_S //]
+  |>bigop_Sfalse [2:@Hb] @H #a #ltab #pa @H1 // @le_S //
+  ]
+qed.
+
+
+(************************* a couple of technical lemmas ***********************)
+lemma minus_to_0: ∀a,b. a ≤ b → minus a b = 0.
+#a elim a // #n #Hind * 
+  [#H @False_ind /2 by absurd/ | #b normalize #H @Hind @le_S_S_to_le /2/]
+qed.
+
+lemma sigma_const: ∀c,a,b. 
+  ∑_{i ∈ [a,b[ }c ≤  (b-a)*c.
+#c #a #b elim (b-a) // #n #Hind normalize @le_plus // 
+qed.
+
+lemma sigma_to_Max:  ∀h,a,b. 
+  ∑_{i ∈ [a,b[ }(h i) ≤  (b-a)*Max_{i ∈ [a,b[ }(h i).
+#h #a #b elim (b-a)
+  [//
+  |#n #Hind >bigop_Strue [2://] whd in ⊢ (??%);
+   @le_plus 
+    [>bigop_Strue // @(le_maxl … (le_n …)) 
+    |@(transitive_le … Hind) @le_times // >bigop_Strue //
+     @(le_maxr … (le_n …))
+    ]
+  ]
+qed.
+
+lemma sigma_bound1:  ∀h,a,b. monotonic nat le h → 
+  ∑_{i ∈ [a,b[ }(h i) ≤  (b-a)*h b.
+#h #a #b #H 
+@(transitive_le … (sigma_to_Max …)) @le_times //
+@Max_le #i #lti #_ @H @lt_to_le @lt_minus_to_plus_r //
+qed.
+
+lemma sigma_bound_decr1:  ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) → 
+  ∑_{i ∈ [a,b[ }(h i) ≤  (b-a)*h a.
+#h #a #b #H 
+@(transitive_le … (sigma_to_Max …)) @le_times //
+@Max_le #i #lti #_ @H // @lt_minus_to_plus_r //
+qed. 
+  
+lemma sigma_bound:  ∀h,a,b. monotonic nat le h → 
+  ∑_{i ∈ [a,S b[ }(h i) ≤  (S b-a)*h b.
+#h #a #b #H cases (decidable_le a b) 
+  [#leab cut (b = pred (S b - a + a)) 
+    [<plus_minus_m_m // @le_S //] #Hb >Hb in match (h b);
+   generalize in match (S b -a);
+   #n elim n 
+    [//
+    |#m #Hind >bigop_Strue [2://] @le_plus 
+      [@H @le_n |@(transitive_le … Hind) @le_times [//] @H //]
+    ]
+  |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
+   cut (S b -a = 0) [@minus_to_0 //] #Hcut >Hcut //
+  ]
+qed.
+
+lemma sigma_bound_decr:  ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) → 
+  ∑_{i ∈ [a,b[ }(h i) ≤  (b-a)*h a.
+#h #a #b #H cases (decidable_le a b) 
+  [#leab cut ((b -a) +a ≤ b) [/2 by le_minus_to_plus_r/] generalize in match (b -a);
+   #n elim n 
+    [//
+    |#m #Hind >bigop_Strue [2://] #Hm 
+     cut (m+a ≤ b) [@(transitive_le … Hm) //] #Hm1
+     @le_plus [@H // |@(transitive_le … (Hind Hm1)) //]
+    ]
+  |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
+   cut (b -a = 0) [@minus_to_0 @lt_to_le @ltba] #Hcut >Hcut //
+  ]
+qed. 
+  
\ No newline at end of file
diff --git a/matita/matita/broken_lib/reverse_complexity/speed_clean.ma b/matita/matita/lib/reverse_complexity/speed_clean.ma
similarity index 100%
rename from matita/matita/broken_lib/reverse_complexity/speed_clean.ma
rename to matita/matita/lib/reverse_complexity/speed_clean.ma
diff --git a/matita/matita/broken_lib/reverse_complexity/speed_def.ma b/matita/matita/lib/reverse_complexity/speed_def.ma
similarity index 100%
rename from matita/matita/broken_lib/reverse_complexity/speed_def.ma
rename to matita/matita/lib/reverse_complexity/speed_def.ma
diff --git a/matita/matita/broken_lib/reverse_complexity/speed_new.ma b/matita/matita/lib/reverse_complexity/speed_new.ma
similarity index 100%
rename from matita/matita/broken_lib/reverse_complexity/speed_new.ma
rename to matita/matita/lib/reverse_complexity/speed_new.ma
diff --git a/matita/matita/broken_lib/reverse_complexity/toolkit.ma b/matita/matita/lib/reverse_complexity/speedup.ma
similarity index 71%
rename from matita/matita/broken_lib/reverse_complexity/toolkit.ma
rename to matita/matita/lib/reverse_complexity/speedup.ma
index 69fa3525a..12605c9e4 100644
--- a/matita/matita/broken_lib/reverse_complexity/toolkit.ma
+++ b/matita/matita/lib/reverse_complexity/speedup.ma
@@ -1,3 +1,5 @@
+(*
+<<<<<<< HEAD:matita/matita/broken_lib/reverse_complexity/toolkit.ma
 include "basics/types.ma".
 include "arithmetics/minimization.ma".
 include "arithmetics/bigops.ma".
@@ -181,6 +183,10 @@ qed.
 lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
 #i #x #r normalize cases (U i x r) normalize >snd_pair //
 qed.
+=======
+include "reverse_complexity/bigops_compl.ma".
+>>>>>>> reverse_complexity lib restored:matita/matita/lib/reverse_complexity/speedup.ma
+*)
 
 (********************************* the speedup ********************************)
 
@@ -282,7 +288,6 @@ lemma condition_1: ∀h,u.g h 0 ≈ g h u.
 qed.
 
 (******************************** Condition 2 *********************************)
-definition total ≝ λf.λx:nat. Some nat (f x).
   
 lemma exists_to_exists_min: ∀h,i. (∃x. i < x ∧ {i ⊙ x} ↓ h (S i) x) → ∃y. min_input h i y = y.
 #h #i * #x * #ltix #Hx %{(min_input h i x)} @min_spec_to_min @found //
@@ -301,200 +306,7 @@ whd in match (out ???); >termy >Hr
 #H @(absurd ? H) @le_to_not_lt @le_n
 qed.
 
-
-(********************************* complexity *********************************)
-
-(* We assume operations have a minimal structural complexity MSC. 
-For instance, for time complexity, MSC is equal to the size of input.
-For space complexity, MSC is typically 0, since we only measure the
-space required in addition to dimension of the input. *)
-
-axiom MSC : nat → nat.
-axiom MSC_le: ∀n. MSC n ≤ n.
-axiom monotonic_MSC: monotonic ? le MSC.
-axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
-
-(* C s i means i is running in O(s) *)
- 
-definition C ≝ λs,i.∃c.∃a.∀x.a ≤ x → ∃y. 
-  U i x (c*(s x)) = Some ? y.
-
-(* C f s means f ∈ O(s) where MSC ∈O(s) *)
-definition CF ≝ λs,f.O s MSC ∧ ∃i.code_for (total f) i ∧ C s i.
- 
-lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
-#f #g #s #Hext * #HO  * #i * #Hcode #HC % // %{i} %
-  [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //] 
-qed.
-
-lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤  s2 x) → CF s1 f → CF s2 f.
-#s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 % 
-  [cases HO #c * #a -HO #HO %{c} %{a} #n #lean @(transitive_le … (HO n lean))
-   @le_times // 
-  |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 %{c} %{a} #n #lean 
-   cases(Hs1 n lean) #y #Hy %{y} @(monotonic_U …Hy) @le_times // 
-  ]
-qed.
-
-lemma O_to_CF: ∀s1,s2,f.O s2 s1 → CF s1 f → CF s2 f.
-#s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 % 
-  [@(O_trans … H) //
-  |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 
-   cases H #c1 * #a1 #Ha1 %{(c*c1)} %{(a+a1)} #n #lean 
-   cases(Hs1 n ?) [2:@(transitive_le … lean) //] #y #Hy %{y} @(monotonic_U …Hy)
-   >associative_times @le_times // @Ha1 @(transitive_le … lean) //
-  ]
-qed.
-
-lemma timesc_CF: ∀s,f,c.CF (λx.c*s x) f → CF s f.
-#s #f #c @O_to_CF @O_times_c 
-qed.
-
-(********************************* composition ********************************)
-axiom CF_comp: ∀f,g,sf,sg,sh. CF sg g → CF sf f → 
-  O sh (λx. sg x + sf (g x)) → CF sh (f ∘ g).
-  
-lemma CF_comp_ext: ∀f,g,h,sh,sf,sg. CF sg g → CF sf f → 
-  (∀x.f(g x) = h x) → O sh (λx. sg x + sf (g x)) → CF sh h.
-#f #g #h #sh #sf #sg #Hg #Hf #Heq #H @(ext_CF (f ∘ g))
-  [#n normalize @Heq | @(CF_comp … H) //]
-qed.
-
-(* primitve recursion *)
-
-let rec prim_rec (k,h:nat →nat) n m on n ≝ 
-  match n with 
-  [ O ⇒ k m
-  | S a ⇒ h 〈a,〈prim_rec k h a m, m〉〉].
-  
-lemma prim_rec_S: ∀k,h,n,m.
-  prim_rec k h (S n) m = h 〈n,〈prim_rec k h n m, m〉〉.
-// qed.
-
-definition unary_pr ≝ λk,h,x. prim_rec k h (fst x) (snd x).
-
-let rec prim_rec_compl (k,h,sk,sh:nat →nat) n m on n ≝ 
-  match n with 
-  [ O ⇒ sk m
-  | S a ⇒ prim_rec_compl k h sk sh a m + sh (prim_rec k h a m)].
-  
-axiom CF_prim_rec: ∀k,h,sk,sh,sf. CF sk k → CF sh h → 
-  O sf (unary_pr sk (λx. fst (snd x) + sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉)) 
-   → CF sf (unary_pr k h).
-
-(* falso ????
-lemma prim_rec_O: ∀k1,h1,k2,h2. O k1 k2 → O h1 h2 → 
-  O (unary_pr k1 h1) (unary_pr k2 h2).
-#k1 #h1 #k2 #h2 #HO1 #HO2 whd *)
-
-  
-(**************************** primitive operations*****************************)
-
-definition id ≝ λx:nat.x.
-
-axiom CF_id: CF MSC id.
-axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
-axiom CF_comp_fst: ∀h,f. CF h f → CF h (fst ∘ f).
-axiom CF_comp_snd: ∀h,f. CF h f → CF h (snd ∘ f). 
-axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉). 
-
-lemma CF_fst: CF MSC fst.
-@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
-qed.
-
-lemma CF_snd: CF MSC snd.
-@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
-qed.
-
-(************************************** eqb ***********************************)
-  
-axiom CF_eqb: ∀h,f,g.
-  CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
-
-(*********************************** maximum **********************************) 
-
-axiom CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
-  CF ha a → CF hb b → CF hp p → CF hf f → 
-  O s (λx.ha x + hb x + ∑_{i ∈[a x ,b x[ }(hp 〈i,x〉 + hf 〈i,x〉)) →
-  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)). 
-
-(******************************** minimization ********************************) 
-
-axiom CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
-  CF sa a → CF sb b → CF sf f → 
-  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
-  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
-  
-(************************************* smn ************************************)
-axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
-
-(****************************** constructibility ******************************)
- 
-definition constructible ≝ λs. CF s s.
-
-lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
-  (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
-#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //] 
-qed.
-
-lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) → 
-  constructible s1 → constructible s2.
-#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1)  #x >Hext //
-qed.
-
-lemma constr_prim_rec: ∀s1,s2. constructible s1 → constructible s2 →
-  (∀n,r,m. 2 * r ≤ s2 〈n,〈r,m〉〉) → constructible (unary_pr s1 s2).
-#s1 #s2 #Hs1 #Hs2 #Hincr @(CF_prim_rec … Hs1 Hs2) whd %{2} %{0} 
-#x #_ lapply (surj_pair x) * #a * #b #eqx >eqx whd in match (unary_pr ???); 
->fst_pair >snd_pair
-whd in match (unary_pr ???); >fst_pair >snd_pair elim a
-  [normalize //
-  |#n #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
-   >prim_rec_S @transitive_le [| @(monotonic_le_plus_l … Hind)]
-   @transitive_le [| @(monotonic_le_plus_l … (Hincr n ? b))] 
-   whd in match (unary_pr ???); >fst_pair >snd_pair //
-  ]
-qed.
-
-(********************************* simulation *********************************)
-
-axiom sU : nat → nat. 
-
-axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
-  sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
-
-lemma monotonic_sU_aux : ∀x1,x2. fst x1 ≤ fst x2 → fst (snd x1) ≤ fst (snd x2) →
-snd (snd x1) ≤ snd (snd x2) → sU x1 ≤ sU x2.
-#x1 #x2 cases (surj_pair x1) #a1 * #y #eqx1 >eqx1 -eqx1 cases (surj_pair y) 
-#b1 * #c1 #eqy >eqy -eqy
-cases (surj_pair x2) #a2 * #y2 #eqx2 >eqx2 -eqx2 cases (surj_pair y2) 
-#b2 * #c2 #eqy2 >eqy2 -eqy2 >fst_pair >snd_pair >fst_pair >snd_pair 
->fst_pair >snd_pair >fst_pair >snd_pair @monotonic_sU
-qed.
-  
-axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
-
-definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
-
-axiom CF_U : CF sU pU_unary.
-
-definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
-definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
-
-lemma CF_termb: CF sU termb_unary. 
-@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
-#n whd in ⊢ (??%?); whd in  ⊢ (??(?%)?); >fst_pair % 
-qed.
-
-lemma CF_out: CF sU out_unary. 
-@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
-#n whd in ⊢ (??%?); whd in  ⊢ (??(?%)?); >snd_pair % 
-qed.
-
-
-(******************** complexity of g ********************)
+(**************************** complexity of g *********************************)
 
 definition unary_g ≝ λh.λux. g h (fst ux) (snd ux).
 definition auxg ≝ 
@@ -519,18 +331,18 @@ lemma compl_g2 : ∀h,s1,s2,s.
     (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
   CF s2
     (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) →
-  O s (λx.MSC x + ∑_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) → 
+  O s (λx.MSC x + (snd x - fst x)*Max_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) → 
   CF s (auxg h).
 #h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h)) 
   [#n whd in ⊢ (??%%); @eq_aux]
-@(CF_max … CF_fst CF_snd Hs1 Hs2 …) @(O_trans … HO) 
+@(CF_max2 … CF_fst CF_snd Hs1 Hs2 …) @(O_trans … HO) 
 @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
 qed.  
 
 lemma compl_g3 : ∀h,s.
   CF s (λp:ℕ.min_input h (fst p) (snd (snd p))) →
   CF s (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))).
-#h #s #H @(CF_eqb … H) @(CF_comp … CF_snd CF_snd) @(O_trans … (proj1 … H))
+#h #s #H @(CF_eqb … H) @(CF_comp … CF_snd CF_snd) @(O_trans … (le_to_O … (MSC_in … H)))
 @O_plus // %{1} %{0} #n #_ >commutative_times <times_n_1 @monotonic_MSC //
 qed.
 
@@ -548,55 +360,20 @@ qed.
 definition termb_aux ≝ λh.
   termb_unary ∘ λp.〈fst (snd p),〈fst p,h (S (fst (snd p))) (fst p)〉〉.
 
-lemma compl_g4 : ∀h,s1,s.
+lemma compl_g4 : ∀h,s1,s. (∀x. 0 < s1 x) →  
   (CF s1 
     (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
-   (O s (λx.MSC x + ∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
+   (O s (λx.MSC x + ((snd(snd x) - (fst x)))*Max_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
   CF s (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
+#h #s1 #s #pos_s1 #Hs1 #HO @(ext_CF (min_input_aux h))
  [#n whd in ⊢ (??%%); @min_input_eq]
-@(CF_mu … MSC MSC … Hs1) 
+@(CF_mu4 … MSC MSC … pos_s1 … Hs1) 
   [@CF_compS @CF_fst 
   |@CF_comp_snd @CF_snd
   |@(O_trans … HO) @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
 qed.
 
-(************************* a couple of technical lemmas ***********************)
-lemma minus_to_0: ∀a,b. a ≤ b → minus a b = 0.
-#a elim a // #n #Hind * 
-  [#H @False_ind /2 by absurd/ | #b normalize #H @Hind @le_S_S_to_le /2/]
-qed.
-
-lemma sigma_bound:  ∀h,a,b. monotonic nat le h → 
-  ∑_{i ∈ [a,S b[ }(h i) ≤  (S b-a)*h b.
-#h #a #b #H cases (decidable_le a b) 
-  [#leab cut (b = pred (S b - a + a)) 
-    [<plus_minus_m_m // @le_S //] #Hb >Hb in match (h b);
-   generalize in match (S b -a);
-   #n elim n 
-    [//
-    |#m #Hind >bigop_Strue [2://] @le_plus 
-      [@H @le_n |@(transitive_le … Hind) @le_times [//] @H //]
-    ]
-  |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
-   cut (S b -a = 0) [@minus_to_0 //] #Hcut >Hcut //
-  ]
-qed.
-
-lemma sigma_bound_decr:  ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) → 
-  ∑_{i ∈ [a,b[ }(h i) ≤  (b-a)*h a.
-#h #a #b #H cases (decidable_le a b) 
-  [#leab cut ((b -a) +a ≤ b) [/2 by le_minus_to_plus_r/] generalize in match (b -a);
-   #n elim n 
-    [//
-    |#m #Hind >bigop_Strue [2://] #Hm 
-     cut (m+a ≤ b) [@(transitive_le … Hm) //] #Hm1
-     @le_plus [@H // |@(transitive_le … (Hind Hm1)) //]
-    ]
-  |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
-   cut (b -a = 0) [@minus_to_0 @lt_to_le @ltba] #Hcut >Hcut //
-  ]
-qed. 
+(* ??? *)
 
 lemma coroll: ∀s1:nat→nat. (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) → 
 O (λx.(snd (snd x)-fst x)*(s1 〈snd (snd x),x〉)) 
@@ -611,15 +388,25 @@ O (λx.(snd x - fst x)*s1 〈fst x,x〉) (λx.∑_{i ∈[fst x,snd x[ }(s1 〈i,
 @(transitive_le … (sigma_bound_decr …)) [2://] @Hs1 
 qed.
 
+lemma coroll3: ∀s1:nat→nat. (∀n,a,b. a ≤ b → b < snd n → s1 〈b,n〉 ≤ s1 〈a,n〉) → 
+O (λx.(snd x - fst x)*s1 〈fst x,x〉) 
+  (λx.(snd x - fst x)*Max_{i ∈[fst x,snd x[ }(s1 〈i,x〉)).
+#s1 #Hs1 @le_to_O #i @le_times // @Max_le #a #lta #_ @Hs1 // 
+@lt_minus_to_plus_r //
+qed.
+
 (**************************** end of technical lemmas *************************)
 
-lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+lemma compl_g5 : ∀h,s1.
+  (∀n. 0 < s1 n) → 
+  (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
   (CF s1 
     (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
   CF (λx.MSC x + (snd (snd x)-fst x)*s1 〈snd (snd x),x〉) 
     (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #s1 #Hmono #Hs1 @(compl_g4 … Hs1) @O_plus 
-[@O_plus_l // |@O_plus_r @coroll @Hmono]
+#h #s1 #Hpos #Hmono #Hs1 @(compl_g4 … Hpos Hs1) @O_plus 
+[@O_plus_l // |@O_plus_r @le_to_O #n @le_times //
+@Max_le #i #lti #_ @Hmono @le_S_S_to_le @lt_minus_to_plus_r @lti
 qed.
 
 lemma compl_g6: ∀h.
@@ -683,7 +470,7 @@ lemma compl_g7: ∀h.
     (λp:ℕ.min_input h (fst p) (snd (snd p))).
 #h #hcostr #hmono @(monotonic_CF … (faux2 h))
   [#n normalize >fst_pair >snd_pair //]
-@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair 
+@compl_g5 [#n @sU_pos | 3:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair 
 >fst_pair >snd_pair @monotonic_sU // @hmono @lexy
 qed.
 
@@ -751,7 +538,7 @@ lemma compl_g9 : ∀h.
 @O_plus 
   [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
     [// | @monotonic_MSC // ]]
-@(O_trans … (coroll2 ??))
+@(O_trans … (coroll3 ??))
   [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
    cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
    cut (max a n = n) 
@@ -772,15 +559,25 @@ lemma compl_g9 : ∀h.
   ]
 qed.
 
+definition c ≝ λx.(S (snd x-fst x))*MSC 〈x,x〉.
+
 definition sg ≝ λh,x.
-  (S (snd x-fst x))*MSC 〈x,x〉 + (snd x-fst x)*(S(snd x-fst x))*sU 〈x,〈snd x,h (S (fst x)) (snd x)〉〉.
+  let a ≝ fst x in
+  let b ≝ snd x in 
+  c x + (b-a)*(S(b-a))*sU 〈x,〈snd x,h (S a) b〉〉.
 
-lemma sg_def : ∀h,a,b. 
-  sg h 〈a,b〉 = (S (b-a))*MSC 〈〈a,b〉,〈a,b〉〉 + 
+lemma sg_def1 : ∀h,a,b. 
+  sg h 〈a,b〉 = c 〈a,b〉 +  
    (b-a)*(S(b-a))*sU 〈〈a,b〉,〈b,h (S a) b〉〉.
 #h #a #b whd in ⊢ (??%?); >fst_pair >snd_pair // 
 qed. 
 
+lemma sg_def : ∀h,a,b. 
+  sg h 〈a,b〉 = 
+   S (b-a)*MSC 〈〈a,b〉,〈a,b〉〉 + (b-a)*(S(b-a))*sU 〈〈a,b〉,〈b,h (S a) b〉〉.
+#h #a #b normalize >fst_pair >snd_pair // 
+qed. 
+
 lemma compl_g11 : ∀h.
   constructible (λx. h (fst x) (snd x)) →
   (∀n. monotonic ? le (h n)) → 
@@ -807,31 +604,6 @@ lemma h_of_aux_S : ∀r,c,d,b.
       (S d)*(S (S d))*sU 〈〈b-(S d),b〉,〈b,r(h_of_aux r c d b)〉〉.
 // qed.
 
-lemma h_of_aux_prim_rec : ∀r,c,n,b. h_of_aux r c n b =
-  prim_rec (λx.c) 
-   (λx.let d ≝ S(fst x) in 
-       let b ≝ snd (snd x) in
-       (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
-        d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉) n b.
-#r #c #n #b elim n
-  [>h_of_aux_O normalize //
-  |#n1 #Hind >h_of_aux_S >prim_rec_S >snd_pair >snd_pair >fst_pair 
-   >fst_pair <Hind //
-  ]
-qed.
-
-lemma h_of_aux_constr : 
-∀r,c. constructible (λx.h_of_aux r c (fst x) (snd x)).
-#r #c 
-  @(ext_constr … 
-    (unary_pr (λx.c) 
-     (λx.let d ≝ S(fst x) in 
-       let b ≝ snd (snd x) in
-       (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
-        d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉)))
-  [#n @sym_eq whd in match (unary_pr ???); @h_of_aux_prim_rec
-  |@constr_prim_rec
-
 definition h_of ≝ λr,p. 
   let m ≝ max (fst p) (snd p) in 
   h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (snd p - fst p) (snd p).
@@ -942,7 +714,7 @@ theorem pseudo_speedup:
 (* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
 #r #Hr #Hmono #Hconstr
 (* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *) 
-%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #i *
 #Hcodei #HCi 
 (* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
 %{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
@@ -967,19 +739,19 @@ theorem pseudo_speedup':
   ∃m,a.∀n. a≤n → r(sg a) < m * sf n. 
 #r #Hr #Hmono #Hconstr 
 (* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *) 
-%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #i *
 #Hcodei #HCi 
 (* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
 %{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
-(* sg is (λx.h_of r 〈i,x〉) *)
+(* sg is (λx.h_of r 〈S i,x〉) *)
 %{(λx. h_of r 〈S i,x〉)}
 lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
 %[%[@condition_1 |@Hg]
  |cases Hg #H1 * #j * #Hcodej #HCj
   lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
   cases HCi #m * #a #Ha
-  %{m} %{(max (S i) a)} #n #ltin @not_le_to_lt @(not_to_not … Hcond2) -Hcond2 #Hlesf 
-  %{n} % [@(transitive_le … ltin) @(le_maxl … (le_n …))]
+  %{m} %{(max (S i) a)} #n #ltin @not_le_to_lt @(not_to_not … Hcond2) -Hcond2 
+  #Hlesf %{n} % [@(transitive_le … ltin) @(le_maxl … (le_n …))]
   cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))] 
   #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf)
   @Hmono @(mono_h_of2 … Hr Hmono … ltin)
diff --git a/matita/matita/lib/reverse_complexity/toolkit.ma b/matita/matita/lib/reverse_complexity/toolkit.ma
new file mode 100644
index 000000000..665df181e
--- /dev/null
+++ b/matita/matita/lib/reverse_complexity/toolkit.ma
@@ -0,0 +1,1623 @@
+include "basics/types.ma".
+include "arithmetics/minimization.ma".
+include "arithmetics/bigops.ma".
+include "arithmetics/sigma_pi.ma".
+include "arithmetics/bounded_quantifiers.ma".
+include "reverse_complexity/big_O.ma".
+
+(************************* notation for minimization **************************)
+notation "μ_{ ident i < n } p" 
+  with precedence 80 for @{min $n 0 (λ${ident i}.$p)}.
+
+notation "μ_{ ident i ≤ n } p" 
+  with precedence 80 for @{min (S $n) 0 (λ${ident i}.$p)}.
+  
+notation "μ_{ ident i ∈ [a,b[ } p"
+  with precedence 80 for @{min ($b-$a) $a (λ${ident i}.$p)}.
+  
+notation "μ_{ ident i ∈ [a,b] } p"
+  with precedence 80 for @{min (S $b-$a) $a (λ${ident i}.$p)}.
+  
+(************************************ MAX *************************************)
+notation "Max_{ ident i < n | p } f"
+  with precedence 80
+for @{'bigop $n max 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
+
+notation "Max_{ ident i < n } f"
+  with precedence 80
+for @{'bigop $n max 0 (λ${ident i}.true) (λ${ident i}. $f)}.
+
+notation "Max_{ ident j ∈ [a,b[ } f"
+  with precedence 80
+for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
+  (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+  
+notation "Max_{ ident j ∈ [a,b[ | p } f"
+  with precedence 80
+for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
+  (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+lemma Max_assoc: ∀a,b,c. max (max a b) c = max a (max b c).
+#a #b #c normalize cases (true_or_false (leb a b)) #leab >leab normalize
+  [cases (true_or_false (leb b c )) #lebc >lebc normalize
+    [>(le_to_leb_true a c) // @(transitive_le ? b) @leb_true_to_le //
+    |>leab //
+    ]
+  |cases (true_or_false (leb b c )) #lebc >lebc normalize //
+   >leab normalize >(not_le_to_leb_false a c) // @lt_to_not_le 
+   @(transitive_lt ? b) @not_le_to_lt @leb_false_to_not_le //
+  ]
+qed.
+
+lemma Max0 : ∀n. max 0 n = n.
+// qed.
+
+lemma Max0r : ∀n. max n 0 = n.
+#n >commutative_max //
+qed.
+
+definition MaxA ≝ 
+  mk_Aop nat 0 max Max0 Max0r (λa,b,c.sym_eq … (Max_assoc a b c)). 
+
+definition MaxAC ≝ mk_ACop nat 0 MaxA commutative_max.
+
+lemma le_Max: ∀f,p,n,a. a < n → p a = true →
+  f a ≤  Max_{i < n | p i}(f i).
+#f #p #n #a #ltan #pa 
+>(bigop_diff p ? 0 MaxAC f a n) // @(le_maxl … (le_n ?))
+qed.
+
+lemma le_MaxI: ∀f,p,n,m,a. m ≤ a → a < n → p a = true →
+  f a ≤  Max_{i ∈ [m,n[ | p i}(f i).
+#f #p #n #m #a #lema #ltan #pa 
+>(bigop_diff ? ? 0 MaxAC (λi.f (i+m)) (a-m) (n-m)) 
+  [<plus_minus_m_m // @(le_maxl … (le_n ?))
+  |<plus_minus_m_m //
+  |/2 by monotonic_lt_minus_l/
+  ]
+qed.
+
+lemma Max_le: ∀f,p,n,b. 
+  (∀a.a < n → p a = true → f a ≤ b) → Max_{i < n | p i}(f i) ≤ b.
+#f #p #n elim n #b #H // 
+#b0 #H1 cases (true_or_false (p b)) #Hb
+  [>bigop_Strue [2:@Hb] @to_max [@H1 // | @H #a #ltab #pa @H1 // @le_S //]
+  |>bigop_Sfalse [2:@Hb] @H #a #ltab #pa @H1 // @le_S //
+  ]
+qed.
+
+(********************************** pairing ***********************************)
+axiom pair: nat → nat → nat.
+axiom fst : nat → nat.
+axiom snd : nat → nat.
+
+interpretation "abstract pair" 'pair f g = (pair f g). 
+
+axiom fst_pair: ∀a,b. fst 〈a,b〉 = a.
+axiom snd_pair: ∀a,b. snd 〈a,b〉 = b.
+axiom surj_pair: ∀x. ∃a,b. x = 〈a,b〉. 
+
+axiom le_fst : ∀p. fst p ≤ p.
+axiom le_snd : ∀p. snd p ≤ p.
+axiom le_pair: ∀a,a1,b,b1. a ≤ a1 → b ≤ b1 → 〈a,b〉 ≤ 〈a1,b1〉.
+
+lemma expand_pair: ∀x. x = 〈fst x, snd x〉. 
+#x lapply (surj_pair x) * #a * #b #Hx >Hx >fst_pair >snd_pair //
+qed.
+
+(************************************* U **************************************)
+axiom U: nat → nat →nat → option nat. 
+
+axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
+  U i x n = Some ? y → U i x m = Some ? y.
+  
+lemma unique_U: ∀i,x,n,m,yn,ym.
+  U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
+#i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
+  [#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
+  |#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
+   >Hn #HS destruct (HS) //
+  ]
+qed.
+
+definition code_for ≝ λf,i.∀x.
+  ∃n.∀m. n ≤ m → U i x m = f x.
+
+definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y. 
+
+notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}. 
+
+lemma terminate_dec: ∀i,x,n. {i ⊙ x} ↓ n ∨ ¬ {i ⊙ x} ↓ n.
+#i #x #n normalize cases (U i x n)
+  [%2 % * #y #H destruct|#y %1 %{y} //]
+qed.
+  
+lemma monotonic_terminate: ∀i,x,n,m.
+  n ≤ m → {i ⊙ x} ↓ n → {i ⊙ x} ↓ m.
+#i #x #n #m #lenm * #z #H %{z} @(monotonic_U … H) //
+qed.
+
+definition termb ≝ λi,x,t. 
+  match U i x t with [None ⇒ false |Some y ⇒ true].
+
+lemma termb_true_to_term: ∀i,x,t. termb i x t = true → {i ⊙ x} ↓ t.
+#i #x #t normalize cases (U i x t) normalize [#H destruct | #y #_ %{y} //]
+qed.    
+
+lemma term_to_termb_true: ∀i,x,t. {i ⊙ x} ↓ t → termb i x t = true.
+#i #x #t * #y #H normalize >H // 
+qed.    
+
+definition out ≝ λi,x,r. 
+  match U i x r with [ None ⇒ 0 | Some z ⇒ z]. 
+
+definition bool_to_nat: bool → nat ≝ 
+  λb. match b with [true ⇒ 1 | false ⇒ 0]. 
+  
+coercion bool_to_nat. 
+
+definition pU : nat → nat → nat → nat ≝ λi,x,r.〈termb i x r,out i x r〉.
+
+lemma pU_vs_U_Some : ∀i,x,r,y. pU i x r = 〈1,y〉 ↔ U i x r = Some ? y.
+#i #x #r #y % normalize
+  [cases (U i x r) normalize 
+    [#H cut (0=1) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H @H] 
+     #H1 destruct
+    |#a #H cut (a=y) [lapply (eq_f … snd … H) >snd_pair >snd_pair #H1 @H1] 
+     #H1 //
+    ]
+  |#H >H //]
+qed.
+  
+lemma pU_vs_U_None : ∀i,x,r. pU i x r = 〈0,0〉 ↔ U i x r = None ?.
+#i #x #r % normalize
+  [cases (U i x r) normalize //
+   #a #H cut (1=0) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H1 @H1] 
+   #H1 destruct
+  |#H >H //]
+qed.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+qed.
+
+
+definition total ≝ λf.λx:nat. Some nat (f x).
+
+
+(********************************* complexity *********************************)
+
+(* We assume operations have a minimal structural complexity MSC. 
+For instance, for time complexity, MSC is equal to the size of input.
+For space complexity, MSC is typically 0, since we only measure the
+space required in addition to dimension of the input. *)
+
+axiom MSC : nat → nat.
+axiom MSC_le: ∀n. MSC n ≤ n.
+axiom monotonic_MSC: monotonic ? le MSC.
+axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
+
+(* C s i means i is running in O(s) *)
+ 
+definition C ≝ λs,i.∃c.∃a.∀x.a ≤ x → ∃y. 
+  U i x (c*(s x)) = Some ? y.
+
+(* C f s means f ∈ O(s) where MSC ∈O(s) *)
+definition CF ≝ λs,f.O s MSC ∧ ∃i.code_for (total f) i ∧ C s i.
+ 
+lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
+#f #g #s #Hext * #HO  * #i * #Hcode #HC % // %{i} %
+  [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //] 
+qed.
+
+lemma ext_CF1 : ∀f,g,s. (∀n. f n = g n) → CF s g → CF s f.
+#f #g #s #Hext * #HO  * #i * #Hcode #HC % // %{i} %
+  [#x cases (Hcode x) #a #H %{a} whd in match (total ??); >Hext @H | //] 
+qed.
+
+lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤  s2 x) → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 % 
+  [cases HO #c * #a -HO #HO %{c} %{a} #n #lean @(transitive_le … (HO n lean))
+   @le_times // 
+  |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 %{c} %{a} #n #lean 
+   cases(Hs1 n lean) #y #Hy %{y} @(monotonic_U …Hy) @le_times // 
+  ]
+qed.
+
+lemma O_to_CF: ∀s1,s2,f.O s2 s1 → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 % 
+  [@(O_trans … H) //
+  |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 
+   cases H #c1 * #a1 #Ha1 %{(c*c1)} %{(a+a1)} #n #lean 
+   cases(Hs1 n ?) [2:@(transitive_le … lean) //] #y #Hy %{y} @(monotonic_U …Hy)
+   >associative_times @le_times // @Ha1 @(transitive_le … lean) //
+  ]
+qed.
+
+lemma timesc_CF: ∀s,f,c.CF (λx.c*s x) f → CF s f.
+#s #f #c @O_to_CF @O_times_c 
+qed.
+
+(********************************* composition ********************************)
+axiom CF_comp: ∀f,g,sf,sg,sh. CF sg g → CF sf f → 
+  O sh (λx. sg x + sf (g x)) → CF sh (f ∘ g).
+  
+lemma CF_comp_ext: ∀f,g,h,sh,sf,sg. CF sg g → CF sf f → 
+  (∀x.f(g x) = h x) → O sh (λx. sg x + sf (g x)) → CF sh h.
+#f #g #h #sh #sf #sg #Hg #Hf #Heq #H @(ext_CF (f ∘ g))
+  [#n normalize @Heq | @(CF_comp … H) //]
+qed.
+
+(* primitve recursion *)
+
+(* no arguments *)
+
+let rec prim_rec0 (k:nat) (h:nat →nat) n on n ≝ 
+  match n with 
+  [ O ⇒ k
+  | S a ⇒ h 〈a, prim_rec0 k h a〉].
+  
+lemma prim_rec0_S: ∀k,h,n.
+  prim_rec0 k h (S n) = h 〈n, prim_rec0 k h n〉.
+// qed.
+
+let rec prim_rec0_compl (k,sk:nat) (h,sh:nat →nat) n on n ≝ 
+  match n with 
+  [ O ⇒ sk
+  | S a ⇒ prim_rec0_compl k sk h sh a + sh (prim_rec0 k h a)].
+
+axiom CF_prim_rec0_gen: ∀k,h,sk,sh,sh1,sf. CF sh h →
+  O sh1 (λx.snd x + sh 〈fst x,prim_rec0 k h (fst x)〉) → 
+  O sf (prim_rec0 sk sh1) →
+   CF sf (prim_rec0 k h).
+
+lemma CF_prim_rec0: ∀k,h,sk,sh,sf. CF sh h → 
+  O sf (prim_rec0 sk (λx. snd x + sh 〈fst x,prim_rec0 k h (fst x)〉))
+   → CF sf (prim_rec0 k h).
+#k #h #sk #sh #sf #Hh #HO @(CF_prim_rec0_gen … Hh … HO) @O_refl
+qed.
+
+(* with argument(s) m *)
+let rec prim_rec (k,h:nat →nat) n m on n ≝ 
+  match n with 
+  [ O ⇒ k m
+  | S a ⇒ h 〈a,〈prim_rec k h a m, m〉〉].
+  
+lemma prim_rec_S: ∀k,h,n,m.
+  prim_rec k h (S n) m = h 〈n,〈prim_rec k h n m, m〉〉.
+// qed.
+
+lemma prim_rec_le: ∀k1,k2,h1,h2. (∀x.k1 x ≤ k2 x) → 
+(∀x,y.x ≤y → h1 x ≤ h2 y) →
+  ∀x,m. prim_rec k1 h1 x m ≤ prim_rec k2 h2 x m.
+#k1 #k2 #h1 #h2 #lek #leh #x #m elim x //
+#n #Hind normalize @leh @le_pair // @le_pair //
+qed.
+ 
+definition unary_pr ≝ λk,h,x. prim_rec k h (fst x) (snd x).
+
+lemma prim_rec_unary: ∀k,h,a,b. 
+  prim_rec k h a b = unary_pr k h 〈a,b〉.
+#k #h #a #b normalize >fst_pair >snd_pair //
+qed.
+  
+
+let rec prim_rec_compl (k,h,sk,sh:nat →nat) n m on n ≝ 
+  match n with 
+  [ O ⇒ sk m
+  | S a ⇒ prim_rec_compl k h sk sh a m + sh (prim_rec k h a m)].
+
+axiom CF_prim_rec_gen: ∀k,h,sk,sh,sh1. CF sk k → CF sh h →  
+  O sh1 (λx. fst (snd x) + sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉) → 
+   CF (unary_pr sk sh1) (unary_pr k h).
+
+lemma CF_prim_rec: ∀k,h,sk,sh,sf. CF sk k → CF sh h → 
+  O sf (unary_pr sk (λx. fst (snd x) + sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉)) 
+   → CF sf (unary_pr k h).
+#k #h #sk #sh #sf #Hk #Hh #Osf @(O_to_CF … Osf) @(CF_prim_rec_gen … Hk Hh) @O_refl
+qed.
+  
+(**************************** primitive operations*****************************)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+axiom CF_const: ∀k.CF MSC (λx.k).
+axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
+axiom CF_comp_fst: ∀h,f. CF h f → CF h (fst ∘ f).
+axiom CF_comp_snd: ∀h,f. CF h f → CF h (snd ∘ f). 
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉). 
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed. 
+
+(************************************** eqb ***********************************)
+  
+axiom CF_eqb: ∀h,f,g.
+  CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
+
+(*********************************** maximum **********************************) 
+
+alias symbol "pair" (instance 13) = "abstract pair".
+alias symbol "pair" (instance 12) = "abstract pair".
+alias symbol "and" (instance 11) = "boolean and".
+alias symbol "plus" (instance 8) = "natural plus".
+alias symbol "pair" (instance 7) = "abstract pair".
+alias symbol "plus" (instance 6) = "natural plus".
+alias symbol "pair" (instance 5) = "abstract pair".
+alias id "max" = "cic:/matita/arithmetics/nat/max#def:2".
+alias symbol "minus" (instance 3) = "natural minus".
+lemma max_gen: ∀a,b,p,f,x. a ≤b → 
+  max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = max_{i < b | leb a i ∧ p 〈i,x〉 }(f 〈i,x〉).
+#a #b #p #f #x @(bigop_I_gen ????? MaxA) 
+qed.
+
+lemma max_prim_rec_base: ∀a,b,p,f,x. a ≤b → 
+  max_{i < b| p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b 
+  [normalize //
+  |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma max_prim_rec: ∀a,b,p,f,x. a ≤b → 
+  max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if leb a (fst i) ∧ p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b 
+  [normalize //
+  |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+   cases (true_or_false (leb a i ∧ p 〈i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma max_prim_rec1: ∀a,b,p,f,x.
+  max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 
+        then max (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma sum_prim_rec1: ∀a,b,p,f,x.
+  ∑_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 
+        then plus (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma bigop_prim_rec: ∀a,b,c,p,f,x.
+  bigop (b x-a x) (λi:ℕ.p 〈i+a x,x〉) ? (c 〈a x,x〉) plus (λi:ℕ.f 〈i+a x,x〉) = 
+    prim_rec c 
+    (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 
+        then plus (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma bigop_prim_rec_dec: ∀a,b,c,p,f,x.
+  bigop (b x-a x) (λi:ℕ.p 〈b x -i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x-i,x〉) = 
+    prim_rec c 
+    (λi.if p 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉 
+        then plus (f 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (b x-a x) 〈b x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+lemma bigop_prim_rec_dec1: ∀a,b,c,p,f,x.
+  bigop (S(b x)-a x) (λi:ℕ.p 〈b x - i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x- i,x〉) = 
+    prim_rec c 
+    (λi.if p 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉 
+        then plus (f 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉) (fst (snd i)) 
+        else fst (snd i)) (S(b x)-a x) 〈b x,x〉.
+#a #b #c #p #f #x elim (S(b x)-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+    [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed.
+
+(*
+lemma bigop_aux_1: ∀k,c,f.
+  bigop (S k) (λi:ℕ.true) ? c plus (λi:ℕ.f i) = 
+    f 0 + bigop k (λi:ℕ.true) ? c plus (λi:ℕ.f (S i)).
+#k #c #f elim k [normalize //] #i #Hind >bigop_Strue //
+
+lemma bigop_prim_rec_dec: ∀a,b,c,f,x.
+  bigop (b x-a x) (λi:ℕ.true) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈i+a x,x〉) = 
+    prim_rec c 
+    (λi. plus (f 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉) (fst (snd i))) 
+    (b x-a x) 〈b x ,x〉.
+#a #b #c #f #x normalize elim (b x-a x) 
+  [normalize //
+  |#i #Hind >prim_rec_S
+   >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+   <Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+  ]
+qed. *)
+
+lemma bigop_plus_c: ∀k,p,f,c.
+  c k + bigop k (λi.p i) ? 0 plus (λi.f i) = 
+    bigop k (λi.p i) ? (c k) plus (λi.f i).
+#k #p #f elim k [normalize //]
+#i #Hind #c cases (true_or_false (p i)) #Hcase
+[>bigop_Strue // >bigop_Strue // <associative_plus >(commutative_plus ? (f i))
+ >associative_plus @eq_f @Hind
+|>bigop_Sfalse // >bigop_Sfalse // 
+]
+qed.
+
+(* the argument is 〈b-a,〈a,x〉〉 *)
+
+
+definition max_unary_pr ≝ λp,f.unary_pr (λx.0) 
+    (λi. 
+      let k ≝ fst i in
+      let r ≝ fst (snd i) in
+      let a ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      if p 〈k + a,x〉 then max (f 〈k+a,x〉) r else r).
+
+lemma max_unary_pr1: ∀a,b,p,f,x.
+  max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    ((max_unary_pr p f) ∘ (λx.〈b x - a x,〈a x,x〉〉)) x.
+#a #b #p #f #x normalize >fst_pair >snd_pair @max_prim_rec1
+qed.
+
+(*
+lemma max_unary_pr: ∀a,b,p,f,x.
+  max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = 
+    prim_rec (λi.0) 
+    (λi.if p 〈fst i +a,x〉 then max (f 〈fst i +a ,snd (snd i)〉) (fst (snd i)) else fst (snd i)) (b-a) x.
+*)
+
+(*
+definition unary_compl ≝ λp,f,hf.
+  unary_pr MSC
+   (λx:ℕ
+    .fst (snd x)
+     +hf
+      〈fst x,
+      〈unary_pr (λx0:ℕ.O)
+       (λi:ℕ
+        .(let (k:ℕ) ≝fst i in 
+          let (r:ℕ) ≝fst (snd i) in 
+          let (a:ℕ) ≝fst (snd (snd i)) in 
+          let (x:ℕ) ≝snd (snd (snd i)) in 
+          if p 〈k+a,x〉 then max (f 〈k+a,x〉) r else r )) 〈fst x,snd (snd x)〉,
+      snd (snd x)〉〉). *)
+      
+definition aux_compl ≝ λhp,hf.λi.
+  let k ≝ fst i in 
+  let r ≝ fst (snd i) in 
+  let a ≝ fst (snd (snd i)) in 
+  let x ≝ snd (snd (snd i)) in 
+  hp 〈k+a,x〉 + hf 〈k+a,x〉 + (* was MSC r*) MSC i .
+  
+definition aux_compl1 ≝ λhp,hf.λi.
+  let k ≝ fst i in 
+  let r ≝ fst (snd i) in 
+  let a ≝ fst (snd (snd i)) in 
+  let x ≝ snd (snd (snd i)) in 
+  hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+
+lemma aux_compl1_def: ∀k,r,m,hp,hf. 
+  aux_compl1 hp hf 〈k,〈r,m〉〉 = 
+  let a ≝ fst m in 
+  let x ≝ snd m in 
+  hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #m #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma aux_compl1_def1: ∀k,r,a,x,hp,hf. 
+  aux_compl1 hp hf 〈k,〈r,〈a,x〉〉〉 = hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #a #x #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair 
+>fst_pair >snd_pair //
+qed.
+
+  
+axiom Oaux_compl: ∀hp,hf. O (aux_compl1 hp hf) (aux_compl hp hf).
+  
+(* 
+definition IF ≝ λb,f,g:nat →option nat. λx.
+  match b x with 
+  [None ⇒ None ?
+  |Some n ⇒ if (eqb n 0) then f x else g x].
+*)
+
+axiom CF_if: ∀b:nat → bool. ∀f,g,s. CF s b → CF s f → CF s g → 
+  CF s (λx. if b x then f x else g x).
+
+(*
+lemma IF_CF: ∀b,f,g,sb,sf,sg. CF sb b → CF sf f → CF sg g → 
+  CF (λn. sb n + sf n + sg n) (IF b f g).
+#b #f #g #sb #sf #sg #Hb #Hf #Hg @IF_CF_new
+  [@(monotonic_CF … Hb) @O_plus_l @O_plus_l //
+  |@(monotonic_CF … Hf) @O_plus_l @O_plus_r //
+  |@(monotonic_CF … Hg) @O_plus_r //
+  ]
+qed.
+*)
+
+axiom CF_le: ∀h,f,g. CF h f → CF h g → CF h (λx. leb (f x) (g x)). 
+axiom CF_max1: ∀h,f,g. CF h f → CF h g → CF h (λx. max (f x) (g x)). 
+axiom CF_plus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x + g x). 
+axiom CF_minus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x - g x). 
+
+axiom MSC_prop: ∀f,h. CF h f → ∀x. MSC (f x) ≤ h x.
+
+lemma MSC_max: ∀f,h,x. CF h f → MSC (max_{i < x}(f i)) ≤ max_{i < x}(h i).
+#f #h #x #hf elim x // #i #Hind >bigop_Strue [|//] >bigop_Strue [|//]
+whd in match (max ??); 
+cases (true_or_false (leb (f i) (bigop i (λi0:ℕ.true) ? 0 max(λi0:ℕ.f i0))))
+#Hcase >Hcase 
+  [@(transitive_le … Hind) @le_maxr //
+  |@(transitive_le … (MSC_prop … hf i)) @le_maxl //
+  ]
+qed.
+  
+lemma CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+  CF ha a → CF hb b → CF hp p → CF hf f → 
+  O s (λx.ha x + hb x + 
+       (∑_{i ∈[a x ,b x[ }
+         (hp 〈i,x〉 + hf 〈i,x〉 + max_{i ∈ [a x, b x [ }(hf 〈i,x〉)))) →
+  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #HO 
+@ext_CF1 [|#x @max_unary_pr1]
+@(CF_comp ??? (λx.ha x + hb x))
+  [|@CF_comp_pair
+    [@CF_minus [@(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+    |@CF_comp_pair 
+      [@(O_to_CF … CFa) @O_plus_l // 
+      | @(O_to_CF … CF_id) @O_plus_r @(proj1 … CFb)
+      ]
+    ]
+  |@(CF_prim_rec … MSC … (aux_compl1 hp hf))
+     [@CF_const
+     |@(O_to_CF … (Oaux_compl … ))
+      @CF_if 
+       [@(CF_comp p … MSC … CFp) 
+         [@CF_comp_pair 
+           [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+           |@CF_comp_snd @CF_comp_snd @CF_snd]
+         |@le_to_O #x normalize >commutative_plus >associative_plus @le_plus //
+         ]
+       |@CF_max1 
+         [@(CF_comp f … MSC … CFf) 
+           [@CF_comp_pair 
+             [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+             |@CF_comp_snd @CF_comp_snd @CF_snd]
+           |@le_to_O #x normalize >commutative_plus // 
+           ]
+         |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+         ]
+       |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+       ]  
+     |@O_refl
+     ]
+  |@(O_trans … HO) -HO
+   @(O_trans ? (λx:ℕ
+   .ha x+hb x
+    +bigop (b x-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus
+     (λi:ℕ
+      .(λi0:ℕ
+        .hp 〈i0,x〉+hf 〈i0,x〉
+         +bigop (b x-a x) (λi1:ℕ.true) ? 0 max
+          (λi1:ℕ.(λi2:ℕ.hf 〈i2,x〉) (i1+a x))) (i+a x))))
+    [
+   @le_to_O #n @le_plus // whd in match (unary_pr ???);
+   >fst_pair >snd_pair >bigop_prim_rec elim (b n - a n)
+    [normalize //
+    |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >aux_compl1_def1
+     >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair 
+     >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+      [-Hind @le_plus // normalize >fst_pair >snd_pair 
+       @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+         (λi1:ℕ.hf 〈i1+a n,n〉)))
+        [elim x [normalize @MSC_le]
+         #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+         >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+          [cases (true_or_false (leb (f 〈x0+a n,n〉)
+            (prim_rec (λx00:ℕ.O)
+             (λi:ℕ
+            .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉 
+             then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+                           (fst (snd i)) 
+                      then fst (snd i) 
+                      else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉  
+             else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+           [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+           |@(transitive_le … (MSC_prop … CFf …)) @(le_maxl … (le_n …))
+           ]
+         |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+         ]
+       |@(le_maxr … (le_n …))
+       ]
+     |@(transitive_le … Hind) -Hind 
+      generalize in match (bigop x (λi:ℕ.true) ? 0 max
+              (λi1:ℕ.(λi2:ℕ.hf 〈i2,n〉) (i1+a n))); #c
+      generalize in match (hf 〈x+a n,n〉); #c1
+      elim x [//] #x0 #Hind 
+      >prim_rec_S >prim_rec_S normalize >fst_pair >fst_pair >snd_pair 
+      >snd_pair >snd_pair >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair 
+      >fst_pair @le_plus 
+       [@le_plus // cases (true_or_false (leb c1 c)) #Hcase 
+        >Hcase normalize // @lt_to_le @not_le_to_lt @(leb_false_to_not_le ?? Hcase)
+       |@Hind
+       ]
+     ]
+   ]
+ |@O_plus [@O_plus_l //] @le_to_O #x 
+  <bigop_plus_c @le_plus // @(transitive_le … (MSC_pair …)) @le_plus 
+   [@MSC_prop @CFa | @MSC_prop @(O_to_CF MSC … (CF_const x)) @(proj1 … CFb)]
+ ]
+qed.
+       
+(* old
+axiom CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+  CF ha a → CF hb b → CF hp p → CF hf f → 
+  O s (λx.ha x + hb x + ∑_{i ∈[a x ,b x[ }(hp 〈i,x〉 + hf 〈i,x〉)) →
+  CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)). *)
+  
+(******************************** minimization ********************************) 
+
+alias symbol "plus" (instance 2) = "natural plus".
+alias symbol "plus" (instance 5) = "natural plus".
+alias symbol "plus" (instance 1) = "natural plus".
+alias symbol "plus" (instance 4) = "natural plus".
+alias symbol "pair" (instance 3) = "abstract pair".
+alias id "O" = "cic:/matita/arithmetics/nat/nat#con:0:1:0".
+let rec my_minim a f x k on k ≝
+  match k with
+  [O ⇒ a
+  |S p ⇒ if eqb (my_minim a f x p) (a+p) 
+         then if f 〈a+p,x〉 then a+p else S(a+p)
+         else (my_minim a f x p) ].
+         
+lemma my_minim_S : ∀a,f,x,k. 
+  my_minim a f x (S k) = 
+    if eqb (my_minim a f x k) (a+k) 
+    then if f 〈a+k,x〉 then a+k else S(a+k)
+    else (my_minim a f x k) .
+// qed.
+  
+lemma my_minim_fa : ∀a,f,x,k. f〈a,x〉 = true → my_minim a f x k = a.
+#a #f #x #k #H elim k // #i #Hind normalize >Hind
+cases (true_or_false (eqb a (a+i))) #Hcase >Hcase normalize //
+<(eqb_true_to_eq … Hcase) >H //
+qed.
+
+lemma my_minim_nfa : ∀a,f,x,k. f〈a,x〉 = false → 
+my_minim a f x (S k) = my_minim (S a) f x k.
+#a #f #x #k #H elim k  
+  [normalize <plus_n_O >H >eq_to_eqb_true // 
+  |#i #Hind >my_minim_S >Hind >my_minim_S <plus_n_Sm //
+  ]
+qed.
+
+lemma my_min_eq : ∀a,f,x,k.
+  min k a (λi.f 〈i,x〉) = my_minim a f x k.
+#a #f #x #k lapply a -a elim k // #i #Hind #a normalize in ⊢ (??%?);
+cases (true_or_false (f 〈a,x〉)) #Hcase >Hcase 
+  [>(my_minim_fa … Hcase) // | >Hind @sym_eq @(my_minim_nfa … Hcase) ]
+qed.
+
+(* cambiare qui: togliere S *)
+
+
+definition minim_unary_pr ≝ λf.unary_pr (λx.S(fst x))
+    (λi. 
+      let k ≝ fst i in
+      let r ≝ fst (snd i) in
+      let b ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      if f 〈b-k,x〉 then b-k else r).
+      
+definition compl_minim_unary ≝ λhf:nat → nat.λi. 
+      let k ≝ fst i in
+      let b ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+
+definition compl_minim_unary_aux ≝ λhf,i. 
+      let k ≝ fst i in
+      let r ≝ fst (snd i) in
+      let b ≝ fst (snd (snd i)) in
+      let x ≝ snd (snd (snd i)) in
+      hf 〈b-k,x〉 + MSC i.
+
+lemma compl_minim_unary_aux_def : ∀hf,k,r,b,x.
+  compl_minim_unary_aux hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈r,〈b,x〉〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def : ∀hf,k,r,b,x.
+  compl_minim_unary hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_aux_def2 : ∀hf,k,r,x.
+  compl_minim_unary_aux hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈r,x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def2 : ∀hf,k,r,x.
+  compl_minim_unary hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈S(fst x),snd x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma min_aux: ∀a,f,x,k. min (S k) (a x) (λi:ℕ.f 〈i,x〉) =
+  ((minim_unary_pr f) ∘ (λx.〈S k,〈a x + k,x〉〉)) x.
+#a #f #x #k whd in ⊢ (???%); >fst_pair >snd_pair
+lapply a -a (* @max_prim_rec1 *)
+elim k
+  [normalize #a >fst_pair >snd_pair >fst_pair >snd_pair >snd_pair >fst_pair 
+   <plus_n_O <minus_n_O >fst_pair //
+  |#i #Hind #a normalize in ⊢ (??%?); >prim_rec_S >fst_pair >snd_pair
+   >fst_pair >snd_pair >snd_pair >fst_pair <plus_n_Sm <(Hind (λx.S (a x)))
+   whd in ⊢ (???%); >minus_S_S <(minus_plus_m_m (a x) i) //
+qed.
+
+lemma minim_unary_pr1: ∀a,b,f,x.
+  μ_{i ∈[a x,b x]}(f 〈i,x〉) = 
+    if leb (a x) (b x) then ((minim_unary_pr f) ∘ (λx.〈S (b x) - a x,〈b x,x〉〉)) x
+    else (a x).
+#a #b #f #x cases (true_or_false (leb (a x) (b x))) #Hcase >Hcase
+  [cut (b x = a x + (b x - a x))
+    [>commutative_plus <plus_minus_m_m // @leb_true_to_le // ]
+   #Hcut whd in ⊢ (???%); >minus_Sn_m [|@leb_true_to_le //]
+   >min_aux whd in ⊢ (??%?); <Hcut //
+  |>eq_minus_O // @not_le_to_lt @leb_false_to_not_le //
+  ]
+qed.
+
+axiom sum_inv: ∀a,b,f.∑_{i ∈ [a,S b[ }(f i) = ∑_{i ∈ [a,S b[ }(f (a + b - i)).
+
+(*
+#a #b #f @(bigop_iso … plusAC) whd %{(λi.b - a - i)} %{(λi.b - a -i)} %
+  [%[#i #lti #_ normalize @eq_f >commutative_plus <plus_minus_associative 
+    [2: @le_minus_to_plus_r //
+     [// @eq_f @@sym_eq @plus_to_minus 
+    |#i #Hi #_ % [% [@le_S_S 
+*)
+
+example sum_inv_check : ∑_{i ∈ [3,6[ }(i*i) = ∑_{i ∈ [3,6[ }((8-i)*(8-i)).
+normalize // qed.
+
+(* provo rovesciando la somma *)
+
+axiom daemon: ∀P:Prop.P.
+
+lemma CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉 + MSC 〈b x - i,〈S(b x),x〉〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #ha #hb #hf #s #CFa #CFb #CFf #HO 
+@ext_CF1 [|#x @minim_unary_pr1]
+@CF_if 
+  [@CF_le @(O_to_CF … HO) 
+    [@(O_to_CF … CFa) @O_plus_l @O_plus_l  @O_refl
+    |@(O_to_CF … CFb) @O_plus_l @O_plus_r @O_refl
+    ]
+  |@(CF_comp ??? (λx.ha x + hb x))
+    [|@CF_comp_pair
+      [@CF_minus [@CF_compS @(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+      |@CF_comp_pair 
+        [@(O_to_CF … CFb) @O_plus_r //
+        |@(O_to_CF … CF_id) @O_plus_r @(proj1 … CFb)
+        ]
+      ]
+    |@(CF_prim_rec_gen … MSC … (compl_minim_unary_aux hf))
+      [@((λx:ℕ.fst (snd x)
+          +compl_minim_unary hf
+          〈fst x,
+          〈unary_pr fst
+            (λi:ℕ
+             .(let (k:ℕ) ≝fst i in 
+               let (r:ℕ) ≝fst (snd i) in 
+               let (b:ℕ) ≝fst (snd (snd i)) in 
+               let (x:ℕ) ≝snd (snd (snd i)) in if f 〈b-S k,x〉 then b-S k else r ))
+          〈fst x,snd (snd x)〉,
+          snd (snd x)〉〉))
+      |@CF_compS @CF_fst
+      |@CF_if 
+        [@(CF_comp f … MSC … CFf) 
+          [@CF_comp_pair 
+            [@CF_minus [@CF_comp_fst @CF_comp_snd @CF_snd|@CF_fst]
+            |@CF_comp_snd @CF_comp_snd @CF_snd]
+          |@le_to_O #x normalize >commutative_plus //
+          ]
+        |@(O_to_CF … MSC)
+          [@le_to_O #x normalize //
+          |@CF_minus
+            [@CF_comp_fst @CF_comp_snd @CF_snd |@CF_fst]
+          ]
+        |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+        ]
+      |@O_plus    
+        [@O_plus_l @O_refl 
+        |@O_plus_r @O_ext2 [||#x >compl_minim_unary_aux_def2 @refl]
+         @O_trans [||@le_to_O #x >compl_minim_unary_def2 @le_n]
+         @O_plus [@O_plus_l //] 
+         @O_plus_r 
+         @O_trans [|@le_to_O #x @MSC_pair] @O_plus 
+           [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_fst …)) 
+            >fst_pair @le_n]
+         @O_trans [|@le_to_O #x @MSC_pair] @O_plus
+           [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …))
+            >snd_pair @(transitive_le ???? (le_fst …)) >fst_pair 
+            normalize >snd_pair >fst_pair lapply (surj_pair x)
+            * #x1 * #x2 #Hx >Hx >fst_pair >snd_pair elim x1 //
+            #i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+            cases (f ?) [@le_S // | //]]
+         @le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …)) >snd_pair
+         >(expand_pair (snd (snd x))) in ⊢ (?%?); @le_pair //
+        ]
+      ] 
+    |cut (O s (λx.ha x + hb x + 
+            ∑_{i ∈[a x ,S(b x)[ }(hf 〈a x+b x-i,x〉 + MSC 〈b x -(a x+b x-i),〈S(b x),x〉〉)))
+      [@(O_ext2 … HO) #x @eq_f @sum_inv] -HO #HO
+     @(O_trans … HO) -HO
+     @(O_trans ? (λx:ℕ.ha x+hb x
+       +bigop (S(b x)-a x) (λi:ℕ.true) ? 
+        (MSC 〈b x,x〉) plus (λi:ℕ.(λj.hf j + MSC 〈b x - fst j,〈S(b (snd j)),snd j〉〉) 〈b x- i,x〉)))
+      [@le_to_O #n @le_plus // whd in match (unary_pr ???); 
+       >fst_pair >snd_pair >(bigop_prim_rec_dec1 a b ? (λi.true)) 
+        (* it is crucial to recall that the index is bound by S(b x) *)
+        cut (S (b n) - a n ≤ S (b n)) [//]
+        elim (S(b n) - a n)
+        [normalize //  
+        |#x #Hind #lex >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair 
+         >compl_minim_unary_def >prim_rec_S >fst_pair >snd_pair >fst_pair 
+         >snd_pair >fst_pair >snd_pair >fst_pair whd in ⊢ (??%); >commutative_plus 
+         @le_plus [2:@Hind @le_S @le_S_S_to_le @lex] -Hind >snd_pair 
+         >minus_minus_associative // @le_S_S_to_le // 
+        ]
+      |@O_plus [@O_plus_l //] @O_ext2 [||#x @bigop_plus_c] 
+       @O_plus 
+        [@O_plus_l @O_trans [|@le_to_O #x @MSC_pair]
+         @O_plus [@O_plus_r @le_to_O @(MSC_prop … CFb)|@O_plus_r @(proj1 … CFb)]
+        |@O_plus_r @(O_ext2 … (O_refl …)) #x @same_bigop
+          [//|#i #H #_ @eq_f2 [@eq_f @eq_f2 //|>fst_pair @eq_f @eq_f2  //]   
+        ]
+      ]
+    ]
+  ] 
+  |@(O_to_CF … CFa) @(O_trans … HO) @O_plus_l @O_plus_l @O_refl
+  ]
+qed.
+
+(*
+lemma CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #ha #hb #hf #s #CFa #CFb #CFf #HO 
+@ext_CF1 [|#x @minim_unary_pr1]
+@(CF_comp ??? (λx.ha x + hb x))
+  [|@CF_comp_pair
+    [@CF_minus [@CF_compS @(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+    |@CF_comp_pair 
+      [@CF_max1 [@(O_to_CF … CFa) @O_plus_l // | @CF_compS @(O_to_CF … CFb) @O_plus_r //]
+      | @(O_to_CF … CF_id) @O_plus_r @(proj1 … CFb)
+      ]
+    ]
+  |@(CF_prim_rec … MSC … (compl_minim_unary_aux hf))
+    [@CF_fst
+    |@CF_if 
+      [@(CF_comp f … MSC … CFf) 
+        [@CF_comp_pair 
+          [@CF_minus [@CF_comp_fst @CF_comp_snd @CF_snd|@CF_compS @CF_fst]
+          |@CF_comp_snd @CF_comp_snd @CF_snd]
+        |@le_to_O #x normalize >commutative_plus //
+        ]
+      |@(O_to_CF … MSC)
+        [@le_to_O #x normalize //
+        |@CF_minus
+          [@CF_comp_fst @CF_comp_snd @CF_snd |@CF_compS @CF_fst]
+        ]
+      |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+      ]
+    |@O_refl
+    ]
+  |@(O_trans … HO) -HO
+   @(O_trans ? (λx:ℕ
+   .ha x+hb x
+    +bigop (S(b x)-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus (λi:ℕ.hf 〈i+a x,x〉)))
+  [@le_to_O #n @le_plus // whd in match (unary_pr ???); 
+   >fst_pair >snd_pair >(bigop_prim_rec ? (λn.S(b n)) ? (λi.true)) elim (S(b n) - a n)
+    [normalize @monotonic_MSC @le_pair
+    |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >compl_minim_unary_def
+     >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair 
+     >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+      [-Hind @le_plus // normalize >fst_pair >snd_pair 
+       @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+         (λi1:ℕ.hf 〈i1+a n,n〉)))
+        [elim x [normalize @MSC_le]
+         #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+         >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+          [cases (true_or_false (leb (f 〈x0+a n,n〉)
+            (prim_rec (λx00:ℕ.O)
+             (λi:ℕ
+            .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉 
+             then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+                           (fst (snd i)) 
+                      then fst (snd i) 
+                      else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉  
+             else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+           [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+           |@(transitive_le … (MSC_prop … CFf …)) @(le_maxl … (le_n …))
+           ]
+         |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+         ]
+       |@(le_maxr … (le_n …))
+       ]
+  
+  
+   @(O_trans ? (λx:ℕ
+   .ha x+hb x
+    +bigop (b x-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus
+     (λi:ℕ
+      .(λi0:ℕ
+        .hp 〈i0,x〉+hf 〈i0,x〉
+         +bigop (b x-a x) (λi1:ℕ.true) ? 0 max
+          (λi1:ℕ.(λi2:ℕ.hf 〈i2,x〉) (i1+a x))) (i+a x))))
+    [
+   @le_to_O #n @le_plus // whd in match (unary_pr ???);
+   >fst_pair >snd_pair >bigop_prim_rec elim (b n - a n)
+    [normalize //
+    |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >aux_compl1_def1
+     >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair 
+     >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+      [-Hind @le_plus // normalize >fst_pair >snd_pair 
+       @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+         (λi1:ℕ.hf 〈i1+a n,n〉)))
+        [elim x [normalize @MSC_le]
+         #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+         >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+          [cases (true_or_false (leb (f 〈x0+a n,n〉)
+            (prim_rec (λx00:ℕ.O)
+             (λi:ℕ
+            .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉 
+             then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+                           (fst (snd i)) 
+                      then fst (snd i) 
+                      else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉  
+             else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+           [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+           |@(transitive_le … (MSC_prop … CFf …)) @(le_maxl … (le_n …))
+           ]
+         |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+         ]
+       |@(le_maxr … (le_n …))
+       ]
+     |@(transitive_le … Hind) -Hind 
+      generalize in match (bigop x (λi:ℕ.true) ? 0 max
+              (λi1:ℕ.(λi2:ℕ.hf 〈i2,n〉) (i1+a n))); #c
+      generalize in match (hf 〈x+a n,n〉); #c1
+      elim x [//] #x0 #Hind 
+      >prim_rec_S >prim_rec_S normalize >fst_pair >fst_pair >snd_pair 
+      >snd_pair >snd_pair >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair 
+      >fst_pair @le_plus 
+       [@le_plus // cases (true_or_false (leb c1 c)) #Hcase 
+        >Hcase normalize // @lt_to_le @not_le_to_lt @(leb_false_to_not_le ?? Hcase)
+       |@Hind
+       ]
+     ]
+   ]
+ |@O_plus [@O_plus_l //] @le_to_O #x 
+  <bigop_plus_c @le_plus // @(transitive_le … (MSC_pair …)) @le_plus 
+   [@MSC_prop @CFa | @MSC_prop @(O_to_CF MSC … (CF_const x)) @(proj1 … CFb)]
+ ]
+qed.
+
+axiom CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+  CF sa a → CF sb b → CF sf f → 
+  O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
+  CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).*)
+  
+(************************************* smn ************************************)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+(****************************** constructibility ******************************)
+ 
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+  (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //] 
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) → 
+  constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1)  #x >Hext //
+qed.
+
+lemma constr_prim_rec: ∀s1,s2. constructible s1 → constructible s2 →
+  (∀n,r,m. 2 * r ≤ s2 〈n,〈r,m〉〉) → constructible (unary_pr s1 s2).
+#s1 #s2 #Hs1 #Hs2 #Hincr @(CF_prim_rec … Hs1 Hs2) whd %{2} %{0} 
+#x #_ lapply (surj_pair x) * #a * #b #eqx >eqx whd in match (unary_pr ???); 
+>fst_pair >snd_pair
+whd in match (unary_pr ???); >fst_pair >snd_pair elim a
+  [normalize //
+  |#n #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
+   >prim_rec_S @transitive_le [| @(monotonic_le_plus_l … Hind)]
+   @transitive_le [| @(monotonic_le_plus_l … (Hincr n ? b))] 
+   whd in match (unary_pr ???); >fst_pair >snd_pair //
+  ]
+qed.
+
+(********************************* simulation *********************************)
+
+axiom sU : nat → nat. 
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+  sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+lemma monotonic_sU_aux : ∀x1,x2. fst x1 ≤ fst x2 → fst (snd x1) ≤ fst (snd x2) →
+snd (snd x1) ≤ snd (snd x2) → sU x1 ≤ sU x2.
+#x1 #x2 cases (surj_pair x1) #a1 * #y #eqx1 >eqx1 -eqx1 cases (surj_pair y) 
+#b1 * #c1 #eqy >eqy -eqy
+cases (surj_pair x2) #a2 * #y2 #eqx2 >eqx2 -eqx2 cases (surj_pair y2) 
+#b2 * #c2 #eqy2 >eqy2 -eqy2 >fst_pair >snd_pair >fst_pair >snd_pair 
+>fst_pair >snd_pair >fst_pair >snd_pair @monotonic_sU
+qed.
+  
+axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom CF_U : CF sU pU_unary.
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary. 
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in  ⊢ (??(?%)?); >fst_pair % 
+qed.
+
+lemma CF_out: CF sU out_unary. 
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in  ⊢ (??(?%)?); >snd_pair % 
+qed.
+
+
+(******************** complexity of g ********************)
+
+(*definition unary_g ≝ λh.λux. g h (fst ux) (snd ux).
+definition auxg ≝ 
+  λh,ux. max_{i ∈[fst ux,snd ux[ | eqb (min_input h i (snd ux)) (snd ux)} 
+    (out i (snd ux) (h (S i) (snd ux))).
+
+lemma compl_g1 : ∀h,s. CF s (auxg h) → CF s (unary_g h).
+#h #s #H1 @(CF_compS ? (auxg h) H1) 
+qed.
+
+definition aux1g ≝ 
+  λh,ux. max_{i ∈[fst ux,snd ux[ | (λp. eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) 〈i,ux〉} 
+    ((λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) 〈i,ux〉).
+
+lemma eq_aux : ∀h,x.aux1g h x = auxg h x.
+#h #x @same_bigop
+  [#n #_ >fst_pair >snd_pair // |#n #_ #_ >fst_pair >snd_pair //]
+qed.
+
+lemma compl_g2 : ∀h,s1,s2,s. 
+  CF s1
+    (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
+  CF s2
+    (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) →
+  O s (λx.MSC x + ∑_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) → 
+  CF s (auxg h).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h)) 
+  [#n whd in ⊢ (??%%); @eq_aux]
+@(CF_max … CF_fst CF_snd Hs1 Hs2 …) @(O_trans … HO) 
+@O_plus [@O_plus @O_plus_l // | @O_plus_r //]
+qed.  
+
+lemma compl_g3 : ∀h,s.
+  CF s (λp:ℕ.min_input h (fst p) (snd (snd p))) →
+  CF s (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))).
+#h #s #H @(CF_eqb … H) @(CF_comp … CF_snd CF_snd) @(O_trans … (proj1 … H))
+@O_plus // %{1} %{0} #n #_ >commutative_times <times_n_1 @monotonic_MSC //
+qed.
+
+definition min_input_aux ≝ λh,p.
+  μ_{y ∈ [S (fst p),snd (snd p)] } 
+    ((λx.termb (fst (snd x)) (fst x) (h (S (fst (snd x))) (fst x))) 〈y,p〉). 
+    
+lemma min_input_eq : ∀h,p. 
+    min_input_aux h p  =
+    min_input h (fst p) (snd (snd p)).
+#h #p >min_input_def whd in ⊢ (??%?); >minus_S_S @min_f_g #i #_ #_ 
+whd in ⊢ (??%%); >fst_pair >snd_pair //
+qed.
+
+definition termb_aux ≝ λh.
+  termb_unary ∘ λp.〈fst (snd p),〈fst p,h (S (fst (snd p))) (fst p)〉〉.
+
+lemma compl_g4 : ∀h,s1,s.
+  (CF s1 
+    (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
+   (O s (λx.MSC x + ∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
+  CF s (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
+ [#n whd in ⊢ (??%%); @min_input_eq]
+@(CF_mu … MSC MSC … Hs1) 
+  [@CF_compS @CF_fst 
+  |@CF_comp_snd @CF_snd
+  |@(O_trans … HO) @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
+qed.*)
+
+(************************* a couple of technical lemmas ***********************)
+lemma minus_to_0: ∀a,b. a ≤ b → minus a b = 0.
+#a elim a // #n #Hind * 
+  [#H @False_ind /2 by absurd/ | #b normalize #H @Hind @le_S_S_to_le /2/]
+qed.
+
+lemma sigma_bound:  ∀h,a,b. monotonic nat le h → 
+  ∑_{i ∈ [a,S b[ }(h i) ≤  (S b-a)*h b.
+#h #a #b #H cases (decidable_le a b) 
+  [#leab cut (b = pred (S b - a + a)) 
+    [<plus_minus_m_m // @le_S //] #Hb >Hb in match (h b);
+   generalize in match (S b -a);
+   #n elim n 
+    [//
+    |#m #Hind >bigop_Strue [2://] @le_plus 
+      [@H @le_n |@(transitive_le … Hind) @le_times [//] @H //]
+    ]
+  |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
+   cut (S b -a = 0) [@minus_to_0 //] #Hcut >Hcut //
+  ]
+qed.
+
+lemma sigma_bound_decr:  ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) → 
+  ∑_{i ∈ [a,b[ }(h i) ≤  (b-a)*h a.
+#h #a #b #H cases (decidable_le a b) 
+  [#leab cut ((b -a) +a ≤ b) [/2 by le_minus_to_plus_r/] generalize in match (b -a);
+   #n elim n 
+    [//
+    |#m #Hind >bigop_Strue [2://] #Hm 
+     cut (m+a ≤ b) [@(transitive_le … Hm) //] #Hm1
+     @le_plus [@H // |@(transitive_le … (Hind Hm1)) //]
+    ]
+  |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
+   cut (b -a = 0) [@minus_to_0 @lt_to_le @ltba] #Hcut >Hcut //
+  ]
+qed. 
+
+lemma coroll: ∀s1:nat→nat. (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) → 
+O (λx.(snd (snd x)-fst x)*(s1 〈snd (snd x),x〉)) 
+  (λx.∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉)).
+#s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1 
+@(transitive_le … (sigma_bound …)) [@Hs1|>minus_S_S //]
+qed.
+
+lemma coroll2: ∀s1:nat→nat. (∀n,a,b. a ≤ b → b < snd n → s1 〈b,n〉 ≤ s1 〈a,n〉) → 
+O (λx.(snd x - fst x)*s1 〈fst x,x〉) (λx.∑_{i ∈[fst x,snd x[ }(s1 〈i,x〉)).
+#s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1 
+@(transitive_le … (sigma_bound_decr …)) [2://] @Hs1 
+qed.
+
+(**************************** end of technical lemmas *************************)
+
+(*lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+  (CF s1 
+    (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
+  CF (λx.MSC x + (snd (snd x)-fst x)*s1 〈snd (snd x),x〉) 
+    (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #Hmono #Hs1 @(compl_g4 … Hs1) @O_plus 
+[@O_plus_l // |@O_plus_r @coroll @Hmono]
+qed.
+
+lemma compl_g6: ∀h.
+  constructible (λx. h (fst x) (snd x)) →
+  (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 
+    (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+#h #hconstr @(ext_CF (termb_aux h))
+  [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb) 
+  [@CF_comp_pair
+    [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
+    |@CF_comp_pair
+      [@(monotonic_CF … CF_fst) #x //
+      |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
+        [#n normalize >fst_pair >snd_pair %]
+       @(CF_comp … MSC …hconstr)
+        [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
+        |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+        ]
+      ]
+    ]
+  |@O_plus
+    [@O_plus
+      [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x))))) 
+        [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+         >fst_pair >snd_pair @(transitive_le … (MSC_pair …)) 
+         >distributive_times_plus @le_plus [//]
+         cases (surj_pair b) #c * #d #eqb >eqb
+         >fst_pair >snd_pair @(transitive_le … (MSC_pair …)) 
+         whd in ⊢ (??%); @le_plus 
+          [@monotonic_MSC @(le_maxl … (le_n …)) 
+          |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))  
+          ]
+        |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
+        ]     
+      |@le_to_O #n @sU_le
+      ] 
+    |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
+  ]
+qed. *)
+
+definition big : nat →nat ≝ λx. 
+  let m ≝ max (fst x) (snd x) in 〈m,m〉.
+
+lemma big_def : ∀a,b. big 〈a,b〉 = 〈max a b,max a b〉.
+#a #b normalize >fst_pair >snd_pair // qed.
+
+lemma le_big : ∀x. x ≤ big x. 
+#x cases (surj_pair x) #a * #b #eqx >eqx @le_pair >fst_pair >snd_pair 
+[@(le_maxl … (le_n …)) | @(le_maxr … (le_n …))]
+qed.
+
+definition faux2 ≝ λh.
+  (λx.MSC x + (snd (snd x)-fst x)*
+    (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+ 
+(*lemma compl_g7: ∀h. 
+  constructible (λx. h (fst x) (snd x)) →
+  (∀n. monotonic ? le (h n)) → 
+  CF (λx.MSC x + (snd (snd x)-fst x)*sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
+    (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (faux2 h))
+  [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair 
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+lemma compl_g71: ∀h. 
+  constructible (λx. h (fst x) (snd x)) →
+  (∀n. monotonic ? le (h n)) → 
+  CF (λx.MSC (big x) + (snd (snd x)-fst x)*sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
+    (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x))) 
+  [#Hle @le_times // @monotonic_sU  
+  |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
+  ]
+qed.*)
+
+definition out_aux ≝ λh.
+  out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
+
+lemma compl_g8: ∀h.
+  constructible (λx. h (fst x) (snd x)) →
+  (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉) 
+    (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h #hconstr @(ext_CF (out_aux h))
+  [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out) 
+  [@CF_comp_pair
+    [@(monotonic_CF … CF_fst) #x //
+    |@CF_comp_pair
+      [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
+      |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
+        [#n normalize >fst_pair >snd_pair %]
+       @(CF_comp … MSC …hconstr)
+        [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
+        |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+        ]
+      ]
+    ]
+  |@O_plus 
+    [@O_plus 
+      [@le_to_O #n @sU_le 
+      |@(O_trans … (λx.MSC (max (fst x) (snd x))))
+        [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+         >fst_pair >snd_pair @(transitive_le … (MSC_pair …)) 
+         whd in ⊢ (??%); @le_plus 
+          [@monotonic_MSC @(le_maxl … (le_n …)) 
+          |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))  
+          ]
+        |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
+        ]
+      ]
+    |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
+  ]
+qed.
+
+(*lemma compl_g9 : ∀h. 
+  constructible (λx. h (fst x) (snd x)) →
+  (∀n. monotonic ? le (h n)) → 
+  (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+  CF (λx. (S (snd x-fst x))*MSC 〈x,x〉 + 
+      (snd x-fst x)*(S(snd x-fst x))*sU 〈x,〈snd x,h (S (fst x)) (snd x)〉〉)
+   (auxg h).
+#h #hconstr #hmono #hantimono 
+@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
+@O_plus 
+  [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
+    [// | @monotonic_MSC // ]]
+@(O_trans … (coroll2 ??))
+  [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+   cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
+   cut (max a n = n) 
+    [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
+   cut (max b n = n) [normalize >le_to_leb_true //] #maxb
+   @le_plus
+    [@le_plus [>big_def >big_def >maxa >maxb //]
+     @le_times 
+      [/2 by monotonic_le_minus_r/ 
+      |@monotonic_sU // @hantimono [@le_S_S // |@ltb] 
+      ]
+    |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
+    ] 
+  |@le_to_O #n >fst_pair >snd_pair
+   cut (max (fst n) n = n) [normalize >le_to_leb_true //] #Hmax >Hmax
+   >associative_plus >distributive_times_plus
+   @le_plus [@le_times [@le_S // |>big_def >Hmax //] |//] 
+  ]
+qed.*)
+
+definition sg ≝ λh,x.
+  (S (snd x-fst x))*MSC 〈x,x〉 + (snd x-fst x)*(S(snd x-fst x))*sU 〈x,〈snd x,h (S (fst x)) (snd x)〉〉.
+
+lemma sg_def : ∀h,a,b. 
+  sg h 〈a,b〉 = (S (b-a))*MSC 〈〈a,b〉,〈a,b〉〉 + 
+   (b-a)*(S(b-a))*sU 〈〈a,b〉,〈b,h (S a) b〉〉.
+#h #a #b whd in ⊢ (??%?); >fst_pair >snd_pair // 
+qed. 
+
+(*lemma compl_g11 : ∀h.
+  constructible (λx. h (fst x) (snd x)) →
+  (∀n. monotonic ? le (h n)) → 
+  (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+  CF (sg h) (unary_g h).
+#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr Hm Ham)
+qed.*)
+
+(**************************** closing the argument ****************************)
+
+let rec h_of_aux (r:nat →nat) (c,d,b:nat) on d : nat ≝ 
+  match d with 
+  [ O ⇒ c 
+  | S d1 ⇒ (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
+     d*(S d)*sU 〈〈b-d,b〉,〈b,r (h_of_aux r c d1 b)〉〉].
+
+lemma h_of_aux_O: ∀r,c,b.
+  h_of_aux r c O b  = c.
+// qed.
+
+lemma h_of_aux_S : ∀r,c,d,b. 
+  h_of_aux r c (S d) b = 
+    (S (S d))*(MSC 〈〈b-(S d),b〉,〈b-(S d),b〉〉) + 
+      (S d)*(S (S d))*sU 〈〈b-(S d),b〉,〈b,r(h_of_aux r c d b)〉〉.
+// qed.
+
+lemma h_of_aux_prim_rec : ∀r,c,n,b. h_of_aux r c n b =
+  prim_rec (λx.c) 
+   (λx.let d ≝ S(fst x) in 
+       let b ≝ snd (snd x) in
+       (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
+        d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉) n b.
+#r #c #n #b elim n
+  [>h_of_aux_O normalize //
+  |#n1 #Hind >h_of_aux_S >prim_rec_S >snd_pair >snd_pair >fst_pair 
+   >fst_pair <Hind //
+  ]
+qed.
+
+(*
+lemma h_of_aux_constr : 
+∀r,c. constructible (λx.h_of_aux r c (fst x) (snd x)).
+#r #c 
+  @(ext_constr … 
+    (unary_pr (λx.c) 
+     (λx.let d ≝ S(fst x) in 
+       let b ≝ snd (snd x) in
+       (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
+        d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉)))
+  [#n @sym_eq whd in match (unary_pr ???); @h_of_aux_prim_rec
+  |@constr_prim_rec*)
+
+definition h_of ≝ λr,p. 
+  let m ≝ max (fst p) (snd p) in 
+  h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (snd p - fst p) (snd p).
+
+lemma h_of_O: ∀r,a,b. b ≤ a → 
+  h_of r 〈a,b〉 = let m ≝ max a b in MSC 〈〈m,m〉,〈m,m〉〉.
+#r #a #b #Hle normalize >fst_pair >snd_pair >(minus_to_0 … Hle) //
+qed.
+
+lemma h_of_def: ∀r,a,b.h_of r 〈a,b〉 = 
+  let m ≝ max a b in 
+  h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (b - a) b.
+#r #a #b normalize >fst_pair >snd_pair //
+qed.
+  
+lemma mono_h_of_aux: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
+  ∀d,d1,c,c1,b,b1.c ≤ c1 → d ≤ d1 → b ≤ b1 → 
+  h_of_aux r c d b ≤ h_of_aux r c1 d1 b1.
+#r #Hr #monor #d #d1 lapply d -d elim d1 
+  [#d #c #c1 #b #b1 #Hc #Hd @(le_n_O_elim ? Hd) #leb 
+   >h_of_aux_O >h_of_aux_O  //
+  |#m #Hind #d #c #c1 #b #b1 #lec #led #leb cases (le_to_or_lt_eq … led)
+    [#ltd @(transitive_le … (Hind … lec ? leb)) [@le_S_S_to_le @ltd]
+     >h_of_aux_S @(transitive_le ???? (le_plus_n …))
+     >(times_n_1 (h_of_aux r c1 m b1)) in ⊢ (?%?); 
+     >commutative_times @le_times [//|@(transitive_le … (Hr ?)) @sU_le]
+    |#Hd >Hd >h_of_aux_S >h_of_aux_S 
+     cut (b-S m ≤ b1 - S m) [/2 by monotonic_le_minus_l/] #Hb1
+     @le_plus [@le_times //] 
+      [@monotonic_MSC @le_pair @le_pair //
+      |@le_times [//] @monotonic_sU 
+        [@le_pair // |// |@monor @Hind //]
+      ]
+    ]
+  ]
+qed.
+
+lemma mono_h_of2: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
+  ∀i,b,b1. b ≤ b1 → h_of r 〈i,b〉 ≤ h_of r 〈i,b1〉.
+#r #Hr #Hmono #i #a #b #leab >h_of_def >h_of_def
+cut (max i a ≤ max i b)
+  [@to_max 
+    [@(le_maxl … (le_n …))|@(transitive_le … leab) @(le_maxr … (le_n …))]]
+#Hmax @(mono_h_of_aux r Hr Hmono) 
+  [@monotonic_MSC @le_pair @le_pair @Hmax |/2 by monotonic_le_minus_l/ |@leab]
+qed.
+
+axiom h_of_constr : ∀r:nat →nat. 
+  (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+  constructible (h_of r).
+
+(*lemma speed_compl: ∀r:nat →nat. 
+  (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+  CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
+#r #Hr #Hmono #Hconstr @(monotonic_CF … (compl_g11 …)) 
+  [#x cases (surj_pair x) #a * #b #eqx >eqx 
+   >sg_def cases (decidable_le b a)
+    [#leba >(minus_to_0 … leba) normalize in ⊢ (?%?);
+     <plus_n_O <plus_n_O >h_of_def 
+     cut (max a b = a) 
+      [normalize cases (le_to_or_lt_eq … leba)
+        [#ltba >(lt_to_leb_false … ltba) % 
+        |#eqba <eqba >(le_to_leb_true … (le_n ?)) % ]]
+     #Hmax >Hmax normalize >(minus_to_0 … leba) normalize
+     @monotonic_MSC @le_pair @le_pair //
+    |#ltab >h_of_def >h_of_def
+     cut (max a b = b) 
+      [normalize >(le_to_leb_true … ) [%] @lt_to_le @not_le_to_lt @ltab]
+     #Hmax >Hmax 
+     cut (max (S a) b = b) 
+      [whd in ⊢ (??%?);  >(le_to_leb_true … ) [%] @not_le_to_lt @ltab]
+     #Hmax1 >Hmax1
+     cut (∃d.b - a = S d) 
+       [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab] 
+     * #d #eqd >eqd  
+     cut (b-S a = d) [//] #eqd1 >eqd1 >h_of_aux_S >eqd1 
+     cut (b - S d = a) 
+       [@plus_to_minus >commutative_plus @minus_to_plus 
+         [@lt_to_le @not_le_to_lt // | //]] #eqd2 >eqd2
+     normalize //
+    ]
+  |#n #a #b #leab #lebn >h_of_def >h_of_def
+   cut (max a n = n) 
+    [normalize >le_to_leb_true [%|@(transitive_le … leab lebn)]] #Hmaxa
+   cut (max b n = n) 
+    [normalize >(le_to_leb_true … lebn) %] #Hmaxb 
+   >Hmaxa >Hmaxb @Hmono @(mono_h_of_aux r … Hr Hmono) // /2 by monotonic_le_minus_r/
+  |#n #a #b #leab @Hmono @(mono_h_of2 … Hr Hmono … leab)
+  |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
+    [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]  
+   @(h_of_constr r Hr Hmono Hconstr)
+  ]
+qed.
+
+lemma speed_compl_i: ∀r:nat →nat. 
+  (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+  ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
+#r #Hr #Hmono #Hconstr #i 
+@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
+  [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
+@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
+qed.*)
+
+(**************************** the speedup theorem *****************************)
+(*theorem pseudo_speedup: 
+  ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+  ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono #Hconstr
+(* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *) 
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
+#Hcodei #HCi 
+(* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
+(* sg is (λx.h_of r 〈i,x〉) *)
+%{(λx. h_of r 〈S i,x〉)}
+lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
+%[%[@condition_1 |@Hg]
+ |cases Hg #H1 * #j * #Hcodej #HCj
+  lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
+  cases HCi #m * #a #Ha %{m} %{(max (S i) a)} #n #ltin @lt_to_le @not_le_to_lt 
+  @(not_to_not … Hcond2) -Hcond2 #Hlesf %{n} % 
+  [@(transitive_le … ltin) @(le_maxl … (le_n …))]
+  cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))] 
+  #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf) //
+ ]
+qed.
+
+theorem pseudo_speedup': 
+  ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+  ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ 
+  (* ¬ O (r ∘ sg) sf. *)
+  ∃m,a.∀n. a≤n → r(sg a) < m * sf n. 
+#r #Hr #Hmono #Hconstr 
+(* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *) 
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
+#Hcodei #HCi 
+(* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
+(* sg is (λx.h_of r 〈i,x〉) *)
+%{(λx. h_of r 〈S i,x〉)}
+lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
+%[%[@condition_1 |@Hg]
+ |cases Hg #H1 * #j * #Hcodej #HCj
+  lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
+  cases HCi #m * #a #Ha
+  %{m} %{(max (S i) a)} #n #ltin @not_le_to_lt @(not_to_not … Hcond2) -Hcond2 #Hlesf 
+  %{n} % [@(transitive_le … ltin) @(le_maxl … (le_n …))]
+  cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))] 
+  #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf)
+  @Hmono @(mono_h_of2 … Hr Hmono … ltin)
+ ]
+qed.*)
+  
\ No newline at end of file
-- 
2.39.2