--- /dev/null
+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
projects / helm.git / blobdiff