[helm.git] / weblib / arithmetics / bigops.ma

(*
    ||M||  This file is part of HELM, an Hypertextual, Electronic        
    ||A||  Library of Mathematics, developed at the Computer Science     
    ||T||  Department of the University of Bologna, Italy.                     
    ||I||                                                                 
    ||T||  
    ||A||  This file is distributed under the terms of the 
    \   /  GNU General Public License Version 2        
     \ /      
      V_______________________________________________________________ *)

include "basics/types.ma".
include "arithmetics/div_and_mod.ma".

\ 5img class="anchor" src="icons/tick.png" id="sameF_upto"\ 6definition sameF_upto: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → ∀A.\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→A)  ≝
λk.λA.λf,g.∀i. i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 k → f i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g i.
     
\ 5img class="anchor" src="icons/tick.png" id="sameF_p"\ 6definition sameF_p: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → (\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) →∀A.\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→A)  ≝
λk,p,A,f,g.∀i. i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 k → p i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → f i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g i.

\ 5img class="anchor" src="icons/tick.png" id="sameF_upto_le"\ 6lemma sameF_upto_le: ∀A,f,g,n,m. 
 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m → \ 5a href="cic:/matita/arithmetics/bigops/sameF_upto.def(2)"\ 6sameF_upto\ 5/a\ 6 m A f g → \ 5a href="cic:/matita/arithmetics/bigops/sameF_upto.def(2)"\ 6sameF_upto\ 5/a\ 6 n A f g.
#A #f #g #n #m #lenm #samef #i #ltin @samef /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt.def(4)"\ 6lt_to_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

\ 5img class="anchor" src="icons/tick.png" id="sameF_p_le"\ 6lemma sameF_p_le: ∀A,p,f,g,n,m. 
 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m → \ 5a href="cic:/matita/arithmetics/bigops/sameF_p.def(2)"\ 6sameF_p\ 5/a\ 6 m p A f g → \ 5a href="cic:/matita/arithmetics/bigops/sameF_p.def(2)"\ 6sameF_p\ 5/a\ 6 n p A f g.
#A #p #f #g #n #m #lenm #samef #i #ltin #pi @samef /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt.def(4)"\ 6lt_to_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

\ 5img class="anchor" src="icons/tick.png" id="prodF"\ 6definition prodF ≝
 λA,B.λf:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→A.λg:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B.λm,x.〈 f(\ 5a href="cic:/matita/arithmetics/div_and_mod/div.def(3)"\ 6div\ 5/a\ 6 x m), g(\ 5a href="cic:/matita/arithmetics/div_and_mod/mod.def(3)"\ 6mod\ 5/a\ 6 x m) \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.

(* bigop *)
\ 5img class="anchor" src="icons/tick.png" id="bigop"\ 6let rec bigop (n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6) (p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (B:Type[0])
   (nil: B) (op: B → B → B)  (f: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → B) ≝
  match n with
   [ O ⇒ nil
   | S k ⇒ 
      match p k with
      [true ⇒ op (f k) (bigop k p B nil op f)
      |false ⇒ bigop k p B nil op f]
   ].
   
notation "\big  [ op , nil ]_{ ident i < n | p } f"
  with precedence 80
for @{'bigop $n $op $nil (λ${ident i}. $p) (λ${ident i}. $f)}.

notation "\big [ op , nil ]_{ ident i < n } f"
  with precedence 80
for @{'bigop $n $op $nil (λ${ident i}.true) (λ${ident i}. $f)}.

interpretation "bigop" 'bigop n op nil p f = (bigop n p ? nil op f).

notation "\big  [ op , nil ]_{ ident j ∈ [a,b[ | p } f"
  with precedence 80
for @{'bigop ($b-$a) $op $nil (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
  (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.

notation "\big  [ op , nil ]_{ ident j ∈ [a,b[ } f"
  with precedence 80
for @{'bigop ($b-$a) $op $nil (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
  (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.  

(* notation "\big  [ op , nil ]_{( term 50) a ≤ ident j < b | p } f"
  with precedence 80
for @{\big[$op,$nil]_{${ident j} < ($b-$a) | ((λ${ident j}.$p) (${ident j}+$a))}((λ${ident j}.$f)(${ident j}+$a))}.
*)
 
interpretation "bigop" 'bigop n op nil p f = (bigop n p ? nil op f).
   
\ 5img class="anchor" src="icons/tick.png" id="bigop_Strue"\ 6lemma bigop_Strue: ∀k,p,B,nil,op.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. p k \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
  \big[op,nil]_{i < \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 k | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
    op (f k) (\big[op,nil]_{i < k | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i)).
#k #p #B #nil #op #f #H normalize >H // qed.

\ 5img class="anchor" src="icons/tick.png" id="bigop_Sfalse"\ 6lemma bigop_Sfalse: ∀k,p,B,nil,op.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. p k \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
  \big[op,nil]_{ i < \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 k | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
    \big[op,nil]_{i < k | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i).
#k #p #B #nil #op #f #H normalize >H // qed. 
 
\ 5img class="anchor" src="icons/tick.png" id="same_bigop"\ 6lemma same_bigop : ∀k,p1,p2,B,nil,op.∀f,g:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. 
  \ 5a href="cic:/matita/arithmetics/bigops/sameF_upto.def(2)"\ 6sameF_upto\ 5/a\ 6 k \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 p1 p2 → \ 5a href="cic:/matita/arithmetics/bigops/sameF_p.def(2)"\ 6sameF_p\ 5/a\ 6 k p1 B f g →
  \big[op,nil]_{i < k | p1 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 
    \big[op,nil]_{i < k | p2 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(g i).
#k #p1 #p2 #B #nil #op #f #g (elim k) // 
#n #Hind #samep #samef normalize >Hind 
  [|@(\ 5a href="cic:/matita/arithmetics/bigops/sameF_p_le.def(5)"\ 6sameF_p_le\ 5/a\ 6 … samef) // |@(\ 5a href="cic:/matita/arithmetics/bigops/sameF_upto_le.def(5)"\ 6sameF_upto_le\ 5/a\ 6 … samep) //]
<(samep … (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6 …)) cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p1 n)) #H1 >H1 
normalize // <(samef … (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6 …) H1) // 
qed.

\ 5img class="anchor" src="icons/tick.png" id="pad_bigop"\ 6theorem pad_bigop: ∀k,n,p,B,nil,op.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 k → 
\big[op,nil]_{i < n | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i)
  \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \big[op,nil]_{i < k | \ 5font class="Apple-style-span" color="#FF0000"\ 6if\ 5/font\ 6 (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n i) then \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 else p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i).
#k #n #p #B #nil #op #f #lenk (elim lenk) 
  [@\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 #i #lti // >(\ 5a href="cic:/matita/arithmetics/nat/not_le_to_leb_false.def(7)"\ 6not_le_to_leb_false\ 5/a\ 6 …) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |#j #leup #Hind >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 … leup) // 
  ] qed.

\ 5img class="anchor" src="icons/tick.png" id="pad_bigop1"\ 6theorem pad_bigop1: ∀k,n,p,B,nil,op.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 k → 
  (∀i. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 k → p i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) →
  \big[op,nil]_{i < n | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) 
    \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \big[op,nil]_{i < k | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i).
#k #n #p #B #nil #op #f #lenk (elim lenk) 
  [#_ @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 #i #lti // 
  |#j #leup #Hind #Hfalse >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 
    [@Hind #i #leni #ltij @Hfalse // @\ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6 //  
    |@Hfalse // 
    ] 
  ] 
qed.

\ 5img class="anchor" src="icons/tick.png" id="bigop_false"\ 6theorem bigop_false: ∀n,B,nil,op.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B.
  \big[op,nil]_{i < n | \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 nil.  
#n #B #nil #op #f elim n // #n1 #Hind 
>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // 
qed.

\ 5img class="anchor" src="icons/tick.png" id="Aop"\ 6record Aop (A:Type[0]) (nil:A) : Type[0] ≝
  {op :2> A → A → A; 
   nill:∀a. op nil a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a; 
   nilr:∀a. op a nil \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
   assoc: ∀a,b,c.op a (op b c) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 op (op a b) c
  }.

\ 5img class="anchor" src="icons/tick.png" id="pad_bigop_nil"\ 6theorem pad_bigop_nil: ∀k,n,p,B,nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 k → 
  (∀i. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 k → p i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 f i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 nil) →
  \big[op,nil]_{i < n | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) 
    \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \big[op,nil]_{i < k | p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i).
#k #n #p #B #nil #op #f #lenk (elim lenk) 
  [#_ @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 #i #lti // 
  |#j #leup #Hind #Hfalse cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p j)) #Hpj
    [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // 
     cut (f j \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 nil) 
      [cases (Hfalse j leup (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6 … )) // >Hpj #H destruct (H)] #Hfj
     >Hfj >\ 5a href="cic:/matita/arithmetics/bigops/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6 @Hind #i #leni #ltij
     cases (Hfalse i leni (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6 … ltij)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
    |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // @Hind #i #leni #ltij
     cases (Hfalse i leni (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6 … ltij)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
    ]  
  ]
qed.

\ 5img class="anchor" src="icons/tick.png" id="bigop_sum"\ 6theorem bigop_sum: ∀k1,k2,p1,p2,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f,g:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B.
op (\big[op,nil]_{i<k1|p1 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i)) \big[op,nil]_{i<k2|p2 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(g i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
      \big[op,nil]_{i<k1\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6k2|\ 5font class="Apple-style-span" color="#FF0000"\ 6 if \ 5/font\ 6\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 k2 i then p1 (i\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6k2) else p2 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6
        (if \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 k2 i then f (i\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6k2) else g i).
#k1 #k2 #p1 #p2 #B #nil #op #f #g (elim k1)
  [normalize >\ 5a href="cic:/matita/arithmetics/bigops/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 #i #lti 
   >(\ 5a href="cic:/matita/arithmetics/nat/lt_to_leb_false.def(8)"\ 6lt_to_leb_false\ 5/a\ 6 … lti) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
  |#i #Hind normalize <\ 5a href="cic:/matita/arithmetics/nat/minus_plus_m_m.def(6)"\ 6minus_plus_m_m\ 5/a\ 6 (cases (p1 i)) 
   >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_n.def(7)"\ 6le_plus_n\ 5/a\ 6 …)) normalize <Hind //
   <\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 //
  ]
qed.

\ 5img class="anchor" src="icons/tick.png" id="plus_minus1"\ 6lemma plus_minus1: ∀a,b,c. c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6c) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6c.
#a #b #c #lecb @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 >(\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 c)
>\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
qed.

\ 5img class="anchor" src="icons/tick.png" id="bigop_I"\ 6theorem bigop_I: ∀n,p,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B.
\big[op,nil]_{i∈[\ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6,n[ |p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \big[op,nil]_{i < n|p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i). 
#n #p #B #nil #op #f <\ 5a href="cic:/matita/arithmetics/nat/minus_n_O.def(3)"\ 6minus_n_O\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 //
qed.

\ 5img class="anchor" src="icons/tick.png" id="bigop_I_gen"\ 6theorem bigop_I_gen: ∀a,b,p,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6b →
\big[op,nil]_{i∈[a,b[ |p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \big[op,nil]_{i < b|\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 a i \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i). 
#a #b elim b // -b #b #Hind #p #B #nil #op #f #lea
cut (∀a,b. a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6a)) 
  [#a #b cases a // #a1 #lta1 normalize >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_S_pred.def(4)"\ 6eq_minus_S_pred\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/nat/S_pred.def(3)"\ 6S_pred\ 5/a\ 6 
   /2 by \ 5a href="cic:/matita/arithmetics/nat/lt_plus_to_minus_r.def(11)"\ 6lt_plus_to_minus_r\ 5/a\ 6/] #Hcut
cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … lea) #Ha
  [cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p b)) #Hcase
    [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 [2: >Hcase >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 a b) // @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 @Ha]
     >(Hcut … (\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 … Ha))
     >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 
      [@\ 5a href="cic:/matita/basics/logic/eq_f2.def(3)"\ 6eq_f2\ 5/a\ 6 
        [@\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 [//|@\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 //] @Hind 
        |@Hind @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 //
        ]
      |<\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 //
      ]
   |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 [2: >Hcase cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 a b)//]
     >(Hcut … (\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 … Ha)) >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6
      [@Hind @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 // | <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 //]
    ]
  |<Ha <\ 5a href="cic:/matita/arithmetics/nat/minus_n_n.def(4)"\ 6minus_n_n\ 5/a\ 6 whd in ⊢ (??%?); <(\ 5a href="cic:/matita/arithmetics/bigops/bigop_false.def(4)"\ 6bigop_false\ 5/a\ 6 a B nil op f) in ⊢ (??%?);
   @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 // #i #ltia >(\ 5a href="cic:/matita/arithmetics/nat/not_le_to_leb_false.def(7)"\ 6not_le_to_leb_false\ 5/a\ 6 a i) // @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 //
  ]
qed.
          
\ 5img class="anchor" src="icons/tick.png" id="bigop_sumI"\ 6theorem bigop_sumI: ∀a,b,c,p,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B.
a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c →
\big[op,nil]_{i∈[a,c[ |p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 
  op (\big[op,nil]_{i ∈ [b,c[ |p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i)) 
      \big[op,nil]_{i ∈ [a,b[ |p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i).
#a #b # c #p #B #nil #op #f #leab #lebc 
>(\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 (c\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6a) (b\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6a)) in ⊢ (??%?); /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
>\ 5a href="cic:/matita/arithmetics/nat/minus_plus.def(13)"\ 6minus_plus\ 5/a\ 6 >(\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 a) <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
>\ 5a href="cic:/matita/arithmetics/bigops/bigop_sum.def(9)"\ 6bigop_sum\ 5/a\ 6 (cut (∀i. b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → i\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 i\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(b\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6a)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6b))
  [#i #lei >\ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6 // <\ 5a href="cic:/matita/arithmetics/bigops/plus_minus1.def(8)"\ 6plus_minus1\ 5/a\ 6 
     [@\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/bigops/plus_minus1.def(8)"\ 6plus_minus1\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]] 
#H @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 #i #ltic @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize // #lei <H //
qed.   

\ 5img class="anchor" src="icons/tick.png" id="bigop_a"\ 6theorem bigop_a: ∀a,b,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B. a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b →
\big[op,nil]_{i∈[a,\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 b[ \ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 
  op (\big[op,nil]_{i ∈ [a,b[ \ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 i))) (f a).
#a #b #B #nil #op #f #leab 
>(\ 5a href="cic:/matita/arithmetics/bigops/bigop_sumI.def(14)"\ 6bigop_sumI\ 5/a\ 6 a (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a) (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 b)) [|@\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 //|//] @\ 5a href="cic:/matita/basics/logic/eq_f2.def(3)"\ 6eq_f2\ 5/a\ 6 
  [@\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 // |<\ 5a href="cic:/matita/arithmetics/nat/minus_Sn_n.def(4)"\ 6minus_Sn_n\ 5/a\ 6 normalize @\ 5a href="cic:/matita/arithmetics/bigops/nilr.fix(0,2,2)"\ 6nilr\ 5/a\ 6]
qed.
  
\ 5img class="anchor" src="icons/tick.png" id="bigop_0"\ 6theorem bigop_0: ∀n,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→B.
\big[op,nil]_{i < \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 
  op (\big[op,nil]_{i < n\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 i))) (f \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6).
#n #B #nil #op #f 
<\ 5a href="cic:/matita/arithmetics/bigops/bigop_I.def(7)"\ 6bigop_I\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/bigops/bigop_a.def(15)"\ 6bigop_a\ 5/a\ 6 [|//] @\ 5a href="cic:/matita/basics/logic/eq_f2.def(3)"\ 6eq_f2\ 5/a\ 6 [|//] <\ 5a href="cic:/matita/arithmetics/nat/minus_n_O.def(3)"\ 6minus_n_O\ 5/a\ 6 
@\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 //
qed.    

\ 5img class="anchor" src="icons/tick.png" id="bigop_prod"\ 6theorem bigop_prod: ∀k1,k2,p1,p2,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → B.
\big[op,nil]_{x<k1|p1 x\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(\big[op,nil]_{i<k2|p2 x i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f x i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
  \big[op,nil]_{i<k1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6k2|\ 5a href="cic:/matita/basics/bool/andb.def(1)"\ 6andb\ 5/a\ 6 (p1 (\ 5a href="cic:/matita/arithmetics/div_and_mod/div.def(3)"\ 6div\ 5/a\ 6 i k2)) (p2 (\ 5a href="cic:/matita/arithmetics/div_and_mod/div.def(3)"\ 6div\ 5/a\ 6 i k2) (i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 k2))\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6
     (f (\ 5a href="cic:/matita/arithmetics/div_and_mod/div.def(3)"\ 6div\ 5/a\ 6 i k2) (i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 k2)).
#k1 #k2 #p1 #p2 #B #nil #op #f (elim k1) //
#n #Hind cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p1 n)) #Hp1
  [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // >Hind >\ 5a href="cic:/matita/arithmetics/bigops/bigop_sum.def(9)"\ 6bigop_sum\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6
   #i #lti @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #lei cut (i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6k2\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(i\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6k2)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/bigops/plus_minus1.def(8)"\ 6plus_minus1\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   #eqi [|#H] >eqi in ⊢ (???%);
     >\ 5a href="cic:/matita/arithmetics/div_and_mod/div_plus_times.def(14)"\ 6div_plus_times\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_plus_to_lt_l.def(6)"\ 6lt_plus_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ >Hp1 >(\ 5a href="cic:/matita/arithmetics/div_and_mod/mod_plus_times.def(14)"\ 6mod_plus_times\ 5/a\ 6 …) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/lt_plus_to_lt_l.def(6)"\ 6lt_plus_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // >Hind >(\ 5a href="cic:/matita/arithmetics/bigops/pad_bigop.def(8)"\ 6pad_bigop\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6k2)) // @\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6
   #i #lti @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #lei cut (i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6k2\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(i\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6k2)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/bigops/plus_minus1.def(8)"\ 6plus_minus1\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   #eqi >eqi in ⊢ (???%); >\ 5a href="cic:/matita/arithmetics/div_and_mod/div_plus_times.def(14)"\ 6div_plus_times\ 5/a\ 6 [ >Hp1 %| /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_minus_l.def(12)"\ 6monotonic_lt_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
  ]
qed.

\ 5img class="anchor" src="icons/tick.png" id="ACop"\ 6record ACop (A:Type[0]) (nil:A) : Type[0] ≝
  {aop :> \ 5a href="cic:/matita/arithmetics/bigops/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 A nil; 
   comm: ∀a,b.aop a b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 aop b a
  }.
 
\ 5img class="anchor" src="icons/tick.png" id="bigop_op"\ 6lemma bigop_op: ∀k,p,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/ACop.ind(1,0,2)"\ 6ACop\ 5/a\ 6 B nil.∀f,g: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → B.
op (\big[op,nil]_{i<k|p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i)) (\big[op,nil]_{i<k|p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(g i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
  \big[op,nil]_{i<k|p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(op (f i) (g i)).
#k #p #B #nil #op #f #g (elim k) [normalize @\ 5a href="cic:/matita/arithmetics/bigops/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6]
-k #k #Hind (cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p k))) #H
  [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 //
   normalize <\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 in ⊢ (???%); @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 
   >\ 5a href="cic:/matita/arithmetics/bigops/comm.fix(0,2,3)"\ 6comm\ 5/a\ 6 in ⊢ (??(????%?)?); <\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @Hind
  |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 //
  ]
qed.

\ 5img class="anchor" src="icons/tick.png" id="bigop_diff"\ 6lemma bigop_diff: ∀p,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/ACop.ind(1,0,2)"\ 6ACop\ 5/a\ 6 B nil.∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → B.∀i,n.
  i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → p i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
  \big[op,nil]_{x<n|p x\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f x)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
    op (f i) (\big[op,nil]_{x<n|\ 5a href="cic:/matita/basics/bool/andb.def(1)"\ 6andb\ 5/a\ 6(\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 i x))(p x)\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f x)).
#p #B #nil #op #f #i #n (elim n) 
  [#ltO @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |#n #Hind #lein #pi cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 …lein)) #Hi
    [cut (\ 5a href="cic:/matita/basics/bool/andb.def(1)"\ 6andb\ 5/a\ 6(\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 i n))(p n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (p n))
      [>(\ 5a href="cic:/matita/arithmetics/nat/not_eq_to_eqb_false.def(6)"\ 6not_eq_to_eqb_false\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 … Hi)) //] #Hcut
     cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p n)) #pn 
      [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 //
       normalize >\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 >(\ 5a href="cic:/matita/arithmetics/bigops/comm.fix(0,2,3)"\ 6comm\ 5/a\ 6 ?? op (f i) (f n)) <\ 5a href="cic:/matita/arithmetics/bigops/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 >Hind //
      |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // >Hind //  
      ]
    |<Hi >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6  
       [@\ 5a href="cic:/matita/arithmetics/bigops/same_bigop.def(6)"\ 6same_bigop\ 5/a\ 6 // #k #ltki >\ 5a href="cic:/matita/arithmetics/nat/not_eq_to_eqb_false.def(6)"\ 6not_eq_to_eqb_false\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
       |>\ 5a href="cic:/matita/arithmetics/nat/eq_to_eqb_true.def(5)"\ 6eq_to_eqb_true\ 5/a\ 6 // 
       ]
     ]
   ]
qed.

(* range *)
\ 5img class="anchor" src="icons/tick.png" id="range"\ 6record range (A:Type[0]): Type[0] ≝
  {enum:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→A; upto:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6; filter:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6}.
  
\ 5img class="anchor" src="icons/tick.png" id="sub_hk"\ 6definition sub_hk: (\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6)→(\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6→\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6)→∀A:Type[0].\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/bigops/range.ind(1,0,1)"\ 6range\ 5/a\ 6 A) ≝
λh,k,A,I,J.∀i.i\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/bigops/upto.fix(0,1,1)"\ 6upto\ 5/a\ 6 A I) → (\ 5a href="cic:/matita/arithmetics/bigops/filter.fix(0,1,1)"\ 6filter\ 5/a\ 6 A I i)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → 
  (h i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/bigops/upto.fix(0,1,1)"\ 6upto\ 5/a\ 6 A J
  \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/bigops/filter.fix(0,1,1)"\ 6filter\ 5/a\ 6 A J (h i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6
  \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 k (h i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 i).

\ 5img class="anchor" src="icons/tick.png" id="iso"\ 6definition iso: ∀A:Type[0].\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/bigops/range.ind(1,0,1)"\ 6range\ 5/a\ 6 A) ≝
  λA,I,J.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6h,k\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6 
    (∀i. i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/bigops/upto.fix(0,1,1)"\ 6upto\ 5/a\ 6 A I) → (\ 5a href="cic:/matita/arithmetics/bigops/filter.fix(0,1,1)"\ 6filter\ 5/a\ 6 A I i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → 
       \ 5a href="cic:/matita/arithmetics/bigops/enum.fix(0,1,1)"\ 6enum\ 5/a\ 6 A I i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/bigops/enum.fix(0,1,1)"\ 6enum\ 5/a\ 6 A J (h i)) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
    \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 h k A I J \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 k h A J I.
  
\ 5img class="anchor" src="icons/tick.png" id="sub_hkO"\ 6lemma sub_hkO: ∀h,k,A,I,J. \ 5a href="cic:/matita/arithmetics/bigops/upto.fix(0,1,1)"\ 6upto\ 5/a\ 6 A I \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 h k A I J.
#h #k #A #I #J #up0 #i #lti >up0 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

\ 5img class="anchor" src="icons/tick.png" id="sub0_to_false"\ 6lemma sub0_to_false: ∀h,k,A,I,J. \ 5a href="cic:/matita/arithmetics/bigops/upto.fix(0,1,1)"\ 6upto\ 5/a\ 6 A I \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 h k A J I → 
  ∀i. i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/bigops/upto.fix(0,1,1)"\ 6upto\ 5/a\ 6 A J → \ 5a href="cic:/matita/arithmetics/bigops/filter.fix(0,1,1)"\ 6filter\ 5/a\ 6 A J i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#h #k #A #I #J #up0 #sub #i #lti cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/bigops/filter.fix(0,1,1)"\ 6filter\ 5/a\ 6 A J i)) //
#ptrue (cases (sub i lti ptrue)) * #hi @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
qed.

\ 5img class="anchor" src="icons/tick.png" id="sub_lt"\ 6lemma sub_lt: ∀A,e,p,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → 
  \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 (λx.x) (λx.x) A (\ 5a href="cic:/matita/arithmetics/bigops/range.con(0,1,1)"\ 6mk_range\ 5/a\ 6 A e n p) (\ 5a href="cic:/matita/arithmetics/bigops/range.con(0,1,1)"\ 6mk_range\ 5/a\ 6 A e m p).
#A #e #f #n #m #lenm #i #lti #fi % // % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt.def(4)"\ 6lt_to_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed. 

\ 5img class="anchor" src="icons/tick.png" id="transitive_sub"\ 6theorem transitive_sub: ∀h1,k1,h2,k2,A,I,J,K. 
  \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 h1 k1 A I J → \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 h2 k2 A J K → 
    \ 5a href="cic:/matita/arithmetics/bigops/sub_hk.def(2)"\ 6sub_hk\ 5/a\ 6 (λx.h2(h1 x)) (λx.k1(k2 x)) A I K.
#h1 #k1 #h2 #k2 #A #I #J #K #sub1 #sub2 #i #lti #fi 
cases(sub1 i lti fi) * #lth1i #fh1i #ei 
cases(sub2 (h1 i) lth1i fh1i) * #H1 #H2 #H3 % // % // 
qed. 

\ 5img class="anchor" src="icons/tick.png" id="bigop_iso"\ 6theorem bigop_iso: ∀n1,n2,p1,p2,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/ACop.ind(1,0,2)"\ 6ACop\ 5/a\ 6 B nil.∀f1,f2.
  \ 5a href="cic:/matita/arithmetics/bigops/iso.def(3)"\ 6iso\ 5/a\ 6 B (\ 5a href="cic:/matita/arithmetics/bigops/range.con(0,1,1)"\ 6mk_range\ 5/a\ 6 B f1 n1 p1) (\ 5a href="cic:/matita/arithmetics/bigops/range.con(0,1,1)"\ 6mk_range\ 5/a\ 6 B f2 n2 p2) →
  \big[op,nil]_{i<n1|p1 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f1 i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \big[op,nil]_{i<n2|p2 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f2 i).
#n1 #n2 #p1 #p2 #B #nil #op #f1 #f2 * #h * #k * * #same
@(\ 5a href="cic:/matita/arithmetics/nat/le_gen.def(1)"\ 6le_gen\ 5/a\ 6 ? n1) #i generalize in match p2; (elim i) 
  [(elim n2) // #m #Hind #p2 #_ #sub1 #sub2
   >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 
    [@(Hind ? (\ 5a href="cic:/matita/arithmetics/nat/le_O_n.def(2)"\ 6le_O_n\ 5/a\ 6 ?)) [@\ 5a href="cic:/matita/arithmetics/bigops/sub_hkO.def(4)"\ 6sub_hkO\ 5/a\ 6 % | @(\ 5a href="cic:/matita/arithmetics/bigops/transitive_sub.def(4)"\ 6transitive_sub\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/bigops/sub_lt.def(5)"\ 6sub_lt\ 5/a\ 6 …) sub2) //]
    |@(\ 5a href="cic:/matita/arithmetics/bigops/sub0_to_false.def(4)"\ 6sub0_to_false\ 5/a\ 6 … sub2) //
    ]
  |#n #Hind #p2 #ltn #sub1 #sub2 (cut (n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6n1)) [/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/] #len
   cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p1 n)) #p1n
    [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // (cases (sub1 n (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6 …) p1n)) * #hn #p2hn #eqn
     >(\ 5a href="cic:/matita/arithmetics/bigops/bigop_diff.def(8)"\ 6bigop_diff\ 5/a\ 6 … (h n) n2) // >same // 
     @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(Hind ? len)
      [#i #ltin #p1i (cases (sub1 i (\ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6 … ltin) p1i)) * 
       #h1i #p2h1i #eqi % // % // >\ 5a href="cic:/matita/arithmetics/nat/not_eq_to_eqb_false.def(6)"\ 6not_eq_to_eqb_false\ 5/a\ 6 normalize // 
       @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 ??? (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 ? ? ltin)) // 
      |#j #ltj #p2j (cases (sub2 j ltj (\ 5a href="cic:/matita/basics/bool/andb_true_r.def(4)"\ 6andb_true_r\ 5/a\ 6 …p2j))) * 
       #ltkj #p1kj #eqj % // % // 
       (cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 …(\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 …ltkj))) //
       #eqkj @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 generalize in match p2j; @\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 
       normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
      ]
    |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // @(Hind ? len) 
      [@(\ 5a href="cic:/matita/arithmetics/bigops/transitive_sub.def(4)"\ 6transitive_sub\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/bigops/sub_lt.def(5)"\ 6sub_lt\ 5/a\ 6 …) sub1) //
      |#i #lti #p2i cases(sub2 i lti p2i) * #ltki #p1ki #eqi
       % // % // cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 …(\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 …ltki)) //
       #eqki @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
      ]
    ]
  ]
qed.

(* commutation *)
\ 5img class="anchor" src="icons/tick.png" id="bigop_commute"\ 6theorem bigop_commute: ∀n,m,p11,p12,p21,p22,B.∀nil.∀op:\ 5a href="cic:/matita/arithmetics/bigops/ACop.ind(1,0,2)"\ 6ACop\ 5/a\ 6 B nil.∀f.
\ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m →
(∀i,j. i \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → j \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → (p11 i \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 p12 i j) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (p21 j \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 p22 i j)) →
\big[op,nil]_{i<n|p11 i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(\big[op,nil]_{j<m|p12 i j\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i j)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
   \big[op,nil]_{j<m|p21 j\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(\big[op,nil]_{i<n|p22 i j\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i j)).
#n #m #p11 #p12 #p21 #p22 #B #nil #op #f #posn #posm #Heq
>\ 5a href="cic:/matita/arithmetics/bigops/bigop_prod.def(15)"\ 6bigop_prod\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/bigops/bigop_prod.def(15)"\ 6bigop_prod\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/bigops/bigop_iso.def(9)"\ 6bigop_iso\ 5/a\ 6 
%{(λi.(i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m)} %{(λi.(i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6n)} % 
  [% 
    [#i #lti #Heq (* whd in ⊢ (???(?(?%?)?)); *) @\ 5a href="cic:/matita/basics/logic/eq_f2.def(3)"\ 6eq_f2\ 5/a\ 6
      [@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/mod_plus_times.def(14)"\ 6mod_plus_times\ 5/a\ 6 /2 by \ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6/
      |@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/div_plus_times.def(14)"\ 6div_plus_times\ 5/a\ 6 /2 by \ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6/
      ]
    |#i #lti #Hi 
     cut ((i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m)\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m)
      [@\ 5a href="cic:/matita/arithmetics/div_and_mod/mod_plus_times.def(14)"\ 6mod_plus_times\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //] #H1
     cut ((i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m)\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m)
      [@\ 5a href="cic:/matita/arithmetics/div_and_mod/div_plus_times.def(14)"\ 6div_plus_times\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //] #H2
     %[%[@(\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt.def(4)"\ 6lt_to_le_to_lt\ 5/a\ 6 ? (i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n))
          [whd >\ 5a href="cic:/matita/arithmetics/nat/plus_n_Sm.def(4)"\ 6plus_n_Sm\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //
          |>\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 @(\ 5a href="cic:/matita/arithmetics/nat/le_times.def(9)"\ 6le_times\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6(i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m)) m n n) // @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 //
          ]
        |lapply (Heq (i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m) (i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m) ??)
          [@\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 // |@\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //|>Hi >H1 >H2 //]
        ]
      |>H1 >H2 //
      ]
    ]
  |#i #lti #Hi 
   cut ((i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6n)\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6n)
    [@\ 5a href="cic:/matita/arithmetics/div_and_mod/mod_plus_times.def(14)"\ 6mod_plus_times\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //] #H1
   cut ((i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6n)\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n)
    [@\ 5a href="cic:/matita/arithmetics/div_and_mod/div_plus_times.def(14)"\ 6div_plus_times\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //] #H2
   %[%[@(\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt.def(4)"\ 6lt_to_le_to_lt\ 5/a\ 6 ? (i\ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m))
        [whd >\ 5a href="cic:/matita/arithmetics/nat/plus_n_Sm.def(4)"\ 6plus_n_Sm\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 //
        |>\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 @(\ 5a href="cic:/matita/arithmetics/nat/le_times.def(9)"\ 6le_times\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6(i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n)) n m m) // @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 //
        ]
      |lapply (Heq (i \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n) (i\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6n) ??)
        [@\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_times_to_lt_div.def(10)"\ 6lt_times_to_lt_div\ 5/a\ 6 // |@\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 // |>Hi >H1 >H2 //]
      ]
    |>H1 >H2 //
    ]
  ]
qed.

(* distributivity *)

\ 5img class="anchor" src="icons/tick.png" id="Dop"\ 6record Dop (A:Type[0]) (nil:A): Type[0] ≝
  {sum : \ 5a href="cic:/matita/arithmetics/bigops/ACop.ind(1,0,2)"\ 6ACop\ 5/a\ 6 A nil; 
   prod: A → A →A;
   null: \forall a. prod a nil \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 nil; 
   distr: ∀a,b,c:A. prod a (sum b c) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 sum (prod a b) (prod a c)
  }.
  
\ 5img class="anchor" src="icons/tick.png" id="bigop_distr"\ 6theorem bigop_distr: ∀n,p,B,nil.∀R:\ 5a href="cic:/matita/arithmetics/bigops/Dop.ind(1,0,2)"\ 6Dop\ 5/a\ 6 B nil.\forall f,a.
  let aop \def  \ 5a href="cic:/matita/arithmetics/bigops/sum.fix(0,2,4)"\ 6sum\ 5/a\ 6 B nil R in
  let mop \def \ 5a href="cic:/matita/arithmetics/bigops/prod.fix(0,2,4)"\ 6prod\ 5/a\ 6 B nil R in
  mop a \big[aop,nil]_{i<n|p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 
   \big[aop,nil]_{i<n|p i\ 5a title="bigop" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6(mop a (f i)).
#n #p #B #nil #R #f #a normalize (elim n) [@\ 5a href="cic:/matita/arithmetics/bigops/null.fix(0,2,5)"\ 6null\ 5/a\ 6]
#n #Hind (cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p n))) #H
  [>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Strue.def(3)"\ 6bigop_Strue\ 5/a\ 6 // >(\ 5a href="cic:/matita/arithmetics/bigops/distr.fix(0,2,5)"\ 6distr\ 5/a\ 6 B nil R) >Hind //
  |>\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 // >\ 5a href="cic:/matita/arithmetics/bigops/bigop_Sfalse.def(3)"\ 6bigop_Sfalse\ 5/a\ 6 //
  ]
qed.