--> ==========================================================
--> Proof scores for verifications of properties
--> including associativity and commutativity of _+_ and _*_
--> about Peano Style natural numbers
--> with named properties
-->                                 kf151027
--> ==========================================================

--> ==========================================================
--> Peano style natural numbers
mod! PNAT {
  [ Nat ]
  op 0 : -> Nat {constr} .
  op s_ : Nat -> Nat {constr} .
  -- equality over the natural numbers
  eq (0 = s(Y:Nat)) = false .
  eq (s(X:Nat) = s(Y:Nat)) = (X = Y) .
}

--> ==========================================================
--> PNAT with plus _+_ operation
mod! PNAT+ {
  pr(PNAT)
  op _+_ : Nat Nat -> Nat {r-assoc} .
  vars X Y : Nat .
  eq 0 + Y = Y .
  eq (s X) + Y = s(X + Y) .
}

--> ==========================================================
--> properties about PNAT+
mod PRED-PNAT+ {
  pr(PNAT+)

  -- declaration of CafeOBJ variables
  vars X Y Z  : Nat .

  -- 0 is right identity of _+_
  op +ri : Nat -> Bool .
  eq +ri(X) = (X + 0 = X) .

  -- associativity of _+_
  op +assoc : Nat Nat Nat -> Bool
  eq +assoc(X,Y,Z) = ((X + Y) + Z = X + (Y + Z)) .

  -- right successor of _+_
  op +rs : Nat Nat -> Bool
  eq +rs(X,Y) = (X + s(Y) = s(X + Y)) .

  -- commutativity of _+_
  op +comm : Nat Nat -> Bool
  eq +comm(X,Y) = (X + Y = Y + X) .
}

--> ==========================================================
--> Property of PRED-PNAT+
-->        \forall X:Nat.(X + 0 = X)
-->        (i.e.  +ri(X:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
select PRED-PNAT+ .
--> I induction base
red +ri(0) .
--> II induction Step
red +ri(`x:Nat) implies +ri(s(`x)) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PRED-PNAT+
-->  \forall X,Y,Z:Nat.((X + Y) + Z = X + (Y + Z))
-->     (i.e. +assoc(X:Nat,Y:Nat,Z:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
select PRED-PNAT+ .
--> I induction Base
red +assoc(0,`y:Nat,`z:Nat) .
--> II induction Step
red +assoc(`x:Nat,`y:Nat,`z:Nat)
    implies +assoc((s `x),`y,`z) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property \forall X,Y:Nat.(X + s(Y) = s(X + Y))
-->  (i.e. +rs(X:Nat,Y:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
select PRED-PNAT+ .
--> I induction base
red +rs(0,`y:Nat) .
--> II induction step
red +rs(`x:Nat,`y:Nat) implies +rs(s(`x),`y) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property \forall X,Y:Nat.(X + Y = Y + X)
-->   (i.e. +comm(X,Y) = true)
--> ==========================================================
-- Proof: By induction on X
--> I base base
select PRED-PNAT+ .
-- uses +ri(X:Nat) as a lemma.
red +ri(`y:Nat) implies +comm(0,`y) .
--> II induction step
open PRED-PNAT+ .
-- +rs(X:Nat,Y:Nat) is already proved
eq X:Nat + (s Y:Nat) = s (X + Y) .
-- check
red +comm(`x:Nat,`y:Nat) implies +comm((s `x),`y) .
close  
--> QED
-- ==========================================================

-- ==========================================================
--> PNAT with _+_ and _*_ operations
mod! PNAT+* {
  pr(PNAT)  

  vars X Y : Nat .

  -- notice that assoc and comm of _+_ are already proved
  op _+_ : Nat Nat -> Nat {assoc comm prec: 30}
  eq 0 + Y = Y .
  eq s(X) + Y = s(X + Y) .

  -- _*_ connects stronger than _+_ 
  -- because it has smaller precedence (prec:)
  op _*_ : Nat Nat -> Nat {prec: 29 r-assoc}
  eq 0 * Y = 0 .
  eq s(X) * Y = Y + (X * Y) .
}

-- ==========================================================
--> properties about PNAT+*
mod PRED-PNAT+*  {
  pr(PNAT+*)

  -- CafeOBJ variables
  vars X Y Z  : Nat

  -- _*_ distributes over _+_ from right
  op *distr : Nat Nat Nat -> Bool
  eq *distr(X,Y,Z) = ((X + Y) * Z = X * Z + Y * Z) .

  -- associativity of _*_
  op *assoc : Nat Nat Nat -> Bool
  eq *assoc(X,Y,Z) = ((X * Y) * Z = X * (Y * Z)) .

  -- 0 is right zero of _*_
  op *rz : Nat -> Bool
  eq *rz(X) = (X * 0 = 0) .

  -- right successor of _*_
  op *rs : Nat Nat -> Bool
  eq *rs(X,Y) = (X * s(Y) = X + (X * Y)) .

  -- commutativity of _*_
  op *comm : Nat Nat -> Bool
  eq *comm(X,Y) = (X * Y = Y * X) .
}

--> ==========================================================
--> Property 
-->    \forall X,Y,Z:Nat.((X + Y) * Z = X * Z + Y * Z)
-->    (i.e. *distr(X:Nat,Y:nat,Z:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
--> I induction base
select PRED-PNAT+* .
red *distr(0,`y:Nat,`z:Nat) .
--> II induction step
open PRED-PNAT+* .
-- induction hypothesis
op x : -> Nat . eq (x + Y:Nat) * Z:Nat = (x * Z) + (Y * Z) .
-- check
red *distr((s x),`y:Nat,`z:Nat) .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PRED-PNAT+*
-->    \forall X,Y,Z:Nat.((X * Y) * Z = X * (Y * Z))
-->     (i.e. *assoc(X:Nat,Y:Nat,Z:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
--> I base base
select PRED-PNAT+* .
red *assoc(0,`y:Nat,`z:Nat) .
--> II induction step
open PRED-PNAT+* .
-- *distr(X:Nat,Y:Nat,Z:Nat) is already proved
eq (X:Nat + Y:Nat) * Z:Nat = (X * Z) + (Y * Z) .
-- induction hypothesis
op x : -> Nat . eq (x * Y:Nat) * Z:Nat = x * (Y * Z) .
-- check
red *assoc(s(x),`y:Nat,`z:Nat) .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property \forall X:Nat.(X * 0 = 0)
-->     (i.e. *rz(X:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
select PRED-PNAT+* .
--> I induction base
red *rz(0) .
--> II induction step
red *rz(`x:Nat) implies *rz(s(`x)) .
--> QED
-- ==========================================================

--> ==========================================================
--> Property \forall X,Y:Nat.(X * s(Y) = X + (X * Y))
-->     (i.e. *rs(X:Nat,Y:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
--> I induction base
select PRED-PNAT+* .
red *rs(0,`y:Nat) .
--> II induction step
open PRED-PNAT+* .
-- induction hypothesis
op x : -> Nat . eq x * s(Y:Nat) = x + (x * Y) .
-- check
red *rs((s x),`y:Nat) .
close
--> QED
-- ==========================================================

--> ==========================================================
--> Property \forall X,Y:Nat.(X * Y = Y * X)
-->     (i.e. *comm(X:Nat,Y:Nat) = true)
--> ==========================================================
-- Proof: By induction on X
--> I induction base
select PRED-PNAT+* .
-- *rz(X:Nat) is already proved
red *rz(`y:Nat) implies *comm(0,`y:Nat) .
--> II induction step
open PRED-PNAT+* .
-- *rs(X:Nat,Y:Nat) is already proved
eq X:Nat * (s Y:Nat) = X + (X * Y) .
-- induction hypothesis
op x : -> Nat . eq x * Y:Nat = Y:Nat * x .
-- check
red *comm((s x),`y:Nat) .
close
--> QED
-- ==========================================================

-- ==========================================================
--> end
-- ==========================================================