--> ==========================================================
--> Proof scores with [CITP for CafeOBJ]
--> for verifications of properties
--> including associativity and commutativity of _+_ and _*_
--> about Peano Style natural numbers
-->                                         kf151028
--> ==========================================================

--> 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) .
}

--> ==========================================================
--> Property of PNAT+: _+_ is associative (+assoc)
-->  eq[+assoc]: (X:Nat + Y:Nat) + Z:Nat = X + (Y + Z) .
--> ==========================================================
:verbose on  . -- to see detailed information
-- proof with CITP
select PNAT+ .
:goal{eq[+assoc]: (X:Nat + Y:Nat) + Z:Nat = X + (Y + Z) .}
-- by induction on X
:ind on (X:Nat) . 
-- check
:apply (SI TC RD) .
--> QED
:show proof  . -- to see structure of proof
:verbose off . -- get back to normal mode
--> ==========================================================

--> ==========================================================
--> Property of PNAT+: 0 is right identity of _+_ (+rz)
-->     eq[+ri]: X:Nat + 0 = X .
--> ==========================================================
-- proof with CITP
select PNAT+ .
:goal{eq[+ri]: X:Nat + 0 = X .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+: right successor of _+_ (+rs)
-->    eq[+rs]: X:Nat + (s Y:Nat) = s (X + Y) .
--> ==========================================================
-- proof with CITP
select PNAT+ .
:goal{eq[+rs]: X:Nat + (s Y:Nat) = s (X + Y) .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+: _+_ is commutative (+comm)
-->   eq[+comm]: X:Nat + Y:Nat = Y + X .
--> ==========================================================
-- proof with CITP
mod PNAT+comm {pr(PNAT+)
-- proved properties
eq[+ri]:  X:Nat + 0 = X .
eq[+rs]: X:Nat + (s Y:Nat) = s (X + Y) .
}
select PNAT+comm .
:goal{eq[+comm]: X:Nat + Y:Nat = Y + X .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> 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) .
}
** "{assoc comm prec: 30}" declares that the operator "_+_"
** is associative and commutative and parsing precedence 30.

--> ==========================================================
--> Property of PNAT+*:
-->     _*_ distributes over _+_ from right (*distr)
-->  eq[*distr]: (X:Nat + Y:Nat) * Z:Nat = X * Z + Y * Z .
--> ==========================================================
-- proof with CITP
select PNAT+* .
:goal{eq[*distr]: (X:Nat + Y:Nat) * Z:Nat = X * Z + Y * Z .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+*: _*_ is associative (*assoc)
-->  eq[*assoc]: (X:Nat * Y:Nat) * Z:Nat = X * (Y * Z) .
--> ==========================================================
-- proof with CITP
mod PNAT+*assoc
{pr(PNAT+*)
-- proved property
eq[*distr]: (X:Nat + Y:Nat) * Z:Nat = (X * Z) + (Y * Z) .
}
select PNAT+*assoc .
:goal{eq[*assoc]: (X:Nat * Y:Nat) * Z:Nat = X * (Y * Z) .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+*: 0 is right-zero of _*_ (*rz)
-->    eq[*rz]: X:Nat * 0 = 0 .
--> ==========================================================
-- proof with CITP
select PNAT+* .
:goal{eq[*rz]: X:Nat * 0 = 0 .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+*: right-successor of _*_ (*rs)
-->  eq[*rs]: X:Nat * (s Y:Nat) = X + (X * Y) .
--> ==========================================================
-- proof with CITP
select PNAT+* .
:goal{eq[*rs]: X:Nat * (s Y:Nat) = X + (X * Y) .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
-- ==========================================================

--> ==========================================================
--> Property of PNAT+*: _*_ is commutative (*comm)
-->    eq[*comm]: X:Nat * Y:Nat = Y * X .
--> ==========================================================
-- proof with CITP
mod PNAT+*comm {pr(PNAT+*)
-- proved properties
eq[*rz]: X:Nat * 0 = 0 .
eq[*rs]: X:Nat * (s Y:Nat) = X + (X * Y) .
}
select PNAT+*comm .
:goal{eq[*comm]: X:Nat * Y:Nat = Y * X .}
-- by induction on X
:ind on (X:Nat) .
-- check
:apply (SI TC RD) .
--> QED
--> ==========================================================

-- ==========================================================
-- end 
-- ==========================================================