--> ==========================================================
-->  Proof scores 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]: eq (X:Nat + Y:Nat) + Z:Nat = X + (Y + Z) .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
select PNAT+ .
red (0 + `y:Nat) + `z:Nat = 0 + (`y + `z) .
--> II Induction Step
open PNAT+ .
-- induction hypothesis
op x : -> Nat . eq (x + Y:Nat) + Z:Nat = x + (Y + Z) .
-- check
red ((s x) + `y:Nat) + `z:Nat = (s x) + (`y + `z) .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+: 0 is right identity of _+_ (+rz)
-->     eq[+ri]: X:Nat + 0 = X .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
select PNAT+ .
red 0 + 0 = 0 .
--> II Induction Step
open PNAT+ .
-- induction hypothesis
op x : -> Nat . eq x + 0 = x .
-- check
red (s x) + 0 = (s x) .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+: right successor of _+_ (+rs)
-->    eq[+rs]: X:Nat + (s Y:Nat) = s (X + Y) .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
select PNAT+ .
red 0 + (s `y:Nat) = s (0 + `y) .
--> II Induction Step
open PNAT+ .
-- induction hypothesis
op x : -> Nat . eq x + (s `y:Nat) = s (x + `y) .
-- check
red (s x) + (s `y:Nat) = s ((s x) + `y) .
close
--
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+: _+_ is commutative (+comm)
-->   eq[+comm]: X:Nat + Y:Nat = Y + X .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
-- This case uses +rz(y) as a lemma.
open PNAT+ .
-- proved property
eq[+ri]:  X:Nat + 0 = X .
-- check
red 0 + `y:Nat = `y + 0 .
close
--> II Induction Step
open PNAT+ .
-- proved property
eq[+rs]: X:Nat + (s Y:Nat) = s (X + Y) .
-- induction hypothesis
op x : -> Nat . eq x + Y:Nat = Y + x .
-- check
red (s x) + `y:Nat = `y + (s x) .
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) .
}
** "{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: By induction on X
--> I Base case 
select PNAT+* .
red (0 + `y:Nat) * `z:Nat = 0 * `z + `y * `z .
--> II Induction step
open PNAT+* .
-- induction hypothesis
op x : -> Nat . eq (x + Y:Nat) * Z:Nat = (x * Z) + (Y * Z) .
-- check
red ((s x) + `y:Nat) * `z:Nat = (s x) * `z + `y * `z .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+*: _*_ is associative (*assoc)
-->  eq[*assoc]: (X:Nat * Y:Nat) * Z:Nat = X * (Y * Z) .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
select PNAT+* .
red (0 * `y:Nat) * `z:Nat = 0 * (`y * `z) .
--> II Induction Step
open PNAT+* .
-- proved property
eq[*distr]: (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 ((s x) * `y:Nat) * `z:Nat = (s x) * (`y * `z) .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+*: 0 is right-zero of _*_ (*rz)
-->    eq[*rz]: X:Nat * 0 = 0 .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
select PNAT+* .
red 0 * 0 = 0 .
--> II Induction Step
open PNAT+* .
-- induction hypothesis
op x : -> Nat . eq x * 0 = 0 .
-- check
red (s x) * 0 = 0 .
close
--> QED
--> ==========================================================

--> ==========================================================
--> Property of PNAT+*: right-successor of _*_ (*rs)
-->  eq[*rs]: X:Nat * (s Y:Nat) = X + (X * Y) .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
select PNAT+* .
red 0 * (s `y:Nat) = 0 + (0 * `y) .
--> II Induction Step
open PNAT+* .
-- induction hypothesis
op x : -> Nat . eq x * (s `y:Nat) = x + (x * `y) .
-- check
red (s x) * (s `y:Nat) = (s x) + ((s x) * `y) .
close
--> QED
-- ==========================================================

--> ==========================================================
--> Property of PNAT+*: _*_ is commutative (*comm)
--> eq[*comm]: X:Nat * Y:Nat = Y * X .
--> ==========================================================
-- Proof: By induction on X
--> I Induction Base
open PNAT+* .
-- proved property
eq[*rz]: X:Nat * 0 = 0 .
-- check
red 0 * `y:Nat = `y * 0 .
close
--> II Induction Step
open PNAT+* .
-- proved property
eq[*rs]: X:Nat * (s Y:Nat) = X + (X * Y) .
-- induction hypothesis
op x : -> Nat . eq x * Y:Nat = Y * x .
-- check
red (s x) * `y:Nat = `y * (s x) .
close
--> QED
--> ==========================================================

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