CafeOBJ> in pnat-ps.cafe

processing input : /Users/kokichi/Documents/d-f090205/lectureF0902/160916+23-NII-Tokyo/homePage/lecture1/pnat-ps.cafe

--> ****************************************************************
--> proof scores for associativity and commutativity of
--> _+_ and _*_ over Nat.PNAT
--> ****************************************************************
--> ----------------------------------------------------------------
--> PNAT: Peano NATural numbers
--> ----------------------------------------------------------------
-- defining module! PNAT
-- reading in file  : bool

processing input : /Users/kokichi/Documents/cafeobjSystemsF0908/cafeobj155-151228/cafeobj-1.5.5-sbcl-x64Darwin/share/cafeobj-1.5/lib/bool.cafe

processing input : /Users/kokichi/Documents/cafeobjSystemsF0908/cafeobj155-151228/cafeobj-1.5.5-sbcl-x64Darwin/share/cafeobj-1.5/lib/base_bool.cafe

processing input : /Users/kokichi/Documents/cafeobjSystemsF0908/cafeobj155-151228/cafeobj-1.5.5-sbcl-x64Darwin/share/cafeobj-1.5/lib/truth.cafe

-- defining module! TRUTH
-- reading in file  : truth

-- done reading in file: truth
.............._...........* done.
-- defining module! BASE-BOOL..
-- reading in file  : eql

processing input : /Users/kokichi/Documents/cafeobjSystemsF0908/cafeobj155-151228/cafeobj-1.5.5-sbcl-x64Darwin/share/cafeobj-1.5/lib/eql.cafe

-- defining module! EQL.._...* done.
-- done reading in file: eql
........._* done.
processing input : /Users/kokichi/Documents/cafeobjSystemsF0908/cafeobj155-151228/cafeobj-1.5.5-sbcl-x64Darwin/share/cafeobj-1.5/lib/sys_bool.cafe

-- defining module! BOOL....._......................* done.
-- done reading in file: bool
..._* done.
--> ----------------------------------------------------------------
--> PNAT with addition _+_
--> ----------------------------------------------------------------
-- defining module! PNAT+.._..* done.

--> ****************************************************************
--> set trace whole on
--> ----------------------------------------------------------------
--> ================================================================
--> proof score for proving right 0 of _+_:
-->   eq X:Nat + 0 = X .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- reduce in PNAT+ : ((0 + 0) = 0):Bool
[1]: ((0 + 0) = 0):Bool
---> (0 = 0):Bool
[2]: (0 = 0):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 2 rewrites + 2 matches)
-- opening module PNAT+.. done.

--> induction hypothesis_
-- reduce in %PNAT+ : (((s n) + 0) = (s n)):Bool
[1]: (((s n) + 0) = (s n)):Bool
---> ((s (n + 0)) = (s n)):Bool
[2]: ((s (n + 0)) = (s n)):Bool
---> ((s n) = (s n)):Bool
[3]: ((s n) = (s n)):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 3 rewrites + 5 matches)

--> QED
--> ================================================================
--> ================================================================
--> proof score for proving right s_ of _+_:
-->   eq X:Nat + s Y:Nat = s (X + Y) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- opening module PNAT+.. done.
_-- reduce in %PNAT+ : ((0 + (s y)) = (s (0 + y))):Bool
[1]: ((0 + (s y)) = (s (0 + y))):Bool
---> ((s y) = (s (0 + y))):Bool
[2]: ((s y) = (s (0 + y))):Bool
---> ((s y) = (s y)):Bool
[3]: ((s y) = (s y)):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 3 rewrites + 3 matches)

--> induction step
-- opening module PNAT+.. done.
__-- reduce in %PNAT+ : (((s n) + (s y)) = (s ((s n) + y))):Bool
[1]: (((s n) + (s y)) = (s ((s n) + y))):Bool
---> ((s (n + (s y))) = (s ((s n) + y))):Bool
[2]: ((s (n + (s y))) = (s ((s n) + y))):Bool
---> ((s (s (n + y))) = (s ((s n) + y))):Bool
[3]: ((s (s (n + y))) = (s ((s n) + y))):Bool
---> ((s (s (n + y))) = (s (s (n + y)))):Bool
[4]: ((s (s (n + y))) = (s (s (n + y)))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 4 rewrites + 14 matches)

--> QED
--> ================================================================
--> ****************************************************************
--> set trace on
--> ----------------------------------------------------------------
--> ================================================================
--> proof score for proving commutativity of _+_:
-->   eq (X:Nat + Y:Nat) = (Y + X) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- opening module PNAT+.. done.
_-- reduce in %PNAT+ : ((0 + y) = (y + 0)):Bool
1>[1] rule: eq (0 + Y:Nat) = Y
    { Y:Nat |-> y }
1<[1] (0 + y):Nat --> (y):Nat
[1]: ((0 + y) = (y + 0)):Bool
---> (y = (y + 0)):Bool
1>[2] rule: eq (X:Nat + 0) = X
    { X:Nat |-> y }
1<[2] (y + 0):Nat --> (y):Nat
[2]: (y = (y + 0)):Bool
---> (y = y):Bool
1>[3] rule: eq (CUX = CUX) = true
    { CUX |-> y }
1<[3] (y = y):Bool --> (true):Bool
[3]: (y = y):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 3 rewrites + 4 matches)

--> induction step
-- opening module PNAT+.. done.
__-- reduce in %PNAT+ : (((s n) + y) = (y + (s n))):Bool
1>[1] rule: eq ((s X:Nat) + Y:Nat) = (s (X + Y))
    { X:Nat |-> n, Y:Nat |-> y }
1<[1] ((s n) + y):Nat --> (s (n + y)):Nat
[1]: (((s n) + y) = (y + (s n))):Bool
---> ((s (n + y)) = (y + (s n))):Bool
1>[2] rule: eq (n + Y:Nat) = (Y + n)
    { Y:Nat |-> y }
1<[2] (n + y):Nat --> (y + n):Nat
[2]: ((s (n + y)) = (y + (s n))):Bool
---> ((s (y + n)) = (y + (s n))):Bool
1>[3] rule: eq (X:Nat + (s Y:Nat)) = (s (X + Y))
    { X:Nat |-> y, Y:Nat |-> n }
1<[3] (y + (s n)):Nat --> (s (y + n)):Nat
[3]: ((s (y + n)) = (y + (s n))):Bool
---> ((s (y + n)) = (s (y + n))):Bool
1>[4] rule: eq (CUX = CUX) = true
    { CUX |-> (s (y + n)) }
1<[4] ((s (y + n)) = (s (y + n))):Bool --> (true):Bool
[4]: ((s (y + n)) = (s (y + n))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 4 rewrites + 16 matches)

--> QED
--> ================================================================
--> ----------------------------------------------------------------
--> set trace off
--> ****************************************************************
--> ================================================================
--> proof score for proving associativity of _+_:
-->   eq (X:Nat + Y:Nat) + Z:Nat = X + (Y + Z) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- opening module PNAT+.. done.
_-- reduce in %PNAT+ : (((0 + y) + z) = (0 + (y + z))):Bool
[1]: (((0 + y) + z) = (0 + (y + z))):Bool
---> ((y + z) = (0 + (y + z))):Bool
[2]: ((y + z) = (0 + (y + z))):Bool
---> ((y + z) = (y + z)):Bool
[3]: ((y + z) = (y + z)):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 3 rewrites + 11 matches)

--> induction step
-- opening module PNAT+.. done.
__-- reduce in %PNAT+ : ((((s n) + y) + z) = ((s n) + (y + z))):Bool
[1]: ((((s n) + y) + z) = ((s n) + (y + z))):Bool
---> (((s (n + y)) + z) = ((s n) + (y + z))):Bool
[2]: (((s (n + y)) + z) = ((s n) + (y + z))):Bool
---> ((s ((n + y) + z)) = ((s n) + (y + z))):Bool
[3]: ((s ((n + y) + z)) = ((s n) + (y + z))):Bool
---> ((s (n + (y + z))) = ((s n) + (y + z))):Bool
[4]: ((s (n + (y + z))) = ((s n) + (y + z))):Bool
---> ((s (n + (y + z))) = (s (n + (y + z)))):Bool
[5]: ((s (n + (y + z))) = (s (n + (y + z)))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 5 rewrites + 47 matches)

--> QED
--> ================================================================
--> ----------------------------------------------------------------
--> PNAT with associative and commutative addition _+_
--> ----------------------------------------------------------------
-- defining module! PNAT+ac.._..* done.

--> ----------------------------------------------------------------
--> PNAT with multiplication _*_
--> ----------------------------------------------------------------
-- defining module! PNAT*.._..* done.

--> ================================================================
--> proof score for proving right 0 of _*_:
-->   eq X:Nat * 0 = X .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- reduce in PNAT* : ((0 * 0) = 0):Bool
[1]: ((0 * 0) = 0):Bool
---> (0 = 0):Bool
[2]: (0 = 0):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 2 rewrites + 2 matches)
--> induction step
-- opening module PNAT*.. done.
_-- reduce in %PNAT* : (((s n) * 0) = 0):Bool
[1]: (((s n) * 0) = 0):Bool
---> ((0 + (n * 0)) = 0):Bool
[2]: ((0 + (n * 0)) = 0):Bool
---> ((0 + 0) = 0):Bool
[3]: ((0 + 0) = 0):Bool
---> (0 = 0):Bool
[4]: (0 = 0):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 4 rewrites + 6 matches)

--> QED
--> ================================================================
--> ================================================================
--> induction base
-- opening module PNAT*.. done.
_-- reduce in %PNAT* : ((0 * (s y)) = (0 + (0 * y))):Bool
[1]: ((0 * (s y)) = (0 + (0 * y))):Bool
---> (0 = (0 + (0 * y))):Bool
[2]: (0 = (0 + (0 * y))):Bool
---> (0 = (0 + 0)):Bool
[3]: (0 = (0 + 0)):Bool
---> (0 = 0):Bool
[4]: (0 = 0):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 4 rewrites + 4 matches)

--> induction step
-- opening module PNAT*.. done.
__-- reduce in %PNAT* : (((s n) * (s y)) = ((s n) + ((s n) * y))):Bool
[1]: (((s n) * (s y)) = ((s n) + ((s n) * y))):Bool
---> (((s y) + (n * (s y))) = ((s n) + ((s n) * y))):Bool
[2]: (((s y) + (n * (s y))) = ((s n) + ((s n) * y))):Bool
---> (((s y) + (n + (n * y))) = ((s n) + ((s n) * y))):Bool
[3]: (((s y) + ((n * y) + n)) = ((s n) + ((s n) * y))):Bool
---> ((s (y + ((n * y) + n))) = ((s n) + ((s n) * y))):Bool
[4]: ((s ((n * y) + (n + y))) = ((s n) + ((s n) * y))):Bool
---> ((s ((n * y) + (n + y))) = ((s n) + (y + (n * y)))):Bool
[5]: ((s ((n * y) + (n + y))) = ((s n) + ((n * y) + y))):Bool
---> ((s ((n * y) + (n + y))) = (s (n + ((n * y) + y)))):Bool
[6]: ((s ((n * y) + (n + y))) = (s ((n * y) + (y + n)))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 6 rewrites + 50 matches)

--> QED
--> ================================================================
--> ****************************************************************
--> set trace on
--> ----------------------------------------------------------------
--> ================================================================
--> proof score for proving commutativity of _*_:
-->   eq (X:Nat * Y:Nat) = (Y * X) .
--> with the induction on X:Nat
--> -----------------------------------------------------------------
--> induction base
-- opening module PNAT*.. done.
_-- reduce in %PNAT* : ((0 * y) = (y * 0)):Bool
1>[1] rule: eq (0 * Y:Nat) = 0
    { Y:Nat |-> y }
1<[1] (0 * y):Nat --> (0):Nat
[1]: ((0 * y) = (y * 0)):Bool
---> (0 = (y * 0)):Bool
1>[2] rule: eq (X:Nat * 0) = 0
    { X:Nat |-> y }
1<[2] (y * 0):Nat --> (0):Nat
[2]: (0 = (y * 0)):Bool
---> (0 = 0):Bool
1>[3] rule: eq (CUX = CUX) = true
    { CUX |-> 0 }
1<[3] (0 = 0):Bool --> (true):Bool
[3]: (0 = 0):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 3 rewrites + 4 matches)

--> induction step
-- opening module PNAT*.. done.
__-- reduce in %PNAT* : (((s n) * y) = (y * (s n))):Bool
1>[1] rule: eq ((s X:Nat) * Y:Nat) = (Y + (X * Y))
    { X:Nat |-> n, Y:Nat |-> y }
1<[1] ((s n) * y):Nat --> (y + (n * y)):Nat
[1]: (((s n) * y) = (y * (s n))):Bool
---> ((y + (n * y)) = (y * (s n))):Bool
1>[2] rule: eq (n * Y:Nat) = (Y * n)
    { Y:Nat |-> y }
1<[2] (n * y):Nat --> (y * n):Nat
[2]: ((y + (n * y)) = (y * (s n))):Bool
---> ((y + (y * n)) = (y * (s n))):Bool
1>[3] rule: eq (X:Nat * (s Y:Nat)) = (X + (X * Y))
    { X:Nat |-> y, Y:Nat |-> n }
1<[3] (y * (s n)):Nat --> (y + (y * n)):Nat
[3]: (((y * n) + y) = (y * (s n))):Bool
---> (((y * n) + y) = (y + (y * n))):Bool
1>[4] rule: eq (CUX = CUX) = true
    { CUX |-> ((y * n) + y) }
1<[4] (((y * n) + y) = ((y * n) + y)):Bool --> (true):Bool
[4]: (((y * n) + y) = ((y * n) + y)):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 4 rewrites + 24 matches)

--> QED
--> ================================================================
--> ----------------------------------------------------------------
--> set trace off
--> ****************************************************************
--> ================================================================
--> proof score for proving distributivity of _*_ over _+_
--> from right:
-->   eq (X:Nat + Y:Nat) * Z:Nat = X * Z + Y * Z .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- opening module PNAT*.. done.
_-- reduce in %PNAT* : (((0 + y) * z) = ((0 * z) + (y * z))):Bool
[1]: (((0 + y) * z) = ((0 * z) + (y * z))):Bool
---> ((y * z) = ((0 * z) + (y * z))):Bool
[2]: ((y * z) = ((0 * z) + (y * z))):Bool
---> ((y * z) = (0 + (y * z))):Bool
[3]: ((y * z) = (0 + (y * z))):Bool
---> ((y * z) = (y * z)):Bool
[4]: ((y * z) = (y * z)):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 4 rewrites + 16 matches)

--> induction step
-- opening module PNAT*.. done.
__-- reduce in %PNAT* : ((((s n) + y) * z) = (((s n) * z) + (y * z))):Bool
[1]: ((((s n) + y) * z) = (((s n) * z) + (y * z))):Bool
---> (((s (n + y)) * z) = (((s n) * z) + (y * z))):Bool
[2]: (((s (y + n)) * z) = (((s n) * z) + (y * z))):Bool
---> ((z + ((y + n) * z)) = (((s n) * z) + (y * z))):Bool
[3]: ((z + ((n + y) * z)) = (((s n) * z) + (y * z))):Bool
---> ((z + ((n * z) + (y * z))) = (((s n) * z) + (y * z))):Bool
[4]: (((n * z) + ((y * z) + z)) = (((s n) * z) + (y * z))):Bool
---> (((n * z) + ((y * z) + z)) = ((z + (n * z)) + (y * z))):Bool
[5]: (((n * z) + ((y * z) + z)) = ((y * z) + (z + (n * z)))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 5 rewrites + 59 matches)

--> QED
--> ================================================================
--> ================================================================
--> proof score for proving associativity of _*_:
-->   eq (X:Nat * Y:Nat) * Z:Nat = X * (Y * Z) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> induction base
-- opening module PNAT*.. done.
_-- reduce in %PNAT* : (((0 * y) * z) = (0 * (y * z))):Bool
[1]: (((0 * y) * z) = (0 * (y * z))):Bool
---> ((0 * z) = (0 * (y * z))):Bool
[2]: ((0 * z) = (0 * (y * z))):Bool
---> (0 = (0 * (y * z))):Bool
[3]: (0 = (0 * (y * z))):Bool
---> (0 = 0):Bool
[4]: (0 = 0):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 4 rewrites + 4 matches)

--> induction step
-- opening module PNAT*.. done.
__-- reduce in %PNAT* : ((((s n) * y) * z) = ((s n) * (y * z))):Bool
[1]: ((((s n) * y) * z) = ((s n) * (y * z))):Bool
---> (((y + (n * y)) * z) = ((s n) * (y * z))):Bool
[2]: ((((n * y) + y) * z) = ((s n) * (y * z))):Bool
---> ((((n * y) * z) + (y * z)) = ((s n) * (y * z))):Bool
[3]: ((((n * y) * z) + (y * z)) = ((s n) * (y * z))):Bool
---> (((n * (y * z)) + (y * z)) = ((s n) * (y * z))):Bool
[4]: (((y * z) + (n * (y * z))) = ((s n) * (y * z))):Bool
---> (((y * z) + (n * (y * z))) = ((y * z) + (n * (y * z)))):Bool
[5]: (((y * z) + (n * (y * z))) = ((n * (y * z)) + (y * z))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0010 sec for 5 rewrites + 80 matches)

--> QED
--> ================================================================
--> ----------------------------------------------------------------
--> Peano style natural numbers with assoc+comm _+_, _*_
--> which satisfy distributive law
--> ----------------------------------------------------------------
-- defining module! PNAT*ac.._...* done.

--> ----------------------------------------------------------------
--> factorial functions on Nat.PNAT*ac
--> ----------------------------------------------------------------
-- defining module! FACT..._....* done.

--> ================================================================
--> proof score for the property:
-->   eq fact2(N1:Nat,N2:Nat) = fact1(N1) * N2 .
--> with the induction on N1:Nat
--> ----------------------------------------------------------------
--> induction base
-- opening module FACT.. done.
_-- reduce in %FACT : (fact2(0,n2) = (fact1(0) * n2)):Bool
[1]: (fact2(0,n2) = (fact1(0) * n2)):Bool
---> (n2 = (fact1(0) * n2)):Bool
[2]: (n2 = (fact1(0) * n2)):Bool
---> (n2 = ((s 0) * n2)):Bool
[3]: (n2 = ((s 0) * n2)):Bool
---> (n2 = (n2 + (0 * n2))):Bool
[4]: (n2 = (n2 + (0 * n2))):Bool
---> (n2 = (n2 + 0)):Bool
[5]: (n2 = (n2 + 0)):Bool
---> (n2 = n2):Bool
[6]: (n2 = n2):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0000 sec for 6 rewrites + 8 matches)

--> induction step
-- opening module FACT.. done.
__-- reduce in %FACT : (fact2((s n1),n2) = (fact1((s n1)) * n2)):Bool
[1]: (fact2((s n1),n2) = (fact1((s n1)) * n2)):Bool
---> (fact2(n1,((s n1) * n2)) = (fact1((s n1)) * n2)):Bool
[2]: (fact2(n1,((s n1) * n2)) = (fact1((s n1)) * n2)):Bool
---> ((fact1(n1) * ((s n1) * n2)) = (fact1((s n1)) * n2)):Bool
[3]: ((fact1(n1) * ((s n1) * n2)) = (fact1((s n1)) * n2)):Bool
---> ((fact1(n1) * (n2 + (n1 * n2))) = (fact1((s n1)) * n2)):Bool
[4]: ((fact1(n1) * ((n2 * n1) + n2)) = (fact1((s n1)) * n2)):Bool
---> (((fact1(n1) * (n2 * n1)) + (fact1(n1) * n2)) = (fact1((s n1)) * n2)):Bool
[5]: (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = (fact1((s n1)) * n2)):Bool
---> (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = (((s n1) * fact1(n1)) * n2)):Bool
[6]: (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = (((s n1) * fact1(n1)) * n2)):Bool
---> (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = ((fact1(n1) + (n1 * fact1(n1))) * n2)):Bool
[7]: (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = (((fact1(n1) * n1) + fact1(n1)) * n2)):Bool
---> (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = ((n2 * fact1(n1)) + (n2 * (fact1(n1) * n1)))):Bool
[8]: (((n2 * fact1(n1)) + (n2 * (n1 * fact1(n1)))) = ((fact1(n1) * (n1 * n2)) + (fact1(n1) * n2))):Bool
---> (true):Bool
(true):Bool
(0.0000 sec for parse, 0.0020 sec for 8 rewrites + 99 matches)

--> QED
--> ================================================================
--> ----------------------------------------------------------------
--> set trace whole off
--> ****************************************************************
--> ****************************************************************
--> end of file
PNAT*>