CafeOBJ> in pnat-ps-citp.cafe 

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

--> ****************************************************************
--> proof scores for associativity and commutativity of
--> _+_ and _*_ over Nat.PNAT with SpecCalc/CITP
--> ****************************************************************
--> -----------------------------------------------------------------
--> 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 the addition _+_
--> ----------------------------------------------------------------
-- defining module! PNAT+.._..* done.

--> ================================================================
--> proof score for proving right 0 of _+_:
-->   eq[+r0]: X:Nat + 0 = X .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
--> proof with CITP
-- reading in file  : int
-- reading in file  : nat
-- reading in file  : nznat

-- done reading in file: nznat

-- done reading in file: nat

-- done reading in file: int

:goal { ** root -----------------------------------------
  -- context module: PNAT+
  -- sentence to be proved
    eq [+r0]: X:Nat.PNAT + 0 = X .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
[rd]=> :goal{1}
[rd] discharged: 
  eq [+r0]: 0 + 0 = 0
[rd] discharged goal "1".
[tc]=> :goal{2}
[rd]=> :goal{2}
[rd] discharged: 
  eq [+r0]: s X#Nat + 0 = s X#Nat
[rd] discharged goal "2".
(consumed 0.0130 sec, including 5 rewrites + 9 matches)
** All goals are successfully discharged.
--> QED
--> ================================================================
--> ================================================================
--> proof score for proving right s_ of _+_:
-->   eq[+rs]: X:Nat + s Y:Nat = s (X + Y) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
:goal { ** root -----------------------------------------
  -- context module: PNAT+
  -- sentence to be proved
    eq [+rs]: X:Nat.PNAT + s Y:Nat.PNAT = s (X + Y) .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC +rs]: 0 + s Y@Nat = s (0 + Y@Nat)
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC +rs]: s X#Nat + s Y@Nat = s (s X#Nat + Y@Nat)
[rd] discharged goal "2-1".
(consumed 0.0290 sec, including 7 rewrites + 25 matches)
** All goals are successfully discharged.
--> QED
--> ================================================================
--> ================================================================
--> proof score for proving commutativity of _+_:
-->   eq (X:Nat + Y:Nat) = (Y + X) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
-- defining module PNAT+comm._..* done.
:goal { ** root -----------------------------------------
  -- context module: PNAT+comm
  -- sentence to be proved
    eq [+comm]: X:Nat.PNAT + Y:Nat.PNAT = Y + X .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC +comm]: 0 + Y@Nat = Y@Nat + 0
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC +comm]: s X#Nat + Y@Nat = Y@Nat + s X#Nat
[rd] discharged goal "2-1".
(consumed 0.0280 sec, including 7 rewrites + 37 matches)
** All goals are successfully discharged.

--> QED
--> ================================================================
--> ================================================================
--> proof score for proving associativity of _+_:
-->   eq[+assoc]: (X:Nat + Y:Nat) + Z:Nat = X + (Y + Z) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
:goal { ** root -----------------------------------------
  -- context module: PNAT+
  -- sentence to be proved
    eq [+assoc]: (X:Nat.PNAT + Y:Nat.PNAT) + Z:Nat.PNAT
        = X + (Y + Z) .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC +assoc]: (0 + Y@Nat) + Z@Nat = 0 + (Y@Nat + Z@Nat)
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC +assoc]: (s X#Nat + Y@Nat) + Z@Nat
      = s X#Nat + (Y@Nat + Z@Nat)
[rd] discharged goal "2-1".
(consumed 0.0280 sec, including 8 rewrites + 92 matches)
** All goals are successfully discharged.
--> 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[*r0] X:Nat * 0 = X .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
:goal { ** root -----------------------------------------
  -- context module: PNAT*
  -- sentence to be proved
    eq [*r0]: X:Nat.PNAT * 0 = 0 .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
[rd]=> :goal{1}
[rd] discharged: 
  eq [*r0]: 0 * 0 = 0
[rd] discharged goal "1".
[tc]=> :goal{2}
[rd]=> :goal{2}
[rd] discharged: 
  eq [*r0]: s X#Nat * 0 = 0
[rd] discharged goal "2".
(consumed 0.0180 sec, including 6 rewrites + 10 matches)
** All goals are successfully discharged.
--> QED
--> ================================================================
--> ================================================================
:goal { ** root -----------------------------------------
  -- context module: PNAT*
  -- sentence to be proved
    eq [*rs]: X:Nat.PNAT * s Y:Nat.PNAT = X + X * Y .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC *rs]: 0 * s Y@Nat = 0 + 0 * Y@Nat
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC *rs]: s X#Nat * s Y@Nat = s X#Nat + s X#Nat * Y@Nat
[rd] discharged goal "2-1".
(consumed 0.0270 sec, including 10 rewrites + 78 matches)
** All goals are successfully discharged.
--> QED
--> ================================================================
--> ================================================================
--> proof score for proving commutativity of _*_:
-->   eq[*comm]: (X:Nat * Y:Nat) = (Y * X) .
--> with the induction on X:Nat
--> -----------------------------------------------------------------
-- defining module PNAT*comm._..* done.
:goal { ** root -----------------------------------------
  -- context module: PNAT*comm
  -- sentence to be proved
    eq [*comm]: X:Nat.PNAT * Y:Nat.PNAT = Y * X .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC *comm]: 0 * Y@Nat = Y@Nat * 0
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC *comm]: s X#Nat * Y@Nat = Y@Nat * s X#Nat
[rd] discharged goal "2-1".
(consumed 0.0300 sec, including 7 rewrites + 53 matches)
** All goals are successfully discharged.

--> QED
--> ================================================================
--> ================================================================
--> proof score for proving distributivity of _*_ over _+_
--> from right:
-->   eq[*distr] (X:Nat + Y:Nat) * Z:Nat = X * Z + Y * Z .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
:goal { ** root -----------------------------------------
  -- context module: PNAT*
  -- sentence to be proved
    eq [*distr]: (X:Nat.PNAT + Y:Nat.PNAT) * Z:Nat.PNAT
        = X * Z + Y * Z .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC *distr]: (0 + Y@Nat) * Z@Nat = 0 * Z@Nat + Y@Nat * Z@Nat
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC *distr]: (s X#Nat + Y@Nat) * Z@Nat
      = s X#Nat * Z@Nat + Y@Nat * Z@Nat
[rd] discharged goal "2-1".
(consumed 0.0280 sec, including 9 rewrites + 125 matches)
** All goals are successfully discharged.
--> QED
--> ================================================================
--> ================================================================
--> proof score for proving associativity of _*_:
-->   eq[*assoc]: (X:Nat * Y:Nat) * Z:Nat = X * (Y * Z) .
--> with the induction on X:Nat
--> ----------------------------------------------------------------
-- defining module PNAT*assoc._.* done.
:goal { ** root -----------------------------------------
  -- context module: PNAT*assoc
  -- sentence to be proved
    eq [*assoc]: (X:Nat.PNAT * Y:Nat.PNAT) * Z:Nat.PNAT
        = X * (Y * Z) .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on X:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC *assoc]: (0 * Y@Nat) * Z@Nat = 0 * (Y@Nat * Z@Nat)
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC *assoc]: (s X#Nat * Y@Nat) * Z@Nat
      = s X#Nat * (Y@Nat * Z@Nat)
[rd] discharged goal "2-1".
(consumed 0.0300 sec, including 9 rewrites + 137 matches)
** All goals are successfully discharged.

--> 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[f2f1]: fact2(N1:Nat,N2:Nat) = fact1(N1) * N2 .
--> with the induction on N1:Nat
--> ----------------------------------------------------------------
:goal { ** root -----------------------------------------
  -- context module: FACT
  -- sentence to be proved
    eq [f2f1]: fact2(N1:Nat.PNAT, N2:Nat.PNAT)
        = fact1(N1) * N2 .
}
** Initial goal (root) is generated. **
**> Induction will be conducted on N1:Nat 
[si]=> :goal{root}
** Generated 2 goals
[tc]=> :goal{1}
** Generated 1 goal
[rd]=> :goal{1-1}
[rd] discharged: 
  eq [TC f2f1]: fact2(0, N2@Nat) = fact1(0) * N2@Nat
[rd] discharged goal "1-1".
[tc]=> :goal{2}
** Generated 1 goal
[rd]=> :goal{2-1}
[rd] discharged: 
  eq [TC f2f1]: fact2(s N1#Nat, N2@Nat) = fact1(s N1#Nat) * N2@Nat
[rd] discharged goal "2-1".
(consumed 0.0320 sec, including 14 rewrites + 159 matches)
** All goals are successfully discharged.
--> QED
--> ================================================================
--> ****************************************************************
--> end of file
FACT>