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|
formula 1 is: true \/ false
formula 2 is: A /\ B -> A
task 1 is:
theory Task
goal goal1 : true \/ false
end
task 2 created:
theory Task
predicate A
predicate B
goal goal2 : A /\ B -> A
end
Versions of Alt-Ergo found: <hidden>
On task 1, Alt-Ergo answers Valid (<hidden>s, <hidden> steps)
On task 2, Alt-Ergo answers Valid in <hidden> seconds, <hidden> steps
task 3 created
On task 3, Alt-Ergo answers Valid (<hidden>s, <hidden> steps)
On task 4, Alt-Ergo answers Valid (<hidden>s, <hidden> steps)
creating theory 'My_theory'
adding goal 1
adding goal 2
adding goal 3
adding goal 4
my new theory is as follows:
theory My_theory
(* use why3.BuiltIn.BuiltIn *)
goal goal1 : true \/ false
predicate A
predicate B
goal goal2 : A /\ B -> A
(* use int.Int *)
goal goal3 : (2 + 2) = 4
goal goal4 : forall x:int. (x * x) >= 0
end
Tasks are:
== Task 1 ==
theory Task
type int
type real
type string
predicate (=) 'a 'a
(* use why3.BuiltIn.BuiltIn *)
predicate A
predicate B
type bool =
| True
| False
(* use why3.Bool.Bool *)
type tuple0 =
| Tuple0
(* use why3.Tuple0.Tuple01 *)
type unit = unit
(* use why3.Unit.Unit *)
constant zero : int = 0
constant one : int = 1
function (-_) int : int
function (+) int int : int
function ( * ) int int : int
predicate (<) int int
function (-) (x:int) (y:int) : int = x + (- y)
predicate (>) (x:int) (y:int) = y < x
predicate (<=) (x:int) (y:int) = x < y \/ x = y
predicate (>=) (x:int) (y:int) = y <= x
axiom Assoc : forall x:int, y:int, z:int. ((x + y) + z) = (x + (y + z))
(* clone algebra.Assoc with type t = int, function op = (+),
prop Assoc1 = Assoc, *)
axiom Unit_def_l : forall x:int. (zero + x) = x
axiom Unit_def_r : forall x:int. (x + zero) = x
(* clone algebra.Monoid with type t1 = int, constant unit = zero,
function op1 = (+), prop Unit_def_r1 = Unit_def_r,
prop Unit_def_l1 = Unit_def_l, prop Assoc2 = Assoc, *)
axiom Inv_def_l : forall x:int. ((- x) + x) = zero
axiom Inv_def_r : forall x:int. (x + (- x)) = zero
(* clone algebra.Group with type t2 = int, function inv = (-_),
constant unit1 = zero, function op2 = (+), prop Inv_def_r1 = Inv_def_r,
prop Inv_def_l1 = Inv_def_l, prop Unit_def_r2 = Unit_def_r,
prop Unit_def_l2 = Unit_def_l, prop Assoc3 = Assoc, *)
axiom Comm : forall x:int, y:int. (x + y) = (y + x)
(* clone algebra.Comm with type t3 = int, function op3 = (+),
prop Comm1 = Comm, *)
(* meta AC function (+) *)
(* clone algebra.CommutativeGroup with type t4 = int, function inv1 = (-_),
constant unit2 = zero, function op4 = (+), prop Comm2 = Comm,
prop Inv_def_r2 = Inv_def_r, prop Inv_def_l2 = Inv_def_l,
prop Unit_def_r3 = Unit_def_r, prop Unit_def_l3 = Unit_def_l,
prop Assoc4 = Assoc, *)
axiom Assoc5 : forall x:int, y:int, z:int. ((x * y) * z) = (x * (y * z))
(* clone algebra.Assoc with type t = int, function op = ( * ),
prop Assoc1 = Assoc5, *)
axiom Mul_distr_l :
forall x:int, y:int, z:int. (x * (y + z)) = ((x * y) + (x * z))
axiom Mul_distr_r :
forall x:int, y:int, z:int. ((y + z) * x) = ((y * x) + (z * x))
(* clone algebra.Ring with type t5 = int, function ( *') = ( * ),
function (-'_) = (-_), function (+') = (+), constant zero1 = zero,
prop Mul_distr_r1 = Mul_distr_r, prop Mul_distr_l1 = Mul_distr_l,
prop Assoc6 = Assoc5, prop Comm3 = Comm, prop Inv_def_r3 = Inv_def_r,
prop Inv_def_l3 = Inv_def_l, prop Unit_def_r4 = Unit_def_r,
prop Unit_def_l4 = Unit_def_l, prop Assoc7 = Assoc, *)
axiom Comm4 : forall x:int, y:int. (x * y) = (y * x)
(* clone algebra.Comm with type t3 = int, function op3 = ( * ),
prop Comm1 = Comm4, *)
(* meta AC function ( * ) *)
(* clone algebra.CommutativeRing with type t6 = int,
function ( *'') = ( * ), function (-''_) = (-_), function (+'') = (+),
constant zero2 = zero, prop Comm5 = Comm4,
prop Mul_distr_r2 = Mul_distr_r, prop Mul_distr_l2 = Mul_distr_l,
prop Assoc8 = Assoc5, prop Comm6 = Comm, prop Inv_def_r4 = Inv_def_r,
prop Inv_def_l4 = Inv_def_l, prop Unit_def_r5 = Unit_def_r,
prop Unit_def_l5 = Unit_def_l, prop Assoc9 = Assoc, *)
axiom Unitary : forall x:int. (one * x) = x
axiom NonTrivialRing : not zero = one
(* clone algebra.UnitaryCommutativeRing with type t7 = int,
constant one1 = one, function ( *''') = ( * ), function (-'''_) = (-_),
function (+''') = (+), constant zero3 = zero,
prop NonTrivialRing1 = NonTrivialRing, prop Unitary1 = Unitary,
prop Comm7 = Comm4, prop Mul_distr_r3 = Mul_distr_r,
prop Mul_distr_l3 = Mul_distr_l, prop Assoc10 = Assoc5,
prop Comm8 = Comm, prop Inv_def_r5 = Inv_def_r,
prop Inv_def_l5 = Inv_def_l, prop Unit_def_r6 = Unit_def_r,
prop Unit_def_l6 = Unit_def_l, prop Assoc11 = Assoc, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Refl : forall x:int. x <= x
(* clone relations.Reflexive with type t9 = int, predicate rel1 = (<=),
prop Refl1 = Refl, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Trans : forall x:int, y:int, z:int. x <= y -> y <= z -> x <= z
(* clone relations.Transitive with type t10 = int, predicate rel2 = (<=),
prop Trans1 = Trans, *)
(* clone relations.PreOrder with type t11 = int, predicate rel3 = (<=),
prop Trans2 = Trans, prop Refl2 = Refl, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Antisymm : forall x:int, y:int. x <= y -> y <= x -> x = y
(* clone relations.Antisymmetric with type t12 = int,
predicate rel4 = (<=), prop Antisymm1 = Antisymm, *)
(* clone relations.PartialOrder with type t13 = int, predicate rel5 = (<=),
prop Antisymm2 = Antisymm, prop Trans3 = Trans, prop Refl3 = Refl, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Total : forall x:int, y:int. x <= y \/ y <= x
(* clone relations.Total with type t14 = int, predicate rel6 = (<=),
prop Total1 = Total, *)
(* clone relations.TotalOrder with type t15 = int, predicate rel7 = (<=),
prop Total2 = Total, prop Antisymm3 = Antisymm, prop Trans4 = Trans,
prop Refl4 = Refl, *)
axiom ZeroLessOne : zero <= one
axiom CompatOrderAdd :
forall x:int, y:int, z:int. x <= y -> (x + z) <= (y + z)
axiom CompatOrderMult :
forall x:int, y:int, z:int. x <= y -> zero <= z -> (x * z) <= (y * z)
(* meta remove_unused:dependency prop CompatOrderMult, function ( * ) *)
(* clone algebra.OrderedUnitaryCommutativeRing with type t16 = int,
predicate (<=') = (<=), constant one2 = one, function ( *'''') = ( * ),
function (-''''_) = (-_), function (+'''') = (+), constant zero4 = zero,
prop CompatOrderMult1 = CompatOrderMult,
prop CompatOrderAdd1 = CompatOrderAdd, prop ZeroLessOne1 = ZeroLessOne,
prop Total3 = Total, prop Antisymm4 = Antisymm, prop Trans5 = Trans,
prop Refl5 = Refl, prop NonTrivialRing2 = NonTrivialRing,
prop Unitary2 = Unitary, prop Comm9 = Comm4,
prop Mul_distr_r4 = Mul_distr_r, prop Mul_distr_l4 = Mul_distr_l,
prop Assoc12 = Assoc5, prop Comm10 = Comm, prop Inv_def_r6 = Inv_def_r,
prop Inv_def_l6 = Inv_def_l, prop Unit_def_r7 = Unit_def_r,
prop Unit_def_l7 = Unit_def_l, prop Assoc13 = Assoc, *)
(* meta remove_unused:keep function (+) *)
(* meta remove_unused:keep function (-) *)
(* meta remove_unused:keep function (-_) *)
(* meta remove_unused:keep predicate (<) *)
(* meta remove_unused:keep predicate (<=) *)
(* meta remove_unused:keep predicate (>) *)
(* meta remove_unused:keep predicate (>=) *)
(* use int.Int *)
goal goal4 : forall x:int. (x * x) >= 0
end
== Task 2 ==
theory Task
type int
type real
type string
predicate (=) 'a 'a
(* use why3.BuiltIn.BuiltIn *)
predicate A
predicate B
type bool =
| True
| False
(* use why3.Bool.Bool *)
type tuple0 =
| Tuple0
(* use why3.Tuple0.Tuple01 *)
type unit = unit
(* use why3.Unit.Unit *)
constant zero : int = 0
constant one : int = 1
function (-_) int : int
function (+) int int : int
function ( * ) int int : int
predicate (<) int int
function (-) (x:int) (y:int) : int = x + (- y)
predicate (>) (x:int) (y:int) = y < x
predicate (<=) (x:int) (y:int) = x < y \/ x = y
predicate (>=) (x:int) (y:int) = y <= x
axiom Assoc : forall x:int, y:int, z:int. ((x + y) + z) = (x + (y + z))
(* clone algebra.Assoc with type t = int, function op = (+),
prop Assoc1 = Assoc, *)
axiom Unit_def_l : forall x:int. (zero + x) = x
axiom Unit_def_r : forall x:int. (x + zero) = x
(* clone algebra.Monoid with type t1 = int, constant unit = zero,
function op1 = (+), prop Unit_def_r1 = Unit_def_r,
prop Unit_def_l1 = Unit_def_l, prop Assoc2 = Assoc, *)
axiom Inv_def_l : forall x:int. ((- x) + x) = zero
axiom Inv_def_r : forall x:int. (x + (- x)) = zero
(* clone algebra.Group with type t2 = int, function inv = (-_),
constant unit1 = zero, function op2 = (+), prop Inv_def_r1 = Inv_def_r,
prop Inv_def_l1 = Inv_def_l, prop Unit_def_r2 = Unit_def_r,
prop Unit_def_l2 = Unit_def_l, prop Assoc3 = Assoc, *)
axiom Comm : forall x:int, y:int. (x + y) = (y + x)
(* clone algebra.Comm with type t3 = int, function op3 = (+),
prop Comm1 = Comm, *)
(* meta AC function (+) *)
(* clone algebra.CommutativeGroup with type t4 = int, function inv1 = (-_),
constant unit2 = zero, function op4 = (+), prop Comm2 = Comm,
prop Inv_def_r2 = Inv_def_r, prop Inv_def_l2 = Inv_def_l,
prop Unit_def_r3 = Unit_def_r, prop Unit_def_l3 = Unit_def_l,
prop Assoc4 = Assoc, *)
axiom Assoc5 : forall x:int, y:int, z:int. ((x * y) * z) = (x * (y * z))
(* clone algebra.Assoc with type t = int, function op = ( * ),
prop Assoc1 = Assoc5, *)
axiom Mul_distr_l :
forall x:int, y:int, z:int. (x * (y + z)) = ((x * y) + (x * z))
axiom Mul_distr_r :
forall x:int, y:int, z:int. ((y + z) * x) = ((y * x) + (z * x))
(* clone algebra.Ring with type t5 = int, function ( *') = ( * ),
function (-'_) = (-_), function (+') = (+), constant zero1 = zero,
prop Mul_distr_r1 = Mul_distr_r, prop Mul_distr_l1 = Mul_distr_l,
prop Assoc6 = Assoc5, prop Comm3 = Comm, prop Inv_def_r3 = Inv_def_r,
prop Inv_def_l3 = Inv_def_l, prop Unit_def_r4 = Unit_def_r,
prop Unit_def_l4 = Unit_def_l, prop Assoc7 = Assoc, *)
axiom Comm4 : forall x:int, y:int. (x * y) = (y * x)
(* clone algebra.Comm with type t3 = int, function op3 = ( * ),
prop Comm1 = Comm4, *)
(* meta AC function ( * ) *)
(* clone algebra.CommutativeRing with type t6 = int,
function ( *'') = ( * ), function (-''_) = (-_), function (+'') = (+),
constant zero2 = zero, prop Comm5 = Comm4,
prop Mul_distr_r2 = Mul_distr_r, prop Mul_distr_l2 = Mul_distr_l,
prop Assoc8 = Assoc5, prop Comm6 = Comm, prop Inv_def_r4 = Inv_def_r,
prop Inv_def_l4 = Inv_def_l, prop Unit_def_r5 = Unit_def_r,
prop Unit_def_l5 = Unit_def_l, prop Assoc9 = Assoc, *)
axiom Unitary : forall x:int. (one * x) = x
axiom NonTrivialRing : not zero = one
(* clone algebra.UnitaryCommutativeRing with type t7 = int,
constant one1 = one, function ( *''') = ( * ), function (-'''_) = (-_),
function (+''') = (+), constant zero3 = zero,
prop NonTrivialRing1 = NonTrivialRing, prop Unitary1 = Unitary,
prop Comm7 = Comm4, prop Mul_distr_r3 = Mul_distr_r,
prop Mul_distr_l3 = Mul_distr_l, prop Assoc10 = Assoc5,
prop Comm8 = Comm, prop Inv_def_r5 = Inv_def_r,
prop Inv_def_l5 = Inv_def_l, prop Unit_def_r6 = Unit_def_r,
prop Unit_def_l6 = Unit_def_l, prop Assoc11 = Assoc, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Refl : forall x:int. x <= x
(* clone relations.Reflexive with type t9 = int, predicate rel1 = (<=),
prop Refl1 = Refl, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Trans : forall x:int, y:int, z:int. x <= y -> y <= z -> x <= z
(* clone relations.Transitive with type t10 = int, predicate rel2 = (<=),
prop Trans1 = Trans, *)
(* clone relations.PreOrder with type t11 = int, predicate rel3 = (<=),
prop Trans2 = Trans, prop Refl2 = Refl, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Antisymm : forall x:int, y:int. x <= y -> y <= x -> x = y
(* clone relations.Antisymmetric with type t12 = int,
predicate rel4 = (<=), prop Antisymm1 = Antisymm, *)
(* clone relations.PartialOrder with type t13 = int, predicate rel5 = (<=),
prop Antisymm2 = Antisymm, prop Trans3 = Trans, prop Refl3 = Refl, *)
(* clone relations.EndoRelation with type t8 = int, predicate rel = (<=),
*)
axiom Total : forall x:int, y:int. x <= y \/ y <= x
(* clone relations.Total with type t14 = int, predicate rel6 = (<=),
prop Total1 = Total, *)
(* clone relations.TotalOrder with type t15 = int, predicate rel7 = (<=),
prop Total2 = Total, prop Antisymm3 = Antisymm, prop Trans4 = Trans,
prop Refl4 = Refl, *)
axiom ZeroLessOne : zero <= one
axiom CompatOrderAdd :
forall x:int, y:int, z:int. x <= y -> (x + z) <= (y + z)
axiom CompatOrderMult :
forall x:int, y:int, z:int. x <= y -> zero <= z -> (x * z) <= (y * z)
(* meta remove_unused:dependency prop CompatOrderMult, function ( * ) *)
(* clone algebra.OrderedUnitaryCommutativeRing with type t16 = int,
predicate (<=') = (<=), constant one2 = one, function ( *'''') = ( * ),
function (-''''_) = (-_), function (+'''') = (+), constant zero4 = zero,
prop CompatOrderMult1 = CompatOrderMult,
prop CompatOrderAdd1 = CompatOrderAdd, prop ZeroLessOne1 = ZeroLessOne,
prop Total3 = Total, prop Antisymm4 = Antisymm, prop Trans5 = Trans,
prop Refl5 = Refl, prop NonTrivialRing2 = NonTrivialRing,
prop Unitary2 = Unitary, prop Comm9 = Comm4,
prop Mul_distr_r4 = Mul_distr_r, prop Mul_distr_l4 = Mul_distr_l,
prop Assoc12 = Assoc5, prop Comm10 = Comm, prop Inv_def_r6 = Inv_def_r,
prop Inv_def_l6 = Inv_def_l, prop Unit_def_r7 = Unit_def_r,
prop Unit_def_l7 = Unit_def_l, prop Assoc13 = Assoc, *)
(* meta remove_unused:keep function (+) *)
(* meta remove_unused:keep function (-) *)
(* meta remove_unused:keep function (-_) *)
(* meta remove_unused:keep predicate (<) *)
(* meta remove_unused:keep predicate (<=) *)
(* meta remove_unused:keep predicate (>) *)
(* meta remove_unused:keep predicate (>=) *)
(* use int.Int *)
goal goal3 : (2 + 2) = 4
end
== Task 3 ==
theory Task
type int
type real
type string
predicate (=) 'a 'a
(* use why3.BuiltIn.BuiltIn *)
predicate A
predicate B
goal goal2 : A /\ B -> A
end
== Task 4 ==
theory Task
type int
type real
type string
predicate (=) 'a 'a
(* use why3.BuiltIn.BuiltIn *)
goal goal1 : true \/ false
end
|
