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

