// This qualifier saves the day; solve constraints WITHOUT IT
// qualif Zog(v:a): (10 <= v)

// But you may use this one
qualif Pog(v:a): (0 <= v)




/* 

-- Version 1 (eliminate k1) --

Rewriting constraints as:

    id 0
    v:int |- (v=0) => k1

    id 3
    v:int |- k0 => k1

Projecting out all variables NOT in the WF of k1

    N/A

Take the \/ of all constraints on k1

    k1 = (v=0) \/ k0

So you get:

  bind 0 x: {v: int | v = 10      }
  bind 1 a: {v: int | (v=0) \/ k0 }
  bind 2 y: {v: int | v = 20      }
  bind 3 b: {v: int | (v=0) \/ k0 }
  bind 4 c: {v: int | k0          }

  constraint:
    env [ 0; 1]
      lhs {v : int | v = x + a}
    rhs {v : int | k0}
    id 1 

  constraint:
    env [2; 3]
      lhs {v : int | v = y + b}
    rhs {v : int | k0}
    id 2 

  constraint:
    env [4]
      lhs {v : int | v = c  }
    rhs {v : int | 10 <= v}
    id 4 

  wf:
    env [ ]
    reft {v: int | k1}




-- Version 2 (eliminate k0) --

Rewriting constraints as:

    id 1
    x:int, a:int, v:int |- (v=10)[x/v] /\ k1[a/v] /\ (v=x+a) => k0
    x:int, a:int, v:int |- (x=10) /\ k1[a/v] /\ (v=x+a) => k0

    id 2
    y:int, b:int, v:int |- (v=20)[y/v] /\ k1[b/v] /\ (v=y+b) => k0
    y:int, b:int, v:int |- (y=20) /\ k1[b/v] /\ (v=y+b) => k0

Projecting out all variables NOT in the WF of k0

    id 1
    v:int |- (exists x:int a:int. (x=10) /\ k1[a/v] /\ (v=x+a)) => k0
    
    id 2
    v:int |- (exists y:int b:int. (y=20) /\ k1[b/v] /\ (v=y+b)) => k0

Take the \/ of all constraints on k0

    k0 = (exists x:int a:int. (x=10) /\ k1[a/v] /\ (v=x+a))
      \/ (exists y:int b:int. (y=20) /\ k1[b/v] /\ (v=y+b))

So you get:

  bind 4 c: {v: int | (exists x:int a:int. (x=10) /\ k1[a/v] /\ (v=x+a))
                   \/ (exists y:int b:int. (y=20) /\ k1[b/v] /\ (v=y+b))    }

  constraint:
    env [ ]
      lhs {v : int | v = 0}
    rhs {v : int | k1 }
    id 0 


  constraint:
    env [ ]
      lhs {v : int | (exists x:int a:int. (x=10) /\ k1[a/v] /\ (v=x+a))
                \/ (exists y:int b:int. (y=20) /\ k1[b/v] /\ (v=y+b))}
    rhs {v : int | k1}
    id 3

  constraint:
    env [4]
      lhs {v : int | v = c  }
    rhs {v : int | 10 <= v}
    id 4