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Add knowledge about floor/ceil bounds #718

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Jul 7, 2023
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Add knowledge about floor/ceil bounds #718

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28 changes: 28 additions & 0 deletions src/preludes/ria.ae
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Expand Up @@ -56,6 +56,34 @@ theory Real_of_Int extends RIA =
{i <= real_of_int(x)}.
k <= x

(* floor(x) ≤ i iff x < i + 1 *)
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I guess you meant floor(x) <= i implies x < i+1 instead of floor(x) <= i iff x < i+1.
Besides, I don't understand why you need to use the semantic trigger not_theory_constant here?

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floor(x) <= i iff x < i + 1 is correct, unless I am mistaken? Even though we only implement the implication.

not_theory_constant is here to avoid adding useless axioms, e.g. 3 < 4 if we know that floor(3.5) <= 3

axiom int_floor_ub:
forall x, y : real, i : int
[ int_floor(x), not_theory_constant(x), int_floor(x) in ]?, i], y |-> real_of_int(i + 1) ]
{ int_floor(x) <= i }.
x < y

(* i <= floor(x) iff i <= x *)
axiom int_floor_lb:
forall x, y : real, i : int
[ int_floor(x), not_theory_constant(x), int_floor(x) in [i, ?[, y |-> real_of_int(i) ]
{ i <= int_floor(x) }.
y <= x

(* ceil(x) ≤ i iff x ≤ i *)
axiom int_ceil_ub:
forall x, y : real, i : int
[ int_ceil(x), not_theory_constant(x), int_ceil(x) in ]?, i], y |-> real_of_int(i) ]
{ int_ceil(x) <= i }.
x <= y

(* i <= ceil(x) iff i - 1 < x *)
axiom int_ceil_lb:
forall x, y : real, i : int
[ int_ceil(x), not_theory_constant(x), int_ceil(x) in [i, ?[, y |-> real_of_int(i - 1) ]
{ i <= int_ceil(x) }.
y < x

(* can add other axioms on strict ineqs on rationals ? *)

end
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2 changes: 2 additions & 0 deletions tests/arith/ceil_floor_propagate.ae
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@@ -0,0 +1,2 @@
logic x: real
goal g: int_ceil(x) = 0 and int_floor(x) = 0 -> x = 0.
2 changes: 2 additions & 0 deletions tests/arith/ceil_floor_propagate.expected
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@@ -0,0 +1,2 @@

unknown
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