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extend results on ceil 'for free' #781

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extend results on ceil 'for free' #781

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16 changes: 13 additions & 3 deletions theories/datatypes/Real.ec
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Original file line number Diff line number Diff line change
Expand Up @@ -218,12 +218,16 @@ instance field with real

(* -------------------------------------------------------------------- *)
op floor : real -> int.
op ceil : real -> int.
op [opaque] ceil : real -> int = fun (x : real)=> -(floor (Self.([-]) x)).

axiom floor_bound (x:real) : x - 1%r < (floor x)%r <= x.
axiom ceil_bound (x:real) : x <= (ceil x)%r < x + 1%r.
axiom from_int_floor n : floor n%r = n.
axiom from_int_ceil n : ceil n%r = n.

lemma ceil_bound (x:real) : x <= (ceil x)%r < x + 1%r.
proof. by rewrite /ceil; smt(floor_bound). qed.

lemma from_int_ceil n : ceil n%r = n.
proof. by rewrite /ceil -fromintN from_int_floor oppzK. qed.

lemma floor_gt x : x - 1%r < (floor x)%r.
proof. by case: (floor_bound x). qed.
Expand All @@ -246,6 +250,12 @@ proof. smt(floor_bound). qed.
lemma from_int_floor_addr n x : floor (x + n%r) = floor x + n.
proof. smt(floor_bound). qed.

lemma from_int_ceil_addl n x : ceil (n%r + x) = n + ceil x.
proof. smt(ceil_bound). qed.

lemma from_int_ceil_addr n x : ceil (x + n%r) = ceil x + n.
proof. smt(ceil_bound). qed.

lemma floor_mono (x y : real) : x <= y => floor x <= floor y.
proof. smt(floor_bound). qed.

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