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Require Import ZArith.
Require Import Lia Psatz.
Open Scope Z_scope.
(* From fiat-crypto Generalized.v *)
Goal forall (x1 : Z) (x2 : Z) (x3 : Z) (x4 : Z) (x5 : Z) (x6 : Z) (x7 : Z) (x8 : Z) (x9 : Z) (x10 : Z) (x11 : Z) (x12 : Z) (x13 : Z) (x14 : Z) (x15 : Z) (x16 : Z) (x17 : Z) (x18 : Z)
(H0 : -1 + -x1^2 + x3*x5 + x1^2*x2 + -x2*x3*x4 >= 0)
(H1 : -1 + x4 >= 0)
(H2 : -1 + x6 >= 0)
(H3 : -1 + -x4 + x1 >= 0)
(H4 : x3 + -x7 = 0)
(H5 : x8 >= 0)
(H6 : -1 + x4 >= 0)
(H7 : x9 >= 0)
(H8 : -x8 + x10 >= 0)
(H9 : -1 + x1^2 + -x9 >= 0)
(H10 : x4 + -x11 >= 0)
(H11 : -x3 + x1*x12 + -x12*x13 >= 0)
(H12 : -1 + -x9 + x1*x4 >= 0)
(H13 : -1 + x4 + -x13 >= 0)
(H14 : x13 >= 0)
(H15 : -1 + x5 >= 0)
(H16 : -1 + x1 + -x2 >= 0)
(H17 : x1^2 + -x13 + -x3*x4 = 0)
(H18 : -1 + x12 + -x14 >= 0)
(H19 : x14 >= 0)
(H20 : x1 + -x14 + -x5*x12 = 0)
(H21 : -1 + x4 + -x15 >= 0)
(H22 : x15 >= 0)
(H23 : x9 + -x15 + -x2*x4 = 0)
(H24 : -x9 + x16 + x4*x17 = 0)
(H25 : x17 + -x18 = 0)
, False
.
Proof.
intros.
Time nia.
Qed.
Goal
forall (x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
x14 x15 x16 : Z)
(H6 : x8 < x10 ^ 2 * x15 ^ 2 + 2 * x10 * x15 * x14 + x14 ^ 2)
(H7 : 0 <= x8)
(H12 : 0 <= x14)
(H0 : x8 = x15 * x11 + x9)
(H14 : x10 ^ 2 * x15 + x10 * x14 < x16)
(H17 : x16 <= 0)
(H15 : 0 <= x9)
(H18 : x9 < x15)
(H16 : 0 <= x12)
(H19 : x12 < (x10 * x15 + x14) * x10)
, False.
Proof.
intros.
Time nia.
Qed.
(* From fiat-crypto Toplevel1.v *)
Goal forall
(x1 x2 x3 x4 x5 x7 : Z)
(H0 : x1 + x2 - x3 = 0) (* substitute x1, nothing happens *)
(H1 : 2 * x2 - x4 - 1 >= 0)
(H2 : - x2 + x4 >= 0)
(H3 : 2 * x2 - x5 - 1 >= 0)
(H5 : x2 - 4 >= 0)
(H7 : - x2 * x7 + x4 * x5 >= 0)
(H6 : x2 * x7 + x2 - x4 * x5 - 1 >= 0)
(H9 : x7 - x2 ^ 2 >= 0), (* x2^2 is *visibly* positive *)
False.
Proof.
intros.
nia.
Qed.
Goal forall
(x1 x2 x3 x4 x5 x7 : Z)
(H0 : x2 + x1 - x3 = 0) (* substitute x2= x3 -x1 ... *)
(H1 : 2 * x2 - x4 - 1 >= 0)
(H2 : - x2 + x4 >= 0)
(H3 : 2 * x2 - x5 - 1 >= 0)
(H5 : x2 - 4 >= 0)
(H7 : - x2 * x7 + x4 * x5 >= 0)
(H6 : x2 * x7 + x2 - x4 * x5 - 1 >= 0)
(H9 : x7 - x2 ^ 2 >= 0), (* (x3 - x1)^2 is not visibly positive *)
False.
Proof.
intros.
nia.
Qed.
(* From bedrock2 FlatToRisc.v *)
(* Variant of the following - omega fails (bad linearisation?)*)
Goal forall
(PXLEN XLEN r : Z)
(q q0 r0 a : Z)
(H3 : 4 * a = 4 * PXLEN * q0 + (4 * q + r))
(H6 : 0 <= 4 * q + r)
(H7 : 4 * q + r < 4 * PXLEN)
(H8 : r <= 3)
(H4 : r >= 1),
False.
Proof.
intros.
Time lia.
Qed.
Goal forall
(PXLEN XLEN r : Z)
(q q0 r0 a : Z)
(H3 : 4 * a = 4 * PXLEN * q0 + (4 * q + r))
(H6 : 0 <= 4 * q + r)
(H7 : 4 * q + r < 4 * PXLEN)
(H8 : r <= 3)
(H4 : r >= 1),
False.
Proof.
intros.
Time nia.
Qed.
(** Very slow *)
Goal forall
(XLEN r : Z)
(H : 4 < 2 ^ XLEN)
(H0 : 8 <= XLEN)
(q q0 r0 a : Z)
(H3 : 4 * a = 4 * 2 ^ (XLEN - 2) * q0 + r0)
(H5 : r0 = 4 * q + r)
(H6 : 0 <= r0)
(H7 : r0 < 4 * 2 ^ (XLEN - 2))
(H2 : 0 <= r)
(H8 : r < 4)
(H4 : r > 0)
(H9 : 0 < 2 ^ (XLEN - 2)),
False.
Proof.
intros.
Time nia.
Qed.
Goal forall
(XLEN r : Z)
(R : r > 0 \/ r < 0)
(H : 4 < 2 ^ XLEN)
(H0 : 8 <= XLEN)
(H1 : ~ (0 <= XLEN - 2) \/ 0 < 2 ^ (XLEN - 2))
(q q0 r0 a : Z)
(H2 : 0 <= r0 < 4 * 2 ^ (XLEN - 2))
(H3 : 4 * a = 4 * 2 ^ (XLEN - 2) * q0 + r0)
(H4 : 0 <= r < 4)
(H5 : r0 = 4 * q + r),
False.
Proof.
intros.
Time nia.
Qed.
Goal forall
(XLEN r : Z)
(R : r > 0 \/ r < 0)
(H : 4 < 2 ^ XLEN)
(H0 : 8 <= XLEN)
(H1 : ~ (0 <= XLEN - 2) \/ 0 < 2 ^ (XLEN - 2))
(q q0 r0 a : Z)
(H2 : 0 <= r0 < 4 * 2 ^ (XLEN - 2))
(H3 : 4 * a = 4 * 2 ^ (XLEN - 2) * q0 + r0)
(H4 : 0 <= r < 4)
(H5 : r0 = 4 * q + r),
False.
Proof.
intros.
intuition idtac.
Time all:nia.
Qed.
Goal forall
(XLEN a q q0 z : Z)
(HR : 4 * a - 4 * z * q0 - 4 * q > 0)
(H0 : 8 <= XLEN)
(H1 : 0 < z)
(H : 0 <= 4 * a - 4 * z * q0 - 4 * q)
(H3 : 4 * a - 4 * z * q0 - 4 * q < 4)
(H4 : 4 * a - 4 * z * q0 < 4 * z),
False.
Proof.
intros.
Time nia.
Qed.
(* From fiat-crypto Modulo.v *)
Goal forall (b : Z)
(H : 0 <> b)
(c r q1 q2 r2 : Z)
(H2 : r2 < c)
(q0 : Z)
(H7 : r < b)
(H5 : 0 <= r)
(H6 : r < b)
(H12 : 0 < c)
(H13 : 0 <> c)
(H0 : 0 <> c * b)
(H1 : 0 <= r2)
(H14 : 0 <= q0)
(H9 : c * q1 + q0 = c * q2 + r2)
(H4 : 0 <= b * q0 + - r)
(H10 : b * q0 + - r < c * b),
q1 = q2.
Proof.
intros.
Fail nia.
Abort.
(* From Sozeau's plugin Equations *)
Goal forall x p2 p1 m,
x <> 0%Z ->
(Z.abs (x * p2 ) > Z.abs (Z.abs p1 + Z.abs m))%Z ->
(Z.abs (x * (p1 + x * p2 )) > Z.abs m)%Z.
Proof.
intros.
Time nia.
Qed.
Goal forall z z0 z1 m
(Heqz0 : z0 = ((1 + z) * z1)%Z)
(H0 : (0 <= m)%Z)
(H3 : z = m)
(H1 : (0 <= z0)%Z)
(H4 : z1 = z0)
(H2 : (z1 > 0)%Z),
(z1 > z)%Z.
Proof.
intros.
Time nia.
Qed.
(* Known issues.
- Found proof may violate Proof using ...
There may be a compliant proof but lia has no way to know.
Proofs could be optimised to minimise the number of hypotheses,
but this seems to costly.
Section S.
Variable z z0 z1 z2 : Z.
Variable H2 : 0 <= z2.
Variable H3 : z2 < z1.
Variable H4 : 0 <= z0.
Variable H5 : z0 < z1.
Variable H6 : z = - z2.
Goal -z1 -z2 >= 0 -> False.
Proof using H2 H3 H6.
intros.
lia.
Qed.
*)
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