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Require Import ZArith.
Require Import Lia.
Section S.
Variables H1 H2 H3 H4 : True.
Lemma bug_9848 : True.
Proof using.
lia.
Qed.
End S.
Lemma concl_in_Type : forall (k : nat)
(H : (k < 0)%nat) (F : k < 0 -> Type),
F H.
Proof.
intros.
lia.
Qed.
Lemma bug_10707 : forall
(T : Type)
(t : nat -> Type)
(k : nat)
(default : T)
(arr : t 0 -> T)
(H : (k < 0)%nat) of_nat_lt,
match k with
| 0 | _ => default
end = arr (of_nat_lt H).
Proof.
intros.
lia.
Qed.
Axiom decompose_nat : nat -> nat -> nat.
Axiom inleft : forall {P}, {m : nat & P m} -> nat.
Axiom foo : nat.
Lemma bug_7886 : forall (x x0 : nat)
(e : 0 = x0 + S x)
(H : decompose_nat x 0 = inleft (existT (fun m : nat => 0 = m + S x) x0 e))
(x1 : nat)
(e0 : 0 = x1 + S (S x))
(H1 : decompose_nat (S x) 0 = inleft (existT (fun m : nat => 0 = m + S (S x)) x1 e0)),
False.
Proof.
intros.
lia.
Qed.
Lemma bug_8898 : forall (p : 0 < 0) (H: p = p), False.
Proof.
intros p H.
lia.
Qed.
Open Scope Z_scope.
Lemma two_x_eq_1 : forall x, 2 * x = 1 -> False.
Proof.
intros.
lia.
Qed.
Lemma two_x_y_eq_1 : forall x y, 2 * x + 2 * y = 1 -> False.
Proof.
intros.
lia.
Qed.
Lemma two_x_y_z_eq_1 : forall x y z, 2 * x + 2 * y + 2 * z= 1 -> False.
Proof.
intros.
lia.
Qed.
Lemma unused : forall x y, y >= 0 /\ x = 1 -> x = 1.
Proof.
intros x y.
lia.
Qed.
Lemma omega_nightmare : forall x y, 27 <= 11 * x + 13 * y <= 45 -> -10 <= 7 * x - 9 * y <= 4 -> False.
Proof.
intros ; intuition auto.
lia.
Qed.
Lemma compact_proof : forall z,
(z < 0) ->
(z >= 0) ->
(0 >= z \/ 0 < z) -> False.
Proof.
intros.
lia.
Qed.
Lemma dummy_ex : exists (x:Z), x = x.
Proof.
eexists.
lia.
Unshelve.
exact Z0.
Qed.
Lemma unused_concl : forall x,
False -> x > 0 -> x < 0.
Proof.
intro.
lia.
Qed.
Lemma unused_concl_match : forall (x:Z),
False -> match x with
| Z0 => True
| _ => x = x
end.
Proof.
intros.
lia.
Qed.
Lemma fresh : forall (__arith : Prop),
__arith -> True.
Proof.
intros.
lia.
Qed.
Lemma fresh1 : forall (__p1 __p2 __p3 __p5:Prop) (x y z:Z), (x = 0 /\ y = 0) /\ z = 0 -> x = 0.
Proof.
intros.
lia.
Qed.
Class Foo {x : Z} := { T : Type ; dec : T -> Z }.
Goal forall bound {F : @Foo bound} (x y : T), 0 <= dec x < bound -> 0 <= dec y
< bound -> dec x + dec y >= bound -> dec x + dec y < 2 * bound.
Proof.
intros.
lia.
Qed.
Section S.
Variables x y: Z.
Variables XGe : x >= 0.
Variables YGt : y > 0.
Variables YLt : y < 0.
Goal False.
Proof using - XGe.
lia.
Qed.
Goal False.
Proof using YGt YLt x y.
lia.
Qed.
End S.
Section S.
Variable x y: Z.
Variable H1 : 1 > 0 -> x = 1.
Variable H2 : x = y.
Goal x = y.
Proof using H2.
lia.
Qed.
End S.
(* Bug 5073 *)
Lemma opp_eq_0_iff a : -a = 0 <-> a = 0.
Proof.
lia.
Qed.
Lemma ex_pos : forall x, exists z t, x = z - t /\ z >= 0 /\ t >= 0.
Proof.
intros.
destruct (dec_Zge x 0).
exists x, 0.
lia.
exists 0, (-x).
lia.
Qed.
Goal forall
(b q r : Z)
(H : b * q + r <= 0)
(H5 : - b < r)
(H6 : r <= 0)
(H2 : 0 <= b),
b = 0 -> False.
Proof.
intros b q r.
lia.
Qed.
Section S.
(* From bedrock2, used to be slow *)
Variables (x3 q r q2 r3 : Z)
(H : 2 ^ 2 <> 0 -> r3 + 3 = 2 ^ 2 * q + r)
(H0 : 0 < 2 ^ 2 -> 0 <= r < 2 ^ 2)
(H1 : 2 ^ 2 < 0 -> 2 ^ 2 < r <= 0)
(H2 : 2 ^ 2 = 0 -> q = 0)
(H3 : 2 ^ 2 = 0 -> r = 0)
(q0 r0 : Z)
(H4 : 4 <> 0 -> 0 = 4 * q0 + r0)
(H5 : 0 < 4 -> 0 <= r0 < 4)
(H6 : 4 < 0 -> 4 < r0 <= 0)
(H7 : 4 = 0 -> q0 = 0)
(H8 : 4 = 0 -> r0 = 0)
(q1 r1 : Z)
(H9 : 4 <> 0 -> q + q + (q + q) = 4 * q1 + r1)
(H10 : 0 < 4 -> 0 <= r1 < 4)
(H11 : 4 < 0 -> 4 < r1 <= 0)
(H12 : 4 = 0 -> q1 = 0)
(H13 : 4 = 0 -> r1 = 0)
(r2 : Z)
(H14 : 2 ^ 16 <> 0 -> x3 = 2 ^ 16 * q2 + r2)
(H15 : 0 < 2 ^ 16 -> 0 <= r2 < 2 ^ 16)
(H16 : 2 ^ 16 < 0 -> 2 ^ 16 < r2 <= 0)
(H17 : 2 ^ 16 = 0 -> q2 = 0)
(H18 : 2 ^ 16 = 0 -> r2 = 0)
(q3 : Z)
(H19 : 16383 + 1 <> 0 -> q2 = (16383 + 1) * q3 + r3)
(H20 : 0 < 16383 + 1 -> 0 <= r3 < 16383 + 1)
(H21 : 16383 + 1 < 0 -> 16383 + 1 < r3 <= 0)
(H22 : 16383 + 1 = 0 -> q3 = 0)
(H23 : 16383 + 1 = 0 -> r3 = 0).
Goal r0 = r1.
Proof using H10 H9 H5 H4.
intros.
lia.
Qed.
End S.
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