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lean4-htt/tests/elab/simp_int_arith.lean
Garmelon 08eb78a5b2
chore: switch to new test/bench suite (#12590) 
This PR sets up the new integrated test/bench suite. It then migrates
all benchmarks and some related tests to the new suite. There's also
some documentation and some linting.

For now, a lot of the old tests are left alone so this PR doesn't become
even larger than it already is. Eventually, all tests should be migrated
to the new suite though so there isn't a confusing mix of two systems.
2026-02-25 13:51:53 +00:00

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example (x y : Int) : x + y + 2 + y = y + 1 + 1 + x + y := by
simp +arith only
example (x y : Int) (h : x + y + 2 + y < y + 1 + 1 + x + y) : False := by
simp +arith only at h
example (x y : Int) (h : x + y + 2 + y > y + 1 + 1 + x + y) : False := by
simp +arith only at h
example (x y : Int) (_h : x + y + 3 + y > y + 1 + 1 + x + y) : True := by
simp +arith only at _h
guard_hyp _h : True
constructor
example (x y : Int) (h : x + y + 2 + y > 1 + 1 + x + x + y + 2*x) : 3*x + -1*y+1 ≤ 0 := by
simp +arith only at h
guard_hyp h : 3 * x + -1*y + 1 ≤ 0
assumption
example (x y : Int) (h : 6*x + y + 3 + y + 1 < y + 1 + 1 + x + 5*y) : 5*x + -4*y + 3 ≤ 0 := by
simp +arith only at h
guard_hyp h : 5*x + -4*y + 3 ≤ 0
assumption
example (x y : Int) : x + y + 2 + y ≤ y + 1 + 1 + x + y := by
simp +arith only
example (x y : Int) : x + y + 2 + y ≤ y + 1 + 1 + 5 + x + y := by
simp +arith only
example (x y z : Int) : x + y + 2 + y + z + z ≤ y + 3*z + 1 + 1 + x + y - z := by
simp +arith only
example (x y : Int) (h : False) : x + y + 20 + y ≤ y + 1 + 1 + 5 + x + y := by
simp +arith only
guard_target = False
assumption
example (x y : Int) (h : False) : x = y := by
fail_if_success simp +arith only
guard_target = x = y
contradiction
example (x : Int) (h : False) : x = 3 := by
fail_if_success simp +arith only
guard_target = x = 3
contradiction
example (x : Int) (h : False) : 3 = x := by
fail_if_success simp +arith only
guard_target = 3 = x
contradiction
example (x : Int) (h : False) : 2*x = x + 3 := by
simp +arith only
guard_target = x = 3
contradiction
example (x y : Int) (h : False) : 2*x = x + y := by
simp +arith only
guard_target = x = y
contradiction
example (x : Int) (h : False) : 2*x + 1 = x := by
simp +arith only
guard_target = x = -1
contradiction
example (x : Int) (h : False) (f : Int → Int) : f (0 + x + x) = 3 := by
simp +arith only
guard_target = f (2*x) = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (x + y - x) = 3 := by
simp +arith only
guard_target = f y = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (x + y - x + 2 - y + 1) = 3 := by
simp +arith only
guard_target = f 3 = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (x + y - x + (2 - y)*2 + 1) = 3 := by
simp +arith only
guard_target = f (-1*y + 5) = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (x + y - x + (2 - y)*(1+1) + 1) = 3 := by
simp +arith only [Int.reduceAdd]
guard_target = f (-1*y + 5) = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (x + y - x + (1-1+2)*(2 - y) + 1) = 3 := by
simp +arith only [Int.reduceAdd, Int.reduceSub]
guard_target = f (-1*y + 5) = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (Int.add x y - x + (2 - y)*2 + 1) = 3 := by
simp +arith only
guard_target = f (-1*y + 5) = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (x + y - x + Int.mul (1-1+2) (2 - y) + 1) = 3 := by
simp +arith only [Int.reduceAdd, Int.reduceSub]
guard_target = f (-1*y + 5) = 3
contradiction
example (x y : Int) (h : False) (f : Int → Int) : f (Int.add x y - x + (2 - y)*(-2) + 1) = 3 := by
simp +arith only
guard_target = f (3*y + -3) = 3
contradiction
example (x : Int) : x > x - 1 := by
simp +arith only
example (x : Int) : x - 1 < x := by
simp +arith only
example (x : Int) : x < x + 1 := by
simp +arith only
example (x : Int) : x ≥ x - 1 := by
simp +arith only
example (x : Int) : x ≤ x := by
simp +arith only
example (x : Int) : x ≤ x + 1 := by
simp +arith only
example (x : Int) (h : False) : x > x := by
simp +arith only
guard_target = False
assumption
theorem ex₁ (x y z : Int) : x + y + 2 + y + z + z ≤ y + 3*z + 1 + 1 + x + y - z := by
simp +arith only
/--
info: theorem ex₁ : ∀ (x y z : Int), x + y + 2 + y + z + z ≤ y + 3 * z + 1 + 1 + x + y - z :=
fun x y z =>
of_eq_true
(id
(le_eq_true
(Lean.RArray.branch 1 (Lean.RArray.leaf z) (Lean.RArray.branch 2 (Lean.RArray.leaf y) (Lean.RArray.leaf x)))
((((((Expr.var 2).add (Expr.var 1)).add (Expr.num 2)).add (Expr.var 1)).add (Expr.var 0)).add (Expr.var 0))
(((((((Expr.var 1).add (Expr.mulL 3 (Expr.var 0))).add (Expr.num 1)).add (Expr.num 1)).add (Expr.var 2)).add
(Expr.var 1)).sub
(Expr.var 0))
(eagerReduce (Eq.refl true))))
-/
#guard_msgs (info) in
open Int.Linear in
#print ex₁
theorem ex₂ (x y z : Int) (f : Int → Int) : x + f y + 2 + f y + z + z ≤ f y + 3*z + 1 + 1 + x + f y - z := by
simp +arith only
/--
info: theorem ex₂ : ∀ (x y z : Int) (f : Int → Int), x + f y + 2 + f y + z + z ≤ f y + 3 * z + 1 + 1 + x + f y - z :=
fun x y z f =>
of_eq_true
(id
(le_eq_true
(Lean.RArray.branch 1 (Lean.RArray.leaf (f y)) (Lean.RArray.branch 2 (Lean.RArray.leaf z) (Lean.RArray.leaf x)))
((((((Expr.var 2).add (Expr.var 0)).add (Expr.num 2)).add (Expr.var 0)).add (Expr.var 1)).add (Expr.var 1))
(((((((Expr.var 0).add (Expr.mulL 3 (Expr.var 1))).add (Expr.num 1)).add (Expr.num 1)).add (Expr.var 2)).add
(Expr.var 0)).sub
(Expr.var 1))
(eagerReduce (Eq.refl true))))
-/
#guard_msgs (info) in
open Int.Linear in
#print ex₂
example (x y : Int) (h : False) : 2*x = x + y := by
simp +arith only
guard_target = x = y
contradiction
example (x y : Int) (h : 2*x + 2*y = 4) : x + y = 2 := by
simp +arith only at h
guard_hyp h : x + y + -2 = 0
simp +arith
assumption
example (x y : Int) (h : 6*x + 3*y = 9) : 2*x + y = 3 := by
simp +arith only at h
guard_hyp h : 2*x + y + -3 = 0
simp +arith
assumption
example (x y : Int) (h : 2*x - 2*y ≤ 4) : x - y ≤ 2 := by
simp +arith only at h
guard_hyp h : x + -1*y + -2 ≤ 0
simp +arith
assumption
example (x y : Int) (h : -6*x + 3*y = -9) : - 2*x = -3 - y := by
simp +arith only at h
guard_hyp h : -2*x + y + 3 = 0
simp +arith
assumption
example (x y : Int) (h : 3*x + 6*y = 2) : False := by
simp +arith only at h
example (x : Int) (h : 3*x = 1) : False := by
simp +arith only at h
example (x : Int) (h : 2*x = 1) : False := by
simp +arith only at h
example (x : Int) (h : x + x = 1) : False := by
simp +arith only at h
example (x y : Int) (h : x + x + x = 1 + 2*y + x) : False := by
simp +arith only at h
example (x : Int) (h : -x - x = 1) : False := by
simp +arith only at h
example (x : Int) (h : 2*x ≤ 1) : x ≤ 0 := by
simp +arith only at h
guard_hyp h : x ≤ 0
assumption
example (x y : Int) (h : 6*x + y + y + y ≤ 7) : 2*x + y + -2 ≤ 0 := by
simp +arith only at h
guard_hyp h : 2*x + y + -2 ≤ 0
assumption
example (x y : Int) (h : 5*x + y + y + y ≤ 7 - x) : 2*x + y + -2 ≤ 0 := by
simp +arith only at h
guard_hyp h : 2*x + y + -2 ≤ 0
assumption
example (x : Int) : (11*x ≤ 10) ↔ (x ≤ 0) := by
simp +arith only
example (x : Int) : (11*x > 10) ↔ (x ≥ 1) := by
simp +arith only
example (x y : Int) : (2*x + y + y = 4) ↔ (y + x = 2) := by
simp +arith
example (x y : Int) : (2*x + y + y ≤ 3) ↔ (y + x ≤ 1) := by
simp +arith
example (f : Int → Int) (x y : Int) : f (2*x + y) = f (y + x + x) := by
simp +arith
example (a b : Int) : ¬ 2 ∣ 2*a + 4*b + 1 := by
simp +arith
example (a b : Int) : ¬ 2 ∣ a + 3*b + 1 + b + a := by
simp +arith
example (a b : Int) : ¬ 2 ∣ a + 3*b + 1 + b + 5*a := by
simp +arith
example (a b : Int) : 2 ∣ 4*a + 6*b + 8 := by
simp +arith
example (a b : Int) : 2 ∣ 2*(a + a) + (3+3)*(b + b) + 8 := by
simp +arith
example (a : Int) : 16 ∣ 4*a + 32 ↔ 4 ∣ a + 8 := by
simp +arith
example (a : Int) : 3 ∣ a + a + 1 + a + 1 + a ↔ 3 ∣ 4*a + 2 := by
simp +arith
example (a : Int) : 2+1 ∣ a + a + 1 - a + 1 + a ↔ 3 ∣ 2*a + 2 := by
simp +arith
example (a b : Int) : 6 ∣ a + 21 - a + 3*a + 6*b + 12 ↔ 2 ∣ a + 2*b + 11 := by
simp +arith
theorem ex3 (a b : Int) : 6 ∣ a + (21 - a) + 3*(a + 2*b) + 12 ↔ 2 ∣ a + 2*b + 11 := by
simp +arith
/--
info: theorem ex3 : ∀ (a b : Int), 6 ∣ a + (21 - a) + 3 * (a + 2 * b) + 12 ↔ 2 ∣ a + 2 * b + 11 :=
fun a b =>
of_eq_true
(Eq.trans
(congrFun'
(congrArg Iff
(id
(norm_dvd_gcd (RArray.branch 1 (RArray.leaf b) (RArray.leaf a)) 6
((((Expr.var 1).add ((Expr.num 21).sub (Expr.var 1))).add
(Expr.mulL 3 ((Expr.var 1).add (Expr.mulL 2 (Expr.var 0))))).add
(Expr.num 12))
2 (Poly.add 1 1 (Poly.add 2 0 (Poly.num 11))) 3 (eagerReduce (Eq.refl true)))))
(2 ∣ a + 2 * b + 11))
(iff_self (2 ∣ a + 2 * b + 11)))
-/
#guard_msgs (info) in
open Lean in open Int.Linear in
#print ex3
theorem ex4 (a b : Int) : 6 ∣ a + (11 - a) + 3*(a + 2*b) - 11 ↔ 2 ∣ a + 2*b := by
simp +arith
/--
info: theorem ex4 : ∀ (a b : Int), 6 ∣ a + (11 - a) + 3 * (a + 2 * b) - 11 ↔ 2 ∣ a + 2 * b :=
fun a b =>
of_eq_true
(Eq.trans
(congr
(congrArg Iff
(Eq.trans
(id
(norm_dvd_gcd (RArray.branch 1 (RArray.leaf b) (RArray.leaf a)) 6
((((Expr.var 1).add ((Expr.num 11).sub (Expr.var 1))).add
(Expr.mulL 3 ((Expr.var 1).add (Expr.mulL 2 (Expr.var 0))))).sub
(Expr.num 11))
2 (Poly.add 1 1 (Poly.add 2 0 (Poly.num 0))) 3 (eagerReduce (Eq.refl true))))
Int.dvd_add_self_mul._simp_1))
Int.dvd_add_self_mul._simp_1)
(iff_self (2 ∣ a)))
-/
#guard_msgs (info) in
open Lean in open Int.Linear in
#print ex4
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