file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Algebra/DirectSum/Finsupp.lean
|
finsuppLEquivDirectSum_single
|
[] |
[
46,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.aposteriori_dist_iterate_fixedPoint_le
|
[
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nx : α\nn : ℕ\n⊢ dist ((f^[n]) x) (fixedPoint f hf) ≤ dist ((f^[n]) x) ((f ∘ f^[n]) x) / (1 - ↑K)",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nx : α\nn : ℕ\n⊢ dist ((f^[n]) x) (fixedPoint f hf) ≤ dist ((f^[n]) x) ((f^[n + 1]) x) / (1 - ↑K)",
"tactic": "rw [iterate_succ']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nx : α\nn : ℕ\n⊢ dist ((f^[n]) x) (fixedPoint f hf) ≤ dist ((f^[n]) x) ((f ∘ f^[n]) x) / (1 - ↑K)",
"tactic": "apply hf.dist_fixedPoint_le"
}
] |
[
331,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Algebra/Invertible.lean
|
invOf_mul_self
|
[] |
[
107,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Dynamics/FixedPoints/Topology.lean
|
isClosed_fixedPoints
|
[] |
[
45,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
Nat.sq_add_sq_zmodEq
|
[
{
"state_after": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = ↑x\n⊢ ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ ↑a ^ 2 + ↑b ^ 2 ≡ x [ZMOD ↑p]",
"state_before": "K : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\n⊢ ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ ↑a ^ 2 + ↑b ^ 2 ≡ x [ZMOD ↑p]",
"tactic": "rcases ZMod.sq_add_sq p x with ⟨a, b, hx⟩"
},
{
"state_after": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = ↑x\n⊢ ↑(Int.natAbs (ZMod.valMinAbs a)) ^ 2 + ↑(Int.natAbs (ZMod.valMinAbs b)) ^ 2 ≡ x [ZMOD ↑p]",
"state_before": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = ↑x\n⊢ ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ ↑a ^ 2 + ↑b ^ 2 ≡ x [ZMOD ↑p]",
"tactic": "refine ⟨a.valMinAbs.natAbs, b.valMinAbs.natAbs, ZMod.natAbs_valMinAbs_le _,\n ZMod.natAbs_valMinAbs_le _, ?_⟩"
},
{
"state_after": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑(ZMod.valMinAbs a) ^ 2 + ↑(ZMod.valMinAbs b) ^ 2 = ↑x\n⊢ ↑(Int.natAbs (ZMod.valMinAbs a)) ^ 2 + ↑(Int.natAbs (ZMod.valMinAbs b)) ^ 2 ≡ x [ZMOD ↑p]",
"state_before": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = ↑x\n⊢ ↑(Int.natAbs (ZMod.valMinAbs a)) ^ 2 + ↑(Int.natAbs (ZMod.valMinAbs b)) ^ 2 ≡ x [ZMOD ↑p]",
"tactic": "rw [← a.coe_valMinAbs, ← b.coe_valMinAbs] at hx"
},
{
"state_after": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑(ZMod.valMinAbs a) ^ 2 + ↑(ZMod.valMinAbs b) ^ 2 = ↑x\n⊢ abs (ZMod.valMinAbs a) ^ 2 + abs (ZMod.valMinAbs b) ^ 2 ≡ x [ZMOD ↑p]",
"state_before": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑(ZMod.valMinAbs a) ^ 2 + ↑(ZMod.valMinAbs b) ^ 2 = ↑x\n⊢ ↑(Int.natAbs (ZMod.valMinAbs a)) ^ 2 + ↑(Int.natAbs (ZMod.valMinAbs b)) ^ 2 ≡ x [ZMOD ↑p]",
"tactic": "push_cast"
},
{
"state_after": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑(ZMod.valMinAbs a) ^ 2 + ↑(ZMod.valMinAbs b) ^ 2 = ↑x\n⊢ ↑(ZMod.valMinAbs a ^ 2 + ZMod.valMinAbs b ^ 2) = ↑x",
"state_before": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑(ZMod.valMinAbs a) ^ 2 + ↑(ZMod.valMinAbs b) ^ 2 = ↑x\n⊢ abs (ZMod.valMinAbs a) ^ 2 + abs (ZMod.valMinAbs b) ^ 2 ≡ x [ZMOD ↑p]",
"tactic": "rw [sq_abs, sq_abs, ← ZMod.int_cast_eq_int_cast_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nK : Type ?u.853209\nR : Type ?u.853212\np : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑(ZMod.valMinAbs a) ^ 2 + ↑(ZMod.valMinAbs b) ^ 2 = ↑x\n⊢ ↑(ZMod.valMinAbs a ^ 2 + ZMod.valMinAbs b ^ 2) = ↑x",
"tactic": "exact_mod_cast hx"
}
] |
[
340,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
BilinearForm.toMatrixAux_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1169905\nM : Type ?u.1169908\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : AddCommMonoid M\ninst✝¹⁵ : Module R M\nR₁ : Type ?u.1169944\nM₁ : Type ?u.1169947\ninst✝¹⁴ : Ring R₁\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹¹ : CommSemiring R₂\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : Module R₂ M₂\nR₃ : Type ?u.1170746\nM₃ : Type ?u.1170749\ninst✝⁸ : CommRing R₃\ninst✝⁷ : AddCommGroup M₃\ninst✝⁶ : Module R₃ M₃\nV : Type ?u.1171337\nK : Type ?u.1171340\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_3\no : Type ?u.1172557\ninst✝² : Fintype n\ninst✝¹ : Fintype o\ninst✝ : DecidableEq n\nb : Basis n R₂ M₂\nB : BilinForm R₂ M₂\ni j : n\n⊢ ↑(toMatrixAux ↑b) B i j = ↑(BilinForm.toMatrix b) B i j",
"tactic": "rw [BilinForm.toMatrix_apply, BilinForm.toMatrixAux_apply]"
}
] |
[
322,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.erase_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.167346\nγ : Type ?u.167349\nι : Type ?u.167352\nM : Type u_2\nM' : Type ?u.167358\nN : Type ?u.167361\nP : Type ?u.167364\nG : Type ?u.167367\nH : Type ?u.167370\nR : Type ?u.167373\nS : Type ?u.167376\ninst✝ : Zero M\na : α\n⊢ erase a 0 = 0",
"tactic": "classical rw [← support_eq_empty, support_erase, support_zero, erase_empty]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.167346\nγ : Type ?u.167349\nι : Type ?u.167352\nM : Type u_2\nM' : Type ?u.167358\nN : Type ?u.167361\nP : Type ?u.167364\nG : Type ?u.167367\nH : Type ?u.167370\nR : Type ?u.167373\nS : Type ?u.167376\ninst✝ : Zero M\na : α\n⊢ erase a 0 = 0",
"tactic": "rw [← support_eq_empty, support_erase, support_zero, erase_empty]"
}
] |
[
676,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.strongInduction_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.74113\ns✝ t : Finset α\nf : α → β\nn : ℕ\np : Finset α → Sort u_2\nH : (s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s\ns : Finset α\n⊢ strongInduction H s = H s fun t x => strongInduction H t",
"tactic": "rw [strongInduction]"
}
] |
[
646,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/RingTheory/LaurentSeries.lean
|
PowerSeries.coe_smul
|
[
{
"state_after": "case coeff.h\nR : Type u\nR' : Type ?u.298758\ninst✝³ : Semiring R\ninst✝² : Ring R'\nf g : PowerSeries R\nf' g' : PowerSeries R'\nS : Type u_1\ninst✝¹ : Semiring S\ninst✝ : Module R S\nr : R\nx : PowerSeries S\nx✝ : ℤ\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ S) (r • x)) x✝ = HahnSeries.coeff (r • ↑(ofPowerSeries ℤ S) x) x✝",
"state_before": "R : Type u\nR' : Type ?u.298758\ninst✝³ : Semiring R\ninst✝² : Ring R'\nf g : PowerSeries R\nf' g' : PowerSeries R'\nS : Type u_1\ninst✝¹ : Semiring S\ninst✝ : Module R S\nr : R\nx : PowerSeries S\n⊢ ↑(ofPowerSeries ℤ S) (r • x) = r • ↑(ofPowerSeries ℤ S) x",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case coeff.h\nR : Type u\nR' : Type ?u.298758\ninst✝³ : Semiring R\ninst✝² : Ring R'\nf g : PowerSeries R\nf' g' : PowerSeries R'\nS : Type u_1\ninst✝¹ : Semiring S\ninst✝ : Module R S\nr : R\nx : PowerSeries S\nx✝ : ℤ\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ S) (r • x)) x✝ = HahnSeries.coeff (r • ↑(ofPowerSeries ℤ S) x) x✝",
"tactic": "simp [coeff_coe, coeff_smul, smul_ite]"
}
] |
[
245,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/LinearAlgebra/TensorProductBasis.lean
|
Basis.tensorProduct_repr_tmul_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_3\nN : Type u_5\nι : Type u_1\nκ : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nb : Basis ι R M\nc : Basis κ R N\nm : M\nn : N\ni : ι\nj : κ\n⊢ ↑(↑(tensorProduct b c).repr (m ⊗ₜ[R] n)) (i, j) = ↑(↑b.repr m) i * ↑(↑c.repr n) j",
"tactic": "simp [Basis.tensorProduct]"
}
] |
[
55,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
commute_star_star
|
[] |
[
160,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.order_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ order 1 = 0",
"tactic": "simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero"
}
] |
[
2486,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2485,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.toWithTop_lt
|
[] |
[
621,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.le_max'
|
[] |
[
1355,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1354,
1
] |
Mathlib/Data/PFunctor/Multivariate/Basic.lean
|
MvPFunctor.const.mk_get
|
[
{
"state_after": "case mk\nn m : ℕ\nP : MvPFunctor n\nA : Type u\nα β : TypeVec n\nfst✝ : (const n A).A\nsnd✝ : B (const n A) fst✝ ⟹ α\n⊢ mk n (get { fst := fst✝, snd := snd✝ }) = { fst := fst✝, snd := snd✝ }",
"state_before": "n m : ℕ\nP : MvPFunctor n\nA : Type u\nα β : TypeVec n\nx : Obj (const n A) α\n⊢ mk n (get x) = x",
"tactic": "cases x"
},
{
"state_after": "case mk\nn m : ℕ\nP : MvPFunctor n\nA : Type u\nα β : TypeVec n\nfst✝ : (const n A).A\nsnd✝ : B (const n A) fst✝ ⟹ α\n⊢ { fst := fst✝, snd := fun x a => PEmpty.elim a } = { fst := fst✝, snd := snd✝ }",
"state_before": "case mk\nn m : ℕ\nP : MvPFunctor n\nA : Type u\nα β : TypeVec n\nfst✝ : (const n A).A\nsnd✝ : B (const n A) fst✝ ⟹ α\n⊢ mk n (get { fst := fst✝, snd := snd✝ }) = { fst := fst✝, snd := snd✝ }",
"tactic": "dsimp [const.get, const.mk]"
},
{
"state_after": "no goals",
"state_before": "case mk\nn m : ℕ\nP : MvPFunctor n\nA : Type u\nα β : TypeVec n\nfst✝ : (const n A).A\nsnd✝ : B (const n A) fst✝ ⟹ α\n⊢ { fst := fst✝, snd := fun x a => PEmpty.elim a } = { fst := fst✝, snd := snd✝ }",
"tactic": "congr with (_⟨⟩)"
}
] |
[
117,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.mem_map
|
[
{
"state_after": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nx : L\nhx : x ∈ I\n⊢ ↑f x ∈ ↑(Submodule.map (↑f) (↑R L I).toSubmodule)",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nx : L\nhx : x ∈ I\n⊢ ↑f x ∈ map f I",
"tactic": "apply LieSubmodule.subset_lieSpan"
},
{
"state_after": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nx : L\nhx : x ∈ I\n⊢ x ∈ ↑(↑R L I).toSubmodule ∧ ↑↑f x = ↑f x",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nx : L\nhx : x ∈ I\n⊢ ↑f x ∈ ↑(Submodule.map (↑f) (↑R L I).toSubmodule)",
"tactic": "use x"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nx : L\nhx : x ∈ I\n⊢ x ∈ ↑(↑R L I).toSubmodule ∧ ↑↑f x = ↑f x",
"tactic": "exact ⟨hx, rfl⟩"
}
] |
[
827,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
824,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.smul_filter_le_smul_filter
|
[] |
[
1227,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1226,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.CompactOpens.coe_inf
|
[] |
[
549,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/CategoryTheory/NatIso.lean
|
CategoryTheory.NatIso.trans_app
|
[] |
[
91,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
mul_lt_of_lt_one_left'
|
[] |
[
470,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean
|
deriv_inv
|
[
{
"state_after": "case inl\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\ns t : Set 𝕜\nL : Filter 𝕜\n⊢ deriv (fun x => x⁻¹) 0 = -(0 ^ 2)⁻¹\n\ncase inr\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhne : x ≠ 0\n⊢ deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\n⊢ deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹",
"tactic": "rcases eq_or_ne x 0 with (rfl | hne)"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\ns t : Set 𝕜\nL : Filter 𝕜\n⊢ deriv (fun x => x⁻¹) 0 = -(0 ^ 2)⁻¹",
"tactic": "simp [deriv_zero_of_not_differentiableAt (mt differentiableAt_inv.1 (not_not.2 rfl))]"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhne : x ≠ 0\n⊢ deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹",
"tactic": "exact (hasDerivAt_inv hne).deriv"
}
] |
[
99,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_nonpos_iff
|
[] |
[
189,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.le_exp_log
|
[
{
"state_after": "case pos\nx✝ y x : ℝ\nh_zero : x = 0\n⊢ x ≤ exp (log x)\n\ncase neg\nx✝ y x : ℝ\nh_zero : ¬x = 0\n⊢ x ≤ exp (log x)",
"state_before": "x✝ y x : ℝ\n⊢ x ≤ exp (log x)",
"tactic": "by_cases h_zero : x = 0"
},
{
"state_after": "case pos\nx✝ y x : ℝ\nh_zero : x = 0\n⊢ 0 ≤ 1",
"state_before": "case pos\nx✝ y x : ℝ\nh_zero : x = 0\n⊢ x ≤ exp (log x)",
"tactic": "rw [h_zero, log, dif_pos rfl, exp_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nx✝ y x : ℝ\nh_zero : x = 0\n⊢ 0 ≤ 1",
"tactic": "exact zero_le_one"
},
{
"state_after": "case neg\nx✝ y x : ℝ\nh_zero : ¬x = 0\n⊢ x ≤ abs x",
"state_before": "case neg\nx✝ y x : ℝ\nh_zero : ¬x = 0\n⊢ x ≤ exp (log x)",
"tactic": "rw [exp_log_eq_abs h_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx✝ y x : ℝ\nh_zero : ¬x = 0\n⊢ x ≤ abs x",
"tactic": "exact le_abs_self _"
}
] |
[
77,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_natCast
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.127287\nα : Type u_1\nβ : Type ?u.127293\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ : α\nn : ℕ\na : ℤ\n⊢ a ≤ ⌊↑n⌋ ↔ a ≤ ↑n",
"tactic": "rw [le_floor, ← cast_ofNat, cast_le]"
}
] |
[
702,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval_comp
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : CommSemiring R\np q : R[X]\nx : R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ eval x (comp p q) = eval (eval x q) p",
"tactic": "induction p using Polynomial.induction_on' with\n| h_add r s hr hs =>\n simp [add_comp, hr, hs]\n| h_monomial n a =>\n simp"
},
{
"state_after": "no goals",
"state_before": "case h_add\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : CommSemiring R\np q : R[X]\nx : R\ninst✝ : CommSemiring S\nf : R →+* S\nr s : R[X]\nhr : eval x (comp r q) = eval (eval x q) r\nhs : eval x (comp s q) = eval (eval x q) s\n⊢ eval x (comp (r + s) q) = eval (eval x q) (r + s)",
"tactic": "simp [add_comp, hr, hs]"
},
{
"state_after": "no goals",
"state_before": "case h_monomial\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na✝ b : R\nm n✝ : ℕ\ninst✝¹ : CommSemiring R\np q : R[X]\nx : R\ninst✝ : CommSemiring S\nf : R →+* S\nn : ℕ\na : R\n⊢ eval x (comp (↑(monomial n) a) q) = eval (eval x q) (↑(monomial n) a)",
"tactic": "simp"
}
] |
[
1082,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1076,
1
] |
Mathlib/Order/RelClasses.lean
|
wellFoundedGT_dual_iff
|
[] |
[
370,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.ext_iff
|
[] |
[
725,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
724,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.addHom_ext'
|
[] |
[
1146,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/Algebra/Group/Conj.lean
|
isConj_comm
|
[] |
[
46,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.shiftl_add
|
[
{
"state_after": "m n✝ k i n : ℕ\n⊢ ↑(Nat.shiftl m i) = ↑(Nat.shiftr (Nat.shiftl m (n + i)) n)",
"state_before": "m n✝ k i n : ℕ\n⊢ (fun n k i => shiftl (↑m) i = ↑(Nat.shiftr (Nat.shiftl m n) k)) (n + i) n ↑i",
"tactic": "dsimp"
},
{
"state_after": "case a\nm n✝ k i n : ℕ\n⊢ n ≤ n + i",
"state_before": "m n✝ k i n : ℕ\n⊢ ↑(Nat.shiftl m i) = ↑(Nat.shiftr (Nat.shiftl m (n + i)) n)",
"tactic": "rw [← Nat.shiftl_sub, add_tsub_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "case a\nm n✝ k i n : ℕ\n⊢ n ≤ n + i",
"tactic": "apply Nat.le_add_right"
},
{
"state_after": "m n✝ k i n : ℕ\n⊢ shiftl ↑m -[i+1] = ↑(Nat.shiftr (Nat.shiftl m n) (n + i + 1))",
"state_before": "m n✝ k i n : ℕ\n⊢ (fun n k i => shiftl (↑m) i = ↑(Nat.shiftr (Nat.shiftl m n) k)) n (n + i + 1) -[i+1]",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "m n✝ k i n : ℕ\n⊢ shiftl ↑m -[i+1] = ↑(Nat.shiftr (Nat.shiftl m n) (n + i + 1))",
"tactic": "rw [add_assoc, Nat.shiftr_add, ← Nat.shiftl_sub, tsub_self] <;> rfl"
},
{
"state_after": "case a\nm n✝ k i n : ℕ\n⊢ n ≤ n + i",
"state_before": "m n✝ k i n : ℕ\n⊢ Nat.shiftl' true m i = Nat.shiftr (Nat.shiftl' true m (n + i)) n",
"tactic": "rw [← Nat.shiftl'_sub, add_tsub_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "case a\nm n✝ k i n : ℕ\n⊢ n ≤ n + i",
"tactic": "apply Nat.le_add_right"
},
{
"state_after": "no goals",
"state_before": "m n✝ k i n : ℕ\n⊢ Nat.shiftr m (Nat.succ i) = Nat.shiftr (Nat.shiftl' true m n) (n + i + 1)",
"tactic": "rw [add_assoc, Nat.shiftr_add, ← Nat.shiftl'_sub, tsub_self] <;> rfl"
}
] |
[
426,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Analysis/PSeries.lean
|
Real.summable_nat_rpow
|
[
{
"state_after": "case intro\np : ℝ\n⊢ (Summable fun n => ↑n ^ (-p)) ↔ -p < -1",
"state_before": "p : ℝ\n⊢ (Summable fun n => ↑n ^ p) ↔ p < -1",
"tactic": "rcases neg_surjective p with ⟨p, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\np : ℝ\n⊢ (Summable fun n => ↑n ^ (-p)) ↔ -p < -1",
"tactic": "simp [rpow_neg]"
}
] |
[
198,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.mk_injective
|
[
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\n⊢ x = y",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx : (i : ↑↑s✝) → β ↑i\ni : ι\ns : Finset ι\n⊢ Function.Injective (mk s)",
"tactic": "intro x y H"
},
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ↑↑s\n⊢ x i = y i",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\n⊢ x = y",
"tactic": "ext i"
},
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ↑↑s\nh1 : ↑(mk s x) ↑i = ↑(mk s y) ↑i\n⊢ x i = y i",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ↑↑s\n⊢ x i = y i",
"tactic": "have h1 : (mk s x : ∀ i, β i) i = (mk s y : ∀ i, β i) i := by rw [H]"
},
{
"state_after": "case h.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ι\nhi : i ∈ s\nh1 : ↑(mk s x) ↑{ val := i, property := hi } = ↑(mk s y) ↑{ val := i, property := hi }\n⊢ x { val := i, property := hi } = y { val := i, property := hi }",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ↑↑s\nh1 : ↑(mk s x) ↑i = ↑(mk s y) ↑i\n⊢ x i = y i",
"tactic": "obtain ⟨i, hi : i ∈ s⟩ := i"
},
{
"state_after": "case h.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ι\nhi : i ∈ s\nh1 : (if H : i ∈ s then x { val := i, property := H } else 0) = if H : i ∈ s then y { val := i, property := H } else 0\n⊢ x { val := i, property := hi } = y { val := i, property := hi }",
"state_before": "case h.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ι\nhi : i ∈ s\nh1 : ↑(mk s x) ↑{ val := i, property := hi } = ↑(mk s y) ↑{ val := i, property := hi }\n⊢ x { val := i, property := hi } = y { val := i, property := hi }",
"tactic": "dsimp only [mk_apply, Subtype.coe_mk] at h1"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ι\nhi : i ∈ s\nh1 : (if H : i ∈ s then x { val := i, property := H } else 0) = if H : i ∈ s then y { val := i, property := H } else 0\n⊢ x { val := i, property := hi } = y { val := i, property := hi }",
"tactic": "simpa only [dif_pos hi] using h1"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns✝ : Finset ι\nx✝ : (i : ↑↑s✝) → β ↑i\ni✝ : ι\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\nH : mk s x = mk s y\ni : ↑↑s\n⊢ ↑(mk s x) ↑i = ↑(mk s y) ↑i",
"tactic": "rw [H]"
}
] |
[
601,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.snd'
|
[] |
[
807,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/Tactic/NormNum/GCD.lean
|
Tactic.NormNum.nat_gcd_helper_1
|
[] |
[
47,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
HasFDerivAt.le_of_lip'
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ 0 ≤ C + ε\n\ncase refine'_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ‖↑f' x‖ ≤ (C + ε) * ‖x‖",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\n⊢ ‖f'‖ ≤ C",
"tactic": "refine' le_of_forall_pos_le_add fun ε ε0 => op_norm_le_of_nhds_zero _ _"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ‖↑f' x‖ ≤ (C + ε) * ‖x‖",
"state_before": "case refine'_1\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ 0 ≤ C + ε\n\ncase refine'_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ‖↑f' x‖ ≤ (C + ε) * ‖x‖",
"tactic": "exact add_nonneg hC₀ ε0.le"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (a : E) in 𝓝 0, ‖f ((fun x x_1 => x + x_1) x₀ a) - f x₀‖ ≤ C * ‖(fun x x_1 => x + x_1) x₀ a - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ‖↑f' x‖ ≤ (C + ε) * ‖x‖",
"state_before": "case refine'_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ‖↑f' x‖ ≤ (C + ε) * ‖x‖",
"tactic": "rw [← map_add_left_nhds_zero x₀, eventually_map] at hlip"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (a : E) in 𝓝 0, ‖f ((fun x x_1 => x + x_1) x₀ a) - f x₀‖ ≤ C * ‖(fun x x_1 => x + x_1) x₀ a - x₀‖\nε : ℝ\nε0 : 0 < ε\ny : E\nhy : ‖f (x₀ + y) - f x₀ - ↑f' y‖ ≤ ε * ‖y‖\nhyC : ‖f (x₀ + y) - f x₀‖ ≤ C * ‖x₀ + y - x₀‖\n⊢ ‖↑f' y‖ ≤ (C + ε) * ‖y‖",
"state_before": "case refine'_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (a : E) in 𝓝 0, ‖f ((fun x x_1 => x + x_1) x₀ a) - f x₀‖ ≤ C * ‖(fun x x_1 => x + x_1) x₀ a - x₀‖\nε : ℝ\nε0 : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ‖↑f' x‖ ≤ (C + ε) * ‖x‖",
"tactic": "filter_upwards [isLittleO_iff.1 (hasFDerivAt_iff_isLittleO_nhds_zero.1 hf) ε0, hlip] with y hy hyC"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (a : E) in 𝓝 0, ‖f ((fun x x_1 => x + x_1) x₀ a) - f x₀‖ ≤ C * ‖(fun x x_1 => x + x_1) x₀ a - x₀‖\nε : ℝ\nε0 : 0 < ε\ny : E\nhy : ‖f (x₀ + y) - f x₀ - ↑f' y‖ ≤ ε * ‖y‖\nhyC : ‖f (x₀ + y) - f x₀‖ ≤ C * ‖y‖\n⊢ ‖↑f' y‖ ≤ (C + ε) * ‖y‖",
"state_before": "case h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (a : E) in 𝓝 0, ‖f ((fun x x_1 => x + x_1) x₀ a) - f x₀‖ ≤ C * ‖(fun x x_1 => x + x_1) x₀ a - x₀‖\nε : ℝ\nε0 : 0 < ε\ny : E\nhy : ‖f (x₀ + y) - f x₀ - ↑f' y‖ ≤ ε * ‖y‖\nhyC : ‖f (x₀ + y) - f x₀‖ ≤ C * ‖x₀ + y - x₀‖\n⊢ ‖↑f' y‖ ≤ (C + ε) * ‖y‖",
"tactic": "rw [add_sub_cancel'] at hyC"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.135450\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.135545\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (a : E) in 𝓝 0, ‖f ((fun x x_1 => x + x_1) x₀ a) - f x₀‖ ≤ C * ‖(fun x x_1 => x + x_1) x₀ a - x₀‖\nε : ℝ\nε0 : 0 < ε\ny : E\nhy : ‖f (x₀ + y) - f x₀ - ↑f' y‖ ≤ ε * ‖y‖\nhyC : ‖f (x₀ + y) - f x₀‖ ≤ C * ‖y‖\n⊢ ‖↑f' y‖ ≤ (C + ε) * ‖y‖",
"tactic": "calc\n ‖f' y‖ ≤ ‖f (x₀ + y) - f x₀‖ + ‖f (x₀ + y) - f x₀ - f' y‖ := norm_le_insert _ _\n _ ≤ C * ‖y‖ + ε * ‖y‖ := (add_le_add hyC hy)\n _ = (C + ε) * ‖y‖ := (add_mul _ _ _).symm"
}
] |
[
340,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
Filter.EventuallyEq.hasFDerivWithinAt_iff
|
[] |
[
859,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Topology/StoneCech.lean
|
denseEmbedding_pure
|
[] |
[
153,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mulVec_transpose
|
[
{
"state_after": "case h\nl : Type ?u.916196\nm : Type u_1\nn : Type u_2\no : Type ?u.916205\nm' : o → Type ?u.916210\nn' : o → Type ?u.916215\nR : Type ?u.916218\nS : Type ?u.916221\nα : Type v\nβ : Type w\nγ : Type ?u.916228\ninst✝¹ : NonUnitalCommSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx : m → α\nx✝ : n\n⊢ mulVec Aᵀ x x✝ = vecMul x A x✝",
"state_before": "l : Type ?u.916196\nm : Type u_1\nn : Type u_2\no : Type ?u.916205\nm' : o → Type ?u.916210\nn' : o → Type ?u.916215\nR : Type ?u.916218\nS : Type ?u.916221\nα : Type v\nβ : Type w\nγ : Type ?u.916228\ninst✝¹ : NonUnitalCommSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx : m → α\n⊢ mulVec Aᵀ x = vecMul x A",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.916196\nm : Type u_1\nn : Type u_2\no : Type ?u.916205\nm' : o → Type ?u.916210\nn' : o → Type ?u.916215\nR : Type ?u.916218\nS : Type ?u.916221\nα : Type v\nβ : Type w\nγ : Type ?u.916228\ninst✝¹ : NonUnitalCommSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx : m → α\nx✝ : n\n⊢ mulVec Aᵀ x x✝ = vecMul x A x✝",
"tactic": "apply dotProduct_comm"
}
] |
[
1932,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1930,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
fderivWithin_sin
|
[] |
[
1026,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
|
Complex.cos_eq_zero_iff
|
[
{
"state_after": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\n⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
"state_before": "θ : ℂ\n⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
"tactic": "have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by\n rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, MulZeroClass.zero_mul,\n add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _), ← exp_sub]\n field_simp only; congr 3; ring_nf"
},
{
"state_after": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\n⊢ (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
"state_before": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\n⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
"tactic": "rw [cos, h, ← exp_pi_mul_I, exp_eq_exp_iff_exists_int, mul_right_comm]"
},
{
"state_after": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\nx : ℤ\n⊢ 2 * I * θ = ↑π * I + ↑x * (2 * ↑π * I) ↔ θ = (2 * ↑x + 1) * ↑π / 2",
"state_before": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\n⊢ (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
"tactic": "refine' exists_congr fun x => _"
},
{
"state_after": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\nx : ℤ\n⊢ ↑π * I + ↑x * (2 * ↑π * I) = 2 * I * ((2 * ↑x + 1) * ↑π / 2)",
"state_before": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\nx : ℤ\n⊢ 2 * I * θ = ↑π * I + ↑x * (2 * ↑π * I) ↔ θ = (2 * ↑x + 1) * ↑π / 2",
"tactic": "refine' (iff_of_eq <| congr_arg _ _).trans (mul_right_inj' <| mul_ne_zero two_ne_zero I_ne_zero)"
},
{
"state_after": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\nx : ℤ\n⊢ (↑π * I + ↑x * (2 * ↑π * I)) * 2 = 2 * I * ((2 * ↑x + 1) * ↑π)",
"state_before": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\nx : ℤ\n⊢ ↑π * I + ↑x * (2 * ↑π * I) = 2 * I * ((2 * ↑x + 1) * ↑π / 2)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "θ : ℂ\nh : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1\nx : ℤ\n⊢ (↑π * I + ↑x * (2 * ↑π * I)) * 2 = 2 * I * ((2 * ↑x + 1) * ↑π)",
"tactic": "ring"
},
{
"state_after": "θ : ℂ\n⊢ exp (θ * I - -θ * I) = -1 ↔ exp (2 * θ * I) = -1",
"state_before": "θ : ℂ\n⊢ (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1",
"tactic": "rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, MulZeroClass.zero_mul,\n add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _), ← exp_sub]"
},
{
"state_after": "θ : ℂ\n⊢ exp (θ * I - -θ * I) = -1 ↔ exp (2 * θ * I) = -1",
"state_before": "θ : ℂ\n⊢ exp (θ * I - -θ * I) = -1 ↔ exp (2 * θ * I) = -1",
"tactic": "field_simp only"
},
{
"state_after": "θ : ℂ\n⊢ exp (θ * I - -θ * I) = -1 ↔ exp (2 * θ * I) = -1",
"state_before": "θ : ℂ\n⊢ exp (θ * I - -θ * I) = -1 ↔ exp (2 * θ * I) = -1",
"tactic": "congr 3"
},
{
"state_after": "no goals",
"state_before": "θ : ℂ\n⊢ exp (θ * I - -θ * I) = -1 ↔ exp (2 * θ * I) = -1",
"tactic": "ring_nf"
}
] |
[
42,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPolynomial.coe_monomial
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\n⊢ (if n = m then a else 0) = if m = n then a else 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\n⊢ ↑(MvPowerSeries.coeff R m) ↑(↑(monomial n) a) = ↑(MvPowerSeries.coeff R m) (↑(MvPowerSeries.monomial R n) a)",
"tactic": "rw [coeff_coe, coeff_monomial, MvPowerSeries.coeff_monomial]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\n⊢ (if n = m then a else 0) = if m = n then a else 0",
"tactic": "split_ifs with h₁ h₂ <;> first |rfl|subst m; contradiction"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nh₁ : ¬n = m\nh✝ : ¬m = n\n⊢ 0 = 0",
"tactic": "rfl"
},
{
"state_after": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nh₁ : ¬n = n\n⊢ 0 = a",
"state_before": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nh₁ : ¬n = m\nh✝ : m = n\n⊢ 0 = a",
"tactic": "subst m"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\nn : σ →₀ ℕ\na : R\nh₁ : ¬n = n\n⊢ 0 = a",
"tactic": "contradiction"
}
] |
[
1084,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1080,
1
] |
Mathlib/Deprecated/Subfield.lean
|
Range.isSubfield
|
[
{
"state_after": "F : Type u_2\ninst✝¹ : Field F\nS : Set F\nK : Type u_1\ninst✝ : Field K\nf : F →+* K\n⊢ IsSubfield (↑f '' Set.univ)",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nS : Set F\nK : Type u_1\ninst✝ : Field K\nf : F →+* K\n⊢ IsSubfield (Set.range ↑f)",
"tactic": "rw [← Set.image_univ]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nS : Set F\nK : Type u_1\ninst✝ : Field K\nf : F →+* K\n⊢ IsSubfield (↑f '' Set.univ)",
"tactic": "apply Image.isSubfield _ Univ.isSubfield"
}
] |
[
79,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Basis.dualBasis_apply_self
|
[
{
"state_after": "case h.e'_3.h₁.a\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ j = i ↔ i = j",
"state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ ↑(↑(dualBasis b) i) (↑b j) = if j = i then 1 else 0",
"tactic": "convert b.toDual_apply i j using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h₁.a\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ j = i ↔ i = j",
"tactic": "rw [@eq_comm _ j i]"
}
] |
[
425,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_congr
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\n⊢ x ∈ f '' s ↔ x ∈ g '' s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\n⊢ f '' s = g '' s",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\n⊢ (∃ x_1, x_1 ∈ s ∧ f x_1 = x) ↔ ∃ x_1, x_1 ∈ s ∧ g x_1 = x",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\n⊢ x ∈ f '' s ↔ x ∈ g '' s",
"tactic": "rw [mem_image, mem_image]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\n⊢ (∃ x_1, x_1 ∈ s ∧ f x_1 = x) ↔ ∃ x_1, x_1 ∈ s ∧ g x_1 = x",
"tactic": "exact {\n mp := by\n rintro ⟨a, ha1, ha2⟩\n exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,\n mpr := by\n rintro ⟨a, ha1, ha2⟩\n exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩\n}"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\na : α\nha1 : a ∈ s\nha2 : f a = x\n⊢ ∃ x_1, x_1 ∈ s ∧ g x_1 = x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\n⊢ (∃ x_1, x_1 ∈ s ∧ f x_1 = x) → ∃ x_1, x_1 ∈ s ∧ g x_1 = x",
"tactic": "rintro ⟨a, ha1, ha2⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\na : α\nha1 : a ∈ s\nha2 : f a = x\n⊢ ∃ x_1, x_1 ∈ s ∧ g x_1 = x",
"tactic": "exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\na : α\nha1 : a ∈ s\nha2 : g a = x\n⊢ ∃ x_1, x_1 ∈ s ∧ f x_1 = x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\n⊢ (∃ x_1, x_1 ∈ s ∧ g x_1 = x) → ∃ x_1, x_1 ∈ s ∧ f x_1 = x",
"tactic": "rintro ⟨a, ha1, ha2⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.27036\nι : Sort ?u.27039\nι' : Sort ?u.27042\nf✝ : α → β\ns✝ t : Set α\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\nx : β\na : α\nha1 : a ∈ s\nha2 : g a = x\n⊢ ∃ x_1, x_1 ∈ s ∧ f x_1 = x",
"tactic": "exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩"
}
] |
[
264,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.NeBot.smul_zero_nonneg
|
[] |
[
1354,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1351,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.image_subset_infs_left
|
[] |
[
253,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.Nodup.of_append_left
|
[] |
[
211,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.image_div_divisors_eq_divisors
|
[
{
"state_after": "case pos\nn✝ n : ℕ\nhn : n = 0\n⊢ image (fun x => n / x) (divisors n) = divisors n\n\ncase neg\nn✝ n : ℕ\nhn : ¬n = 0\n⊢ image (fun x => n / x) (divisors n) = divisors n",
"state_before": "n✝ n : ℕ\n⊢ image (fun x => n / x) (divisors n) = divisors n",
"tactic": "by_cases hn : n = 0"
},
{
"state_after": "case neg.a\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∈ image (fun x => n / x) (divisors n) ↔ a ∈ divisors n",
"state_before": "case neg\nn✝ n : ℕ\nhn : ¬n = 0\n⊢ image (fun x => n / x) (divisors n) = divisors n",
"tactic": "ext a"
},
{
"state_after": "case neg.a.mp\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∈ image (fun x => n / x) (divisors n) → a ∈ divisors n\n\ncase neg.a.mpr\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∈ divisors n → a ∈ image (fun x => n / x) (divisors n)",
"state_before": "case neg.a\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∈ image (fun x => n / x) (divisors n) ↔ a ∈ divisors n",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case pos\nn✝ n : ℕ\nhn : n = 0\n⊢ image (fun x => n / x) (divisors n) = divisors n",
"tactic": "simp [hn]"
},
{
"state_after": "case neg.a.mp\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ (∃ a_1, a_1 ∈ divisors n ∧ n / a_1 = a) → a ∈ divisors n",
"state_before": "case neg.a.mp\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∈ image (fun x => n / x) (divisors n) → a ∈ divisors n",
"tactic": "rw [mem_image]"
},
{
"state_after": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∈ divisors n\nhx2 : n / x = a\n⊢ a ∈ divisors n",
"state_before": "case neg.a.mp\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ (∃ a_1, a_1 ∈ divisors n ∧ n / a_1 = a) → a ∈ divisors n",
"tactic": "rintro ⟨x, hx1, hx2⟩"
},
{
"state_after": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∣ n ∧ n ≠ 0\nhx2 : n / x = a\n⊢ a ∣ n ∧ n ≠ 0",
"state_before": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∈ divisors n\nhx2 : n / x = a\n⊢ a ∈ divisors n",
"tactic": "rw [mem_divisors] at *"
},
{
"state_after": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∣ n ∧ n ≠ 0\nhx2 : n / x = a\n⊢ a ∣ n",
"state_before": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∣ n ∧ n ≠ 0\nhx2 : n / x = a\n⊢ a ∣ n ∧ n ≠ 0",
"tactic": "refine' ⟨_, hn⟩"
},
{
"state_after": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∣ n ∧ n ≠ 0\nhx2 : n / x = a\n⊢ n / x ∣ n",
"state_before": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∣ n ∧ n ≠ 0\nhx2 : n / x = a\n⊢ a ∣ n",
"tactic": "rw [← hx2]"
},
{
"state_after": "no goals",
"state_before": "case neg.a.mp.intro.intro\nn✝ n : ℕ\nhn : ¬n = 0\na x : ℕ\nhx1 : x ∣ n ∧ n ≠ 0\nhx2 : n / x = a\n⊢ n / x ∣ n",
"tactic": "exact div_dvd_of_dvd hx1.1"
},
{
"state_after": "case neg.a.mpr\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∣ n ∧ n ≠ 0 → ∃ a_2, a_2 ∈ divisors n ∧ n / a_2 = a",
"state_before": "case neg.a.mpr\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∈ divisors n → a ∈ image (fun x => n / x) (divisors n)",
"tactic": "rw [mem_divisors, mem_image]"
},
{
"state_after": "case neg.a.mpr.intro\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\nh1 : a ∣ n\n⊢ ∃ a_1, a_1 ∈ divisors n ∧ n / a_1 = a",
"state_before": "case neg.a.mpr\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\n⊢ a ∣ n ∧ n ≠ 0 → ∃ a_2, a_2 ∈ divisors n ∧ n / a_2 = a",
"tactic": "rintro ⟨h1, -⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.a.mpr.intro\nn✝ n : ℕ\nhn : ¬n = 0\na : ℕ\nh1 : a ∣ n\n⊢ ∃ a_1, a_1 ∈ divisors n ∧ n / a_1 = a",
"tactic": "exact ⟨n / a, mem_divisors.mpr ⟨div_dvd_of_dvd h1, hn⟩, Nat.div_div_self h1 hn⟩"
}
] |
[
508,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Init/Logic.lean
|
or_self_iff
|
[] |
[
187,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.sub_comp
|
[] |
[
1281,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1280,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.exists_primitive_lcm_of_isPrimitive
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\n⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\n⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s",
"tactic": "have h : ∃ (n : ℕ)(r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=\n ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\n⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\n⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s",
"tactic": "rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ns : R[X]\n⊢ p ∣ s ∧ q ∣ s → r ∣ s",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\n⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s",
"tactic": "refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ns : R[X]\nhs : ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s\n⊢ p ∣ s ∧ q ∣ s → r ∣ s\n\ncase hs\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ns : R[X]\n⊢ ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ns : R[X]\n⊢ p ∣ s ∧ q ∣ s → r ∣ s",
"tactic": "suffices hs : ∀ (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s"
},
{
"state_after": "case hs\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\n⊢ ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s",
"state_before": "case hs\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ns : R[X]\n⊢ ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s",
"tactic": "clear s"
},
{
"state_after": "case hs\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\n⊢ False",
"state_before": "case hs\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\n⊢ ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s",
"tactic": "by_contra' con"
},
{
"state_after": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\n⊢ False",
"state_before": "case hs\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\n⊢ False",
"tactic": "rcases Nat.find_spec con with ⟨s, sdeg, ⟨ps, qs⟩, rs⟩"
},
{
"state_after": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\n⊢ False",
"state_before": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\n⊢ False",
"tactic": "have s0 : s ≠ 0 := by\n contrapose! rs\n simp [rs]"
},
{
"state_after": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : Nat.find h ≤ natDegree s\n⊢ False",
"state_before": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\n⊢ False",
"tactic": "have hs :=\n Nat.find_min' h\n ⟨_, s.natDegree_primPart, s.isPrimitive_primPart, (hp.dvd_primPart_iff_dvd s0).2 ps,\n (hq.dvd_primPart_iff_dvd s0).2 qs⟩"
},
{
"state_after": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\n⊢ False",
"state_before": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : Nat.find h ≤ natDegree s\n⊢ False",
"tactic": "rw [← rdeg] at hs"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : natDegree s ≤ 0\n⊢ False\n\ncase neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\n⊢ False",
"state_before": "case hs.intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\n⊢ False",
"tactic": "by_cases sC : s.natDegree ≤ 0"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < natDegree s\n⊢ False",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\n⊢ False",
"tactic": "have hcancel := natDegree_cancelLeads_lt_of_natDegree_le_natDegree hs (lt_of_not_ge sC)"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\n⊢ False",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < natDegree s\n⊢ False",
"tactic": "rw [sdeg] at hcancel"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\n⊢ ∃ s_1, natDegree s_1 = natDegree (cancelLeads r s) ∧ (p ∣ s_1 ∧ q ∣ s_1) ∧ ¬r ∣ s_1",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\n⊢ False",
"tactic": "apply Nat.find_min con hcancel"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ cancelLeads r s\n⊢ r ∣ s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\n⊢ ∃ s_1, natDegree s_1 = natDegree (cancelLeads r s) ∧ (p ∣ s_1 ∧ q ∣ s_1) ∧ ¬r ∣ s_1",
"tactic": "refine'\n ⟨_, rfl, ⟨dvd_cancelLeads_of_dvd_of_dvd pr ps, dvd_cancelLeads_of_dvd_of_dvd qr qs⟩,\n fun rcs => rs _⟩"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ cancelLeads r s\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ cancelLeads r s\n⊢ r ∣ s",
"tactic": "rw [← rprim.dvd_primPart_iff_dvd s0]"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ cancelLeads r s\n⊢ r ∣ primPart s",
"tactic": "rw [cancelLeads, tsub_eq_zero_iff_le.mpr hs, pow_zero, mul_one] at rcs"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh :\n r ∣\n ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +\n ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\n⊢ r ∣ primPart s",
"tactic": "have h :=\n dvd_add rcs (Dvd.intro_left (C (leadingCoeff s) * X ^ (natDegree s - natDegree r)) rfl)"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh :\n r ∣\n ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +\n ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nhC0 : r ≠ 0\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh :\n r ∣\n ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +\n ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\n⊢ r ∣ primPart s",
"tactic": "have hC0 := rprim.ne_zero"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh :\n r ∣\n ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +\n ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nhC0 : ¬↑C (leadingCoeff r) = 0\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh :\n r ∣\n ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +\n ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nhC0 : r ≠ 0\n⊢ r ∣ primPart s",
"tactic": "rw [Ne.def, ← leadingCoeff_eq_zero, ← C_eq_zero] at hC0"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh : r ∣ primPart (↑C (leadingCoeff r) * s)\nhC0 : ¬↑C (leadingCoeff r) = 0\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh :\n r ∣\n ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +\n ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nhC0 : ¬↑C (leadingCoeff r) = 0\n⊢ r ∣ primPart s",
"tactic": "rw [sub_add_cancel, ← rprim.dvd_primPart_iff_dvd (mul_ne_zero hC0 s0)] at h"
},
{
"state_after": "case neg.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh : r ∣ primPart (↑C (leadingCoeff r) * s)\nhC0 : ¬↑C (leadingCoeff r) = 0\nu : R[X]ˣ\nhu : ↑u = primPart (↑C (leadingCoeff r))\n⊢ r ∣ primPart s",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh : r ∣ primPart (↑C (leadingCoeff r) * s)\nhC0 : ¬↑C (leadingCoeff r) = 0\n⊢ r ∣ primPart s",
"tactic": "rcases isUnit_primPart_C r.leadingCoeff with ⟨u, hu⟩"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh : r ∣ primPart (↑C (leadingCoeff r) * s)\nhC0 : ¬↑C (leadingCoeff r) = 0\nu : R[X]ˣ\nhu : ↑u = primPart (↑C (leadingCoeff r))\n⊢ primPart s * ↑u = primPart (↑C (leadingCoeff r) * s)",
"state_before": "case neg.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh : r ∣ primPart (↑C (leadingCoeff r) * s)\nhC0 : ¬↑C (leadingCoeff r) = 0\nu : R[X]ˣ\nhu : ↑u = primPart (↑C (leadingCoeff r))\n⊢ r ∣ primPart s",
"tactic": "apply h.trans (Associated.symm ⟨u, _⟩).dvd"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh✝ : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h✝\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : ¬natDegree s ≤ 0\nhcancel : natDegree (cancelLeads r s) < Nat.find con\nrcs : r ∣ ↑C (leadingCoeff r) * s - ↑C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r\nh : r ∣ primPart (↑C (leadingCoeff r) * s)\nhC0 : ¬↑C (leadingCoeff r) = 0\nu : R[X]ˣ\nhu : ↑u = primPart (↑C (leadingCoeff r))\n⊢ primPart s * ↑u = primPart (↑C (leadingCoeff r) * s)",
"tactic": "rw [primPart_mul (mul_ne_zero hC0 s0), hu, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ns : R[X]\nhs : ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s\n⊢ p ∣ s ∧ q ∣ s → r ∣ s",
"tactic": "apply hs s.natDegree s rfl"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nps : p ∣ s\nqs : q ∣ s\nrs : s = 0\n⊢ r ∣ s",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\n⊢ s ≠ 0",
"tactic": "contrapose! rs"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nps : p ∣ s\nqs : q ∣ s\nrs : s = 0\n⊢ r ∣ s",
"tactic": "simp [rs]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim✝ : IsPrimitive (↑C (coeff r 0))\nrprim : IsUnit (coeff r 0)\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : natDegree s ≤ 0\n⊢ False",
"state_before": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim : IsPrimitive r\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : natDegree s ≤ 0\n⊢ False",
"tactic": "rw [eq_C_of_natDegree_le_zero (le_trans hs sC), isPrimitive_iff_content_eq_one, content_C,\n normalize_eq_one] at rprim"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim✝ : IsPrimitive (↑C (coeff r 0))\nrprim : IsUnit (coeff r 0)\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs✝ : ¬↑C (coeff r 0) ∣ s\nrs : ¬coeff r 0 ∣ content s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : natDegree s ≤ 0\n⊢ False",
"state_before": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim✝ : IsPrimitive (↑C (coeff r 0))\nrprim : IsUnit (coeff r 0)\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs : ¬r ∣ s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : natDegree s ≤ 0\n⊢ False",
"tactic": "rw [eq_C_of_natDegree_le_zero (le_trans hs sC), ← dvd_content_iff_C_dvd] at rs"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : IsPrimitive q\nh : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : natDegree r = Nat.find h\nrprim✝ : IsPrimitive (↑C (coeff r 0))\nrprim : IsUnit (coeff r 0)\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s\ns : R[X]\nsdeg : natDegree s = Nat.find con\nrs✝ : ¬↑C (coeff r 0) ∣ s\nrs : ¬coeff r 0 ∣ content s\nps : p ∣ s\nqs : q ∣ s\ns0 : s ≠ 0\nhs : natDegree r ≤ natDegree s\nsC : natDegree s ≤ 0\n⊢ False",
"tactic": "apply rs rprim.dvd"
}
] |
[
476,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
isLittleO_rpow_exp_atTop
|
[
{
"state_after": "no goals",
"state_before": "s : ℝ\n⊢ (fun x => x ^ s) =o[atTop] exp",
"tactic": "simpa only [one_mul] using isLittleO_rpow_exp_pos_mul_atTop s one_pos"
}
] |
[
269,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
deriv_clm_comp
|
[] |
[
380,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.preimage_Icc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ↑e ⁻¹' Icc a b = Icc (↑(symm e) a) (↑(symm e) b)",
"tactic": "simp [← Ici_inter_Iic]"
}
] |
[
51,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.linear_bijective_iff
|
[] |
[
490,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean
|
CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_rightAdd
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nh₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f\n⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0",
"tactic": "exact ⟨⟨fun f => by simp [h₂ f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]⟩,\n ⟨fun f => by simp [h₁ f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]⟩⟩"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.desc 0 f = biprod.snd ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\n⊢ ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f",
"tactic": "intro f"
},
{
"state_after": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.inl ≫ biprod.desc 0 f = biprod.inl ≫ biprod.snd ≫ f\n\ncase h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.inr ≫ biprod.desc 0 f = biprod.inr ≫ biprod.snd ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.desc 0 f = biprod.snd ≫ f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.inl ≫ biprod.desc 0 f = biprod.inl ≫ biprod.snd ≫ f",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "case h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.inr ≫ biprod.desc 0 f = biprod.inr ≫ biprod.snd ≫ f",
"tactic": "simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nf : X ⟶ Y\n⊢ biprod.desc f 0 = biprod.fst ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\n⊢ ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f",
"tactic": "intro f"
},
{
"state_after": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nf : X ⟶ Y\n⊢ biprod.inl ≫ biprod.desc f 0 = biprod.inl ≫ biprod.fst ≫ f\n\ncase h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nf : X ⟶ Y\n⊢ biprod.inr ≫ biprod.desc f 0 = biprod.inr ≫ biprod.fst ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nf : X ⟶ Y\n⊢ biprod.desc f 0 = biprod.fst ≫ f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nf : X ⟶ Y\n⊢ biprod.inl ≫ biprod.desc f 0 = biprod.inl ≫ biprod.fst ≫ f",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "case h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nf : X ⟶ Y\n⊢ biprod.inr ≫ biprod.desc f 0 = biprod.inr ≫ biprod.fst ≫ f",
"tactic": "simp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nh₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f\nf : X ⟶ Y\n⊢ rightAdd X Y 0 f = f",
"tactic": "simp [h₂ f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f\nh₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f\nf : X ⟶ Y\n⊢ rightAdd X Y f 0 = f",
"tactic": "simp [h₁ f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]"
}
] |
[
86,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.symm_apply_apply
|
[] |
[
345,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
ModelWithCorners.continuousAt
|
[] |
[
227,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
11
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.untrop_pow
|
[] |
[
450,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.to_rightInvOn
|
[] |
[
156,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/Group/WithOne/Defs.lean
|
WithOne.coe_unone
|
[] |
[
144,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Combinatorics/Configuration.lean
|
Configuration.ProjectivePlane.two_lt_pointCount
|
[
{
"state_after": "no goals",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Finite P\ninst✝ : Finite L\nl : L\n⊢ 2 < pointCount P l",
"tactic": "simpa only [pointCount_eq P l, Nat.succ_lt_succ_iff] using one_lt_order P L"
}
] |
[
487,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.Nonempty.of_sUnion
|
[] |
[
1123,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1121,
1
] |
src/lean/Init/Data/Array/Basic.lean
|
Array.extLit
|
[] |
[
605,
71
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
601,
1
] |
Mathlib/Topology/Inseparable.lean
|
Continuous.specialization_monotone
|
[] |
[
246,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/RingTheory/RootsOfUnity/Complex.lean
|
IsPrimitiveRoot.arg_ext
|
[] |
[
118,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.comp_sub
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝¹⁶ : Ring R\nR₂ : Type u_2\ninst✝¹⁵ : Ring R₂\nR₃ : Type u_3\ninst✝¹⁴ : Ring R₃\nM : Type u_6\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : AddCommGroup M\nM₂ : Type u_4\ninst✝¹¹ : TopologicalSpace M₂\ninst✝¹⁰ : AddCommGroup M₂\nM₃ : Type u_5\ninst✝⁹ : TopologicalSpace M₃\ninst✝⁸ : AddCommGroup M₃\nM₄ : Type ?u.856610\ninst✝⁷ : TopologicalSpace M₄\ninst✝⁶ : AddCommGroup M₄\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : TopologicalAddGroup M₂\ninst✝ : TopologicalAddGroup M₃\ng : M₂ →SL[σ₂₃] M₃\nf₁ f₂ : M →SL[σ₁₂] M₂\nx✝ : M\n⊢ ↑(comp g (f₁ - f₂)) x✝ = ↑(comp g f₁ - comp g f₂) x✝",
"state_before": "R : Type u_1\ninst✝¹⁶ : Ring R\nR₂ : Type u_2\ninst✝¹⁵ : Ring R₂\nR₃ : Type u_3\ninst✝¹⁴ : Ring R₃\nM : Type u_6\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : AddCommGroup M\nM₂ : Type u_4\ninst✝¹¹ : TopologicalSpace M₂\ninst✝¹⁰ : AddCommGroup M₂\nM₃ : Type u_5\ninst✝⁹ : TopologicalSpace M₃\ninst✝⁸ : AddCommGroup M₃\nM₄ : Type ?u.856610\ninst✝⁷ : TopologicalSpace M₄\ninst✝⁶ : AddCommGroup M₄\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : TopologicalAddGroup M₂\ninst✝ : TopologicalAddGroup M₃\ng : M₂ →SL[σ₂₃] M₃\nf₁ f₂ : M →SL[σ₁₂] M₂\n⊢ comp g (f₁ - f₂) = comp g f₁ - comp g f₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝¹⁶ : Ring R\nR₂ : Type u_2\ninst✝¹⁵ : Ring R₂\nR₃ : Type u_3\ninst✝¹⁴ : Ring R₃\nM : Type u_6\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : AddCommGroup M\nM₂ : Type u_4\ninst✝¹¹ : TopologicalSpace M₂\ninst✝¹⁰ : AddCommGroup M₂\nM₃ : Type u_5\ninst✝⁹ : TopologicalSpace M₃\ninst✝⁸ : AddCommGroup M₃\nM₄ : Type ?u.856610\ninst✝⁷ : TopologicalSpace M₄\ninst✝⁶ : AddCommGroup M₄\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : TopologicalAddGroup M₂\ninst✝ : TopologicalAddGroup M₃\ng : M₂ →SL[σ₂₃] M₃\nf₁ f₂ : M →SL[σ₁₂] M₂\nx✝ : M\n⊢ ↑(comp g (f₁ - f₂)) x✝ = ↑(comp g f₁ - comp g f₂) x✝",
"tactic": "simp"
}
] |
[
1415,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1412,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.sameCycle_conj
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.37340\nα : Type u_1\nβ : Type ?u.37346\nf g : Perm α\np : α → Prop\nx y z : α\ni : ℤ\n⊢ ↑((g * f * g⁻¹) ^ i) x = y ↔ ↑(f ^ i) (↑g⁻¹ x) = ↑g⁻¹ y",
"tactic": "simp [conj_zpow, eq_inv_iff_eq]"
}
] |
[
121,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.Equiv.rfl
|
[] |
[
176,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
11
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocMod_eq_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocMod hp a b = toIocMod hp 0 (b - a) + a",
"tactic": "rw [toIocMod_sub_eq_sub, zero_add, sub_add_cancel]"
}
] |
[
781,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
hasStrictDerivAt_zpow
|
[
{
"state_after": "case inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\n\ncase inr.inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m = 0\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\n\ncase inr.inr\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : 0 < m\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"state_before": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "rcases lt_trichotomy m 0 with (hm | hm | hm)"
},
{
"state_after": "case intro\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ HasStrictDerivAt (fun x => x ^ ↑m) (↑↑m * x ^ (↑m - 1)) x",
"state_before": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℤ\nhm : 0 < m\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "lift m to ℕ using hm.le"
},
{
"state_after": "case intro\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (↑m - 1)) x",
"state_before": "case intro\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ HasStrictDerivAt (fun x => x ^ ↑m) (↑↑m * x ^ (↑m - 1)) x",
"tactic": "simp only [zpow_ofNat, Int.cast_ofNat]"
},
{
"state_after": "case h.e'_7.h.e'_6\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ x ^ (↑m - 1) = x ^ (m - 1)",
"state_before": "case intro\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (↑m - 1)) x",
"tactic": "convert hasStrictDerivAt_pow m x using 2"
},
{
"state_after": "case h.e'_7.h.e'_6\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ 1 ≤ m",
"state_before": "case h.e'_7.h.e'_6\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ x ^ (↑m - 1) = x ^ (m - 1)",
"tactic": "rw [← Int.ofNat_one, ← Int.ofNat_sub, zpow_ofNat]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h.e'_6\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝¹ m✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ 1 ≤ m",
"tactic": "norm_cast at hm"
},
{
"state_after": "case inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"state_before": "case inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "have hx : x ≠ 0 := h.resolve_right hm.not_le"
},
{
"state_after": "case inl.refine_2\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (Inv.inv ∘ fun x => x ^ (-m)) ((↑(-m) * x ^ (-m - 1)) • -((x ^ (-m)) ^ 2)⁻¹) x\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"state_before": "case inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "have := (hasStrictDerivAt_inv ?_).scomp _ (this (-m) (neg_pos.2 hm)) <;>\n [skip; exact zpow_ne_zero_of_ne_zero hx _]"
},
{
"state_after": "case h.e'_7\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ ↑m * x ^ (m - 1) = ↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹",
"state_before": "case inl.refine_2\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "convert this using 1"
},
{
"state_after": "case h.e'_7\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ ↑m * x ^ (m - 1) = ↑m * x ^ (-m - 1 + (m + m))",
"state_before": "case h.e'_7\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ ↑m * x ^ (m - 1) = ↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹",
"tactic": "rw [sq, mul_inv, inv_inv, Int.cast_neg, neg_mul, neg_mul_neg, ← zpow_add₀ hx, mul_assoc, ←\n zpow_add₀ hx]"
},
{
"state_after": "case h.e'_7.e_a.e_a\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ m - 1 = -m - 1 + (m + m)",
"state_before": "case h.e'_7\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ ↑m * x ^ (m - 1) = ↑m * x ^ (-m - 1 + (m + m))",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.e_a.e_a\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m < 0\nhx : x ≠ 0\nthis : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x\n⊢ m - 1 = -m - 1 + (m + m)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : m = 0\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "simp only [hm, zpow_zero, Int.cast_zero, zero_mul, hasStrictDerivAt_const]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm✝ m : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m\nthis : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x\nhm : 0 < m\n⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x",
"tactic": "exact this m hm"
}
] |
[
64,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Finsupp/ToDfinsupp.lean
|
finsuppLequivDfinsupp_symm_apply
|
[] |
[
272,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
lt_inv
|
[] |
[
296,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_const_mul_zpow_atTop_zero
|
[] |
[
173,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_id
|
[
{
"state_after": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, x ^ 1) = (b ^ (1 + 1) - a ^ (1 + 1)) / (↑1 + 1)\n⊢ (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2",
"tactic": "have := @integral_pow a b 1"
},
{
"state_after": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2\n⊢ (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2",
"state_before": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, x ^ 1) = (b ^ (1 + 1) - a ^ (1 + 1)) / (↑1 + 1)\n⊢ (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2",
"tactic": "norm_num at this"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2\n⊢ (∫ (x : ℝ) in a..b, x) = (b ^ 2 - a ^ 2) / 2",
"tactic": "exact this"
}
] |
[
414,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/Analysis/Complex/ReImTopology.lean
|
Complex.frontier_preimage_im
|
[] |
[
93,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.image_smul_comm
|
[] |
[
759,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_insert_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.48083\ninst✝ : PartialOrder α\na b c : α\nh : a ≤ b\n⊢ insert b (Ico a b) = Icc a b",
"tactic": "rw [insert_eq, union_comm, Ico_union_right h]"
}
] |
[
877,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
876,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_eq_zero_iff_isEmpty
|
[] |
[
232,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Topology/Maps.lean
|
OpenEmbedding.open_iff_image_open
|
[
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236323\nδ : Type ?u.236326\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : OpenEmbedding f\ns : Set α\nh : IsOpen (f '' s)\n⊢ f ⁻¹' (f '' s) = s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.236323\nδ : Type ?u.236326\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : OpenEmbedding f\ns : Set α\nh : IsOpen (f '' s)\n⊢ IsOpen s",
"tactic": "convert ← h.preimage hf.toEmbedding.continuous"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236323\nδ : Type ?u.236326\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : OpenEmbedding f\ns : Set α\nh : IsOpen (f '' s)\n⊢ f ⁻¹' (f '' s) = s",
"tactic": "apply preimage_image_eq _ hf.inj"
}
] |
[
565,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.inv_le_inv'
|
[] |
[
1504,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1503,
21
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.l_iSup
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.18544\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (l '' range f) (l (sSup (range f)))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.18544\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (range (l ∘ f)) (l (iSup f))",
"tactic": "rw [range_comp, ← sSup_range]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.18544\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (l '' range f) (l (sSup (range f)))",
"tactic": "exact gc.isLUB_l_image (isLUB_sSup _)"
}
] |
[
292,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.Step.diamond
|
[] |
[
223,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.none_le
|
[] |
[
212,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.multiset_sum_mem
|
[] |
[
347,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
11
] |
Std/Logic.lean
|
not_exists
|
[] |
[
419,
67
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
419,
9
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
Ideal.dvd_span_singleton
|
[] |
[
703,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.abs_top
|
[] |
[
1056,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1056,
9
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.IsImage.iff_preimage_eq'
|
[] |
[
536,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/LinearAlgebra/TensorPower.lean
|
PiTensorProduct.gradedMonoid_eq_of_reindex_cast
|
[
{
"state_after": "R : Type u_4\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nιι : Type u_1\nι : ιι → Type u_2\nai : ιι\na b : ⨂[R] (x : ι ai), M\nh : ↑(reindex R M (Equiv.cast (_ : ι { fst := ai, snd := a }.fst = ι { fst := ai, snd := b }.fst))) a = b\n⊢ { fst := ai, snd := a } = { fst := ai, snd := b }",
"state_before": "R : Type u_4\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nιι : Type u_1\nι : ιι → Type u_2\nai : ιι\na : ⨂[R] (x : ι ai), M\nbi : ιι\nb : ⨂[R] (x : ι bi), M\nhi : ai = bi\nh : ↑(reindex R M (Equiv.cast (_ : ι { fst := ai, snd := a }.fst = ι { fst := bi, snd := b }.fst))) a = b\n⊢ { fst := ai, snd := a } = { fst := bi, snd := b }",
"tactic": "subst hi"
},
{
"state_after": "no goals",
"state_before": "R : Type u_4\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nιι : Type u_1\nι : ιι → Type u_2\nai : ιι\na b : ⨂[R] (x : ι ai), M\nh : ↑(reindex R M (Equiv.cast (_ : ι { fst := ai, snd := a }.fst = ι { fst := ai, snd := b }.fst))) a = b\n⊢ { fst := ai, snd := a } = { fst := ai, snd := b }",
"tactic": "simp_all"
}
] |
[
60,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
IsEquivalent.summable_iff_nat
|
[] |
[
717,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/Data/W/Constructions.lean
|
WType.rightInverse_list
|
[
{
"state_after": "no goals",
"state_before": "γ : Type u\nhd : γ\ntl : List γ\n⊢ toList γ (ofList γ (hd :: tl)) = hd :: tl",
"tactic": "simp only [toList, rightInverse_list tl]"
}
] |
[
178,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
PadicSeq.norm_one
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nh1 : ¬1 ≈ 0\n⊢ norm 1 = 1",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ norm 1 = 1",
"tactic": "have h1 : ¬(1 : PadicSeq p) ≈ 0 := one_not_equiv_zero _"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nh1 : ¬1 ≈ 0\n⊢ norm 1 = 1",
"tactic": "simp [h1, norm, hp.1.one_lt]"
}
] |
[
325,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.pi_univ_Ioc_subset
|
[] |
[
80,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.le_inv_iff_mul_le
|
[
{
"state_after": "no goals",
"state_before": "r p : ℝ≥0\nh : p ≠ 0\n⊢ r ≤ p⁻¹ ↔ r * p ≤ 1",
"tactic": "rw [← mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]"
}
] |
[
791,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Odd.mod_even_iff
|
[] |
[
347,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
|
CategoryTheory.Functor.mapBicone_whisker
|
[] |
[
71,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
norm_expSeries_summable'
|
[] |
[
417,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_lt_inter_lt
|
[
{
"state_after": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝¹ : LinearOrder α\ninst✝ : NoMaxOrder α\na : α\n⊢ Bounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Bounded (fun x x_1 => x < x_1) s",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝¹ : LinearOrder α\ninst✝ : NoMaxOrder α\na : α\n⊢ Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Unbounded (fun x x_1 => x < x_1) s",
"tactic": "rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝¹ : LinearOrder α\ninst✝ : NoMaxOrder α\na : α\n⊢ Bounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Bounded (fun x x_1 => x < x_1) s",
"tactic": "exact bounded_lt_inter_lt a"
}
] |
[
389,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.map_pure
|
[] |
[
1034,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1033,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.