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/Data/Vector/Basic.lean
|
Vector.cons_val
|
[] |
[
58,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.mem_of_mem
|
[] |
[
296,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocMod_eq_self
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c : α\nn : ℤ\n⊢ ∃ z, b = b + z • p",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c : α\nn : ℤ\n⊢ toIocMod hp a b = b ↔ b ∈ Set.Ioc a (a + p)",
"tactic": "rw [toIocMod_eq_iff, and_iff_left]"
},
{
"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\n⊢ ∃ z, b = b + z • p",
"tactic": "exact ⟨0, by simp⟩"
},
{
"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\n⊢ b = b + 0 • p",
"tactic": "simp"
}
] |
[
726,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
724,
1
] |
Mathlib/Logic/Denumerable.lean
|
Denumerable.encode_ofNat
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.516\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nn : ℕ\na : ?m.551\nh : a ∈ decode n\ne : encode a = n\n⊢ encode (ofNat α n) = n",
"state_before": "α : Type u_1\nβ : Type ?u.516\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nn : ℕ\n⊢ encode (ofNat α n) = n",
"tactic": "obtain ⟨a, h, e⟩ := decode_inv n"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.516\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nn : ℕ\na : ?m.551\nh : a ∈ decode n\ne : encode a = n\n⊢ encode (ofNat α n) = n",
"tactic": "rwa [ofNat_of_decode h]"
}
] |
[
70,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Logic/Nontrivial.lean
|
subsingleton_or_nontrivial
|
[
{
"state_after": "α✝ : Type ?u.3498\nβ : Type ?u.3501\nα : Type u_1\n⊢ Nontrivial α ∨ ¬Nontrivial α",
"state_before": "α✝ : Type ?u.3498\nβ : Type ?u.3501\nα : Type u_1\n⊢ Subsingleton α ∨ Nontrivial α",
"tactic": "rw [← not_nontrivial_iff_subsingleton, or_comm]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.3498\nβ : Type ?u.3501\nα : Type u_1\n⊢ Nontrivial α ∨ ¬Nontrivial α",
"tactic": "exact Classical.em _"
}
] |
[
133,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
sInf_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\n⊢ sInf (s / t) = sInf s / sSup t",
"tactic": "simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]"
}
] |
[
97,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Data/Set/Intervals/WithBotTop.lean
|
WithBot.preimage_coe_Ico
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ some ⁻¹' Ico ↑a ↑b = Ico a b",
"tactic": "simp [← Ici_inter_Iio]"
}
] |
[
172,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.llift_symm_apply
|
[] |
[
434,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.or
|
[] |
[
724,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
11
] |
Mathlib/Algebra/GroupWithZero/Defs.lean
|
mul_right_cancel₀
|
[] |
[
86,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Tendsto.neBot
|
[] |
[
2932,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2930,
1
] |
Mathlib/MeasureTheory/Group/Action.lean
|
MeasureTheory.measure_smul
|
[] |
[
186,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDeriv_succ_eq_comp_left
|
[] |
[
1542,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1538,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_quadratic_lt_degree_C_mul_X_cb
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.1327898\nha : a ≠ 0\n⊢ degree (↑C b * X ^ 2 + ↑C c * X + ↑C d) < degree (↑C a * X ^ 3)",
"tactic": "simpa only [degree_C_mul_X_pow 3 ha] using degree_quadratic_lt"
}
] |
[
1222,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1220,
1
] |
Std/Data/List/Lemmas.lean
|
List.range'_sublist_right
|
[
{
"state_after": "no goals",
"state_before": "step s m n : Nat\nh : range' s m step <+ range' s n step\n⊢ m ≤ n",
"tactic": "simpa only [length_range'] using h.length_le"
},
{
"state_after": "step s m n : Nat\nh : m ≤ n\n⊢ range' s m step <+ range' s m step ++ range' (s + step * m) (n - m) step",
"state_before": "step s m n : Nat\nh : m ≤ n\n⊢ range' s m step <+ range' s n step",
"tactic": "rw [← Nat.sub_add_cancel h, ← range'_append]"
},
{
"state_after": "no goals",
"state_before": "step s m n : Nat\nh : m ≤ n\n⊢ range' s m step <+ range' s m step ++ range' (s + step * m) (n - m) step",
"tactic": "apply sublist_append_left"
}
] |
[
1859,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1857,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.tail_map
|
[] |
[
156,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Part.lean
|
Part.some_dom
|
[] |
[
143,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/MvPolynomial/Expand.lean
|
MvPolynomial.rename_comp_expand
|
[
{
"state_after": "case hf\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type ?u.347378\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\np : ℕ\nφ : σ\n⊢ ↑(AlgHom.comp (rename f) (expand p)) (X φ) = ↑(AlgHom.comp (expand p) (rename f)) (X φ)",
"state_before": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type ?u.347378\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\np : ℕ\n⊢ AlgHom.comp (rename f) (expand p) = AlgHom.comp (expand p) (rename f)",
"tactic": "ext1 φ"
},
{
"state_after": "no goals",
"state_before": "case hf\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type ?u.347378\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\np : ℕ\nφ : σ\n⊢ ↑(AlgHom.comp (rename f) (expand p)) (X φ) = ↑(AlgHom.comp (expand p) (rename f)) (X φ)",
"tactic": "simp only [rename_expand, AlgHom.comp_apply]"
}
] |
[
97,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.toEnd_nat_cast
|
[] |
[
392,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
lt_of_mul_lt_mul_left
|
[] |
[
172,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.mapIdx_nil
|
[] |
[
45,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.lt_rpow_iff_log_lt
|
[
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\n⊢ x < y ^ z ↔ log x < z * log y",
"tactic": "rw [← Real.log_lt_log_iff hx (Real.rpow_pos_of_pos hy z), Real.log_rpow hy]"
}
] |
[
603,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
PowerBasis.equivAdjoinSimple_aeval
|
[] |
[
1288,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1286,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero
|
[
{
"state_after": "𝕜 : Type ?u.3112345\nE : Type ?u.3112348\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ 2 = 0 ∨ ↑re (inner x y) = 0 ↔ inner x y = 0",
"state_before": "𝕜 : Type ?u.3112345\nE : Type ?u.3112348\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ inner x y = 0",
"tactic": "rw [@norm_sub_mul_self ℝ, add_right_cancel_iff, sub_eq_add_neg, add_right_eq_self, neg_eq_zero,\n mul_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.3112345\nE : Type ?u.3112348\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ 2 = 0 ∨ ↑re (inner x y) = 0 ↔ inner x y = 0",
"tactic": "norm_num"
}
] |
[
1493,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1489,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.nontrivial_iff_card
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype G\nhGnt : Nontrivial G\nk : ℕ\nhk : card G = 1\nhk0 : k = 0\n⊢ False",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype G\nhGnt : Nontrivial G\nk : ℕ\nhk : card G = p ^ k\nhk0 : k = 0\n⊢ False",
"tactic": "rw [hk0, pow_zero] at hk"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype G\nhGnt : Nontrivial G\nk : ℕ\nhk : card G = 1\nhk0 : k = 0\n⊢ False",
"tactic": "exact Fintype.one_lt_card.ne' hk"
}
] |
[
168,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.coe_rTensorHom
|
[] |
[
1033,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1032,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_eq_single
|
[
{
"state_after": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nthis : mulSupport (f ∘ PLift.down) ⊆ ↑{{ down := a }}\n⊢ (∏ᶠ (x : α), f x) = f a",
"state_before": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\n⊢ (∏ᶠ (x : α), f x) = f a",
"tactic": "have : mulSupport (f ∘ PLift.down) ⊆ ({PLift.up a} : Finset (PLift α)) := by\n intro x\n contrapose\n simpa [PLift.eq_up_iff_down_eq] using ha x.down"
},
{
"state_after": "no goals",
"state_before": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nthis : mulSupport (f ∘ PLift.down) ⊆ ↑{{ down := a }}\n⊢ (∏ᶠ (x : α), f x) = f a",
"tactic": "rw [finprod_eq_prod_pLift_of_mulSupport_subset this, Finset.prod_singleton]"
},
{
"state_after": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nx : PLift α\n⊢ x ∈ mulSupport (f ∘ PLift.down) → x ∈ ↑{{ down := a }}",
"state_before": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\n⊢ mulSupport (f ∘ PLift.down) ⊆ ↑{{ down := a }}",
"tactic": "intro x"
},
{
"state_after": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nx : PLift α\n⊢ ¬x ∈ ↑{{ down := a }} → ¬x ∈ mulSupport (f ∘ PLift.down)",
"state_before": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nx : PLift α\n⊢ x ∈ mulSupport (f ∘ PLift.down) → x ∈ ↑{{ down := a }}",
"tactic": "contrapose"
},
{
"state_after": "no goals",
"state_before": "G : Type ?u.34693\nM : Type u_2\nN : Type ?u.34699\nα : Sort u_1\nβ : Sort ?u.34705\nι : Sort ?u.34708\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nx : PLift α\n⊢ ¬x ∈ ↑{{ down := a }} → ¬x ∈ mulSupport (f ∘ PLift.down)",
"tactic": "simpa [PLift.eq_up_iff_down_eq] using ha x.down"
}
] |
[
224,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Order/Disjointed.lean
|
preimage_find_eq_disjointed
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.14611\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\nx : α\n⊢ x ∈ (fun x => Nat.find (_ : ∃ n, x ∈ s n)) ⁻¹' {n} ↔ x ∈ disjointed s n",
"state_before": "α : Type u_1\nβ : Type ?u.14611\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\n⊢ (fun x => Nat.find (_ : ∃ n, x ∈ s n)) ⁻¹' {n} = disjointed s n",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.14611\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\nx : α\n⊢ x ∈ (fun x => Nat.find (_ : ∃ n, x ∈ s n)) ⁻¹' {n} ↔ x ∈ disjointed s n",
"tactic": "simp [Nat.find_eq_iff, disjointed_eq_inter_compl]"
}
] |
[
181,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.Pairwise.of_map
|
[] |
[
191,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.supp_eq_of_isUniform
|
[
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\n⊢ supp (abs { fst := a, snd := f }) i✝ = f i✝ '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ ∀ (i : Fin2 n), supp (abs { fst := a, snd := f }) i = f i '' univ",
"tactic": "intro"
},
{
"state_after": "case h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ u ∈ supp (abs { fst := a, snd := f }) i✝ ↔ u ∈ f i✝ '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\n⊢ supp (abs { fst := a, snd := f }) i✝ = f i✝ '' univ",
"tactic": "ext u"
},
{
"state_after": "case h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ (∀ (a_1 : (P F).A) (f_1 : MvPFunctor.B (P F) a_1 ⟹ α),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ) ↔\n u ∈ f i✝ '' univ",
"state_before": "case h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ u ∈ supp (abs { fst := a, snd := f }) i✝ ↔ u ∈ f i✝ '' univ",
"tactic": "rw [mem_supp]"
},
{
"state_after": "case h.mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ (∀ (a_1 : (P F).A) (f_1 : MvPFunctor.B (P F) a_1 ⟹ α),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ) →\n u ∈ f i✝ '' univ\n\ncase h.mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ u ∈ f i✝ '' univ →\n ∀ (a_2 : (P F).A) (f_1 : MvPFunctor.B (P F) a_2 ⟹ α),\n abs { fst := a_2, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ",
"state_before": "case h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ (∀ (a_1 : (P F).A) (f_1 : MvPFunctor.B (P F) a_1 ⟹ α),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ) ↔\n u ∈ f i✝ '' univ",
"tactic": "constructor"
},
{
"state_after": "case h.mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\nh' : u ∈ f i✝ '' univ\na' : (P F).A\nf' : MvPFunctor.B (P F) a' ⟹ α\ne : abs { fst := a', snd := f' } = abs { fst := a, snd := f }\n⊢ u ∈ f' i✝ '' univ",
"state_before": "case h.mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ u ∈ f i✝ '' univ →\n ∀ (a_2 : (P F).A) (f_1 : MvPFunctor.B (P F) a_2 ⟹ α),\n abs { fst := a_2, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ",
"tactic": "intro h' a' f' e"
},
{
"state_after": "case h.mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\nh' : u ∈ f i✝ '' univ\na' : (P F).A\nf' : MvPFunctor.B (P F) a' ⟹ α\ne : abs { fst := a', snd := f' } = abs { fst := a, snd := f }\n⊢ u ∈ f i✝ '' univ",
"state_before": "case h.mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\nh' : u ∈ f i✝ '' univ\na' : (P F).A\nf' : MvPFunctor.B (P F) a' ⟹ α\ne : abs { fst := a', snd := f' } = abs { fst := a, snd := f }\n⊢ u ∈ f' i✝ '' univ",
"tactic": "rw [← h _ _ _ _ e.symm]"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\nh' : u ∈ f i✝ '' univ\na' : (P F).A\nf' : MvPFunctor.B (P F) a' ⟹ α\ne : abs { fst := a', snd := f' } = abs { fst := a, snd := f }\n⊢ u ∈ f i✝ '' univ",
"tactic": "apply h'"
},
{
"state_after": "case h.mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\nh' :\n ∀ (a_1 : (P F).A) (f_1 : MvPFunctor.B (P F) a_1 ⟹ α),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ\n⊢ u ∈ f i✝ '' univ",
"state_before": "case h.mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\n⊢ (∀ (a_1 : (P F).A) (f_1 : MvPFunctor.B (P F) a_1 ⟹ α),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ) →\n u ∈ f i✝ '' univ",
"tactic": "intro h'"
},
{
"state_after": "no goals",
"state_before": "case h.mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\ni✝ : Fin2 n\nu : α i✝\nh' :\n ∀ (a_1 : (P F).A) (f_1 : MvPFunctor.B (P F) a_1 ⟹ α),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } → u ∈ f_1 i✝ '' univ\n⊢ u ∈ f i✝ '' univ",
"tactic": "apply h' _ _ rfl"
}
] |
[
236,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.pi_div_two_pos
|
[] |
[
174,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Order/Grade.lean
|
grade_self
|
[] |
[
238,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/CategoryTheory/Abelian/Exact.lean
|
CategoryTheory.Abelian.exact_iff_image_eq_kernel
|
[
{
"state_after": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g → imageSubobject f = kernelSubobject g\n\ncase mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ imageSubobject f = kernelSubobject g → Exact f g",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g ↔ imageSubobject f = kernelSubobject g",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Exact f g\n⊢ imageSubobject f = kernelSubobject g",
"state_before": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g → imageSubobject f = kernelSubobject g",
"tactic": "intro h"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Exact f g\nthis : IsIso (imageToKernel f g (_ : f ≫ g = 0))\n⊢ imageSubobject f = kernelSubobject g",
"state_before": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Exact f g\n⊢ imageSubobject f = kernelSubobject g",
"tactic": "have : IsIso (imageToKernel f g h.w) := have := h.epi; isIso_of_mono_of_epi _"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Exact f g\nthis : IsIso (imageToKernel f g (_ : f ≫ g = 0))\n⊢ (asIso (imageToKernel f g (_ : f ≫ g = 0))).hom ≫ Subobject.arrow (kernelSubobject g) =\n Subobject.arrow (imageSubobject f)",
"state_before": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Exact f g\nthis : IsIso (imageToKernel f g (_ : f ≫ g = 0))\n⊢ imageSubobject f = kernelSubobject g",
"tactic": "refine Subobject.eq_of_comm (asIso (imageToKernel _ _ h.w)) ?_"
},
{
"state_after": "no goals",
"state_before": "case mp\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Exact f g\nthis : IsIso (imageToKernel f g (_ : f ≫ g = 0))\n⊢ (asIso (imageToKernel f g (_ : f ≫ g = 0))).hom ≫ Subobject.arrow (kernelSubobject g) =\n Subobject.arrow (imageSubobject f)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ imageSubobject f = kernelSubobject g → Exact f g",
"tactic": "apply exact_of_image_eq_kernel"
}
] |
[
66,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.coprod_apply
|
[] |
[
688,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.node3L
|
[] |
[
1133,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1130,
1
] |
Mathlib/Order/Interval.lean
|
Interval.dual_top
|
[] |
[
445,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.productMap_left
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_1\nB : Type u_4\nS : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ ∀ (x : A), ↑(AlgHom.comp (productMap f g) includeLeft) x = ↑f x",
"tactic": "simp"
}
] |
[
975,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
974,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
id_eq_sum_orthogonalProjection_self_orthogonalComplement
|
[
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.968584\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace { x // x ∈ K }\nw : E\n⊢ ↑(ContinuousLinearMap.id 𝕜 E) w =\n ↑(ContinuousLinearMap.comp (Submodule.subtypeL K) (orthogonalProjection K) +\n ContinuousLinearMap.comp (Submodule.subtypeL Kᗮ) (orthogonalProjection Kᗮ))\n w",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.968584\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace { x // x ∈ K }\n⊢ ContinuousLinearMap.id 𝕜 E =\n ContinuousLinearMap.comp (Submodule.subtypeL K) (orthogonalProjection K) +\n ContinuousLinearMap.comp (Submodule.subtypeL Kᗮ) (orthogonalProjection Kᗮ)",
"tactic": "ext w"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.968584\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace { x // x ∈ K }\nw : E\n⊢ ↑(ContinuousLinearMap.id 𝕜 E) w =\n ↑(ContinuousLinearMap.comp (Submodule.subtypeL K) (orthogonalProjection K) +\n ContinuousLinearMap.comp (Submodule.subtypeL Kᗮ) (orthogonalProjection Kᗮ))\n w",
"tactic": "exact eq_sum_orthogonalProjection_self_orthogonalComplement K w"
}
] |
[
1046,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.modByMonic_eq_sub_mul_div
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\n⊢ p %ₘ q = p - q * (p /ₘ q)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n⊢ p %ₘ q = p - q * (p /ₘ q)",
"tactic": "have _wf := div_wf_lemma h hq"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ p %ₘ q = p - q * (p /ₘ q)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\n⊢ p %ₘ q = p - q * (p /ₘ q)",
"tactic": "have ih :=\n modByMonic_eq_sub_mul_div (p - C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ p %ₘ q = p - q * (p /ₘ q)",
"tactic": "unfold modByMonic divByMonic divModByMonicAux"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).fst\n else 0",
"tactic": "dsimp"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd).snd =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"tactic": "rw [dif_pos hq, if_pos h]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).snd =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd).snd =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) %ₘ q =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd).snd =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"tactic": "rw [modByMonic, dif_pos hq] at ih"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).snd =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).snd =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd).snd =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"tactic": "refine' ih.trans _"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).snd =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q *\n if hq : Monic q then (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).fst\n else 0) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).snd =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"tactic": "unfold divByMonic"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : degree q ≤ degree p ∧ p ≠ 0\n_wf : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p\nih :\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).snd =\n p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q * ((p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) /ₘ q)\n⊢ (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q -\n q *\n if hq : Monic q then (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) hq).fst\n else 0) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"tactic": "rw [dif_pos hq, dif_pos hq, if_pos h, mul_add, sub_add_eq_sub_sub, mul_comm]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : ¬(degree q ≤ degree p ∧ p ≠ 0)\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : ¬(degree q ≤ degree p ∧ p ≠ 0)\n⊢ p %ₘ q = p - q * (p /ₘ q)",
"tactic": "unfold modByMonic divByMonic divModByMonicAux"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : ¬(degree q ≤ degree p ∧ p ≠ 0)\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : ¬(degree q ≤ degree p ∧ p ≠ 0)\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).fst\n else 0",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhq : Monic q\nh : ¬(degree q ≤ degree p ∧ p ≠ 0)\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).snd\n else p) =\n p -\n q *\n if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0",
"tactic": "rw [dif_pos hq, if_neg h, dif_pos hq, if_neg h, MulZeroClass.mul_zero, sub_zero]"
}
] |
[
249,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
ContinuousLinearMap.isBoundedLinearMap_comp_right
|
[] |
[
464,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.mapFun.one
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_2\nS : Type u_1\nT : Type ?u.54735\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.54750\nβ : Type ?u.54753\nf : R →+* S\nx y : 𝕎 R\n⊢ mapFun (↑f) 1 = 1",
"tactic": "map_fun_tac"
}
] |
[
112,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean
|
Polynomial.prod_X_sub_C_coeff_card_pred
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type w\ns✝ : Finset ι\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\nhs : 0 < Finset.card s\n⊢ coeff (∏ i in s, (X - ↑C (f i))) (Finset.card s - 1) = -∑ i in s, f i",
"tactic": "simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type w\ns✝ : Finset ι\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\nhs : 0 < Finset.card s\n⊢ 0 < ↑Multiset.card (Multiset.map f s.val)",
"tactic": "simpa using hs"
}
] |
[
283,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.eq_bot_iff_forall
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.160395\nP : Type ?u.160398\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.160419\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n⊢ (∀ (x : M), x ∈ S ↔ x ∈ ⊥) ↔ ∀ (x : M), x ∈ S → x = 1",
"tactic": "simp (config := { contextual := true }) [iff_def, S.one_mem]"
}
] |
[
1364,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1363,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.mk_coe
|
[] |
[
176,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.tendsto_zero_of_tendsto
|
[
{
"state_after": "α✝ : Type ?u.639926\nβ : Type ?u.639929\nE✝ : Type ?u.639932\nF : Type ?u.639935\nG : Type ?u.639938\nE' : Type ?u.639941\nF' : Type ?u.639944\nG' : Type ?u.639947\nE'' : Type ?u.639950\nF'' : Type ?u.639953\nG'' : Type ?u.639956\nR : Type ?u.639959\nR' : Type ?u.639962\n𝕜✝ : Type ?u.639965\n𝕜' : Type ?u.639968\ninst✝¹⁴ : Norm E✝\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜✝\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α✝ → E✝\ng : α✝ → F\nk : α✝ → G\nf' : α✝ → E'\ng' : α✝ → F'\nk' : α✝ → G'\nf'' : α✝ → E''\ng'' : α✝ → F''\nk'' : α✝ → G''\nl✝ l' : Filter α✝\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedField 𝕜\nu : α → E\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\nh : u =o[l] fun _x => 1\n⊢ Tendsto u l (𝓝 0)\n\ncase h\nα✝ : Type ?u.639926\nβ : Type ?u.639929\nE✝ : Type ?u.639932\nF : Type ?u.639935\nG : Type ?u.639938\nE' : Type ?u.639941\nF' : Type ?u.639944\nG' : Type ?u.639947\nE'' : Type ?u.639950\nF'' : Type ?u.639953\nG'' : Type ?u.639956\nR : Type ?u.639959\nR' : Type ?u.639962\n𝕜✝ : Type ?u.639965\n𝕜' : Type ?u.639968\ninst✝¹⁴ : Norm E✝\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜✝\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α✝ → E✝\ng : α✝ → F\nk : α✝ → G\nf' : α✝ → E'\ng' : α✝ → F'\nk' : α✝ → G'\nf'' : α✝ → E''\ng'' : α✝ → F''\nk'' : α✝ → G''\nl✝ l' : Filter α✝\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedField 𝕜\nu : α → E\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\n⊢ u =o[l] fun _x => 1",
"state_before": "α✝ : Type ?u.639926\nβ : Type ?u.639929\nE✝ : Type ?u.639932\nF : Type ?u.639935\nG : Type ?u.639938\nE' : Type ?u.639941\nF' : Type ?u.639944\nG' : Type ?u.639947\nE'' : Type ?u.639950\nF'' : Type ?u.639953\nG'' : Type ?u.639956\nR : Type ?u.639959\nR' : Type ?u.639962\n𝕜✝ : Type ?u.639965\n𝕜' : Type ?u.639968\ninst✝¹⁴ : Norm E✝\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜✝\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α✝ → E✝\ng : α✝ → F\nk : α✝ → G\nf' : α✝ → E'\ng' : α✝ → F'\nk' : α✝ → G'\nf'' : α✝ → E''\ng'' : α✝ → F''\nk'' : α✝ → G''\nl✝ l' : Filter α✝\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedField 𝕜\nu : α → E\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\n⊢ Tendsto u l (𝓝 0)",
"tactic": "suffices h : u =o[l] fun _x => (1 : 𝕜)"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type ?u.639926\nβ : Type ?u.639929\nE✝ : Type ?u.639932\nF : Type ?u.639935\nG : Type ?u.639938\nE' : Type ?u.639941\nF' : Type ?u.639944\nG' : Type ?u.639947\nE'' : Type ?u.639950\nF'' : Type ?u.639953\nG'' : Type ?u.639956\nR : Type ?u.639959\nR' : Type ?u.639962\n𝕜✝ : Type ?u.639965\n𝕜' : Type ?u.639968\ninst✝¹⁴ : Norm E✝\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜✝\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α✝ → E✝\ng : α✝ → F\nk : α✝ → G\nf' : α✝ → E'\ng' : α✝ → F'\nk' : α✝ → G'\nf'' : α✝ → E''\ng'' : α✝ → F''\nk'' : α✝ → G''\nl✝ l' : Filter α✝\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedField 𝕜\nu : α → E\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\n⊢ u =o[l] fun _x => 1",
"tactic": "exact huv.trans_isBigO (hv.isBigO_one 𝕜)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.639926\nβ : Type ?u.639929\nE✝ : Type ?u.639932\nF : Type ?u.639935\nG : Type ?u.639938\nE' : Type ?u.639941\nF' : Type ?u.639944\nG' : Type ?u.639947\nE'' : Type ?u.639950\nF'' : Type ?u.639953\nG'' : Type ?u.639956\nR : Type ?u.639959\nR' : Type ?u.639962\n𝕜✝ : Type ?u.639965\n𝕜' : Type ?u.639968\ninst✝¹⁴ : Norm E✝\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜✝\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α✝ → E✝\ng : α✝ → F\nk : α✝ → G\nf' : α✝ → E'\ng' : α✝ → F'\nk' : α✝ → G'\nf'' : α✝ → E''\ng'' : α✝ → F''\nk'' : α✝ → G''\nl✝ l' : Filter α✝\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedField 𝕜\nu : α → E\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\nh : u =o[l] fun _x => 1\n⊢ Tendsto u l (𝓝 0)",
"tactic": "rwa [isLittleO_one_iff] at h"
}
] |
[
2018,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2013,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.isPwo_of_finite
|
[] |
[
447,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
9
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.le_of_exists
|
[] |
[
797,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_mono
|
[] |
[
1319,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1319,
1
] |
Mathlib/Data/List/Lex.lean
|
List.Lex.to_ne
|
[] |
[
153,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Data/Finset/Functor.lean
|
Finset.map_comp_coe
|
[] |
[
213,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.fuzzy_irrefl
|
[] |
[
916,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
916,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Subspace.lean
|
Projectivization.Subspace.mem_carrier_iff
|
[] |
[
67,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.map_surjective
|
[
{
"state_after": "case intro.inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nc : γ\na : α\nh : Sum.map f g (inl a) = inl c\n⊢ ∃ a, f a = c\n\ncase intro.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nc : γ\nb : β\nh : Sum.map f g (inr b) = inl c\n⊢ ∃ a, f a = c",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh : Surjective (Sum.map f g)\nc : γ\n⊢ ∃ a, f a = c",
"tactic": "obtain ⟨a | b, h⟩ := h (inl c)"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nc : γ\na : α\nh : Sum.map f g (inl a) = inl c\n⊢ ∃ a, f a = c",
"tactic": "exact ⟨a, inl_injective h⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nc : γ\nb : β\nh : Sum.map f g (inr b) = inl c\n⊢ ∃ a, f a = c",
"tactic": "cases h"
},
{
"state_after": "case intro.inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nd : δ\na : α\nh : Sum.map f g (inl a) = inr d\n⊢ ∃ a, g a = d\n\ncase intro.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nd : δ\nb : β\nh : Sum.map f g (inr b) = inr d\n⊢ ∃ a, g a = d",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh : Surjective (Sum.map f g)\nd : δ\n⊢ ∃ a, g a = d",
"tactic": "obtain ⟨a | b, h⟩ := h (inr d)"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nd : δ\na : α\nh : Sum.map f g (inl a) = inr d\n⊢ ∃ a, g a = d",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_2\nδ : Type u_1\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nd : δ\nb : β\nh : Sum.map f g (inr b) = inr d\n⊢ ∃ a, g a = d",
"tactic": "exact ⟨b, inr_injective h⟩"
}
] |
[
614,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.div
|
[
{
"state_after": "F : Type ?u.56333\nG : Type ?u.56336\nα : Type u_1\nM : Type ?u.56342\nN : Type ?u.56345\ninst✝ : DivisionMonoid α\na b c : α\nha : IsUnit a\nhb : IsUnit b\n⊢ IsUnit (a * b⁻¹)",
"state_before": "F : Type ?u.56333\nG : Type ?u.56336\nα : Type u_1\nM : Type ?u.56342\nN : Type ?u.56345\ninst✝ : DivisionMonoid α\na b c : α\nha : IsUnit a\nhb : IsUnit b\n⊢ IsUnit (a / b)",
"tactic": "rw [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.56333\nG : Type ?u.56336\nα : Type u_1\nM : Type ?u.56342\nN : Type ?u.56345\ninst✝ : DivisionMonoid α\na b c : α\nha : IsUnit a\nhb : IsUnit b\n⊢ IsUnit (a * b⁻¹)",
"tactic": "exact ha.mul hb.inv"
}
] |
[
399,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.SuperpolynomialDecay.const_mul
|
[] |
[
104,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
bddAbove_singleton
|
[] |
[
633,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_add_cancel_iff_le
|
[
{
"state_after": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ a + (b - a) = b ↔ a ≤ b",
"state_before": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ b - a + a = b ↔ a ≤ b",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ a + (b - a) = b ↔ a ≤ b",
"tactic": "exact add_tsub_cancel_iff_le"
}
] |
[
323,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
LaurentPolynomial.exists_T_pow
|
[
{
"state_after": "case refine_1\nR : Type u_1\ninst✝ : Semiring R\n⊢ ∀ (p q : R[T;T⁻¹]),\n (∃ n f', ↑toLaurent f' = p * T ↑n) → (∃ n f', ↑toLaurent f' = q * T ↑n) → ∃ n f', ↑toLaurent f' = (p + q) * T ↑n\n\ncase refine_2\nR : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\n⊢ ∃ n_1 f', ↑toLaurent f' = ↑C a * T n * T ↑n_1",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\n⊢ ∃ n f', ↑toLaurent f' = f * T ↑n",
"tactic": "refine f.induction_on' ?_ fun n a => ?_ <;> clear f"
},
{
"state_after": "case refine_1.intro.intro.intro.intro\nR : Type u_1\ninst✝ : Semiring R\nf g : R[T;T⁻¹]\nm : ℕ\nfn : R[X]\nhf : ↑toLaurent fn = f * T ↑m\nn : ℕ\ngn : R[X]\nhg : ↑toLaurent gn = g * T ↑n\n⊢ ∃ n f', ↑toLaurent f' = (f + g) * T ↑n",
"state_before": "case refine_1\nR : Type u_1\ninst✝ : Semiring R\n⊢ ∀ (p q : R[T;T⁻¹]),\n (∃ n f', ↑toLaurent f' = p * T ↑n) → (∃ n f', ↑toLaurent f' = q * T ↑n) → ∃ n f', ↑toLaurent f' = (p + q) * T ↑n",
"tactic": "rintro f g ⟨m, fn, hf⟩ ⟨n, gn, hg⟩"
},
{
"state_after": "case refine_1.intro.intro.intro.intro\nR : Type u_1\ninst✝ : Semiring R\nf g : R[T;T⁻¹]\nm : ℕ\nfn : R[X]\nhf : ↑toLaurent fn = f * T ↑m\nn : ℕ\ngn : R[X]\nhg : ↑toLaurent gn = g * T ↑n\n⊢ ↑toLaurent (fn * X ^ n + gn * X ^ m) = (f + g) * T ↑(m + n)",
"state_before": "case refine_1.intro.intro.intro.intro\nR : Type u_1\ninst✝ : Semiring R\nf g : R[T;T⁻¹]\nm : ℕ\nfn : R[X]\nhf : ↑toLaurent fn = f * T ↑m\nn : ℕ\ngn : R[X]\nhg : ↑toLaurent gn = g * T ↑n\n⊢ ∃ n f', ↑toLaurent f' = (f + g) * T ↑n",
"tactic": "refine' ⟨m + n, fn * X ^ n + gn * X ^ m, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_1.intro.intro.intro.intro\nR : Type u_1\ninst✝ : Semiring R\nf g : R[T;T⁻¹]\nm : ℕ\nfn : R[X]\nhf : ↑toLaurent fn = f * T ↑m\nn : ℕ\ngn : R[X]\nhg : ↑toLaurent gn = g * T ↑n\n⊢ ↑toLaurent (fn * X ^ n + gn * X ^ m) = (f + g) * T ↑(m + n)",
"tactic": "simp only [hf, hg, add_mul, add_comm (n : ℤ), map_add, map_mul, Polynomial.toLaurent_X_pow,\n mul_T_assoc, Int.ofNat_add]"
},
{
"state_after": "case refine_2.ofNat\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', ↑toLaurent f' = ↑C a * T (Int.ofNat n) * T ↑n_1\n\ncase refine_2.negSucc\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', ↑toLaurent f' = ↑C a * T (Int.negSucc n) * T ↑n_1",
"state_before": "case refine_2\nR : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\n⊢ ∃ n_1 f', ↑toLaurent f' = ↑C a * T n * T ↑n_1",
"tactic": "cases' n with n n"
},
{
"state_after": "no goals",
"state_before": "case refine_2.ofNat\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', ↑toLaurent f' = ↑C a * T (Int.ofNat n) * T ↑n_1",
"tactic": "exact ⟨0, Polynomial.C a * X ^ n, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ↑toLaurent (↑Polynomial.C a * X ^ n) = ↑C a * T (Int.ofNat n) * T ↑0",
"tactic": "simp"
},
{
"state_after": "case refine_2.negSucc\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ↑toLaurent (↑Polynomial.C a) = ↑C a * T (Int.negSucc n) * T ↑(n + 1)",
"state_before": "case refine_2.negSucc\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', ↑toLaurent f' = ↑C a * T (Int.negSucc n) * T ↑n_1",
"tactic": "refine' ⟨n + 1, Polynomial.C a, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_2.negSucc\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ↑toLaurent (↑Polynomial.C a) = ↑C a * T (Int.negSucc n) * T ↑(n + 1)",
"tactic": "simp only [Int.negSucc_eq, Polynomial.toLaurent_C, Int.ofNat_succ, mul_T_assoc, add_left_neg,\n T_zero, mul_one]"
}
] |
[
403,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Algebra/Quandle.lean
|
Rack.PreEnvelGroupRel.trans
|
[] |
[
658,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
one_div_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.27916\nM₀ : Type ?u.27919\nG₀ : Type u_1\nM₀' : Type ?u.27925\nG₀' : Type ?u.27928\nF : Type ?u.27931\nF' : Type ?u.27934\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a ≠ 0\n⊢ 1 / a ≠ 0",
"tactic": "simpa only [one_div] using inv_ne_zero h"
}
] |
[
395,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.prod_div_prod_filter
|
[] |
[
962,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
960,
1
] |
Mathlib/Data/Nat/Totient.lean
|
Nat.filter_coprime_Ico_eq_totient
|
[
{
"state_after": "case pp\na n : ℕ\n⊢ Function.Periodic (coprime a) a",
"state_before": "a n : ℕ\n⊢ card (filter (coprime a) (Ico n (n + a))) = φ a",
"tactic": "rw [totient, filter_Ico_card_eq_of_periodic, count_eq_card_filter_range]"
},
{
"state_after": "no goals",
"state_before": "case pp\na n : ℕ\n⊢ Function.Periodic (coprime a) a",
"tactic": "exact periodic_coprime a"
}
] |
[
82,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableEquiv.apply_symm_apply
|
[] |
[
1264,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1263,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.sum
|
[] |
[
261,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Std/Data/Array/Lemmas.lean
|
getElem?_neg
|
[] |
[
27,
90
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
26,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sub_nonempty
|
[] |
[
534,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Surjective.forall₂
|
[] |
[
200,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
11
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.biInter_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nf g : Filter α\ns✝ t : Set α\nβ : Type v\ns : β → Set α\nis : Set β\nhf : Set.Finite is\n⊢ (⋂ (i : β) (_ : i ∈ ∅), s i) ∈ f ↔ ∀ (i : β), i ∈ ∅ → s i ∈ f",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nf g : Filter α\ns✝¹ t : Set α\nβ : Type v\ns : β → Set α\nis : Set β\nhf : Set.Finite is\na✝ : β\ns✝ : Set β\nx✝¹ : ¬a✝ ∈ s✝\nx✝ : Set.Finite s✝\nhs : (⋂ (i : β) (_ : i ∈ s✝), s i) ∈ f ↔ ∀ (i : β), i ∈ s✝ → s i ∈ f\n⊢ (⋂ (i : β) (_ : i ∈ insert a✝ s✝), s i) ∈ f ↔ ∀ (i : β), i ∈ insert a✝ s✝ → s i ∈ f",
"tactic": "simp [hs]"
}
] |
[
187,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.atTop_basis
|
[] |
[
133,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLinearMapRight'_one
|
[
{
"state_after": "case h.h.h\nR : Type u_1\ninst✝² : Semiring R\nl : Type ?u.170495\nm : Type u_2\nn : Type ?u.170501\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\ni✝ x✝ : m\n⊢ ↑(comp (↑toLinearMapRight' 1) (single i✝)) 1 x✝ = ↑(comp LinearMap.id (single i✝)) 1 x✝",
"state_before": "R : Type u_1\ninst✝² : Semiring R\nl : Type ?u.170495\nm : Type u_2\nn : Type ?u.170501\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\n⊢ ↑toLinearMapRight' 1 = LinearMap.id",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nR : Type u_1\ninst✝² : Semiring R\nl : Type ?u.170495\nm : Type u_2\nn : Type ?u.170501\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\ni✝ x✝ : m\n⊢ ↑(comp (↑toLinearMapRight' 1) (single i✝)) 1 x✝ = ↑(comp LinearMap.id (single i✝)) 1 x✝",
"tactic": "simp [LinearMap.one_apply, stdBasis_apply]"
}
] |
[
169,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
exists_measurable_piecewise_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.100202\nδ : Type ?u.100205\nδ' : Type ?u.100208\nι : Sort uι\ns t✝ u : Set α\nm✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nm : MeasurableSpace α\nt : ℕ → Set β\nt_meas : ∀ (n : ℕ), MeasurableSet (t n)\nt_disj : Pairwise (Disjoint on t)\ng : ℕ → β → α\nhg : ∀ (n : ℕ), Measurable (g n)\ni j : ℕ\nh : (Disjoint on t) i j\n⊢ EqOn (g i) (g j) (t i ∩ t j)",
"tactic": "simp only [h.inter_eq, eqOn_empty]"
}
] |
[
825,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Multiset.toFinset_zero
|
[] |
[
3143,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3142,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
ball_mul_singleton
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ ball x δ * {y} = ball (x * y) δ",
"tactic": "rw [mul_comm, singleton_mul_ball, mul_comm y]"
}
] |
[
118,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasureTheory.IsFundamentalDomain.map_restrict_quotient
|
[
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"tactic": "let π : G →* G ⧸ Γ := QuotientGroup.mk' Γ"
},
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"tactic": "have meas_π : Measurable π := continuous_quotient_mk'.measurable"
},
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"tactic": "have 𝓕meas : NullMeasurableSet 𝓕 μ := h𝓕.nullMeasurableSet"
},
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\nthis : Fact (↑↑μ 𝓕 < ⊤)\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"tactic": "haveI := Fact.mk h𝓕_finite"
},
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\nthis✝ : Fact (↑↑μ 𝓕 < ⊤)\nthis : IsMulLeftInvariant (map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕))\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\nthis : Fact (↑↑μ 𝓕 < ⊤)\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"tactic": "haveI : (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)).IsMulLeftInvariant :=\n h𝓕.isMulLeftInvariant_map"
},
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\nthis✝ : Fact (↑↑μ 𝓕 < ⊤)\nthis : IsMulLeftInvariant (map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕))\n⊢ MeasurableSet ↑K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\nthis✝ : Fact (↑↑μ 𝓕 < ⊤)\nthis : IsMulLeftInvariant (map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕))\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) • haarMeasure K",
"tactic": "rw [Measure.haarMeasure_unique (Measure.map (QuotientGroup.mk' Γ) (μ.restrict 𝓕)) K,\n Measure.map_apply meas_π, Measure.restrict_apply₀' 𝓕meas, inter_comm]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nπ : G →* G ⧸ Γ := QuotientGroup.mk' Γ\nmeas_π : Measurable ↑π\n𝓕meas : NullMeasurableSet 𝓕\nthis✝ : Fact (↑↑μ 𝓕 < ⊤)\nthis : IsMulLeftInvariant (map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕))\n⊢ MeasurableSet ↑K",
"tactic": "exact K.isCompact.measurableSet"
}
] |
[
142,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Topology/UniformSpace/Compact.lean
|
nhdsSet_diagonal_eq_uniformity
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\n⊢ 𝓤 α ≤ 𝓝ˢ (diagonal α)",
"state_before": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\n⊢ 𝓝ˢ (diagonal α) = 𝓤 α",
"tactic": "refine' nhdsSet_diagonal_le_uniformity.antisymm _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\nthis :\n HasBasis (𝓤 (α × α)) (fun U => U ∈ 𝓤 α) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U\n⊢ 𝓤 α ≤ 𝓝ˢ (diagonal α)",
"state_before": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\n⊢ 𝓤 α ≤ 𝓝ˢ (diagonal α)",
"tactic": "have :\n (𝓤 (α × α)).HasBasis (fun U => U ∈ 𝓤 α) fun U =>\n (fun p : (α × α) × α × α => ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U := by\n rw [uniformity_prod_eq_comap_prod]\n exact (𝓤 α).basis_sets.prod_self.comap _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\nthis :\n HasBasis (𝓤 (α × α)) (fun U => U ∈ 𝓤 α) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ (⋃ (x : α × α) (_ : x ∈ diagonal α), ball x ((fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U)) ∈\n 𝓤 α",
"state_before": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\nthis :\n HasBasis (𝓤 (α × α)) (fun U => U ∈ 𝓤 α) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U\n⊢ 𝓤 α ≤ 𝓝ˢ (diagonal α)",
"tactic": "refine' (isCompact_diagonal.nhdsSet_basis_uniformity this).ge_iff.2 fun U hU => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\nthis :\n HasBasis (𝓤 (α × α)) (fun U => U ∈ 𝓤 α) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ (⋃ (x : α × α) (_ : x ∈ diagonal α), ball x ((fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U)) ∈\n 𝓤 α",
"tactic": "exact mem_of_superset hU fun ⟨x, y⟩ hxy => mem_iUnion₂.2\n ⟨(x, x), rfl, refl_mem_uniformity hU, hxy⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\n⊢ HasBasis (Filter.comap (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) (𝓤 α ×ˢ 𝓤 α)) (fun U => U ∈ 𝓤 α)\n fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U",
"state_before": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\n⊢ HasBasis (𝓤 (α × α)) (fun U => U ∈ 𝓤 α) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U",
"tactic": "rw [uniformity_prod_eq_comap_prod]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20\nγ : Type ?u.23\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompactSpace α\n⊢ HasBasis (Filter.comap (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) (𝓤 α ×ˢ 𝓤 α)) (fun U => U ∈ 𝓤 α)\n fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹' U ×ˢ U",
"tactic": "exact (𝓤 α).basis_sets.prod_self.comap _"
}
] |
[
62,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean
|
CategoryTheory.CommSq.left_adjoint
|
[
{
"state_after": "C : Type u_4\nD : Type u_1\ninst✝¹ : Category C\ninst✝ : Category D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\n⊢ G.map u ≫ adj.counit.app X ≫ p = G.map (u ≫ F.map p) ≫ adj.counit.app Y",
"state_before": "C : Type u_4\nD : Type u_1\ninst✝¹ : Category C\ninst✝ : Category D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\n⊢ ↑(Adjunction.homEquiv adj A X).symm u ≫ p = G.map i ≫ ↑(Adjunction.homEquiv adj B Y).symm v",
"tactic": "simp only [Adjunction.homEquiv_counit, assoc, ← G.map_comp_assoc, ← sq.w]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_4\nD : Type u_1\ninst✝¹ : Category C\ninst✝ : Category D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\n⊢ G.map u ≫ adj.counit.app X ≫ p = G.map (u ≫ F.map p) ≫ adj.counit.app Y",
"tactic": "rw [G.map_comp, assoc, Adjunction.counit_naturality]"
}
] |
[
92,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/CategoryTheory/Monoidal/Limits.lean
|
CategoryTheory.Limits.limLax_obj
|
[] |
[
114,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/CategoryTheory/Idempotents/HomologicalComplex.lean
|
CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comm_f
|
[] |
[
53,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.inner_eq_neg_mul_norm_of_angle_eq_pi
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nh : angle x y = π\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"tactic": "simp [← cos_angle_mul_norm_mul_norm, h]"
}
] |
[
242,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Linear.lean
|
ContinuousLinearMap.fderivWithin
|
[
{
"state_after": "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.38206\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.38301\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\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (↑e) x = e",
"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.38206\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.38301\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\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderivWithin 𝕜 (↑e) s x = e",
"tactic": "rw [DifferentiableAt.fderivWithin e.differentiableAt hxs]"
},
{
"state_after": "no goals",
"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.38206\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.38301\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\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (↑e) x = e",
"tactic": "exact e.fderiv"
}
] |
[
98,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
11
] |
Mathlib/Topology/Algebra/Monoid.lean
|
Submonoid.isClosed_topologicalClosure
|
[] |
[
451,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
tendsto_const_uniformity
|
[] |
[
506,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.vector_head
|
[] |
[
1301,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1300,
1
] |
Mathlib/Data/W/Cardinal.lean
|
WType.cardinal_mk_eq_sum
|
[
{
"state_after": "α : Type u\nβ : α → Type u\n⊢ (#WType β) = (#(i : α) × (β i → WType β))",
"state_before": "α : Type u\nβ : α → Type u\n⊢ (#WType β) = sum fun a => (#WType β) ^ (#β a)",
"tactic": "simp only [Cardinal.power_def, ← Cardinal.mk_sigma]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type u\n⊢ (#WType β) = (#(i : α) × (β i → WType β))",
"tactic": "exact mk_congr (equivSigma β)"
}
] |
[
46,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/FieldTheory/Normal.lean
|
Normal.of_isSplittingField
|
[
{
"state_after": "case inl\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\nhFEp : IsSplittingField F E 0\n⊢ Normal F E\n\ncase inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\n⊢ Normal F E",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\n⊢ Normal F E",
"tactic": "rcases eq_or_ne p 0 with (rfl | hp)"
},
{
"state_after": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\n⊢ IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)",
"state_before": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\n⊢ Normal F E",
"tactic": "refine' normal_iff.2 fun x => _"
},
{
"state_after": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\n⊢ IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)",
"state_before": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\n⊢ IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)",
"tactic": "have hFE : FiniteDimensional F E := IsSplittingField.finiteDimensional E p"
},
{
"state_after": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\n⊢ IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)",
"state_before": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\n⊢ IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)",
"tactic": "have Hx : IsIntegral F x := isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x"
},
{
"state_after": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\n⊢ ∀ {g : E[X]}, Irreducible g → g ∣ map (algebraMap F E) (minpoly F x) → degree g = 1",
"state_before": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\n⊢ IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)",
"tactic": "refine' ⟨Hx, Or.inr _⟩"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\n⊢ degree q = 1",
"state_before": "case inr\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\n⊢ ∀ {g : E[X]}, Irreducible g → g ∣ map (algebraMap F E) (minpoly F x) → degree g = 1",
"tactic": "rintro q q_irred ⟨r, hr⟩"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\n⊢ degree q = 1",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\n⊢ degree q = 1",
"tactic": "let D := AdjoinRoot q"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis : Fact (Irreducible q)\n⊢ degree q = 1",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\n⊢ degree q = 1",
"tactic": "haveI := Fact.mk q_irred"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\n⊢ degree q = 1",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis : Fact (Irreducible q)\n⊢ degree q = 1",
"tactic": "let pbED := AdjoinRoot.powerBasis q_irred.ne_zero"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis : FiniteDimensional E D\n⊢ degree q = 1",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\n⊢ degree q = 1",
"tactic": "haveI : FiniteDimensional E D := PowerBasis.finiteDimensional pbED"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\n⊢ degree q = 1",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis : FiniteDimensional E D\n⊢ degree q = 1",
"tactic": "have finrankED : FiniteDimensional.finrank E D = q.natDegree := by\n rw [PowerBasis.finrank pbED, AdjoinRoot.powerBasis_dim]"
},
{
"state_after": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\n⊢ degree q = 1",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\n⊢ degree q = 1",
"tactic": "haveI : FiniteDimensional F D := FiniteDimensional.trans F E D"
},
{
"state_after": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ degree q = 1\n\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "case inr.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\n⊢ degree q = 1",
"tactic": "rsuffices ⟨ϕ⟩ : Nonempty (D →ₐ[F] E)"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "let C := AdjoinRoot (minpoly F x)"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "haveI Hx_irred := Fact.mk (minpoly.irreducible Hx)"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "have heval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0 := by\n rw [algebraMap_eq F E D, ← eval₂_map, hr, AdjoinRoot.algebraMap_eq, eval₂_mul,\n AdjoinRoot.eval₂_root, MulZeroClass.zero_mul]"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝² : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝¹ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "letI : Algebra C D :=\n RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝³ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝² : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝¹ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝² : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝¹ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "letI : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (minpoly.aeval F x))"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁴ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝³ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝² : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝¹ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis : IsScalarTower F C D\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝³ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝² : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝¹ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "haveI : IsScalarTower F C D := of_algebraMap_eq fun y => (AdjoinRoot.lift_of heval).symm"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\n⊢ Nonempty (D →ₐ[F] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁴ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝³ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝² : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝¹ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis : IsScalarTower F C D\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "haveI : IsScalarTower F C E := by\n refine' of_algebraMap_eq fun y => (AdjoinRoot.lift_of _).symm\nrw [← aeval_def, minpoly.aeval]"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\n⊢ Nonempty (D →ₐ[C] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "suffices Nonempty (D →ₐ[C] E) by exact Nonempty.map (AlgHom.restrictScalars F) this"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Nonempty (D →ₐ[C] E)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\n⊢ Nonempty (D →ₐ[C] E)",
"tactic": "let S : Set D := ((p.map (algebraMap F E)).roots.map (algebraMap E D)).toFinset"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ ⊤ ≤ (IntermediateField.adjoin C S).toSubalgebra",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ ⊤ ≤ IntermediateField.adjoin C S",
"tactic": "rw [← IntermediateField.toSubalgebra_le_toSubalgebra, IntermediateField.top_toSubalgebra]"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Algebra.adjoin C S ≥ ⊤",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ ⊤ ≤ (IntermediateField.adjoin C S).toSubalgebra",
"tactic": "apply ge_trans (IntermediateField.algebra_adjoin_le_adjoin C S)"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Subalgebra.restrictScalars F (Algebra.adjoin C S) =\n Subalgebra.restrictScalars F (Algebra.adjoin E {AdjoinRoot.root q})",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Algebra.adjoin C S ≥ ⊤",
"tactic": "suffices\n (Algebra.adjoin C S).restrictScalars F =\n (Algebra.adjoin E {AdjoinRoot.root q}).restrictScalars F by\n rw [AdjoinRoot.adjoinRoot_eq_top, Subalgebra.restrictScalars_top, ←\n @Subalgebra.restrictScalars_top F C] at this\n exact top_le_iff.mpr (Subalgebra.restrictScalars_injective F this)"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Subalgebra.restrictScalars F\n (Algebra.adjoin C ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))) =\n Subalgebra.restrictScalars F (Algebra.adjoin E {AdjoinRoot.root q})",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Subalgebra.restrictScalars F (Algebra.adjoin C S) =\n Subalgebra.restrictScalars F (Algebra.adjoin E {AdjoinRoot.root q})",
"tactic": "change Subalgebra.restrictScalars F (Algebra.adjoin C\n (((p.map (algebraMap F E)).roots.map (algebraMap E D)).toFinset : Set D)) = _"
},
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"tactic": "rw [← Finset.image_toFinset, Finset.coe_image]"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁵ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁴ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝³ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝² : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝¹ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝ : IsScalarTower F C D\nthis : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\n⊢ Subalgebra.restrictScalars F (Algebra.adjoin E (↑(algebraMap C D) '' {AdjoinRoot.root (minpoly F x)})) =\n Subalgebra.restrictScalars F (Algebra.adjoin E {AdjoinRoot.root q})",
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"tactic": "apply\n Eq.trans\n (Algebra.adjoin_res_eq_adjoin_res F E C D hFEp.adjoin_roots AdjoinRoot.adjoinRoot_eq_top)"
},
{
"state_after": "no goals",
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"tactic": "rw [Set.image_singleton, RingHom.algebraMap_toAlgebra, AdjoinRoot.lift_root]"
},
{
"state_after": "case inl\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\nhFEp : IsSplittingField F E 0\nthis : Algebra.adjoin F ↑(Multiset.toFinset (roots (map (algebraMap F E) 0))) = ⊤\n⊢ Normal F E",
"state_before": "case inl\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\nhFEp : IsSplittingField F E 0\n⊢ Normal F E",
"tactic": "have := hFEp.adjoin_roots"
},
{
"state_after": "case inl\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\nhFEp : IsSplittingField F E 0\nthis : ⊥ = ⊤\n⊢ Normal F E",
"state_before": "case inl\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\nhFEp : IsSplittingField F E 0\nthis : Algebra.adjoin F ↑(Multiset.toFinset (roots (map (algebraMap F E) 0))) = ⊤\n⊢ Normal F E",
"tactic": "simp only [Polynomial.map_zero, roots_zero, Multiset.toFinset_zero, Finset.coe_empty,\n Algebra.adjoin_empty] at this"
},
{
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"state_before": "case inl\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\nhFEp : IsSplittingField F E 0\nthis : ⊥ = ⊤\n⊢ Normal F E",
"tactic": "exact\n Normal.of_algEquiv\n (AlgEquiv.ofBijective (Algebra.ofId F E) (Algebra.bijective_algebraMap_iff.2 this.symm))"
},
{
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"tactic": "rw [PowerBasis.finrank pbED, AdjoinRoot.powerBasis_dim]"
},
{
"state_after": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ degree q = ↑1",
"state_before": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ degree q = 1",
"tactic": "change degree q = ↑(1 : ℕ)"
},
{
"state_after": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ FiniteDimensional.finrank E D = 1",
"state_before": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ degree q = ↑1",
"tactic": "rw [degree_eq_iff_natDegree_eq q_irred.ne_zero, ← finrankED]"
},
{
"state_after": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\nnat_lemma : ∀ (a b c : ℕ), a * b = c → c ≤ a → 0 < c → b = 1\n⊢ FiniteDimensional.finrank E D = 1",
"state_before": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ FiniteDimensional.finrank E D = 1",
"tactic": "have nat_lemma : ∀ a b c : ℕ, a * b = c → c ≤ a → 0 < c → b = 1 := by\n intro a b c h1 h2 h3\n nlinarith"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\nnat_lemma : ∀ (a b c : ℕ), a * b = c → c ≤ a → 0 < c → b = 1\n⊢ FiniteDimensional.finrank E D = 1",
"tactic": "exact\n nat_lemma _ _ _ (FiniteDimensional.finrank_mul_finrank F E D)\n (LinearMap.finrank_le_finrank_of_injective\n (show Function.Injective ϕ.toLinearMap from ϕ.toRingHom.injective))\n FiniteDimensional.finrank_pos"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\na b c : ℕ\nh1 : a * b = c\nh2 : c ≤ a\nh3 : 0 < c\n⊢ b = 1",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\n⊢ ∀ (a b c : ℕ), a * b = c → c ≤ a → 0 < c → b = 1",
"tactic": "intro a b c h1 h2 h3"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nϕ : D →ₐ[F] E\na b c : ℕ\nh1 : a * b = c\nh2 : c ≤ a\nh3 : 0 < c\n⊢ b = 1",
"tactic": "nlinarith"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝¹ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\n⊢ eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0",
"tactic": "rw [algebraMap_eq F E D, ← eval₂_map, hr, AdjoinRoot.algebraMap_eq, eval₂_mul,\n AdjoinRoot.eval₂_root, MulZeroClass.zero_mul]"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁴ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝³ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝² : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝¹ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis : IsScalarTower F C D\ny : F\n⊢ eval₂ (algebraMap F E) x (minpoly F x) = 0",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁴ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝³ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝² : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝¹ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis : IsScalarTower F C D\n⊢ IsScalarTower F C E",
"tactic": "refine' of_algebraMap_eq fun y => (AdjoinRoot.lift_of _).symm"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁴ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝³ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝² : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝¹ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝ : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis : IsScalarTower F C D\ny : F\n⊢ eval₂ (algebraMap F E) x (minpoly F x) = 0",
"tactic": "rw [← aeval_def, minpoly.aeval]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nthis : Nonempty (D →ₐ[C] E)\n⊢ Nonempty (D →ₐ[F] E)",
"tactic": "exact Nonempty.map (AlgHom.restrictScalars F) this"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\n⊢ IsIntegral C y ∧ Splits (algebraMap C E) (minpoly C y)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\n⊢ Nonempty (D →ₐ[C] E)",
"tactic": "refine' IntermediateField.algHom_mk_adjoin_splits' (top_le_iff.mp this) fun y hy => _"
},
{
"state_after": "case intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\n⊢ IsIntegral C y ∧ Splits (algebraMap C E) (minpoly C y)",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\n⊢ IsIntegral C y ∧ Splits (algebraMap C E) (minpoly C y)",
"tactic": "rcases Multiset.mem_map.mp (Multiset.mem_toFinset.mp hy) with ⟨z, hz1, hz2⟩"
},
{
"state_after": "case intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ IsIntegral C y ∧ Splits (algebraMap C E) (minpoly C y)",
"state_before": "case intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\n⊢ IsIntegral C y ∧ Splits (algebraMap C E) (minpoly C y)",
"tactic": "have Hz : IsIntegral F z := isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) z"
},
{
"state_after": "case intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ Splits (algebraMap C E) (minpoly C y)",
"state_before": "case intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ IsIntegral C y ∧ Splits (algebraMap C E) (minpoly C y)",
"tactic": "use\n show IsIntegral C y from\n isIntegral_of_noetherian (IsNoetherian.iff_fg.2 (FiniteDimensional.right F C D)) y"
},
{
"state_after": "case intro.intro.hf\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ Splits (algebraMap C E) (map ?m.151496 (minpoly F z))\n\ncase intro.intro.hgf\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ minpoly C y ∣ map ?m.151496 (minpoly F z)\n\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ F →+* C",
"state_before": "case intro.intro\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ Splits (algebraMap C E) (minpoly C y)",
"tactic": "apply splits_of_splits_of_dvd (algebraMap C E) (map_ne_zero (minpoly.ne_zero Hz))"
},
{
"state_after": "case intro.intro.hf\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ Splits (algebraMap F E) (minpoly F z)",
"state_before": "case intro.intro.hf\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ Splits (algebraMap C E) (map ?m.151496 (minpoly F z))",
"tactic": "rw [splits_map_iff, ← algebraMap_eq F C E]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hf\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ Splits (algebraMap F E) (minpoly F z)",
"tactic": "exact\n splits_of_splits_of_dvd _ hp hFEp.splits\n (minpoly.dvd F z (Eq.trans (eval₂_eq_eval_map _) ((mem_roots (map_ne_zero hp)).mp hz1)))"
},
{
"state_after": "case intro.intro.hgf.hp\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ ↑(aeval y) (map (algebraMap F C) (minpoly F z)) = 0",
"state_before": "case intro.intro.hgf\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ minpoly C y ∣ map (algebraMap F C) (minpoly F z)",
"tactic": "apply minpoly.dvd"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hgf.hp\nF : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : ⊤ ≤ IntermediateField.adjoin C S\ny : D\nhy : y ∈ S\nz : E\nhz1 : z ∈ roots (map (algebraMap F E) p)\nhz2 : ↑(algebraMap E D) z = y\nHz : IsIntegral F z\n⊢ ↑(aeval y) (map (algebraMap F C) (minpoly F z)) = 0",
"tactic": "rw [← hz2, aeval_def, eval₂_map, ← algebraMap_eq F C D, algebraMap_eq F E D, ← hom_eval₂, ←\n aeval_def, minpoly.aeval F z, RingHom.map_zero]"
},
{
"state_after": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : Subalgebra.restrictScalars F (Algebra.adjoin C S) = Subalgebra.restrictScalars F ⊤\n⊢ Algebra.adjoin C S ≥ ⊤",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis :\n Subalgebra.restrictScalars F (Algebra.adjoin C S) =\n Subalgebra.restrictScalars F (Algebra.adjoin E {AdjoinRoot.root q})\n⊢ Algebra.adjoin C S ≥ ⊤",
"tactic": "rw [AdjoinRoot.adjoinRoot_eq_top, Subalgebra.restrictScalars_top, ←\n @Subalgebra.restrictScalars_top F C] at this"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type ?u.87123\ninst✝¹¹ : Field F\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra F K\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra K E\ninst✝⁵ : IsScalarTower F K E\nE' : Type ?u.87603\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Field E'\ninst✝ : Algebra F E'\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nhFE : FiniteDimensional F E\nHx : IsIntegral F x\nq : E[X]\nq_irred : Irreducible q\nr : E[X]\nhr : map (algebraMap F E) (minpoly F x) = q * r\nD : Type u_2 := AdjoinRoot q\nthis✝⁶ : Fact (Irreducible q)\npbED : PowerBasis E (AdjoinRoot q) := AdjoinRoot.powerBasis (_ : q ≠ 0)\nthis✝⁵ : FiniteDimensional E D\nfinrankED : FiniteDimensional.finrank E D = natDegree q\nthis✝⁴ : FiniteDimensional F D\nC : Type u_1 := AdjoinRoot (minpoly F x)\nHx_irred : Fact (Irreducible (minpoly F x))\nheval : eval₂ (algebraMap F D) (AdjoinRoot.root q) (minpoly F x) = 0\nthis✝³ : Algebra C D := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F D) (AdjoinRoot.root q) heval)\nthis✝² : Algebra C E := RingHom.toAlgebra (AdjoinRoot.lift (algebraMap F E) x (_ : ↑(aeval x) (minpoly F x) = 0))\nthis✝¹ : IsScalarTower F C D\nthis✝ : IsScalarTower F C E\nS : Set D := ↑(Multiset.toFinset (Multiset.map (↑(algebraMap E D)) (roots (map (algebraMap F E) p))))\nthis : Subalgebra.restrictScalars F (Algebra.adjoin C S) = Subalgebra.restrictScalars F ⊤\n⊢ Algebra.adjoin C S ≥ ⊤",
"tactic": "exact top_le_iff.mpr (Subalgebra.restrictScalars_injective F this)"
}
] |
[
245,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
bit0_mul
|
[
{
"state_after": "α : Type ?u.268299\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : NonUnitalNonAssocRing R\nn r : R\n⊢ (n + n) * r = 2 • (n * r)",
"state_before": "α : Type ?u.268299\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : NonUnitalNonAssocRing R\nn r : R\n⊢ bit0 n * r = 2 • (n * r)",
"tactic": "dsimp [bit0]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.268299\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : NonUnitalNonAssocRing R\nn r : R\n⊢ (n + n) * r = 2 • (n * r)",
"tactic": "rw [add_mul, ← one_add_one_eq_two, add_zsmul, one_zsmul]"
}
] |
[
562,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivWithinAt.cos
|
[] |
[
809,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Data/Nat/Choose/Multinomial.lean
|
Nat.binomial_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ multinomial {a, b} f = (f a + f b)! / ((f a)! * (f b)!)",
"tactic": "simp [multinomial, Finset.sum_pair h, Finset.prod_pair h]"
}
] |
[
108,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/CategoryTheory/Limits/Lattice.lean
|
CategoryTheory.Limits.CompleteLattice.finite_product_eq_finset_inf
|
[
{
"state_after": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ (∏ f) = ?m.26818\n\nα : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ ?m.26818 = Finset.inf Fintype.elems f\n\nα : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ α",
"state_before": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ (∏ f) = Finset.inf Fintype.elems f",
"tactic": "trans"
},
{
"state_after": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ (finiteLimitCone (Discrete.functor f)).cone.pt = Finset.inf Fintype.elems f",
"state_before": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ (∏ f) = ?m.26818\n\nα : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ ?m.26818 = Finset.inf Fintype.elems f\n\nα : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ α",
"tactic": "exact\n (IsLimit.conePointUniqueUpToIso (limit.isLimit _)\n (finiteLimitCone (Discrete.functor f)).isLimit).to_eq"
},
{
"state_after": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ Finset.inf Finset.univ (f ∘ ↑(Equiv.toEmbedding discreteEquiv)) = Finset.inf Fintype.elems f",
"state_before": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ (finiteLimitCone (Discrete.functor f)).cone.pt = Finset.inf Fintype.elems f",
"tactic": "change Finset.univ.inf (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.inf f"
},
{
"state_after": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ Finset.inf Finset.univ f = Finset.inf Fintype.elems f",
"state_before": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ Finset.inf Finset.univ (f ∘ ↑(Equiv.toEmbedding discreteEquiv)) = Finset.inf Fintype.elems f",
"tactic": "simp only [← Finset.inf_map, Finset.univ_map_equiv_to_embedding]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nJ : Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : FinCategory J\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\nι : Type u\ninst✝ : Fintype ι\nf : ι → α\n⊢ Finset.inf Finset.univ f = Finset.inf Fintype.elems f",
"tactic": "rfl"
}
] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.congr_arg
|
[] |
[
137,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
8
] |
Mathlib/Algebra/PUnitInstances.lean
|
PUnit.gcd_eq
|
[] |
[
102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.LHom.sumElim_inl_inr
|
[] |
[
193,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Topology/Constructions.lean
|
isOpenMap_fst
|
[] |
[
709,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.fromEdgeSet_sdiff
|
[
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.86958\n𝕜 : Type ?u.86961\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\ns✝ s t : Set (Sym2 V)\nv w : V\n⊢ Adj (fromEdgeSet s \\ fromEdgeSet t) v w ↔ Adj (fromEdgeSet (s \\ t)) v w",
"state_before": "ι : Sort ?u.86958\n𝕜 : Type ?u.86961\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns✝ s t : Set (Sym2 V)\n⊢ fromEdgeSet s \\ fromEdgeSet t = fromEdgeSet (s \\ t)",
"tactic": "ext (v w)"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nι : Sort ?u.86958\n𝕜 : Type ?u.86961\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\ns✝ s t : Set (Sym2 V)\nv w : V\n⊢ Adj (fromEdgeSet s \\ fromEdgeSet t) v w ↔ Adj (fromEdgeSet (s \\ t)) v w",
"tactic": "constructor <;> simp (config := { contextual := true })"
}
] |
[
667,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.lt_of_add_lt_add_right
|
[
{
"state_after": "no goals",
"state_before": "a b c : Nat\nh : a + b < c + b\n⊢ b + a < b + c",
"tactic": "rwa [Nat.add_comm b a, Nat.add_comm b c]"
}
] |
[
78,
91
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
77,
11
] |
Mathlib/Analysis/Normed/MulAction.lean
|
lipschitzWith_smul
|
[] |
[
50,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Finsupp/Interval.lean
|
Finsupp.card_Ico
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : Zero α\ninst✝ : LocallyFiniteOrder α\nf g : ι →₀ α\n⊢ card (Ico f g) = ∏ i in f.support ∪ g.support, card (Icc (↑f i) (↑g i)) - 1",
"tactic": "rw [card_Ico_eq_card_Icc_sub_one, card_Icc]"
}
] |
[
116,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/ModelTheory/FinitelyGenerated.lean
|
FirstOrder.Language.Structure.cg_iff
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\n⊢ CG L M ↔ ∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = ⊤",
"tactic": "rw [cg_def, Substructure.cg_def]"
}
] |
[
235,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Topology/Connected.lean
|
isPreconnected_sUnion
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.2906\nπ : ι → Type ?u.2911\ninst✝ : TopologicalSpace α\ns t u v : Set α\nx : α\nc : Set (Set α)\nH1 : ∀ (s : Set α), s ∈ c → x ∈ s\nH2 : ∀ (s : Set α), s ∈ c → IsPreconnected s\n⊢ ∀ (y : α), y ∈ ⋃₀ c → ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.2906\nπ : ι → Type ?u.2911\ninst✝ : TopologicalSpace α\ns t u v : Set α\nx : α\nc : Set (Set α)\nH1 : ∀ (s : Set α), s ∈ c → x ∈ s\nH2 : ∀ (s : Set α), s ∈ c → IsPreconnected s\n⊢ IsPreconnected (⋃₀ c)",
"tactic": "apply isPreconnected_of_forall x"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.2906\nπ : ι → Type ?u.2911\ninst✝ : TopologicalSpace α\ns✝ t u v : Set α\nx : α\nc : Set (Set α)\nH1 : ∀ (s : Set α), s ∈ c → x ∈ s\nH2 : ∀ (s : Set α), s ∈ c → IsPreconnected s\ny : α\ns : Set α\nsc : s ∈ c\nys : y ∈ s\n⊢ ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.2906\nπ : ι → Type ?u.2911\ninst✝ : TopologicalSpace α\ns t u v : Set α\nx : α\nc : Set (Set α)\nH1 : ∀ (s : Set α), s ∈ c → x ∈ s\nH2 : ∀ (s : Set α), s ∈ c → IsPreconnected s\n⊢ ∀ (y : α), y ∈ ⋃₀ c → ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t",
"tactic": "rintro y ⟨s, sc, ys⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.2906\nπ : ι → Type ?u.2911\ninst✝ : TopologicalSpace α\ns✝ t u v : Set α\nx : α\nc : Set (Set α)\nH1 : ∀ (s : Set α), s ∈ c → x ∈ s\nH2 : ∀ (s : Set α), s ∈ c → IsPreconnected s\ny : α\ns : Set α\nsc : s ∈ c\nys : y ∈ s\n⊢ ∃ t, t ⊆ ⋃₀ c ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t",
"tactic": "exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩"
}
] |
[
134,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.const_smul'
|
[] |
[
472,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
11
] |
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