file_path
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Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.div_spanSingleton
|
[
{
"state_after": "R : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"state_before": "R : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\n⊢ J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J",
"tactic": "rw [← one_div_spanSingleton]"
},
{
"state_after": "case pos\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : d = 0\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J\n\ncase neg\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"state_before": "R : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"tactic": "by_cases hd : d = 0"
},
{
"state_after": "case neg\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"state_before": "case neg\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"tactic": "have h_spand : spanSingleton R₁⁰ d ≠ 0 := mt spanSingleton_eq_zero_iff.mp hd"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ J / spanSingleton R₁⁰ d ≤ 1 / spanSingleton R₁⁰ d * J\n\ncase neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ 1 / spanSingleton R₁⁰ d * J ≤ J / spanSingleton R₁⁰ d",
"state_before": "case neg\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : d = 0\n⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J",
"tactic": "simp only [hd, spanSingleton_zero, div_zero, MulZeroClass.zero_mul]"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x ∈ (fun a => ↑a) (J / spanSingleton R₁⁰ d)\n⊢ x ∈ (fun a => ↑a) (1 / spanSingleton R₁⁰ d * J)",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ J / spanSingleton R₁⁰ d ≤ 1 / spanSingleton R₁⁰ d * J",
"tactic": "intro x hx"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x ∈ ↑(J / spanSingleton R₁⁰ d)\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x ∈ (fun a => ↑a) (J / spanSingleton R₁⁰ d)\n⊢ x ∈ (fun a => ↑a) (1 / spanSingleton R₁⁰ d * J)",
"tactic": "dsimp only [val_eq_coe] at hx ⊢"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : ∀ (y : K), y ∈ ↑(spanSingleton R₁⁰ d) → x * y ∈ ↑J\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x ∈ ↑(J / spanSingleton R₁⁰ d)\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"tactic": "rw [coe_div h_spand, Submodule.mem_div_iff_forall_mul_mem] at hx"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : ∀ (y : K), y ∈ ↑(spanSingleton R₁⁰ d) → x * y ∈ ↑J\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"tactic": "specialize hx d (mem_spanSingleton_self R₁⁰ d)"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\nh_xd : x = d⁻¹ * (x * d)\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"tactic": "have h_xd : x = d⁻¹ * (x * d) := by field_simp"
},
{
"state_after": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\nh_xd : x = d⁻¹ * (x * d)\n⊢ d⁻¹ * (x * d) ∈ ↑(spanSingleton R₁⁰ d⁻¹) * ↑J",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\nh_xd : x = d⁻¹ * (x * d)\n⊢ x ∈ ↑(1 / spanSingleton R₁⁰ d * J)",
"tactic": "rw [coe_mul, one_div_spanSingleton, h_xd]"
},
{
"state_after": "no goals",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\nh_xd : x = d⁻¹ * (x * d)\n⊢ d⁻¹ * (x * d) ∈ ↑(spanSingleton R₁⁰ d⁻¹) * ↑J",
"tactic": "exact Submodule.mul_mem_mul (mem_spanSingleton_self R₁⁰ _) hx"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\nx : K\nhx : x * d ∈ ↑J\n⊢ x = d⁻¹ * (x * d)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case neg.a\nR : Type ?u.1525795\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1526002\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ 1 / spanSingleton R₁⁰ d * J ≤ J / spanSingleton R₁⁰ d",
"tactic": "rw [le_div_iff_mul_le h_spand, mul_assoc, mul_left_comm, one_div_spanSingleton,\n spanSingleton_mul_spanSingleton, inv_mul_cancel hd, spanSingleton_one, mul_one]"
}
] |
[
1486,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1471,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasurableEmbedding.snormEssSup_map_measure
|
[] |
[
926,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
924,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castSucc_castPred
|
[
{
"state_after": "n m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑castSucc (castLT i (_ : ↑i < n + 1)) = i\n\ncase hnc\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ¬↑castSucc (last n) < i",
"state_before": "n m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑castSucc (castPred i) = i",
"tactic": "rw [castPred, predAbove, dif_neg]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑castSucc (castLT i (_ : ↑i < n + 1)) = i",
"tactic": "simp [Fin.eq_iff_veq]"
},
{
"state_after": "no goals",
"state_before": "case hnc\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ¬↑castSucc (last n) < i",
"tactic": "exact h.not_le"
}
] |
[
2459,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2456,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
DirectSum.coe_mul_of_apply
|
[
{
"state_after": "case inl\nι : Type u_1\nσ : Type u_3\nS : Type ?u.414941\nR : Type u_2\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : i ≤ n\n⊢ ↑(↑(r * ↑(of (fun i => { x // x ∈ A i }) i) r') n) = ↑(↑r (n - i)) * ↑r'\n\ncase inr\nι : Type u_1\nσ : Type u_3\nS : Type ?u.414941\nR : Type u_2\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : ¬i ≤ n\n⊢ ↑(↑(r * ↑(of (fun i => { x // x ∈ A i }) i) r') n) = 0",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.414941\nR : Type u_2\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nn : ι\ninst✝ : Decidable (i ≤ n)\n⊢ ↑(↑(r * ↑(of (fun i => { x // x ∈ A i }) i) r') n) = if i ≤ n then ↑(↑r (n - i)) * ↑r' else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u_1\nσ : Type u_3\nS : Type ?u.414941\nR : Type u_2\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : i ≤ n\n⊢ ↑(↑(r * ↑(of (fun i => { x // x ∈ A i }) i) r') n) = ↑(↑r (n - i)) * ↑r'\n\ncase inr\nι : Type u_1\nσ : Type u_3\nS : Type ?u.414941\nR : Type u_2\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : ¬i ≤ n\n⊢ ↑(↑(r * ↑(of (fun i => { x // x ∈ A i }) i) r') n) = 0",
"tactic": "exacts [coe_mul_of_apply_of_le _ _ _ n h, coe_mul_of_apply_of_not_le _ _ _ n h]"
}
] |
[
271,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Order/MinMax.lean
|
min_lt_iff
|
[] |
[
67,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.constants_mem
|
[] |
[
163,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
HasStrictFDerivAt.map_nhds_eq_of_equiv
|
[] |
[
612,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_const
|
[] |
[
1294,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1293,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/Over/Products.lean
|
CategoryTheory.Over.ConstructProducts.over_products_of_widePullbacks
|
[] |
[
157,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.coe_iInf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.52313\nγ : Type ?u.52316\nι : Sort u_2\nκ : ι → Sort ?u.52324\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : ι → UpperSet α\n⊢ ↑(⨅ (i : ι), f i) = ⋃ (i : ι), ↑(f i)",
"tactic": "simp [iInf]"
}
] |
[
557,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMap.isOpenMap
|
[] |
[
173,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
11
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.isTopologicalBasis_basic_opens
|
[
{
"state_after": "case h_open\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ ∀ (u : Set (ProjectiveSpectrum 𝒜)), (u ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) → IsOpen u\n\ncase h_nhds\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ ∀ (a : ProjectiveSpectrum 𝒜) (u : Set (ProjectiveSpectrum 𝒜)),\n a ∈ u → IsOpen u → ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ a ∈ v ∧ v ⊆ u",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ IsTopologicalBasis (Set.range fun r => ↑(basicOpen 𝒜 r))",
"tactic": "apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds"
},
{
"state_after": "case h_open.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nr : A\n⊢ IsOpen ((fun r => ↑(basicOpen 𝒜 r)) r)",
"state_before": "case h_open\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ ∀ (u : Set (ProjectiveSpectrum 𝒜)), (u ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) → IsOpen u",
"tactic": "rintro _ ⟨r, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h_open.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nr : A\n⊢ IsOpen ((fun r => ↑(basicOpen 𝒜 r)) r)",
"tactic": "exact isOpen_basicOpen 𝒜"
},
{
"state_after": "case h_nhds.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\nhp : p ∈ U\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ p ∈ v ∧ v ⊆ U",
"state_before": "case h_nhds\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ ∀ (a : ProjectiveSpectrum 𝒜) (u : Set (ProjectiveSpectrum 𝒜)),\n a ∈ u → IsOpen u → ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ a ∈ v ∧ v ⊆ u",
"tactic": "rintro p U hp ⟨s, hs⟩"
},
{
"state_after": "case h_nhds.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhp : ∃ a, a ∈ s ∧ ¬a ∈ ↑p.asHomogeneousIdeal\nhs : zeroLocus 𝒜 s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ p ∈ v ∧ v ⊆ U",
"state_before": "case h_nhds.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\nhp : p ∈ U\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ p ∈ v ∧ v ⊆ U",
"tactic": "rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zeroLocus, Set.not_subset] at hp"
},
{
"state_after": "case h_nhds.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\nf : A\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asHomogeneousIdeal\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ p ∈ v ∧ v ⊆ U",
"state_before": "case h_nhds.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhp : ∃ a, a ∈ s ∧ ¬a ∈ ↑p.asHomogeneousIdeal\nhs : zeroLocus 𝒜 s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ p ∈ v ∧ v ⊆ U",
"tactic": "obtain ⟨f, hfs, hfp⟩ := hp"
},
{
"state_after": "case h_nhds.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\nf : A\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asHomogeneousIdeal\n⊢ ↑(basicOpen 𝒜 f) ⊆ U",
"state_before": "case h_nhds.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\nf : A\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asHomogeneousIdeal\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen 𝒜 r)) ∧ p ∈ v ∧ v ⊆ U",
"tactic": "refine' ⟨basicOpen 𝒜 f, ⟨f, rfl⟩, hfp, _⟩"
},
{
"state_after": "case h_nhds.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\nf : A\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asHomogeneousIdeal\n⊢ zeroLocus 𝒜 s ⊆ zeroLocus 𝒜 {f}",
"state_before": "case h_nhds.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\nf : A\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asHomogeneousIdeal\n⊢ ↑(basicOpen 𝒜 f) ⊆ U",
"tactic": "rw [← Set.compl_subset_compl, ← hs, basicOpen_eq_zeroLocus_compl, compl_compl]"
},
{
"state_after": "no goals",
"state_before": "case h_nhds.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\np : ProjectiveSpectrum 𝒜\nU : Set (ProjectiveSpectrum 𝒜)\ns : Set A\nhs : zeroLocus 𝒜 s = Uᶜ\nf : A\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asHomogeneousIdeal\n⊢ zeroLocus 𝒜 s ⊆ zeroLocus 𝒜 {f}",
"tactic": "exact zeroLocus_anti_mono 𝒜 (Set.singleton_subset_iff.mpr hfs)"
}
] |
[
468,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Submonoid.FG.map
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN : Type ?u.12734\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\nh : FG P\ne : M →* M'\n⊢ FG (Submonoid.map e P)",
"tactic": "classical\n obtain ⟨s, rfl⟩ := h\n exact ⟨s.image e, by rw [Finset.coe_image, MonoidHom.map_mclosure]⟩"
},
{
"state_after": "case intro\nM : Type u_2\nN : Type ?u.12734\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\ne : M →* M'\ns : Finset M\n⊢ FG (Submonoid.map e (closure ↑s))",
"state_before": "M : Type u_2\nN : Type ?u.12734\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\nh : FG P\ne : M →* M'\n⊢ FG (Submonoid.map e P)",
"tactic": "obtain ⟨s, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type u_2\nN : Type ?u.12734\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\ne : M →* M'\ns : Finset M\n⊢ FG (Submonoid.map e (closure ↑s))",
"tactic": "exact ⟨s.image e, by rw [Finset.coe_image, MonoidHom.map_mclosure]⟩"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN : Type ?u.12734\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\ne : M →* M'\ns : Finset M\n⊢ closure ↑(Finset.image (↑e) s) = Submonoid.map e (closure ↑s)",
"tactic": "rw [Finset.coe_image, MonoidHom.map_mclosure]"
}
] |
[
150,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
dist_le_of_le_geometric_of_tendsto₀
|
[] |
[
392,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Topology/Inseparable.lean
|
inseparable_pi
|
[
{
"state_after": "no goals",
"state_before": "X : Type ?u.34050\nY : Type ?u.34053\nZ : Type ?u.34056\nα : Type ?u.34059\nι : Type u_1\nπ : ι → Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf✝ : X → Y\nf g : (i : ι) → π i\n⊢ (f ~ᵢ g) ↔ ∀ (i : ι), f i ~ᵢ g i",
"tactic": "simp only [Inseparable, nhds_pi, funext_iff, pi_inj]"
}
] |
[
335,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.zipWith_swap_prod_support
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.716493\ninst✝¹ : DecidableEq α\nl✝ : List α\nx✝ : α\ninst✝ : Fintype α\nl l' : List α\nx : α\nhx : x ∈ support (prod (zipWith swap l l'))\n⊢ x ∈ toFinset l ⊔ toFinset l'",
"state_before": "α : Type u_1\nβ : Type ?u.716493\ninst✝¹ : DecidableEq α\nl✝ : List α\nx : α\ninst✝ : Fintype α\nl l' : List α\n⊢ support (prod (zipWith swap l l')) ≤ toFinset l ⊔ toFinset l'",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.716493\ninst✝¹ : DecidableEq α\nl✝ : List α\nx✝ : α\ninst✝ : Fintype α\nl l' : List α\nx : α\nhx : x ∈ support (prod (zipWith swap l l'))\nhx' : x ∈ {x | ↑(prod (zipWith swap l l')) x ≠ x}\n⊢ x ∈ toFinset l ⊔ toFinset l'",
"state_before": "α : Type u_1\nβ : Type ?u.716493\ninst✝¹ : DecidableEq α\nl✝ : List α\nx✝ : α\ninst✝ : Fintype α\nl l' : List α\nx : α\nhx : x ∈ support (prod (zipWith swap l l'))\n⊢ x ∈ toFinset l ⊔ toFinset l'",
"tactic": "have hx' : x ∈ { x | (zipWith swap l l').prod x ≠ x } := by simpa using hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.716493\ninst✝¹ : DecidableEq α\nl✝ : List α\nx✝ : α\ninst✝ : Fintype α\nl l' : List α\nx : α\nhx : x ∈ support (prod (zipWith swap l l'))\nhx' : x ∈ {x | ↑(prod (zipWith swap l l')) x ≠ x}\n⊢ x ∈ toFinset l ⊔ toFinset l'",
"tactic": "simpa using zipWith_swap_prod_support' _ _ hx'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.716493\ninst✝¹ : DecidableEq α\nl✝ : List α\nx✝ : α\ninst✝ : Fintype α\nl l' : List α\nx : α\nhx : x ∈ support (prod (zipWith swap l l'))\n⊢ x ∈ {x | ↑(prod (zipWith swap l l')) x ≠ x}",
"tactic": "simpa using hx"
}
] |
[
197,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_inf_eq_map_inf_comap
|
[
{
"state_after": "case intro.intro.intro\nF : Type u_5\nR : Type u_1\nR₁ : Type ?u.928463\nR₂ : Type u_2\nR₃ : Type ?u.928469\nR₄ : Type ?u.928472\nS : Type ?u.928475\nK : Type ?u.928478\nK₂ : Type ?u.928481\nM : Type u_3\nM' : Type ?u.928487\nM₁ : Type ?u.928490\nM₂ : Type u_4\nM₃ : Type ?u.928496\nM₄ : Type ?u.928499\nN : Type ?u.928502\nN₂ : Type ?u.928505\nι : Type ?u.928508\nV : Type ?u.928511\nV₂ : Type ?u.928514\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np✝ p'✝ : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx✝ y : M\nsc : SemilinearMapClass F σ₁₂ M M₂\ninst✝ : RingHomSurjective σ₁₂\nf : F\np : Submodule R M\np' : Submodule R₂ M₂\nx : M\nh₁ : x ∈ ↑p\nh₂ : ↑f x ∈ ↑p'\n⊢ ↑f x ∈ map f (p ⊓ comap f p')",
"state_before": "F : Type u_5\nR : Type u_1\nR₁ : Type ?u.928463\nR₂ : Type u_2\nR₃ : Type ?u.928469\nR₄ : Type ?u.928472\nS : Type ?u.928475\nK : Type ?u.928478\nK₂ : Type ?u.928481\nM : Type u_3\nM' : Type ?u.928487\nM₁ : Type ?u.928490\nM₂ : Type u_4\nM₃ : Type ?u.928496\nM₄ : Type ?u.928499\nN : Type ?u.928502\nN₂ : Type ?u.928505\nι : Type ?u.928508\nV : Type ?u.928511\nV₂ : Type ?u.928514\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np✝ p'✝ : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F σ₁₂ M M₂\ninst✝ : RingHomSurjective σ₁₂\nf : F\np : Submodule R M\np' : Submodule R₂ M₂\n⊢ map f p ⊓ p' ≤ map f (p ⊓ comap f p')",
"tactic": "rintro _ ⟨⟨x, h₁, rfl⟩, h₂⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nF : Type u_5\nR : Type u_1\nR₁ : Type ?u.928463\nR₂ : Type u_2\nR₃ : Type ?u.928469\nR₄ : Type ?u.928472\nS : Type ?u.928475\nK : Type ?u.928478\nK₂ : Type ?u.928481\nM : Type u_3\nM' : Type ?u.928487\nM₁ : Type ?u.928490\nM₂ : Type u_4\nM₃ : Type ?u.928496\nM₄ : Type ?u.928499\nN : Type ?u.928502\nN₂ : Type ?u.928505\nι : Type ?u.928508\nV : Type ?u.928511\nV₂ : Type ?u.928514\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np✝ p'✝ : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx✝ y : M\nsc : SemilinearMapClass F σ₁₂ M M₂\ninst✝ : RingHomSurjective σ₁₂\nf : F\np : Submodule R M\np' : Submodule R₂ M₂\nx : M\nh₁ : x ∈ ↑p\nh₂ : ↑f x ∈ ↑p'\n⊢ ↑f x ∈ map f (p ⊓ comap f p')",
"tactic": "exact ⟨_, ⟨h₁, h₂⟩, rfl⟩"
}
] |
[
1017,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.degree_of_d_eq_zero'
|
[] |
[
376,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
vadd_segment
|
[] |
[
253,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.add_right_cancel_iff
|
[
{
"state_after": "case intro\na b : PartENat\nc : ℕ\nhc : ↑c ≠ ⊤\n⊢ a + ↑c = b + ↑c ↔ a = b",
"state_before": "a b c : PartENat\nhc : c ≠ ⊤\n⊢ a + c = b + c ↔ a = b",
"tactic": "rcases ne_top_iff.1 hc with ⟨c, rfl⟩"
},
{
"state_after": "case intro.refine_2.refine_2\na b : PartENat\nc : ℕ\nhc : ↑c ≠ ⊤\n⊢ ∀ (n n_1 : ℕ), ↑n_1 + ↑c = ↑n + ↑c ↔ n_1 = n",
"state_before": "case intro\na b : PartENat\nc : ℕ\nhc : ↑c ≠ ⊤\n⊢ a + ↑c = b + ↑c ↔ a = b",
"tactic": "refine PartENat.casesOn a ?_ ?_\n<;> refine PartENat.casesOn b ?_ ?_\n<;> simp [add_eq_top_iff, natCast_ne_top, @eq_comm _ (⊤ : PartENat)]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine_2.refine_2\na b : PartENat\nc : ℕ\nhc : ↑c ≠ ⊤\n⊢ ∀ (n n_1 : ℕ), ↑n_1 + ↑c = ↑n + ↑c ↔ n_1 = n",
"tactic": "simp only [←Nat.cast_add, add_left_cancel_iff, PartENat.natCast_inj, add_comm, forall_const]"
}
] |
[
537,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
11
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.sub_apply
|
[] |
[
316,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Std/Data/Fin/Lemmas.lean
|
Fin.mod_val
|
[] |
[
12,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
11,
9
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformContinuous.prod_mk
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.156682\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf₁ : α → β\nf₂ : α → γ\nh₁ : UniformContinuous f₁\nh₂ : UniformContinuous f₂\n⊢ Tendsto (fun x => ((f₁ x.fst, f₂ x.fst), f₁ x.snd, f₂ x.snd)) (𝓤 α)\n (comap (fun p => (p.fst.fst, p.snd.fst)) (𝓤 β) ⊓ comap (fun p => (p.fst.snd, p.snd.snd)) (𝓤 γ))",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.156682\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf₁ : α → β\nf₂ : α → γ\nh₁ : UniformContinuous f₁\nh₂ : UniformContinuous f₂\n⊢ UniformContinuous fun a => (f₁ a, f₂ a)",
"tactic": "rw [UniformContinuous, uniformity_prod]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.156682\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf₁ : α → β\nf₂ : α → γ\nh₁ : UniformContinuous f₁\nh₂ : UniformContinuous f₂\n⊢ Tendsto (fun x => ((f₁ x.fst, f₂ x.fst), f₁ x.snd, f₂ x.snd)) (𝓤 α)\n (comap (fun p => (p.fst.fst, p.snd.fst)) (𝓤 β) ⊓ comap (fun p => (p.fst.snd, p.snd.snd)) (𝓤 γ))",
"tactic": "exact tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩"
}
] |
[
1625,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1622,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
Bornology.IsVonNBounded.smul_tendsto_zero
|
[
{
"state_after": "𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\n⊢ ∀ (s : Set E), s ∈ 𝓝 0 → (ε • x) ⁻¹' s ∈ l",
"state_before": "𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : Tendsto ε l (𝓝 0)\n⊢ Tendsto (ε • x) l (𝓝 0)",
"tactic": "rw [tendsto_def] at *"
},
{
"state_after": "𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\n⊢ (ε • x) ⁻¹' V ∈ l",
"state_before": "𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\n⊢ ∀ (s : Set E), s ∈ 𝓝 0 → (ε • x) ⁻¹' s ∈ l",
"tactic": "intro V hV"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\n⊢ (ε • x) ⁻¹' V ∈ l",
"state_before": "𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\n⊢ (ε • x) ⁻¹' V ∈ l",
"tactic": "rcases hS hV with ⟨r, r_pos, hrS⟩"
},
{
"state_after": "case h\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : n ∈ ε ⁻¹' Metric.ball 0 r⁻¹\n⊢ n ∈ (ε • x) ⁻¹' V",
"state_before": "case intro.intro\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\n⊢ (ε • x) ⁻¹' V ∈ l",
"tactic": "filter_upwards [hxS, hε _ (Metric.ball_mem_nhds 0 <| inv_pos.mpr r_pos)] with n hnS hnr"
},
{
"state_after": "case pos\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : n ∈ ε ⁻¹' Metric.ball 0 r⁻¹\nhε : ε n = 0\n⊢ n ∈ (ε • x) ⁻¹' V\n\ncase neg\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : n ∈ ε ⁻¹' Metric.ball 0 r⁻¹\nhε : ¬ε n = 0\n⊢ n ∈ (ε • x) ⁻¹' V",
"state_before": "case h\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : n ∈ ε ⁻¹' Metric.ball 0 r⁻¹\n⊢ n ∈ (ε • x) ⁻¹' V",
"tactic": "by_cases hε : ε n = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : n ∈ ε ⁻¹' Metric.ball 0 r⁻¹\nhε : ε n = 0\n⊢ n ∈ (ε • x) ⁻¹' V",
"tactic": "simp [hε, mem_of_mem_nhds hV]"
},
{
"state_after": "case neg\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : r < ‖(ε n)⁻¹‖\nhε : ¬ε n = 0\n⊢ n ∈ (ε • x) ⁻¹' V",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : n ∈ ε ⁻¹' Metric.ball 0 r⁻¹\nhε : ¬ε n = 0\n⊢ n ∈ (ε • x) ⁻¹' V",
"tactic": "rw [mem_preimage, mem_ball_zero_iff, lt_inv (norm_pos_iff.mpr hε) r_pos, ← norm_inv] at hnr"
},
{
"state_after": "case neg\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : r < ‖(ε n)⁻¹‖\nhε : ¬ε n = 0\n⊢ x n ∈ (ε n)⁻¹ • V",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : r < ‖(ε n)⁻¹‖\nhε : ¬ε n = 0\n⊢ n ∈ (ε • x) ⁻¹' V",
"tactic": "rw [mem_preimage, Pi.smul_apply', ← Set.mem_inv_smul_set_iff₀ hε]"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜' : Type ?u.156377\nE : Type u_1\nE' : Type ?u.156383\nF : Type ?u.156386\nι : Type u_2\n𝕝 : Type ?u.156392\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕝 E\nS : Set E\nε : ι → 𝕜\nx : ι → E\nl : Filter ι\nhS : IsVonNBounded 𝕜 S\nhxS : ∀ᶠ (n : ι) in l, x n ∈ S\nhε✝ : ∀ (s : Set 𝕜), s ∈ 𝓝 0 → ε ⁻¹' s ∈ l\nV : Set E\nhV : V ∈ 𝓝 0\nr : ℝ\nr_pos : 0 < r\nhrS : ∀ (a : 𝕜), r ≤ ‖a‖ → S ⊆ a • V\nn : ι\nhnS : x n ∈ S\nhnr : r < ‖(ε n)⁻¹‖\nhε : ¬ε n = 0\n⊢ x n ∈ (ε n)⁻¹ • V",
"tactic": "exact hrS _ hnr.le hnS"
}
] |
[
161,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.smul_opposite_image_mul_preimage
|
[] |
[
248,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/RingTheory/Adjoin/Basic.lean
|
AlgHom.ext_of_adjoin_eq_top
|
[] |
[
430,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
LinearIsometryEquiv.norm_iteratedFDeriv_comp_left
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.282885\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F ≃ₗᵢ[𝕜] G\nf : E → F\nx : E\ni : ℕ\n⊢ ‖iteratedFDerivWithin 𝕜 i (↑g ∘ f) univ x‖ = ‖iteratedFDerivWithin 𝕜 i f univ x‖",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.282885\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F ≃ₗᵢ[𝕜] G\nf : E → F\nx : E\ni : ℕ\n⊢ ‖iteratedFDeriv 𝕜 i (↑g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i f x‖",
"tactic": "rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.282885\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F ≃ₗᵢ[𝕜] G\nf : E → F\nx : E\ni : ℕ\n⊢ ‖iteratedFDerivWithin 𝕜 i (↑g ∘ f) univ x‖ = ‖iteratedFDerivWithin 𝕜 i f univ x‖",
"tactic": "apply g.norm_iteratedFDerivWithin_comp_left f uniqueDiffOn_univ (mem_univ x) i"
}
] |
[
321,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.add_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.723564\nn : ZNum\n⊢ n + 0 = n",
"tactic": "cases n <;> rfl"
}
] |
[
1164,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1164,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_const'
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n✝ : Nontrivial β\n⊢ Measurable f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n⊢ Measurable f",
"tactic": "nontriviality β"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n✝ : Nontrivial β\ninhabited_h : Inhabited β\n⊢ Measurable f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n✝ : Nontrivial β\n⊢ Measurable f",
"tactic": "inhabit β"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n✝ : Nontrivial β\ninhabited_h : Inhabited β\nx✝ : β\n⊢ f x✝ = f default",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n✝ : Nontrivial β\ninhabited_h : Inhabited β\n⊢ Measurable f",
"tactic": "convert @measurable_const α β ‹_› ‹_› (f default) using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.9600\nδ : Type ?u.9603\nδ' : Type ?u.9606\nι : Sort uι\ns t u : Set α\nf✝ g : α → β\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : β → α\nhf : ∀ (x y : β), f x = f y\n✝ : Nontrivial β\ninhabited_h : Inhabited β\nx✝ : β\n⊢ f x✝ = f default",
"tactic": "apply hf"
}
] |
[
277,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ici_diff_Ici
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.60813\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ici a \\ Ici b = Ico a b",
"tactic": "rw [diff_eq, compl_Ici, Ici_inter_Iio]"
}
] |
[
1080,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1080,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_quot_lift
|
[] |
[
1154,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1152,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
map_exp_of_mem_ball
|
[
{
"state_after": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ ↑f ((fun x => ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) x) = (fun x_1 => ∑' (n : ℕ), (↑n !)⁻¹ • x_1 ^ n) (↑f x)",
"state_before": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ ↑f (exp 𝕂 x) = exp 𝕂 (↑f x)",
"tactic": "rw [exp_eq_tsum, exp_eq_tsum]"
},
{
"state_after": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (b : ℕ), (↑f ∘ fun n => (↑n !)⁻¹ • x ^ n) b) = (fun x_1 => ∑' (n : ℕ), (↑n !)⁻¹ • x_1 ^ n) (↑f x)",
"state_before": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ ↑f ((fun x => ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) x) = (fun x_1 => ∑' (n : ℕ), (↑n !)⁻¹ • x_1 ^ n) (↑f x)",
"tactic": "refine' ((expSeries_summable_of_mem_ball' _ hx).hasSum.map f hf).tsum_eq.symm.trans _"
},
{
"state_after": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (b : ℕ), ↑f ((↑b !)⁻¹ • x ^ b)) = ∑' (n : ℕ), (↑n !)⁻¹ • ↑f x ^ n",
"state_before": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (b : ℕ), (↑f ∘ fun n => (↑n !)⁻¹ • x ^ n) b) = (fun x_1 => ∑' (n : ℕ), (↑n !)⁻¹ • x_1 ^ n) (↑f x)",
"tactic": "dsimp only [Function.comp]"
},
{
"state_after": "no goals",
"state_before": "𝕂 : Type u_4\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\nF : Type u_1\ninst✝ : RingHomClass F 𝔸 𝔹\nf : F\nhf : Continuous ↑f\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (b : ℕ), ↑f ((↑b !)⁻¹ • x ^ b)) = ∑' (n : ℕ), (↑n !)⁻¹ • ↑f x ^ n",
"tactic": "simp_rw [one_div, map_inv_nat_cast_smul f 𝕂 𝕂, map_pow]"
}
] |
[
318,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.const_congr
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf₁ g₁ : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ ∈ Ideal.primeCompl x.asIdeal\n⊢ const R f₁ g₁ U hu = const R f₁ g₁ U (_ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ ∈ Ideal.primeCompl x.asIdeal)",
"state_before": "R : Type u\ninst✝ : CommRing R\nf₁ f₂ g₁ g₂ : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ ∈ Ideal.primeCompl x.asIdeal\nhf : f₁ = f₂\nhg : g₁ = g₂\n⊢ const R f₁ g₁ U hu = const R f₂ g₂ U (_ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₂ ∈ Ideal.primeCompl x.asIdeal)",
"tactic": "substs hf hg"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf₁ g₁ : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ ∈ Ideal.primeCompl x.asIdeal\n⊢ const R f₁ g₁ U hu = const R f₁ g₁ U (_ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ ∈ Ideal.primeCompl x.asIdeal)",
"tactic": "rfl"
}
] |
[
393,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.cast_natAdd_zero
|
[] |
[
1454,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1452,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.nthHomSeq_add
|
[
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\n⊢ nthHomSeq f_compat (r + s) ≈ nthHomSeq f_compat r + nthHomSeq f_compat s",
"tactic": "intro ε hε"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"tactic": "obtain ⟨n, hn⟩ := exists_pow_neg_lt_rat p hε"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∀ (j : ℕ), j ≥ n → padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"tactic": "use n"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∀ (j : ℕ), j ≥ n → padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"tactic": "intro j hj"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p\n (↑(nthHom (fun k2 => f k2) (r + s) j) - (↑(nthHom (fun k2 => f k2) r j) + ↑(nthHom (fun k2 => f k2) s j))) <\n ε",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHomSeq f_compat (r + s) - (nthHomSeq f_compat r + nthHomSeq f_compat s)) j) < ε",
"tactic": "dsimp [nthHomSeq]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p\n (↑(nthHom (fun k2 => f k2) (r + s) j) - (↑(nthHom (fun k2 => f k2) r j) + ↑(nthHom (fun k2 => f k2) s j))) ≤\n ↑p ^ (-↑n)",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p\n (↑(nthHom (fun k2 => f k2) (r + s) j) - (↑(nthHom (fun k2 => f k2) r j) + ↑(nthHom (fun k2 => f k2) s j))) <\n ε",
"tactic": "apply lt_of_le_of_lt _ hn"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(nthHom (fun k2 => f k2) (r + s) j - (nthHom (fun k2 => f k2) r j + nthHom (fun k2 => f k2) s j)) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p\n (↑(nthHom (fun k2 => f k2) (r + s) j) - (↑(nthHom (fun k2 => f k2) r j) + ↑(nthHom (fun k2 => f k2) s j))) ≤\n ↑p ^ (-↑n)",
"tactic": "rw [← Int.cast_add, ← Int.cast_sub, ← padicNorm.dvd_iff_norm_le, ←\n ZMod.int_cast_zmod_eq_zero_iff_dvd]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(ZMod.val (↑(f j) (r + s))) - ↑(ZMod.val (↑(f j) r) + ZMod.val (↑(f j) s))) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(nthHom (fun k2 => f k2) (r + s) j - (nthHom (fun k2 => f k2) r j + nthHom (fun k2 => f k2) s j)) = 0",
"tactic": "dsimp [nthHom]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(f j) r + ↑(f j) s) - ↑↑(ZMod.val (↑(f j) r) + ZMod.val (↑(f j) s)) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(ZMod.val (↑(f j) (r + s))) - ↑(ZMod.val (↑(f j) r) + ZMod.val (↑(f j) s))) = 0",
"tactic": "simp only [ZMod.nat_cast_val, RingHom.map_add, Int.cast_sub, ZMod.int_cast_cast, Int.cast_add]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(f j) r) + ↑(↑(f j) s) - ↑↑(ZMod.val (↑(f j) r) + ZMod.val (↑(f j) s)) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(f j) r + ↑(f j) s) - ↑↑(ZMod.val (↑(f j) r) + ZMod.val (↑(f j) s)) = 0",
"tactic": "rw [ZMod.cast_add (show p ^ n ∣ p ^ j from pow_dvd_pow _ hj)]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(f j) r) + ↑(↑(f j) s) - ↑↑(ZMod.val (↑(f j) r) + ZMod.val (↑(f j) s)) = 0",
"tactic": "simp only [cast_add, ZMod.nat_cast_val, Int.cast_add, ZMod.int_cast_cast, sub_self]"
}
] |
[
552,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.of_denote_eq_cancelAux
|
[
{
"state_after": "no goals",
"state_before": "case zero\nctx : Context\nm₁ m₂ r₁ r₂ : Poly\nh : denote_eq ctx (cancelAux zero m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)",
"tactic": "assumption"
},
{
"state_after": "case succ\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nm₁ m₂ r₁ r₂ : Poly\nh :\n denote_eq ctx\n (match m₁, m₂ with\n | m₁, [] => (List.reverse r₁ ++ m₁, List.reverse r₂)\n | [], m₂ => (List.reverse r₁, List.reverse r₂ ++ m₂)\n | (k₁, v₁) :: m₁, (k₂, v₂) :: m₂ =>\n bif blt v₁ v₂ then cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂\n else\n bif blt v₂ v₁ then cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)\n else\n bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)",
"state_before": "case succ\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nm₁ m₂ r₁ r₂ : Poly\nh : denote_eq ctx (cancelAux (succ fuel) m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)",
"tactic": "simp at h"
},
{
"state_after": "case succ.h_3\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝¹ m₂✝¹ : Poly\nk₁✝ : Nat\nv₁✝ : Var\nm₁✝ : List (Nat × Var)\nk₂✝ : Nat\nv₂✝ : Var\nm₂✝ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt v₁✝ v₂✝ then cancelAux fuel m₁✝ ((k₂✝, v₂✝) :: m₂✝) ((k₁✝, v₁✝) :: r₁) r₂\n else\n bif blt v₂✝ v₁✝ then cancelAux fuel ((k₁✝, v₁✝) :: m₁✝) m₂✝ r₁ ((k₂✝, v₂✝) :: r₂)\n else\n bif blt k₁✝ k₂✝ then cancelAux fuel m₁✝ m₂✝ r₁ ((k₂✝ - k₁✝, v₁✝) :: r₂)\n else bif blt k₂✝ k₁✝ then cancelAux fuel m₁✝ m₂✝ ((k₁✝ - k₂✝, v₁✝) :: r₁) r₂ else cancelAux fuel m₁✝ m₂✝ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁✝, v₁✝) :: m₁✝, List.reverse r₂ ++ (k₂✝, v₂✝) :: m₂✝)",
"state_before": "case succ\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nm₁ m₂ r₁ r₂ : Poly\nh :\n denote_eq ctx\n (match m₁, m₂ with\n | m₁, [] => (List.reverse r₁ ++ m₁, List.reverse r₂)\n | [], m₂ => (List.reverse r₁, List.reverse r₂ ++ m₂)\n | (k₁, v₁) :: m₁, (k₂, v₂) :: m₂ =>\n bif blt v₁ v₂ then cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂\n else\n bif blt v₂ v₁ then cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)\n else\n bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)",
"tactic": "split at h <;> simp <;> try assumption"
},
{
"state_after": "case succ.h_3\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt v₁ v₂ then cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂\n else\n bif blt v₂ v₁ then cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)\n else\n bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"state_before": "case succ.h_3\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝¹ m₂✝¹ : Poly\nk₁✝ : Nat\nv₁✝ : Var\nm₁✝ : List (Nat × Var)\nk₂✝ : Nat\nv₂✝ : Var\nm₂✝ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt v₁✝ v₂✝ then cancelAux fuel m₁✝ ((k₂✝, v₂✝) :: m₂✝) ((k₁✝, v₁✝) :: r₁) r₂\n else\n bif blt v₂✝ v₁✝ then cancelAux fuel ((k₁✝, v₁✝) :: m₁✝) m₂✝ r₁ ((k₂✝, v₂✝) :: r₂)\n else\n bif blt k₁✝ k₂✝ then cancelAux fuel m₁✝ m₂✝ r₁ ((k₂✝ - k₁✝, v₁✝) :: r₂)\n else bif blt k₂✝ k₁✝ then cancelAux fuel m₁✝ m₂✝ ((k₁✝ - k₂✝, v₁✝) :: r₁) r₂ else cancelAux fuel m₁✝ m₂✝ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁✝, v₁✝) :: m₁✝, List.reverse r₂ ++ (k₂✝, v₂✝) :: m₂✝)",
"tactic": "rename_i k₁ v₁ m₁ k₂ v₂ m₂"
},
{
"state_after": "case succ.h_3.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : blt v₁ v₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)\n\ncase succ.h_3.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nh :\n denote_eq ctx\n (bif blt v₂ v₁ then cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)\n else\n bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"state_before": "case succ.h_3\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt v₁ v₂ then cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂\n else\n bif blt v₂ v₁ then cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)\n else\n bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "by_cases hltv : Nat.blt v₁ v₂ <;> simp [hltv] at h"
},
{
"state_after": "case succ.h_3\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝¹ m₂✝¹ : Poly\nk₁✝ : Nat\nv₁✝ : Var\nm₁✝ : List (Nat × Var)\nk₂✝ : Nat\nv₂✝ : Var\nm₂✝ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt v₁✝ v₂✝ then cancelAux fuel m₁✝ ((k₂✝, v₂✝) :: m₂✝) ((k₁✝, v₁✝) :: r₁) r₂\n else\n bif blt v₂✝ v₁✝ then cancelAux fuel ((k₁✝, v₁✝) :: m₁✝) m₂✝ r₁ ((k₂✝, v₂✝) :: r₂)\n else\n bif blt k₁✝ k₂✝ then cancelAux fuel m₁✝ m₂✝ r₁ ((k₂✝ - k₁✝, v₁✝) :: r₂)\n else bif blt k₂✝ k₁✝ then cancelAux fuel m₁✝ m₂✝ ((k₁✝ - k₂✝, v₁✝) :: r₁) r₂ else cancelAux fuel m₁✝ m₂✝ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁✝, v₁✝) :: m₁✝, List.reverse r₂ ++ (k₂✝, v₂✝) :: m₂✝)",
"state_before": "case succ.h_3\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝¹ m₂✝¹ : Poly\nk₁✝ : Nat\nv₁✝ : Var\nm₁✝ : List (Nat × Var)\nk₂✝ : Nat\nv₂✝ : Var\nm₂✝ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt v₁✝ v₂✝ then cancelAux fuel m₁✝ ((k₂✝, v₂✝) :: m₂✝) ((k₁✝, v₁✝) :: r₁) r₂\n else\n bif blt v₂✝ v₁✝ then cancelAux fuel ((k₁✝, v₁✝) :: m₁✝) m₂✝ r₁ ((k₂✝, v₂✝) :: r₂)\n else\n bif blt k₁✝ k₂✝ then cancelAux fuel m₁✝ m₂✝ r₁ ((k₂✝ - k₁✝, v₁✝) :: r₂)\n else bif blt k₂✝ k₁✝ then cancelAux fuel m₁✝ m₂✝ ((k₁✝ - k₂✝, v₁✝) :: r₁) r₂ else cancelAux fuel m₁✝ m₂✝ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁✝, v₁✝) :: m₁✝, List.reverse r₂ ++ (k₂✝, v₂✝) :: m₂✝)",
"tactic": "assumption"
},
{
"state_after": "case succ.h_3.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : blt v₁ v₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂)\nih : denote_eq ctx (List.reverse ((k₁, v₁) :: r₁) ++ m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"state_before": "case succ.h_3.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : blt v₁ v₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "have ih := ih (h := h)"
},
{
"state_after": "case succ.h_3.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : blt v₁ v₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂)\nih : denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)",
"state_before": "case succ.h_3.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : blt v₁ v₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂)\nih : denote_eq ctx (List.reverse ((k₁, v₁) :: r₁) ++ m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "simp [denote_eq] at ih ⊢"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : blt v₁ v₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂)\nih : denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)",
"tactic": "assumption"
},
{
"state_after": "case succ.h_3.inr.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : blt v₂ v₁ = true\nh : denote_eq ctx (cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂))\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)\n\ncase succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\nh :\n denote_eq ctx\n (bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"state_before": "case succ.h_3.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nh :\n denote_eq ctx\n (bif blt v₂ v₁ then cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)\n else\n bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "by_cases hgtv : Nat.blt v₂ v₁ <;> simp [hgtv] at h"
},
{
"state_after": "case succ.h_3.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : blt v₂ v₁ = true\nh : denote_eq ctx (cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂))\nih : denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse ((k₂, v₂) :: r₂) ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"state_before": "case succ.h_3.inr.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : blt v₂ v₁ = true\nh : denote_eq ctx (cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂))\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "have ih := ih (h := h)"
},
{
"state_after": "case succ.h_3.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : blt v₂ v₁ = true\nh : denote_eq ctx (cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂))\nih : denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)",
"state_before": "case succ.h_3.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : blt v₂ v₁ = true\nh : denote_eq ctx (cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂))\nih : denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse ((k₂, v₂) :: r₂) ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "simp [denote_eq] at ih ⊢"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : blt v₂ v₁ = true\nh : denote_eq ctx (cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂))\nih : denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₂)",
"tactic": "assumption"
},
{
"state_after": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\nh :\n denote_eq ctx\n (bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\nheqv : v₁ = v₂\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"state_before": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\nh :\n denote_eq ctx\n (bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "have heqv : v₁ = v₂ := eq_of_not_blt_eq_true hltv hgtv"
},
{
"state_after": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\nhltv hgtv : ¬blt v₁ v₁ = true\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\nm₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\nh :\n denote_eq ctx\n (bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\nheqv : v₁ = v₂\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₂) :: m₂)",
"tactic": "subst heqv"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)\n\ncase succ.h_3.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nh : denote_eq ctx (bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nh :\n denote_eq ctx\n (bif blt k₁ k₂ then cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)\n else bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\nhltv hgtv : ¬blt v₁ v₁ = true\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "by_cases hltk : Nat.blt k₁ k₂ <;> simp [hltk] at h"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse ((k₂ - k₁, v₁) :: r₂) ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "have ih := ih (h := h)"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + (denote ctx m₂ + (k₂ - k₁) * Var.denote ctx v₁)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse ((k₂ - k₁, v₁) :: r₂) ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "simp [denote_eq] at ih ⊢"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + (denote ctx m₂ + (k₂ - k₁) * Var.denote ctx v₁)\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + (denote ctx m₂ + (k₂ - k₁) * Var.denote ctx v₁)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "have haux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁ := Nat.mul_le_mul_right _ (Nat.le_of_lt (Nat.blt_eq.mp hltk))"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ - k₁ * Var.denote ctx v₁\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + (denote ctx m₂ + (k₂ - k₁) * Var.denote ctx v₁)\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "rw [Nat.mul_sub_right_distrib, ← Nat.add_assoc, ← Nat.add_sub_assoc haux] at ih"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih✝ : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ - k₁ * Var.denote ctx v₁\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\nih : denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ = denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ - k₁ * Var.denote ctx v₁\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "have ih := Nat.eq_add_of_sub_eq (Nat.le_trans haux (Nat.le_add_left ..)) ih.symm"
},
{
"state_after": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih✝ : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ - k₁ * Var.denote ctx v₁\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\nih : denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁) = denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih✝ : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ - k₁ * Var.denote ctx v₁\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\nih : denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ = denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "simp at ih"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : blt k₁ k₂ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂))\nih✝ : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁ - k₁ * Var.denote ctx v₁\nhaux : k₁ * Var.denote ctx v₁ ≤ k₂ * Var.denote ctx v₁\nih : denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁) = denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "rw [ih]"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)\n\ncase succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : ¬blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nh : denote_eq ctx (bif blt k₂ k₁ then cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂ else cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "by_cases hgtk : Nat.blt k₂ k₁ <;> simp [hgtk] at h"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote_eq ctx (List.reverse ((k₁ - k₂, v₁) :: r₁) ++ m₁, List.reverse r₂ ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "have ih := ih (h := h)"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote ctx r₁ + (denote ctx m₁ + (k₁ - k₂) * Var.denote ctx v₁) = denote ctx r₂ + denote ctx m₂\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote_eq ctx (List.reverse ((k₁ - k₂, v₁) :: r₁) ++ m₁, List.reverse r₂ ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "simp [denote_eq] at ih ⊢"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote ctx r₁ + (denote ctx m₁ + (k₁ - k₂) * Var.denote ctx v₁) = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote ctx r₁ + (denote ctx m₁ + (k₁ - k₂) * Var.denote ctx v₁) = denote ctx r₂ + denote ctx m₂\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "have haux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁ := Nat.mul_le_mul_right _ (Nat.le_of_lt (Nat.blt_eq.mp hgtk))"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ - k₂ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote ctx r₁ + (denote ctx m₁ + (k₁ - k₂) * Var.denote ctx v₁) = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "rw [Nat.mul_sub_right_distrib, ← Nat.add_assoc, ← Nat.add_sub_assoc haux] at ih"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih✝ : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ - k₂ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\nih : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ - k₂ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "have ih := Nat.eq_add_of_sub_eq (Nat.le_trans haux (Nat.le_add_left ..)) ih"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih✝ : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ - k₂ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\nih : denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih✝ : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ - k₂ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\nih : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂ + k₂ * Var.denote ctx v₁\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "simp at ih"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3.inr.inr.inr.inl\nctx : Context\nfuel : Nat\nih✝¹ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂)\nih✝ : denote ctx r₁ + denote ctx m₁ + k₁ * Var.denote ctx v₁ - k₂ * Var.denote ctx v₁ = denote ctx r₂ + denote ctx m₂\nhaux : k₂ * Var.denote ctx v₁ ≤ k₁ * Var.denote ctx v₁\nih : denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₂ * Var.denote ctx v₁)",
"tactic": "rw [ih]"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : ¬blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nheqk : k₁ = k₂\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : ¬blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "have heqk : k₁ = k₂ := eq_of_not_blt_eq_true hltk hgtk"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ m₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nhltk hgtk : ¬blt k₁ k₁ = true\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₁, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ : List (Nat × Var)\nk₂ : Nat\nm₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nhltk : ¬blt k₁ k₂ = true\nhgtk : ¬blt k₂ k₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nheqk : k₁ = k₂\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₂, v₁) :: m₂)",
"tactic": "subst heqk"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ m₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nhltk hgtk : ¬blt k₁ k₁ = true\nih : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₁, v₁) :: m₂)",
"state_before": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ m₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nhltk hgtk : ¬blt k₁ k₁ = true\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₁, v₁) :: m₂)",
"tactic": "have ih := ih (h := h)"
},
{
"state_after": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ m₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nhltk hgtk : ¬blt k₁ k₁ = true\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₁ * Var.denote ctx v₁)",
"state_before": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ m₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nhltk hgtk : ¬blt k₁ k₁ = true\nih : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\n⊢ denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m₁, List.reverse r₂ ++ (k₁, v₁) :: m₂)",
"tactic": "simp [denote_eq] at ih ⊢"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3.inr.inr.inr.inr\nctx : Context\nfuel : Nat\nih✝ :\n ∀ (m₁ m₂ r₁ r₂ : Poly),\n denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)\nr₁ r₂ m₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\nm₁ m₂ : List (Nat × Var)\nhltv hgtv : ¬blt v₁ v₁ = true\nh : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)\nhltk hgtk : ¬blt k₁ k₁ = true\nih : denote ctx r₁ + denote ctx m₁ = denote ctx r₂ + denote ctx m₂\n⊢ denote ctx r₁ + (denote ctx m₁ + k₁ * Var.denote ctx v₁) = denote ctx r₂ + (denote ctx m₂ + k₁ * Var.denote ctx v₁)",
"tactic": "rw [← Nat.add_assoc, ih, Nat.add_assoc]"
}
] |
[
413,
52
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
384,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.deriv_mul_zero
|
[] |
[
650,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.ofNat'_bit
|
[] |
[
242,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.ext
|
[] |
[
201,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM0.univ_supports
|
[
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : Machine₀\n⊢ Supports M Set.univ",
"tactic": "constructor <;> intros <;> apply Set.mem_univ"
}
] |
[
1121,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1120,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.prod_mapRange_index
|
[
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ prod (mk (support g) fun i => f (↑i) (↑g ↑i)) h = prod g fun i b => h i (f i b)",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ prod (mapRange f hf g) h = prod g fun i b => h i (f i b)",
"tactic": "rw [mapRange_def]"
},
{
"state_after": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ ∀ (x : ι),\n x ∈ support g →\n ¬x ∈ support (mk (support g) fun i => f (↑i) (↑g ↑i)) → h x (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) x) = 1\n\ncase refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ ∏ x in support g, h x (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) x) = prod g fun i b => h i (f i b)",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ prod (mk (support g) fun i => f (↑i) (↑g ↑i)) h = prod g fun i b => h i (f i b)",
"tactic": "refine' (Finset.prod_subset support_mk_subset _).trans _"
},
{
"state_after": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : i ∈ support g\nh2 : ¬i ∈ support (mk (support g) fun i => f (↑i) (↑g ↑i))\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = 1",
"state_before": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ ∀ (x : ι),\n x ∈ support g →\n ¬x ∈ support (mk (support g) fun i => f (↑i) (↑g ↑i)) → h x (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) x) = 1",
"tactic": "intro i h1 h2"
},
{
"state_after": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh2 : ¬i ∈ support (mk (support g) fun i => f (↑i) (↑g ↑i))\nh1 : ¬↑g i = 0\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = 1",
"state_before": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : i ∈ support g\nh2 : ¬i ∈ support (mk (support g) fun i => f (↑i) (↑g ↑i))\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = 1",
"tactic": "simp only [mem_support_toFun, ne_eq] at h1"
},
{
"state_after": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : ¬↑g i = 0\nh2 : f i (↑g i) = 0\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = 1",
"state_before": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh2 : ¬i ∈ support (mk (support g) fun i => f (↑i) (↑g ↑i))\nh1 : ¬↑g i = 0\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = 1",
"tactic": "simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,\n dite_eq_ite, ite_true, not_not] at h2"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : ¬↑g i = 0\nh2 : f i (↑g i) = 0\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = 1",
"tactic": "simp [h2, h0]"
},
{
"state_after": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ ∀ (x : ι), x ∈ support g → h x (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) x) = (fun i b => h i (f i b)) x (↑g x)",
"state_before": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ ∏ x in support g, h x (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) x) = prod g fun i b => h i (f i b)",
"tactic": "refine' Finset.prod_congr rfl _"
},
{
"state_after": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : i ∈ support g\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = (fun i b => h i (f i b)) i (↑g i)",
"state_before": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\n⊢ ∀ (x : ι), x ∈ support g → h x (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) x) = (fun i b => h i (f i b)) x (↑g x)",
"tactic": "intro i h1"
},
{
"state_after": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : ¬↑g i = 0\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = (fun i b => h i (f i b)) i (↑g i)",
"state_before": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : i ∈ support g\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = (fun i b => h i (f i b)) i (↑g i)",
"tactic": "simp only [mem_support_toFun, ne_eq] at h1"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁✝ : ι → Type v₁\nβ₂✝ : ι → Type v₂\ndec : DecidableEq ι\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝⁴ : (i : ι) → Zero (β₁ i)\ninst✝³ : (i : ι) → Zero (β₂ i)\ninst✝² : (i : ι) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → (x : β₂ i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ng : Π₀ (i : ι), β₁ i\nh : (i : ι) → β₂ i → γ\nh0 : ∀ (i : ι), h i 0 = 1\ni : ι\nh1 : ¬↑g i = 0\n⊢ h i (↑(mk (support g) fun i => f (↑i) (↑g ↑i)) i) = (fun i b => h i (f i b)) i (↑g i)",
"tactic": "simp [h1]"
}
] |
[
1723,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1709,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
Set.PairwiseDisjoint.elim
|
[] |
[
324,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.cos_eq_one_iff_angle_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = cos 0 ↔ angle x y = 0",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = 1 ↔ angle x y = 0",
"tactic": "rw [← cos_zero]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = cos 0 ↔ angle x y = 0",
"tactic": "exact injOn_cos.eq_iff ⟨angle_nonneg x y, angle_le_pi x y⟩ (left_mem_Icc.2 pi_pos.le)"
}
] |
[
348,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
Submodule.iInf_comap_proj
|
[
{
"state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝² : Semiring R\nφ : ι → Type u_1\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ x : (i : ι) → φ i\n⊢ (x ∈ ⨅ (i : ι), comap (proj i) (p i)) ↔ x ∈ pi Set.univ p",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝² : Semiring R\nφ : ι → Type u_1\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\n⊢ (⨅ (i : ι), comap (proj i) (p i)) = pi Set.univ p",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝² : Semiring R\nφ : ι → Type u_1\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ x : (i : ι) → φ i\n⊢ (x ∈ ⨅ (i : ι), comap (proj i) (p i)) ↔ x ∈ pi Set.univ p",
"tactic": "simp"
}
] |
[
309,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Algebra/Quandle.lean
|
Rack.self_invAct_eq_iff_eq
|
[
{
"state_after": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : op x ◃ op x = op y ◃ op y ↔ op x = op y\n⊢ x ◃⁻¹ x = y ◃⁻¹ y ↔ x = y",
"state_before": "R : Type u_1\ninst✝ : Rack R\nx y : R\n⊢ x ◃⁻¹ x = y ◃⁻¹ y ↔ x = y",
"tactic": "have h := @self_act_eq_iff_eq _ _ (op x) (op y)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : op x ◃ op x = op y ◃ op y ↔ op x = op y\n⊢ x ◃⁻¹ x = y ◃⁻¹ y ↔ x = y",
"tactic": "simpa using h"
}
] |
[
320,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
lt_div_comm
|
[] |
[
1008,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.coe_to_submodule_eq_iff
|
[] |
[
225,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/Closure.lean
|
ClosureOperator.closure_sup_closure_left
|
[] |
[
259,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.mem_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7802\nγ : Type ?u.7805\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na : α\n⊢ a ∈ sᶜ ↔ ¬a ∈ s",
"tactic": "simp [compl_eq_univ_sdiff]"
}
] |
[
170,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.IsMultiplicative.map_mul_of_coprime
|
[] |
[
597,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.set_integral_eq_zero_of_ae_eq_zero
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0\n\ncase neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : ¬AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"state_before": "α : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"tactic": "by_cases hf : AEStronglyMeasurable f (μ.restrict t)"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : ¬AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0\n\ncase pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0\n\ncase neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : ¬AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"tactic": "swap"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nthis : (∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ) = 0\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"tactic": "have : (∫ x in t, hf.mk f x ∂μ) = 0 := by\n refine' integral_eq_zero_of_ae _\n rw [EventuallyEq,\n ae_restrict_iff (hf.stronglyMeasurable_mk.measurableSet_eq_fun stronglyMeasurable_zero)]\n filter_upwards [ae_imp_of_ae_restrict hf.ae_eq_mk, ht_eq] with x hx h'x h''x\n rw [← hx h''x]\n exact h'x h''x"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nthis : (∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ) = 0\n⊢ (∫ (x : α) in t, f x ∂μ) = ∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nthis : (∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ) = 0\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"tactic": "rw [← this]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nthis : (∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ) = 0\n⊢ (∫ (x : α) in t, f x ∂μ) = ∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ",
"tactic": "exact integral_congr_ae hf.ae_eq_mk"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : ¬AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ ¬Integrable fun x => f x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : ¬AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, f x ∂μ) = 0",
"tactic": "rw [integral_undef]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : Integrable fun x => f x\n⊢ AEStronglyMeasurable f (Measure.restrict μ t)",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : ¬AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ ¬Integrable fun x => f x",
"tactic": "contrapose! hf"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : Integrable fun x => f x\n⊢ AEStronglyMeasurable f (Measure.restrict μ t)",
"tactic": "exact hf.1"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (fun x => AEStronglyMeasurable.mk f hf x) =ᵐ[Measure.restrict μ t] 0",
"state_before": "α : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (∫ (x : α) in t, AEStronglyMeasurable.mk f hf x ∂μ) = 0",
"tactic": "refine' integral_eq_zero_of_ae _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ t → AEStronglyMeasurable.mk f hf x = OfNat.ofNat 0 x",
"state_before": "α : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ (fun x => AEStronglyMeasurable.mk f hf x) =ᵐ[Measure.restrict μ t] 0",
"tactic": "rw [EventuallyEq,\n ae_restrict_iff (hf.stronglyMeasurable_mk.measurableSet_eq_fun stronglyMeasurable_zero)]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nx : α\nhx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x\nh'x : x ∈ t → f x = 0\nh''x : x ∈ t\n⊢ AEStronglyMeasurable.mk f hf x = OfNat.ofNat 0 x",
"state_before": "α : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ t → AEStronglyMeasurable.mk f hf x = OfNat.ofNat 0 x",
"tactic": "filter_upwards [ae_imp_of_ae_restrict hf.ae_eq_mk, ht_eq] with x hx h'x h''x"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nx : α\nhx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x\nh'x : x ∈ t → f x = 0\nh''x : x ∈ t\n⊢ f x = OfNat.ofNat 0 x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nx : α\nhx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x\nh'x : x ∈ t → f x = 0\nh''x : x ∈ t\n⊢ AEStronglyMeasurable.mk f hf x = OfNat.ofNat 0 x",
"tactic": "rw [← hx h''x]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.72393\nE : Type u_2\nF : Type ?u.72399\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0\nhf : AEStronglyMeasurable f (Measure.restrict μ t)\nx : α\nhx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x\nh'x : x ∈ t → f x = 0\nh''x : x ∈ t\n⊢ f x = OfNat.ofNat 0 x",
"tactic": "exact h'x h''x"
}
] |
[
280,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
HomogeneousIdeal.toIdeal_sInf
|
[] |
[
390,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.Embedding.isElementary_of_exists
|
[
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nh :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty n) (xs : Fin n → M),\n BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\n⊢ ∀ {n : ℕ} (φ : Formula L (Fin n)) (x : Fin n → M), Formula.Realize φ (↑f ∘ x) ↔ Formula.Realize φ x\n\ncase h\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\n⊢ ∀ (n : ℕ) (φ : BoundedFormula L Empty n) (xs : Fin n → M),\n BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\n⊢ ∀ {n : ℕ} (φ : Formula L (Fin n)) (x : Fin n → M), Formula.Realize φ (↑f ∘ x) ↔ Formula.Realize φ x",
"tactic": "suffices h :\n ∀ (n : ℕ) (φ : L.BoundedFormula Empty n) (xs : Fin n → M),\n φ.Realize (f ∘ default) (f ∘ xs) ↔ φ.Realize default xs"
},
{
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"tactic": "refine' fun n φ => φ.recOn _ _ _ _ _"
},
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"tactic": "intro n φ x"
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{
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"tactic": "refine' φ.realize_relabel_sum_inr.symm.trans (_root_.trans (h n _ _) φ.realize_relabel_sum_inr)"
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{
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"tactic": "exact fun {_} _ => Iff.rfl"
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{
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"tactic": "intros"
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{
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"tactic": "simp [BoundedFormula.Realize, ← Sum.comp_elim, Embedding.realize_term]"
},
{
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"tactic": "simp [BoundedFormula.Realize, ← Sum.comp_elim, Embedding.realize_term]"
},
{
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"tactic": "erw [map_rel f]"
},
{
"state_after": "case h.refine'_4\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn : ℕ\nφ : BoundedFormula L Empty n\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L Empty n✝\nih1 : ∀ (xs : Fin n✝ → M), BoundedFormula.Realize f₁✝ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize f₁✝ default xs\nih2 : ∀ (xs : Fin n✝ → M), BoundedFormula.Realize f₂✝ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize f₂✝ default xs\nxs✝ : Fin n✝ → M\n⊢ BoundedFormula.Realize (f₁✝ ⟹ f₂✝) (↑f ∘ default) (↑f ∘ xs✝) ↔ BoundedFormula.Realize (f₁✝ ⟹ f₂✝) default xs✝",
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"tactic": "intro _ _ _ ih1 ih2 _"
},
{
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"tactic": "simp [ih1, ih2]"
},
{
"state_after": "case h.refine'_5\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ BoundedFormula.Realize (∀'φ) (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize (∀'φ) default xs",
"state_before": "case h.refine'_5\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn : ℕ\nφ : BoundedFormula L Empty n\n⊢ ∀ {n : ℕ} (f_1 : BoundedFormula L Empty (n + 1)),\n (∀ (xs : Fin (n + 1) → M),\n BoundedFormula.Realize f_1 (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize f_1 default xs) →\n ∀ (xs : Fin n → M),\n BoundedFormula.Realize (∀'f_1) (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize (∀'f_1) default xs",
"tactic": "intro n φ ih xs"
},
{
"state_after": "case h.refine'_5\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ (∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)) ↔\n ∀ (a : M), BoundedFormula.Realize φ default (Fin.snoc xs a)",
"state_before": "case h.refine'_5\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ BoundedFormula.Realize (∀'φ) (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize (∀'φ) default xs",
"tactic": "simp only [BoundedFormula.realize_all]"
},
{
"state_after": "case h.refine'_5.refine'_1\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\nh : ∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\na : M\n⊢ BoundedFormula.Realize φ default (Fin.snoc xs a)\n\ncase h.refine'_5.refine'_2\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ (∀ (a : M), BoundedFormula.Realize φ default (Fin.snoc xs a)) →\n ∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)",
"state_before": "case h.refine'_5\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ (∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)) ↔\n ∀ (a : M), BoundedFormula.Realize φ default (Fin.snoc xs a)",
"tactic": "refine' ⟨fun h a => _, _⟩"
},
{
"state_after": "case h.refine'_5.refine'_1\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\nh : ∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\na : M\n⊢ BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) (↑f a))",
"state_before": "case h.refine'_5.refine'_1\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\nh : ∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\na : M\n⊢ BoundedFormula.Realize φ default (Fin.snoc xs a)",
"tactic": "rw [← ih, Fin.comp_snoc]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_5.refine'_1\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\nh : ∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\na : M\n⊢ BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) (↑f a))",
"tactic": "exact h (f a)"
},
{
"state_after": "case h.refine'_5.refine'_2\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ (∃ a, ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)) →\n ∃ a, ¬BoundedFormula.Realize φ default (Fin.snoc xs a)",
"state_before": "case h.refine'_5.refine'_2\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ (∀ (a : M), BoundedFormula.Realize φ default (Fin.snoc xs a)) →\n ∀ (a : N), BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)",
"tactic": "contrapose!"
},
{
"state_after": "case h.refine'_5.refine'_2.intro\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\n⊢ ∃ a, ¬BoundedFormula.Realize φ default (Fin.snoc xs a)",
"state_before": "case h.refine'_5.refine'_2\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\n⊢ (∃ a, ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)) →\n ∃ a, ¬BoundedFormula.Realize φ default (Fin.snoc xs a)",
"tactic": "rintro ⟨a, ha⟩"
},
{
"state_after": "case h.refine'_5.refine'_2.intro.intro\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\nb : M\nhb : BoundedFormula.Realize (∼φ) default (Fin.snoc (↑f ∘ xs) (↑f b))\n⊢ ∃ a, ¬BoundedFormula.Realize φ default (Fin.snoc xs a)",
"state_before": "case h.refine'_5.refine'_2.intro\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\n⊢ ∃ a, ¬BoundedFormula.Realize φ default (Fin.snoc xs a)",
"tactic": "obtain ⟨b, hb⟩ := htv n φ.not xs a (by\n rw [BoundedFormula.realize_not, ← Unique.eq_default (f ∘ default)]\n exact ha)"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\n⊢ ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\n⊢ BoundedFormula.Realize (∼φ) default (Fin.snoc (↑f ∘ xs) a)",
"tactic": "rw [BoundedFormula.realize_not, ← Unique.eq_default (f ∘ default)]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\n⊢ ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)",
"tactic": "exact ha"
},
{
"state_after": "case h.refine'_5.refine'_2.intro.intro\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\nb : M\nhb : BoundedFormula.Realize (∼φ) default (Fin.snoc (↑f ∘ xs) (↑f b))\nh : BoundedFormula.Realize φ default (Fin.snoc xs b)\n⊢ BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ Fin.snoc xs b) =\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ xs) (↑f b))",
"state_before": "case h.refine'_5.refine'_2.intro.intro\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\nb : M\nhb : BoundedFormula.Realize (∼φ) default (Fin.snoc (↑f ∘ xs) (↑f b))\n⊢ ∃ a, ¬BoundedFormula.Realize φ default (Fin.snoc xs a)",
"tactic": "refine' ⟨b, fun h => hb (Eq.mp _ ((ih _).2 h))⟩"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_5.refine'_2.intro.intro\nL : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.353970\nQ : Type ?u.353973\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪[L] N\nhtv :\n ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → M) (a : N),\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ x) (↑f b))\nn✝ : ℕ\nφ✝ : BoundedFormula L Empty n✝\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nih : ∀ (xs : Fin (n + 1) → M), BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ xs) ↔ BoundedFormula.Realize φ default xs\nxs : Fin n → M\na : N\nha : ¬BoundedFormula.Realize φ (↑f ∘ default) (Fin.snoc (↑f ∘ xs) a)\nb : M\nhb : BoundedFormula.Realize (∼φ) default (Fin.snoc (↑f ∘ xs) (↑f b))\nh : BoundedFormula.Realize φ default (Fin.snoc xs b)\n⊢ BoundedFormula.Realize φ (↑f ∘ default) (↑f ∘ Fin.snoc xs b) =\n BoundedFormula.Realize φ default (Fin.snoc (↑f ∘ xs) (↑f b))",
"tactic": "rw [Unique.eq_default (f ∘ default), Fin.comp_snoc]"
}
] |
[
313,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean
|
FiniteDimensional.finrank_eq_card_chooseBasisIndex
|
[
{
"state_after": "R : Type u\nM : Type v\nN : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : StrongRankCondition R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module.Free R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ finrank R M = card (ChooseBasisIndex R M)",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : StrongRankCondition R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module.Free R N\ninst✝ : Module.Finite R N\n⊢ finrank R M = card (ChooseBasisIndex R M)",
"tactic": "letI := nontrivial_of_invariantBasisNumber R"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : StrongRankCondition R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module.Free R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ finrank R M = card (ChooseBasisIndex R M)",
"tactic": "simp [finrank, rank_eq_card_chooseBasisIndex]"
}
] |
[
93,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean
|
CategoryTheory.MonoidalCategory.id_tensor_associator_inv_naturality
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\n𝒞 : Category C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z✝ X Y Z X' : C\nf : X ⟶ X'\n⊢ (f ⊗ 𝟙 (Y ⊗ Z)) ≫ (α_ X' Y Z).inv = (α_ X Y Z).inv ≫ ((f ⊗ 𝟙 Y) ⊗ 𝟙 Z)",
"tactic": "rw [← tensor_id, associator_inv_naturality]"
}
] |
[
350,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_compl
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.90549\nγ : Type ?u.90552\ns t : Set α\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype ↑s\ninst✝² : Fintype ↑t\ninst✝¹ : Fintype α\ninst✝ : Fintype ↑(sᶜ)\na✝ : α\n⊢ a✝ ∈ toFinset (sᶜ) ↔ a✝ ∈ toFinset sᶜ",
"state_before": "α : Type u_1\nβ : Type ?u.90549\nγ : Type ?u.90552\ns t : Set α\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype ↑s\ninst✝² : Fintype ↑t\ninst✝¹ : Fintype α\ninst✝ : Fintype ↑(sᶜ)\n⊢ toFinset (sᶜ) = toFinset sᶜ",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.90549\nγ : Type ?u.90552\ns t : Set α\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype ↑s\ninst✝² : Fintype ↑t\ninst✝¹ : Fintype α\ninst✝ : Fintype ↑(sᶜ)\na✝ : α\n⊢ a✝ ∈ toFinset (sᶜ) ↔ a✝ ∈ toFinset sᶜ",
"tactic": "simp"
}
] |
[
724,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
722,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.one_lt_ncard
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.156100\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ 1 < ncard s ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ s ∧ a ≠ b",
"tactic": "simp_rw [ncard_eq_toFinset_card _ hs, Finset.one_lt_card, Finite.mem_toFinset]"
}
] |
[
699,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
697,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
finrank_eq_zero_of_not_exists_basis_finset
|
[] |
[
209,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.lastFun_from_append1_drop_last
|
[] |
[
766,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Topology/Order.lean
|
induced_iInf
|
[] |
[
444,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Data/Finset/Pairwise.lean
|
Set.PairwiseDisjoint.image_finset_of_le
|
[
{
"state_after": "α : Type u_2\nι : Type u_1\nι' : Type ?u.1728\ninst✝² : DecidableEq ι\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\nhs : PairwiseDisjoint (↑s) f\ng : ι → ι\nhf : ∀ (a : ι), f (g a) ≤ f a\n⊢ PairwiseDisjoint (g '' ↑s) f",
"state_before": "α : Type u_2\nι : Type u_1\nι' : Type ?u.1728\ninst✝² : DecidableEq ι\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\nhs : PairwiseDisjoint (↑s) f\ng : ι → ι\nhf : ∀ (a : ι), f (g a) ≤ f a\n⊢ PairwiseDisjoint (↑(Finset.image g s)) f",
"tactic": "rw [coe_image]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nι : Type u_1\nι' : Type ?u.1728\ninst✝² : DecidableEq ι\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\nhs : PairwiseDisjoint (↑s) f\ng : ι → ι\nhf : ∀ (a : ι), f (g a) ≤ f a\n⊢ PairwiseDisjoint (g '' ↑s) f",
"tactic": "exact hs.image_of_le hf"
}
] |
[
47,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Algebra/Invertible.lean
|
mul_left_inj_of_invertible
|
[] |
[
305,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Order/MinMax.lean
|
min_max_distrib_left
|
[] |
[
124,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Topology/Inseparable.lean
|
Inseparable.of_eq
|
[] |
[
350,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Embedding.equivRange_apply
|
[] |
[
983,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
982,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_image_ofInjective
|
[] |
[
270,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Topology/Bornology/Hom.lean
|
LocallyBoundedMap.ext
|
[] |
[
101,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_le_iff_le_comap
|
[] |
[
828,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasDerivAt_exp_smul_const'
|
[] |
[
430,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.exists_not_mem_col
|
[
{
"state_after": "case h.e'_2\nμ : YoungDiagram\nj : ℕ\n⊢ (fun i => ¬(i, j) ∈ μ.cells) = fun j_1 => ¬(j, j_1) ∈ transpose μ",
"state_before": "μ : YoungDiagram\nj : ℕ\n⊢ ∃ i, ¬(i, j) ∈ μ.cells",
"tactic": "convert μ.transpose.exists_not_mem_row j using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nμ : YoungDiagram\nj : ℕ\n⊢ (fun i => ¬(i, j) ∈ μ.cells) = fun j_1 => ¬(j, j_1) ∈ transpose μ",
"tactic": "simp"
}
] |
[
360,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
11
] |
Mathlib/Topology/Order/Basic.lean
|
Filter.Tendsto.atBot_add
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\nC : α\nhf : Tendsto f l atBot\nhg : Tendsto g l (𝓝 C)\n⊢ Tendsto (fun x => g x + f x) l atBot",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\nC : α\nhf : Tendsto f l atBot\nhg : Tendsto g l (𝓝 C)\n⊢ Tendsto (fun x => f x + g x) l atBot",
"tactic": "conv in _ + _ => rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\nC : α\nhf : Tendsto f l atBot\nhg : Tendsto g l (𝓝 C)\n⊢ Tendsto (fun x => g x + f x) l atBot",
"tactic": "exact hg.add_atBot hf"
}
] |
[
1907,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1904,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.apply_inv_apply
|
[] |
[
215,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.IsPeriodicPt.comp_lcm
|
[] |
[
151,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.domRestrict_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.448336\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nS : Submodule R E\n⊢ (domRestrict f S).domain ≤ f.domain",
"tactic": "simp"
}
] |
[
712,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
711,
1
] |
Mathlib/Data/Prod/Lex.lean
|
Prod.Lex.lt_iff
|
[] |
[
67,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
Submodule.exists_lieSubmodule_coe_eq_iff
|
[
{
"state_after": "case mp\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\n⊢ (∃ N, ↑N = p) → ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p\n\ncase mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\n⊢ (∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p) → ∃ N, ↑N = p",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\n⊢ (∃ N, ↑N = p) ↔ ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx✝ : L\nm✝ : M\n⊢ m✝ ∈ ↑N → ⁅x✝, m✝⁆ ∈ ↑N",
"state_before": "case mp\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\n⊢ (∃ N, ↑N = p) → ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p",
"tactic": "rintro ⟨N, rfl⟩ _ _"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx✝ : L\nm✝ : M\n⊢ m✝ ∈ ↑N → ⁅x✝, m✝⁆ ∈ ↑N",
"tactic": "exact N.lie_mem"
},
{
"state_after": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\nh : ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p\n⊢ ∃ N, ↑N = p",
"state_before": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\n⊢ (∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p) → ∃ N, ↑N = p",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\np : Submodule R M\nh : ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p\n⊢ ∃ N, ↑N = p",
"tactic": "use { p with lie_mem := @h }"
}
] |
[
297,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.mem_closedBall
|
[] |
[
481,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
9
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.ContinuousHom.ext
|
[
{
"state_after": "case mk\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ng : α →𝒄 β\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh : ∀ (x : α), ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x = ↑g.toOrderHom x\n⊢ { toOrderHom := toOrderHom✝, cont := cont✝ } = g",
"state_before": "α : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nf g : α →𝒄 β\nh : ∀ (x : α), ↑f.toOrderHom x = ↑g.toOrderHom x\n⊢ f = g",
"tactic": "cases f"
},
{
"state_after": "case mk.mk\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ntoOrderHom✝¹ : α →o β\ncont✝¹ : ∀ {Monotone' : Monotone ↑toOrderHom✝¹}, Continuous { toFun := ↑toOrderHom✝¹, monotone' := Monotone' }\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh :\n ∀ (x : α),\n ↑{ toOrderHom := toOrderHom✝¹, cont := cont✝¹ }.toOrderHom x =\n ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x\n⊢ { toOrderHom := toOrderHom✝¹, cont := cont✝¹ } = { toOrderHom := toOrderHom✝, cont := cont✝ }",
"state_before": "case mk\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ng : α →𝒄 β\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh : ∀ (x : α), ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x = ↑g.toOrderHom x\n⊢ { toOrderHom := toOrderHom✝, cont := cont✝ } = g",
"tactic": "cases g"
},
{
"state_after": "case mk.mk.e_toOrderHom\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ntoOrderHom✝¹ : α →o β\ncont✝¹ : ∀ {Monotone' : Monotone ↑toOrderHom✝¹}, Continuous { toFun := ↑toOrderHom✝¹, monotone' := Monotone' }\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh :\n ∀ (x : α),\n ↑{ toOrderHom := toOrderHom✝¹, cont := cont✝¹ }.toOrderHom x =\n ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x\n⊢ toOrderHom✝¹ = toOrderHom✝",
"state_before": "case mk.mk\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ntoOrderHom✝¹ : α →o β\ncont✝¹ : ∀ {Monotone' : Monotone ↑toOrderHom✝¹}, Continuous { toFun := ↑toOrderHom✝¹, monotone' := Monotone' }\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh :\n ∀ (x : α),\n ↑{ toOrderHom := toOrderHom✝¹, cont := cont✝¹ }.toOrderHom x =\n ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x\n⊢ { toOrderHom := toOrderHom✝¹, cont := cont✝¹ } = { toOrderHom := toOrderHom✝, cont := cont✝ }",
"tactic": "congr"
},
{
"state_after": "case mk.mk.e_toOrderHom.h.h\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ntoOrderHom✝¹ : α →o β\ncont✝¹ : ∀ {Monotone' : Monotone ↑toOrderHom✝¹}, Continuous { toFun := ↑toOrderHom✝¹, monotone' := Monotone' }\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh :\n ∀ (x : α),\n ↑{ toOrderHom := toOrderHom✝¹, cont := cont✝¹ }.toOrderHom x =\n ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x\nx✝ : α\n⊢ ↑toOrderHom✝¹ x✝ = ↑toOrderHom✝ x✝",
"state_before": "case mk.mk.e_toOrderHom\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ntoOrderHom✝¹ : α →o β\ncont✝¹ : ∀ {Monotone' : Monotone ↑toOrderHom✝¹}, Continuous { toFun := ↑toOrderHom✝¹, monotone' := Monotone' }\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh :\n ∀ (x : α),\n ↑{ toOrderHom := toOrderHom✝¹, cont := cont✝¹ }.toOrderHom x =\n ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x\n⊢ toOrderHom✝¹ = toOrderHom✝",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.e_toOrderHom.h.h\nα : Type u\nα' : Type ?u.64625\nβ : Type v\nβ' : Type ?u.64630\nγ : Type ?u.64633\nφ : Type ?u.64636\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\ntoOrderHom✝¹ : α →o β\ncont✝¹ : ∀ {Monotone' : Monotone ↑toOrderHom✝¹}, Continuous { toFun := ↑toOrderHom✝¹, monotone' := Monotone' }\ntoOrderHom✝ : α →o β\ncont✝ : ∀ {Monotone' : Monotone ↑toOrderHom✝}, Continuous { toFun := ↑toOrderHom✝, monotone' := Monotone' }\nh :\n ∀ (x : α),\n ↑{ toOrderHom := toOrderHom✝¹, cont := cont✝¹ }.toOrderHom x =\n ↑{ toOrderHom := toOrderHom✝, cont := cont✝ }.toOrderHom x\nx✝ : α\n⊢ ↑toOrderHom✝¹ x✝ = ↑toOrderHom✝ x✝",
"tactic": "apply h"
}
] |
[
713,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
11
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
Set.Finite.nullMeasurableSet_sUnion
|
[] |
[
387,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Data/Quot.lean
|
Quot.induction_on₃
|
[] |
[
185,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
11
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.le_of_forall_lf
|
[] |
[
442,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.isSuccLimit_bot
|
[] |
[
68,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Measurable.sin
|
[] |
[
158,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.mulEquivOfQuotient_monoidOf
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type ?u.3281425\ninst✝ : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\n⊢ ↑(mulEquivOfQuotient f) (↑(Submonoid.LocalizationMap.toMap (monoidOf S)) x) = ↑(Submonoid.LocalizationMap.toMap f) x",
"tactic": "simp"
}
] |
[
1710,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1709,
1
] |
Mathlib/Topology/Sheaves/Presheaf.lean
|
TopCat.Presheaf.Pushforward.comp_eq
|
[] |
[
252,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.hasBasis_uniformity_of_basis_aux₁
|
[
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type ?u.59454\nι : Type u_2\ns✝ s' : Set α\nx : α\np✝ : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\np : ι → Prop\ns : ι → Set (β × β)\nhb : HasBasis (𝓤 β) p s\nS : Set α\n⊢ HasBasis (comap (Prod.map (restrict S) (restrict S)) (𝓤 (↑S → β))) p fun i =>\n Prod.map (restrict S ∘ ↑UniformFun.toFun) (restrict S ∘ ↑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β (s i)",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type ?u.59454\nι : Type u_2\ns✝ s' : Set α\nx : α\np✝ : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\np : ι → Prop\ns : ι → Set (β × β)\nhb : HasBasis (𝓤 β) p s\nS : Set α\n⊢ HasBasis (𝓤 (α →ᵤ[𝔖] β)) p fun i => UniformOnFun.gen 𝔖 S (s i)",
"tactic": "simp_rw [UniformOnFun.gen_eq_preimage_restrict, uniformity_comap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type ?u.59454\nι : Type u_2\ns✝ s' : Set α\nx : α\np✝ : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\np : ι → Prop\ns : ι → Set (β × β)\nhb : HasBasis (𝓤 β) p s\nS : Set α\n⊢ HasBasis (comap (Prod.map (restrict S) (restrict S)) (𝓤 (↑S → β))) p fun i =>\n Prod.map (restrict S ∘ ↑UniformFun.toFun) (restrict S ∘ ↑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β (s i)",
"tactic": "exact (UniformFun.hasBasis_uniformity_of_basis S β hb).comap _"
}
] |
[
641,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
11
] |
Mathlib/Topology/Order/Basic.lean
|
IsGLB.mem_upperBounds_of_tendsto
|
[] |
[
2077,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2074,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Finset.lean
|
Finset.Nonempty.cInf_mem
|
[] |
[
58,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.isLittleO_of_lt_radius
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\n⊢ ∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\n⊢ ∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x",
"tactic": "have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\n⊢ ∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x",
"tactic": "rw [this]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nh : ∃ i i_1 i_2, ↑r < ↑i\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"tactic": "simp only [radius, lt_iSup_iff] at h"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nh : ∃ i i_1 i_2, ↑r < ↑i\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"tactic": "rcases h with ⟨t, C, hC, rt⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"tactic": "rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r < ↑t\nthis : 0 < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"tactic": "have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r < ↑t\nthis : 0 < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"tactic": "rw [← div_lt_one this] at rt"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\nn : ℕ\n⊢ abs (‖p n‖ * ↑r ^ n) ≤ C * (↑r / ↑t) ^ n",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\n⊢ ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n",
"tactic": "refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\nn : ℕ\n⊢ abs (‖p n‖ * ↑r ^ n) ≤ C * (↑r / ↑t) ^ n",
"tactic": "calc\n |‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by\n field_simp [mul_right_comm, abs_mul, this.ne']\n _ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\nn : ℕ\n⊢ abs (‖p n‖ * ↑r ^ n) = ‖p n‖ * ↑t ^ n * (↑r / ↑t) ^ n",
"tactic": "field_simp [mul_right_comm, abs_mul, this.ne']"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\nn : ℕ\n⊢ ‖p n‖ * ↑t ^ n ≤ C",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\nn : ℕ\n⊢ ‖p n‖ * ↑t ^ n * (↑r / ↑t) ^ n ≤ C * (↑r / ↑t) ^ n",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.114222\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nthis✝ :\n (∃ a, a ∈ Ioo 0 1 ∧ (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔\n ∃ a, a < 1 ∧ ∃ C x, ∀ (n : ℕ), abs (‖p n‖ * ↑r ^ n) ≤ C * a ^ n\nt : ℝ≥0\nC : ℝ\nhC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C\nrt : ↑r / ↑t < 1\nthis : 0 < ↑t\nn : ℕ\n⊢ ‖p n‖ * ↑t ^ n ≤ C",
"tactic": "apply hC"
}
] |
[
209,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Std/Data/Int/DivMod.lean
|
Int.fdiv_one
|
[] |
[
127,
47
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
124,
9
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_le_div_left
|
[] |
[
378,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.constantCoeff_comp_map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\n⊢ RingHom.comp constantCoeff (map f) = RingHom.comp f constantCoeff",
"tactic": "ext <;> simp"
}
] |
[
1345,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1343,
1
] |
Mathlib/Data/List/Zip.lean
|
List.zipWith_zipWith_left
|
[] |
[
303,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/RingTheory/MvPolynomial/Symmetric.lean
|
MvPolynomial.IsSymmetric.smul
|
[] |
[
132,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Topology/Instances/Real.lean
|
Real.Continuous.inv
|
[] |
[
122,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/CategoryTheory/Adjunction/Opposites.lean
|
CategoryTheory.Adjunction.rightAdjointUniq_trans_app
|
[
{
"state_after": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' G'' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nadj3 : F ⊣ G''\nx : D\n⊢ ((rightAdjointUniq adj1 adj2).hom.app x ≫ (rightAdjointUniq adj2 adj3).hom.app x).op =\n ((rightAdjointUniq adj1 adj3).hom.app x).op",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' G'' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nadj3 : F ⊣ G''\nx : D\n⊢ (rightAdjointUniq adj1 adj2).hom.app x ≫ (rightAdjointUniq adj2 adj3).hom.app x =\n (rightAdjointUniq adj1 adj3).hom.app x",
"tactic": "apply Quiver.Hom.op_inj"
},
{
"state_after": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' G'' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nadj3 : F ⊣ G''\nx : D\n⊢ (leftAdjointUniq (opAdjointOpOfAdjoint G'' F adj3) (opAdjointOpOfAdjoint G' F adj2)).hom.app x.op ≫\n (leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op =\n (leftAdjointUniq (opAdjointOpOfAdjoint G'' F adj3) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op",
"state_before": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' G'' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nadj3 : F ⊣ G''\nx : D\n⊢ ((rightAdjointUniq adj1 adj2).hom.app x ≫ (rightAdjointUniq adj2 adj3).hom.app x).op =\n ((rightAdjointUniq adj1 adj3).hom.app x).op",
"tactic": "dsimp [rightAdjointUniq]"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' G'' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nadj3 : F ⊣ G''\nx : D\n⊢ (leftAdjointUniq (opAdjointOpOfAdjoint G'' F adj3) (opAdjointOpOfAdjoint G' F adj2)).hom.app x.op ≫\n (leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op =\n (leftAdjointUniq (opAdjointOpOfAdjoint G'' F adj3) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op",
"tactic": "simp"
}
] |
[
301,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Ioc_ofDual
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\n⊢ Ioc (↑ofDual a) (↑ofDual b) = Ico b a",
"state_before": "α : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\n⊢ Ioc (↑ofDual a) (↑ofDual b) = map (Equiv.toEmbedding ofDual) (Ico b a)",
"tactic": "refine' Eq.trans _ map_refl.symm"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ c ∈ Ioc (↑ofDual a) (↑ofDual b) ↔ c ∈ Ico b a",
"state_before": "α : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\n⊢ Ioc (↑ofDual a) (↑ofDual b) = Ico b a",
"tactic": "ext c"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ ↑ofDual a < c ∧ c ≤ ↑ofDual b ↔ b ≤ c ∧ c < a",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ c ∈ Ioc (↑ofDual a) (↑ofDual b) ↔ c ∈ Ico b a",
"tactic": "rw [mem_Ioc, mem_Ico (α := αᵒᵈ)]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.115387\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ ↑ofDual a < c ∧ c ≤ ↑ofDual b ↔ b ≤ c ∧ c < a",
"tactic": "exact and_comm"
}
] |
[
873,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/CategoryTheory/Abelian/Exact.lean
|
CategoryTheory.Abelian.exact_iff
|
[
{
"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 → f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n\ncase mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0 → 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 ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0",
"tactic": "constructor"
},
{
"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\n⊢ Exact f g → f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0",
"tactic": "exact fun h ↦ ⟨h.1, kernel_comp_cokernel f g h⟩"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))",
"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⊢ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0 → Exact f g",
"tactic": "refine fun h ↦ ⟨h.1, ?_⟩"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))\n\ncase hl\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))",
"tactic": "suffices hl :\n IsLimit (KernelFork.ofι (imageSubobject f).arrow (imageSubobject_arrow_comp_eq_zero h.1))"
},
{
"state_after": "case hl.refine_1\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nu : W'✝ ⟶ Y\nhu : u ≫ g = 0\n⊢ W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\n\ncase hl.refine_2\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ ∀ {W' : C} (g' : W' ⟶ Y) (eq' : g' ≫ g = 0),\n (fun {W'} u hu => ?m.8507 u hu) g' eq' ≫ Subobject.arrow (imageSubobject f) = g'\n\ncase hl.refine_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh✝ : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nx✝² : W'✝ ⟶ Y\nx✝¹ : x✝² ≫ g = 0\nx✝ : W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\nh : x✝ ≫ Subobject.arrow (imageSubobject f) = x✝²\n⊢ x✝ = (fun {W'} u hu => ?m.8507 u hu) x✝² x✝¹",
"state_before": "case hl\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))",
"tactic": "refine KernelFork.IsLimit.ofι _ _ (fun u hu ↦ ?_) ?_ (fun _ _ _ h ↦ ?_)"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\nthis :\n imageToKernel f g (_ : f ≫ g = 0) =\n (IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))",
"tactic": "have : imageToKernel f g h.1 = (hl.conePointUniqueUpToIso (limit.isLimit _)).hom ≫\n (kernelSubobjectIso _).inv := by ext; simp"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\nthis :\n imageToKernel f g (_ : f ≫ g = 0) =\n (IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv\n⊢ Epi ((IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv)",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\nthis :\n imageToKernel f g (_ : f ≫ g = 0) =\n (IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))",
"tactic": "rw [this]"
},
{
"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\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\nthis :\n imageToKernel f g (_ : f ≫ g = 0) =\n (IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv\n⊢ Epi ((IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv)",
"tactic": "infer_instance"
},
{
"state_after": "case h\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\n⊢ imageToKernel f g (_ : f ≫ g = 0) ≫ Subobject.arrow (kernelSubobject g) =\n ((IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv) ≫\n Subobject.arrow (kernelSubobject g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\n⊢ imageToKernel f g (_ : f ≫ g = 0) =\n (IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nhl : IsLimit (KernelFork.ofι (Subobject.arrow (imageSubobject f)) (_ : Subobject.arrow (imageSubobject f) ≫ g = 0))\n⊢ imageToKernel f g (_ : f ≫ g = 0) ≫ Subobject.arrow (kernelSubobject g) =\n ((IsLimit.conePointUniqueUpToIso hl (limit.isLimit (parallelPair g 0))).hom ≫ (kernelSubobjectIso g).inv) ≫\n Subobject.arrow (kernelSubobject g)",
"tactic": "simp"
},
{
"state_after": "case hl.refine_1\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nu : W'✝ ⟶ Y\nhu : u ≫ g = 0\n⊢ u ≫ cokernel.π f = 0",
"state_before": "case hl.refine_1\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nu : W'✝ ⟶ Y\nhu : u ≫ g = 0\n⊢ W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)",
"tactic": "refine kernel.lift (cokernel.π f) u ?_ ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso _).inv"
},
{
"state_after": "no goals",
"state_before": "case hl.refine_1\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nu : W'✝ ⟶ Y\nhu : u ≫ g = 0\n⊢ u ≫ cokernel.π f = 0",
"tactic": "rw [← kernel.lift_ι g u hu, Category.assoc, h.2, comp_zero]"
},
{
"state_after": "no goals",
"state_before": "case hl.refine_2\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ ∀ {W' : C} (g' : W' ⟶ Y) (eq' : g' ≫ g = 0),\n (fun {W'} u hu =>\n kernel.lift (cokernel.π f) u (_ : u ≫ cokernel.π f = 0) ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso f).inv)\n g' eq' ≫\n Subobject.arrow (imageSubobject f) =\n g'",
"tactic": "aesop_cat"
},
{
"state_after": "case hl.refine_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh✝ : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nx✝² : W'✝ ⟶ Y\nx✝¹ : x✝² ≫ g = 0\nx✝ : W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\nh : x✝ ≫ Subobject.arrow (imageSubobject f) = x✝²\n⊢ x✝ =\n (fun {W'} u hu =>\n kernel.lift (cokernel.π f) u (_ : u ≫ cokernel.π f = 0) ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso f).inv)\n x✝² x✝¹",
"state_before": "case hl.refine_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh✝ : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nx✝² : W'✝ ⟶ Y\nx✝¹ : x✝² ≫ g = 0\nx✝ : W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\nh : x✝ ≫ Subobject.arrow (imageSubobject f) = x✝²\n⊢ x✝ =\n (fun {W'} u hu =>\n kernel.lift (cokernel.π f) u (_ : u ≫ cokernel.π f = 0) ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso f).inv)\n x✝² x✝¹",
"tactic": "intros"
},
{
"state_after": "case hl.refine_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh✝ : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nx✝² : W'✝ ⟶ Y\nx✝¹ : x✝² ≫ g = 0\nx✝ : W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\nh : x✝ ≫ Subobject.arrow (imageSubobject f) = x✝²\n⊢ x✝² =\n (fun {W'} u hu =>\n kernel.lift (cokernel.π f) u (_ : u ≫ cokernel.π f = 0) ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso f).inv)\n x✝² x✝¹ ≫\n Subobject.arrow (imageSubobject f)",
"state_before": "case hl.refine_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh✝ : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nx✝² : W'✝ ⟶ Y\nx✝¹ : x✝² ≫ g = 0\nx✝ : W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\nh : x✝ ≫ Subobject.arrow (imageSubobject f) = x✝²\n⊢ x✝ =\n (fun {W'} u hu =>\n kernel.lift (cokernel.π f) u (_ : u ≫ cokernel.π f = 0) ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso f).inv)\n x✝² x✝¹",
"tactic": "rw [← cancel_mono (imageSubobject f).arrow, h]"
},
{
"state_after": "no goals",
"state_before": "case hl.refine_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh✝ : f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\nW'✝ : C\nx✝² : W'✝ ⟶ Y\nx✝¹ : x✝² ≫ g = 0\nx✝ : W'✝ ⟶ Subobject.underlying.obj (imageSubobject f)\nh : x✝ ≫ Subobject.arrow (imageSubobject f) = x✝²\n⊢ x✝² =\n (fun {W'} u hu =>\n kernel.lift (cokernel.π f) u (_ : u ≫ cokernel.π f = 0) ≫ (imageIsoImage f).hom ≫ (imageSubobjectIso f).inv)\n x✝² x✝¹ ≫\n Subobject.arrow (imageSubobject f)",
"tactic": "simp"
}
] |
[
85,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Data/Matrix/PEquiv.lean
|
PEquiv.equiv_toPEquiv_toMatrix
|
[] |
[
185,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.mem_factors_iff_dvd
|
[] |
[
148,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
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