Datasets:

l3lab
/

ntp-mathlib

state stringlengths 0 159k	srcUpToTactic stringlengths 387 167k	nextTactic stringlengths 3 9k	declUpToTactic stringlengths 22 11.5k	declId stringlengths 38 95	decl stringlengths 16 1.89k	file_tag stringlengths 17 73
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R ⊢ r ∈ annihilator (span R s) ↔ ∀ (n : ↑s), r • ↑n = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [Submodule.mem_annihilator]	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R ⊢ (∀ n ∈ span R s, r • n = 0) ↔ ∀ (n : ↑s), r • ↑n = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	constructor	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator]	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mp R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R ⊢ (∀ n ∈ span R s, r • n = 0) → ∀ (n : ↑s), r • ↑n = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	intro h n	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor ·	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mp R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ n ∈ span R s, r • n = 0 n : ↑s ⊢ r • ↑n = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact h _ (Submodule.subset_span n.prop)	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R ⊢ (∀ (n : ↑s), r • ↑n = 0) → ∀ n ∈ span R s, r • n = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	intro h n hn	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) ·	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s ⊢ r • n = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	refine Submodule.span_induction hn ?_ ?_ ?_ ?_	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s ⊢ ∀ x ∈ s, r • x = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	intro x hx	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ ·	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s x : M hx : x ∈ s ⊢ r • x = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact h ⟨x, hx⟩	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx ...	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_2 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s ⊢ r • 0 = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact smul_zero _	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx ...	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_3 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s ⊢ ∀ (x y : M), r • x = 0 → r • y = 0 → r • (x + y) = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	intro x y hx hy	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx ...	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_3 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s x y : M hx : r • x = 0 hy : r • y = 0 ⊢ r • (x + y) = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [smul_add, hx, hy, zero_add]	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx ...	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_4 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s ⊢ ∀ (a : R) (x : M), r • x = 0 → r • a • x = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	intro a x hx	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx ...	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
case mpr.refine_4 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M s : Set M r : R h : ∀ (n : ↑s), r • ↑n = 0 n : M hn : n ∈ span R s a : R x : M hx : r • x = 0 ⊢ r • a • x = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [smul_comm, hx, smul_zero]	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx ...	Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY	theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M g : M r : R ⊢ r ∈ annihilator (span R {g}) ↔ r • g = 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	simp [mem_annihilator_span]	theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by	Mathlib.RingTheory.Ideal.Operations.77_0.5qK551sG47yBciY	theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M p : M → Prop x : M H : x ∈ I • N Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n) H1 : ∀ (x y : M), p x → p y → p (x + y) ⊢ p x	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by	Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M p : M → Prop x : M H : x ∈ I • N Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n) H1 : ∀ (x y : M), p x → p y → p (x + y) ⊢ p 0	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by	Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M p : M → Prop x : M H : x ∈ I • N Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n) H1 : ∀ (x y : M), p x → p y → p (x + y) H0 : p 0 ⊢ p x	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem	Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M p : M → Prop x : M H : x ∈ I • N Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n) H1 : ∀ (x y : M), p x → p y → p (x + y) H0 : p 0 ⊢ ∀ (i : ↥I), ∀ x ∈ map ((LinearMap.lsmul R M) ↑i) N, p...	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1	Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x	Mathlib_RingTheory_Ideal_Operations
case mk.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M p : M → Prop x : M H : x ∈ I • N Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n) H1 : ∀ (x y : M), p x → p y → p (x + y) H0 : p 0 i : R hi : i ∈ I m j : M hj : j ∈ ↑...	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [← hj']	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨...	Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x	Mathlib_RingTheory_Ideal_Operations
case mk.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M p : M → Prop x : M H : x ∈ I • N Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n) H1 : ∀ (x y : M), p x → p y → p (x + y) H0 : p 0 i : R hi : i ∈ I m j : M hj : j ∈ ↑...	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact Hb _ hi _ hj	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨...	Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY	@[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M x : M hx : x ∈ I • N p : (x : M) → x ∈ I • N → Prop Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N) H1 : ∀ (x : M) (hx : x ∈ I • N) (y : M...	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H	/-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹...	Mathlib.RingTheory.Ideal.Operations.123_0.5qK551sG47yBciY	/-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹...	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N P : Submodule R M x : M hx : x ∈ I • N p : (x : M) → x ∈ I • N → Prop Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N) H1 : ∀ (x : M) (hx : x ∈ I • N) (y : M...	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩	/-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹...	Mathlib.RingTheory.Ideal.Operations.123_0.5qK551sG47yBciY	/-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹...	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I✝ J : Ideal R N P : Submodule R M I : Ideal R m x : M hx : x ∈ I • span R {m} m1 m2 : M x✝¹ : ∃ y ∈ I, y • m = m1 x✝ : ∃ y ∈ I, y • m = m2 y1 : R hyi1 : y1 ∈ I hy1 : y1 • m = m1 y2 : R hyi2 : y2 ∈ I hy2 ...	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [add_smul, hy1, hy2]	theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1,...	Mathlib.RingTheory.Ideal.Operations.134_0.5qK551sG47yBciY	theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I✝ J : Ideal R N P : Submodule R M I : Ideal R f : R →ₗ[R] M ⊢ map f I ≤ I • ⊤	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rintro _ ⟨y, hy, rfl⟩	theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by	Mathlib.RingTheory.Ideal.Operations.162_0.5qK551sG47yBciY	theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M)	Mathlib_RingTheory_Ideal_Operations
case intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I✝ J : Ideal R N P : Submodule R M I : Ideal R f : R →ₗ[R] M y : R hy : y ∈ ↑I ⊢ f y ∈ I • ⊤	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [← mul_one y, ← smul_eq_mul, f.map_smul]	theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩	Mathlib.RingTheory.Ideal.Operations.162_0.5qK551sG47yBciY	theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M)	Mathlib_RingTheory_Ideal_Operations
case intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I✝ J : Ideal R N P : Submodule R M I : Ideal R f : R →ₗ[R] M y : R hy : y ∈ ↑I ⊢ y • f 1 ∈ I • ⊤	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact smul_mem_smul hy mem_top	theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul]	Mathlib.RingTheory.Ideal.Operations.162_0.5qK551sG47yBciY	theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M)	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I✝ J : Ideal R N P : Submodule R M I : Ideal R ⊢ I * annihilator I = ⊥	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [mul_comm, annihilator_mul]	@[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by	Mathlib.RingTheory.Ideal.Operations.179_0.5qK551sG47yBciY	@[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N✝ P : Submodule R M S : Set R T : Set M r : R N : Submodule R M ⊢ Ideal.span {r} • N = r • N	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by	Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N✝ P : Submodule R M S : Set R T : Set M r : R N : Submodule R M ⊢ span R (⋃ t ∈ N, {r • t}) = r • N	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	convert span_eq (r • N)	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by	Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N	Mathlib_RingTheory_Ideal_Operations
case h.e'_2.h.e'_6 R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N✝ P : Submodule R M S : Set R T : Set M r : R N : Submodule R M ⊢ ⋃ t ∈ N, {r • t} = ↑(r • N)	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	exact (Set.image_eq_iUnion _ (N : Set M)).symm	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N)	Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N✝ P : Submodule R M S : Set R T : Set M r : R N : Submodule R M this : span R (⋃ t ∈ N, {r • t}) = r • N ⊢ Ideal.span {r} • N = r • N	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	conv_lhs => rw [← span_eq N, span_smul_span]	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm	Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N	Mathlib_RingTheory_Ideal_Operations
R : Type u M : Type v F : Type u_1 G : Type u_2 inst✝² : CommSemiring R inst✝¹ : AddCommMonoid M inst✝ : Module R M I J : Ideal R N✝ P : Submodule R M S : Set R T : Set M r : R N : Submodule R M this : span R (⋃ t ∈ N, {r • t}) = r • N \| Ideal.span {r} • N	/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTh...	rw [← span_eq N, span_smul_span]	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs =>	Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY	theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N	Mathlib_RingTheory_Ideal_Operations

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miniCTX: Neural Theorem Proving with (Long-)Contexts

Lean 4 tactic prediction examples extracted from Mathlib.

These examples have not been formatted for instruction tuning (including data splits).

Please see l3lab/ntp-mathlib-instruct-* for datasets with instruction tuning examples.

Version

Generated using ntptoolkit's ntp-training-data.

It used the following config for ntp-training-data:

{
    "repo": "https://github.com/leanprover-community/mathlib4",
    "commit": "cf8e23a62939ed7cc530fbb68e83539730f32f86",
    "lean": "leanprover/lean4:v4.4.0",
    "name": "mathlib",
    "import_file": "Mathlib.lean",
    "imports": ["Mathlib"]
}

Example usage:

ds = datasets.load_dataset('l3lab/ntp-mathlib')

print(len(ds['train']))
# ==> 307049

Format:

{
  'state': 'proof state',
  'srcUpToTactic': 'source up to tactic invocation',
  'nextTactic': 'tactic',
  'declUpToTactic': 'declariation up to tactic invocation',
  'declId': 'unique ID for declaration',
  'decl': 'declaration',
  'file_tag': 'file ID'
}

Citation

Please cite:

@misc{hu2024minictx,
  title={miniCTX: Neural Theorem Proving with (Long-)Contexts}, 
  author={Jiewen Hu and Thomas Zhu and Sean Welleck},
  year={2024},
  eprint={2408.03350},
  archivePrefix={arXiv},
  primaryClass={cs.AI},
  url={https://arxiv.org/abs/2408.03350}, 
}

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Size of the auto-converted Parquet files (First 5GB):

1.09 GB

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213,206

Collection including l3lab/ntp-mathlib

miniCTX

Collection

miniCTX: Neural Theorem Proving with (Long-)Contexts (ICLR 2025 Oral) • 8 items • Updated Mar 19, 2025 • 2

Paper for l3lab/ntp-mathlib

miniCTX: Neural Theorem Proving with (Long-)Contexts

Paper • 2408.03350 • Published Aug 5, 2024 • 1