internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	centralBinomialLower_bound	[152, 1]	[157, 23]	rwa [sqrt_mul, ← div_div, div_le_iff, div_eq_mul_one_div, ← le_div_iff', one_div (sqrt π)]	n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 4 ^ n / Real.sqrt (π * (↑n + 1 / 3)) ≤ ↑(centralBinom n)	n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < 4 ^ n n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < Real.sqrt (↑n + 1 / 3) case hx n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 ≤ π
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	centralBinomialLower_bound	[152, 1]	[157, 23]	all_goals positivity	n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < 4 ^ n n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < Real.sqrt (↑n + 1 / 3) case hx n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 ≤ π	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	centralBinomialLower_bound	[152, 1]	[157, 23]	positivity	case hx n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 ≤ π	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	cexp_eq_tsum	[159, 1]	[160, 45]	rw [Complex.exp_eq_exp_ℂ, exp_eq_tsum_div]	x : ℂ ⊢ Complex.exp x = ∑' (i : ℕ), x ^ i / ↑i !	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	rexp_eq_tsum	[162, 1]	[163, 37]	rw [exp_eq_exp_ℝ, exp_eq_tsum_div]	x : ℝ ⊢ rexp x = ∑' (i : ℕ), x ^ i / ↑i !	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound	[165, 1]	[167, 78]	rw [exp_eq_exp_ℝ]	x : ℝ hx : 0 ≤ x n : ℕ ⊢ x ^ n / ↑n ! ≤ rexp x	x : ℝ hx : 0 ≤ x n : ℕ ⊢ x ^ n / ↑n ! ≤ _root_.exp ℝ x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound	[165, 1]	[167, 78]	exact le_hasSum (expSeries_div_hasSum_exp ℝ x) n (fun _ _ => by positivity)	x : ℝ hx : 0 ≤ x n : ℕ ⊢ x ^ n / ↑n ! ≤ _root_.exp ℝ x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound	[165, 1]	[167, 78]	positivity	x : ℝ hx : 0 ≤ x n x✝¹ : ℕ x✝ : x✝¹ ≠ n ⊢ 0 ≤ x ^ x✝¹ / ↑x✝¹!	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound_of_ne_zero	[169, 1]	[176, 7]	rw [exp_eq_exp_ℝ]	n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < rexp x	n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < _root_.exp ℝ x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound_of_ne_zero	[169, 1]	[176, 7]	refine (sum_le_hasSum {n, 0} ?_ (expSeries_div_hasSum_exp ℝ x)).trans_lt' ?_	n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < _root_.exp ℝ x	case refine_1 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ ∀ i ∉ {n, 0}, 0 ≤ x ^ i / ↑i ! case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < Finset.sum {n, 0} fun i => x ^ i / ↑i !
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound_of_ne_zero	[169, 1]	[176, 7]	rw [sum_pair hn]	case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < Finset.sum {n, 0} fun i => x ^ i / ↑i !	case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < x ^ n / ↑n ! + x ^ 0 / ↑0!
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound_of_ne_zero	[169, 1]	[176, 7]	simp	case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < x ^ n / ↑n ! + x ^ 0 / ↑0!	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound_of_ne_zero	[169, 1]	[176, 7]	intro x _	case refine_1 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ ∀ i ∉ {n, 0}, 0 ≤ x ^ i / ↑i !	case refine_1 n : ℕ x✝ : ℝ hx : 0 ≤ x✝ hn : n ≠ 0 x : ℕ a✝ : x ∉ {n, 0} ⊢ 0 ≤ x✝ ^ x / ↑x !
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	exp_factorial_bound_of_ne_zero	[169, 1]	[176, 7]	positivity	case refine_1 n : ℕ x✝ : ℝ hx : 0 ≤ x✝ hn : n ≠ 0 x : ℕ a✝ : x ∉ {n, 0} ⊢ 0 ≤ x✝ ^ x / ↑x !	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp	[178, 1]	[182, 15]	rw [div_pow, ← rpow_nat_cast (exp 1), exp_one_rpow, div_le_iff, ← div_le_iff']	n : ℕ ⊢ (↑n / rexp 1) ^ n ≤ ↑n !	n : ℕ ⊢ ↑n ^ n / ↑n ! ≤ rexp ↑n n : ℕ ⊢ 0 < ↑n ! n : ℕ ⊢ 0 < rexp ↑n
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp	[178, 1]	[182, 15]	exact exp_factorial_bound (Nat.cast_nonneg _)	n : ℕ ⊢ ↑n ^ n / ↑n ! ≤ rexp ↑n	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp	[178, 1]	[182, 15]	positivity	n : ℕ ⊢ 0 < ↑n !	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp	[178, 1]	[182, 15]	positivity	n : ℕ ⊢ 0 < rexp ↑n	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp_of_ne_zero	[184, 1]	[188, 15]	rw [div_pow, ← rpow_nat_cast (exp 1), exp_one_rpow, div_lt_iff, ← div_lt_iff']	n : ℕ hn : n ≠ 0 ⊢ (↑n / rexp 1) ^ n < ↑n !	n : ℕ hn : n ≠ 0 ⊢ ↑n ^ n / ↑n ! < rexp ↑n n : ℕ hn : n ≠ 0 ⊢ 0 < ↑n ! n : ℕ hn : n ≠ 0 ⊢ 0 < rexp ↑n
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp_of_ne_zero	[184, 1]	[188, 15]	exact exp_factorial_bound_of_ne_zero (Nat.cast_nonneg _) hn	n : ℕ hn : n ≠ 0 ⊢ ↑n ^ n / ↑n ! < rexp ↑n	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp_of_ne_zero	[184, 1]	[188, 15]	positivity	n : ℕ hn : n ≠ 0 ⊢ 0 < ↑n !	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	factorial_bound_exp_of_ne_zero	[184, 1]	[188, 15]	positivity	n : ℕ hn : n ≠ 0 ⊢ 0 < rexp ↑n	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	Nat.choose_le_pow'	[191, 1]	[193, 45]	simpa using Nat.choose_le_pow (α := α) r n	α : Type u_1 inst✝ : LinearOrderedSemifield α r n : ℕ ⊢ ↑(choose n r) ≤ ↑n ^ r / ↑r !	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	cases' Nat.eq_zero_or_pos t with h h	n t : ℕ ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t	case inl n t : ℕ h : t = 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t case inr n t : ℕ h : t > 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	refine' (Nat.choose_le_pow' t n).trans _	case inr n t : ℕ h : t > 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t	case inr n t : ℕ h : t > 0 ⊢ ↑n ^ t / ↑t ! ≤ (rexp 1 * ↑n / ↑t) ^ t
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	refine' (div_le_div_of_le_left _ _ factorial_bound_exp).trans _	case inr n t : ℕ h : t > 0 ⊢ ↑n ^ t / ↑t ! ≤ (rexp 1 * ↑n / ↑t) ^ t	case inr.refine'_1 n t : ℕ h : t > 0 ⊢ 0 ≤ ↑n ^ t case inr.refine'_2 n t : ℕ h : t > 0 ⊢ 0 < (↑t / rexp 1) ^ t case inr.refine'_3 n t : ℕ h : t > 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t ≤ (rexp 1 * ↑n / ↑t) ^ t
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	rw [← div_pow, div_div_eq_mul_div, mul_comm]	case inr.refine'_3 n t : ℕ h : t > 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t ≤ (rexp 1 * ↑n / ↑t) ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	simp [h]	case inl n t : ℕ h : t = 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	positivity	case inr.refine'_1 n t : ℕ h : t > 0 ⊢ 0 ≤ ↑n ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound	[195, 1]	[202, 47]	positivity	case inr.refine'_2 n t : ℕ h : t > 0 ⊢ 0 < (↑t / rexp 1) ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound_of_pos	[204, 1]	[210, 47]	refine' (Nat.choose_le_pow' t n).trans_lt _	n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑(Nat.choose n t) < (rexp 1 * ↑n / ↑t) ^ t	n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / ↑t ! < (rexp 1 * ↑n / ↑t) ^ t
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound_of_pos	[204, 1]	[210, 47]	refine' (div_lt_div_of_lt_left _ _ (factorial_bound_exp_of_ne_zero ht)).trans_eq _	n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / ↑t ! < (rexp 1 * ↑n / ↑t) ^ t	case refine'_1 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < ↑n ^ t case refine'_2 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < (↑t / rexp 1) ^ t case refine'_3 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t = (rexp 1 * ↑n / ↑t) ^ t
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound_of_pos	[204, 1]	[210, 47]	rw [← div_pow, div_div_eq_mul_div, mul_comm]	case refine'_3 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t = (rexp 1 * ↑n / ↑t) ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound_of_pos	[204, 1]	[210, 47]	positivity	case refine'_1 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < ↑n ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound_of_pos	[204, 1]	[210, 47]	positivity	case refine'_2 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < (↑t / rexp 1) ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean	choose_upper_bound'	[213, 1]	[214, 97]	rw [mul_div_assoc, mul_pow, ← exp_one_rpow t, rpow_nat_cast]	n t : ℕ ⊢ (rexp 1 * ↑n / ↑t) ^ t = rexp ↑t * (↑n / ↑t) ^ t	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.card_top_edgeSet	[38, 1]	[40, 70]	rw [Fintype.card_congr topEdgeSetEquiv, Sym2.card_subtype_not_diag]	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : DecidableEq V inst✝ : Fintype V ⊢ Fintype.card ↑(edgeSet ⊤) = Nat.choose (Fintype.card V) 2	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.edgeSet_eq_empty_iff	[42, 1]	[43, 34]	rw [← edgeSet_bot, edgeSet_inj]	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G : SimpleGraph V ⊢ edgeSet G = ∅ ↔ G = ⊥	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.disjoint_edgeSet	[45, 1]	[47, 90]	rw [Set.disjoint_iff_inter_eq_empty, disjoint_iff, ← edgeSet_inf, edgeSet_eq_empty_iff]	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V ⊢ Disjoint (edgeSet G) (edgeSet H) ↔ Disjoint G H	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.disjoint_left	[49, 1]	[50, 78]	simp only [← disjoint_edgeSet, Set.disjoint_left, Sym2.forall, mem_edgeSet]	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V ⊢ Disjoint G H ↔ ∀ (x y : V), Adj G x y → ¬Adj H x y	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.adj_sup_iff	[53, 1]	[58, 39]	induction' s using Finset.cons_induction_on with a s has ih	V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 s : Finset ι f : ι → SimpleGraph V x y : V ⊢ Adj (sup s f) x y ↔ ∃ i ∈ s, Adj (f i) x y	case h₁ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V ⊢ Adj (sup ∅ f) x y ↔ ∃ i ∈ ∅, Adj (f i) x y case h₂ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V a : ι s : Finset ι has : a ∉ s ih : Adj (sup s f) x y ↔ ∃ i ∈ s, Adj (f i) x y ⊢ Adj (sup (cons a s has) f) x y ↔ ∃ i ∈ cons a s has, Adj (f i) x y
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.adj_sup_iff	[53, 1]	[58, 39]	simp	case h₁ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V ⊢ Adj (sup ∅ f) x y ↔ ∃ i ∈ ∅, Adj (f i) x y	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.adj_sup_iff	[53, 1]	[58, 39]	simp [or_and_right, exists_or, ih]	case h₂ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V a : ι s : Finset ι has : a ∉ s ih : Adj (sup s f) x y ↔ ∃ i ∈ s, Adj (f i) x y ⊢ Adj (sup (cons a s has) f) x y ↔ ∃ i ∈ cons a s has, Adj (f i) x y	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_bot	[71, 1]	[71, 90]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x : V ⊢ neighborSet ⊥ x = ∅	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊥ x ↔ y ∈ ∅
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_bot	[71, 1]	[71, 90]	simp	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊥ x ↔ y ∈ ∅	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_top	[73, 1]	[75, 70]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x : V ⊢ neighborSet ⊤ x = {x}ᶜ	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊤ x ↔ y ∈ {x}ᶜ
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_top	[73, 1]	[75, 70]	rw [mem_neighborSet, top_adj, Set.mem_compl_singleton_iff, ne_comm]	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊤ x ↔ y ∈ {x}ᶜ	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_sup	[77, 1]	[78, 80]	ext y	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V ⊢ neighborSet (G ⊔ H) x = neighborSet G x ∪ neighborSet H x	case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊔ H) x ↔ y ∈ neighborSet G x ∪ neighborSet H x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_sup	[77, 1]	[78, 80]	simp	case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊔ H) x ↔ y ∈ neighborSet G x ∪ neighborSet H x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_inf	[80, 1]	[83, 58]	ext y	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V ⊢ neighborSet (G ⊓ H) x = neighborSet G x ∩ neighborSet H x	case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊓ H) x ↔ y ∈ neighborSet G x ∩ neighborSet H x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_inf	[80, 1]	[83, 58]	simp only [mem_neighborSet, inf_adj, Set.mem_inter_iff]	case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊓ H) x ↔ y ∈ neighborSet G x ∩ neighborSet H x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iSup	[85, 1]	[86, 74]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 s : ι → SimpleGraph V x : V ⊢ neighborSet (⨆ i, s i) x = ⋃ i, neighborSet (s i) x	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨆ i, s i) x ↔ y ∈ ⋃ i, neighborSet (s i) x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iSup	[85, 1]	[86, 74]	simp	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨆ i, s i) x ↔ y ∈ ⋃ i, neighborSet (s i) x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iInf	[88, 1]	[96, 23]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x : V ⊢ neighborSet (⨅ i, s i) x = ⋂ i, neighborSet (s i) x	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨅ i, s i) x ↔ y ∈ ⋂ i, neighborSet (s i) x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iInf	[88, 1]	[96, 23]	simp only [mem_neighborSet, iInf_adj, Ne.def, Set.iInf_eq_iInter, Set.mem_iInter, and_iff_left_iff_imp]	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨅ i, s i) x ↔ y ∈ ⋂ i, neighborSet (s i) x	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ (∀ (i : ι), Adj (s i) x y) → ¬x = y
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iInf	[88, 1]	[96, 23]	intro h	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ (∀ (i : ι), Adj (s i) x y) → ¬x = y	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y ⊢ ¬x = y
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iInf	[88, 1]	[96, 23]	inhabit ι	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y ⊢ ¬x = y	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y inhabited_h : Inhabited ι ⊢ ¬x = y
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_iInf	[88, 1]	[96, 23]	exact (h default).ne	case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y inhabited_h : Inhabited ι ⊢ ¬x = y	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborSet_disjoint	[98, 1]	[100, 85]	rw [Set.disjoint_iff_inter_eq_empty, ← neighborSet_inf, h.eq_bot, neighborSet_bot]	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V h : Disjoint G H ⊢ Disjoint (neighborSet G x) (neighborSet H x)	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_bot	[107, 1]	[107, 96]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x : V ⊢ neighborFinset ⊥ x = ∅	case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborFinset ⊥ x ↔ y ∈ ∅
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_bot	[107, 1]	[107, 96]	simp	case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborFinset ⊥ x ↔ y ∈ ∅	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_top	[109, 1]	[112, 77]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : Fintype V inst✝ : DecidableEq V x : V ⊢ neighborFinset ⊤ x = {x}ᶜ	case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : Fintype V inst✝ : DecidableEq V x y : V ⊢ y ∈ neighborFinset ⊤ x ↔ y ∈ {x}ᶜ
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_top	[109, 1]	[112, 77]	rw [mem_neighborFinset, top_adj, Finset.mem_compl, mem_singleton, ne_comm]	case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : Fintype V inst✝ : DecidableEq V x y : V ⊢ y ∈ neighborFinset ⊤ x ↔ y ∈ {x}ᶜ	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_sup	[114, 1]	[116, 89]	ext y	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊔ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) ⊢ neighborFinset (G ⊔ H) x = neighborFinset G x ∪ neighborFinset H x	case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊔ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊔ H) x ↔ y ∈ neighborFinset G x ∪ neighborFinset H x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_sup	[114, 1]	[116, 89]	simp	case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊔ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊔ H) x ↔ y ∈ neighborFinset G x ∪ neighborFinset H x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_inf	[118, 1]	[120, 89]	ext y	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊓ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) ⊢ neighborFinset (G ⊓ H) x = neighborFinset G x ∩ neighborFinset H x	case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊓ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊓ H) x ↔ y ∈ neighborFinset G x ∩ neighborFinset H x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_inf	[118, 1]	[120, 89]	simp	case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊓ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊓ H) x ↔ y ∈ neighborFinset G x ∩ neighborFinset H x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_supr	[126, 1]	[128, 93]	ext y	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝² : DecidableEq V ι : Type u_5 s : Finset ι f : ι → SimpleGraph V x : V inst✝¹ : (i : ι) → Fintype ↑(neighborSet (f i) x) inst✝ : Fintype ↑(neighborSet (sup s f) x) ⊢ neighborFinset (sup s f) x = Finset.biUnion s fun i => neighborFinset (f i) x	case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝² : DecidableEq V ι : Type u_5 s : Finset ι f : ι → SimpleGraph V x : V inst✝¹ : (i : ι) → Fintype ↑(neighborSet (f i) x) inst✝ : Fintype ↑(neighborSet (sup s f) x) y : V ⊢ y ∈ neighborFinset (sup s f) x ↔ y ∈ Finset.biUnion s fun i => neighborFinset (f i) x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_supr	[126, 1]	[128, 93]	simp	case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝² : DecidableEq V ι : Type u_5 s : Finset ι f : ι → SimpleGraph V x : V inst✝¹ : (i : ι) → Fintype ↑(neighborSet (f i) x) inst✝ : Fintype ↑(neighborSet (sup s f) x) y : V ⊢ y ∈ neighborFinset (sup s f) x ↔ y ∈ Finset.biUnion s fun i => neighborFinset (f i) x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.coe_neighborFinset	[131, 1]	[132, 99]	rw [neighborFinset_def, Set.coe_toFinset]	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G : SimpleGraph V x : V inst✝ : Fintype ↑(neighborSet G x) ⊢ ↑(neighborFinset G x) = neighborSet G x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_disjoint	[134, 1]	[137, 92]	rw [← disjoint_coe, coe_neighborFinset, coe_neighborFinset]	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) h : Disjoint G H ⊢ Disjoint (neighborFinset G x) (neighborFinset H x)	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) h : Disjoint G H ⊢ Disjoint (neighborSet G x) (neighborSet H x)
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.neighborFinset_disjoint	[134, 1]	[137, 92]	exact neighborSet_disjoint h	V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) h : Disjoint G H ⊢ Disjoint (neighborSet G x) (neighborSet H x)	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean	SimpleGraph.degree_eq_zero_iff	[141, 1]	[142, 73]	rw [← not_exists, ← degree_pos_iff_exists_adj, not_lt, le_zero_iff]	V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 v : V inst✝ : Fintype ↑(neighborSet G v) ⊢ degree G v = 0 ↔ ∀ (w : V), ¬Adj G v w	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	rw [Ne.def, filter_eq_self] at this	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) this : filter (fun x => Odd (degree G x)) univ ≠ univ ⊢ ∃ v, Even (degree G v)	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) this : ¬∀ x ∈ univ, Odd (degree G x) ⊢ ∃ v, Even (degree G v)
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	simpa using this	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) this : ¬∀ x ∈ univ, Odd (degree G x) ⊢ ∃ v, Even (degree G v)	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	rw [←card_lt_iff_ne_univ, (card_le_univ _).lt_iff_ne]	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) ⊢ filter (fun x => Odd (degree G x)) univ ≠ univ	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) ⊢ Finset.card (filter (fun x => Odd (degree G x)) univ) ≠ Fintype.card V
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	intro h	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) ⊢ Finset.card (filter (fun x => Odd (degree G x)) univ) ≠ Fintype.card V	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V ⊢ False
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	have h' := even_card_odd_degree_vertices G	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V ⊢ False	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : Even (Finset.card (filter (fun v => Odd (degree G v)) univ)) ⊢ False
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	rw [h, Nat.even_iff_not_odd] at h'	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : Even (Finset.card (filter (fun v => Odd (degree G v)) univ)) ⊢ False	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : ¬Odd (Fintype.card V) ⊢ False
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean	SimpleGraph.exists_even_degree	[27, 1]	[36, 19]	exact h' hV	V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : ¬Odd (Fintype.card V) ⊢ False	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean	Nat.choose_le_two_pow	[19, 1]	[25, 17]	cases' le_or_lt k n with h h	n k : ℕ ⊢ choose n k ≤ 2 ^ n	case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ 2 ^ n case inr n k : ℕ h : n < k ⊢ choose n k ≤ 2 ^ n
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean	Nat.choose_le_two_pow	[19, 1]	[25, 17]	rw [choose_eq_zero_of_lt h]	case inr n k : ℕ h : n < k ⊢ choose n k ≤ 2 ^ n	case inr n k : ℕ h : n < k ⊢ 0 ≤ 2 ^ n
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean	Nat.choose_le_two_pow	[19, 1]	[25, 17]	exact zero_le'	case inr n k : ℕ h : n < k ⊢ 0 ≤ 2 ^ n	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean	Nat.choose_le_two_pow	[19, 1]	[25, 17]	rw [← sum_range_choose n]	case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ 2 ^ n	case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ Finset.sum (range (n + 1)) fun m => choose n m
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean	Nat.choose_le_two_pow	[19, 1]	[25, 17]	refine' single_le_sum (fun _ _ => zero_le') _	case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ Finset.sum (range (n + 1)) fun m => choose n m	case inl n k : ℕ h : k ≤ n ⊢ k ∈ range (n + 1)
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean	Nat.choose_le_two_pow	[19, 1]	[25, 17]	rwa [mem_range_succ_iff]	case inl n k : ℕ h : k ≤ n ⊢ k ∈ range (n + 1)	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	symmetric_isSquare	[22, 1]	[23, 76]	simpa using h.mul (FiniteField.isSquare_neg_one_iff.2 hF)	F : Type u_1 inst✝¹ : Fintype F inst✝ : Field F hF : Fintype.card F % 4 ≠ 3 x✝¹ x✝ : F h : (fun x y => IsSquare (x - y)) x✝¹ x✝ ⊢ (fun x y => IsSquare (x - y)) x✝ x✝¹	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	have : (univ.filter fun x : F => ¬IsSquare x) = univ.filter fun x : F => x ≠ 0 ∧ ¬IsSquare x := by refine' filter_congr _ simp (config := { contextual := true }) [not_imp_not]	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => ¬IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => ¬IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	rw [this]	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => ¬IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	have cf := quadraticChar_sum_zero hF	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun a => ↑(quadraticChar F) a) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	simp only [quadraticChar_apply, quadraticCharFun] at cf	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun a => ↑(quadraticChar F) a) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun x => if x = 0 then 0 else if IsSquare x then 1 else -1) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	rw [sum_ite, sum_const_zero, zero_add, sum_ite, sum_const, sum_const, nsmul_eq_mul, nsmul_eq_mul, mul_neg, mul_one, mul_one, add_neg_eq_zero, Nat.cast_inj, filter_filter, filter_filter] at cf	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun x => if x = 0 then 0 else if IsSquare x then 1 else -1) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	rw [← cf, and_self_iff]	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	have : ((univ.filter fun x : F => x ≠ 0 ∧ IsSquare x) ∪ univ.filter fun x : F => x ≠ 0 ∧ ¬IsSquare x) ∪ {0} = univ := by simp only [← filter_or, ← and_or_left, em, and_true_iff, filter_ne'] rw [union_comm, ← insert_eq, insert_erase] exact mem_univ _	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	have h' := congr_arg Finset.card this	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0}) = Finset.card univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	rw [card_disjoint_union, card_disjoint_union, card_singleton, card_univ, ← cf, ← two_mul, ← bit0_eq_two_mul, ← bit1] at h'	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0}) = Finset.card univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : bit1 (Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ)) = Fintype.card F ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) + Finset.card {0} = Finset.card univ ⊢ Disjoint (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0}) = Finset.card univ ⊢ Disjoint (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) {0}
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	refine' filter_congr _	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ ∀ x ∈ univ, ¬IsSquare x ↔ x ≠ 0 ∧ ¬IsSquare x
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	simp (config := { contextual := true }) [not_imp_not]	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ ∀ x ∈ univ, ¬IsSquare x ↔ x ≠ 0 ∧ ¬IsSquare x	no goals
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	simp only [← filter_or, ← and_or_left, em, and_true_iff, filter_ne']	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ erase univ 0 ∪ {0} = univ
https://github.com/b-mehta/ExponentialRamsey.git	7e17b629a915a082869f494c8afa56a3e1c7a88d	ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean	card_non_zero_square_non_square	[25, 1]	[53, 9]	rw [union_comm, ← insert_eq, insert_erase]	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ erase univ 0 ∪ {0} = univ	F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ 0 ∈ univ

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

centralBinomialLower_bound

[152, 1]

[157, 23]

rwa [sqrt_mul, ← div_div, div_le_iff, div_eq_mul_one_div, ← le_div_iff', one_div (sqrt π)]

n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 4 ^ n / Real.sqrt (π * (↑n + 1 / 3)) ≤ ↑(centralBinom n)

n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < 4 ^ n n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < Real.sqrt (↑n + 1 / 3) case hx n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 ≤ π

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

centralBinomialLower_bound

[152, 1]

[157, 23]

all_goals positivity

n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < 4 ^ n n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 < Real.sqrt (↑n + 1 / 3) case hx n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 ≤ π

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

centralBinomialLower_bound

[152, 1]

[157, 23]

positivity

case hx n : ℕ this✝ : 0 < π this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n ⊢ 0 ≤ π

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

cexp_eq_tsum

[159, 1]

[160, 45]

rw [Complex.exp_eq_exp_ℂ, exp_eq_tsum_div]

x : ℂ ⊢ Complex.exp x = ∑' (i : ℕ), x ^ i / ↑i !

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

rexp_eq_tsum

[162, 1]

[163, 37]

rw [exp_eq_exp_ℝ, exp_eq_tsum_div]

x : ℝ ⊢ rexp x = ∑' (i : ℕ), x ^ i / ↑i !

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound

[165, 1]

[167, 78]

rw [exp_eq_exp_ℝ]

x : ℝ hx : 0 ≤ x n : ℕ ⊢ x ^ n / ↑n ! ≤ rexp x

x : ℝ hx : 0 ≤ x n : ℕ ⊢ x ^ n / ↑n ! ≤ _root_.exp ℝ x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound

[165, 1]

[167, 78]

exact le_hasSum (expSeries_div_hasSum_exp ℝ x) n (fun _ _ => by positivity)

x : ℝ hx : 0 ≤ x n : ℕ ⊢ x ^ n / ↑n ! ≤ _root_.exp ℝ x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound

[165, 1]

[167, 78]

positivity

x : ℝ hx : 0 ≤ x n x✝¹ : ℕ x✝ : x✝¹ ≠ n ⊢ 0 ≤ x ^ x✝¹ / ↑x✝¹!

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound_of_ne_zero

[169, 1]

[176, 7]

rw [exp_eq_exp_ℝ]

n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < rexp x

n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < _root_.exp ℝ x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound_of_ne_zero

[169, 1]

[176, 7]

refine (sum_le_hasSum {n, 0} ?_ (expSeries_div_hasSum_exp ℝ x)).trans_lt' ?_

n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < _root_.exp ℝ x

case refine_1 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ ∀ i ∉ {n, 0}, 0 ≤ x ^ i / ↑i ! case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < Finset.sum {n, 0} fun i => x ^ i / ↑i !

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound_of_ne_zero

[169, 1]

[176, 7]

rw [sum_pair hn]

case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < Finset.sum {n, 0} fun i => x ^ i / ↑i !

case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < x ^ n / ↑n ! + x ^ 0 / ↑0!

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound_of_ne_zero

[169, 1]

[176, 7]

simp

case refine_2 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ x ^ n / ↑n ! < x ^ n / ↑n ! + x ^ 0 / ↑0!

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound_of_ne_zero

[169, 1]

[176, 7]

intro x _

case refine_1 n : ℕ x : ℝ hx : 0 ≤ x hn : n ≠ 0 ⊢ ∀ i ∉ {n, 0}, 0 ≤ x ^ i / ↑i !

case refine_1 n : ℕ x✝ : ℝ hx : 0 ≤ x✝ hn : n ≠ 0 x : ℕ a✝ : x ∉ {n, 0} ⊢ 0 ≤ x✝ ^ x / ↑x !

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

exp_factorial_bound_of_ne_zero

[169, 1]

[176, 7]

positivity

case refine_1 n : ℕ x✝ : ℝ hx : 0 ≤ x✝ hn : n ≠ 0 x : ℕ a✝ : x ∉ {n, 0} ⊢ 0 ≤ x✝ ^ x / ↑x !

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp

[178, 1]

[182, 15]

rw [div_pow, ← rpow_nat_cast (exp 1), exp_one_rpow, div_le_iff, ← div_le_iff']

n : ℕ ⊢ (↑n / rexp 1) ^ n ≤ ↑n !

n : ℕ ⊢ ↑n ^ n / ↑n ! ≤ rexp ↑n n : ℕ ⊢ 0 < ↑n ! n : ℕ ⊢ 0 < rexp ↑n

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp

[178, 1]

[182, 15]

exact exp_factorial_bound (Nat.cast_nonneg _)

n : ℕ ⊢ ↑n ^ n / ↑n ! ≤ rexp ↑n

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp

[178, 1]

[182, 15]

positivity

n : ℕ ⊢ 0 < ↑n !

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp

[178, 1]

[182, 15]

positivity

n : ℕ ⊢ 0 < rexp ↑n

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp_of_ne_zero

[184, 1]

[188, 15]

rw [div_pow, ← rpow_nat_cast (exp 1), exp_one_rpow, div_lt_iff, ← div_lt_iff']

n : ℕ hn : n ≠ 0 ⊢ (↑n / rexp 1) ^ n < ↑n !

n : ℕ hn : n ≠ 0 ⊢ ↑n ^ n / ↑n ! < rexp ↑n n : ℕ hn : n ≠ 0 ⊢ 0 < ↑n ! n : ℕ hn : n ≠ 0 ⊢ 0 < rexp ↑n

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp_of_ne_zero

[184, 1]

[188, 15]

exact exp_factorial_bound_of_ne_zero (Nat.cast_nonneg _) hn

n : ℕ hn : n ≠ 0 ⊢ ↑n ^ n / ↑n ! < rexp ↑n

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp_of_ne_zero

[184, 1]

[188, 15]

positivity

n : ℕ hn : n ≠ 0 ⊢ 0 < ↑n !

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

factorial_bound_exp_of_ne_zero

[184, 1]

[188, 15]

positivity

n : ℕ hn : n ≠ 0 ⊢ 0 < rexp ↑n

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

Nat.choose_le_pow'

[191, 1]

[193, 45]

simpa using Nat.choose_le_pow (α := α) r n

α : Type u_1 inst✝ : LinearOrderedSemifield α r n : ℕ ⊢ ↑(choose n r) ≤ ↑n ^ r / ↑r !

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

cases' Nat.eq_zero_or_pos t with h h

n t : ℕ ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t

case inl n t : ℕ h : t = 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t case inr n t : ℕ h : t > 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

refine' (Nat.choose_le_pow' t n).trans _

case inr n t : ℕ h : t > 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t

case inr n t : ℕ h : t > 0 ⊢ ↑n ^ t / ↑t ! ≤ (rexp 1 * ↑n / ↑t) ^ t

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

refine' (div_le_div_of_le_left _ _ factorial_bound_exp).trans _

case inr n t : ℕ h : t > 0 ⊢ ↑n ^ t / ↑t ! ≤ (rexp 1 * ↑n / ↑t) ^ t

case inr.refine'_1 n t : ℕ h : t > 0 ⊢ 0 ≤ ↑n ^ t case inr.refine'_2 n t : ℕ h : t > 0 ⊢ 0 < (↑t / rexp 1) ^ t case inr.refine'_3 n t : ℕ h : t > 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t ≤ (rexp 1 * ↑n / ↑t) ^ t

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

rw [← div_pow, div_div_eq_mul_div, mul_comm]

case inr.refine'_3 n t : ℕ h : t > 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t ≤ (rexp 1 * ↑n / ↑t) ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

simp [h]

case inl n t : ℕ h : t = 0 ⊢ ↑(Nat.choose n t) ≤ (rexp 1 * ↑n / ↑t) ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

positivity

case inr.refine'_1 n t : ℕ h : t > 0 ⊢ 0 ≤ ↑n ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound

[195, 1]

[202, 47]

positivity

case inr.refine'_2 n t : ℕ h : t > 0 ⊢ 0 < (↑t / rexp 1) ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound_of_pos

[204, 1]

[210, 47]

refine' (Nat.choose_le_pow' t n).trans_lt _

n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑(Nat.choose n t) < (rexp 1 * ↑n / ↑t) ^ t

n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / ↑t ! < (rexp 1 * ↑n / ↑t) ^ t

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound_of_pos

[204, 1]

[210, 47]

refine' (div_lt_div_of_lt_left _ _ (factorial_bound_exp_of_ne_zero ht)).trans_eq _

n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / ↑t ! < (rexp 1 * ↑n / ↑t) ^ t

case refine'_1 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < ↑n ^ t case refine'_2 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < (↑t / rexp 1) ^ t case refine'_3 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t = (rexp 1 * ↑n / ↑t) ^ t

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound_of_pos

[204, 1]

[210, 47]

rw [← div_pow, div_div_eq_mul_div, mul_comm]

case refine'_3 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ ↑n ^ t / (↑t / rexp 1) ^ t = (rexp 1 * ↑n / ↑t) ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound_of_pos

[204, 1]

[210, 47]

positivity

case refine'_1 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < ↑n ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound_of_pos

[204, 1]

[210, 47]

positivity

case refine'_2 n t : ℕ hn : n ≠ 0 ht : t ≠ 0 ⊢ 0 < (↑t / rexp 1) ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean

choose_upper_bound'

[213, 1]

[214, 97]

rw [mul_div_assoc, mul_pow, ← exp_one_rpow t, rpow_nat_cast]

n t : ℕ ⊢ (rexp 1 * ↑n / ↑t) ^ t = rexp ↑t * (↑n / ↑t) ^ t

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.card_top_edgeSet

[38, 1]

[40, 70]

rw [Fintype.card_congr topEdgeSetEquiv, Sym2.card_subtype_not_diag]

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : DecidableEq V inst✝ : Fintype V ⊢ Fintype.card ↑(edgeSet ⊤) = Nat.choose (Fintype.card V) 2

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.edgeSet_eq_empty_iff

[42, 1]

[43, 34]

rw [← edgeSet_bot, edgeSet_inj]

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G : SimpleGraph V ⊢ edgeSet G = ∅ ↔ G = ⊥

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.disjoint_edgeSet

[45, 1]

[47, 90]

rw [Set.disjoint_iff_inter_eq_empty, disjoint_iff, ← edgeSet_inf, edgeSet_eq_empty_iff]

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V ⊢ Disjoint (edgeSet G) (edgeSet H) ↔ Disjoint G H

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.disjoint_left

[49, 1]

[50, 78]

simp only [← disjoint_edgeSet, Set.disjoint_left, Sym2.forall, mem_edgeSet]

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V ⊢ Disjoint G H ↔ ∀ (x y : V), Adj G x y → ¬Adj H x y

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.adj_sup_iff

[53, 1]

[58, 39]

induction' s using Finset.cons_induction_on with a s has ih

V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 s : Finset ι f : ι → SimpleGraph V x y : V ⊢ Adj (sup s f) x y ↔ ∃ i ∈ s, Adj (f i) x y

case h₁ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V ⊢ Adj (sup ∅ f) x y ↔ ∃ i ∈ ∅, Adj (f i) x y case h₂ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V a : ι s : Finset ι has : a ∉ s ih : Adj (sup s f) x y ↔ ∃ i ∈ s, Adj (f i) x y ⊢ Adj (sup (cons a s has) f) x y ↔ ∃ i ∈ cons a s has, Adj (f i) x y

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.adj_sup_iff

[53, 1]

[58, 39]

simp

case h₁ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V ⊢ Adj (sup ∅ f) x y ↔ ∃ i ∈ ∅, Adj (f i) x y

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.adj_sup_iff

[53, 1]

[58, 39]

simp [or_and_right, exists_or, ih]

case h₂ V✝ : Type u_1 V' : Type u_2 G : SimpleGraph V✝ K : Type u_3 K' : Type u_4 ι : Type u_5 V : Type u_6 f : ι → SimpleGraph V x y : V a : ι s : Finset ι has : a ∉ s ih : Adj (sup s f) x y ↔ ∃ i ∈ s, Adj (f i) x y ⊢ Adj (sup (cons a s has) f) x y ↔ ∃ i ∈ cons a s has, Adj (f i) x y

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_bot

[71, 1]

[71, 90]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x : V ⊢ neighborSet ⊥ x = ∅

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊥ x ↔ y ∈ ∅

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_bot

[71, 1]

[71, 90]

simp

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊥ x ↔ y ∈ ∅

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_top

[73, 1]

[75, 70]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x : V ⊢ neighborSet ⊤ x = {x}ᶜ

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊤ x ↔ y ∈ {x}ᶜ

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_top

[73, 1]

[75, 70]

rw [mem_neighborSet, top_adj, Set.mem_compl_singleton_iff, ne_comm]

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborSet ⊤ x ↔ y ∈ {x}ᶜ

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_sup

[77, 1]

[78, 80]

ext y

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V ⊢ neighborSet (G ⊔ H) x = neighborSet G x ∪ neighborSet H x

case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊔ H) x ↔ y ∈ neighborSet G x ∪ neighborSet H x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_sup

[77, 1]

[78, 80]

simp

case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊔ H) x ↔ y ∈ neighborSet G x ∪ neighborSet H x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_inf

[80, 1]

[83, 58]

ext y

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V ⊢ neighborSet (G ⊓ H) x = neighborSet G x ∩ neighborSet H x

case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊓ H) x ↔ y ∈ neighborSet G x ∩ neighborSet H x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_inf

[80, 1]

[83, 58]

simp only [mem_neighborSet, inf_adj, Set.mem_inter_iff]

case h V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x y : V ⊢ y ∈ neighborSet (G ⊓ H) x ↔ y ∈ neighborSet G x ∩ neighborSet H x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iSup

[85, 1]

[86, 74]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 s : ι → SimpleGraph V x : V ⊢ neighborSet (⨆ i, s i) x = ⋃ i, neighborSet (s i) x

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨆ i, s i) x ↔ y ∈ ⋃ i, neighborSet (s i) x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iSup

[85, 1]

[86, 74]

simp

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨆ i, s i) x ↔ y ∈ ⋃ i, neighborSet (s i) x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iInf

[88, 1]

[96, 23]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x : V ⊢ neighborSet (⨅ i, s i) x = ⋂ i, neighborSet (s i) x

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨅ i, s i) x ↔ y ∈ ⋂ i, neighborSet (s i) x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iInf

[88, 1]

[96, 23]

simp only [mem_neighborSet, iInf_adj, Ne.def, Set.iInf_eq_iInter, Set.mem_iInter, and_iff_left_iff_imp]

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ y ∈ neighborSet (⨅ i, s i) x ↔ y ∈ ⋂ i, neighborSet (s i) x

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ (∀ (i : ι), Adj (s i) x y) → ¬x = y

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iInf

[88, 1]

[96, 23]

intro h

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V ⊢ (∀ (i : ι), Adj (s i) x y) → ¬x = y

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y ⊢ ¬x = y

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iInf

[88, 1]

[96, 23]

inhabit ι

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y ⊢ ¬x = y

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y inhabited_h : Inhabited ι ⊢ ¬x = y

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_iInf

[88, 1]

[96, 23]

exact (h default).ne

case h V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 ι : Type u_5 inst✝ : Nonempty ι s : ι → SimpleGraph V x y : V h : ∀ (i : ι), Adj (s i) x y inhabited_h : Inhabited ι ⊢ ¬x = y

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborSet_disjoint

[98, 1]

[100, 85]

rw [Set.disjoint_iff_inter_eq_empty, ← neighborSet_inf, h.eq_bot, neighborSet_bot]

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V h : Disjoint G H ⊢ Disjoint (neighborSet G x) (neighborSet H x)

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_bot

[107, 1]

[107, 96]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x : V ⊢ neighborFinset ⊥ x = ∅

case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborFinset ⊥ x ↔ y ∈ ∅

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_bot

[107, 1]

[107, 96]

simp

case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 x y : V ⊢ y ∈ neighborFinset ⊥ x ↔ y ∈ ∅

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_top

[109, 1]

[112, 77]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : Fintype V inst✝ : DecidableEq V x : V ⊢ neighborFinset ⊤ x = {x}ᶜ

case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : Fintype V inst✝ : DecidableEq V x y : V ⊢ y ∈ neighborFinset ⊤ x ↔ y ∈ {x}ᶜ

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_top

[109, 1]

[112, 77]

rw [mem_neighborFinset, top_adj, Finset.mem_compl, mem_singleton, ne_comm]

case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝¹ : Fintype V inst✝ : DecidableEq V x y : V ⊢ y ∈ neighborFinset ⊤ x ↔ y ∈ {x}ᶜ

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_sup

[114, 1]

[116, 89]

ext y

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊔ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) ⊢ neighborFinset (G ⊔ H) x = neighborFinset G x ∪ neighborFinset H x

case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊔ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊔ H) x ↔ y ∈ neighborFinset G x ∪ neighborFinset H x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_sup

[114, 1]

[116, 89]

simp

case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊔ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊔ H) x ↔ y ∈ neighborFinset G x ∪ neighborFinset H x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_inf

[118, 1]

[120, 89]

ext y

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊓ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) ⊢ neighborFinset (G ⊓ H) x = neighborFinset G x ∩ neighborFinset H x

case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊓ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊓ H) x ↔ y ∈ neighborFinset G x ∩ neighborFinset H x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_inf

[118, 1]

[120, 89]

simp

case a V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝³ : DecidableEq V G H : SimpleGraph V x : V inst✝² : Fintype ↑(neighborSet (G ⊓ H) x) inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) y : V ⊢ y ∈ neighborFinset (G ⊓ H) x ↔ y ∈ neighborFinset G x ∩ neighborFinset H x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_supr

[126, 1]

[128, 93]

ext y

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝² : DecidableEq V ι : Type u_5 s : Finset ι f : ι → SimpleGraph V x : V inst✝¹ : (i : ι) → Fintype ↑(neighborSet (f i) x) inst✝ : Fintype ↑(neighborSet (sup s f) x) ⊢ neighborFinset (sup s f) x = Finset.biUnion s fun i => neighborFinset (f i) x

case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝² : DecidableEq V ι : Type u_5 s : Finset ι f : ι → SimpleGraph V x : V inst✝¹ : (i : ι) → Fintype ↑(neighborSet (f i) x) inst✝ : Fintype ↑(neighborSet (sup s f) x) y : V ⊢ y ∈ neighborFinset (sup s f) x ↔ y ∈ Finset.biUnion s fun i => neighborFinset (f i) x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_supr

[126, 1]

[128, 93]

simp

case a V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 inst✝² : DecidableEq V ι : Type u_5 s : Finset ι f : ι → SimpleGraph V x : V inst✝¹ : (i : ι) → Fintype ↑(neighborSet (f i) x) inst✝ : Fintype ↑(neighborSet (sup s f) x) y : V ⊢ y ∈ neighborFinset (sup s f) x ↔ y ∈ Finset.biUnion s fun i => neighborFinset (f i) x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.coe_neighborFinset

[131, 1]

[132, 99]

rw [neighborFinset_def, Set.coe_toFinset]

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G : SimpleGraph V x : V inst✝ : Fintype ↑(neighborSet G x) ⊢ ↑(neighborFinset G x) = neighborSet G x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_disjoint

[134, 1]

[137, 92]

rw [← disjoint_coe, coe_neighborFinset, coe_neighborFinset]

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) h : Disjoint G H ⊢ Disjoint (neighborFinset G x) (neighborFinset H x)

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) h : Disjoint G H ⊢ Disjoint (neighborSet G x) (neighborSet H x)

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.neighborFinset_disjoint

[134, 1]

[137, 92]

exact neighborSet_disjoint h

V : Type u_1 V' : Type u_2 G✝ : SimpleGraph V K : Type u_3 K' : Type u_4 G H : SimpleGraph V x : V inst✝¹ : Fintype ↑(neighborSet G x) inst✝ : Fintype ↑(neighborSet H x) h : Disjoint G H ⊢ Disjoint (neighborSet G x) (neighborSet H x)

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/Basic.lean

SimpleGraph.degree_eq_zero_iff

[141, 1]

[142, 73]

rw [← not_exists, ← degree_pos_iff_exists_adj, not_lt, le_zero_iff]

V : Type u_1 V' : Type u_2 G : SimpleGraph V K : Type u_3 K' : Type u_4 v : V inst✝ : Fintype ↑(neighborSet G v) ⊢ degree G v = 0 ↔ ∀ (w : V), ¬Adj G v w

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

rw [Ne.def, filter_eq_self] at this

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) this : filter (fun x => Odd (degree G x)) univ ≠ univ ⊢ ∃ v, Even (degree G v)

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) this : ¬∀ x ∈ univ, Odd (degree G x) ⊢ ∃ v, Even (degree G v)

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

simpa using this

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) this : ¬∀ x ∈ univ, Odd (degree G x) ⊢ ∃ v, Even (degree G v)

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

rw [←card_lt_iff_ne_univ, (card_le_univ _).lt_iff_ne]

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) ⊢ filter (fun x => Odd (degree G x)) univ ≠ univ

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) ⊢ Finset.card (filter (fun x => Odd (degree G x)) univ) ≠ Fintype.card V

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

intro h

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) ⊢ Finset.card (filter (fun x => Odd (degree G x)) univ) ≠ Fintype.card V

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V ⊢ False

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

have h' := even_card_odd_degree_vertices G

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V ⊢ False

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : Even (Finset.card (filter (fun v => Odd (degree G v)) univ)) ⊢ False

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

rw [h, Nat.even_iff_not_odd] at h'

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : Even (Finset.card (filter (fun v => Odd (degree G v)) univ)) ⊢ False

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : ¬Odd (Fintype.card V) ⊢ False

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean

SimpleGraph.exists_even_degree

[27, 1]

[36, 19]

exact h' hV

V : Type u_1 G : SimpleGraph V inst✝¹ : Fintype V inst✝ : DecidableRel G.Adj hV : Odd (Fintype.card V) h : Finset.card (filter (fun x => Odd (degree G x)) univ) = Fintype.card V h' : ¬Odd (Fintype.card V) ⊢ False

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean

Nat.choose_le_two_pow

[19, 1]

[25, 17]

cases' le_or_lt k n with h h

n k : ℕ ⊢ choose n k ≤ 2 ^ n

case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ 2 ^ n case inr n k : ℕ h : n < k ⊢ choose n k ≤ 2 ^ n

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean

Nat.choose_le_two_pow

[19, 1]

[25, 17]

rw [choose_eq_zero_of_lt h]

case inr n k : ℕ h : n < k ⊢ choose n k ≤ 2 ^ n

case inr n k : ℕ h : n < k ⊢ 0 ≤ 2 ^ n

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean

Nat.choose_le_two_pow

[19, 1]

[25, 17]

exact zero_le'

case inr n k : ℕ h : n < k ⊢ 0 ≤ 2 ^ n

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean

Nat.choose_le_two_pow

[19, 1]

[25, 17]

rw [← sum_range_choose n]

case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ 2 ^ n

case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ Finset.sum (range (n + 1)) fun m => choose n m

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean

Nat.choose_le_two_pow

[19, 1]

[25, 17]

refine' single_le_sum (fun _ _ => zero_le') _

case inl n k : ℕ h : k ≤ n ⊢ choose n k ≤ Finset.sum (range (n + 1)) fun m => choose n m

case inl n k : ℕ h : k ≤ n ⊢ k ∈ range (n + 1)

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Sum.lean

Nat.choose_le_two_pow

[19, 1]

[25, 17]

rwa [mem_range_succ_iff]

case inl n k : ℕ h : k ≤ n ⊢ k ∈ range (n + 1)

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

symmetric_isSquare

[22, 1]

[23, 76]

simpa using h.mul (FiniteField.isSquare_neg_one_iff.2 hF)

F : Type u_1 inst✝¹ : Fintype F inst✝ : Field F hF : Fintype.card F % 4 ≠ 3 x✝¹ x✝ : F h : (fun x y => IsSquare (x - y)) x✝¹ x✝ ⊢ (fun x y => IsSquare (x - y)) x✝ x✝¹

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

have : (univ.filter fun x : F => ¬IsSquare x) = univ.filter fun x : F => x ≠ 0 ∧ ¬IsSquare x := by refine' filter_congr _ simp (config := { contextual := true }) [not_imp_not]

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => ¬IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => ¬IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

rw [this]

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => ¬IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

have cf := quadraticChar_sum_zero hF

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun a => ↑(quadraticChar F) a) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

simp only [quadraticChar_apply, quadraticCharFun] at cf

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun a => ↑(quadraticChar F) a) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun x => if x = 0 then 0 else if IsSquare x then 1 else -1) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

rw [sum_ite, sum_const_zero, zero_add, sum_ite, sum_const, sum_const, nsmul_eq_mul, nsmul_eq_mul, mul_neg, mul_one, mul_one, add_neg_eq_zero, Nat.cast_inj, filter_filter, filter_filter] at cf

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : (Finset.sum univ fun x => if x = 0 then 0 else if IsSquare x then 1 else -1) = 0 ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

rw [← cf, and_self_iff]

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 ∧ Finset.card (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

have : ((univ.filter fun x : F => x ≠ 0 ∧ IsSquare x) ∪ univ.filter fun x : F => x ≠ 0 ∧ ¬IsSquare x) ∪ {0} = univ := by simp only [← filter_or, ← and_or_left, em, and_true_iff, filter_ne'] rw [union_comm, ← insert_eq, insert_erase] exact mem_univ _

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

have h' := congr_arg Finset.card this

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0}) = Finset.card univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

rw [card_disjoint_union, card_disjoint_union, card_singleton, card_univ, ← cf, ← two_mul, ← bit0_eq_two_mul, ← bit1] at h'

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0}) = Finset.card univ ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : bit1 (Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ)) = Fintype.card F ⊢ Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) = Fintype.card F / 2 F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) + Finset.card {0} = Finset.card univ ⊢ Disjoint (filter (fun x => x ≠ 0 ∧ IsSquare x) univ) (filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this✝ : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) this : filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ h' : Finset.card (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0}) = Finset.card univ ⊢ Disjoint (filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ) {0}

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

refine' filter_congr _

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ ∀ x ∈ univ, ¬IsSquare x ↔ x ≠ 0 ∧ ¬IsSquare x

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

simp (config := { contextual := true }) [not_imp_not]

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 ⊢ ∀ x ∈ univ, ¬IsSquare x ↔ x ≠ 0 ∧ ¬IsSquare x

no goals

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

simp only [← filter_or, ← and_or_left, em, and_true_iff, filter_ne']

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ filter (fun x => x ≠ 0 ∧ IsSquare x) univ ∪ filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ ∪ {0} = univ

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ erase univ 0 ∪ {0} = univ

https://github.com/b-mehta/ExponentialRamsey.git

7e17b629a915a082869f494c8afa56a3e1c7a88d

ExponentialRamsey/Prereq/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean

card_non_zero_square_non_square

[25, 1]

[53, 9]

rw [union_comm, ← insert_eq, insert_erase]

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ erase univ 0 ∪ {0} = univ

F : Type u_1 inst✝² : Fintype F inst✝¹ : Field F inst✝ : DecidableEq F hF : ringChar F ≠ 2 this : filter (fun x => ¬IsSquare x) univ = filter (fun x => x ≠ 0 ∧ ¬IsSquare x) univ cf : Finset.card (filter (fun a => ¬a = 0 ∧ IsSquare a) univ) = Finset.card (filter (fun a => ¬a = 0 ∧ ¬IsSquare a) univ) ⊢ 0 ∈ univ