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linter?
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b-mehta committed Sep 29, 2023
1 parent bd68196 commit 0dc9f82
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8 changes: 8 additions & 0 deletions src/analysis/special_functions/explicit_stirling.lean
Original file line number Diff line number Diff line change
Expand Up @@ -55,6 +55,10 @@ begin
all_goals { positivity },
end

/--
an explicit function of the central binomial coefficient which we will show is always less than
√π, to get a lower bound on the central binomial coefficient
-/
def central_binomial_lower (n : ℕ) : ℝ := central_binom n * sqrt (n + 1 / 4) / 4 ^ n

lemma central_binomial_lower_monotone : monotone central_binomial_lower :=
Expand All @@ -72,6 +76,10 @@ begin
{ positivity }
end

/--
an explicit function of the central binomial coefficient which we will show is always more than
√π, to get an upper bound on the central binomial coefficient
-/
def central_binomial_upper (n : ℕ) : ℝ := central_binom n * sqrt (n + 1 / 3) / 4 ^ n

lemma central_binomial_upper_monotone : antitone central_binomial_upper :=
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6 changes: 6 additions & 0 deletions src/combinatorics/simple_graph/basic_mathlib.lean
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Expand Up @@ -26,6 +26,11 @@ begin
cases hi.ne (subsingleton.elim _ _)
end

/--
The edge set of the complete graph on V is in bijection with the off-diagonal elements of sym2 V
TODO: maybe this should be an equality of sets?
and then turn it into a bijection of types
-/
def top_edge_set_equiv : (⊤ : simple_graph V).edge_set ≃ {a : sym2 V // ¬a.is_diag} :=
equiv.subtype_equiv_right $ λ i, sym2.induction_on i $ λ x y, iff.rfl

Expand Down Expand Up @@ -54,6 +59,7 @@ lemma top_hom_injective (f : (⊤ : simple_graph V') →g G) :
function.injective f :=
λ x y h, by_contra (λ z, (f.map_rel z).ne h)

/-- graph embeddings from a complete graph are in bijection with graph homomorphisms from it -/
def top_hom_graph_equiv :
((⊤ : simple_graph V') ↪g G) ≃ ((⊤ : simple_graph V') →g G) :=
{ to_fun := λ f, f,
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