# Theory Girth_Chromatic.Girth_Chromatic_Misc

```theory Girth_Chromatic_Misc
imports
Main
"HOL-Library.Extended_Real"
begin

section ‹Auxilliary lemmas and setup›

text ‹
This section contains facts about general concepts which are not directly
connected to the proof of the Chromatic-Girth theorem. At some point in time,
most of them could be moved to the Isabelle base library.

Also, a little bit of setup happens.
›

subsection ‹Numbers›

lemma enat_in_Inf:
fixes S :: "enat set"
assumes "Inf S ≠ top"
shows "Inf S ∈ S"
proof (rule ccontr)
assume A: "~?thesis"

obtain n where Inf_conv: "Inf S = enat n" using assms by (auto simp: top_enat_def)
{ fix s assume "s ∈ S"
then have "Inf S ≤ s" by (rule complete_lattice_class.Inf_lower)
moreover have "Inf S ≠ s" using A ‹s ∈ S› by auto
ultimately have "Inf S < s" by simp
with Inf_conv have "enat (Suc n) ≤ s" by (cases s) auto
}
then have "enat (Suc n) ≤ Inf S" by (simp add: le_Inf_iff)
with Inf_conv show False by auto
qed

lemma enat_in_INF:
fixes f :: "'a ⇒ enat"
assumes "(INF x∈ S. f x) ≠ top"
obtains x where "x ∈ S" and "(INF x∈ S. f x) = f x"
proof -
from assms have "(INF x∈ S. f x) ∈ f ` S"
using enat_in_Inf [of "f ` S"] by auto
then obtain x where "x ∈ S" "(INF x∈ S. f x) = f x" by auto
then show ?thesis ..
qed

lemma enat_less_INF_I:
fixes f :: "'a ⇒ enat"
assumes not_inf: "x ≠ ∞" and less: "⋀y. y ∈ S ⟹ x < f y"
shows "x < (INF y∈S. f y)"
using assms by (auto simp: Suc_ile_eq[symmetric] INF_greatest)

lemma enat_le_Sup_iff:
"enat k ≤ Sup M ⟷ k = 0 ∨ (∃m ∈ M. enat k ≤ m)" (is "?L ⟷ ?R")
proof cases
assume "k = 0" then show ?thesis by (auto simp: enat_0)
next
assume "k ≠ 0"
show ?thesis
proof
assume ?L
then have "⟦enat k ≤ (if finite M then Max M else ∞); M ≠ {}⟧ ⟹ ∃m∈M. enat k ≤ m"
by (metis Max_in Sup_enat_def finite_enat_bounded linorder_linear)
with ‹k ≠ 0› and ‹?L› show ?R
unfolding Sup_enat_def
by (cases "M={}") (auto simp add: enat_0[symmetric])
next
assume ?R then show ?L
by (auto simp: enat_0 intro: complete_lattice_class.Sup_upper2)
qed
qed

lemma enat_neq_zero_cancel_iff[simp]:
"0 ≠ enat n ⟷ 0 ≠ n"
"enat n ≠ 0 ⟷ n ≠ 0"
by (auto simp: enat_0[symmetric])

lemma natceiling_lessD: "nat(ceiling x) < n ⟹ x < real n"
by linarith

lemma le_natceiling_iff:
fixes n :: nat and r :: real
shows "n ≤ r ⟹ n ≤ nat(ceiling r)"
by linarith

lemma natceiling_le_iff:
fixes n :: nat and r :: real
shows "r ≤ n ⟹ nat(ceiling r) ≤ n"
by linarith

lemma dist_real_noabs_less:
fixes a b c :: real assumes "dist a b < c" shows "a - b < c"
using assms by (simp add: dist_real_def)

lemma n_choose_2_nat:
fixes n :: nat shows "(n choose 2) = (n * (n - 1)) div 2"
proof -
show ?thesis
proof (cases "2 ≤ n")
case True
then obtain m where "n = Suc (Suc m)"
moreover have "(n choose 2) = (fact n div fact (n - 2)) div 2"
using ‹2 ≤ n› by (simp add: binomial_altdef_nat
div_mult2_eq[symmetric] mult.commute numeral_2_eq_2)
ultimately show ?thesis by (simp add: algebra_simps)
qed (auto simp: binomial_eq_0)
qed

lemma powr_less_one:
fixes x::real
assumes "1 < x" "y < 0"
shows "x powr y < 1"
using assms less_log_iff by force

lemma powr_le_one_le: "⋀x y::real. 0 < x ⟹ x ≤ 1 ⟹ 1 ≤ y ⟹ x powr y ≤ x"
proof -
fix x y :: real
assume "0 < x" "x ≤ 1" "1 ≤ y"
have "x powr y = (1 / (1 / x)) powr y" using ‹0 < x› by (simp add: field_simps)
also have "… = 1 / (1 / x) powr y" using ‹0 < x› by (simp add: powr_divide)
also have "… ≤ 1 / (1 / x) powr 1" proof -
have "1 ≤ 1 / x" using ‹0 < x› ‹x ≤ 1› by (auto simp: field_simps)
then have "(1 / x) powr 1  ≤ (1 / x) powr y" using ‹0 < x›
using ‹1 ≤ y› by ( simp only: powr_mono)
then show ?thesis
by (metis ‹1 ≤ 1 / x› ‹1 ≤ y› neg_le_iff_le powr_minus_divide powr_mono)
qed
also have "… ≤ x" using ‹0 < x› by (auto simp: field_simps)
finally show "?thesis x y" .
qed

subsection ‹Lists and Sets›

lemma list_set_tl: "x ∈ set (tl xs) ⟹ x ∈ set xs"
by (cases xs) auto

lemma list_exhaust3:
obtains "xs = []" | x where "xs = [x]" | x y ys where "xs = x # y # ys"
by (metis list.exhaust)

lemma card_Ex_subset:
"k ≤ card M ⟹ ∃N. N ⊆ M ∧ card N = k"
by (induct rule: inc_induct) (auto simp: card_Suc_eq)

subsection ‹Limits and eventually›

text ‹
We employ filters and the @{term eventually} predicate to deal with the
@{term "∃N. ∀n≥N. P n"} cases. To make this more convenient, introduce
a shorter syntax.
›

abbreviation evseq :: "(nat ⇒ bool) ⇒ bool" (binder "∀⇧∞" 10) where
"evseq P ≡ eventually P sequentially"

lemma eventually_le_le:
fixes P :: "'a => ('b :: preorder)"
assumes "eventually (λx. P x ≤ Q x) net"
assumes "eventually (λx. Q x ≤ R  x) net"
shows "eventually (λx. P x ≤ R x) net"
using assms by eventually_elim (rule order_trans)

lemma LIMSEQ_neg_powr:
assumes s: "s < 0"
shows "(%x. (real x) powr s) ⇢ 0"
by (rule tendsto_neg_powr[OF assms filterlim_real_sequentially])

lemma LIMSEQ_inv_powr:
assumes "0 < c" "0 < d"
shows "(λn :: nat. (c / n) powr d) ⇢ 0"
proof (rule tendsto_zero_powrI)
from ‹0 < c› have "⋀x. 0 < x ⟹ 0 < c / x" by simp
then show "∀⇧∞n. 0 ≤ c / real n"
using assms(1) by auto
show "(λx. c / real x) ⇢ 0"
by (intro tendsto_divide_0[OF tendsto_const] filterlim_at_top_imp_at_infinity
filterlim_real_sequentially tendsto_divide_0)
show "0 < d" by (rule assms)
show "(λx. d) ⇢ d" by auto
qed

end
```