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23 May 2024: thesis. qed lemma order_less_subst1: "(a::'a::f b (b::'b::c (!x y. x < y f x < f y) a < f c" proof - assume r: "!x y. ... thesis. qed lemma order_subst1: "(a::'a::f b (b::'b::c (!x y. x <= y f x <= f y) a <= f c" proof - assume r: "!x y.

23 Feb 2024: Campus./p pOn the International Women’s Day, the book featured in the Guardian.b/b/p pa

23 May 2024: zero [simp]: "begin subclass ring_1_no_zero_divisors proof fix a b :: 'a assume a: "aand b: "bshow "a bproof assume ab: "a bhence "a (a b)b" by simp ... thesis by (qed lemma division_ring_inverse_add: "ababa (a b)b" by (simp add: algebra_simps) lemma

23 May 2024: subset: "A = B A B B A" by blast lemma subset_iff: "A Bt. ... vimageI [intro]: "f a = b b B a f - B" unfolding vimage_def by blast lemma vimageI2: "f a A a f - A" unfolding vimage_def by fast lemma vimageE [elim!]:

23 May 2024: be f surj(A,B), b B for symmetry with left_inverse? ... b,B))" unfolding inj_def apply (force intro: apply_type simp add: fun_extend) done end.

23 Feb 2024: p class="p1"span class="s1"bDo you have a/b/span span class="s1"bpunchy ‘elevator pitch’ to sell your idea to a collaborator, colleagues or potential ... We’ll use the same format that worked so well during the recent “Selling to industry for

23 May 2024: bB. x b j) (x = t) (x - D = t))" "⟦Dt B⟧ ((x::jD}. ... bB. x b j) (x t) (x - D t))" "⟦Dt B⟧ ((x::jD}.

23 Feb 2024: D, Lopez A, Saez B, Taylor-King JP, Prosper F, Fortelny Nsup#/sup, Huntly BJPsup#/sup. ... to other functional changes associated with ageing.”/p pbSearching for cellular changes/b/p pAll human cells acquire genetic changes throughout life, known as

23 May 2024: a, b], [b, b], [c, b], [a, c], [b, c], [c, c]]" by (simp add: eval_nat_numeral)} @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp} @{lemma ... b 'b list 'b list" where "insort_insert_key f x xsf x fxs then xsf x xs)" abbreviation "sort sort_key (λx.

23 May 2024: syntax "_quote" :: "'b ('a 'b)" "_antiquote" :: "('a 'b) 'b" ("_" [1000] 1000) "_Subst" :: "'a bexp 'b idt 'a bexp" ("_[_'/_]" [1000] 999) "_Assert" :: "'a 'a set" ("(⦃_⦄)" [0] 1000) "_Assign" :: "idt 'b 'a com" ("(_ :=/ _)" ... INV _ //DO _ /OD)" [0,

23 May 2024: b F} = {a b |b. b = x b F}" by blast have "a (xF) = a x aF" by (simp add: inf_sup_distrib1) also have ". = a xa b |b. b F}" ... by simp also have ". a b |b. b = x b F}" by (unfold Sup_insert[THEN sym], simp) finally show "a (xF)a b |b.

23 May 2024: a band a: "a = a'a b" and b: "b = b'a b" shows "coprime a' b'" proof - from c have "abby simp with div_gcd_coprime have "coprime (aa b) ... a = a'a b b = b'a ba' b'" proof - have "coprime (aa b) (ba b)" usingby auto then show?

23 May 2024: lemma wf_onI_pf: assumes "B. B A B R B Bshows "wf_on A R" unfolding wf_on_def proof (introfix P :: "'a bool" and x :: 'a let? ... a 'a) set ('b 'b) set (('a 'b) ('a 'b)) set" (infixr "<lex>" 80) where "ra <lex> rb = {((a, b), (a', b')).

23 May 2024: A)" ― ‹legacy› section ‹Quantifier elimination for ‹ (0, 1, , floor, <)›› declare of_int_floor_cancel [simp del] lemma myle: fixes a b :: "'a::{ordered_ab_group_add}" shows "(a b)b - a)" ... by (lemma myless: fixes a b ::

23 May 2024: axiomatization where extension: "A = B A B B A" and Union_iff: "AC) (BC. ... lemma ReplaceI [intro]: "⟦P(x,b); x: A; y. P(x,y) y=b⟧ b {y.

23 Feb 2024: b>. B Cell Fcγ Receptor IIb Modulates Atherosclerosis in Male and Female Mice by Controlling Adaptive Germinal Center and Innate B1-Cell Responses. ... b><b>AP,</b>Nus M, Bagchi Chakraborty J, Tsiantoulas D, Newland SA, Finigan AJ, Masters L, Binder CJ,

23 Feb 2024: do</div></div></div> Fri, 24 Mar 2023 16:30:05 0000 Dr Hayley B. ... JR, Bliss JM and <b>Coles CE</b> on behalf of the IMPORT Trialists.

23 May 2024: B x) a A b B a" by auto lemma INT_E [elim]: "b (xA. ... A B))" "AA else (B. B A))" by auto lemma Un_Union_image: "(xC.

23 May 2024: a. a A - b! b'. b' B - {b} a b'" using unique_class by fastforce next show "b'. ... b' B - {b} b' A - b" usingby fastforce qed lemma disjoint_sum: ‹contributor ‹Paulo Emílio de Vilhena›› "⟦ finite B; finite A; partition A B⟧ (bB.

23 May 2024: T (Rep b) b) R = TT" unfoldingby auto lemma Quotient_alt_def5: "Quotient R Abs Rep T TAbsRep TR = T OO T" unfoldingby blast lemma fun_quotient: assumes 1: "Quotient R1 ... correspondence relation› definition POS :: "('a 'b bool) ('a 'bwhere "POS A B A B