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23 May 2024: thesis by (simp add:qed theorem n_subsets: assumes "finite A" shows "card {B. ... g? B? B'" apply (where f' = "λl. (ll"]) using assms by (auto simp: 2 simp flip: length_0_conv intro!: 3) have fin: "finite {xs.

23 May 2024: simp]: "(vc t) = (v «| t)" by (cases t) auto lemma paint_rbt_less[simp]: "(paint c t |« v) = (t |« v)" by (cases t) auto fun rbt_ins :: "('a 'b 'b 'b) ... f k v t rule:end context ord begin definition rbt_insert_with_key :: "('a 'b 'b 'b) 'a 'b ('a,'b)

23 Feb 2024: 335559740":259}"u00a0/span/pnpbspan data-contrast="auto"The Malaysian Acute Vascular Events RIsK (MAVERIK)/span/b bspan data-contrast="auto"study/span/bspan ... data-contrast="auto"The Malaysian Acute Vascular Events RIsK (MAVERIK)/span/b bspan

23 May 2024: P x)xP x)› by auto lemma verit_ite_simplify: ‹(B C) = B› ‹(B C) = C› ‹(If A' B B) = B› ‹(A') B C)A' C B)› ‹(If c (If c ... by auto lemmas verit_eq_simplify =lemma verit_minus_simplify: ‹(a :: 'a ::a› ‹(a :: 'a ::a› ‹b :: 'b :

23 May 2024: by (simp add:next show "a a" for a by (simp add: order_iff) next fix a b assume "a b" "b a" then have "a = a b" "a b = b" ... by (simp_all add: order_iff commute) then show "a = b" by simp next fix a b c assume "a b" "b c" then have "a = a b" "b = b c"

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: A" by (blast dest: fun_is_rel) lemma range_type: "⟦⟨a,b⟩ f; fA,B)⟧ b B(a)" by (blast dest: fun_is_rel) lemma Pair_mem_PiD: "⟦⟨a,b⟩: f; ... fA,B)⟧ a A b B(a) fa = b" by (blast intro:subsection‹Lambda Abstraction› lemma lamI: "a A <a,b(a

23 May 2024: of forward proof!) lemma well_ord_iso_preserving: "⟦well_ord(A,r); well_ord(B,s); ⟨a,c⟩: r; fA,a,r), r, pred(B,b,s), s); gA,c,r), ... r, pred(B,d,s), s); a A; c A; b B; d B⟧ ⟨b,d⟩: s" apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN

23 May 2024: EX a:A. xa)) | (EX b:B. xb))}" definition Pi :: "[where "Pi(A,B) == {x. ... pair: "⟦a = a'; b = b'; <a',b'A⟧ <a,bA" by simp ML ‹ val coinduct3_tac = SUBPROOF (fn {context = ctxt, prems = mono :: prems,. } =>

23 Feb 2024: Health Foundation./bb/b/p pThe Clinician Scientist Fellowship enables clinicians to conduct research within a clinical setting in order to improve the long-term quality of healthcare. ... b/p pThe award is for study into how protein breakdown is

23 May 2024: simp lemma set_mset_empty [simp]: "by (simp add: set_mset_def) lemma set_mset_single: "set_mset {#b#} = {b}" by (simp add: set_mset_def) lemma ... Mx A" by (where x = "Mx#}"]) simp lemma multiset_add_sub_el_shuffle: assumes "c # B" and "b c" shows

23 May 2024: BA. f B B})" proof (rule order.antisym) show "A)f A |f. ... B A f B B" "B A" for f B using that by (auto intro: SUP_upper2 INF_lower2) then show "(x?F.

23 Feb 2024: class="authors"><span class="author">Finucane HK</span>, <span class="author">Bulik-Sullivan B</span>, <span class="author">Gusev A</span>, <span class="author">Trynka G</span>, <span class="author">Reshef ... b></p> <p>I strongly believe that

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: 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: 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: 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

23 May 2024: a,a'ra; pa,ba',b'⟧ R" and 3: "a b b'. ⟦<b,b'rb; pa,ba,b'⟧ R" shows R apply (THEN lexXH [THEN iffD1], THEN exE]) usingapply blast done ... induct]) apply (apply (apply (erule 2) apply blast done lemma SPLITB: "a,b>,B) = B(a,b)" unfolding SPLIT_def

23 Feb 2024: JR, Bliss JM and <b>Coles CE</b> on behalf of the IMPORT Trialists. ... Hak C, Qian W, Twyman N, Burnet NG, Wishart GC and <b>Coles CE</b>.

23 May 2024: and (Haskell) infixr 9 "." subsection ‹The Forward Composition Operator ‹fcomp›› definition fcomp :: "('a 'b) ('b 'c) 'a 'c" (infixl ">" 60) where "f > g = (λx.

23 May 2024: b B ⟨b',b⟩:s" by (unfoldlemma radd_Inr_Inl_iff [simp]: "⟨Inr(b), Inl(a)⟩A,r,B,s)by (unfolddeclare radd_Inr_Inl_iff [THENsubsubsection‹Elimination Rule› lemma raddE: ... Can be used to obtain introduction rules› lemma rmult_iff [iff]:

23 May 2024: transfer by simp lift_definition ffold :: "('a 'b 'b) 'b 'a fset 'b" is Finite_Set.fold. ... A || C" by (lemma fsubset_pfsubset_trans: "A || B B || C A || C" by (lemma pfsubset_imp_ex_fmem: "A || B b.

23 May 2024: a B b B xB. F xF a) (F b)) eventually P (bB. ... F. B) B" unfoldingby (force intro: eventually_True) lemma prod_filter_INF: assumes "Iand "Jshows "(iI.

23 Feb 2024: 11),14878-14891.</p> <p><i>Heritability of Haemodynamics in the Ascending Aorta</i>.McGurk KA, Owen B,<b>Watson WD</b>, Nethonoda RM, Cordell HJ, Farrall M, Rider OJ, Watkins ... JJ, Sayeed RA, Petrou M, Krasopoulos G, Lake HA, Raman B,<b>Watson WD</b>,

23 Feb 2024: fibrinogen: A randomised double-blind placebo-controlled trial.</a> <b>PLoS One</b>. ... Chest</b><b></b><b> </b>2017 Mar;151(3):555-563.

23 May 2024: b B⟧ cons(a,A)b,B)" apply (unfoldapply (rule_tac x = "λycons (a,A). ... B; a A; b B⟧ cons(a,A)b,B)" by (simp add:lemma cons_lepoll_cons_iff: "⟦a A; b B⟧ cons(a,A)b,B) A B" by (blast

23 May 2024: Pz = x) (Pz = y))" definition Let :: "'a ('a 'b) 'b" where "Let s f f s" translations "_Let (_binds b bs) e" "_Let b (_Let bs e)" "let x = a in ... lemma not_sym: "t s s t" by (erule contrapos_nn) (erule sym) lemma eq_neq_eq_imp_neq: "⟦x = a; a b; b =

23 May 2024: thesis by (qed show? thesis usingby (qed lemma SR_B_inj: "inj_on (λx. ... E: "using finite_E by (simp add:lemma C_card_eq_B: "proof - have "inj_on (λx.

23 May 2024: a A b B}" definition refl' :: "('a 'a)where "refl' r x. ... set xs = Axs))inductive fold_graph' :: "('a 'b 'b) 'b 'a set 'b bool" where "fold_graph' f z {} z" | "⟦x A; fold_graph' f z (A - {x}) y⟧ fold_graph'

23 Feb 2024: 2012;7(12)</span></p> <p class=" "><em>S</em>chwarz E, Guest PC, Steiner J, Bogerts B, <b>Bahn S.</b> (2012)<span>Identification of blood-based molecular signatures for ... sup>, David Niebuhr¹, David Cowan, Fuller Torrey E, Robert H Yolken,

23 May 2024: simp]: "⟦b; b⟧ b m b n n m" using power_strict_decreasing [of m n b] by (auto intro:lemma power_strict_decreasing_iff [simp]: "⟦b; b⟧ b m < b ... le) qed lemma power_increasing_iff [simp]: "b b x b y x y" by (blast intro:less_imp_le) lemma

23 May 2024: FROM the responder) | K6: "⟦ evs6 kerbIV; Says A' B ⦃ (B) ⦃Agent A, Agent B, Key servK, Number Ts⦄), (Crypt servK ⦃Agent A, Number T3⦄)⦄evs6;Ts evs6;T3 evs6 ⟧ Says B ... evs); Key SesKeyevs); evs⟧ K=K' B=B' T=T' Ticket=Ticket'" apply

23 May 2024: def) done lemma pred_Inr_bij: "b B id(AB,b,s))AB,b,s), pred(AB, Inr(b), radd(A,r,B,s)))" unfoldingapply (rule_tac d = "λz. ... z" indone lemma ordertype_pred_Inr_eq: "⟦b B; well_ord(A,r); well_ord(B,s)⟧AB, Inr(b), radd(A,r,B,s)),

23 May 2024: iso_tuple_fst_update :: "('a, 'b, 'c) tuple_isomorphism ('b 'b) ('a 'a)" where "iso_tuple_fst_update isom fisomfisom" definition iso_tuple_snd_update :: "('a, 'b, 'c) tuple_isomorphism ('c ... b 'b) ('a 'a)) ('a 'b) bool" where