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23 May 2024: xBB x bmx" proof fix x assume x: "xB" have "busing ‹independent B› ‹b B› dependent_zero by blast have [simp]: "b b'b' = bb)if "b B" "b' B" for ... lemma continuous_on_representation: fixes B :: "'N::assumes "finite B" "independent B" "b B" "B"

23 May 2024: S::'a :: {b (b'<b. ... xS. b' < x) Sup S = b" by (simp: not_le[symmetric] setle_def) lemma cInf_unique: "b <= (S::'a :: {b'>b.

23 May 2024: and g :: "'b 'a" and A :: "'a set" and B :: "'b set" assumes inj1: "inj_on f A" and sub1: "f A B" and inj2: "inj_on g B" and sub2: ... a: "a X" and b: "b A - X" for a b proof - from a have fa: "f a f X" by (from b have "g' b g' (A - X)" by (with

23 May 2024: and "a b" "b a" shows "P" using assms by (elimlemma (in weak_partial_order) lless_trans [trans]: assumes "a b" "b c" and carr[simp]: "aL" "bL" "cL" shows "a ... add:subsubsection ‹Idempotent functions› definition idempotent :: "('a, 'b) gorder_scheme

23 May 2024: mk_disjoint_insert by fastforce lemma fst_Pi: "A B A" and snd_Pi: "A B B" by auto subsection ‹Composition With a Restricted Domain: <term>‹compose›› lemma funcset_compose: "f ... E. x{a}. B x) = (b B a. {λx {a}. b})" apply (auto simp: PiE_iff

23 May 2024: b xb (x / b b)" using that by simp also have "…b (x / b)b b" using that by (also have "…b (x / b)using that by simp also have "b ... x / b)b (x div b (x / b - x div b))" by simp also have "…b (x div b)" usingby (introfinally show?

23 May 2024: based on Lawrence Paulson's book Logic and Computation.› subsection ‹Natural Deduction Rules for LCF› class cpo = "term" default_sort cpo typedecl tr typedecl void typedecl ('a,'b) prod (infixl "" 6) ... typedecl ('a,'b) sum (infixl "" 5) instance

23 May 2024: f b b}" by auto interpret bijection f by‹bij f›) from fin show "f)a. ... f b b}" then have "bij f" by simp interpret bijection f by‹bij f›) show "{a.

23 May 2024: uniq_in_ya2'_parts [rule_format]: "⟦evsBB'⟧ Ciph B ⦃Agent A, Nonce NA, Nonce NB⦄evs)B' ⦃Agent A', Nonce NA', Nonce NB⦄evs) A=A' B=B' NA=NA'" apply ... simp) lemma NB_is_uniq_in_ya2': "⟦ya2' C A B NA NBevs; ya2' C' A' B' NA' NBevs; evsBB'⟧

23 May 2024: inner a x < b}" unfolding convex_def by (auto simp:lemma convex_halfspace_gt: "convex {x. ... b}" and "a<.b}" and "a.<b}" and "a<.<b}" proof - have "{ax.

23 May 2024: a. f aa A) ( a A. case f a of Some b b B |None True)}" lemma card_of_Func_option_Func: "A BA B|" proof (unfold card_of_ordIso[symmetric], ... case f a of None True | Some b b B)}" lemma Func_Pfunc: "Func_option A BA B" unfoldingby auto lemma

3 May 2024: Now let Z_n be bin(n, p), independent of B which is Bernoulli ber(p). ... By the result of Exercise 3.6.5,. H(Z_n1) = H(Z_n B) >= H(Z_n B|B) = H(Z_n|B) = H(Z_n).

23 May 2024: k K open k" unfolding eq by auto from ex_countable_basis obtain B :: "'awhere B: "b. ... fix X::"'a set" assume "open X" from open_countable_basisE[OF this] obtain B' where B': "B' B" "XB'".

23 May 2024: a - b)x = a (ax2 b) a x b b x" by (simp add:have "(kn. ... norm(f x - g x) < e)" proof - { fix b :: 'b assume "bhave "pp (x S.

23 May 2024: B (B) ⦃Nonce NA, Agent A⦄)" abbreviation (input) ns1' :: "where "ns1' A' A B NA == Says A' B (B) ⦃Nonce NA, Agent A⦄)" abbreviation (input) ns2 :: "where "ns2 B A NA ... NB == Says B A (A) ⦃Nonce NA, Nonce NB, Agent B⦄)" abbreviation (input)

23 May 2024: h xa.b} g (h x) = x" assumes range: "{a.bh" shows "f) lborel h) {a.bλx. ... where B=B])(auto simp add: f_bounded) then show "enn2real (. - f x M') = x. f x M'" by(simp add: real_lebesgue_integral_def) qed qed simp_all finally show?

23 May 2024: bool: "C) = C(C(1)" by (unfoldIntroduction rules for the injections ) lemma InlI [intro!,simp,TC]: "a A Inl(a) AB" by (unfoldlemma InrI [intro!,simp,TC]: "b B Inr(b) ... b): AB b B" by blast lemma sum_iff: "u ABx.

23 May 2024: B b C))" | "list_of (a A)a (list_of A)" | "list_ofbytermination proof let? ... t rule: rotR_poss.induct) case (1 A a B b C) then show?

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: nm#k))" by (auto intro: less_imp_succ_add) lemma add_lt_elim2: "⟦a # d = b # c; a < b; bcd⟧ c < d" by (drulelemma add_le_elim2: "⟦a # d = b # c; ... add:lemma raw_nat_diff_split: "⟦ab⟧ (P(a #- b)) ((a < b P(dnat.

4 Jun 2024: p style="text-align: left;"b[beep]Generation–the new Polish Poets, /b/p p style="text-align: left;"Monday 10th February, 2020, 18.00–19.30/p p ... Wilson Drama Studio, Faculty of English, 9 West Road, Cambridge CB3 9DP/p p /p p style="text-align: left

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 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: B AA)" proof safe fix B assume B: "B Ashow "A)" unfolding inj_on_def by (next assume "A)" from vec.linear_injective_left_inverse[OFobtain g where "g" and "gAby ... A unfolding invertible_def by blast obtain B' where BB': "B B'and B'B: "B' Busing inv_B

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: B B'B'B'lemma assumes B: "independent B" shows extend_basis_superset: "BB" and independent_extend_basis: "B)" and span_extend_basis[simp]: "B)proof - define p where "p B' ... B B'B'B'for B' obtain B' where "p B'" using

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: 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 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: case using A by (auto simp:intro!: bexI[where x="λi. f i g i"]) next case (UN B) then obtain b where "x b" "b B" by auto hence ... A. a b" using UN by simp thus? case using ‹b B› by (next case (Basis s) then obtain a b where xs: "xa b" "sa b" "i.

23 May 2024: a'' b'" by auto also have "b (a'' b') = b' (a'' b)" by simp finally have "b' (a b'') = b' (a'' b)". ... a = "aa b" let? b = "ba b" from b have "?ba b = b" by simp with b have "?bby fastforce withhave q2: "qa?

23 May 2024: f y g)" by (simp add:end inductive fold_graph :: "('a 'b 'b) 'b 'a set 'b bool" for f :: "'a 'b 'b" and z :: 'b where emptyI [intro]: "fold_graph ... that by auto definition fold :: "('a 'b 'b) 'b 'a set 'b" where "fold f z AA then (THE y.

23 May 2024: R v bby (intro finite_subset[OF ]) simp_all have "(b |? R u b -? R v b) s b) = (b{b? R u bb? R v bR u b -? R v ... b) s b)" by (introalso have ". = (b{b? R u bb? R v bR u b s b) - (b{b?

23 May 2024: x A" and "Ashows "x A) = x F A" proof - from ‹A› obtain b where "b A" by blast then obtain B where : "Ab B" "b B" by (blast dest: ... a set 'b 'b" ("(3MAX __./ _)" [0, 0, 10] 10) translations "MIN x y.

23 May 2024: aba $< b | b $ #0) P(a, #2 $ b)⟧ P(a,b)" shows "P(u,v)" apply (subgoal_tac "(λ⟨x,y⟩. ... ab0 $ a $+ b | b $ #0) P(a, #2 $ b)⟧ P(a,b)" shows "P(u,v)" apply (subgoal_tac " (λ⟨x,y⟩.

23 May 2024: definition SIM :: "[where "SIM(t,t',R) == (tt'tt'a a' b b'. ... rews => fn ctxt => SIMPLE_METHOD' (inj_rl_tac ctxt rews)) › lemma ccl_injs: "<a,ba',b'a=a' b=b')" "b b'.

23 May 2024: b T j (i b) = b" "b. b T i b S" assumes eq: "a. ... b T - T' j (i b) = b" "b. b T - T' i b S - S'" assumes nn: "a.

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: 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: bB r a b" "a1 a2 b. ⟦ a1 A; a2 A; b B; r a1 b; r a2 b ⟧ a1 = a2" shows "card AB" proof - let? ... P = "λa b. b B r a b" let? f = "λa.

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: 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 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 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 May 2024: thesis by auto next case False from ‹a dvd b› obtain c where b: "b = a c". ... b b" by simp then have "normalize aa div b b)" by simp then show?

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: 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 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 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}.