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23 May 2024: f x - g x) a - b) F" using tendsto_add [of f a F "λx. - ... norm (x y)xy K" using nonneg_bounded by blast obtain B where B: "B" and norm_g: "eventually (λx.

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

26 Jun 2024: Typically AB is not the same as BA. The commutator [A, B] captures. ... the dierence between these. Clearly [A, B] = [B, A] and the same also holds foroperators: [Ô, M̂] = [M̂, Ô].

23 May 2024: b) 'a 'b bool" for e :: 'b and r :: "'a 'b 'b" where Rec_zero: "Rec e r zero e" | Rec_succ: "Rec e r m n Rec e r (succ ... by (simp only:) qed qed qed text ‹ The recursion operator -- polymorphic!› definition rec :: "'b ('a 'b 'b) 'a 'b" where "rec e

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: x A" | A B where "x D" "A face_of B" "B " "B 𝒢" "aff_dim AT" "x A" using x by (auto simp: _def) then have xrel: "xD" proof cases ... b" using bounded_subset_cbox_symmetric by blast define bbox where "bboxbbhave "b) b bbox" by (auto simp:intro!:

23 May 2024: a 'bwhere "Inr_Rep b x y p y = bp" definition "sum = {f. ... A <> BAB" hide_const (open) Plus ― ‹Valuable identifier› lemma InlI [intro!]: "a A Inl a A <> B" by (simp add: Plus_def) lemma InrI [intro!]: "b B Inr b A <> B"

23 May 2024: Theory Sym_Groups. ( Title: HOL/Algebra/Sym_Groups.thy Author: Paulo Emílio de Vilhena ) theory Sym_Groups imports "begin section ‹Symmetric Groups› subsection ‹Definitions› abbreviation inv' :: "('a 'b) ('b 'a)" where ... by auto obtain a b c

23 May 2024: X Y) T'" for T' proof (clarsimp; intro conjI) fix a :: "'a" and b :: "'b" assume "(a, b) T'" with that obtain U V where UV: "openin X U" "openin Y ... AX) BY)}" if "finite T" "T 𝒳 𝒴" for T using that proof induction case (insert B ) then show?

23 May 2024: 𝟬)" by (induction n) (auto) lemma scalar_coeff: "aRλb. a b) p) = (λi. ... a_p2 = "(map (λb. a b) p2)a # p1)) 𝟬)" have "a # p1))λ_.

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: B b RB" apply blast done lemma oneStep_reachable: "b RB b RB" apply blast done lemma still_reachable: "⟦BRaA; (x,y) Rb-Ra. ... x R m x x reachable? Rb? B›.› let? T = "{t, p.r}" have "?RaA?

23 May 2024: a A}" definition "Grp A f = (λa b. b = f a a A)" definition vimage2p where "vimage2p f g R = (λx y. ... y, x)) AR)" by auto lemma predicate2_eqD: "A = B A a b B a b" by simp lemma case_sum_o_inj: "case_sum f gf" "case_sum f gg"

23 May 2024: y,x⟩: r y B) x B A<=B›› lemma wf_onI2: assumes prem: "y B. ... well-founded induction then we have <term>‹wf(r)›.› lemma wfI: "⟦field(r)<=A; y B.

23 May 2024: scheme A and show the existence of an isomorphism between A and B. ... f b = f a } H #> a" proof fix b assume b: "b { b carrier G.

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: inf (infixl "" 70) and sup (infixl "" 65) and Inf (" _" [900] 900) and Sup (" _" [900] 900) syntax "_INF1" :: "pttrns 'b 'b" ("(3_./ _)" [0, 10] 10) "_INF" :: "pttrn 'a set 'b 'b" ("(3__./ _)" ... 0, 0, 10] 10) "_SUP1" :: "pttrns 'b 'b" ("(3_./ _)" [0, 10

3 Jun 2024: Thus for large B we can expect to see thatFB(x) F(x). ... B <- length(f_stat). theoretical_quantiles <- qf((1:B) / (B 1), df1, df2).

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: 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: 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: 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 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 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: 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: 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: 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: 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: 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 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: 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: 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: 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: 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. 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: 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 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 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: 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.