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23 May 2024: Q(x))› apply (eruleTHEN mp] | assumption | ruleerule (1)done subsection ‹Equality rules› lemma sym: ‹a = b b = a› apply (erule subst) apply (rule refl) done lemma trans: ‹⟦a = b; b = c⟧ ... for the equality predicate!› lemma eq_cong:

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: combine_options (f x) (m1 x) (m2 x))" unfolding combine_options_def by transfer_prover lemma combine_parametric: "((B ===> B ===> B) ===> (AB) ===> (AB) ===> (AB)) (λf m1 m2 x. ... lift_definition combine :: "('b 'b 'b) ('a,'b) mapping ('a,'b) mapping

23 May 2024: thesis using rel_interior_sing by auto next case False obtain B where B: "independent B B S SBBS" using basis_exists[of S] by metis then have "Busing‹S› ... empty by auto have "BB" using subspace_span[of B] subspace_0[of "span B"] span_superset by

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: card the session key and various verifiers) | SR_U7: "⟦ evs7 srb; Nonce Nbevs7;B); BKNb,pairK(A,B)); Key Kevs7; Inputs B (Card B) ⦃Agent A, Nonce Na⦄evs7⟧B) B ... sesK_authentic) done lemma Confidentiality_B: "⟦B) B ⦃Nonce Nb, Agent A, Key K,

23 May 2024: ng ((n2)) sums x" using sums_if'[OF ‹g sums x›]. have if_eq: "B T E. ... xa <.< b} ya <.< bf x n - f y nL nx - y" shows "DERIV (λ x.

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: subset.chain 𝒜 𝒞 = (𝒞 𝒜 (X𝒞. Y𝒞. X Y Y X))" by (auto simp: subset.chain_def) lemma subset_chain_insert: "subset.chain 𝒜 (insert B ) B 𝒜 (X. ... S" and 2: "AC. BC. A B B A" for C proof - let?

23 May 2024: definition length :: "'awhere "length dxsdxs)" qualified definition fold :: "('a 'b 'b) 'a dlist 'b 'b" where "fold f dxsf (list_of_dlist dxs)" qualified definition foldr :: "('a 'b 'b) 'a dlist ... b 'b" where "foldr f dxsf (list_of_dlist dxs)" end

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: c)(Split(d))" definition List_rec :: "['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b" where "List_rec M c dg. ... ML ) list_rec :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b" where "list_rec l c dl) c (%x y r.

23 May 2024: iff: "principal a = principal b a b b a" unfolding po_eq_conv [where 'a='b] principal_below_iff. ... principal a = Abs {b. b a}" assumes countable: "f::'af" shows "ideal_completion r principal Rep" proof interpret type_definition Rep Abs "{S.

23 May 2024: assumes "bound0 p" shows "Ifm bbs (b # bs) pbbs (b' # bs) p" usingwhere b="b" and bs="bs" and b'="b'"] by (induct p rule:simp add: gr0_conv_Suc) fun ... I': "numbound0 a Inum (b#bs) (numsubst0 a t)b'#bs) a)#bs) t" by (induct t rule:simp:where b="b" and b'

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: locale semigroup = fixes f :: "'a 'a 'a" (infixl "" 70) assumes assoc [ac_simps]: "a b c = a (b c)" locale abel_semigroup = semigroup assumes commute [ac_simps]: "a b = b a" ... standard (fact add_assoc) declareend hide_fact add_assoc class

23 May 2024: ASCII) "_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10) syntax "_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3__. _)" ... b" ("(3PROD _<-_. _)" [0, 51, 10] 10) syntax "_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3__.

23 May 2024: assumes group_hom: "group_hom G (BijGroup E) φ" definition orbit :: "[_, 'a 'b 'b, 'b] 'b set" where "orbit G φ x = {(φ g) x | g. ... x E}" definition stabilizer :: "[_, 'a 'b 'b, 'b] 'a set" where "stabilizer G φ x = {g carrier G. (

23 May 2024: A a u B b) apply (case_tac "A = B") apply simp apply simp apply (apply clarify apply (rename_tac A a u B b) apply (case_tac "A = B") apply ... rotate_tac -1) apply simp apply (rotate_tac -1) apply (case_tac "B = C") apply simp apply (rotate_tac -1) apply

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. 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: 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: B; u A; v B⟧ A B" by (lemma insert_lepoll_cong: assumes "A B" "b B" shows "insert a Ab B" proof - obtain f where f: "inj_on f A" ... where "inj_on f A" "f A B" using assms by (then obtain b where b: "b B" "b f A" by auto show?

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: f aa" definition zero_l_neutral :: "('a::zero 'b 'b) bool" where "zero_l_neutral f == a. ... fmul fadd A B" by simp qed definition r_distributive :: "('a 'b 'b) ('b 'b 'b) bool" where "r_distributive fmul fadd == a u v.

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: b1 'a 'b2 ('a, 'b1, 'b2) F2 = unit 'b1 'b2 ) locale loc = fixes c :: 'a and d :: 'a assumes "c d" begin datatype (discs_sels) 'b I1 = I11 'b "'b I1" | ... I12 'b "'b I2" and 'b I2 = I21 | I22 "'b I1" "'b I2" datatype (discs_sels) 'b tree = TEmpty | TNode

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: 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: 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: 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: thesis unfolding mset_less_eq_Bag0 by auto qed declare inter_mset_def [code] declare union_mset_def [code] declare mset.simps [code] fun fold_impl :: "('a nat 'b 'b) 'b ... ab" where "fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)" |

23 May 2024: auto simp add: set_plus_def) lemma set_plus_elim: assumes "x A B" obtains a b where "x = a b" and "a A" and "b B" using assms unfolding ... b" and "a A" and "b B" using assms unfolding set_times_def by fast lemma set_times_intro2 [intro!]: "b C a b a o C"

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.

23 May 2024: g = "gcd (a b' b a') (b b')" have gz: "?gusing False by simp show? ... g = "gcd (a a') (b b')" have gz: "?gusing neq by simp fromwhere?

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: 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: syntax begin lemma transfer_rule_numeral: ‹(R)› if [transfer_rule]: ‹R› ‹R› ‹(R ===> R ===> R)› for R :: ‹'a::{b::{› proof - have "(R) (λk. ... b)k) 0)› using numeral_add_unfold_funpow [where? 'a = 'b, of _ 0] by (simp add:

23 May 2024: a, 's2)where "fair_implements C ACACACA" lemma implements_trans: "A =<| B B =<| C A =<| C" by (auto simp add: ioa_implements_def) subsection ‹Modules› subsubsection ‹Execution, schedule and trace modules› ... mk_trace›› (alternative

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_.