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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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: b 'b) ('a 'b 'b)" where F_strict: "F(and K: "K = (λh x y. ... a'b'b)") apply simp apply (simp split: COND_cases_iff) done end.

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: 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: by blast lemma QSigmaD2: "<a;bA,B) b B(a)" by blast lemma QSigma_cong: "⟦A=A'; x. ... lemma QInrD: "QInr(b): A<>B b B" by blast ( <> is itself injective.

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: 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: f (A <> B) (B <> A)" unfoldingby force thus? thesis using card_of_ordIso by blast qed lemma card_of_Plus_assoc: fixes A :: "'a set" and B :: "'b set" and C :: ... thesis using inj_on_iff_surj[of B "B A"]by blast qed (simp add: card_of_empty) lemma card_of

23 May 2024: aR" proof assume "carrier Ra" hence "a" by auto then obtain b where "bR" "a b" "b a" unfolding cgenideal_def using m_comm[OF assms] by auto thus "aR" using ... b Iφ a φ b" using phi_img_def by blast have "Ia" proof (assume "Ia" then obtain b where 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: i) b = f b" by (simp add: inner_sum_leftlemma (in euclidean_space) euclidean_eqI: assumes b: "b. ... b) = f" by (simp add: euclidean_representation_sum) lemma euclidean_isCont: assumes "b.

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: inverse (X n))" proof (fix r :: rat assume "r" obtain b i where b: "b" and i: "ni. ... r› b) show "rb (b r b)b" using b by simp qed obtain j where j: "mj.

23 May 2024: based on the three cases of ordinals ) lemma transrec2_0 [simp]: "a,b) = a" by (THEN def_transrec, THEN trans], simp) lemma transrec2_succ [simp]: "i),a,b) = b(i, ... m),a,b) = b(m, rec(m,a,b))" unfolding rec_def apply (done lemma rec_type: "⟦na C(0);

23 May 2024: LeafI: "Leaf(a) sexp" | NumbI: "Numb(i) sexp" | SconsI: "[| M sexp; N sexp |] ==> Scons M N sexp" definition sexp_case :: "['a=>'b, nat=>'b, ['a item, 'a item]=>'b, 'a item] => ... M, Scons M N), (N, Scons M N)})" definition sexp_rec :: "['a item, 'a=>'b,

23 May 2024: lift_definition bulkload :: "('a::b) list ('a, 'b) rbt" is "rbt_bulkload". lift_definition map_entry :: "'a ('b 'b) ('a::linorder, 'b) rbt ('a, 'b) rbt" is rbt_map_entry by ... lift_definition combine_with_key :: "('a 'b 'b 'b) ('a::linorder, 'b) rbt ('a,

30 Apr 2024: Alternatively, you can book a hotel or B&B room via the Visit Cambridge website or visit the Cambridge Rooms site that includes accommodation at other Colleges.

23 May 2024: R. b" using assms by (auto simp: in_segment) have xb: "x b < b b" usingby auto assume "norm cc x" then have "c c (c - x) (c - x)" by (simp ... b = b a=b uby (smt (verit, best)lemma segment_degen_0: fixes a :: "'a :: real_vector" shows "(u).

23 May 2024: a'b'b" assume "module_on S s" then interpret module_on S s. ... on S2 s2 for S1 :: "'b::and S2 :: "'c::and s1 :: "'a::comm_ring_1 'b 'b" (infixr "a" 75) and s2 :: "'a::comm_ring_1 'c 'c" (infixr "b" 75)

23 May 2024: l i<.<r i delta i})" proof (show "a'.b}" by (show "i. ... D (box l u) = (b?B. (u - l) b)" by (auto simp:symmetric]symmetric] intro!: prod.cong) qed simp lemma lborel_affine: fixes t :: "'a::euclidean_space" shows "cλx.

23 May 2024: m adefinition ran :: "('a 'b) 'b set" where "ran m = {b. ... S'" hence 1: "dom m = A" by force hence 2: "ran m B" using ‹m?

23 May 2024: SesKeyevs); evs⟧ K=K' B=B' T=T'" apply (erule rev_mp) apply (erule rev_mp) apply (erule rev_mp) apply (eruleapply (frule_tac [7] Says_ticket_parts) apply (frule_tac ... lemma unique_CryptKey: "⟦B) ⦃Agent A, Agent B, Key SesKey, T⦄evs);B')

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 May 2024: pmult q b) r) r' =? pminus (?padd (?pmult q b') r') r'" using ldiv and that unfolding long_divides_def by auto hence eq: "?padd (?pmult q (?pminus b b')) (?pminus ... r r')R" using in_carrier by algebra have "b = b'" proof (assume "b b'" hence pminus:

23 May 2024: subsection ‹Simple types› schematic_goal "A: AA :? T" by (schematic_goal "A: λa:A. a :? T" by (schematic_goal "A: B: b:B λx:A. ... B :? T" by (schematic_goal (in Lomega) "B: b:B (λy:B.

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: b' B (b, b') r" proof fix b' show "b' B (b, b') r" proof assume As: "b' B" hence : "br b'r" usingby auto fromhave "b' = b (b',b) r" ... by auto moreover have "b' = b (b, b') r" usingby (auto simp add: refl_on_def) moreover have "b' b (b',b) r (b,b') r"

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: b" "(b, a) r" proof - from that have "br a" unfolding underS_def by blast with have "br b" by blast then show? ... b, a) r phi b) phi a" define chi where "chi b (b, a) r phi b" for b with have "wf (R a)" by auto then have "(bc.

23 May 2024: rArB" usingby blast have "(A < B)B A)" usingof r A B] by blast also have "…r Br A)" usingby blast also have "…r Ar B)" usingby blast finally show? ... 1 have "(a',a')r f (b',b')r f" unfolding dir_image_def by auto } thus?

23 May 2024: s" inductive_set setr :: "'a 'b 'b set" for s :: "'a 'b" where "sx x setr s" lemma sum_set_defs[code]: "λx. ... set fsts :: "'a 'b 'a set" for p :: "'a 'b" where "fst p fsts p" inductive_set snds :: "'a 'b 'b set" for p :: "'a 'b" where "snd p snds p"

23 May 2024: b B (b,a) r)⟧ (supr B, a) r" by(auto simp add:lemma equals_supr_Above: assumes "a Above B" " a'. ... b. b B a b (b,a) r" and MINIM: " a'. ⟦a'r; b.

23 May 2024: a' s a a'" assumes b: "b t" "b'. b' t b b'" assumes eq: "s - {a} = t - {b}" shows "s = t" proof cases assume "nwithshow? ... a' s a' a" assumes b: "b t" "b'. b' t b' b" assumes eq: "s - {a} = t - {b}" shows "s = t" proof (cases n) assume "nwithshow?

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: simp next assume "euclidean_size bshow "bproof (assume "bwith mod_size_less have "euclidean_size (b mod b)b". ... b dvd f a}› with ‹finite A› have ‹finite B› and ‹a B b dvd f a› for a by simp_all then have ‹(aB.

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