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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: 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: 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: 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: 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 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 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: 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: 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: 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: 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,

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: finite b b B}. B') = k" by auto then obtain k where "kaK. ... finite b b B}. B')K" by (intro exI[of _ "kK)"]) auto next case (Basis S) then show?

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: notation (ASCII) not_below (infix "<<" 50) lemma below_eq_trans: "a b b = c a c" by (lemma eq_below_trans: "a = b b c a c" by (end class po = ... S <<| x)" end syntax (ASCII) "_BLub" :: "[pttrn, 'a set, 'b] 'b" ("(3LUB _:_./ _)" [0,0, 10] 10) syntax

23 May 2024: UNIV)" bynote B' = ‹B B'› withshow? case by auto next case CondOr thus? ... UNIV)" by (elimsimp add: inj_term_simps) ( inj_term_simps needed to handle wt (defined without ⟨⟩) ) note B' = ‹B B'› withshow?

23 May 2024: A x y Q) (a b. B (a :: 'a) (b :: 'a) Q) (x y. ... A x B" "x. A x" assume Y: "A B" have "B B B B B B" apply (intro conjI) apply (in H[OF X(2)]:"x.

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

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: 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: 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: fixes f::"_ _ :: {assumes [M f" "AM" "BM" and null: "(A - B) (B - A)M" shows "(x A. ... a (A - B) (B - A)" usingby blast then show "AE x in M.

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: 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: 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; B = C⟧ A = C" and trans_elem: "a b c A. ... A type; b : B⟧ inr(b) : AB" and PlusI_inrL: "b d A B.

23 May 2024: session key and various verifiers) | SR7: "⟦ evs7 sr; Nonce Nbevs7;B); BKNb,pairK(A,B)); Key Kevs7; Inputs B (Card B) ⦃Agent A, Nonce Na⦄evs7⟧B) B ⦃Nonce Nb, Key ... lemma Confidentiality_B: "⟦B) B ⦃Nonce Nb, Key K, Certificate,A,B)) (Nonce

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: bvusing ‹T› by auto have [simp]: " T {v. bvusing ‹norm b› ‹b T› by auto define f where "f λx. ... R. ba} (- U)for a::'a and b::'b by force have : "r.

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: apply (fast intro!: FiniteFun.intros) done lemma FiniteFun_subset: "⟦c<=b; b A-||>B⟧ c A-||>B" by (blast intro: FiniteFun_subset_lemma) ( Some further results by Sidi O. ... f A->B f A-||>B" apply (erule Fin.induct) apply (simp add:apply (case_tac "a

23 May 2024: case by (auto intro!: exI[of _ "min a b"]) next case UN then show? ... Q b b < a" and P: "eventually P at_top" shows "filterlim fa)" proof - from P obtain x where x: "y.

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: a = x) = a" by blast subsection‹Conditional Terms: ‹if-then-else›› lemma if_true [simp]: "(a else b) = a" by (unfoldlemma if_false [simp]: "(a else b) = b" by (unfoldNever use ... lemma misc_simps [simp]: "A = A" "AA" "AAAAA" "b,A)) = bA)" "({b})

23 May 2024: thesis. qed lemma sigma_sets_UNION: "countable B (b. b B bX A) BX A" using from_nat_into [of B] range_from_nat_into [of B] sigma_sets.Union [of ... a A aM B" assumes B: "b. b B bM A" shows "sigma_sets M AM B" proof (introfix a assume "aM A" fromshow "aM B

23 May 2024: lemma SigmaD1: "(a, b)A B a A" by blast lemma SigmaD2: "(a, b)A B b B a" by blast lemma SigmaE2: "(a, b)A B (a A b B ... j). (j, i)) (A B) = B A" by (auto simp add: set_eq_iff) lemma image_split_eq_Sigma: "(λx.

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: a = "aλb. (a b / (b b)). R. ... a b / (b b)). R.

23 May 2024: finally have "b<": "b r (a f b) f c". fromhave "c<": "c r (a f b) f c". ... also fromhave "… r a f …". finally have "b<": "b r a f (b f c)".

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: 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"