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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: 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: 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: 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: 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: 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: 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: 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: 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: 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.