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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: 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"