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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 =