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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.