hcju/isabelleps huggingface dataset:
set_step2: fixes s :: "'b list"   and r :: "'b"   and mf :: "nat"   and sws :: "nat list" shows "set (step s r (mf, sws)) = set s"
in_Keys: fixes s :: "'d"   and F :: "('d \<Rightarrow>\<^sub>0 'e) set" shows "(s \<in> Keys F) = (\<exists>f\<in>F. s \<in> keys f)"
LtK_synth_eq: fixes K :: "ltkey"   and H :: "msg set" shows "(LtK K \<in> synth H) = (LtK K \<in> H)"
hyps: assumes "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<leadsto>[Rel] \<lparr>\<nu>*xvec\<rparr>Q"
CartProd: fixes x :: "ZF"   and A :: "ZF"   and B :: "ZF" shows "Elem x (CartProd A B) = (\<exists>a b. Elem a A \<and> Elem b B \<and> x = Opair a b)"
IH_2: fixes k :: "('fun, 'var) Term.term" assumes "k \<in> set K"   and "M \<subseteq> M'" shows "\<langle>M';P\<rangle> \<turnstile>\<^sub>r k"
det_h: shows "det (M_mat_R signs subsets) \<noteq> 0"
conjugate_exponent_real_1: fixes p :: "real" assumes "1 < p" shows "1 / p + 1 / conjugate_exponent p = 1"
pmf_transition_of_st_mat: fixes a :: "'i"   and i :: "'i" shows "pmf (transition_of_st_mat A a) i = st_mat A $h i $h a"
set_less_eq_aux_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<sqsubseteq>' B"   and "B \<sqsubseteq>' C" shows "A \<sqsubseteq>' C"
ide_hcomp: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.ide \<nu>"   and "D.ide \<mu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.ide (\<nu> \<star>\<^sub>D \<mu>)"
sinit_len: fixes i :: "nat"   and xs :: "'b seq" shows "length (sinit i xs) = i"
f_Exec_Stream_Acc_LocalState_nth_eq_i_nth: fixes k :: "nat"   and n :: "nat"   and n' :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" assumes "0 < k"   and "n < n'" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun (input \<Down> n') c ! n = i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c n"
list_list_eqI: fixes xs :: "('b \<times> 'c list) list"   and ys :: "('b \<times> 'c list) list" assumes "\<forall>(uu_, x)\<in>set xs. x \<noteq> []"   and "\<forall>(uu_, y)\<in>set ys. y \<noteq> []"   and "map (\<lambda>(uu_, x). hd x) xs = map (\<lambda>(uu_, x). hd x) ys"   and "map \<pi> xs = map \<pi> ys" shows "xs = ys"
Riemann_localization_integral_range: fixes f :: "real \<Rightarrow> real"   and d :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "0 < d"   and "d \<le> pi" shows "(\<lambda>n. (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f x) - (LINT x|lebesgue_on {- d..d}. Dirichlet_kernel n x * f x)) \<longlonglongrightarrow> 0"
step_3_a_dims_1: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_3_a i j A) = dim_row A"
all_inferred_clause_sets_are_finite: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set"   and N :: "nat" assumes "derived_clauses_are_finite R"   and "all_fulfill finite S" shows "all_fulfill finite (inferred_clause_sets R S N)"
edge_pred_def: fixes M :: "'e pre_map" shows "edge_pred M = inv (edge_succ M)"
ilength_def: shows "ilength \<equiv> map_fun Rep_fin_string id length"
eventually_subadditive_ineq: fixes u :: "nat \<Rightarrow> real"   and N0 :: "nat"   and e :: "real"   and n :: "nat" assumes "eventually_subadditive u N0"   and "0 < e"   and "N0 < n" shows "\<exists>N>N0. \<forall>m\<ge>N. u m / real m < u n / real n + e"
pgen_lossless_gpvI: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1" shows "pgen_lossless_gpv fail \<I> gpv"
tendsto_def: fixes "open" :: "'d set \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and l :: "'d"   and F :: "'e filter" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
negation_constant_def: fixes a :: "'a" shows "negation_constant a \<equiv> \<forall>x. -' x = inv_resr x a \<and> - x = inv_resl a x"
rec_mod_def: shows "rec_mod = Cn (Suc (Suc 0)) rec_minus [recf.id (Suc (Suc 0)) 0, Cn (Suc (Suc 0)) rec_mult [rec_quo, recf.id (Suc (Suc 0)) (Suc 0)]]"
le_minus_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (uminus b) = less_eq b (uminus a)"
assms_2: shows "(state0, state) \<in> transitionRelation F0 decisionVars"
FunctorId2: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and X :: "'a" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "F @@ X \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
cltn2_id_def: shows "cltn2_id \<equiv> cltn2_abs (mat 1)"
cvalid_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A \<equiv> (\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P Call p Q,A) \<longrightarrow> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
sameDom_sym: fixes inp :: "'b \<Rightarrow> 'c option"   and inp' :: "'b \<Rightarrow> 'd option" shows "sameDom inp inp' = sameDom inp' inp"
that: fixes a :: "'a" assumes "R = {x. a \<sim> x}" shows "thesis"
Map_reflects_hhom: fixes a :: "'a"   and b :: "'a"   and g :: "('a E.term, 'a) arr" assumes "B.obj a"   and "B.obj b"   and "ide g"   and "\<guillemotleft>g : UP.map\<^sub>0 a \<rightarrow> UP.map\<^sub>0 b\<guillemotright>" shows "\<guillemotleft>Map g : a \<rightarrow>\<^sub>B b\<guillemotright>"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>acc xs. (\<And>x. \<lbrakk>xs \<noteq> []; x \<in> set xs\<rbrakk> \<Longrightarrow> P (x # acc) (remove1 x xs)) \<Longrightarrow> P acc xs" shows "P a0 a1"
subset_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (\<forall>t. s2p A t \<longrightarrow> s2p B t)"
prod_dfs_bij: fixes x :: "nat \<times> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "prod_is_node A B x" shows "(fst (prod_dfs A B x) ! i ! j = Some k \<and> dfa_is_node A i \<and> dfa_is_node B j) = (k < length (snd (prod_dfs A B x)) \<and> snd (prod_dfs A B x) ! k = (i, j))"
t_Node_2: fixes p :: "ref" assumes "Dag p low high rt"   and "st \<le> rt" shows "\<exists>stp. Dag stp low high st"
conj_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x AND y) AND z = x AND y AND z"
le_nofailI: fixes M' :: "'a nres"   and M :: "'a nres" assumes "nofail M' \<Longrightarrow> M \<le> M'" shows "M \<le> M'"
Collect_restrict: fixes X :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "{x \<in> X. P x} \<subseteq> X"
rpo_optim: assumes "\<And>f. ext_snoc len_lexext" shows "rpo_optim ground_head_var local.gt_sym (\<lambda>f. len_lexext) arity_sym arity_var"
sum_squares_gt_zero_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (plus (times x x) (times y y)) = (x \<noteq> zero \<or> y \<noteq> zero)"
lepoll_def: fixes A :: "'a set"   and B :: "'b set" shows "A \<lesssim> B \<equiv> \<exists>f. inj_on f A \<and> f ` A \<subseteq> B"
abs_eq: fixes xa :: "'a \<Rightarrow>\<^sub>0 'b"   and x :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pp (PP xa) (PP x) = lex_pm xa x"
inverse_unique: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "Groups.group f z inverse"   and "f a b = z" shows "inverse a = b"
simps_1: fixes a :: "'a" shows "BVP ($\<alpha> a) = UNIV"
sum_representation_eq: fixes basis :: "'d set"   and v :: "'d"   and B :: "'d set" assumes "independent basis"   and "v \<in> span basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. representation basis v b *\<^sub>R b) = v"
simps: fixes g :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'a \<times> 'b)"   and s :: "'b" shows "local.filter_next g s = (let (x, s') = snd g s in if P x then (x, s') else local.filter_next g s')"
rel_funI: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
ann_matches_weaken: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and P :: "('e, 'd, 'f) ann"   and P' :: "'e set"   and c :: "('e, 'd, 'f) com" assumes "ann_matches \<Gamma> \<Theta> (weaken_pre P P') c" shows "ann_matches \<Gamma> \<Theta> P c"
close_ab: fixes m :: "nat"   and n :: "nat" assumes "odd m" shows "\<bar>a (real m / 2 ^ n) - b (real m / 2 ^ n)\<bar> \<le> 2 / 2 ^ n"
add_edge_valid: fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "valid_graph g" shows "valid_graph (add_edge v e v' g)"
E_1: fixes n :: "nat" shows "E n \<in> sets M"
fds_converges_cong: fixes f :: "'b fds"   and g :: "'b fds"   and s :: "'b"   and s' :: "'b" assumes "\<forall>\<^sub>F n in sequentially. fds_nth f n = fds_nth g n"   and "s = s'" shows "fds_converges f s = fds_converges g s'"
Nil_change_css_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and ass :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and ass' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and xss :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> ([], ass @ css, s) \<rightarrow> (cs', ass' @ css, t)"   and "ass \<noteq> []" shows "\<Gamma>\<turnstile> ([], ass @ xss, s) \<rightarrow> (cs', ass' @ xss, t)"
pcr_fset_def: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_fset T \<equiv> list_all2 T OO cr_fset"
matrix_sum_positive: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and d :: "nat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> positive (f k)" shows "positive (matrix_sum d f n)"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(g \<circ> f \<down> A) (addA x y) = addC ((g \<circ> f \<down> A) x) ((g \<circ> f \<down> A) y)"
that: fixes c :: "(_) object_ptr list" assumes "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c"   and "set c \<subseteq> set sc" shows "thesis"
map_transform_Cons: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>"   and x :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "map_transform t ae ((x, e) # \<Gamma>) = (x, lift_transform t (ae x) e) # map_transform t ae \<Gamma>"
maximal_ends_on_deadend: fixes P :: "'a llist" assumes "G'.maximal_path P"   and "lfinite P"   and "\<not> lnull P" shows "G'.deadend (llast P)"
abs_summable_on_Times_swap: fixes f :: "'a \<times> 'b \<Rightarrow> 'c"   and A :: "'a set"   and B :: "'b set" shows "(f abs_summable_on A \<times> B) = ((\<lambda>(x, y). f (y, x)) abs_summable_on B \<times> A)"
boxInit_stp: fixes F :: "state \<Rightarrow> bool" shows "\<box>F \<equiv> \<box>Init F"
VcompNml_Cod_Nml: fixes t :: "'a local.term" assumes "Nml t" shows "Cod t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
inverse: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)"   and "f \<in> O[F](g)" shows "(\<lambda>x. inverse (g x)) \<in> O[F](\<lambda>x. inverse (f x))"
polypow_normh: fixes p :: "'b poly"   and n :: "nat"   and k :: "nat" assumes "isnpolyh p n" shows "isnpolyh (p ^\<^sub>p k) n"
Abs_Assertion_inject: fixes x :: "'c"   and y :: "'c" assumes "x \<in> assertion"   and "y \<in> assertion" shows "(Abs_Assertion x = Abs_Assertion y) = (x = y)"
th3_6_1: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and c :: "(real, 'b) vec"   and d :: "(real, 'b) vec" assumes "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> a c d" shows "B\<^sub>\<real> b c d"
holo: fixes b :: "real"   and c :: "real"   and e :: "real" assumes "a \<le> b"   and "c \<le> x" shows "(\<lambda>z. integral (cbox b c) (\<lambda>t. complex_of_real (bernpoly n (t - e)) * f z t)) holomorphic_on U"
wellfounded_relation_axioms: shows "wellfounded_relation scut\<inverse>\<inverse>"
interval_integrable_isCont: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "\<And>x. \<lbrakk>min a b \<le> x; x \<le> max a b\<rbrakk> \<Longrightarrow> isCont f x" shows "interval_lebesgue_integrable lborel (ereal a) (ereal b) f"
deterministic_def: fixes P :: "'e process" shows "deterministic P \<equiv> \<forall>xs\<in>traces P. \<forall>X. (X \<in> refusals P xs) = (X \<inter> next_events P xs = {})"
orderE: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" obtains "a = max a b"
monom_mult_in_pmdl: fixes b :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c"   and t :: "'b" assumes "b \<in> B" shows "MPoly_Type_Class.punit.monom_mult c t b \<in> MPoly_Type_Class.punit.pmdl B"
no_cycles_implies_unique_paths: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "\<And>xs. \<not> Graph.cycle G xs"   and "Graph.connected G v w" shows "\<exists>!xs. v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
undefined: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" assumes "rh a \<noteq> rh b"   and "a \<in> A" shows "m a b = None"
impr_impl_times: fixes a :: "'a N"   and b :: "'a N" shows "(a l\<rightarrow> b) * a = a * (a r\<rightarrow> b)"
step_invariant_weakenE: fixes A :: "('d, 'e) automaton"   and PI :: "'e \<Rightarrow> bool"   and P :: "'d \<times> 'e \<times> 'd \<Rightarrow> bool"   and Q :: "'d \<times> 'e \<times> 'd \<Rightarrow> bool"   and QI :: "'e \<Rightarrow> bool" assumes "A \<TTurnstile>\<^sub>A (PI \<rightarrow>) P"   and "\<And>t. P t \<Longrightarrow> Q t"   and "\<And>a. QI a \<Longrightarrow> PI a" shows "A \<TTurnstile>\<^sub>A (QI \<rightarrow>) Q"
ca_lf_rep_def: shows "ca_lf_rep \<equiv> \<lambda>(rm, rp). {\<bottom>} \<times> UNIV \<union> {(d, P) |d P. (\<exists>n. d = ValN\<cdot>n \<and> unProg P \<Down> DBNum n) \<or> d = ValTT \<and> unProg P \<Down> DBtt \<or> d = ValFF \<and> unProg P \<Down> DBff \<or> (\<exists>f M. d = ValF\<cdot>f \<and> unProg P \<Down> DBAbsN M \<and> (\<forall>(x, X)\<in>unsynlr (undual rm). (f\<cdot>x, mkProg (M<unProg X/0>)) \<in> unsynlr rp)) \<or> (\<exists>f M. d = ValF\<cdot>f \<and> unProg P \<Down> DBAbsV M \<and> f\<cdot>\<bottom> = \<bottom> \<and> (\<forall>(x, X)\<in>unsynlr (undual rm). \<forall>V. unProg X \<Down> V \<longrightarrow> (f\<cdot>x, mkProg (M<V/0>)) \<in> unsynlr rp))}"
intro: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and labelsIn :: "'a \<Rightarrow> 'b \<Rightarrow> 'e"   and labelsOut :: "'a \<Rightarrow> 'c \<Rightarrow> 'e"   and freshenLC :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and renameLCs :: "('f \<Rightarrow> 'f) \<Rightarrow> 'e \<Rightarrow> 'e"   and lconsts :: "'e \<Rightarrow> 'f set"   and closed :: "'e \<Rightarrow> bool"   and subst :: "'g \<Rightarrow> 'e \<Rightarrow> 'e"   and subst_lconsts :: "'g \<Rightarrow> 'f set"   and subst_renameLCs :: "('f \<Rightarrow> 'f) \<Rightarrow> 'g \<Rightarrow> 'g"   and anyP :: "'e"   and vidx :: "'d \<Rightarrow> nat"   and inst :: "'d \<Rightarrow> 'g" assumes "Well_Shaped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"   and "Solution inPorts outPorts nodeOf hyps vertices labelsIn labelsOut freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx inst edges" shows "Proof_Graph nodes inPorts outPorts vertices nodeOf edges hyps labelsIn labelsOut freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx inst"
sorted_iff_nth_mono_less: fixes xs :: "'a list" shows "sorted xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le> xs ! j)"
coprime_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a b = (\<forall>c. dvd.dvd times c a \<longrightarrow> dvd.dvd times c b \<longrightarrow> dvd.dvd times c one)"
irreducible_mult_unit_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and p :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "comm_semiring_1.irreducible times one zero (times a p) = comm_semiring_1.irreducible times one zero p"
coms_choice_inv: fixes G :: "diagram"   and H :: "diagram"   and c :: "command" assumes "coms_com (Choose_dia G H) c" obtains ca :: "command" and d :: "command" where "c = Choose ca d" and "coms_dia G ca" and "coms_dia H d"
eq: fixes i :: "nat" shows "{..n} \<inter> {i} = (if i \<le> n then {i} else {})"
INF_lower: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "i \<in> A" shows "\<Sqinter>(f ` A) \<le> f i"
deg_le_sect_Suc: fixes X :: "'b set"   and d :: "nat" shows "deg_le_sect X (Suc d) = deg_le_sect X d \<union> deg_sect X (Suc d)"
Initial: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and qs :: "('b, 'c) fm list"   and i :: "'c"   and v' :: "nat"   and k :: "'c" assumes "branch !. v = Some (qs, i)"   and "qs !. v' = Some (\<^bold>\<diamond> Nom k)" shows "descendants k i branch {(v, v')}"
mem_imp_VWO: fixes x :: "V"   and y :: "V" assumes "x \<in> elts y" shows "(x, y) \<in> VWO"
ucast_less: assumes "LENGTH('b) < LENGTH('a)" shows "UCAST('b \<rightarrow> 'a) x < 2 ^ LENGTH('b)"
fderiv: fixes x :: "'d" assumes "x \<in> Csafe" shows "(ode has_derivative blinfun_apply (ode_d1 x)) (at x)"
is_processT7: fixes P :: "'a process" shows "\<forall>s t. s \<in> D P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> D P"
squarefree_decompose: fixes n :: "'b" shows "n = squarefree_part n * (square_part n)\<^sup>2"
prime_elem_mult_unit_left: fixes a :: "'a"   and p :: "'a" assumes "is_unit a" shows "prime_elem (a * p) = prime_elem p"
typeII_cube_line_integral_exists_boundary: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain_typeII. line_integral_exists F {j} \<gamma>"
BfunI: fixes f :: "'a \<Rightarrow> 'b"   and K :: "real"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> K" shows "Bfun f F"
span_superset: fixes S :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "S \<subseteq> local.punit.pmdl S"
antisym: fixes x :: "'a\<^sub>m"   and y :: "'a\<^sub>m" assumes "bo x y"   and "bo y x" shows "x = y"
l_zero: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<zero> \<oplus> x = x"
map_ms_aux_MSLNil: fixes f :: "'b \<Rightarrow> 'a" shows "map_ms_aux f MSLNil = MSLNil"
pseudo_mod_def: fixes f :: "'c poly"   and g :: "'c poly" shows "pseudo_mod f g = snd (pseudo_divmod f g)"
ha: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"
is_pkt_def: fixes \<xi> :: "'a state_scheme" shows "is_pkt \<xi> \<equiv> case msg \<xi> of Pkt num' sid' \<Rightarrow> {\<xi>\<lparr>num := num', sid := sid'\<rparr>} | Newpkt nat1 nat2 \<Rightarrow> {}"
scaleC_scaleC: fixes a :: "complex"   and b :: "complex"   and x :: "'a" shows "a *\<^sub>C b *\<^sub>C x = (a * b) *\<^sub>C x"
offs_length: fixes ns :: "nat list"   and i :: "nat" shows "length (offs ns i) = length ns"
faceSquanderLowerBound_def: fixes g :: "graph" shows "faceSquanderLowerBound g \<equiv> \<Sum>\<^bsub>f\<in>finals g\<^esub> \<d> |vertices f|"
HFinite_add: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "x + y \<in> HFinite"
match_vars: fixes p :: "term"   and t :: "'a"   and env :: "(name, 'a) fmap" assumes "match p t = Some env" shows "fmpred (\<lambda>_ u. frees u |\<subseteq>| frees t) env"
A3implT2_global: shows "\<lfloor>(\<lambda>w. \<forall>x xa. (\<forall>xa. (x xa \<^bold>\<rightarrow> \<P> xa) w) \<and> (\<^bold>\<box>(\<lambda>v. \<forall>xb. xa xb v = (\<forall>xa. (x xa \<^bold>\<rightarrow> xa xb) v))) w \<longrightarrow> \<P> xa w)\<rfloor> \<longrightarrow> \<lfloor>\<P> (\<lambda>x w. \<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w)\<rfloor>"
intros_1: fixes a :: "'g"   and r :: "('g \<times> 'g) set" shows "(a, a) \<in> r\<^sup>*"
assms_1: shows "wf\<^sub>l\<^sub>s\<^sub>t X A"
locate_a_2_locate_a: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and xs :: "cell list"   and ires :: "cell list" assumes "inv_locate_a (as, am) (q, aaa, Bk # xs) ires" shows "inv_locate_a (as, am) (q, aaa, Oc # xs) ires"
reachableHD: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>\<^sup>*\<^bsub>H\<^esub> y" shows "proj_verts_H x \<rightarrow>\<^sup>*\<^bsub>G\<^esub> proj_verts_H y"
imp: shows "{x \<in> space M. P x \<and> \<not> Q x} \<subseteq> B"
is_chain: shows "wf_infpath (buildchain (\<lambda>ds. if infinite (subtree T (ds @ [Left])) then ds @ [Left] else ds @ [Right]))"
alpha_par_prod: fixes x :: "'a"   and y :: "'a" shows "\<nu> (x \<parallel> \<nu> y) = x \<parallel> \<nu> y"
top_on_step': fixes C :: "'av st option acom" assumes "top_on\<^sub>c C (- vars C)" shows "top_on\<^sub>c (step' \<top> C) (- vars C)"
Says_to_Gets: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X # evs \<in> set_cr" shows "Gets B X # Says A B X # evs \<in> set_cr"
AE_return: fixes x :: "'c"   and M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> space M"   and "Measurable.pred M P" shows "almost_everywhere (return M x) P = P x"
a2_eq': fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> apd_d y) = ad (x \<cdot> y)"
segment_degen_0: fixes u :: "real"   and a :: "'a"   and b :: "'a" shows "((1 - u) *\<^sub>R a + u *\<^sub>R b = a) = (a = b \<or> u = 0)"
lossless2: fixes \<sigma> :: "'grp \<times> bool list"   and cipher :: "'a" shows "lossless_gpv \<I>_full (\<A>\<^sub>2 \<sigma> cipher)"
w2Less: shows "sint (SCAST(32 \<rightarrow> 64) w2) < 9223372039002259455"
mem\<^sub>A_of_def: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" shows "mem\<^sub>A_of mem\<^sub>C \<equiv> \<lambda>x\<^sub>A. mem\<^sub>C (Eg2_var\<^sub>C_of_Eg1 x\<^sub>A)"
gt: shows "\<forall>i. f i $$ (0, 0) \<succ> f (Suc i) $$ (0, 0)"