hcju/isabelleps huggingface dataset:
sat_Since_point: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula" assumes "MFOTL.sat \<sigma> v i (formula.Since \<phi> I \<psi>)" obtains j :: "nat" where "j \<le> i" and "mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I" and "MFOTL.sat \<sigma> v i (formula.Since \<phi> (point (\<tau> \<sigma> i - \<tau> \<sigma> j)) \<psi>)"
simps_2: fixes v :: "ref" shows "reduced (Node Tip v Tip) = True"
card_of_ordLeqI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A"   and "\<And>a. a \<in> A \<Longrightarrow> f a \<in> B" shows "|A| \<le>o |B|"
expr_typing_randE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and dst :: "pdf_dist"   and e :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> Random dst e : t" obtains "t = dist_result_type dst" and "\<Gamma> \<turnstile> e : dist_param_type dst"
conversep_iff: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'b" shows "r\<inverse>\<inverse> a b = r b a"
mgu_wf_trms: fixes s :: "('fun, 'a) Term.term"   and t :: "('fun, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('fun, 'a) Term.term" assumes "mgu s t = Some \<sigma>"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
tdghm_NTMap_in_Vset: fixes \<beta> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<NN>\<lparr>NTMap\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
mset_subset_eq_insertD: fixes x :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "add_mset x A \<subseteq># B" shows "x \<in># B \<and> A \<subset># B"
ProcModifyReturnSameFaults: 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 init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return c Q,A"
orbit_nonempty: fixes f :: "'c \<Rightarrow> 'c"   and s :: "'c" shows "orbit f s \<noteq> {}"
with_def: fixes zero :: "'c"   and zeroa :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "dependent.with zero zeroa plus \<equiv> \<lambda>scale s. \<exists>t u. finite t \<and> t \<subseteq> s \<and> sum_with plus zeroa (\<lambda>v. scale (u v) v) t = zeroa \<and> (\<exists>v\<in>t. u v \<noteq> zero)"
simps_10: fixes x :: "nat"   and vs :: "real list" shows "isDERIV x Pi vs = True"
assms_2: shows "\<forall>c. c el formula \<longrightarrow> c el formula'"
delete_edge_set_edges: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "set (edgesL (FiniteListGraph.delete_edge v v' G)) = {(a, b). (a, b) \<in> set (edgesL G) \<and> (a, b) \<noteq> (v, v')}"
eq_inc_simp: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and n :: "Arity"   and v1 :: "Value"   and v2 :: "Value" assumes "ArityAnalysisLetSafe Aexp Aheap" shows "eq (inc\<cdot>n) v1 v2 = (\<forall>v. eq n (v1 \<down>Fn v) (v2 \<down>Fn v))"
alpha_Tree_relE: fixes x :: "('b, 'c, 'd) Tree"   and y :: "('b, 'c, 'd) Tree" assumes "(x, y) \<in> Validity.alpha_Tree_rel" obtains "x =\<^sub>\<alpha> y"
fmfilter_false: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x y. m $$ x = Some y \<Longrightarrow> \<not> P x" shows "fmfilter P m = {$$}"
collapse_2: fixes sum :: "'b + 'c" assumes "\<not> isl sum" shows "Inr (projr sum) = sum"
wp_mono2: fixes p :: "'b Assertion"   and q :: "'b Assertion"   and x :: "'b" assumes "p \<le> q" shows "wp x p \<le> wp x q"
compl_less_compl_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less (uminus x) (uminus y) = less y x"
fmdom_map: fixes f :: "'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fmdom (fmmap f m) = fmdom m"
\<T>_def: shows "\<T> = 1\<^sub>\<nat>"
D: shows "\<forall>p\<in>set (p\<^sub>2 # ps). real_of_int l - \<Delta> < real_of_int (fst p) \<and> real_of_int (fst p) < real_of_int l + \<Delta>"
well_def_signs: shows "well_def_signs (length qs) [[1], [- 1]]"
subspace_span: fixes S :: "('b, 'c) vec set" shows "vec.subspace (vec.span S)"
carrier_quotient_ring_iff: fixes X :: "('a \<times> 'a) set" shows "(X \<in> carrier_quotient_ring) = valid_frac X"
final_finished: fixes s :: "'state" assumes "final s" shows "finished (\<rightarrow>) s"
to_bl_ucast: fixes w :: "'b word" shows "to_bl (ucast w) = replicate (LENGTH('a) - LENGTH('b)) False @ drop (LENGTH('b) - LENGTH('a)) (to_bl w)"
in_homE: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" obtains "C.arr f" and "C.dom f = a" and "C.cod f = b"
det_transpose: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det (Finite_Cartesian_Product.transpose A) = det A"
simps: fixes i :: "'a \<times> nat" shows "state i = foldM integrate [] (received_messages i)"
net_ips_netgmap: fixes sr :: "'e \<Rightarrow> 'd \<times> 'c"   and s :: "'e net_state" shows "net_ips (snd (netgmap sr s)) = net_ips s"
assms: shows "a + (1::'a) \<noteq> (0::'a)"
default_rep_def: fixes d :: "'c"   and X :: "'c" shows "default_rep d X = SPEC (\<lambda>x. case x of None \<Rightarrow> X = d | Some r \<Rightarrow> X = r)"
divmod_def: fixes m :: "num"   and n :: "num" shows "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
finite_rows: fixes A :: "(('a, 'b) vec, 'c) vec" shows "finite (rows A)"
"0_7": shows "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. i \<noteq> j \<longrightarrow> mgu (pair p) (pair q \<cdot> \<alpha> p) = None"
aform_to_aform_err_def: fixes X :: "real \<times> real pdevs"   and n :: "nat" shows "aform_to_aform_err X n = ((fst X, pdev_upd (snd X) n 0), \<bar>pdevs_apply (snd X) n\<bar>)"
lessalive_refl: fixes s :: "Store" shows "s \<lless> s"
bis_Union: fixes I :: "'d set"   and B1 :: "'e set"   and B2 :: "'f set"   and s1 :: "'e \<Rightarrow> ('g, 'e, 'f) F1"   and s2 :: "'f \<Rightarrow> ('h, 'e, 'f) F2"   and R1i :: "'d \<Rightarrow> ('e \<times> 'e) set"   and R2i :: "'d \<Rightarrow> ('f \<times> 'f) set" assumes "\<forall>i\<in>I. sbis B1 B2 s1 s2 (R1i i) (R2i i)" shows "sbis B1 B2 s1 s2 (\<Union> (R1i ` I)) (\<Union> (R2i ` I))"
simps_3: fixes a' :: "'a" shows "Def a' \<noteq> \<bottom>"
tranclp_into_tranclp2: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
pequiv_trans: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<simeq> b"   and "b \<simeq> c" shows "a \<simeq> c"
neg_def: shows "neg \<equiv> \<lambda>x i. - x i"
Sup_set_def: fixes A :: "'a set set" shows "\<Union> A = {x. \<Squnion> ((\<in>) x ` A)}"
type_to_set_1: fixes p :: "nat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<int p}"   and "1 < p" shows "class.nontriv TYPE('b)"
vars2'_def: shows "vars2' = ind_in_set vars0 ` vars2"
qqpp_tendsto: shows "((\<lambda>x. qq x / pp x) \<longlongrightarrow> lead_coeff q / lead_coeff p) at_bot"
OAlist_tc_min_key_val_in: fixes xs :: "('a, 'b) oalist_tc" assumes "list_of_oalist_tc xs \<noteq> []" shows "OAlist_tc_min_key_val xs \<in> set (list_of_oalist_tc xs)"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> M"   and "b \<in> M" shows "a \<cdot> b \<in> M"
eqvt_fresh_cong1: fixes f :: "'b \<Rightarrow> 'c"   and a :: "atom"   and x :: "'b" assumes "\<And>p x. p \<bullet> f x = f (p \<bullet> x)"   and "a \<sharp> x" shows "a \<sharp> f x"
PolysD_1: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "p \<in> P[X]" shows "keys p \<subseteq> .[X]"
not_empty_eq_Ici_eq_empty: fixes l :: "'c list" shows "{} \<noteq> prefix_order.atLeast l"
statImpWeakStatImp: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"   and "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', P, Q) \<in> Rel" shows "\<Psi> \<rhd> P \<lessapprox><Rel> Q"
homotopic_loops_refl: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a" shows "homotopic_loops s p p = (path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p)"
poly_ops_def: fixes ops :: "'b arith_ops_record" shows "poly_ops ops \<equiv> Arith_Ops_Record [] (one_poly_i ops) (plus_poly_i ops) (times_poly_i ops) (minus_poly_i ops) (uminus_poly_i ops) (div_field_poly_i ops) (\<lambda>_. []) (mod_field_poly_i ops) (normalize_poly_i ops) (unit_factor_poly_i ops) (\<lambda>i. if i = 0 then [] else [arith_ops_record.of_int ops i]) (\<lambda>_. 0) (is_poly ops)"
ofilter_underS_Field: fixes A :: "'a set" shows "r.ofilter A = ((\<exists>a\<in>Field r. A = r.underS a) \<or> A = Field r)"
empty_subsetI: fixes A :: "'f set" shows "{} \<subseteq> A"
ORRIGHT1_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set" shows "pi \<bullet> ORRIGHT1 (A OR B) X = ORRIGHT1 (A OR B) (pi \<bullet> X)"
outside_disj: shows "x \<notin> S \<or> s x \<noteq> 0 \<or> blinfun_apply (Ds x) (f x) \<noteq> 0"
validG: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/G \<^esub>P' c R,X"
code: fixes x :: "quat" shows "inverse x = Quat (quat.Re x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)) (- Im1 x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)) (- Im2 x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)) (- Im3 x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2))"
def: shows "usubstr \<sigma> (A # B, C) \<noteq> None"
eq_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
moebius_similarity_oriented_lines_to_oriented_lines: fixes a :: "complex"   and H :: "ocircline"   and b :: "complex" assumes "a \<noteq> 0" shows "(\<infinity>\<^sub>h \<in> ocircline_set H) = (\<infinity>\<^sub>h \<in> ocircline_set (moebius_ocircline (moebius_similarity a b) H))"
antisym_conv: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
B_def: shows "B = {x. \<exists>a D. nonneg_lincomb a D x \<and> D \<subseteq> C \<and> lin_indpt D \<and> (\<forall>d\<in>D. a d \<le> (1::'a))}"
A_modifies_vars: shows "\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x"
Iod_Order: fixes T :: "'a set" assumes "T \<subseteq> carrier D" shows "Order (Iod D T)"
enn2ereal_top: shows "enn2ereal top = \<infinity>"
bal_i_insert: fixes t :: "'a tree"   and x :: "'a" assumes "bal_i (size t) (height t)" shows "bal_i (size (local.insert x t)) (height (local.insert x t))"
vint_of_int_in_vintE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" obtains b :: "int" where "b\<^sub>\<int> = a"
list_all_simps_1: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "list_all P (x # xs) = (P x \<and> list_all P xs)"
unit_unique_upto_unique_iso: fixes a :: "'d"   and w :: "'d"   and \<omega> :: "'d" assumes "D.obj a"   and "D.isomorphic a w"   and "\<guillemotleft>\<omega> : w \<star>\<^sub>D w \<Rightarrow>\<^sub>D w\<guillemotright>"   and "D.iso \<omega>" shows "\<exists>!\<psi>. \<guillemotleft>\<psi> : a \<Rightarrow>\<^sub>D w\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[a] = \<omega> \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
smaps_over_Union: fixes ms :: "('a \<Rightarrow> 'b set) list"   and A :: "'a set"   and B :: "'b set" assumes "set ms \<subseteq> smaps_over A B" shows "\<Union>.ms \<in> smaps_over A B"
point_measure_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" shows "point_measure A f = density (count_space A) f"
i_2: shows "\<forall>x. eval r_phi [i, x] = eval r_psi [i, x]"
close_consistency: fixes C :: "('a, 'b) form set set" assumes "consistency C" shows "consistency (close C)"
bv_sub_def: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_sub w1 w2 = bv_sadd w1 (bv_uminus w2)"
circ_sup_sub: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y)\<^sup>\<circ> * x\<^sup>\<circ> \<le> (x \<squnion> y)\<^sup>\<circ>"
simps_1: fixes a :: "'b set"   and b :: "'b set" shows "trivial_series a b 0 = a"
time_merge_tm: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "time (merge_tm f xs ys) \<le> length xs + length ys + 1"
continuous_ge_on_closure: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real"   and x :: "'a"   and a :: "real" assumes "continuous_on (closure s) f"   and "x \<in> closure s"   and "\<And>x. x \<in> s \<Longrightarrow> a \<le> f x" shows "a \<le> f x"
sf_c_sum_le_arg: fixes u :: "nat" shows "sf (c_sum u) \<le> u"
LLL_inv: shows "LLL.LLL_invariant_weak n m fs_init fs"
HcompNml_Trg_Nml: fixes t :: "'c local.term" assumes "Nml t" shows "Trg t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> t = t"
nonprimitive_imp_conductor_less: assumes "\<not> primitive_dchar d \<Phi>" shows "\<Phi>.conductor < d"
not_square_less_zero: 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 a :: "'a" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (times a a) zero"
return_def: fixes result :: "'g" shows "return result = Prog (\<lambda>h. Inr (result, h))"
intro: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a x y. scale a (x + y) = scale a x + scale a y"   and "\<And>a b x. scale (a + b) x = scale a x + scale b x"   and "\<And>a b x. scale a (scale b x) = scale (a * b) x"   and "\<And>x. scale (1::'b) x = x" shows "module scale"
j_def: shows "j \<equiv> \<lambda>l. case H l of (n, M, j) \<Rightarrow> j"
axioms_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "fully_faithful_functor A B F" shows "faithful_functor A B F"
if_eval_true: fixes mem :: "'Var \<Rightarrow> 'Val"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "ev\<^sub>B mem e" shows "\<langle>Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1, mds \<oplus> annos, mem\<rangle>"
dsum_sprod_mem: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'a, 'h) Module_scheme"   and a :: "'a"   and b :: "'f \<Rightarrow> 'g" assumes "\<forall>i\<in>I. R module M i"   and "a \<in> carrier R"   and "b \<in> carr_dsumag I M" shows "prodM_sprod R I M a b \<in> carr_dsumag I M"
Inf_comp: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and post :: "'c \<Rightarrow> 'c"   and X :: "'c set"   and z :: "'c" assumes "class.complete_mbt_algebra Inf Sup inf less_eq less sup bot top one times dual dual_star omega star post" shows "times (Inf X) z = Inf ((\<lambda>x. times x z) ` X)"
cases: fixes x :: "'a ifex" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e"   | "x = Trueif"   | "x = Falseif"
Standard_starfun2: fixes x :: "'b star"   and y :: "'c star"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "x \<in> Standard"   and "y \<in> Standard" shows "(*f2* f) x y \<in> Standard"
is_zero_row_def': fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec" shows "is_zero_row i A = (\<forall>j. A $ i $ j = (0::'b))"
FAccReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<rangle> \<rightarrow>* \<langle>e'\<bullet>F{Cs},s'\<rangle>"
simps_2: fixes t :: "'a Trace"   and s :: "'a" shows "tLast (t \<leadsto> s) = s"
rep_eq: fixes x :: "('d, 'c) lgenerator'" shows "lunstream' x = lunstream (lgenerator x)"
bit_seq_of_nat_cset_bij: shows "bij bit_seq_of_nat_set"
isOK_check_allm: fixes f :: "'b \<Rightarrow> 'a + unit"   and xs :: "'b list" shows "isOK (try forallM f xs catch (\<lambda>x. Inl (snd x))) = (\<forall>x\<in>set xs. isOK (f x))"
nb\<^sub>2: fixes op' :: "('a \<times> 'b) strips_operator" assumes "op' \<in> set ops'" shows "\<exists>op\<in>set (\<Psi>\<^sub>\<O>\<^sub>+). op' = \<phi>\<^sub>O \<Psi> op"
AbGroup: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FGModule G smult V" shows "AbGroup V"
valid_G_delete: shows "wf_graph \<lparr>nodes = nodes G, edges = edges G - X\<rparr>"
punit_dgrad_max_set_subset_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and m :: "nat" assumes "dickson_grading d"   and "set fs \<subseteq> punit.dgrad_p_set d m"   and "\<not> set fs \<subseteq> {0}" shows "punit.dgrad_p_set d (qpm_inf_term.dgrad_max fs d) \<subseteq> punit.dgrad_p_set d m"
of_int_less_1_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) one = (z < 1)"
chain_map_def: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" shows "chain_map p g c \<equiv> frag_extend (frag_of \<circ> simplex_map p g) c"
cancels_to_1_at_def: fixes i :: "nat"   and l1 :: "(bool \<times> 'c) list"   and l2 :: "(bool \<times> 'c) list" shows "cancels_to_1_at i l1 l2 = (0 \<le> i \<and> 1 + i < length l1 \<and> canceling (l1 ! i) (l1 ! (1 + i)) \<and> l2 = cancel_at i l1)"
Memoryidle: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals" shows "\<turnstile> \<not> $Calling ch p \<longrightarrow> \<not> RNext ch mm rs p"
ordertype_append_image_IJ: fixes \<I> :: "'a set set"   and J :: "'a set \<Rightarrow> 'a set"   and B :: "'a \<Rightarrow> nat list"   and c :: "nat"   and A :: "'a set \<Rightarrow> nat list"   and \<beta> :: "V" assumes "\<And>i j. \<lbrakk>i \<in> \<I>; j \<in> J i\<rbrakk> \<Longrightarrow> length (B j) = c"   and "\<And>i j. \<lbrakk>i \<in> \<I>; j \<in> J i\<rbrakk> \<Longrightarrow> A i < B j"   and "\<And>i. i \<in> \<I> \<Longrightarrow> i \<lless> J i \<and> finite i"   and "\<And>i. i \<in> \<I> \<Longrightarrow> ordertype (B ` J i) (lenlex less_than) = \<beta>"   and "inj_on A \<I>" shows "ordertype (usplit (\<lambda>i j. A i @ B j) ` USigma \<I> J) (lenlex less_than) = \<beta> * ordertype (A ` \<I>) (lenlex less_than)"
exec_meth_drop: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and b :: "nat" assumes "exec_meth ci P (ins @ ins') (shift (length ins) xt) t h (stk, loc, length ins + pc, xcp) ta h' (stk', loc', pc', xcp')"   and "jump_ok ins' 0 b" shows "exec_meth ci P ins' xt t h (stk, loc, pc, xcp) ta h' (stk', loc', pc' - length ins, xcp')"
it_step_insert_iff: fixes it :: "'c set"   and S :: "'c set"   and x :: "'c" assumes "it \<subseteq> S"   and "x \<in> it" shows "S - (it - {x}) = insert x (S - it)"
iiaE: fixes swf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and S :: "'b set"   and Is :: "'a set"   and x :: "'b"   and y :: "'b"   and a :: "'b"   and b :: "'b"   and P' :: "'a \<Rightarrow> ('b \<times> 'b) set"   and P :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "iia swf S Is"   and "{x, y} \<subseteq> S"   and "a \<in> {x, y}"   and "b \<in> {x, y}"   and "\<And>i a b. \<lbrakk>a \<in> {x, y}; b \<in> {x, y}; i \<in> Is\<rbrakk> \<Longrightarrow> (a \<^bsub>(P' i)\<^esub>\<preceq> b) = (a \<^bsub>(P i)\<^esub>\<preceq> b)"   and "profile S Is P"   and "profile S Is P'" shows "(a \<^bsub>(swf P)\<^esub>\<preceq> b) = (a \<^bsub>(swf P')\<^esub>\<preceq> b)"
Sign_Exp: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Sign X Y \<noteq> Exp X' Y'"