hcju/isabelleps huggingface dataset:
method_def: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list" shows "method P C M \<equiv> THE (D, b, Ts, T, m). P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"
l: shows "content (reflect_poly f) = gcd_list (dropWhile ((=) (0::'a)) (coeffs f))"
arrI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr"   and "Arr f" shows "partial_magma.arr (subcategory.comp C Arr) f"
bisims1_Val_\<tau>Reds1r: fixes P :: "'addr J1_prog"   and h :: "'heap"   and Es :: "'addr expr1 list"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and vs :: "'addr val list"   and loc :: "'addr val list"   and n :: "nat"   and t :: "'thread_id" assumes "P,Es,h \<turnstile> (es, xs) [\<leftrightarrow>] (rev vs, loc, length (compEs2 Es), None)"   and "n + max_varss es \<le> length xs"   and "\<B>s Es n" shows "\<tau>reds1r P t h (es, xs) (map Val vs, loc)"
kop_galois: fixes f :: "'d \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'd set" shows "(op\<^sub>K f = g) = (op\<^sub>K g = f)"
simps_2: fixes u :: "real"   and d :: "real"   and v :: "real"   and n :: "nat" shows "geom_rand_walk u d v (Suc n) = (\<lambda>w. (case w !! n of True \<Rightarrow> u | False \<Rightarrow> d) * geom_rand_walk u d v n w)"
CH_range': fixes c :: "'a"   and B :: "'a set" assumes "c \<in> ManiPref'.CH B" shows "c \<in> B"
degen_run_find_acc_aux: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat"   and i :: "nat"   and q :: "'Q"   and n :: "nat" assumes "num_acc \<noteq> 0"   and "b_graph.is_acc_run (degeneralize_ext m) r"   and "r i = (q, 0)"   and "0 \<in> acc q"   and "n < num_acc" shows "\<exists>j qj. i \<le> j \<and> r j = (qj, n) \<and> n \<in> acc qj"
\<phi>a: shows "\<phi> (a, a) a \<in> Hom.set (a, a)"
eq_fin_symp: fixes u :: "'action list"   and v :: "'action list" assumes "u =\<^sub>F v" shows "v =\<^sub>F u"
FW_word_def: fixes p :: "nat"   and q :: "nat" shows "FW_word p q = (if q < p then FW_word q p else if p = 0 \<or> p = q then \<epsilon> else if gcd p q = q - p then fw_base p q else take p (FW_word p (q - p)) \<cdot> FW_word p (q - p))"
lexext_rev_simps_3: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and y :: "'b"   and xs :: "'b list"   and x :: "'b" shows "lexext_rev gt (ys @ [y]) (xs @ [x]) = (gt y x \<or> x = y \<and> lexext_rev gt ys xs)"
fixate_def: fixes a :: "'b"   and f :: "'b perm" shows "fixate a f = (if a \<in> affected f then f * \<langle>inverse f \<langle>$\<rangle> a \<leftrightarrow> a\<rangle> else f)"
main_5: fixes i' :: "nat"   and j' :: "nat" assumes "i' < m"   and "j' < m" shows "gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs') i' j' = (if i' = i \<and> j' \<le> j then gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j' - rat_of_int c * gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) j j' else gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i' j')"
pos_decomp_subset: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "set (ps\<^sub>+) \<subseteq> set ps"
linorder_rank_set_sorted_wrt: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b" assumes "linorder_on B R"   and "set xs \<subseteq> B"   and "Linorder_Relations.sorted_wrt R xs"   and "x \<in> set xs"   and "distinct xs" shows "linorder_rank R (set xs) x = index xs x"
h1: shows "\<forall>x\<in>set les. case x of (d, e, f) \<Rightarrow> d * (c / b)\<^sup>2 + e * - (c / b) + f < 0"
height\<^sub>F_insortTree: fixes t :: "'a tree"   and ts :: "'a tree list" shows "height\<^sub>F (insortTree t ts) = max (height t) (height\<^sub>F ts)"
state_update_vs_allInstances_empty: shows "OclAllInstances_generic pre_post Type (mk \<lparr>heap = Map.empty, assocs = A\<rparr>) = Set{} (mk \<lparr>heap = Map.empty, assocs = A\<rparr>)"
xor_left_commute: fixes x :: "'g"   and y :: "'g"   and z :: "'g" shows "x \<oplus> y \<oplus> z = y \<oplus> x \<oplus> z"
BP: fixes i :: "nat" assumes "i < n" shows "0 \<le> B $$ (i, i)"
elims: fixes x :: "rat mat"   and xa :: "rat vec"   and y :: "constraint list" assumes "mat_x_leq_vec x xa = y" obtains A :: "rat mat" and b :: "rat vec" where "x = A" and "xa = b" and "y = map (\<lambda>i. LEQ (matrix_to_lpolies A ! i) (b $ i)) [0..<dim_vec b]"
Derived: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and qs :: "('b, 'c) fm list"   and a :: "'c"   and v' :: "nat"   and k :: "'c"   and p :: "('b, 'c) fm"   and i :: "'c"   and xs :: "(nat \<times> nat) set"   and w :: "nat"   and w' :: "nat"   and rs :: "('b, 'c) fm list" assumes "branch !. v = Some (qs, a)"   and "qs !. v' = Some (\<^bold>\<not> (\<^bold>@ k p))"   and "descendants k i branch xs"   and "(w, w') \<in> xs"   and "branch !. w = Some (rs, a)"   and "rs !. w' = Some (\<^bold>\<diamond> Nom k)" shows "descendants k i branch ({(v, v')} \<union> xs)"
ccompare_sum_def: shows "ccompare \<equiv> case ID ccompare of None \<Rightarrow> None | Some comp_'a \<Rightarrow> case ID ccompare of None \<Rightarrow> None | Some comp_'b \<Rightarrow> Some (comparator_sum comp_'a comp_'b)"
map_of_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "('c \<times> 'e) list" shows "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
assms_3: shows "\<lparr>p \<cdot> b\<rparr>x\<lparr>x \<cdot> p\<rparr>"
le_Store_def: fixes s :: "Store"   and t :: "Store" shows "(s \<le> t) = s \<lless> t"
collapse_5: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" assumes "is_Delete stateful_strand_step" shows "delete\<langle>the_elem_term stateful_strand_step,the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
semi_confluence: shows "R\<inverse> O R\<^sup>* \<subseteq> R\<^sup>\<down>"
that: fixes as :: "'edge list"   and m' :: "'node"   and cfs' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "m -as\<rightarrow>\<^sub>\<surd>* m'"   and "m' \<triangleq> c'"   and "preds (slice_kinds {CFG_node m'} as) [(cf, undefined)]"   and "\<forall>V\<in>Use m'. state_val (transfers (slice_kinds {CFG_node m'} as) [(cf, undefined)]) V = state_val cfs' V"   and "map fst cfs' = s'" shows "thesis"
InitBlockRed: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and V :: "char list"   and v :: "val"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v' :: "val"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V \<mapsto> v), sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "l' V = \<lfloor>v'\<rfloor>" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l, sh),b\<rangle> \<rightarrow> \<langle>{V:T; V:=Val v';; e'},(h', l'(V := l V), sh'),b'\<rangle>"
integrable_discrete_difference: fixes X :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "integrable M f = integrable M g"
exhaust: fixes y :: "bal" obtains "y = Lh"   | "y = Bal"   | "y = Rh"
lookup_parametric: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_fun A B) (rel_fun A B) (\<lambda>m. m) (\<lambda>m. m)"
infinite: fixes A :: "'a set" assumes "infinite A" shows "Gcd\<^sub>f\<^sub>i\<^sub>n A = (1::'a)"
rel_compp: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "fmrel (R OO S) = fmrel R OO fmrel S"
zor_poly_eq: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z"   and "\<exists>\<^sub>F w in at z. f w \<noteq> 0" shows "\<forall>\<^sub>F w in at z. zor_poly f z w = f w * (w - z) powr of_int (- zorder f z)"
cases: fixes a :: "'b"   and r :: "('b \<times> 'd) set" assumes "a \<in> Domain r" obtains aa :: "'b" and b :: "'d" where "a = aa" and "(aa, b) \<in> r"
auto_weaken_pre_uncurry_finish: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "PROTECT f x \<equiv> f x"
simps_37: shows "cospan_shape.comp cospan_shape.Null cospan_shape.AT = cospan_shape.Null"
fixedaction_wf_ruleset: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and a :: "action"   and ms :: "'b match_expr list" shows "wf_ruleset \<gamma> p (map (\<lambda>m. Rule m a) ms) = (\<not> match_list \<gamma> ms a p \<or> (\<nexists>chain. a = Call chain) \<and> a \<noteq> Return \<and> (\<nexists>chain. a = Goto chain) \<and> a \<noteq> Unknown)"
env_restr_eqI: fixes S :: "'a set"   and m\<^sub>1 :: "'a \<Rightarrow> 'b"   and m\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> m\<^sub>1 x = m\<^sub>2 x" shows "m\<^sub>1 f|` S = m\<^sub>2 f|` S"
Confidentiality_B: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Key K \<notin> analz (knows Spy evs)"
positions_of_an_atomic_term: fixes t :: "'b trm" assumes "is_a_variable t \<or> is_a_constant t" shows "pos_of t = {[]}"
is_val_iff: fixes e :: "('a, 'b, 'c) exp" shows "is_val e = (\<exists>v. e = Val v)"
degree_eq_0_mpoly_to_nested_polyI: fixes mp :: "'b mpoly"   and v :: "nat" assumes "mpoly_to_nested_poly mp v = 0" shows "MPoly_Type.degree mp v = 0"
left_neutral: fixes a :: "'d" shows "(0::'d) + a = a"
projection_eqI': fixes M :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "convex M"   and "is_projection_on f M" shows "projection M = f"
classes_above_has_fields: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C has_fields FDTs" shows "P' \<turnstile> C has_fields FDTs"
aHom_inv_inv: fixes F :: "('d, 'e) aGroup_scheme"   and G :: "('f, 'g) aGroup_scheme"   and f :: "'d \<Rightarrow> 'f"   and a :: "'d" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F" shows "f (-\<^sub>a\<^bsub>F\<^esub> a) = -\<^sub>a\<^bsub>G\<^esub> f a"
transfer_generic_Assert_remove_axioms: shows "transfer_generic_Assert_remove (\<bind>) RETURN ASSERT ASSUME"
abs_eq: fixes xa :: "'d set"   and x :: "'d \<Rightarrow> 'e" shows "finmap_of xa x = Abs_fmap (\<lambda>xb. if xb \<in> xa \<and> finite xa then Some (x xb) else None)"
s_s'_cfg_on: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.cfg_on s" shows "absc cfg \<in> R_G.cfg_on s'"
assms_1: shows "threshold \<phi> w \<G> = Some k"
nsum_mem1Tr: fixes J :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A +> J" shows "(\<forall>j\<le>n. f j \<in> J) \<longrightarrow> \<Sigma>\<^sub>e A f n \<in> J"
that_2: fixes x :: "'a" assumes "x \<in> P" shows "(x \<in> S) = (x \<in> R)"
frag_cmul_distrib: fixes c :: "int"   and d :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul (c + d) a = frag_cmul c a + frag_cmul d a"
SeqCond: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P :: "'b set"   and b :: "'b set"   and P\<^sub>1 :: "('b, 'a, 'c) ann"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set"   and P\<^sub>2 :: "('b, 'a, 'c) ann"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and r :: "'b set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (P \<inter> b) P\<^sub>1 c\<^sub>1 Q, A"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (P \<inter> - b) P\<^sub>2 c\<^sub>2 Q, A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnBin r P\<^sub>1 P\<^sub>2) (Cond b c\<^sub>1 c\<^sub>2) Q, A"
nat_of_inject: fixes x :: "'b sat"   and y :: "'b sat" shows "(nat_of x = nat_of y) = (x = y)"
set_integrable_def: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
atLeastAtMost_iff: fixes i :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and l :: "'e"   and u :: "'e" shows "(i \<in> ord.atLeastAtMost less_eq l u) = (less_eq l i \<and> less_eq i u)"
eq_imp_below: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.po below"   and "x = y" shows "below x y"
min_MInf: fixes x :: "ereal" shows "min (- \<infinity>) x = - \<infinity>"
assms_5: shows "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"
nP: shows "nP = nm_node_props SINVAR_LIB_BLPtrusted \<lparr>node_properties = BLP_security_levels\<rparr>"
aux_tj: fixes x :: "'v" assumes "x \<in> set (tl (stack s))"   and "u \<in> set (tj_stack s)" shows "x \<in> set (tl (dropWhile ((\<noteq>) u) (tj_stack s)))"
assms_1: shows "x \<cdot> y \<le> y \<cdot> x \<cdot> y"
RETURN_refine_iff: fixes x :: "'b"   and R :: "('b \<times> 'c) set"   and y :: "'c" shows "(RETURN x \<le> \<Down> R (RETURN y)) = ((x, y) \<in> R)"
False: shows "demand (C_case\<cdot>f) \<noteq> C\<^sup>\<infinity>"
lsided_le2: fixes x :: "'a"   and y :: "'a" assumes "lsd x" shows "x \<cdot> y \<le> x \<sqinter> (y \<cdot> \<top>)"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (add_monoid R)"   and "a \<in> H" shows "\<ominus> a \<in> H"
gmnl_correct: fixes xs :: "('c, 'd) Node list" assumes "\<forall>x\<in>set xs. is_measured_node x" shows "gmnl xs = sum_list (map snd (nlistToList xs))"
nts: shows "wf_ts \<sigma> j \<phi>'' \<psi>'' nts"
induct: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x1 :: "'e"   and x2 :: "'e"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* x1 x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>r\<^sup>*\<^sup>* a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
binary_functor_Prod_2: shows "binary_functor (\<cdot>) (\<cdot>) (\<cdot>) CCC.Prod'"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd"   and xa :: "'c \<Rightarrow>\<^sub>0 'd" shows "(x < xa) = less_fun (lookup x) (lookup xa)"
inj: fixes f :: "'d \<Rightarrow> 'd" assumes "bijection f" shows "inj f"
higher_differentiable_on_snd_comp: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e \<times> 'f"   and k :: "nat" assumes "higher_differentiable_on S f k"   and "open S" shows "higher_differentiable_on S (\<lambda>x. snd (f x)) k"
awlast_if_cas: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.cas u p v" shows "G.awlast u p = v"
uw\<theta>_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" shows "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<equiv> tabulation_data V H \<a> \<i> src trg r \<rho> f g \<and> uw\<theta>_axioms V H f u w \<theta>"
swap_sum_swap_sum: fixes x :: "'h + 'i" shows "swap_sum (swap_sum x) = x"
i_set_cont_FROM: fixes n :: "nat" shows "i_set_contp [n\<dots>]"
if_distrib: fixes f :: "'e \<Rightarrow> 'd"   and c :: "bool"   and x :: "'e"   and y :: "'e" shows "f (if c then x else y) = (if c then f x else f y)"
Joints_set_zip_subset: fixes xs :: "'b list"   and XS :: "('b \<times> 'b pdevs) list"   and ys :: "'b list"   and YS :: "('b \<times> 'b pdevs) list" assumes "xs \<in> Joints XS"   and "length xs = length XS"   and "length ys = length YS"   and "set (zip ys YS) \<subseteq> set (zip xs XS)" shows "ys \<in> Joints YS"
assms: shows "0 < cos (Re z) \<or> cos (Re z) = 0 \<and> Im z * sin (Re z) \<le> 0"
div_0: 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" assumes "class.semidom_divide divide plus minus zero times one" shows "divide zero a = zero"
document_ptr_casts_commute3: fixes ptr :: "(_) object_ptr" assumes "is_document_ptr_kind ptr" shows "cast (the (cast ptr)) = ptr"
eff_Some: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list" shows "eff i P pc xt \<lfloor>\<tau>\<rfloor> = norm_eff i P pc \<tau> @ xcpt_eff i P pc \<tau> xt"
inf_absorb1: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x" shows "x \<squnion> y = x"
satC_subst: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and c :: "('fsym, 'psym) lit list" shows "satC \<xi> (substC \<rho> c) = satC (int \<xi> \<circ> \<rho>) c"
ordinal_number_def: fixes S :: "'c Order" shows "ordinal_number S = {X. Worder X \<and> ord_equiv X S}"
strategy_attractsE: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy_attracts G p \<sigma> A W"   and "v0 \<in> A" shows "visits_via P A W"
assms: shows "circuit_in \<E> C \<and> C \<subseteq> X"
big_forest_def: fixes H :: "'a"   and d :: "'a" shows "big_forest H d \<equiv> equivalence H \<and> d \<le> - H \<and> coreflexive (H * d[[H * d]]) \<and> coreflexive (H \<sqinter> d * d\<^sup>T) \<and> (H * d)\<^sup>+ \<le> - H"
varSort_sort_lt_var: shows "|UNIV| <o |UNIV|"
plus_period: fixes x :: "'a" shows "frac (x + (1::'a)) = frac x"
bigo_abs: fixes f :: "'a \<Rightarrow> 'b" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> O(f)"
sep_globalise: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool"   and h :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P R h"   and "\<And>h. P h \<Longrightarrow> Q h" shows "sep_algebra.sep_conj plus sep_disj Q R h"
while_def: fixes p :: "'a"   and x :: "'a" shows "antidomain_kleene_algebra.while antirange_op (\<lambda>x y. y \<cdot> x) star p x = ar p \<cdot> (x \<cdot> antidomain_semiringl.ads_d antirange_op p)\<^sup>\<star>"
supr_least: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> (b, a) \<in> r" shows "(supr B, a) \<in> r"
countable_bipartite_web_of: shows "countable_bipartite_web (bipartite_web_of \<Gamma>)"
compare_double_simps_2: fixes x :: "double"   and y :: "double" shows "(compare_double x y = 0) = (Code_Double.is_nan x \<and> Code_Double.is_nan y \<or> fcompare_double x y = ccode.Eq)"
simps_1: fixes t :: "('a, 'b) Term.term"   and y :: "'b" shows "subterm_of t (Var y) = (t = Var y)"
isMaxFlow_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a" shows "isMaxFlow f \<equiv> Flow c s t f \<and> (\<forall>f'. Flow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f)"
check_tableau_equiv: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and v :: "nat \<Rightarrow> 'b" assumes "Check check"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> (check s)"
replace_coeff_extract_var_cong: fixes f :: "nat \<Rightarrow> 'a"   and v :: "nat"   and g :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly" assumes "f v = g v" shows "replace_coeff (insertion f) (extract_var p v) = replace_coeff (insertion g) (extract_var p v)"
Allowed_lift: fixes i :: "nat"   and F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program" shows "Allowed (lift i F) = lift i ` Allowed F"
seq_mono: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice"   and c\<^sub>2 :: "'b::refinement_lattice"   and d\<^sub>2 :: "'b::refinement_lattice" assumes "seq_distrib (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(c\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'b::refinement_lattice)"   and "(c\<^sub>2::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>2::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c\<^sub>1::'b::refinement_lattice) (c\<^sub>2::'b::refinement_lattice) \<sqsubseteq> seq (d\<^sub>1::'b::refinement_lattice) (d\<^sub>2::'b::refinement_lattice)"
simps_2: fixes x :: "atom \<times> atom"   and xs :: "(atom \<times> atom) list"   and a :: "atom" shows "atom_prm_atom (x # xs) a = swap_atom x (atom_prm_atom xs a)"
zeros: shows "\<forall>x\<in>D. Abstract_Linear_Poly.coeff v x = 0"
wt_cont_eff_ParT_notWtFT_notFinished: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and i :: "nat"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "properL cl"   and "n < length cl"   and "i < PL.brn (cl ! n)"   and "WtFT cl \<noteq> 1"   and "\<not> finished (cl ! n)" shows "PL.wt aval tval cval (ParT cl) s (PL.brnL cl n + i) = 1 / real (length cl) / (1 - WtFT cl) * PL.wt aval tval cval (cl ! n) s i"
atLeastAtMost_iff: fixes i :: "'b word"   and l :: "'b word"   and u :: "'b word" shows "(i \<in> signed.atLeastAtMost l u) = (l \<le>s i \<and> i \<le>s u)"
Contra: fixes A :: "fm"   and H :: "fm set" assumes "insert (Neg A) H \<turnstile> A" shows "H \<turnstile> A"
eq_swap_sym: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a list"   and w :: "'a list" assumes "traces ind"   and "traces.eq_swap ind v w" shows "traces.eq_swap ind w v"
cases: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and a :: "('e \<times> 'd) stream" assumes "RuleSystem_Defs.epath eff rules a" obtains steps :: "('e \<times> 'd) stream" and sl :: "'e fset" where "a = steps" and "snd (shd steps) \<in> i.R rules" and "fst (shd (stl steps)) |\<in>| sl" and "eff (snd (shd steps)) (fst (shd steps)) sl" and "RuleSystem_Defs.epath eff rules (stl steps)"
rep_eq: fixes x :: "String.literal" shows "size x = length (literal.explode x)"
monotone_onI: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X; r x y\<rbrakk> \<Longrightarrow> s (f x) (f y)" shows "monotone_on X r s f"
filtermap_id': shows "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
less_add_order_eq_0: fixes a :: "'a"   and k :: "nat" assumes "a +^ k = (0::'a)"   and "order.greater (add_order a) k" shows "k = 0"
fun_cong: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "f = g" shows "f x = g x"
hd_dropWhile: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "dropWhile P xs \<noteq> []" shows "\<not> P (hd (dropWhile P xs))"
cimage_iff: fixes z :: "'f"   and f :: "'g \<Rightarrow> 'f"   and A :: "'g cset" shows "cin z (cimage f A) = cBex A (\<lambda>x. z = f x)"
express_p: shows "p = A * (Var var)\<^sup>2 + B * Var var + C"