hcju/isabelleps huggingface dataset:
merge_coeffs0_alt_def: fixes p :: "(char list list \<times> int) list" shows "(RETURN \<circ> merge_coeffs0) p = REC\<^sub>T (\<lambda>f p. case p of [] \<Rightarrow> RETURN [] | [(aa, b)] \<Rightarrow> if snd (aa, b) = 0 then RETURN [] else RETURN [(aa, b)] | (aa, b) # (ys, m) # p \<Rightarrow> if aa = ys then if b + m \<noteq> 0 then f ((aa, b + m) # p) else f p else if b = 0 then f ((ys, m) # p) \<bind> RETURN else f ((ys, m) # p) \<bind> (\<lambda>p. RETURN ((aa, b) # p))) p"
pos_le_minus_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq a (uminus (divide b c)) = less_eq (times a c) (uminus b)"
assms_3: assumes "x \<sqsubseteq> y" shows "P"
hyps: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set" assumes "non_volatile_owned_or_read_only pending_write \<S> \<O> sb"   and "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb" shows "a \<in> acquired_reads True sb \<O> \<or> a \<in> read_only_reads \<O> sb"
induct: fixes P :: "('b, 'c) RBT_Impl.rbt \<Rightarrow> bool"   and a0 :: "('b, 'c) RBT_Impl.rbt" assumes "\<And>c lt k v. P (Branch c lt k v rbt.Empty)"   and "\<And>c lt k v rc rlc rk rv rrt. P (Branch rc rlc rk rv rrt) \<Longrightarrow> P (Branch c lt k v (Branch rc rlc rk rv rrt))"   and "P rbt.Empty" shows "P a0"
0: shows "rawpsubst (all y \<phi>) (zip (map Var ws) (map snd txs)) = all y (rawpsubst \<phi> (zip (map Var ws) (map snd txs)))"
finiteVarsFormula: fixes F :: "Literal list list" shows "finite (vars F)"
C1: fixes \<phi> :: "'c"   and P :: "('a, 'b, 'c) psi" assumes "(\<phi>, P) mem CsP" shows "\<exists>Q. (\<phi>, Q) mem CsQ \<and> guarded Q \<and> \<Psi> \<rhd> P \<sim>\<^sub>s Q"
op_mset_insert_def: shows "op_mset_insert \<equiv> add_mset"
identity_map: fixes U :: "'a set"   and x :: "'b" assumes "is_open U"   and "x \<in> local.im_sheaf U" shows "im_sheaf_morphisms U U x = x"
optimize_matches_fst: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr"   and r :: "'b rule"   and rs :: "'b rule list" shows "optimize_matches f (r # rs) = optimize_matches f [r] @ optimize_matches f rs"
simps_2: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and v :: "Vars.var"   and e :: "exp"   and "binds" :: "(Vars.var \<times> exp) list" shows "ArityAnalysis.ABinds Aexp ((v, e) # binds) = ArityAnalysis.ABind Aexp v e \<squnion> ArityAnalysis.ABinds Aexp (delete v binds)"
scale_cancel_right: fixes a :: "'b"   and x :: "('b, 'c) vec"   and b :: "'b" shows "(a *s x = b *s x) = (a = b \<or> x = 0)"
not_none_is_obs: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "local.att (\<pi> i) \<noteq> None" shows "is_kth_obs \<pi> (card (obs_ids \<pi> \<inter> {..<i})) i"
product_in_sets_PiFI: fixes J :: "'d set"   and I :: "'d set set"   and S :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<in> I"   and "\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)" shows "Pi' J S \<in> sets (Pi\<^sub>F I M)"
candidate_invariant_def: fixes n :: "nat" shows "candidate_invariant n = (n = 0 \<or> n mod 30 = 11)"
DomainE: fixes a :: "'c"   and r :: "('c \<times> 'd) set" assumes "a \<in> Domain r" obtains b :: "'d" where "(a, b) \<in> r"
ag_pOp_assoc: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "z \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<plusminus>\<^bsub>A\<^esub> z = x \<plusminus>\<^bsub>A\<^esub> (y \<plusminus>\<^bsub>A\<^esub> z)"
circ_back_loop_fixpoint: fixes z :: "'a"   and y :: "'a" shows "z \<sqinter> top \<sqinter> y \<squnion> z = z \<sqinter> top"
tinstT_imp_ex_raw_match': fixes T1 :: "typ"   and T2 :: "typ" assumes "tinstT T1 T2" shows "\<exists>subs. raw_match' T2 T1 Map.empty = Some subs"
abs_eq: fixes x :: "int" shows "uint32_of_int x = Abs_uint32 (word_of_int x)"
assms_1: shows "is_induced_by_cell w r\<^sub>0 \<omega>"
Conj_def: fixes xset :: "('a, 'b, 'c) formula set['a]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
elims: fixes x :: "'a simplex_state"   and y :: "nat \<Rightarrow> rat" assumes "solution_simplex x = y" obtains cs :: "QDelta ns_constraint list" and asi :: "('a, ('a \<times> QDelta atom) list) mapping" and tv :: "(nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping" and ui :: "'a list" and s :: "('a, QDelta) state" where "x = Simplex_State (cs, (asi, tv, ui), s)" and "y = \<langle>from_ns (tv (\<V> s)) cs\<rangle>"
ld1: shows "LeftDerives1 (\<alpha> @ \<delta>) i r \<beta>"
summable_comparison_test': fixes g :: "nat \<Rightarrow> real"   and N :: "nat" assumes "summable g"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n" shows "summable f"
surjective_pairing: fixes t :: "'a \<times> 'b" shows "t = (cc t, snd t)"
getRel_dom_1: fixes G :: "('d, 'e) labeled_graph"   and a :: "'e"   and b :: "'e"   and l :: "'d" assumes "graph G"   and "(a, b) \<in> getRel l G" shows "a \<in> vertices G"
mfinal_def: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set" shows "mfinal s = (\<forall>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<longrightarrow> final x \<and> ln = no_wait_locks \<and> wset s t = None)"
comp_MkArr: fixes A :: "'j \<Rightarrow> 'c"   and B :: "'j \<Rightarrow> 'c"   and F :: "'j \<Rightarrow> 'c"   and C :: "'j \<Rightarrow> 'c"   and G :: "'j \<Rightarrow> 'c" assumes "J_C.arr (J_C.MkArr A B F)"   and "J_C.arr (J_C.MkArr B C G)" shows "J_C.MkArr B C G \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] J_C.MkArr A B F = J_C.MkArr A C (vertical_composite.map (\<cdot>\<^sub>J) (\<cdot>) F G)"
mbisim_deadlocked_preserve1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s1 \<approx>m s2"   and "t \<in> r1.deadlocked s1" shows "t \<in> r2.deadlocked s2"
cmp_simps_3: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "trg\<^sub>D (\<Phi> (f, g)) = trg\<^sub>D (F f)"
the_f: shows "(THE f. \<forall>i. X $ j $ i = (\<Sum>x\<in>UNIV. f x * X $ x $ i)) = f"
matchT_ZO_def2: fixes d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchT_ZO d c = (\<forall>s t t'. s \<approx> t \<and> (d, t) \<rightarrow>t t' \<longrightarrow> s \<approx> t' \<and> discr c \<or> (\<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> s' \<approx> t' \<and> discr c') \<or> (\<exists>s'. (c, s) \<rightarrow>t s' \<and> s' \<approx> t'))"
ereal_open_cont_interval: fixes S :: "ereal set"   and x :: "ereal" assumes "open S"   and "x \<in> S"   and "\<bar>x\<bar> \<noteq> \<infinity>" obtains e :: "ereal" where "0 < e" and "{x - e<..<x + e} \<subseteq> S"
ring_mat: fixes n :: "nat"   and b :: "'d" shows "ring (ring_mat TYPE('c) n b)"
EncE: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" assumes "[\<lbrace>x,F\<rbrace> in v]" shows "\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.en r"
Qg_iop: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set" assumes "Group G"   and "G \<triangleright> N" shows "c_iop G N \<in> set_rcs G N \<rightarrow> set_rcs G N"
e2: assumes "a \<in> Units G"   and "irreducible G b" shows "P"
SUP_const: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and f :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Sup ((\<lambda>i. f) ` A) = f"
bind_def2: fixes m :: "'g\<cdot>'f"   and k :: "'g \<rightarrow> 'e\<cdot>'f" shows "m \<bind> k = join\<cdot>(fmap\<cdot>k\<cdot>m)"
is_empty_set: fixes xs :: "'b list" shows "Set.is_empty (set xs) = List.null xs"
fp_cop_F_range_inv': fixes x :: "'a"   and ds :: "'b set" assumes "x \<in> TruePref.fp_cop_F ds" shows "x \<in> Field (Pd' (Xd x)) \<and> Xd x \<in> ds"
Un_infinite: fixes S :: "'a set"   and T :: "'a set" assumes "infinite S" shows "infinite (S \<union> T)"
holomorphic_on_eval_fps: fixes A :: "complex set"   and f :: "complex fps" assumes "A \<subseteq> eball 0 (fps_conv_radius f)" shows "eval_fps f holomorphic_on A"
lincomb_conv_take_right: fixes smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and rs :: "'b list"   and ms :: "'a list" shows "scalar_mult.lincomb smult rs ms = scalar_mult.lincomb smult rs (take (length rs) ms)"
assms: shows "f \<approx>>\<^bsub>CatDom F\<^esub> g"
left_gpv_Inl_transfer: fixes gpv :: "('a, 'b, 'd) gpv" shows "rel_gpv'' (=) (\<lambda>l r. l = Inl r) (\<lambda>l r. l = Inl r) (left_gpv gpv) gpv"
simps_2: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.seqseq P (Suc n) = subseqs.seqseq P n \<circ> subseqs.reduce P (subseqs.seqseq P n) n"
check_del_def: fixes A :: "(nat, int mpoly) fmap"   and p :: "nat" shows "check_del A p = SPEC (\<lambda>b. b \<longrightarrow> True)"
pigeonhole_card: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "finite A"   and "finite B"   and "B \<noteq> {}" shows "\<exists>y\<in>B. card A \<le> card (f -` {y} \<inter> A) * card B"
Mp_Mp: fixes m :: "int"   and f :: "int poly" shows "poly_mod.Mp m (poly_mod.Mp m f) = poly_mod.Mp m f"
complex_of_def: fixes x :: "'a" shows "complex_of x = complex_of_real (x $\<^sub>e 0) + \<i> * complex_of_real (x $\<^sub>e 1)"
measurable_stream_space2: fixes f :: "'b \<Rightarrow> 'c stream"   and N :: "'b measure"   and M :: "'c measure" assumes "\<And>n. (\<lambda>x. f x !! n) \<in> N \<rightarrow>\<^sub>M M" shows "f \<in> N \<rightarrow>\<^sub>M stream_space M"
horner_sum_of_bool_2_less: fixes bs :: "bool list" shows "horner_sum of_bool 2 bs < 2 ^ length bs"
lstutter_extend: shows "lbisimulation (br pidgc_\<alpha> (\<lambda>_. True)) (stutter_extend pid_system_automaton) (stutter_extend system_automaton)"
empty_path_nodes: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n [] n'" shows "n = n'"
fst1: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P >> P'" shows "Fst P >> Fst P'"
hom_connected_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (H \<nu> (partial_magma.cod V \<mu>) \<noteq> partial_magma.null V)"
sel: fixes x :: "nat" shows "natOf (Pvariable x) = x"
zero_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
C'leqD: shows "P \<turnstile> C' \<prec>\<^sup>1 D"
isometry_preserves_bounded: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "isometry_on X f"   and "A \<subseteq> X" shows "bounded (f ` A) = bounded A"
apart: shows "vars\<^sub>l\<^sub>s C\<^sub>1 \<inter> vars\<^sub>l\<^sub>s C\<^sub>2 = {}"
assms_2: shows "constructive_security_obsf' real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 (\<lambda>\<eta>. absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (converter_of_resource (real1 \<eta>) |\<^sub>\<propto> 1\<^sub>C))))"
magQ_uminus: fixes x :: "'e['f, 'g]" shows "\<lbrakk>- x\<rbrakk>\<^sub>Q = - \<lbrakk>x\<rbrakk>\<^sub>Q"
stream_sel_2: fixes t :: "'a tree" shows "stl (stream t) = stream (tree_chop t)"
range_pos: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and pos :: "'a \<Rightarrow> int" assumes "class.spark_enum less_eq less pos" shows "range pos = {0..<int CARD('a)}"
\<O>b_def: shows "cring0.\<O>b = (\<lambda>\<pp>. undefined)"
rn_price_def: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" shows "rn_price N der matur n w = expl_cond_expect N (proj_stoch_proc geom_proc n) (discounted_value r (\<lambda>m. der) (matur - n)) w"
not_Low_High: fixes V :: "'var" assumes "V \<notin> L" shows "V \<in> H"
sparsegrid'_start: fixes b :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set" assumes "b \<in> grid (start dm) ds" shows "b \<in> sparsegrid' dm"
uniform_limit_flow: fixes a :: "real"   and x0 :: "'a"   and b :: "real" assumes "a \<in> existence_ivl0 x0"   and "b \<in> existence_ivl0 x0"   and "a \<le> b" shows "uniform_limit {a..b} flow0 (flow0 x0) (at x0)"
AE_1_3: shows "AE x in M. 0 \<le> - subcocycle_lim_ereal u x"
assms_3: shows "ccw'.sortedP 0 (P # QRRs)"
degree_aform_approx_un: fixes p :: "nat"   and ivl :: "float interval \<Rightarrow> float interval option"   and X :: "((real \<times> real pdevs) \<times> real) option"   and Y :: "(real \<times> real pdevs) \<times> real"   and d1 :: "nat" assumes "approx_un p ivl X = Some Y"   and "case X of None \<Rightarrow> True | Some X \<Rightarrow> degree_aform_err X \<le> d1" shows "degree_aform_err Y \<le> d1"
bind_def: fixes xs :: "'e list"   and f :: "'e \<Rightarrow> 'd list" shows "xs \<bind> f = concat (map f xs)"
t_find_closest_pair_bound: fixes ps :: "(int \<times> int) list"   and \<delta> :: "real"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int"   and \<Delta> :: "real" assumes "distinct ps"   and "sorted_snd ps"   and "\<delta> = dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"   and "set ps = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p\<in>set ps. real_of_int l - \<Delta> < real_of_int (fst p) \<and> real_of_int (fst p) < real_of_int l + \<Delta>"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse \<Delta> ps\<^sub>L"   and "sparse \<Delta> ps\<^sub>R"   and "\<delta> \<le> \<Delta>" shows "t_find_closest_pair (c\<^sub>0, c\<^sub>1) ps \<le> 9 * length ps + 1"
continuous_on_generated_topo: fixes S :: "'c set set"   and T1 :: "'d topology"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>U. U \<in> S \<Longrightarrow> openin T1 (f -` U \<inter> topspace T1)"   and "f ` topspace T1 \<subseteq> \<Union> S" shows "continuous_map T1 (topology_generated_by S) f"
matrix_mult_index: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and m1 :: "'a list list"   and nr :: "nat"   and n :: "nat"   and nc :: "nat"   and m2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "m1 \<noteq> []"   and "Matrix_Legacy.mat nr n m1"   and "Matrix_Legacy.mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI zer g f (mult.row_length m1) m1 m2 ! j ! i = plus_mult.scalar_product f zer g (Matrix_Legacy.row m1 i) (Matrix_Legacy.col m2 j)"
assms_22: shows "InvariantVarsQ (getQ state) F0 Vbl"
context_left: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and u :: "('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "App t u \<rightarrow>\<^sub>e\<^sub>m\<^sub>b App s u"
nn_integral_bind: fixes f :: "'c \<Rightarrow> ennreal"   and B :: "'c measure"   and N :: "'d \<Rightarrow> 'c measure"   and M :: "'d measure" assumes "f \<in> borel_measurable B"   and "N \<in> M \<rightarrow>\<^sub>M subprob_algebra B" shows "integral\<^sup>N (M \<bind> N) f = \<integral>\<^sup>+ x. integral\<^sup>N (N x) f \<partial>M"
a1: shows "q \<cdot> \<langle>x| p = (zero_class.zero::'a)"
distinct_83: fixes pi1 :: "pi"   and pi2 :: "pi"   and pi' :: "pi" shows "pi1 \<oplus> pi2 \<noteq> !pi'"
domains_def: fixes f :: "'a arrow_of_spans_data" shows "V.domains f \<equiv> {a. V.ide a \<and> f \<bullet> a \<noteq> V.null}"
swap_ident: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "X #[x \<and> x]_xs = X"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_4: fixes t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
divide_eq_1_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(a / b = 1) = (b \<noteq> \<top> \<and> b \<noteq> 0 \<and> b = a)"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1"
cases: fixes a :: "'c"   and r :: "('c \<times> 'd) set" assumes "a \<in> Domain r" obtains aa :: "'c" and b :: "'d" where "a = aa" and "(aa, b) \<in> r"
may_lockI: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "l = None \<or> (\<exists>n. l = \<lfloor>(t, n)\<rfloor>)" shows "may_lock l t"
es: shows "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:])"
a2: shows "take (length i2 - Suc 0) i1 = butlast i2"
gfp_unfold: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "gfp f = f (gfp f)"
ptrm_prv_exu: fixes \<sigma> :: "'fmla"   and n :: "nat" assumes "\<sigma> \<in> ptrm n" shows "prv (exu out \<sigma>)"
Tn_integral_preserving_2: fixes f :: "'a \<Rightarrow> 'c"   and n :: "nat" assumes "integrable M f" shows "LINT x|M. f ((T ^^ n) x) = integral\<^sup>L M f"
assms: shows "\<forall>x. f x = \<infinity> \<or> f x = - \<infinity>"
left_ideal_eq_right_ideal: fixes I :: "'a set" shows "left_ideal I = right_ideal I"
finite_imp_not_open: fixes S :: "'b set" assumes "finite S"   and "open S" shows "S = {}"
ord_\<Gamma>_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "gd.ord_\<Gamma> Sts = {uu_. \<exists>CAs DA AAs As E. uu_ = old_Infer (mset CAs) DA E \<and> gd.ord_resolve Sts CAs DA AAs As E}"
refine: fixes succ :: "nat \<Rightarrow> nat dlist"   and src :: "nat"   and dst :: "nat" shows "nres_of (bfs_code succ src dst) \<le> bfs_impl succ src dst"
simps_2: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "insert_in_list_at_arbitrary_pos x (y # ys) = insert (x # y # ys) ((#) y ` insert_in_list_at_arbitrary_pos x ys)"
wf_imp_irrefl: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "irrefl r"
vfsequence_axioms: shows "vfsequence (xs #\<^sub>\<circ> x)"
mat_inverse_simps_2: fixes c :: "'c" assumes "c \<noteq> (0::'c)" shows "mat c ** mat (inverse c) = mat (1::'c)"
starfun_divide: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b star"   and g :: "'b \<Rightarrow> 'a" shows "(*f* f) x / (*f* g) x = (*f* (\<lambda>x. f x / g x)) x"
sel_10: fixes x51 :: "nat"   and x52 :: "'d"   and x53 :: "'d" shows "getId (RecvMarker x51 x52 x53) = x51"
i_set_singleton: fixes a :: "nat" shows "{a} \<in> i_set"
d: fixes w :: "complex" assumes "w \<in> ball x d - {a}" shows "g w = (w - a) * f w"
half_space: shows "\<exists>v. norm v = 1 \<and> (\<forall>l. v \<bullet> (T ^^ l) (0::'a) \<le> - A * real l)"
assms: shows "\<langle>\<langle>r, A\<rangle>, s\<rangle> \<in>\<^sub>\<circ> vrestriction D"
gt_sub_fun: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "rpo_optim ground_heads_var gt_sym extf arity_sym arity_var" shows "rpo_optim.gt ground_heads_var gt_sym extf (App s t) s"
cases: fixes a1 :: "'a sequent"   and a2 :: "nat"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "(a1, a2) \<in> derivable R" obtains C :: "'a sequent" where "a1 = C" and "a2 = 0" and "([], C) \<in> R"   | r :: "'a sequent list \<times> 'a sequent" and m :: "nat" where "a1 = snd r" and "a2 = m + 1" and "r \<in> R" and "fst r \<noteq> []" and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R"
Semi: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c1 :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c2 :: "com"   and R :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<turnstile>\<^sub>t {P} c1 {Q}"   and "C \<turnstile>\<^sub>t {Q} c2 {R}" shows "C \<turnstile>\<^sub>t {P} c1; c2 {R}"
sym_Un_converse: fixes r :: "('b \<times> 'b) set" shows "sym (r \<union> r\<inverse>)"
q2: shows "\<forall>x. \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel> \<le> M * \<parallel>(A\<^sup>\<dagger>) x\<parallel>"
simps_1: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c" shows "insort_key f x [] = [x]"
surjective_sum: fixes f :: "'c + 'd \<Rightarrow> 'e" shows "case_sum (\<lambda>x. f (Inl x)) (\<lambda>y. f (Inr y)) = f"
hbex_triv_one_point2: fixes A :: "hf"   and a :: "hf" shows "HBex A ((=) a) = (a \<^bold>\<in> A)"
connectedD_interval: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and U :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.linorder_topology less_eq less open"   and "topological_space.connected open U"   and "x \<in> U"   and "y \<in> U"   and "less_eq x z"   and "less_eq z y" shows "z \<in> U"
point_rep_singleton_val: fixes x :: "'b"   and b :: "'b set" assumes "x \<in> b" shows "{#b#} rep x = 1"
execs_Fault_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>cs,css,s\<rangle> \<Rightarrow> t"   and "s = Fault f" shows "t = Fault f"
AB: shows "PTA.rel_set_strong (stream_all2 R) A B"
brnL_mono: fixes n1 :: "nat"   and n2 :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list" assumes "n1 \<le> n2" shows "brnL cl n1 \<le> brnL cl n2"
Nneg_def: shows "~\<^sub>N = (\<lambda>(a, b). (- a, b))"
finsum_empty: fixes f :: "'b \<Rightarrow> 'a vec" shows "finsum V f {} = 0\<^sub>v n"