hcju/isabelleps huggingface dataset:
ltlc_to_rabin_def: fixes eager :: "bool"   and mode :: "mode"   and \<phi>\<^sub>c :: "String.literal ltlc" shows "ltlc_to_rabin eager mode \<phi>\<^sub>c \<equiv> let \<phi>\<^sub>n = ltlc_to_ltln \<phi>\<^sub>c; \<Sigma> = map set (subseqs (atoms_list \<phi>\<^sub>n)); \<phi> = ltln_to_ltl (simplify mode \<phi>\<^sub>n) in if eager then ltl_to_generalized_rabin\<^sub>C_af\<^sub>\<UU> \<Sigma> \<phi> else ltl_to_generalized_rabin\<^sub>C_af \<Sigma> \<phi>"
subst_cls_id_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and C :: "'b literal multiset" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_cls subst_atm C id_subst = C"
con_rews_1: fixes e :: "'b" shows "Err\<cdot>e \<noteq> \<bottom>"
Main_no_return_source: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>Main\<^esub>f'" shows "False"
wp3_is_pre: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3 {wp\<^sub>3 c Q} c { Q}"
assms_1: shows "succ_rel (insert_spec (b # list) (oid, Some ref)) = insert_alt (succ_rel (b # list)) (oid, ref)"
rep_eq: fixes x :: "'d" shows "apply_bcontfun (const_bcontfun x) = (\<lambda>_. x)"
exhaust: fixes y :: "'b point" obtains "y = Infinity"   | x21 :: "'b" and x22 :: "'b" where "y = Point x21 x22"
mult_sign_intros_5: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) < a"   and "(0::'b) < b" shows "(0::'b) < a * b"
remove1Prio_remove1: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.remove1Prio (BinomialHeapStruc.prio (BinomialHeapStruc.getMinTree bq)) bq = remove1 (BinomialHeapStruc.getMinTree bq) bq"
2: shows "avl \<langle>l, (ab, h), r\<rangle>"
pair_list_eqI: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "map fst xs = map fst ys"   and "map snd xs = map snd ys" shows "xs = ys"
append_child_impl: shows "Shadow_DOM.append_child = Shadow_DOM.i_append_child.a_append_child"
is_path_split: fixes G :: "('a, 'b) graph"   and v :: "'a"   and p1 :: "('a \<times> 'b \<times> 'a) list"   and p2 :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_graph G" shows "valid_graph.is_path G v (p1 @ p2) v' = (\<exists>u. valid_graph.is_path G v p1 u \<and> valid_graph.is_path G u p2 v')"
steps_epsclosure: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and w :: "'d list" shows "(eps A)\<^sup>* O NAe.steps A w = NAe.steps A w"
relation_subset_crossp: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "relation \<subseteq> carrier \<times> carrier"
strict_trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b <p a"   and "c <p b" shows "c <p a"
obj_src: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.obj (src\<^sub>D \<mu>)"
SWF_rpr: fixes swf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and Pcond :: "'b set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "SWF swf A Is Pcond"   and "Pcond A Is P" shows "rpr A (swf P)"
comm: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "Matrix_Tensor.mult id f" shows "f a b = f b a"
power3_eq_cube: fixes a :: "'b list" shows "a \<^sup>@ 3 = (a \<cdot> a) \<cdot> a"
map_idI: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
max_Var_floatariths_euler_err_fas: fixes xs :: "floatarith list"   and cxs :: "floatarith list"   and h :: "floatarith"   and z :: "nat" assumes "0 < length ode_e"   and "max_Var_floatariths ode_e \<le> length ode_e"   and "length xs = length ode_e"   and "length cxs = length ode_e"   and "max (max_Var_floatariths xs) (max (max_Var_floatarith h) (max_Var_floatariths cxs)) \<le> z" shows "max_Var_floatariths (euler_err_fas xs h cxs) \<le> z"
one_step_equiv: fixes T1 :: "'f inference set"   and N1 :: "('f \<times> 'l) set"   and T2 :: "'f inference set"   and N2 :: "('f \<times> 'l) set" assumes "(T1, N1) \<leadsto>LGC (T2, N2)" shows "N1 \<rhd>L N2"
measurable_pair_swap': fixes M1 :: "'c measure"   and M2 :: "'d measure" shows "(\<lambda>(x, y). (y, x)) \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M2 \<Otimes>\<^sub>M M1"
subcls1_induct_aux: fixes P :: "'b prog"   and C :: "char list"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and Q :: "char list \<Rightarrow> bool" assumes "is_class P C"   and "wf_prog wf_md P"   and "Q Object"   and "\<And>C D fs ms. \<lbrakk>C \<noteq> Object; is_class P C; class P C = \<lfloor>(D, fs, ms)\<rfloor> \<and> wf_cdecl wf_md P (C, D, fs, ms) \<and> P \<turnstile> C \<prec>\<^sup>1 D \<and> is_class P D \<and> Q D\<rbrakk> \<Longrightarrow> Q C" shows "Q C"
homeomorphic_interiors: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "(interior S = {}) = (interior T = {})" shows "interior S homeomorphic interior T"
radical_sqrt_simultaneous_quadratic_quadratic: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and e :: "real"   and f :: "real"   and x :: "real"   and y :: "real" assumes "a \<in> radical_sqrt"   and "b \<in> radical_sqrt"   and "c \<in> radical_sqrt"   and "d \<in> radical_sqrt"   and "e \<in> radical_sqrt"   and "f \<in> radical_sqrt"   and "\<not> (a = d \<and> b = e \<and> c = f)"   and "(x - a)\<^sup>2 + (y - b)\<^sup>2 = c"   and "(x - d)\<^sup>2 + (y - e)\<^sup>2 = f" shows "x \<in> radical_sqrt \<and> y \<in> radical_sqrt"
tauTransitionChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
f_assoc: fixes x :: "'a"   and z :: "'a" shows "f x (f x z) = f (f x x) z"
range_from_nat_into_subset: fixes A :: "'b set" assumes "A \<noteq> {}" shows "range (from_nat_into A) \<subseteq> A"
gga_from_list_correct: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and e :: "unit \<Rightarrow> 'd"   and a :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and u :: "'e \<Rightarrow> 'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "graph_empty \<alpha> invar e"   and "graph_add_node \<alpha> invar a"   and "graph_add_edge \<alpha> invar u" shows "graph_from_list \<alpha> invar (gga_from_list e a u)"
obj_simps_1: fixes a :: "'d" assumes "D.obj a" shows "D.arr a"
gf_U_eq_U: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" shows "MC_syntax.gf_U K x y 1 = MC_syntax.U K x y"
mset_nondetMs_1: fixes return :: "'a multiset \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a multiset \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind" shows "monad (nondetM_base.return_nondet return (\<lambda>x. {#x#})) (nondetM_base.bind_nondet bind (mmerge return bind))"
cfg_onD_action: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "cfg \<in> R_G.cfg_on s" shows "action cfg \<in> \<K> s"
isomorphic_reflexive: fixes f :: "'a" assumes "ide f" shows "f \<cong> f"
restrict_map_def: fixes m :: "'d \<Rightarrow> 'e option"   and A :: "'d set" shows "m |` A = (\<lambda>x. if x \<in> A then m x else None)"
cases: fixes x :: "'c list \<times> 'c list" obtains ys :: "'c list" where "x = ([], ys__)"   | xa :: "'c" and xs :: "'c list" and ys :: "'c list" where "x = (xa # xs, ys)"
max_ip_addr_max: shows "\<forall>a. a \<le> max_ip_addr"
rec_conf_def: shows "rec_conf = Pr 2 (Cn 2 rec_trpl [Cn 2 (constn 0) [recf.id 2 0], Cn 2 (constn (Suc 0)) [recf.id 2 0], recf.id 2 1]) (Cn 4 rec_newconf [recf.id 4 0, recf.id 4 3])"
fresh_ssubst_dbfm: fixes i :: "'b"   and V :: "name set"   and A :: "dbfm" assumes "atom i \<sharp> p \<bullet> V"   and "V \<subseteq> Vs" shows "atom i \<sharp> ssubst (vquot_dbfm V A) V F"
chine_assoc_props_1: shows "\<guillemotleft>\<mu>\<nu>_\<pi>_\<rho>.chine_assoc : HHH\<mu>\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C HH\<mu>\<nu>H\<pi>\<rho>.chine\<guillemotright>"
hyps_free_limited: fixes v :: "'vertex"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "terminal_path v v' pth" shows "length pth \<le> fcard vertices"
take_length: fixes xs :: "'a list" shows "take (length xs) xs = xs"
seqs_Suc: fixes n :: "nat" shows "seqs (Suc n) = (#) True ` seqs n \<union> (#) False ` seqs n"
Max_by_UNION_distrib: fixes A :: "'a set"   and f :: "'b \<Rightarrow> 'a set"   and S :: "'b set"   and cmp_f :: "'a \<Rightarrow> 'c" assumes "finite A"   and "A = \<Union> (f ` S)"   and "S \<noteq> {}"   and "{} \<notin> f ` S"   and "inj_on cmp_f A" shows "Max_by cmp_f A = Max_by cmp_f (Max_by cmp_f ` f ` S)"
sints_num: fixes n :: "nat" shows "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
R_finrank_def: fixes M :: "'v set" shows "R_finrank M = (\<exists>n. \<forall>ms. set ms \<subseteq> M \<and> R_lin_independent ms \<longrightarrow> length ms \<le> n)"
type_stable_is_sufficient: fixes t :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set" assumes "type_stable \<S> t" shows "\<forall>mem mem'. (\<forall>x. stable \<S> x \<longrightarrow> mem x = mem' x) \<longrightarrow> ev\<^sub>B mem ` t = ev\<^sub>B mem' ` t"
setUp_program_inv': fixes ast :: "AST.module list" shows "program_inv (fst (setUp ast))"
cases: fixes r :: "state" obtains heap :: "heap" and locals :: "locals" where "r = \<lparr>heap = heap, locals = locals\<rparr>"
d2: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "- - (p \<sqinter> x) \<le> p"
emeasure_eq_0_AE: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "AE x in M. \<not> P x" shows "emeasure M {x \<in> space M. P x} = 0"
add_mult_one_time_pad: fixes s0 :: "nat"   and b :: "nat"   and r0 :: "nat" assumes "s0 < order \<G>"   and "s0 \<noteq> 0" shows "map_spmf (\<lambda>c\<^sub>v'. (b * r0 + s0 * c\<^sub>v') mod order \<G>) (sample_uniform (order \<G>)) = sample_uniform (order \<G>)"
simps_3: fixes x :: "'b" shows "New_max.merge_option (None, Some x) = Some x"
subprob_space_sigma: fixes \<Omega> :: "'c set"   and X :: "'c set set" assumes "\<Omega> \<noteq> {}" shows "subprob_space (sigma \<Omega> X)"
nonneg_cexpr_int: fixes V :: "nat set"   and t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e" shows "nonneg_cexpr V \<Gamma> (\<integral>\<^sub>c e \<partial>t)"
closed_hterm_1: fixes ht :: "'a hterm" shows "closedt 0 (term_of_hterm ht)"
Fourier_sum_offset_Fejer_kernel_half: fixes f :: "real \<Rightarrow> real"   and n :: "nat"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "0 < n" shows "(\<Sum>r<n. \<Sum>k\<le>2 * r. Fourier_coefficient f k * trigonometric_set k t) / real n - l = (LINT x|lebesgue_on {0..pi}. Fejer_kernel n x * (f (t + x) + f (t - x) - 2 * l)) / pi"
phis_finite: fixes g :: "'g" shows "finite (dom (Mapping.lookup (phis g)))"
return_returns_result: fixes h :: "'x"   and x :: "'z"   and y :: "'z" shows "h \<turnstile> return x \<rightarrow>\<^sub>r y = (x = y)"
redT_updW_simps_5: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option" shows "redT_updW t ws WokenUp ws' = (ws' = ws(t := None))"
map_pmf_def: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
lookup_correct: fixes m :: "'b"   and k :: "'a" assumes "P m" shows "local.map_of (snd (run_state (lookup k) m)) \<subseteq>\<^sub>m local.map_of m"
ne_matchers_imp_matchrel: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option"   and e :: "('e, 'd) Term.term \<times> ('e, 'f) Term.term" assumes "matchers_map \<sigma> \<inter> matchers {e} \<noteq> {}" shows "\<exists>y. (({#e#}, \<sigma>), y) \<in> matchrel"
Mismatch_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and b :: "name"   and x :: "name"   and P' :: "pi"   and a :: "name"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>b<x> \<prec> P'"   and "a \<noteq> c" shows "[a\<noteq>c]P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>b<x> \<prec> P'"
abs_eq: fixes xa :: "'a set"   and x :: "('a \<times> 'a) set" shows "Rel xa x = rel_of_set (Id_on xa \<union> x)"
map_filter_M_pure: fixes xs :: "'l list"   and f :: "'l \<Rightarrow> ('m, 'n, 'o option) prog"   and h :: "'m" assumes "\<And>x h. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_filter_M f xs) h"
rel_set_Grp: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "rel_set (BNF_Def.Grp A f) = BNF_Def.Grp {X. X \<subseteq> A} ((`) f)"
that: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "valid_decomp X ps"   and "cone_decomp (cone (h, U)) ps"   and "standard_decomp (Suc (poly_deg h)) ps"   and "\<lbrakk>is_monomial h; lcf h = (1::'a)\<rbrakk> \<Longrightarrow> monomial_decomp ps"   and "homogeneous h \<Longrightarrow> hom_decomp ps" shows "thesis"
inf1I: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a"   and B :: "'a \<Rightarrow> bool" assumes "A x"   and "B x" shows "inf A B x"
emeasure_UN_eq_0: fixes M :: "'b measure"   and N :: "nat \<Rightarrow> 'b set" assumes "\<And>i. emeasure M (N i) = 0"   and "range N \<subseteq> sets M" shows "emeasure M (\<Union> (range N)) = 0"
rb_aux_inv_is_RB_upt_Cons: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv (bs, ss, p # ps)" shows "is_RB_upt dgrad rword (set bs) (sig_of_pair p)"
impl_of: fixes x :: "('a, 'b) alist" shows "alist.impl_of x \<in> {xs. (distinct \<circ> map fst) xs}"
tr_def_1: shows "path M (w1 @ w2 || tr) q"
simulation1_aux: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl1 :: "'tl1"   and s1' :: "'s1" assumes "s1 \<approx> s2"   and "s1 -1-tl1\<rightarrow> s1'"   and "\<not> \<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. s2 -\<tau>2\<rightarrow>* s2' \<and> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> s1' \<approx> s2'' \<and> tl1 \<sim> tl2"
fps_inverse_eq_0_iff: fixes f :: "'a fps" shows "(inverse f = 0) = (fps_nth f 0 = (0::'a))"
combined_restrict_perm: fixes \<pi> :: "perm"   and S :: "Vars.var set"   and p :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" assumes "supp \<pi> \<sharp>* S"   and "finite S" shows "combined_restrict S (\<pi> \<bullet> p) = combined_restrict S p"
coeff_sym_mpoly_monom_of_set: fixes X :: "nat set"   and Y :: "nat set"   and k :: "nat" assumes "finite X"   and "Y \<subseteq> X"   and "card Y = k" shows "MPoly_Type.coeff (sym_mpoly X k) (monom_of_set Y) = (1::'b)"
cupcake_match_result_preserve: fixes envC :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and pes :: "(pat \<times> exp0) list"   and v :: "v"   and err_v :: "v" assumes "cupcake_c_ns envC"   and "cupcake_clauses pes"   and "is_cupcake_value v" shows "if_rval (\<lambda>(e, p, env'). is_cupcake_pat p \<and> is_cupcake_exp e \<and> list_all (is_cupcake_value \<circ> snd) env') (cupcake_match_result envC v pes err_v)"
ex_min_if_finite: fixes S :: "'b set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>m\<in>S. \<not> (\<exists>x\<in>S. x < m)"
ta_reduce_def: fixes TA :: "('d, 'e) tree_automaton_rec"   and P :: "'d set" shows "ta_reduce TA P \<equiv> \<lparr>ta_initial = ta_initial TA \<inter> P, ta_rules = reduce_rules (ta_rules TA) P\<rparr>"
cofactorM_def: fixes A :: "(('c, 'd) vec, 'd) vec" shows "cofactorM A = (\<chi>i j. cofactor A i j)"
wf_digraph_axioms: shows "wf_digraph (del_arc u)"
filter_empty_conv: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
find_lambda_measures_decreases1: fixes c :: "nat"   and i :: "nat"   and c' :: "nat" assumes "c = 2 ^ i"   and "mu \<le> carrots_total c \<longrightarrow> c \<le> lambda"   and "seq (carrots_total c) \<noteq> seq (carrots_total c + c)" shows "((c', 2 * c), c, c) \<in> find_lambda_measures"
finite_subcls1: fixes P :: "'a prog" shows "finite {(C, D). P \<turnstile> C \<prec>\<^sup>1 D}"
sel_1: fixes x :: "complex"   and y :: "complex" shows "Re (x - y) = Re x - Re y"
rel_pmfD_measure: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf"   and y :: "'c"   and x :: "'b" assumes "rel_pmf R p q"   and "\<And>a b. R a b \<Longrightarrow> R a y = R x b"   and "x \<in> set_pmf p"   and "y \<in> set_pmf q" shows "measure_pmf.prob p {x. R x y} = measure_pmf.prob q {y. R x y}"
fun_upd_triv: fixes f :: "'f \<Rightarrow> 'g"   and x :: "'f" shows "f(x := f x) = f"
is_ground_lit_def: fixes L :: "('a, 'b) Term.term literal" shows "is_ground_lit L = is_ground_atm (atm_of L)"
measurable_T2: fixes M :: "'b measure"   and s :: "'s" shows "M \<rightarrow>\<^sub>M T s = M \<rightarrow>\<^sub>M stream_space (count_space UNIV)"
assms_1: shows "h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h'"
matrix_row_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "'a list list" assumes "mat nr nc M" shows "mat (row_length M) (length M) M"
valid_pub: shows "(x0, x1) \<in> valid_pub_AND"
chamber_in_opp_half_apartment: fixes C :: "'a set" assumes "SimplicialComplex.maxsimp folding.Y C" shows "folding.chamber C"
hom_induced_compose: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and Z :: "'d topology"   and g :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and p :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y Z g"   and "g ` T \<subseteq> U" shows "hom_induced p X S Z U (g \<circ> f) = hom_induced p Y T Z U g \<circ> hom_induced p X S Y T f"
def_wfrec: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a \<times> 'a) set"   and F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a" assumes "f \<equiv> wfrec R F"   and "wf R" shows "f a = F (cut f R a) a"
reflpI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
simps_2: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and x :: "'b" shows "ptrm_infer_type \<Gamma> (PVar x) = \<Gamma> x"
in_hhomE: fixes \<mu> :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>" obtains "D.arr \<mu>" and "D.obj a" and "D.obj b" and "src\<^sub>D \<mu> = a" and "trg\<^sub>D \<mu> = b"
transaction_selects_def: fixes prot_transaction :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_selects prot_transaction = (case prot_transaction of Transaction x1 x2 x3 x4 x5 x6 \<Rightarrow> x3)"
aleb: fixes x :: "'a" assumes "x \<in> S" shows "a \<bullet> x \<le> b"
hyps_3: assumes "S - {Q} \<noteq> {}" shows "\<exists>Q\<in>S - {Q}. depth P \<le> depth Q \<and> (\<forall>R\<in>S - {Q}. depth R \<le> depth Q)"
scene_union_annhil: fixes X :: "'a scene" assumes "idem_scene X" shows "X \<squnion>\<^sub>S \<top>\<^sub>S = \<top>\<^sub>S"
finite_state_imp_restriction_finite_1: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and r :: "'d" assumes "substitution subst"   and "finite (RID\<^sub>G s)" shows "finite (RID\<^sub>G (s(r := None)))"
bo: shows "bounded {integral (cbox c d) f |c d. cbox c d \<subseteq> box a b}"
vu: shows "w \<le>p (v \<cdot> u) \<cdot> w"
frontier_of_Un_subset: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X frontier_of (S \<union> T) \<subseteq> X frontier_of S \<union> X frontier_of T"
SucState_Conf_Source_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "S \<notin> Target TS"   and "A \<in> the (CompFun (HA ST) S)"   and "S \<in> Conf ST"   and "States A \<inter> ChiStar (HA ST) `` Source TS \<noteq> {}"   and "S \<notin> ChiStar (HA ST) `` Source TS" shows "\<exists>x. x \<in> States A \<and> x \<in> StepConf (HA ST) (Conf ST) TS"
high_proc\<^sub>4_s\<^sub>H\<^sub>0: shows "high_proc\<^sub>4 s\<^sub>H\<^sub>0"
chain_map_diff: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" shows "chain_map p g (a - b) = chain_map p g a - chain_map p g b"
option2elem: fixes opt :: "ZF"   and g :: "ZF" assumes "(opt, g) \<in> is_option_of" shows "\<exists>u v. Elem opt u \<and> Elem u v \<and> Elem v g"
1: shows "w\<^sub>1 @ w\<^sub>2 =\<^sub>F w"
curry_case_prod: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd" shows "curry (\<lambda>(x, y). f x y) = f"
wt_elim_cases_25: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and l :: "label"   and e :: "expr"   and c :: "stmt"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (l\<bullet> While(e) c)\<Colon>x" obtains "x = Inl (PrimT Void)" and "E,dt\<Turnstile>e\<Colon>-PrimT Boolean" and "E,dt\<Turnstile>c\<Colon>\<surd>"
dghm_proj_snd_def: fixes \<AA> :: "V"   and \<BB> :: "V" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>2 \<AA> \<BB> = \<pi>\<^sub>D\<^sub>G (2\<^sub>\<nat>) (\<lambda>i. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>) (1\<^sub>\<nat>)"
ideD_2: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.dom C a = a"
Base_foo_no_hide: fixes sig :: "sig"   and old :: "qtname \<times> methd"   and P :: "bool" assumes "tprg,sig\<turnstile> (Base, snd Base_foo) hides old" shows "P"
isubexprs_def: shows "isubexprs \<equiv> {(xa, x). isubexprsp xa x}"
rcos_self: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "subgroup H (G\<lparr>carrier := S\<rparr>)" shows "x \<in> H #>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x"
coprime_common_divisor: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "algebraic_semidom.coprime times one a b"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "dvd.dvd times c one"
induced_automorphism_half_chamber_system_pgallery_map_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and Cs :: "'b set list" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "order.greater_eq (f \<turnstile> ChamberComplex.chamber_system X) (set Cs)"   and "SimplicialComplex.pmaxsimpchain X Cs" shows "SimplicialComplex.pmaxsimpchain X (OpposedThinChamberComplexFoldings.induced_automorphism X f g \<Turnstile> Cs)"