hcju/isabelleps huggingface dataset:
swap_preserves_fresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and x :: "'var" assumes "good X" shows "fresh xs (x @xs[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys) = fresh xs x X"
lift_stop_oracle_def: fixes "oracle" :: "'h \<Rightarrow> 'i \<Rightarrow> ('g \<times> 'h) spmf"   and s :: "'h"   and x :: "'i" shows "lift_stop_oracle oracle s x = map_spmf (apfst Some) (oracle s x)"
seq_trel: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and P :: "'b hrel"   and \<rho> :: "'b \<Rightarrow> 'b"   and Q :: "'b hrel"   and R :: "'b hrel" assumes "(\<sigma>, P) \<rightarrow>\<^sub>u (\<rho>, Q)" shows "(\<sigma>, P ;; R) \<rightarrow>\<^sub>u (\<rho>, Q ;; R)"
S: shows "space (return_val (expr_sem_rf \<sigma> e1)) = type_universe t1"
ucast_distrib: fixes M :: "'b word \<Rightarrow> 'b word \<Rightarrow> 'b word"   and L :: "int \<Rightarrow> int \<Rightarrow> int"   and M' :: "'c word \<Rightarrow> 'c word \<Rightarrow> 'c word"   and a :: "'b word"   and b :: "'b word" assumes "\<And>x y. uint (M x y) = L (uint x) (uint y) mod 2 ^ LENGTH('b)"   and "\<And>x y. uint (M' x y) = L (uint x) (uint y) mod 2 ^ LENGTH('c)"   and "\<And>x y. L (x mod 2 ^ LENGTH('c)) (y mod 2 ^ LENGTH('c)) mod 2 ^ LENGTH('c) = L x y mod 2 ^ LENGTH('c)"   and "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) (M a b) = M' (UCAST('b \<rightarrow> 'c) a) (UCAST('b \<rightarrow> 'c) b)"
open_openin_trans: fixes S :: "'c set"   and T :: "'c set" assumes "open S"   and "open T"   and "T \<subseteq> S" shows "openin (top_of_set S) T"
sorted_upper: shows "sorted_sublist_map R h xs'' (Suc p) hi"
sliceable: fixes v :: "'b list"   and S :: "'b list set"   and \<sigma> :: "'a trace"   and i :: "nat" assumes "v \<in> S" shows "sat (slice S \<sigma>) v i = sat \<sigma> v i"
Arr_rep: fixes f :: "'c local.term set" assumes "arr f" shows "Arr (rep f)"
path_singleton: fixes v :: "Vertices" assumes "v \<in> V\<^bsub>T2\<^esub>" shows "path [v]"
lowEquivSymmetric: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option" assumes "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2" shows "\<Gamma> \<turnstile> s2 \<approx>\<^sub>L s1"
trans: fixes b :: "'d list"   and a :: "'d list"   and c :: "'d list" assumes "prefix b a"   and "prefix c b" shows "prefix c a"
Cdom: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
pred_def: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" shows "(P \<in> M \<rightarrow>\<^sub>M BOOL) = ({x \<in> space M. P x} \<in> sets M)"
Times_mset_single_right: fixes A :: "'b multiset"   and b :: "'c" shows "A \<times># {#b#} = {#(a, b). a \<in># A#}"
stable_no_blocking_onI2: fixes ds :: "'b set"   and X :: "'a set" assumes "\<And>h X''. PdXXX.blocking_on ds X h X'' \<Longrightarrow> False" shows "PdXXX.stable_no_blocking_on ds X"
ys: shows "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) [Monitor.progress \<sigma> \<psi> j..<Monitor.progress \<sigma> \<psi> j'] ys"
canceled_rev: fixes l :: "(bool \<times> 'd) list" assumes "canceled l" shows "canceled (rev l)"
h2: shows "well_def_signs (length (qs1 @ qs2)) (get_signs_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
float_of_inject: fixes x :: "real"   and y :: "real" assumes "x \<in> float"   and "y \<in> float" shows "(float_of x = float_of y) = (x = y)"
semilattice_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semilattice f \<equiv> abel_semigroup f \<and> semilattice_axioms f"
in_set_zip: fixes p :: "'b \<times> 'c"   and xs :: "'b list"   and ys :: "'c list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
scaleR2_rep_def: fixes X :: "('a \<times> 'b) set" shows "scaleR2_rep X = SPEC (\<lambda>((l, u), Y). ereal -` {l..u} \<noteq> {} \<and> X = scaleR2 l u Y)"
good_interp: shows "\<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x)"
digraph_def: fixes \<alpha> :: "V"   and \<CC> :: "V" shows "digraph \<alpha> \<CC> \<equiv> (\<Z> \<alpha> \<and> vfsequence \<CC>) \<and> vsv (\<CC>\<lparr>Dom\<rparr>) \<and> vsv (\<CC>\<lparr>Cod\<rparr>) \<and> digraph_axioms \<alpha> \<CC>"
is_unit_prod_mset_iff: fixes A :: "'a multiset" shows "is_unit (\<Prod>\<^sub># A) = (\<forall>x\<in>#A. is_unit x)"
countable_UN': fixes X :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "countable X"   and "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
up_eqI: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "\<And>n. coeff P p n = coeff P q n"   and "p \<in> carrier P"   and "q \<in> carrier P" shows "p = q"
singleton_inject: fixes a :: "'a"   and b :: "'a" assumes "{a} = {b}" shows "a = b"
closed_map_on_empty: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "topspace X = {}" shows "closed_map X Y f"
usubstsize_def: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)" shows "usubstsize \<sigma> = (if dom (SFuncs \<sigma>) = {} \<and> dom (SPreds \<sigma>) = {} then 1 else 2)"
cfun_cong: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "f = g"   and "x = y" shows "f\<cdot>x = g\<cdot>y"
emp_back: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and II :: "nat set" shows "(back cl n II = {}) = (II = {})"
perm_simps_1: fixes p :: "perm" shows "p \<bullet> zer = zer"
l2_inv7_aux: fixes s :: "l2_state"   and x :: "msg" assumes "s \<in> l2_inv7"   and "x \<in> secret s" shows "x \<notin> synth (analz (generators s))"
I_def: shows "I = (\<lambda>\<chi>. {z. z ^ h = \<chi> c})"
paths_edge_finite: shows "finite (\<Union> (H.edges_of_walk ` Q))"
gauss_jordan_single_rank: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "n \<le> nc" shows "vec_space.rank n (take_cols A (map snd (pivot_positions (gauss_jordan_single A)))) = vec_space.rank n A"
TESL_sem_decreases_tail: fixes \<Gamma> :: "'a constr list"   and \<gamma> :: "'a constr" shows "\<lbrakk>\<lbrakk> \<Gamma> @ [\<gamma>] \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<subseteq> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
xx: fixes x :: "'b" shows "summable (\<lambda>n. of_real (inverse (fact n)) * x ^ n)"
F_3: fixes i :: "'i" shows "\<Union> (range (F i)) = space (M i)"
abs_eq: fixes x :: "Vars.var list" shows "ccFromList x = Abs_CoCall {(x, y). list_pairs x (x, y) \<or> list_pairs x (y, x)}"
p: shows "p = monomial (1::'b) (term_of_pair (0::'a, component_of_term v))"
dist_les_2: fixes cfun :: "ValD \<rightarrow> ValD" shows "ValF\<cdot>cfun \<notsqsubseteq> ValFF"
subpath_smooth: fixes \<gamma> :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" assumes "\<gamma> C1_differentiable_on {0..1}"   and "0 \<le> a"   and "a < b"   and "b \<le> 1" shows "subpath a b \<gamma> C1_differentiable_on {0..1}"
af\<^sub>G_eval\<^sub>G: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl"   and w :: "'b set list" assumes "Only_G \<G>" shows "\<G> \<Turnstile>\<^sub>P af\<^sub>G (eval\<^sub>G \<G> \<phi>) w = \<G> \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w)"
r_code_const1: fixes c :: "nat" shows "eval r_code_const1 [c] \<down>= code_const1 c"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int" assumes "\<And>a n. \<lbrakk>\<lbrakk>n \<noteq> 0; n \<noteq> 1; a \<noteq> 1; n < 0\<rbrakk> \<Longrightarrow> P a (- n); \<lbrakk>n \<noteq> 0; n \<noteq> 1; a \<noteq> 1; \<not> n < 0; even n; odd a\<rbrakk> \<Longrightarrow> P a (n div 2); \<lbrakk>n \<noteq> 0; n \<noteq> 1; a \<noteq> 1; \<not> n < 0; odd n; a < 0\<rbrakk> \<Longrightarrow> P (- a) n; \<lbrakk>n \<noteq> 0; n \<noteq> 1; a \<noteq> 1; \<not> n < 0; odd n; \<not> a < 0; a \<noteq> 0; n \<le> a\<rbrakk> \<Longrightarrow> P (a mod n) n; \<lbrakk>n \<noteq> 0; n \<noteq> 1; a \<noteq> 1; \<not> n < 0; odd n; \<not> a < 0; a \<noteq> 0; \<not> n \<le> a; even a\<rbrakk> \<Longrightarrow> P (a div 2) n; \<lbrakk>n \<noteq> 0; n \<noteq> 1; a \<noteq> 1; \<not> n < 0; odd n; \<not> a < 0; a \<noteq> 0; \<not> n \<le> a; odd a; coprime a n\<rbrakk> \<Longrightarrow> P n a\<rbrakk> \<Longrightarrow> P a n" shows "P a0 a1"
dom_object: fixes CC :: "('d, 'e, 'f) category_scheme"   and f :: "'e" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "Dom\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>"
di: fixes x :: "'a set" assumes "x \<in> p" shows "\<exists>d. d division_of \<Union> (q x - {x})"
assms: shows "S \<^bold>\<or> T, \<Gamma> \<Rightarrow>\<^sub>n"
list_all2_shortcircuit_induct: fixes P :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "('b, 'c) result list"   and R :: "'a list \<Rightarrow> ('b, 'c) result list \<Rightarrow> bool" assumes "list_all2_shortcircuit P xs ys"   and "R [] []"   and "\<And>x xs y ys. \<lbrakk>P x (Rval y); list_all2_shortcircuit P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (Rval y # ys)"   and "\<And>x xs y ys. P x (Rerr y) \<Longrightarrow> R (x # xs) (Rerr y # ys)" shows "R xs ys"
k: shows "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
simps_2: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and r :: "'b memref"   and rs :: "'b memref list" shows "reads_consistent pending_write \<O> m (r # rs) = (case r of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> if volatile then outstanding_refs is_volatile_Read\<^sub>s\<^sub>b rs = {} \<and> reads_consistent True (\<O> \<union> A - R) (m(a := v)) rs else reads_consistent pending_write \<O> (m(a := v)) rs | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> (\<not> volatile \<longrightarrow> pending_write \<or> a \<in> \<O> \<longrightarrow> v = m a) \<and> reads_consistent pending_write \<O> m rs | Prog\<^sub>s\<^sub>b p1 p2 list \<Rightarrow> reads_consistent pending_write \<O> m rs | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> reads_consistent pending_write (\<O> \<union> A - R) m rs)"
mor_cong: fixes f' :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and g' :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g"   and B1 :: "'d set"   and B2 :: "'f set"   and s1 :: "('h, 'd, 'f) F1 \<Rightarrow> 'd"   and s2 :: "('h, 'd, 'f) F2 \<Rightarrow> 'f"   and B1' :: "'e set"   and B2' :: "'g set"   and s1' :: "('h, 'e, 'g) F1 \<Rightarrow> 'e"   and s2' :: "('h, 'e, 'g) F2 \<Rightarrow> 'g" assumes "f' = f"   and "g' = g"   and "mor B1 B2 s1 s2 B1' B2' s1' s2' f g" shows "mor B1 B2 s1 s2 B1' B2' s1' s2' f' g'"
tmboundslt_split0: fixes n :: "nat"   and t :: "tm"   and vs :: "'b list" assumes "tmboundslt n t" shows "\<lparr>fst (split0 t)\<rparr>\<^sub>p\<^bsup>vs\<^esup> = (0::'b) \<or> 0 < n"
pivotandupdate_rvars: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "rvars (\<T> (pivot_and_update x\<^sub>i x\<^sub>j c s)) = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i}"
ih: fixes x :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat"   and y :: "'a literal list list" assumes "self_call x = Some y" shows "\<exists>N' P' Q' n'. (\<exists>k. (deterministic_RP_step ^^ k) x = (N', P', Q', n')) \<and> is_final_dstate (N', P', Q', n') \<and> y = map fst Q'"
qty_replace_comp_def: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "qty_replace_comp pf1 x pf2 = qty_sum (qty_rem_comp pf1 x) (qty_mult_comp pf2 (pf1 x))"
try_unbox_def: fixes \<tau> :: "'d"   and x :: "'e"   and \<Sigma> :: "'d option list"   and unbox :: "'e \<Rightarrow> 'f option"   and mk_instr :: "'f \<Rightarrow> 'c" shows "try_unbox \<tau> x \<Sigma> unbox mk_instr \<equiv> case unbox x of None \<Rightarrow> Error () | Some n \<Rightarrow> Ok (mk_instr n, Some \<tau> # \<Sigma>)"
inject_2: fixes x2 :: "char list \<times> expr list \<times> char list list"   and y2 :: "char list \<times> expr list \<times> char list list" shows "(CEdge x2 = CEdge y2) = (x2 = y2)"
div_add_self2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a b) b = plus (divide a b) one"
vec_comp_all_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a vec"   and w :: "'a vec" shows "vec_comp_all r v w \<equiv> \<forall>i<dim_vec v. r (v $ i) (w $ i)"
in_homE: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>C.in_hom f a b" obtains "\<F>C.arr f" and "\<F>C.dom f = a" and "\<F>C.cod f = b"
rsp: shows "((=) ===> (\<approx>) ===> (\<approx>)) removeAll removeAll"
ord_option_mono: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "ord_option A x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "ord_option B x y"
cases: fixes a1 :: "('c, 'd) tm"   and a2 :: "('c, 'd) tm" assumes "a1 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b a2" obtains t1 :: "('c, 'd) tm" and t2 :: "('c, 'd) tm" where "a1 = App t1__ t2__" and "a2 = t1__"   | t1 :: "('c, 'd) tm" and t2 :: "('c, 'd) tm" where "a1 = App t1 t2" and "a2 = t2"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and u :: "('c, 'd) tm" where "a1 = App t__ u__" and "a2 = App sa__ u__" and "t__ \<rightarrow>\<^sub>e\<^sub>m\<^sub>b sa__"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and u :: "('c, 'd) tm" where "a1 = App u t" and "a2 = App u s" and "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s"
image_to_nat_on: fixes A :: "'b set" assumes "countable A"   and "infinite A" shows "to_nat_on A ` A = UNIV"
generator_generator: fixes H :: "'a set"   and H1 :: "'a set" assumes "generator R M H"   and "H1 \<subseteq> carrier M"   and "H \<subseteq> linear_span R M (carrier R) H1" shows "generator R M H1"
divides_mult_l: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "monoid_cancel G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "c \<otimes>\<^bsub>G\<^esub> a divides\<^bsub>G\<^esub> c \<otimes>\<^bsub>G\<^esub> b = a divides\<^bsub>G\<^esub> b"
P_def: fixes insert_message :: "('a, 'b) insert_message" shows "P insert_message = (case insert_message of InsertMessage x1 x2 x3 x4 \<Rightarrow> x1)"
strip_all_body_single_simp': fixes t :: "Core.term" shows "strip_all_single_body (strip_all_body t) = strip_all_body t"
normalize_match_already_normalized: fixes m :: "'b match_expr" assumes "normalized_nnf_match m" shows "normalize_match m = [m]"
directly_attracted_contains_no_deadends: fixes v :: "'a"   and p :: "Player"   and W :: "'a set" assumes "v \<in> directly_attracted p W" shows "\<not> deadend v"
dvd_mult_cancel_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a * c dvd b * c) = (c = (0::'a) \<or> a dvd b)"
continuous_at_compose: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "isCont f a"   and "isCont g (f a)" shows "isCont (g \<circ> f) a"
simps_1: fixes uu :: "('b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>h" shows "is_blinded (Tree\<^sub>m (Blinded uu)) = True"
scaleR1_eq: fixes e :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real" assumes "0 < e" shows "ccw' 0 (e *\<^sub>R a) b = ccw' 0 a b"
axioms_3: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog" assumes "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove" shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs"
lbs_ex_Zleast: fixes A :: "int set"   and n :: "int" assumes "A \<noteq> {}"   and "A \<subseteq> Zset"   and "LB A n" shows "\<exists>!m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)"
P_k_pre_not_Nil: shows "P_k_pre \<noteq> []"
negligible_translation: fixes S :: "'a set"   and c :: "'a" assumes "negligible S" shows "negligible ((+) c ` S)"
bounded_clinear_right: fixes a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "bounded_clinear ((*\<^sub>V) a)"
functor_is_transformation: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "functor A B F" shows "natural_transformation A B F F F"
segment_furthest_le: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and y :: "'b" assumes "x \<in> closed_segment a b" shows "norm (y - x) \<le> norm (y - a) \<or> norm (y - x) \<le> norm (y - b)"
Collect_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x::'a. (P::'a \<Rightarrow> bool) x \<longrightarrow> (Q::'a \<Rightarrow> bool) x" shows "Collect (P::'a \<Rightarrow> bool) \<subseteq> Collect (Q::'a \<Rightarrow> bool)"
k_def: shows "k < length xs \<and> \<not> xs ! k \<le> maxne0 ys b"
sin_product_formula_real: fixes x :: "real" shows "(\<lambda>n. pi * x * (\<Prod>k = 1..n. 1 - x\<^sup>2 / (real k)\<^sup>2)) \<longlonglongrightarrow> sin (pi * x)"
z_in_aug_minf: fixes z :: "int" shows "ant z \<in> Z\<^sub>-\<^sub>\<infinity>"
simps: fixes x :: "'b extB"   and y :: "'b extB" shows "x * y = x \<sqinter> y"
delete_edges_correct: fixes G :: "'a list_graph"   and E :: "('a \<times> 'a) list" shows "FiniteGraph.delete_edges (list_graph_to_graph G) (set E) = list_graph_to_graph (FiniteListGraph.delete_edges G E)"
obj_def: fixes a :: "'b" shows "D.obj a \<equiv> D.arr a \<and> src\<^sub>D a = a"
path_pd_ipd0: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> 0 \<noteq> local.return"   and "\<pi> n \<noteq> \<pi> 0"   and "\<pi> n pd\<rightarrow> \<pi> 0" obtains k :: "nat" where "k \<le> n" and "\<pi> k = ipd (\<pi> 0)"
2: shows "Fun f = restrict (Fun (incl_of (img f) (cod f)) \<circ> Fun (corestr f)) (Dom f)"
face_cycle_set_altdef: fixes a :: "'b" shows "face_cycle_set a = {(face_cycle_succ ^^ n) a |n. True}"
less_irrefl: fixes x :: "'c DBMEntry" shows "\<not> x \<prec> x"
ltl_prop_implies_equiv: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "(\<phi> \<longrightarrow>\<^sub>P \<psi> \<and> \<psi> \<longrightarrow>\<^sub>P \<phi>) = \<phi> \<equiv>\<^sub>P \<psi>"
circle_equation: fixes r :: "real"   and \<mu> :: "complex" assumes "0 \<le> r" shows "circle \<mu> r = {z. z * cnj z - z * cnj \<mu> - cnj z * \<mu> + \<mu> * cnj \<mu> - cor (r * r) = 0}"
cases: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'c" assumes "(r OO s) a1 a2" obtains a :: "'a" and b :: "'b" and c :: "'c" where "a1 = a" and "a2 = c" and "r a b" and "s b c"
fractrel_normalize_quot: fixes p :: "'b \<times> 'b"   and q :: "'b \<times> 'b" assumes "fractrel p p"   and "fractrel q q" shows "fractrel (normalize_quot p) (normalize_quot q) = fractrel p q"
tc_implies_pc: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" assumes "[p]   c   [q]" shows "{p}   c   {q}"
Red_I_of_subset: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N :: "'a set"   and N' :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "N \<subseteq> N'" shows "Red_I N \<subseteq> Red_I N'"
conj_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma> \<and>\<^sup>b \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<and> \<gamma>' t n"
Rep_G_induct: fixes y :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "y \<in> G"   and "\<And>x. P (Rep_G x)" shows "P y"
trans_allrel: fixes A :: "'b set" shows "trans (A \<times> A)"
arev_arev: fixes G :: "('b, 'c) pre_digraph"   and arev :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "bidirected_digraph G arev" shows "arev (arev a) = a"
AE_I': fixes N :: "'a set"   and M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "N \<in> null_sets M"   and "{x \<in> space M. \<not> P x} \<subseteq> N" shows "almost_everywhere M P"
commute: fixes a :: "'c"   and b :: "'c" shows "a AND b = b AND a"
berlekamp_basis_code: fixes u :: "'a mod_ring poly" shows "berlekamp_basis u = map (poly_of_list \<circ> list_of_vec) (find_base_vectors (berlekamp_resulting_mat u))"
boundedE: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}"   and "x \<le> Min A" obtains "\<And>a. a \<in> A \<Longrightarrow> x \<le> a"
kpower_inductr: fixes g :: "'a \<Rightarrow> 'b set"   and f :: "'b \<Rightarrow> 'b set"   and i :: "nat" assumes "g \<circ>\<^sub>K f \<le> g" shows "g \<circ>\<^sub>K kpower f i \<le> g"