hcju/isabelleps huggingface dataset:
Uvariation_repv: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and d :: "real" shows "Uvariation \<omega> (repv \<omega> x d) {x}"
parCompose: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and R :: "pi"   and Rel' :: "(pi \<times> pi) set"   and T :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto>[Rel] Q"   and "R \<leadsto>[Rel'] T"   and "(P, Q) \<in> Rel"   and "(R, T) \<in> Rel'"   and "\<And>P Q R T. \<lbrakk>(P, Q) \<in> Rel; (R, T) \<in> Rel'\<rbrakk> \<Longrightarrow> (P \<parallel> R, Q \<parallel> T) \<in> Rel''"   and "\<And>P Q a. (P, Q) \<in> Rel'' \<Longrightarrow> (<\<nu>a>P, <\<nu>a>Q) \<in> Rel''"   and "eqvt Rel"   and "eqvt Rel'"   and "eqvt Rel''" shows "P \<parallel> R \<leadsto>[Rel''] Q \<parallel> T"
negligible_iff_measure0: fixes S :: "'a set" assumes "S \<in> lmeasurable" shows "negligible S = (Sigma_Algebra.measure lebesgue S = 0)"
get_option_def: fixes get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> ('a, 'b) optionT" shows "get_option get f = OptionT (get (\<lambda>s. run_option (f s)))"
simps: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" shows "alw \<phi> a = (\<exists>xs. a = xs \<and> \<phi> xs \<and> alw \<phi> (stl xs))"
finc: shows "\<forall>S'\<subseteq>S. finite S' \<longrightarrow> S' \<in> C"
constant_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
id_arrow: fixes CC :: "('d, 'e, 'f) category_scheme"   and A :: "'d" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Id\<^bsub>CC\<^esub> A \<in> Ar\<^bsub>CC\<^esub>"
hyps_2: shows "init_conf prb_ \<sqsubseteq> confs prb_ (root (red prb_))"
ring_pow_apow: fixes I :: "'a set"   and n :: "nat" assumes "ideal R I" shows "I \<^bsup>\<diamondsuit>R n\<^esup> = I\<^bsup> R an n\<^esup>"
subst_comp: fixes t :: "'b trm"   and r :: "('b \<times> 'b trm) list"   and s :: "('b \<times> 'b trm) list" shows "t \<lhd> r \<lozenge> s = t \<lhd> r \<lhd> s"
subset'_code_3: fixes xs :: "'a list"   and ys :: "'a list" shows "subset' (List.coset xs) (set ys) = (let n = CARD('a) in 0 < n \<and> card (set (xs @ ys)) = n)"
det_upper_triangular: fixes A :: "'a mat"   and n :: "nat" assumes "upper_triangular A"   and "A \<in> carrier_mat n n" shows "det A = prod_list (diag_mat A)"
assms: shows "\<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> t \<cdot> \<delta> \<cdot> \<I> \<noteq> t' \<cdot> \<delta> \<cdot> \<I>"
simps_17: fixes f1 :: "ipc_direction_t \<Rightarrow> ipc_stage_t \<Rightarrow> thread_id_t \<Rightarrow> page_t \<Rightarrow> 'a"   and f2 :: "ev_wait_stage_t \<Rightarrow> ev_consume_t \<Rightarrow> 'a"   and f3 :: "ev_signal_stage_t \<Rightarrow> thread_id_t \<Rightarrow> 'a"   and f4 :: "'a"   and x21 :: "ev_wait_stage_t"   and x22 :: "ev_consume_t" shows "(case SK_EV_WAIT x21 x22 of SK_IPC x xa xb xc \<Rightarrow> f1 x xa xb xc | SK_EV_WAIT x xa \<Rightarrow> f2 x xa | SK_EV_SIGNAL x xa \<Rightarrow> f3 x xa | NONE \<Rightarrow> f4) = f2 x21 x22"
max_def: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
Fvars_PPf: fixes t1 :: "'b"   and t2 :: "'b" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "Fvars (PPf t1 t2) = FvarsT t1 \<union> FvarsT t2"
dw: shows "relation.dw_closed (red F) (dgrad_p_set d m)"
lassoc: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "xs \<cdot> ys \<cdot> zs = (xs \<cdot> ys) \<cdot> zs"
map_bnds_def: fixes lb :: "nat \<Rightarrow> float \<Rightarrow> float"   and ub :: "nat \<Rightarrow> float \<Rightarrow> float"   and prec :: "nat" shows "map_bnds lb ub prec = (\<lambda>(l, u). (lb prec l, ub prec u))"
d_s_prod_closed: fixes x :: "'a"   and y :: "'a" shows "d (d x \<cdot> d y) = d x \<cdot> d y"
these_insert_Some: fixes x :: "'a"   and A :: "'a option set" shows "Option.these (insert (Some x) A) = insert x (Option.these A)"
inj: shows "inj_on (\<lambda>i. g ^ i mod m) {..<totient m}"
less_Suc_eq_le: fixes m :: "nat"   and n :: "nat" shows "((m::nat) < Suc (n::nat)) = (m \<le> n)"
pred_iffI: fixes A :: "'b Predicate.pred"   and B :: "'b Predicate.pred" assumes "\<And>x. pred.eval A x \<Longrightarrow> pred.eval B x"   and "\<And>x. pred.eval B x \<Longrightarrow> pred.eval A x" shows "A = B"
cnj_units_2: shows "Quaternions.cnj \<j> = - \<j>"
order_less_le: fixes x :: "'c"   and y :: "'c" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
non_Col_example: shows "\<not> real_euclid.Col 0 (vector [1 / 2, 0]) (vector [0, 1 / 2])"
pcr_cr_eq: shows "pcr_set_dlist (=) = cr_set_dlist"
abs_eq: fixes xa :: "int"   and x :: "nat" shows "arith_prog_fb xa x = fbint ` arith_prog xa x"
plus_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "f + g = (\<lambda>x. f x + g x)"
label_tree_list: fixes t :: "'b Tree_Relabelling.tree" shows "State_Monad.ap (State_Monad.return labels) (label_tree t) = label_list (labels t)"
hausdorff_distance_closure: fixes A :: "'b set" shows "hausdorff_distance A (closure A) = 0"
zero_above: fixes qs :: "rat poly list"   and x1 :: "real" assumes "has_no_zeros (sign_vec qs x1)" shows "\<forall>x2>x1. sign_vec qs x1 \<noteq> sign_vec qs x2 \<longrightarrow> (\<exists>r>x1. r \<le> x2 \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
bin_cat_foldl_lem: fixes n :: "nat"   and x :: "int"   and xs :: "int list"   and y :: "int" shows "foldl (\<lambda>u l. concat_bit n l u) x xs = concat_bit (length xs * n) (foldl (\<lambda>u l. concat_bit n l u) y xs) x"
intros_1: fixes valid_edge :: "'a \<Rightarrow> bool"   and a :: "'a"   and src :: "'a \<Rightarrow> 'b"   and E :: "'b"   and trg :: "'a \<Rightarrow> 'b"   and X :: "'b"   and e :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node"   and knd :: "'a \<Rightarrow> 'c edge_kind" assumes "valid_edge a"   and "src a \<noteq> E \<or> trg a \<noteq> X"   and "e = (LDCFG_node.Node (src a), knd a, LDCFG_node.Node (trg a))" shows "lift_valid_edge valid_edge src trg knd E X e"
DiffD2: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
disc_lbk_option_lgeom: fixes T :: "nat"   and w :: "bool stream" shows "discounted_value r (\<lambda>m. lbk_option T) T w = det_discount r T * ((MAX i\<in>{0..T}. lgeom_proc u d init (take i (stake T w))) - lgeom_proc u d init (stake T w))"
nasty_split_lt: fixes x :: "'b word"   and m :: "nat"   and n :: "nat" assumes "x < 2 ^ (m - n)"   and "n \<le> m"   and "m < LENGTH('b)" shows "x * 2 ^ n + (2 ^ n - 1) \<le> 2 ^ m - 1"
load_sub1_result: fixes i :: "sparc_operation \<times> inst_operand list"   and rd :: "word5"   and s_val :: "word1"   and s :: "'b sparc_state" shows "snd (load_sub1 i rd s_val s) = False"
worklist_tree_def: fixes succs :: "'d \<Rightarrow> 'd list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and ws :: "'d list"   and s :: "'c" shows "worklist_tree succs f ws s = (case worklist_tree_aux succs f (ws, s) of None \<Rightarrow> None | Some (ws, s) \<Rightarrow> Some s)"
declassified_append: fixes A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and I :: "'var \<Rightarrow> ('fun, 'b) Term.term" shows "declassified\<^sub>l\<^sub>s\<^sub>t (A @ B) I = declassified\<^sub>l\<^sub>s\<^sub>t A I \<union> declassified\<^sub>l\<^sub>s\<^sub>t B I"
strip_eq_If: fixes c :: "'c acom"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com" shows "(strip c = IF b THEN c1 ELSE c2) = (\<exists>d1 d2 P. c = IF b THEN d1 ELSE d2 {P} \<and> strip d1 = c1 \<and> strip d2 = c2)"
ChamberComplexIsomorphism_axioms_def: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" shows "ChamberComplexIsomorphism_axioms X Y f \<equiv> bij_betw f (\<Union> X) (\<Union> Y) \<and> f \<turnstile> X = Y"
AE_mp: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure"   and Q :: "'c \<Rightarrow> bool" assumes "almost_everywhere M P"   and "AE x in M. P x \<longrightarrow> Q x" shows "almost_everywhere M Q"
cr_sq_matrix_def: shows "cr_sq_matrix \<equiv> \<lambda>x y. x = to_fun y"
pivot_fun_zero_row_iff: fixes A :: "'b mat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and nr :: "nat"   and i :: "nat" assumes "pivot_fun A f nc"   and "A \<in> carrier_mat nr nc"   and "i < nr" shows "(f i = nc) = (row A i = 0\<^sub>v nc)"
span_zero: fixes S :: "'a set" shows "(0::'a) \<in> More_Modules.ideal S"
reads_restrict2: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'b"   and \<sigma>' :: "'var \<Rightarrow> 'b" assumes "\<forall>x\<in>local.reads n. \<sigma> x = \<sigma>' x" shows "\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n"
rel_alt: shows "rel = (\<Union>(t, rel)\<in>set aux'. if left I \<le> \<tau> \<sigma> ne - t then rel else empty_table)"
that: fixes s :: "'c" assumes "t = Fun (Set s) []" shows "thesis"
eq: shows "\<b> ps (Suc 0) = Suc (Max (poly_deg ` fst ` set (ps\<^sub>+)))"
not_tendsto_and_filterlim_at_infinity: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> c) F"   and "filterlim f at_infinity F" shows "False"
coboundedI2: fixes b :: "'d"   and c :: "'d"   and a :: "'d" assumes "b \<le> c" shows "a \<sqinter> b \<le> c"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case \<one> of \<zero> \<Rightarrow> f1 | \<one> \<Rightarrow> f2) = f2"
OSou_iff_inverse_of_indRelRTPO_is_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_sound (TRel\<^sup>*) \<and> weak_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target) = weak_reduction_simulation ((indRelRTPO TRel)\<inverse>) (STCal Source Target)"
gdpr_Kripke_def: shows "gdpr_Kripke \<equiv> Kripke gdpr_states {gdpr_scenario}"
ide_map\<^sub>0_obj: fixes a :: "'d" assumes "D.obj a" shows "D.ide (\<eta>'.map\<^sub>0 a)"
that: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set" assumes "(h, U) \<in> set (ps\<^sub>+)"   and "poly_deg h = d"   and "card U = i" shows "thesis"
hom_add_eq_zero: fixes x :: "'a poly"   and y :: "'a poly" assumes "x + y = 0" shows "map_poly hom x + map_poly hom y = 0"
tendsto_ennrealD: fixes f :: "'b \<Rightarrow> real"   and x :: "real"   and F :: "'b filter" assumes "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 \<le> x" shows "(f \<longlongrightarrow> x) F"
subset_empty: fixes A :: "'a set" shows "(A \<subseteq> {}) = (A = {})"
parent_child_rel_acyclic: fixes h :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h" shows "acyclic (parent_child_rel h)"
reachable_steps_append: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c"   and a :: "'c"   and xs :: "'c list"   and b :: "'c" assumes "Graph_Start_Defs.reachable E s\<^sub>0 a"   and "Graph_Defs.steps E xs"   and "hd xs = a"   and "last xs = b" shows "Graph_Start_Defs.reachable E s\<^sub>0 b"
i_sccs_finished_stack_ss_tj_stack: shows "is_invar (\<lambda>s. \<Union> (sccs s) \<subseteq> dom (finished s) \<and> set (stack s) \<subseteq> set (tj_stack s))"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" obtains c :: "('Var, 'AExp, 'BExp) Stmt" and c'a :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E__ cb__" and "eval\<^sub>w_simple_abv (cb__, mem) (c'a__, mem')" and "c' = cxt_to_stmt E__ c'a__" and "mds' = mds"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = c\<^sub>1''__ ;; c\<^sub>2" and "c' = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1''__, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem'\<rangle>"   | c :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c'a :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E (c@[mu])" and "c' = cxt_to_stmt E c'a" and "\<langle>c, update_modes mu mds, mem\<rangle> \<leadsto> \<langle>c'a, mds', mem'\<rangle>"   | b :: "'BExp" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = Await b c\<^sub>1" and "ev\<^sub>B mem b" and "no_await c\<^sub>1" and "\<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto>\<^sup>+ \<langle>c', mds', mem'\<rangle>" and "is_final c'"
gcd_abs_mult_right_int: fixes x :: "int"   and a :: "int"   and y :: "int" shows "gcd x (\<bar>a\<bar> * y) = gcd x (a * y)"
step: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"
minus_mult_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = times a (uminus b)"
InvariantUniqAfterUnitPropagate: fixes M :: "(Literal \<times> bool) list"   and uClause :: "Literal list"   and uLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantUniq M"   and "isUnitClause uClause uLiteral (elements M)"   and "M' = M @ [(uLiteral, False)]" shows "InvariantUniq M'"
transitive_rule_def: fixes t :: "'c" shows "transitive_rule t = transl_rule (A_Cmp (A_Lbl t) (A_Lbl t) \<sqsubseteq> A_Lbl t)"
ldropn_eq_LNil: fixes n :: "nat"   and xs :: "'c llist" shows "(ldropn n xs = LNil) = (llength xs \<le> enat n)"
IHc: fixes t' :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> t'" shows "t' = s\<^sub>1"
simps: fixes a1 :: "'a \<Rightarrow> PreSimplyTyped.type option"   and a2 :: "'a trm"   and a3 :: "PreSimplyTyped.type" shows "a1 \<turnstile> a2 : a3 = ((\<exists>\<Gamma>. a1 = \<Gamma> \<and> a2 = Unit \<and> a3 = TUnit) \<or> (\<exists>\<Gamma> x \<tau>. a1 = \<Gamma> \<and> a2 = Var x \<and> a3 = \<tau> \<and> \<Gamma> x = Some \<tau>) \<or> (\<exists>\<Gamma> f \<tau> \<sigma> x. a1 = \<Gamma> \<and> a2 = App f x \<and> a3 = \<sigma> \<and> \<Gamma> \<turnstile> f : TArr \<tau> \<sigma> \<and> \<Gamma> \<turnstile> x : \<tau>) \<or> (\<exists>\<Gamma> x \<tau> A \<sigma>. a1 = \<Gamma> \<and> a2 = Fn x \<tau> A \<and> a3 = TArr \<tau> \<sigma> \<and> \<Gamma>(x \<mapsto> \<tau>) \<turnstile> A : \<sigma>) \<or> (\<exists>\<Gamma> A \<tau> B \<sigma>. a1 = \<Gamma> \<and> a2 = SimplyTyped.Pair A B \<and> a3 = TPair \<tau> \<sigma> \<and> \<Gamma> \<turnstile> A : \<tau> \<and> \<Gamma> \<turnstile> B : \<sigma>) \<or> (\<exists>\<Gamma> P \<tau> \<sigma>. a1 = \<Gamma> \<and> a2 = Fst P \<and> a3 = \<tau> \<and> \<Gamma> \<turnstile> P : TPair \<tau> \<sigma>) \<or> (\<exists>\<Gamma> P \<tau> \<sigma>. a1 = \<Gamma> \<and> a2 = Snd P \<and> a3 = \<sigma> \<and> \<Gamma> \<turnstile> P : TPair \<tau> \<sigma>))"
rjoin\<^sub>\<epsilon>_deterministic: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r' :: "'r"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = None" shows "revision_step r s s' = (s' = \<epsilon>)"
is_unit_pCons_iff: fixes a :: "'b"   and p :: "'b poly" shows "is_unit (pCons a p) = (p = 0 \<and> a \<noteq> (0::'b))"
qtable_r\<delta>: fixes \<phi>s :: "Formula.formula list"   and mr :: "mregex"   and A :: "nat set"   and n :: "nat"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and rels :: "event_data option list set list"   and Q :: "event_data list \<Rightarrow> Formula.formula Regex.regex \<Rightarrow> bool"   and rel :: "(mregex, event_data option list set) mapping" assumes "ok (length \<phi>s) mr"   and "fv_regex (from_mregex mr \<phi>s) \<subseteq> A"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) j \<phi>)) \<phi>s rels"   and "\<forall>ms\<in>RPD mr. qtable n A (mem_restr R) (\<lambda>v. Q (map the v) (from_mregex ms \<phi>s)) (lookup rel ms)" shows "qtable n A (mem_restr R) (\<lambda>v. \<exists>s\<in>Regex.rpd (Formula.sat \<sigma> V (map the v)) j (from_mregex mr \<phi>s). Q (map the v) s) (r\<delta> (\<lambda>x. x) rel rels mr)"
bind_id_def: shows "bind_id \<equiv> rec_id (\<lambda>x f. f x)"
hyps_3: fixes i :: "nat" assumes "i < length T"   and "\<forall>f\<in>funs_term (S ! i). \<not> is_Abs f" shows "T ! i = S ! i"
err_step_def: fixes n :: "nat"   and app :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and p :: "nat"   and t :: "'b err" shows "err_step n app step p t = (case t of Err \<Rightarrow> error n | OK \<tau> \<Rightarrow> if app p \<tau> then map_snd OK (step p \<tau>) else error n)"
safeInits_def: fixes s :: "('a, 'b, 'c) SLin_state" shows "safeInits s \<equiv> if initVals s = {} then {} else {d. \<exists>ivs\<in>initSets s. \<exists>rs\<in>pendingSeqs s. d = \<Sqinter> ivs \<star> rs \<and> (\<forall>av\<in>abortVals s. d \<preceq> av)}"
normalized_cidr_ip_def: fixes ip :: "'a ipt_iprange" shows "normalized_cidr_ip ip \<equiv> case ip of IpAddrNetmask x xa \<Rightarrow> True | _ \<Rightarrow> False"
AllS: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<forall>x. \<phi> x in v] = (\<forall>x. [\<phi> x in v])"
mono2mono_ennreal_add: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> ennreal"   and t' :: "'a \<Rightarrow> ennreal" assumes "monotone orda (\<le>) t"   and "monotone orda (\<le>) t'" shows "monotone orda (\<le>) (\<lambda>x. t x + t' x)"
assms_2: fixes i :: "'c" assumes "i \<in> I" shows "subgroup (S i) G"
fcd: fixes x :: "complex" assumes "x \<in> interior S - K" shows "f field_differentiable at x"
Var: shows "\<forall>x\<in>fv (Var x). (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)"
iarray_exhaust2: fixes xs :: "'c iarray"   and ys :: "'c iarray" shows "(xs = ys) = (IArray.list_of xs = IArray.list_of ys)"
assms: shows "ok (e\<lparr>a:X\<rparr>)"
assoc_mult_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and t :: "'a"   and u :: "'a"   and v :: "'a"   and w :: "'a" assumes "(r, s) \<in> carrier rel"   and "(t, u) \<in> carrier rel"   and "(v, w) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (t |\<^bsub>rel\<^esub> u) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (v |\<^bsub>rel\<^esub> w) = (r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> ((t |\<^bsub>rel\<^esub> u) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (v |\<^bsub>rel\<^esub> w))"
Abs_sat'_def: fixes n :: "nat" shows "Abs_sat' n = Abs_sat (min LENGTH('b) n)"
image_eqI: fixes b :: "'x"   and f :: "'y \<Rightarrow> 'x"   and x :: "'y"   and A :: "'y set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
eq_iff_diff_eq_0: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (a - b = (0::'a))"
enumerate_Suc'': fixes S :: "'a set"   and n :: "nat" assumes "infinite S" shows "enum S (Suc n) = (LEAST s. s \<in> S \<and> enum S n < s)"
complete_imp_closed: fixes S :: "'b set" assumes "complete S" shows "closed S"
\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c_def: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c \<gamma> \<equiv> \<lambda>a p. if \<gamma> a p then TernaryTrue else TernaryFalse"
interaction_any_bounded_converter_start_2: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
eq_\<I>_gpv_coinduct_bind: fixes X :: "('a, 'b, 'c) gpv \<Rightarrow> ('d, 'b, 'c) gpv \<Rightarrow> bool"   and gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('d, 'b, 'c) gpv"   and A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>" assumes "X gpv gpv'"   and "\<And>gpv gpv'. X gpv gpv' \<Longrightarrow> rel_spmf (eq_\<I>_generat A \<I> (\<lambda>gpv gpv'. X gpv gpv' \<or> eq_\<I>_gpv A \<I> gpv gpv' \<or> (\<exists>gpv'' gpv''' B f g. gpv = gpv'' \<bind> f \<and> gpv' = gpv''' \<bind> g \<and> eq_\<I>_gpv B \<I> gpv'' gpv''' \<and> rel_fun B X f g))) (the_gpv gpv) (the_gpv gpv')" shows "eq_\<I>_gpv A \<I> gpv gpv'"
pred_spmf_return_pmf_None: fixes P :: "'c \<Rightarrow> bool" shows "pred_spmf P (return_pmf None)"
smult_minus_left: fixes a :: "'a"   and p :: "'a poly" shows "Polynomial.smult (- a) p = - Polynomial.smult a p"
cl_augment: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "X \<subseteq> carrier"   and "x \<in> carrier"   and "y \<in> cl (insert x X) - cl X" shows "x \<in> cl (insert y X)"
assms_3: shows "\<forall>b\<^sub>\<sigma> n. etp_rsa.HCP_adv n etp.\<A> m2 b\<^sub>\<sigma> D \<le> hcp_advantage n"
preserve_agree1: assumes "vid1 \<notin> ODE_vars I (OVar vid1)" shows "VSagree (sol 0) (fst (mk_v I (OVar vid1) (sol 0, b) (sol t))) {vid1}"
qmax_def: shows "qmax = Max (insert 0 (abs ` set (coeffs q)))"
grab_Un_eq: fixes N :: "nat set"   and n :: "nat" assumes "infinite N" shows "fst (grab N n) \<union> snd (grab N n) = N"
AnalBinds_delete_bot: fixes ae :: "Vars.var \<Rightarrow> 'c\<^sub>\<bottom>"   and x :: "Vars.var"   and exp :: "exp \<Rightarrow> 'c \<rightarrow> 'd"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "ae x = \<bottom>" shows "ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae = ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae"
rel_interior_maximal: fixes T :: "'a set"   and S :: "'a set" assumes "T \<subseteq> S"   and "openin (top_of_set (affine hull S)) T" shows "T \<subseteq> rel_interior S"
list_update_code_3: fixes x :: "'k"   and xs :: "'k list"   and i :: "nat"   and y :: "'k" shows "(x # xs)[Suc i := y] = x # xs[i := y]"
tauChainCons2: fixes P' :: "ccs"   and P'' :: "ccs"   and P :: "ccs" assumes "P' \<longmapsto>\<tau> \<prec> P''"   and "P \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P''"
counit_in_vhom: shows "\<guillemotleft>\<epsilon>' : F f \<star>\<^sub>D F g \<Rightarrow>\<^sub>D src\<^sub>D (F g)\<guillemotright>"
size_update1: fixes t :: "'b tree"   and n :: "nat"   and x :: "'b" assumes "braun t"   and "n \<in> {1..size t}" shows "size (update1 n x t) = size t"
HOMSetIsSet: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "HOMCollection C X Y \<in> range explode"
tee_local: fixes v :: "e"   and i :: "nat" assumes "is_const v" shows "\<lparr>[v, $Tee_local i]\<rparr> \<leadsto> \<lparr>[v, v, $Set_local i]\<rparr>"
Red_F_of_Red_F_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> Red_F N" shows "Red_F N \<subseteq> Red_F (N - N')"
add_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "a \<oplus>\<^sub>o\<^sub>w b = b \<oplus>\<^sub>o\<^sub>w a"