hcju/isabelleps huggingface dataset:
simps: fixes t :: "name"   and v :: "tm"   and i :: "tm"   and y :: "tm"   and y' :: "tm"   and t' :: "name"   and u :: "name"   and u' :: "name" assumes "atom t \<sharp> (v, i, y, y', t', u, u')"   and "atom t' \<sharp> (v, i, y, y', u, u')"   and "atom u \<sharp> (v, i, y, y', u')"   and "atom u' \<sharp> (v, i, y, y')" shows "AbstAtomicP v i y y' = SyntaxN.Ex t (SyntaxN.Ex u (SyntaxN.Ex t' (SyntaxN.Ex u' (AbstTermP v i (Var t) (Var t') AND AbstTermP v i (Var u) (Var u') AND (y EQ Q_Eq (Var t) (Var u) AND y' EQ Q_Eq (Var t') (Var u') OR y EQ Q_Mem (Var t) (Var u) AND y' EQ Q_Mem (Var t') (Var u'))))))"
IH: shows "(\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>) \<or> t \<in> SMP (subterms\<^sub>s\<^sub>e\<^sub>t M \<union> \<Union> ((set \<circ> fst \<circ> Ana) ` M) - M)"
snd_rep_of_\<psi>\<^sub>2_1: fixes i :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))" shows "real_of_int ((int (1 - f (i div 2)) + - int (f (i div 2))) * 1) / sqrt 2 ^ (n + 1) = (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1)"
filterlim_principal: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and F :: "'c filter" shows "filterlim f (principal S) F = (\<forall>\<^sub>F x in F. f x \<in> S)"
is_map_group_by: fixes f :: "'c \<Rightarrow> 'a \<times> 'b"   and M :: "'c set" shows "is_map (group_by f M)"
SCRS3: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "singleCombinators p" shows "singleCombinators (rm_MT_rules Cp p)"
is_nth_power_conv_multiplicity_nat: fixes n :: "nat"   and x :: "nat" assumes "0 < n" shows "is_nth_power n x = (\<forall>p. prime p \<longrightarrow> n dvd multiplicity p x)"
V_def: shows "V \<equiv> {if learn_lim \<phi> {g i} (\<phi> i) then prefix (g i) (LEAST n. \<phi> (the (\<phi> i (g i \<triangleright> n))) = g i) @ [1] \<odot> 0\<^sup>\<infinity> else g i |i. i \<in> UNIV}"
scalar_mult_eq_scaleR: shows "(*s) \<equiv> (*\<^sub>R)"
axioms_1: fixes return :: "'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and sample :: "'e pmf \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_state_prob return bind get put sample" shows "monad_state return bind get put"
higher_differentiable_on_snd_comp: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd"   and k :: "nat" assumes "higher_differentiable_on S f k"   and "open S" shows "higher_differentiable_on S (\<lambda>x. snd (f x)) k"
excess_u_pos: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "push_effect_locale c s t f l u v" shows "(0::'a) < Network.excess c f u"
add_mset_lt_lt_le: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a < b"   and "A \<le> B" shows "add_mset a A < add_mset b B"
30411: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<squnion> x \<sqinter> y) = bot"
t_def: shows "t = of_int x + gauss_i * of_int y"
DIM_cart: shows "DIM(('a, 'b) vec) = CARD('b) * DIM('a)"
trnr\<^sub>\<epsilon>_eq: fixes u :: "'a"   and v :: "'a"   and \<nu> :: "'a" assumes "ide u"   and "ide v"   and "src v = trg g"   and "src u = trg f"   and "\<nu> \<in> hom v (u \<star> f)" shows "trnr\<^sub>\<epsilon> u \<nu> = (u \<star> \<epsilon>) \<cdot> \<a>[u, f, g] \<cdot> (\<nu> \<star> g)"
nonpos_Ints_nonpos: fixes x :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0" shows "x \<le> (0::'a)"
distinct_if_sorted: fixes xs :: "'c list" assumes "Sorted_Less.sorted xs" shows "distinct xs"
f_bound_def: fixes m :: "nat"   and fs_init :: "int vec list"   and outside :: "bool"   and ii :: "nat"   and fs :: "int vec list" shows "LLL.f_bound m fs_init outside ii fs = (\<forall>i<m. \<parallel>fs ! i\<parallel>\<^sup>2 \<le> (if i \<noteq> ii \<or> outside then int (LLL.N fs_init * m) else int (4 ^ (m - 1) * LLL.N fs_init ^ m * m * m)))"
elims_2: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "pred x xa" obtains f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = \<Up>f__" and "xa = cfa__ # cfsa__"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Qa__)\<^sub>\<surd>" and "xa = cfb__ # cfsb__" and "Qa__ (fst cfb__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qb__:r\<hookrightarrow>\<^bsub>pa__\<^esub>fs" and "xa = cfc__ # cfsc__" and "Qb__ (fst cfc__, r)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and p :: "'pname" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "Q cf \<and> cfs \<noteq> []"
set_vars_mset: fixes t :: "('b, 'a) tm" shows "set_mset (vars_mset t) = vars t"
asian_option_lgeom: fixes T :: "nat"   and w :: "bool stream" shows "asian_option T w = (\<Sum>i = 1..T. lgeom_proc u d init (stake i w)) / real T"
span_Gram_Schmidt_column_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k < ncols A" shows "span (columns A) = span (columns (Gram_Schmidt_column_k A k))"
weak_reduction_coupled_simulation_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_coupled_simulation Rel Cal" shows "weak_reduction_coupled_simulation (Rel\<^sup>=) Cal"
map_minus_common_subset: fixes h :: "'a \<Rightarrow> 'b option"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "h \<subseteq>\<^sub>m f"   and "h \<subseteq>\<^sub>m g" shows "(f -- h = g -- h) = (f = g)"
IMPRIGHT_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set"   and Y :: "ctrm set"   and Z :: "ntrm set"   and U :: "ctrm set" shows "pi \<bullet> IMPRIGHT (A IMP B) X Y Z U = IMPRIGHT (A IMP B) (pi \<bullet> X) (pi \<bullet> Y) (pi \<bullet> Z) (pi \<bullet> U)"
insertI_2_3: shows "\<forall>t\<in>{s<..<1}. Re (g t - z) \<noteq> 0"
pddl_opr_to_act_def: fixes g_opr :: "ground_action"   and s :: "predicate \<times> object list \<Rightarrow> bool" shows "pddl_opr_to_act g_opr s = (let M = state_to_wm s in if close_eq (wm_to_state (close_world M)) \<Turnstile> ground_action.precondition g_opr then Some (wm_to_state (apply_effect (ground_action.effect g_opr) M)) else None)"
flip_set_both_not_in: fixes x :: "'c"   and S :: "'c set"   and x' :: "'c" assumes "x \<notin> S"   and "x' \<notin> S" shows "(x' \<leftrightarrow> x) \<bullet> S = S"
NSsums_def: fixes f :: "nat \<Rightarrow> real"   and s :: "real" shows "f NSsums s = (\<lambda>n. sum f {..<n}) \<longlonglongrightarrow>\<^sub>N\<^sub>S s"
empty_not_add_mset: fixes a :: "'k"   and A :: "'k multiset" shows "{#} \<noteq> add_mset a A"
ex1: assumes "\<exists>v\<in>W. c v \<noteq> (0::'a)" shows "False"
pmf_not_neg: fixes p :: "'c pmf"   and x :: "'c" shows "\<not> pmf p x < 0"
bounded_by_o: fixes b :: "real"   and P :: "'c \<Rightarrow> real"   and f :: "'d \<Rightarrow> 'c" assumes "bounded_by b P" shows "bounded_by b (P \<circ> f)"
UnionP_Mem1: fixes x :: "tm"   and y :: "tm"   and z :: "tm"   and a :: "tm" shows "{UnionP x y z, a IN x} \<turnstile> a IN z"
hol_pal_lem4: fixes S :: "complex set"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and f :: "complex \<Rightarrow> complex"   and k :: "real" assumes "convex S"   and "open S"   and "a \<in> S"   and "b \<in> S"   and "c \<in> S"   and "d \<noteq> 0"   and "f holomorphic_on {z \<in> S. d \<bullet> z < k}"   and "f holomorphic_on {z \<in> S. k < d \<bullet> z}"   and "continuous_on S f" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
intros_1: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and s :: "('a, 'b, 'c) state" shows "GM_sec_model.wffp do [s]"
box_below: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.po below"   and "below a b"   and "below c a"   and "below b d" shows "below c d"
assms_5: fixes \<rho>\<^sub>1 :: "complex mat"   and \<rho>\<^sub>2 :: "complex mat" assumes "\<rho>\<^sub>1 \<in> carrier_mat d d"   and "\<rho>\<^sub>2 \<in> carrier_mat d d"   and "partial_density_operator \<rho>\<^sub>1"   and "partial_density_operator \<rho>\<^sub>2"   and "trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1" shows "denote x2 (\<rho>\<^sub>1 + \<rho>\<^sub>2) = denote x2 \<rho>\<^sub>1 + denote x2 \<rho>\<^sub>2"
exhaust: fixes y :: "metaop" obtains "y = AllInstancesOp"
assms_2: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and J :: "'a set"   and c :: "'a"   and d :: "'a"   and x :: "'a" assumes "I = interval a b"   and "J = interval c d" shows "(x = a \<or> x = c \<longrightarrow> P x I J) \<and> (\<not> (x = a \<or> x = b \<or> x = c \<or> x = d) \<longrightarrow> P x I J)"
err2_def: shows "err2 \<equiv> \<lambda>s r. cnj (contour_integral (part_circlepath 0 r pi 0) (f (cnj s)))"
smap_cong: fixes xs :: "[:'r:]"   and xs' :: "[:'r:]"   and f :: "'r \<rightarrow> 's"   and f' :: "'r \<rightarrow> 's" assumes "xs = xs'"   and "\<And>x. x \<in> sset xs \<Longrightarrow> f\<cdot>x = f'\<cdot>x" shows "smap\<cdot>f\<cdot>xs = smap\<cdot>f'\<cdot>xs'"
SystemDiffAssignCut_def: shows "SystemDiffAssignCut = ([[DiffAssign vid1 ($f fid1 local.empty)]]Geq ($' vid1) (Const 0) \<leftrightarrow> Geq ($f fid1 local.empty) (Const 0))"
ostep_arrive_invariantD: fixes p :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f node_action) automaton"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool"   and P :: "((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and IPS :: "nat set"   and a :: "'f node_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e" assumes "p \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable p (otherwith S IPS (oarrivemsg I)) U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans p"   and "oarrivemsg I \<sigma> a" shows "P ((\<sigma>, s), a, \<sigma>', s')"
And_ltlp_def: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "\<phi> and\<^sub>p \<psi> \<equiv> not\<^sub>p (not\<^sub>p \<phi> or\<^sub>p not\<^sub>p \<psi>)"
leg1_simps_3: shows "src (tab\<^sub>1 r) = src (tab\<^sub>0 r)"
divisors_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors plus zero times"   and "times a b = zero" shows "a = zero \<or> b = zero"
degree_unique: fixes p :: "'a list"   and q :: "'a list" assumes "poly p = poly q" shows "degree p = degree q"
pw_sup_nofail: fixes a :: "'b nres"   and b :: "'b nres" shows "nofail (sup a b) = (nofail a \<and> nofail b)"
sel: fixes r :: "('k, 'i, 'j) gpv"   and f :: "'k \<Rightarrow> ('h, 'i, 'j) gpv" shows "the_gpv (r \<bind> f) = map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. r \<bind> f)))) (the_gpv r \<bind> case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))"
member_of: shows "G \<turnstile>Methd sig m member_of C"
diff_Suc: shows "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"
select_convs_2: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'a" shows "cls \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = cls"
set_map0_1: fixes f1 :: "'o \<Rightarrow> 'q"   and f2 :: "'p \<Rightarrow> 'r" shows "F'set3 \<circ> Fmap f1 f2 = (`) f1 \<circ> F'set3"
len_paid_A: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list" assumes "BIT_Off acts qs init" shows "length (BIT_Off.paid_A acts init) = length qs"
r_one: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
thetaSeqZOD_ZOretr: shows "thetaSeqZOD \<subseteq> ZOretr (thetaSeqZOD \<union> ZObis)"
scaleRat_left_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "rat"   and b :: "rat"   and x :: "'a" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat (a + b) x = plus (scaleRat a x) (scaleRat b x)"
Min_le_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Min less_eq A) x = (\<exists>a\<in>A. less_eq a x)"
big_or: shows "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<or> (\<exists>(a', b', c')\<in>set les. a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<or> (\<exists>(a', b', c')\<in>set les. a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0))"
squarefree_prime: fixes p :: "'b" assumes "prime p" shows "squarefree p"
vars_clause_mono: fixes S :: "('a, 'b) Term.term literal multiset"   and C :: "('a, 'b) Term.term literal multiset" assumes "S \<subseteq># C" shows "vars_clause S \<subseteq> vars_clause C"
nop_bij_axioms_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_bij_axioms A n f \<equiv> n \<in>\<^sub>\<circ> \<omega> \<and> \<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n \<and> \<R>\<^sub>\<circ> f = A"
coeff_map_poly: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly"   and n :: "nat" assumes "f (0::'d) = (0::'c)" shows "poly.coeff (map_poly f p) n = f (poly.coeff p n)"
sortedConcEnd: fixes a :: "('a, 'b) Combinators"   and p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "FWNormalisationCore.sorted (a # p) l"   and "in_list a l"   and "all_in_list p l"   and "singleCombinators [a]"   and "singleCombinators p" shows "FWNormalisationCore.sorted p l"
assms: shows "(f, h) \<in> [comp_PRE S1 Q (\<lambda>x _. P x) X]\<^sub>a R \<rightarrow> S"
Quotient_complex_homo: shows "Quotient (\<approx>) abs_complex_homo rep_complex_homo cr_complex_homo"
conv: fixes e :: "real" assumes "0 < e" shows "\<exists>N. \<forall>n x. N \<le> n \<longrightarrow> x \<in> S \<longrightarrow> cmod (f' n x - g' x) \<le> e"
symmetric_reachable': fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w" shows "w \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v"
Abs_comp_hom: fixes f :: "'b \<Rightarrow> 'b set"   and g :: "'b \<Rightarrow> 'b set" shows "Abs_nd_fun (f \<circ>\<^sub>K g) = Abs_nd_fun f \<cdot> Abs_nd_fun g"
sibling_ne_imp_sibling_in_alphabet: fixes t :: "'a tree"   and a :: "'a" assumes "sibling t a \<noteq> a" shows "sibling t a \<in> alphabet t"
coeff_of_sum_diff_degree1: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "(p \<oplus>\<^bsub>P\<^esub> q) (deg R p) = p (deg R p)"
pmf_join: fixes N :: "'b pmf pmf"   and i :: "'b" shows "pmf (join_pmf N) i = measure_pmf.expectation N (\<lambda>M. pmf M i)"
Y_def: shows "Y \<equiv> abs_var (PMF (pmf_of_set {(True, undefined), (False, undefined)}))"
runProc_pState_id: fixes name :: "String.literal"   and args :: "expr list"   and prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState" shows "snd (runProc name args prog g p) = p"
reaches: fixes x :: "'b"   and y :: "'b" assumes "mut_m.reachable m x s"   and "(x points_to y) s"   and "P x" shows "P y"
error_free_abrupt_if_Xcpt: fixes s :: "state"   and p :: "bool"   and x :: "xcpt" assumes "error_free s" shows "error_free (abupd (abrupt_if p (Some (Xcpt x))) s)"
linorder_linorder_of_list: fixes xs :: "'b list" assumes "distinct xs" shows "linorder_on (set xs) (linorder_of_list xs)"
direct_arc_chooses: fixes u :: "'a"   and v :: "'a" shows "direct_arc (u, v) = (u, v) \<or> direct_arc (u, v) = (v, u)"
no_trailing_unfold: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "no_trailing P xs = (xs \<noteq> [] \<longrightarrow> \<not> P (last xs))"
dgr_demonic_def: fixes Q :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and ij :: "'c" shows "dgr_demonic Q ij = [: Q ij :]"
assms: shows "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
collapse: fixes stateT :: "('d, 'e) stateT" shows "StateT (run_state stateT) = stateT"
rel_compp_Grp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R = (BNF_Def.Grp {x. set_option x \<subseteq> {(x, y). R x y}} (map_option fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_option x \<subseteq> {(x, y). R x y}} (map_option snd)"
hermitian_exproj_psi: shows "hermitian (tensor_P proj_psi (1\<^sub>m K))"
assms: shows "indep_events (\<lambda>m. {x \<in> space M. P m x}) UNIV"
TC_small: fixes A :: "'a set" assumes "class.small TYPE('a)" shows "small A"
id_trans: fixes a :: "ref \<Rightarrow> ref"   and b :: "ref \<Rightarrow> ref"   and c :: "'d \<Rightarrow> ref"   and p :: "'d" shows "(a \<propto> id) (b (c p)) = (a \<propto> b) (c p)"
least_non_zero_position_of_vector_from_index_def: fixes A :: "'a iarray"   and i :: "nat" shows "least_non_zero_position_of_vector_from_index A i = the (find (\<lambda>x. A !! x \<noteq> (0::'a)) [i..<IArray.length A])"
mod_mod_cancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c b" shows "modulo (modulo a b) c = modulo a c"
qml_act_2: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<A>(\<^bold>\<box>\<phi>)]]"
DT_T: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>D\<^sub>T Q" shows "P \<sqsubseteq>\<^sub>T Q"
ord_le_eq_subst: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a \<le> b"   and "f b = c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
Out_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Out Bet P A B \<equiv> A \<noteq> P \<and> B \<noteq> P \<and> (Bet P A B \<or> Bet P B A)"
\<sigma>_dom_bvars_disj: shows "\<forall>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> subst_domain \<sigma> \<inter> set X = {}"
u: fixes i :: "'a" assumes "i \<in> I"   and "a \<le> i"   and "i < b" shows "(f \<longlongrightarrow> u i) (at_right i)"
fmfilter_alt_defs_4: fixes A :: "'d set" shows "fmrestrict_set A = fmfilter (\<lambda>a. a \<in> A)"
bdd_above_bot: fixes A :: "'b multiset set" shows "subset_mset.bdd_below A"
assms_1: shows "prob_space (density M (\<lambda>x. ennreal (D x)))"
B_meas: fixes i :: "nat" shows "B i \<in> sets M"
i_dom_j: shows "i \<in> set (dom_kildall start ! j)"
hidesI: fixes new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl"   and G :: "prog" assumes "is_static new"   and "msig new = msig old"   and "G\<turnstile>declclass new\<prec>\<^sub>C declclass old"   and "G\<turnstile>Method new declared_in declclass new"   and "G\<turnstile>Method old declared_in declclass old"   and "G \<turnstile>Method old inheritable_in pid (declclass new) " shows "G\<turnstile> new hides old"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b fundef"   and b :: "'c fundef" shows "rel_fundef R a b = (list_all2 R (body a) (body b) \<and> arity a = arity b)"
opt_obs_safe_def: fixes obs_f :: "'c \<Rightarrow> 'd option"   and v :: "'d" shows "opt_obs_safe obs_f v \<equiv> \<exists>p. obs_f p \<in> {None, Some v}"
Succeed: fixes m :: "nat"   and x :: "nat"   and j :: "nat" assumes "n \<le> m"   and "token_run x m \<notin> F - {q\<^sub>0}"   and "token_run x (Suc m) \<in> F"   and "rank x m = Some j" shows "i \<le> j"
rel_prime_equation: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "Corps S"   and "f \<in> carrier R"   and "g \<in> carrier R"   and "0 < deg R S X f"   and "0 < deg R S X g"   and "rel_prime_pols R S X f g"   and "h \<in> carrier R" shows "\<exists>u\<in>carrier R. \<exists>v\<in>carrier R. deg R S X u \<le> amax (deg R S X h - deg R S X f) (deg R S X g) \<and> deg R S X v \<le> deg R S X f \<and> u \<cdot>\<^sub>r f \<plusminus> v \<cdot>\<^sub>r g = h"
assms: shows "cr_envT_stateT s M m1 m2"
rel_resource_pos_distr: fixes A :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and B' :: "'h \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_resource A B OO rel_resource A' B' \<le> rel_resource (A OO A') (B OO B')"
cod_char: fixes f :: "'a" shows "hhom\<^sub>C.cod f = (if hhom\<^sub>C.arr f then C.cod f else C.null)"
Suc: shows "Brouwer_degree2 p (\<lambda>x i. if i = 0 then - x i else x i) = - 1"