hcju/isabelleps huggingface dataset:
in_span_adjust: fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "w \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (insert (adjuster n w us + w) (set us))"
IH: fixes y :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "y < n"   and "y \<le> length xs"   and "bal y xs = (t, zs)" shows "xs = inorder t @ zs \<and> size t = y"
mult_1_left: fixes a :: "'a" shows "1' ; a = a"
circuitI: fixes X :: "'a set" assumes "X \<subseteq> carrier"   and "\<not> indep X"   and "\<And>x. x \<in> X \<Longrightarrow> indep (X - {x})" shows "circuit X"
RUV_split_ee: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and q :: "'b"   and q' :: "'b"   and Ml :: "'d set"   and Me :: "'d set"   and h :: "'d \<Rightarrow> 'd set"   and U :: "'a set"   and Ml' :: "'d set"   and Me' :: "'d set"   and h' :: "'d \<Rightarrow> 'd set"   and V :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 2"   and "{#q#} + {#q'#} \<subseteq># P"   and "(entry fg q, Ml, Me, h) \<in> RU_cs fg U"   and "(entry fg q', Ml', Me', h') \<in> RU_cs fg V"   and "(mon_n fg u \<union> mon fg p) \<inter> (Ml \<union> Me \<union> Ml' \<union> Me') = {}"   and "h [*] h'" shows "(u, mon fg p \<union> M, Ml \<union> Me \<union> Ml' \<union> Me') \<in> RUV_cs fg U V"
isometryD_1: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry f" shows "isometry_on UNIV f"
cSUP_upper: fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "x \<in> A"   and "bdd_above (f ` A)" shows "f x \<le> Sup (f ` A)"
add_mset_replicate_mset_safe: fixes M :: "'c multiset"   and a :: "'c" assumes "NO_MATCH {#} M" shows "a, M = (a, {#}) + M"
2: shows "Right.in_hom \<r>[Right.dom \<mu>] (local.dom \<mu> \<star> src \<mu>) (local.dom \<mu>)"
simps: fixes x :: "indices list" shows "left_app x = indices.Left # x"
add: shows "count (image_mset f X) y = (\<Sum>x | x \<in># X \<and> y = f x. count X x)"
cut_ge_Min_greater: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>\<ge> t \<noteq> {}" shows "t \<le> iMin (I \<down>\<ge> t)"
in_on_out_univ: fixes H :: "ocircline" shows "disc H \<union> disc_compl H \<union> ocircline_set H = UNIV"
Cons_3: fixes u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and m :: "'node"   and v :: "'var" assumes "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "(if m \<in> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys p) (\<alpha>n g)) n \<and> v \<in> set us then Some (map (\<lambda>m. lookupDef g m v) (predecessors g m)) else Mapping.lookup p (m, v)) = Mapping.lookup p' (m, v)"
intS': shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S'"
leq_def: fixes x :: "'a"   and y :: "'a" shows "(x \<le>\<^sub>\<S> y) = (x \<oplus> y = y)"
sets: fixes J :: "'b set"   and X :: "'b \<Rightarrow> 'c set" assumes "J \<noteq> {} \<or> I = {}"   and "finite J"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" shows "{\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
a_Cut_l: fixes a :: "coname"   and N :: "trm"   and x :: "name"   and M :: "trm"   and M' :: "trm" assumes "a \<sharp> N"   and "x \<sharp> M"   and "M \<longrightarrow>\<^sub>a M'" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a Cut <a>.M' x.N"
bounded_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<le> b \<sqinter> c) = (a \<le> b \<and> a \<le> c)"
meld_queue_invar: fixes q :: "('c, 'd) BinomialTree list"   and q' :: "('c, 'd) BinomialTree list" assumes "queue_invar q"   and "queue_invar q'" shows "queue_invar (meld q q')"
HPhase0Read_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "Inv2b s"   and "HPhase0Read s s' p d" shows "Inv2b s'"
not_ev_iff: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" shows "(\<not> ev P \<omega>) = alw (\<lambda>xs. \<not> P xs) \<omega>"
list_before_rel_cons: fixes x :: "'b"   and l :: "'b list" shows "list_before_rel (x # l) = {x} \<times> set l \<union> list_before_rel l"
no_zero_height_derivsAux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and MM :: "Var list \<times> (int \<times> Instr) list \<times> int"   and lll :: "int"   and ss :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and m :: "nat"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" shows "((M, l, s, n, ll, t) \<in> Step \<longrightarrow> n = 0 \<longrightarrow> False) \<and> ((MM, lll, ss, m, h, v) \<in> Exec \<longrightarrow> m = 0 \<longrightarrow> False)"
4: shows "Right.R \<mu> \<in> Right.hom (local.dom \<mu> \<star> src \<mu>) (cod \<mu> \<star> src \<mu>)"
b: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
basis_in_supI: fixes \<E> :: "'a set"   and B :: "'a set"   and \<E>' :: "'a set" assumes "\<E> \<subseteq> carrier"   and "B \<subseteq> \<E>'"   and "\<E>' \<subseteq> \<E>"   and "\<E> \<subseteq> carrier"   and "basis_in \<E> B" shows "basis_in \<E>' B"
leq_refl: fixes x :: "'c llist" shows "lprefix x x"
prv_neg_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "prv \<phi>"   and "prv (neg \<phi>)" shows "prv fls"
map_id: fixes p :: "'a spmf" shows "map_spmf id p = p"
Suc_1: fixes E :: "'a set" assumes "dimension m F E" shows "dimension (n * m) K E"
reachable_trans_4: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and l :: "'d"   and sh :: "'b"   and s' :: "'b" assumes "step s l sh"   and "LTS.reachable step sh s'" shows "LTS.reachable step s s'"
less_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs < ys) = (xs \<le> ys \<and> \<not> ys \<le> xs)"
cr_uint16_def: shows "cr_uint16 \<equiv> \<lambda>x y. x = Rep_uint16 y"
degree_eq: shows "degree (\<Prod>x\<in>UNIV. [:0, 1:] - [:x:]) = degree (monom 1 CARD('a) - monom 1 1)"
xney: fixes j :: "'a" assumes "j \<in> K"   and "j \<noteq> i" shows "x i \<noteq> x j \<and> y i \<noteq> y j"
p_add: fixes x :: "'s"   and y :: "'s"   and n :: "nat"   and m :: "nat" shows "ennreal (p x y (n + m)) = \<integral>\<^sup>+ x. ennreal (p x x n * p x y m) \<partial>count_space UNIV"
Rep_range: shows "range to_bl = {bl. length bl = LENGTH('b)}"
line_inter_2: fixes l :: "Lines"   and m :: "Lines"   and l' :: "Lines" assumes "Pascal_Property.inter l m \<noteq> Pascal_Property.inter l' m" shows "line (Pascal_Property.inter l m) (Pascal_Property.inter l' m) = m"
remove: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<in> A" shows "Max A = (if A - {x} = {} then x else max x (Max (A - {x})))"
simps: fixes a1 :: "('e, 'f, 'g) Tree"   and a2 :: "('e, 'f, 'g) Tree" shows "((a1, a2) \<in> Tree_wf) = ((\<exists>t tset. a1 = t \<and> a2 = tConj tset \<and> t \<in> set_bset tset) \<or> (\<exists>t. a1 = t \<and> a2 = tNot t) \<or> (\<exists>t \<alpha>. a1 = t \<and> a2 = tAct \<alpha> t))"
unitary_hadamard_on_i: fixes k :: "nat" assumes "k < n" shows "unitary (hadamard_on_i k)"
mod_mult_cong_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and c :: "'d"   and b :: "'d"   and a :: "'d"   and d :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo c (times b a)) d a = unique_euclidean_semiring.cong modulo c d a"
simps_2: fixes r :: "'b rexp"   and c :: "'b"   and s :: "'b list" shows "lexer r (c # s) = (case lexer (deriv c r) s of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c v))"
hnorm_le_Infinitesimal: fixes e :: "real star"   and x :: "'a star" assumes "e \<in> Infinitesimal"   and "hnorm x \<le> e" shows "x \<in> Infinitesimal"
m1_inv0_finI: fixes x :: "'a m1_state_scheme" assumes "finite (dom (runs x))" shows "x \<in> m1_inv0_fin"
nop_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop A n f \<equiv> vsv f \<and> nop_axioms A n f"
card_Diff_singleton: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) = card A - 1"
src_src: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.src (Span.src \<mu>) = Span.src \<mu>"
concat_injective: fixes xs :: "'a list list"   and ys :: "'a list list" assumes "concat xs = concat ys"   and "length xs = length ys"   and "\<forall>(x, y)\<in>set (zip xs ys). length x = length y" shows "xs = ys"
bound_nat_induct: fixes n :: "nat"   and l :: "nat"   and u :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "n \<in> {l..u}"   and "P l"   and "\<And>n. \<lbrakk>P n; n \<in> {l..<u}\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P n"
simps_4: fixes x1 :: "nat"   and x2 :: "nat" shows "Const x1 \<noteq> Intv x2"
r_univuniv_converg: fixes x :: "nat"   and y :: "nat" assumes "eval r_phi [x, x] \<down>" shows "eval r_univuniv [x, y] = eval r_phi [the (eval r_phi [x, x]), y]"
False: shows "mod_type_class.to_nat i \<noteq> nrows A - 1"
order_gt_0: fixes \<G> :: "'a cyclic_group" assumes "dis_log \<G>" shows "0 < order \<G>"
3: shows "src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine"
nonFinals_def: fixes g :: "graph" shows "nonFinals g \<equiv> filter (\<lambda>f. \<not> final f) (faces g)"
of_nat_ne_\<omega>\<^sub>z: fixes n :: "nat" shows "of_nat n \<noteq> \<omega>\<^sub>z"
span_mem: fixes E :: "'a vec set"   and u :: "'a vec" assumes "E \<subseteq> carrier_vec n"   and "u \<in> E" shows "u \<in> local.span E"
rs_def: shows "rs = arg_min length (word_for (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (sum_list ss))"
these_insert_None: fixes A :: "'c option set" shows "Option.these (insert None A) = Option.these A"
smult_minus_left: fixes a :: "'b"   and p :: "'b poly" shows "Polynomial.smult (- a) p = - Polynomial.smult a p"
matrix_vector_all_zero_from_index2: fixes i :: "'c"   and A :: "(('d, 'e) vec, 'c) vec"   and k :: "'e" shows "(\<forall>m>i. A $ m $ k = (0::'d)) = vector_all_zero_from_index (mod_type_class.to_nat i + 1, vec_to_iarray (column k A))"
FrpR: shows "extractFrame (p \<bullet> R) = \<langle>A\<^sub>R, p \<bullet> \<Psi>\<^sub>R\<rangle>"
nb\<^sub>1: shows "\<phi>\<^sub>O\<inverse> \<Psi> op' = op"
inf1D1: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "(A \<sqinter> B) x" shows "A x"
Cons_2: fixes u :: "'t" assumes "u \<in> set vs" shows "u \<prec>\<^sub>t v"
num_params_tm_norm_poly_le: fixes t :: "taylor_model"   and X :: "nat" assumes "num_params (tm_poly t) \<le> X" shows "num_params (tm_poly (tm_norm_poly t)) \<le> X"
IH1: assumes "(_Entry_) \<in> set (sourcenodes as)" shows "n'' = (_Entry_)"
simps_1: fixes p :: "'b \<Rightarrow> bool"   and i :: "'b"   and a :: "('a, 'b) Node" shows "FingerTreeStruc.splitNlist p i [a] = ([], a, [])"
eq_\<I>_attach: fixes \<I>' :: "('a, 'b) \<I>"   and res :: "('a, 'b) resource"   and conv :: "('c, 'd, 'a, 'b) converter"   and conv' :: "('c, 'd, 'a, 'b) converter" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>_full,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'" shows "conv \<rhd> res = conv' \<rhd> res"
axioms: shows "class.enum enum_class.enum enum_class.enum_all enum_class.enum_ex"
compact_lfiniteI: fixes xs :: "'c llist" assumes "lfinite xs" shows "ccpo.compact lSup (\<sqsubseteq>) xs"
neqs1_6: shows "(\<lambda>x. (diamond_x (x / 2), diamond_x (x / 2) + d / 2)) \<noteq> (\<lambda>x. (diamond_x (x / 2 + 1 / 2), d / 2 - diamond_x (x / 2 + 1 / 2)))"
ask_writer_def: fixes ask :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) writerT" shows "ask_writer ask f = WriterT (ask (\<lambda>r. run_writer (f r)))"
assoc_in_hom_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<a>[f, g, h] : (local.dom f \<star> local.dom g) \<star> local.dom h \<Rightarrow> cod f \<star> cod g \<star> cod h\<guillemotright>"
CACH_SUB: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and n :: "nat"   and k :: "nat" assumes "deriv G C m l (Cachera n)"   and "n \<le> k" shows "deriv G C m l (Cachera k)"
pos_poly_def: fixes p :: "'b poly" shows "pos_poly p = ((0::'b) < lead_coeff p)"
omax_isOmax: fixes R :: "('b \<times> 'b) set set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. Well_order r" shows "isOmax R (omax R)"
pred_mono_strong: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool"   and x :: "('d, 'e, 'f) gpv"   and P1a :: "'d \<Rightarrow> bool"   and P2a :: "'e \<Rightarrow> bool" assumes "pred_gpv P1 P2 x"   and "\<And>z1. \<lbrakk>z1 \<in> results'_gpv x; P1 z1\<rbrakk> \<Longrightarrow> P1a z1"   and "\<And>z2. \<lbrakk>z2 \<in> outs'_gpv x; P2 z2\<rbrakk> \<Longrightarrow> P2a z2" shows "pred_gpv P1a P2a x"
l_galois_connection: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice"   and y :: "'b::refinement_lattice" assumes "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "(lower_galois_connections.l_adjoint (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (x::'b::refinement_lattice) \<sqsubseteq> (y::'b::refinement_lattice)) = (x \<sqsubseteq> F y)"
mult_oner: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod x t_one = x"
distinct_2: fixes x21 :: "variable"   and x22 :: "String.literal set"   and x11 :: "String.literal"   and x12 :: "typ list" shows "Tv x21 x22 \<noteq> Ty x11 x12"
functor_ToTC: shows "functor C.CCC.comp (\<cdot>) C.T.ToTC"
StrictRefEq\<^sub>S\<^sub>e\<^sub>t: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "x \<doteq> y \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then (x \<triangleq> y) \<tau> else invalid \<tau>"
ran_distinct: fixes al :: "('e \<times> 'f) list" assumes "distinct (map fst al)" shows "ran (map_of al) = snd ` set al"
islimpt_Un: fixes x :: "'a"   and S :: "'a set"   and T :: "'a set" shows "x islimpt S \<union> T = (x islimpt S \<or> x islimpt T)"
new_redundant_infer_imp_old_redundant_infer: fixes N :: "'a literal multiset set"   and \<iota> :: "'a literal multiset inference" assumes "G.redundant_infer N \<iota>" shows "src.redundant_infer N (old_infer_of \<iota>)"
sylvester_mat_sub_map: fixes f :: "'c \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and p :: "'c poly"   and q :: "'c poly" assumes "f (0::'c) = (0::'b)" shows "map_mat f (sylvester_mat_sub m n p q) = sylvester_mat_sub m n (map_poly f p) (map_poly f q)"
mapOf_lookup4: fixes t :: "(int \<times> 'b) Tree"   and i :: "int"   and a :: "'b" assumes "valid_tmap t"   and "mapOf t i = Some a" shows "tlookup fst i t = Some (i, a)"
incseq_pow: shows "incseq (\<lambda>i. (f ^^ i) bot)"
0: shows "\<forall>j. j \<noteq> i \<and> (j, i) \<in> r \<longrightarrow> P j"
degree_mod_less': fixes b :: "'b poly"   and a :: "'b poly" assumes "b \<noteq> 0"   and "a mod b \<noteq> 0" shows "degree (a mod b) < degree b"
nth_list_lookup1: fixes t :: "'b tree"   and i :: "nat" assumes "braun t"   and "i < size t" shows "list t ! i = lookup1 t (i + 1)"
distributed_borel_measurable: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> ennreal" assumes "distributed M N X f" shows "f \<in> borel_measurable N"
emeasure_std_simplex_aux_step: fixes b :: "'a"   and A :: "'a set"   and x :: "'a \<Rightarrow> real"   and y :: "real"   and t :: "real" assumes "b \<notin> A"   and "finite A" shows "(x(b := y) \<in> S (insert b A) t) = (y \<in> {0..t} \<and> x \<in> S A (t - y))"
norm_cblinfun_compose: fixes f :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and g :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "norm (f o\<^sub>C\<^sub>L g) \<le> norm f * norm g"
congruence_nonzero: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "H \<noteq> mat_zero"   and "mat_det M \<noteq> 0" shows "congruence M H \<noteq> mat_zero"
msbw: shows "bv_msb (norm_unsigned w) = \<one>"
assumption_L: fixes Cn :: "'b set \<Rightarrow> 'b set"   and \<phi> :: "'b"   and A :: "'b set" assumes "Tarskian_logic Cn"   and "\<phi> \<in> A" shows "\<phi> \<in> Cn A"
iT_Minus_icard: fixes k :: "nat"   and I :: "nat set" shows "icard (k \<ominus> I) = icard (I \<down>\<le> k)"
has_horizontal_composition: shows "horizontal_composition (\<bullet>) (\<star>) src trg"
bij_betw_inv_into_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and B :: "'a set"   and B' :: "'b set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "bij_betw (inv_into A f) B' B"
fundfoldpairs_fundchamber_image: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "Abs_induced_automorph f g `\<rightarrow> C0 = g ` C0"
simps_18: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Powr a b) vs = approx_floatarith p a vs \<bind> (\<lambda>ae1. approx_floatarith p b vs \<bind> powr_aform_err p ae1)"
assms_3: shows "ipresIGAbs h' hA' MOD' MOD''"
deadlock_free_implies_non_terminating: fixes P :: "'b process" assumes "deadlock_free P" shows "\<forall>s\<in>T P. tickFree s"
simps_1: fixes x21 :: "'a"   and x22 :: "'a msllist"   and y21 :: "'a"   and y22 :: "'a msllist" shows "(MSLCons x21 x22 = MSLCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
run_one_step_basic_return_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Return) = (s', vs', res)" shows "\<exists>vrs. res = RSReturn vrs"
\<phi>_def: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "local.\<phi> t1 t2 = \<beta> * log \<alpha> (real (size1 t1 + size1 t2))"
replicate_0: fixes x :: "'m" shows "replicate 0 x = []"
PopElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some pop" shows "\<exists>v ops. os = v # ops \<and> n = Suc 0 \<and> t = (ops, S, h) \<and> ll = l + 1"
trms_of_deg_leq_degree_f: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "trms_of_deg_leq (deg R f) f = f"
safety_prop_INTER1: fixes X :: "'c \<Rightarrow> 'd program set" assumes "\<And>i. safety_prop (X i)" shows "safety_prop (\<Inter> (range X))"
bind_ask2: fixes m :: "'m"   and f :: "'a \<times> 'w list \<Rightarrow> 'r \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
support: shows "pmf (sds R27) c = 0 \<or> pmf (sds R27) b = 0"
fbd_inj_iff: fixes f :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd set" shows "(bd\<^sub>\<F> f = bd\<^sub>\<F> g) = (f = g)"
View'_def: shows "View' = (\<lambda>((vm, vd), vs). View vm vd vs)"
landau_gt_1_in_cong: fixes L :: "real filter \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) set"   and L' :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) set"   and Lr :: "real filter \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) set"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and h :: "real \<Rightarrow> 'a" assumes "landau_symbol L L' Lr"   and "\<And>x. 1 < x \<Longrightarrow> f x = g x" shows "(f \<in> L at_top h) = (g \<in> L at_top h)"
sbh_step_preserves_store_sops_bound: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and t :: "nat" assumes "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')"   and "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t" shows "\<forall>i\<in>\<Union> (fst ` store_sops is'). i < t"
equiv_Int: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and B :: "'a set"   and s :: "('a \<times> 'a) set" assumes "equiv A r"   and "equiv B s" shows "equiv (A \<inter> B) (r \<inter> s)"
take_nTVL: shows "take n (natToVertexList v f es) = natToVertexList v f es"
MkArr_Map: fixes f :: "('o, 'a) arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
less_SUP_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"