hcju/isabelleps huggingface dataset:
neutral': fixes \<sigma> :: "'alt \<Rightarrow> 'alt"   and R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and a :: "'alt" assumes "\<sigma> permutes alts"   and "is_pref_profile R"   and "a \<in> alts" shows "(\<sigma> a \<in> scf (permute_profile \<sigma> R)) = (a \<in> scf R)"
parallel_1: fixes i :: "nat"   and x :: "nat"   and j :: "nat" assumes "\<phi> i x \<up> \<and> \<phi> j x \<up>" shows "parallel i j x \<up>"
bin_nth_ops_4: fixes x :: "int"   and n :: "nat" shows "bit (NOT x) n = (\<not> bit x n)"
assms: shows "f \<in> space\<^sub>N (\<LL> \<infinity> M)"
bdd_map_bdd_lookup: fixes ws :: "bool list"   and bdd :: "'a bdd"   and f :: "'a \<Rightarrow> 'b" assumes "bddh (length ws) bdd" shows "bdd_lookup (bdd_map f bdd) ws = f (bdd_lookup bdd ws)"
span_raw_def: shows "span \<equiv> module.span (*\<^sub>R)"
inv_unique_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "f \<circ> g = id"   and "g \<circ> f = id" shows "inv f = g"
bezw_aux: fixes x :: "nat"   and y :: "nat" shows "int (gcd x y) = Number_Theory_Aux.inverse x y * int x + snd (bezw x y) * int y"
clock_submodel_aux: fixes t :: "'s Trace"   and s :: "'s set \<times> 's" assumes "t \<in> Clock.jkbpC"   and "s \<in> worlds (clock_simMCt t)" shows "gen_model clock_simMC s = gen_model (clock_simMCt t) s"
intros: fixes a :: "'b"   and b :: "'d"   and r :: "('b \<times> 'd) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
ap_stream_def: shows "(\<diamondop>) \<equiv> \<lambda>uu uua. corec_stream (\<lambda>(f, x). shd f (shd x)) (\<lambda>(f, x). False) (\<lambda>(f, x). undefined) (\<lambda>(f, x). (stl f, stl x)) (uu, uua)"
delta: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and n1 :: "nat"   and n2 :: "nat" shows "EPrim f (ENat n1) (ENat n2) \<longrightarrow> ENat (f n1 n2)"
elims_2: fixes x :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "wf_theory x" obtains \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" and axs :: "Core.term set" where "x = (\<Sigma>, axs)" and "(\<forall>p\<in>axs. wt_term \<Sigma> p \<and> \<turnstile>\<^sub>\<tau> p : propT) \<and> is_std_sig \<Sigma> \<and> wf_sig \<Sigma> \<and> eq_axs \<subseteq> axs"
nonpos_Reals_mult_of_nat_iff: fixes a :: "'a"   and n :: "nat" shows "(a * of_nat n \<in> \<real>\<^sub>\<le>\<^sub>0) = (a \<in> \<real>\<^sub>\<le>\<^sub>0 \<or> n = 0)"
A21_NotSource_level1: shows "isNotDSource level1 sA21"
agreement_def: fixes c :: "('p, 'v, 's) configuration" shows "agreement c \<equiv> \<forall>v1. <\<bottom>, outM v1> \<in># msgs c \<longrightarrow> (\<forall>v2. <\<bottom>, outM v2> \<in># msgs c = (v2 = v1))"
OneState_HPT_Target_ChiPlus: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and U :: "'d"   and T :: "'d"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "(U, T) \<in> ChiPlus (HA ST)"   and "S \<in> Target TS"   and "A \<in> SAs (HA ST)"   and "S \<in> States A"   and "T \<in> States A" shows "U \<notin> Target TS"
s1_s2: shows "[s2 + s1 = x * y + q] (mod q)"
FinFunc_osum: fixes fg :: "'b + 'c \<Rightarrow> 'a" shows "(fg \<in> rst.FINFUNC) = (fg \<circ> Inl \<in> rs.FINFUNC \<and> fg \<circ> Inr \<in> rt.FINFUNC)"
index_col: fixes i :: "nat"   and A :: "'a Matrix.mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "Matrix.col A j $ i = A $$ (i, j)"
simps: fixes P1 :: "'c \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> bool"   and a :: "'c \<times> 'd" shows "pred_prod P1 P2 a = (\<exists>aa b. a = (aa, b) \<and> P1 aa \<and> P2 b)"
quasi_measure_preservingI: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "\<And>A. A \<in> sets N \<Longrightarrow> (f -` A \<inter> space M \<in> null_sets M) = (A \<in> null_sets N)" shows "f \<in> quasi_measure_preserving M N"
partition_on_the_part_mem: fixes A :: "'c set"   and P :: "'c set set"   and x :: "'c" assumes "partition_on A P"   and "x \<in> A" shows "(THE X. x \<in> X \<and> X \<in> P) \<in> P"
terminates_on_generator_eq_UNIV: fixes g :: "('b, 'a) generator" shows "terminates_on (generator g) = UNIV"
upper_asymptotic_density_in_01_2: fixes A :: "nat set" shows "upper_asymptotic_density A \<le> 1"
hcomp_isomorphic_ide: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.isomorphic f g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D h" shows "D.isomorphic (f \<star>\<^sub>D h) (g \<star>\<^sub>D h)"
star_trans: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "star r x y"   and "star r y z" shows "star r x z"
bres_galois_imp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqinter> y \<le> z" shows "y \<le> proto_near_quantale.bres Sup (\<le>) (\<sqinter>) x z"
add: fixes x :: "complex"   and y :: "complex"   and xa :: "'b" shows "(x + y) *\<^sub>C xa = x *\<^sub>C xa + y *\<^sub>C xa"
mk_eqcl_len_snd: fixes zs :: "nat list"   and xs :: "nat option list"   and i :: "nat"   and T :: "bool list list" shows "length zs \<le> length (snd (mk_eqcl xs zs i T))"
assms_2: shows "action_needed_asses a s1 \<subseteq>\<^sub>f action_needed_asses a s2"
sets_Collect_const: fixes P :: "bool" shows "{x \<in> \<Omega>. P} \<in> M"
last_appendR: fixes ys :: "'b list"   and xs :: "'b list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
inj_imp_bij_betw_inv: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b set" assumes "inj f" shows "bij_betw (inv f) (f ` M) M"
Runit_in_runit: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "arr f"   and "t \<in> f" shows "\<^bold>\<r>\<^bold>[t\<^bold>] \<in> \<r>[f]"
ps: shows "\<forall>P. local_peak lrs P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD lrs ((ars\<^sup>+)\<inverse>) (fst P, snd P, \<sigma>', \<tau>'))"
gt_embedding_property: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "t \<noteq> s" shows "t >\<^sub>t s"
mustT_Atm: fixes atm :: "'atom"   and s :: "'state" shows "mustT (Atm atm) s"
simps_22: fixes va :: "(val \<times> val) fset" shows "Wrong \<sqsubseteq> val.Fun va = False"
degree_add_eq_right: fixes p :: "'a poly"   and q :: "'a poly" assumes "degree p < degree q" shows "degree (p + q) = degree q"
HFinite_bounded: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<le> x"   and "0 \<le> y" shows "y \<in> HFinite"
eq_sym: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D"   and "y \<in> D"   and "cmp x y = EQUAL" shows "cmp y x = EQUAL"
hdomainI: fixes x :: "hf"   and y :: "hf"   and f :: "hf" assumes "\<langle>x, y\<rangle> \<^bold>\<in> f" shows "x \<^bold>\<in> hdomain f"
assms: fixes p :: "rat poly" assumes "p \<in> set (poly_list fml)" shows "ls ! index_of ps p = rpoly p x"
simps_1: fixes n :: "nat" shows "to_mregex (regex.Skip n) = (MSkip n, [])"
level_approx: fixes S :: "real set"   and f :: "real \<Rightarrow> real"   and x :: "real"   and n :: "nat" assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x \<and> f x \<le> 1"   and "x \<in> S"   and "n \<noteq> 0" shows "\<bar>f x - (\<Sum>k = Suc 0..n. if real k / real n \<le> f x then inverse (real n) else 0)\<bar> < inverse (real n)"
linear_subspace_vimage: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'c set" assumes "clinear f"   and "csubspace S" shows "csubspace (f -` S)"
identity_implies_part: fixes x :: "i"   and y :: "i" assumes "x = y" shows "P x y"
module_hom_zero: shows "module_hom s1 s2 (\<lambda>x. 0::'c)"
square_free_prodD: fixes fs :: "'b poly set"   and f :: "'b poly"   and g :: "'b poly" assumes "square_free (\<Prod>fs)"   and "finite fs"   and "f \<in> fs"   and "g \<in> fs"   and "f \<noteq> g" shows "algebraic_semidom_class.coprime f g"
representant_exists: fixes A :: "'a set" assumes "A \<in> contained.Partition" shows "\<exists>a\<in>G. a \<in> A \<and> A = contained.Class a"
simps_3: fixes b1 :: "bexp"   and b2 :: "bexp"   and res :: "bool"   and S :: "'av st option" shows "inv_bval' (And b1 b2) res S = (if res then inv_bval' b1 True (inv_bval' b2 True S) else inv_bval' b1 False S \<squnion> inv_bval' b2 False S)"
hom_add: fixes x :: "'a poly"   and y :: "'a poly"   and i :: "nat" shows "coeff (x + y) i = coeff x i + coeff y i"
frontier_of_restrict: fixes X :: "'b topology"   and S :: "'b set" shows "X frontier_of S = X frontier_of (topspace X \<inter> S)"
handle_append: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d :: "nat"   and vs :: "val list"   and pc :: "nat"   and xa :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ws :: "val list"   and ls :: "val list"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P,C,M \<rhd> xt / I,d"   and "d \<le> length vs"   and "pc \<in> I"   and "pc \<notin> pcs xt" shows "handle P C M xa h (ws @ vs) ls pc frs = handle P C M xa h vs ls pc frs"
zeta_partial_sum_le_pos: assumes "0 < s"   and "s \<noteq> 1" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>x. real x powr - s) x - (x powr (1 - s) / (1 - s) + Re (zeta (complex_of_real s)))\<bar> \<le> c * x powr - s"
preserves_seq: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "D.seq (src\<^sub>D g) (src\<^sub>D f)"
simps_4: fixes \<phi> :: "atomUni fmUni"   and \<psi> :: "atomUni fmUni" shows "negUni (AndUni \<phi> \<psi>) = OrUni (negUni \<phi>) (negUni \<psi>)"
mult_noteq_inf: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h"   and "w \<noteq> \<infinity>\<^sub>h" shows "z *\<^sub>h w \<noteq> \<infinity>\<^sub>h"
iffb_inj: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and \<psi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>"   and "Inf_pres \<psi>"   and "fb\<^sup>-\<^sub>\<F> \<phi> = fb\<^sup>-\<^sub>\<F> \<psi>" shows "\<phi> = \<psi>"
scale_eq_0_iff: fixes a :: "'b"   and x :: "('b, 'c) Finite_Cartesian_Product.vec" shows "(a *s x = 0) = (a = (0::'b) \<or> x = 0)"
fls_subdegree_eqI: fixes f :: "'a fls"   and n :: "int" assumes "fls_nth f n \<noteq> (0::'a)"   and "\<And>k. k < n \<Longrightarrow> fls_nth f k = (0::'a)" shows "fls_subdegree f = n"
undefined: fixes a :: "'b\<^sub>m"   and b :: "'b\<^sub>m" assumes "rhb a \<noteq> rhb b"   and "a \<in> B" shows "mb a b = None"
continuous_on_eq: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "continuous_on s g"
simps_4: fixes v :: "val" shows "fv (Val v) = {}"
interior: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and t :: "'a" assumes "distinct4 p q r t"   and "in4 S p q r t"   and "ccw t q r"   and "ccw p t r"   and "ccw p q t" shows "ccw p q r"
eval_Init: fixes C :: "qtname"   and s0 :: "st"   and s3 :: "state"   and G :: "prog"   and s1 :: "state"   and s2 :: "state" assumes "if inited C (globs s0) then s3 = Norm s0 else G\<turnstile>Norm (init_class_obj G C s0) \<midarrow>(if C = Object then Skip else Init (super (the (class G C))))\<rightarrow> s1 \<and> G\<turnstile>set_lvars Map.empty s1 \<midarrow>init (the (class G C))\<rightarrow> s2 \<and> s3 = set_lvars (locals (snd s1)) s2" shows "G\<turnstile>Norm s0 \<midarrow>Init C\<rightarrow> s3"
cnj_rGamma: fixes z :: "complex" shows "cnj (rGamma z) = rGamma (cnj z)"
class_add_hconf: fixes P :: "jvm_prog"   and h :: "heap"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> h \<surd>"   and "\<not> is_class P C"   and "\<And>a obj D'. \<lbrakk>h a = \<lfloor>obj\<rfloor>; P \<turnstile> fst obj \<preceq>\<^sup>* D'\<rbrakk> \<Longrightarrow> D' \<noteq> C" shows "class_add P (C, cdec) \<turnstile> h \<surd>"
concat_eq_appendD: fixes xss :: "'c list list"   and ys :: "'c list"   and zs :: "'c list" assumes "concat xss = ys @ zs"   and "xss \<noteq> []" shows "\<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2"
vdomain_vempty: shows "\<D>\<^sub>\<circ> []\<^sub>\<circ> = []\<^sub>\<circ>"
linear_compose: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and s3 :: "'b \<Rightarrow> 'e \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "Vector_Spaces.linear s1 s2 f"   and "Vector_Spaces.linear s2 s3 g" shows "Vector_Spaces.linear s1 s3 (g \<circ> f)"
simps: fixes a1 :: "nat \<times> nat \<times> thread"   and a2 :: "nat \<times> nat \<times> thread" shows "astep_sng a1 a2 = ((\<exists>c n. a1 = (c, n, INIT) \<and> a2 = (c, n + 1, WAIT n)) \<or> (\<exists>c k n. a1 = (c, n, WAIT k) \<and> a2 = (c, n, WAIT k) \<and> c \<noteq> k) \<or> (\<exists>c n. a1 = (c, n, WAIT c) \<and> a2 = (c, n, HOLD c)) \<or> (\<exists>c n k. a1 = (c, n, HOLD k) \<and> a2 = (c, n, REL k)) \<or> (\<exists>c n k. a1 = (c, n, REL k) \<and> a2 = (k + 1, n, INIT)))"
unit_wfactors_empty: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and fs :: "'c list" assumes "comm_monoid_cancel G"   and "a \<in> Units G"   and "wfactors G fs a"   and "set fs \<subseteq> carrier G" shows "fs = []"
funs_term_subst: fixes t :: "('c, 'e) Term.term"   and \<theta> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "funs_term (t \<cdot> \<theta>) = funs_term t \<union> (\<Union>x\<in>fv t. funs_term (\<theta> x))"
l12_eq: shows "(l1 = l2) = (subset_sorted l1 l2 \<and> subset_sorted l2 l1)"
simps_2: fixes x :: "dB" shows "CF_pure (Pure x) = x"
lexord_Nil_right: fixes x :: "'b list"   and r :: "('b \<times> 'b) set" shows "(x, []) \<notin> lexord r"
cond_def: fixes b :: "'a upred"   and P :: "'a upred"   and Q :: "'a upred" shows "P \<triangleleft> b \<triangleright> Q = (b \<and> P \<or> \<not> b \<and> Q)"
vec_eq_iff: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" shows "(x = y) = (\<forall>i. x $ i = y $ i)"
new_document_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h h'"
only_new_introduces_lids': fixes 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"   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"   and l :: "'l" assumes "revision_step r s s'"   and "\<And>\<sigma> \<tau> \<E> v. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"   and "l \<notin> LID\<^sub>G s" shows "l \<notin> LID\<^sub>G s'"
fmdrop_lookup: fixes a :: "'d"   and m :: "'d \<rightharpoonup>\<^sub>f 'c" shows "fmlookup (fmdrop a m) a = None"
assms_2: shows "GE.satC \<xi> (gC c)"
filter_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_fun A (=)) (rel_fun (rel_set A) (rel_set A)) Set.filter Set.filter"
topological_basis_def: fixes B :: "'b set set" shows "topological_basis B = ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
bind_spmf_const: fixes p :: "'h spmf"   and q :: "'g spmf" shows "p \<bind> (\<lambda>x. q) = scale_spmf (weight_spmf p) q"
b_gt_1: fixes M :: "'c measure"   and b :: "real" assumes "information_space M b" shows "1 < b"
rel_intros_1: fixes R2 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a"   and y1 :: "'b"   and R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R2 x1 y1" shows "tllist_all2 R1 R2 (TNil x1) (TNil y1)"
mod_add_self2: fixes divide :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g"   and modulo :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and euclidean_size :: "'g \<Rightarrow> nat"   and a :: "'g"   and b :: "'g" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a b) b = modulo a b"
simps_15: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a gexp"   and va :: "'a gexp" shows "local.less_gexp_aux (Gt e1 e2) (Nor v va) = True"
abs_eq: fixes x :: "'b set" shows "singles x = Abs_ttree {xs. \<forall>x\<in>x. length (filter (\<lambda>x'. x' = x) xs) \<le> 1}"
3: fixes d :: "'a" assumes "d \<in> S" shows "\<exists>U. openin (top_of_set S) U \<and> d \<in> U \<and> (\<forall>x\<in>U. \<exists>f g. homeomorphism T T f g \<and> f d = x \<and> {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)})"
\<theta>_def: shows "\<theta> \<equiv> Var(z := Var x)"
le_supI1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)"   and "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (a::'b)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) ((sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (a::'b) (b::'b))"
div_mult1_eq: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and euclidean_size :: "'m \<Rightarrow> nat"   and division_segment :: "'m \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m"   and c :: "'m" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (times a b) c = plus (times a (divide b c)) (divide (times a (modulo b c)) c)"
vars_monom_subset: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b" shows "vars (monom m a) \<subseteq> keys m"
rwp_on_refl_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f"   and "decisive_on A f" shows "refl_on A (rwp_on A f)"
pwd_spy_con_def: fixes n :: "nat" shows "pwd_spy_con n \<equiv> let (s, S, A, B, C, D) = pwd_spy_enc_sign n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, \<lbrace>Crypt (SesK SK) (Auth_PriK n \<otimes> B), Crypt (SesK SK) (Sign n (Auth_PriK n))\<rbrace>) s, S, A, B, C, D)"
fset_equiv: fixes f1 :: "'a fset"   and f2 :: "'a fset" shows "(f1 = f2) = (fset f1 = fset f2)"
lex_list_wfP: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP order" shows "wfP (lexp order)"
prems_1: shows "\<forall>A\<in>F. finite A \<and> card A = Suc k"
R\<^sub>C_of_sym: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" assumes "sym \<R>\<^sub>A"   and "sym P"   and "secure_refinement \<R>\<^sub>A \<R> P"   and "\<And>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mdsa mem\<^sub>2. (\<langle>c\<^sub>1, mdsa, mem\<^sub>1\<rangle>\<^sub>A, \<langle>c\<^sub>2, mdsa, mem\<^sub>2\<rangle>\<^sub>A) \<in> \<R>\<^sub>A \<Longrightarrow> abs.low_mds_eq mdsa mem\<^sub>1 mem\<^sub>2" shows "sym (R\<^sub>C_of \<R>\<^sub>A \<R> P)"
Lazy_Sequence_def: fixes f :: "unit \<Rightarrow> ('a \<times> 'a lazy_sequence) option" shows "Lazy_Sequence f = lazy_sequence_of_list (case f () of None \<Rightarrow> [] | Some (x, xq) \<Rightarrow> x # list_of_lazy_sequence xq)"
psubst_nsubst: fixes x :: "name"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and a :: "coname"   and P :: "trm"   and M :: "trm" assumes "x \<sharp> (\<theta>_n, \<theta>_c)" shows "((x, a, P) # \<theta>_n),\<theta>_c<M> = \<theta>_n,\<theta>_c<M>{x:=<a>.P}"
intros: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "wt t = wt s"   and "head t >\<^sub>h\<^sub>d head s" shows "gt_diff t s"
context_right: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and u :: "('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "App u t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b App u s"
indRel_cond_respection_iff_indRelR_cond_respection: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_respects_pred Rel Pred) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_pred Rel Pred)"
poly_add: fixes q :: "'a poly"   and p :: "'a poly"   and x :: "'a"   and y :: "'a" assumes "q \<noteq> 0"   and "poly p x = (0::'a)"   and "poly q y = (0::'a)" shows "poly (poly_add p q) (x + y) = (0::'a)"
ChamberComplexAutomorphism_axioms: shows "ChamberComplexAutomorphism X f"
plossless_map_converter: fixes f :: "'m \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'n"   and \<I> :: "('m, 'n) \<I>"   and h :: "'k \<Rightarrow> 'o"   and k :: "'p \<Rightarrow> 'l"   and \<I>' :: "('o, 'p) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter" assumes "plossless_converter (map_\<I> (inv f) (inv g) \<I>) (map_\<I> h k \<I>') conv"   and "inj f" shows "plossless_converter \<I> \<I>' (map_converter f g h k conv)"
S_transform_weak_transition_iff: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" shows "P \<Rightarrow>\<langle>S_action.Act \<alpha>\<rangle>\<^sub>S P' = P \<Rightarrow>\<langle>\<alpha>\<rangle> P'"
nodisc: shows "\<not> has_disc_negated disc2 neg m"
sg_i_closed: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and x :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "x \<in> H" shows "\<rho>\<^bsub>G\<^esub> x \<in> H"
maximal_independent_subset_extend: fixes S :: "'b set"   and V :: "'b set" assumes "S \<subseteq> V"   and "vs1.independent S" obtains B :: "'b set" where "S \<subseteq> B" and "B \<subseteq> V" and "vs1.independent B" and "V \<subseteq> vs1.span B"
residue_ring_def: fixes m :: "int" shows "residue_ring m = \<lparr>carrier = {0..m - 1}, monoid.mult = \<lambda>x y. x * y mod m, one = 1, zero = 0, add = \<lambda>x y. (x + y) mod m\<rparr>"