hcju/isabelleps huggingface dataset:
real_sqrt_power_even: fixes n :: "nat"   and x :: "real" assumes "even n"   and "0 \<le> x" shows "sqrt x ^ n = x ^ (n div 2)"
frechet_derivative_zero_fun: fixes a :: "'c" shows "frechet_derivative 0 (at a) = 0"
generator_induct0: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> carrier G"   and "P \<one>"   and "\<And>x. \<lbrakk>x \<in> carrier G; P x\<rbrakk> \<Longrightarrow> P (x \<otimes> gen)" shows "P x"
simps_2: fixes c :: "int \<times> int"   and c' :: "poly"   and n' :: "nat"   and p' :: "poly" shows "C c *\<^sub>p CN c' n' p' = (if c = 0\<^sub>N then 0\<^sub>p else CN (C c *\<^sub>p c') n' (C c *\<^sub>p p'))"
strict_boundedE: fixes a :: "'g"   and b :: "'g"   and c :: "'g" assumes "a < min b c" obtains "a < b" and "a < c"
in_subterms_subset_Union: fixes t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "subterms t \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t M"
eval_fm_Iff: fixes e :: "(name, hf) finfun"   and A :: "fm"   and B :: "fm" shows "eval_fm e (A IFF B) = (eval_fm e A = eval_fm e B)"
sup_level: fixes l1 :: "level"   and l2 :: "level" shows "l1 \<squnion> l2 \<equiv> if Hi \<in> {l1, l2} then Hi else Lo"
length_0_conv: fixes xs :: "'a list" shows "(\<parallel>xs\<parallel> = 0) = (xs = [])"
n_def: shows "n = Max {n\<^sub>1, n\<^sub>2, n\<^sub>3}"
equiintegrable_sum: fixes F :: "('a \<Rightarrow> 'b) set"   and a :: "'a"   and b :: "'a" assumes "F equiintegrable_on cbox a b" shows "(\<Union>I\<in>Collect finite. \<Union>c\<in>{c. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1}. \<Union>f\<in>I \<rightarrow> F. {\<lambda>x. \<Sum>i\<in>I. c i *\<^sub>R f i x}) equiintegrable_on cbox a b"
locally_compatible_l_def: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset" shows "locally_compatible_l R1 R2 A B C \<equiv> \<forall>t u v. t \<in># A \<longrightarrow> u \<in># B \<longrightarrow> v \<in># C \<longrightarrow> (t, u) \<in> R1 \<longrightarrow> (u, v) \<in> R2 \<longrightarrow> (t, v) \<in> R2"
maxExpandDepth: fixes R :: "pi"   and P :: "pi"   and Q :: "pi" assumes "R \<in> expandSet P Q"   and "hnf P"   and "hnf Q" shows "depth R \<le> depth (P \<parallel> Q)"
qVar_alpha_iff: fixes xs :: "'e"   and x :: "'f"   and X' :: "('c, 'd, 'e, 'f, 'g) qTerm" shows "(qVar xs x #= X') = (X' = qVar xs x)"
rbt_less_nit_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_less less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less x k)"
L_infinity_2: fixes M :: "'a measure" shows "defect (\<LL> \<infinity> M) = 1"
access_def: fixes s :: "Store"   and l :: "Location" shows "s@@l \<equiv> vals (Rep_Store s) l"
antisym_conv: fixes ko :: "'b key_order"   and y :: "'b"   and x :: "'b" assumes "ko.le ko y x" shows "ko.le ko x y = (x = y)"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "gcd b (gcd a c) = gcd a (gcd b c)"
seqsimilar_def: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<tau> :: "nat \<Rightarrow> 'a" shows "\<sigma> \<approx> \<tau> \<equiv> \<natural> \<sigma> = \<natural> \<tau>"
beukers_nn_integral3_finite: fixes n :: "nat" shows "beukers_nn_integral3 n < \<infinity>"
subst_def: fixes xs :: "'varSort"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and x :: "'var" shows "subst xs X x \<equiv> psubst (idEnv [x \<leftarrow> X]_xs)"
f\<^sub>2: shows "f\<^sub>2 = snd (split_face f u v vs)"
ccpo_spmf: shows "class.ccpo lub_spmf (ord_spmf (=)) (mk_less (ord_spmf (=)))"
gcd_nnn: fixes n :: "nat" assumes "0 < n" shows "n = Arith2.gcd n n"
inverse_exists: fixes a :: "'a" assumes "a \<in> carrier K"   and "a \<noteq> \<zero>\<^bsub>K\<^esub>" shows "inv\<^bsub>K\<^esub> a \<in> carrier K"
openE: fixes m :: "'a set" assumes "m open" obtains "m \<in> S"
induct: fixes x :: "nat \<Rightarrow> nat"   and P :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" assumes "x \<in> PrimRec1'"   and "P (\<lambda>x. 0)"   and "P Suc"   and "P c_fst"   and "P c_snd"   and "\<And>f g. \<lbrakk>f \<in> PrimRec1'; P f; g \<in> PrimRec1'; P g\<rbrakk> \<Longrightarrow> P (\<lambda>x. f (g x))"   and "\<And>f g. \<lbrakk>f \<in> PrimRec1'; P f; g \<in> PrimRec1'; P g\<rbrakk> \<Longrightarrow> P (c_f_pair f g)"   and "\<And>f g. \<lbrakk>f \<in> PrimRec1'; P f; g \<in> PrimRec1'; P g\<rbrakk> \<Longrightarrow> P (UnaryRecOp f g)" shows "P x"
separate_aEval: fixes L :: "atomUni list"   and a :: "(real \<times> real \<times> real) list"   and b :: "(real \<times> real \<times> real) list"   and c :: "(real \<times> real \<times> real) list"   and d :: "(real \<times> real \<times> real) list"   and x :: "real" assumes "separateAtoms L = (a, b, c, d)" shows "(\<forall>l\<in>set L. aEvalUni l x) = ((\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
Rw: shows "A = \<Union> (SN_rel_ext_step P Pw R Rw ` {top_s, top_ns, normal_s, normal_ns}) \<inter> M'"
comp_in_homI': fixes f :: "'b"   and g :: "'b"   and a :: "'b"   and c :: "'b" assumes "D.arr f"   and "D.arr g"   and "D.dom f = a"   and "D.cod g = c"   and "D.dom g = D.cod f" shows "\<guillemotleft>g \<cdot>\<^sub>D f : a \<Rightarrow>\<^sub>D c\<guillemotright>"
f: shows "integrable (M1 \<Otimes>\<^sub>M M2) f"
similar_mat_wit_char_matrix: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat"   and ev :: "'b" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit (char_matrix A ev) (char_matrix B ev) P Q"
unlabel_append: fixes A :: "('b \<times> 'a) list"   and B :: "('b \<times> 'a) list" shows "unlabel (A @ B) = unlabel A @ unlabel B"
the_inverseD_1: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and g :: "'a" assumes "BinOpSetGroup G binop e"   and "g \<in> G" shows "BinOpSetGroup.the_inverse G binop e g \<in> G"
algebraic_mult_cancel_right: fixes y :: "'b"   and x :: "'b" assumes "algebraic y"   and "y \<noteq> (0::'b)" shows "algebraic (x * y) = algebraic x"
tr: shows "(st, a, SubnetS s' t') \<in> automaton.trans (pnet np (p1 \<parallel> p2))"
mrDenyAll_is_unique: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "wellformed_policy1_strong p"   and "applied_rule_rev Cp x p = \<lfloor>DenyAll\<rfloor>"   and "r \<in> set (tl p)" shows "x \<notin> dom (Cp r)"
Span_eq_generate: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R" shows "ring.Span R K Us = generate (add_monoid R) (K <#>\<^bsub>R\<^esub> set Us)"
exec_gpv_bind_lift_spmf: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and p :: "'e spmf"   and f :: "'e \<Rightarrow> ('a, 'c, 'd) gpv"   and s :: "'b" shows "exec_gpv callee (lift_spmf p \<bind> f) s = p \<bind> (\<lambda>x. exec_gpv callee (f x) s)"
elims_2: fixes x :: "nat \<times> nat list"   and xa :: "nat \<times> cell list \<times> cell list"   and xb :: "cell list" assumes "inv_check_left_moving x xa xb" obtains as :: "nat" and lm :: "nat list" and s :: "nat" and l :: "cell list" and r :: "cell list" and ires :: "cell list" where "x = (as, lm)" and "xa = (s, l, r)" and "xb = ires" and "inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires \<or> inv_check_left_moving_in_middle (as, lm) (s, l, r) ires"
nz: fixes ka :: "nat"   and kb :: "nat" assumes "l = ka + kb - 1"   and "0 < kb"   and "kb \<le> ka"   and "ka \<le> Suc kb" shows "thesis"
App_def: fixes x2 :: "lam"   and x1 :: "lam" shows "App x2 x1 \<equiv> Abs_lam (App_Rep (Rep_lam x2) (Rep_lam x1))"
sat_prefix_conv: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<sigma>' :: "(String.literal \<times> event_data list) trace"   and i :: "nat"   and \<phi> :: "Formula.formula"   and v :: "event_data list" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'"   and "i < Monitor.progress \<sigma> Map.empty \<phi> (plen \<pi>)" shows "Formula.sat \<sigma> Map.empty v i \<phi> = Formula.sat \<sigma>' Map.empty v i \<phi>"
HSigmaI: fixes a :: "hf"   and A :: "hf"   and b :: "hf"   and B :: "hf \<Rightarrow> hf" assumes "a \<^bold>\<in> A"   and "b \<^bold>\<in> B a" shows "\<langle>a, b\<rangle> \<^bold>\<in> HSigma A B"
tagged_division_ofD_4: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "s tagged_division_of i"   and "(x, K) \<in> s" shows "\<exists>a b. K = cbox a b"
rtranclp_imp_Sup_relpowp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "P\<^sup>*\<^sup>* x y" shows "Sup (range ((^^) P)) x y"
emeasure_finite: fixes M :: "'c measure"   and A :: "'c set" assumes "finite_measure M" shows "emeasure M A \<noteq> top"
continuous_on_eq: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "continuous_on s g"
diffLessSuc: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "Suc (n - m) = Suc n - m"
Rep_ratfps: fixes x :: "'b ratfps" shows "Rep_ratfps x \<in> {x. constant_term (denominator x) \<noteq> (0::'b)}"
simps: fixes a :: "'d"   and r :: "('d \<times> 'e) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
from_nat_into_inj: fixes A :: "'c set"   and m :: "nat"   and n :: "nat" assumes "countable A"   and "m \<in> to_nat_on A ` A"   and "n \<in> to_nat_on A ` A" shows "(from_nat_into A m = from_nat_into A n) = (m = n)"
zoom_def: fixes l :: "'e \<Longrightarrow> 'c"   and m :: "('e, 'd) state" shows "zoom l m = State (\<lambda>b. case run_state m (get\<^bsub>l\<^esub> b) of (c, a) \<Rightarrow> (c, put\<^bsub>l\<^esub> b a))"
Struct_axioms: shows "CU.Struct wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) (intT any) intF intP"
zero_le_floor: fixes x :: "'c" shows "(0 \<le> \<lfloor>x\<rfloor>) = ((0::'c) \<le> x)"
Int_insert_right_if1: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "a \<in> A" shows "A \<inter> insert a B = insert a (A \<inter> B)"
poly_condD_5: fixes p :: "int poly" assumes "poly_cond p" shows "p \<noteq> 0"
of_nat_simps_4: fixes m :: "nat"   and n :: "nat" shows "of_nat (m + n) = of_nat m + of_nat n"
hyps: assumes "k < n" shows "exH_k k = st.pmat_extension (H_k k)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "semiring_hom hom" shows "ring_hom hom"
filterlim_tendsto_neg_mult_at_top_iff: fixes f :: "'c \<Rightarrow> real"   and c :: "real"   and F :: "'c filter"   and g :: "'c \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c < 0" shows "(LIM x F. f x * g x :> at_top) = filterlim g at_bot F"
neg_lit_in_atms_of: fixes A :: "'b"   and C :: "'b literal multiset" assumes "Neg A \<in># C" shows "A \<in> atms_of C"
assoc: fixes a :: "'h"   and b :: "'h"   and c :: "'h" shows "a * b * c = a * (b * c)"
app_vrrestrictionD_1: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> A"
oc_projection_exist: fixes i :: "nat" assumes "i < length fs" shows "fs ! i - gso i \<in> local.span (gso ` {0..<i})"
shd_sset: fixes a :: "'d stream" shows "shd a \<in> sset a"
assms_2: fixes x :: "'a" assumes "\<Phi> x" shows "\<Psi> x"
assms_6: shows "\<exists>As. CT;\<Gamma>1 \<turnstile>+ ds : As \<and> CT \<turnstile>+ As <: Bs"
assms_3: shows "i < Monitor.progress \<sigma> \<phi> (plen \<pi>)"
compile_append_dummy_correct: fixes f :: "recf"   and ap :: "abc_inst list"   and ary :: "nat"   and fp :: "nat"   and args :: "nat list" assumes "rec_ci f = (ap, ary, fp)"   and "terminate f args" shows "{\<lambda>nl. nl = args} ap [+] dummy_abc (length args) {\<lambda>nl. \<exists>m. nl = args @ rec_exec f args # 0 \<up> m}"
rComm1: fixes M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and P'' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "\<Psi> \<rhd> !P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''"   and "\<And>C. Prop C \<Psi> (!P) (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) P''"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop C \<Psi> (P \<parallel> !P) (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"
dual_dual: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual (dual x) = x"
lem1: shows "((\<lambda>x. \<Sum>b\<in>basis. f ((g1 +++ g2) x) \<bullet> b * (vector_derivative (g1 +++ g2) (at x within {0..1}) \<bullet> b)) has_integral i1) {0..1 / 2}"
simps: fixes s1 :: "('e, 'b, 'c, 'f) SLin_state_scheme"   and s2 :: "('a, 'b, 'c, 'g) SLin_state_scheme" shows "P26 (s1, s2) = (\<forall>p. status s1 p = Aborted \<and> \<not> contains (dstate s2) (pending s1 p) \<longrightarrow> status s2 p \<in> {Pending, Aborted} \<and> pending s1 p = pending s2 p)"
bind_split_asm: fixes P :: "'b option \<Rightarrow> bool"   and m :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "P (m \<bind> f) = (\<not> (m = None \<and> \<not> P None \<or> (\<exists>x. m = Some x \<and> \<not> P (f x))))"
equivs_2: shows "mylens.base\<^sub>L +\<^sub>L mylens.more\<^sub>L \<approx>\<^sub>L 1\<^sub>L"
ih_18: shows "InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
one_carrier_mat: fixes n :: "nat" shows "1\<^sub>m n \<in> carrier_mat n n"
hyperUn_def: fixes A :: "char list set option"   and B :: "char list set option" shows "A \<squnion> B \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> case B of None \<Rightarrow> None | \<lfloor>B\<rfloor> \<Rightarrow> \<lfloor>A \<union> B\<rfloor>"
one_less_numeral_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less one (numeral.numeral one plus n) = (num.One < n)"
wp2_is_weakestprePotential1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<Turnstile>\<^sub>2 {P} c {Q}" shows "\<forall>s. wp\<^sub>2 c Q s \<le> P s"
relpow_1: fixes R :: "('b \<times> 'b) set" shows "R ^^ 1 = R"
degree_lhs_card: shows "degree (monom 1 CARD('a) - monom 1 1) = CARD('a)"
disc_2: fixes x2 :: "'f" shows "is_Skip (Skip x2)"
ideal_points_different_2: fixes u :: "complex_homo"   and v :: "complex_homo"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "ideal_points (poincare_line u v) = {i1, i2}" shows "u \<noteq> i1"
globalasimp: fixes P :: "(nat \<Rightarrow> 'a) \<times> 'b \<times> (nat \<Rightarrow> 'a) \<Rightarrow> bool"   and s :: "((nat \<Rightarrow> 'a) \<times> 'c) \<times> 'b \<times> (nat \<Rightarrow> 'a) \<times> 'c" shows "globala P s = P (fst (fst s), fst (snd s), fst (snd (snd s)))"
AX_int: fixes p :: "'b set"   and q :: "'b set" shows "\<^bold>A\<^bold>X (p \<inter> q) = \<^bold>A\<^bold>X p \<inter> \<^bold>A\<^bold>X q"
b: shows "Ide b \<and> Arr b \<and> Dom b = b \<and> Cod b = b \<and> ide \<lbrace>b\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>b\<^bold>\<rfloor>\<rbrace> \<and> \<guillemotleft>\<lbrace>b\<^bold>\<down>\<rbrace> : \<lbrace>b\<rbrace> \<rightarrow> \<lbrace>\<^bold>\<lfloor>b\<^bold>\<rfloor>\<rbrace>\<guillemotright>"
simps_3: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "lhd (lalternate xs) = lhd xs"
assms_2: shows "sset (gtrace r v) \<subseteq> U"
rcos_const_imp_mem: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme"   and i :: "'e" assumes "ideal I R"   and "i \<in> carrier R"   and "I +>\<^bsub>R\<^esub> i = I" shows "i \<in> I"
lsc_hull_greatest: fixes g :: "'b \<Rightarrow> ereal"   and f :: "'b \<Rightarrow> ereal" assumes "lsc g"   and "\<forall>x. g x \<le> f x" shows "\<forall>x. g x \<le> lsc_hull f x"
const_lt_component_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "const_lt_component p = (let v = lt p in if pp_of_term v = (0::'a) then Some (component_of_term v) else None)"
thmD7: fixes N :: "symbol"   and \<gamma> :: "symbol list"   and \<alpha> :: "symbol list" assumes "is_derivation (N # \<gamma>)"   and "is_nonterminal N"   and "(N, \<alpha>) \<in> \<RR>" shows "init_item (N, \<alpha>) 0 \<in> \<pi> 0 {} Init"
actionFresh: fixes x :: "name"   and \<alpha> :: "'d action" shows "x \<sharp> \<alpha> = (x \<sharp> subject \<alpha> \<and> x \<sharp> bn \<alpha> \<and> x \<sharp> object \<alpha>)"
order_on_defs_1: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
Pi_Int: fixes I :: "'c set"   and E :: "'c \<Rightarrow> 'd set"   and F :: "'c \<Rightarrow> 'd set" shows "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
intro: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "agents \<noteq> {}"   and "alts \<noteq> {}"   and "\<And>i. i \<in> agents \<Longrightarrow> finite_total_preorder_on alts (R i)"   and "\<And>i x y. i \<notin> agents \<Longrightarrow> \<not> R i x y" shows "pref_profile_wf agents alts R"
Finite_V: fixes X :: "V set" assumes "finite X" shows "small X"
divide_uexpr_def: fixes u :: "('a, 'b) uexpr"   and v :: "('a, 'b) uexpr" shows "u div v = bop (div) u v"
lebesgue_sets_translation: fixes S :: "'b set"   and a :: "'b" assumes "S \<in> sets lebesgue" shows "(+) a ` S \<in> sets lebesgue"
checkmem_def: fixes param :: "'k"   and calc :: "(heap, 'v) state" shows "state_dp_consistency.checkmem param calc \<equiv> lookup_st param \<bind> (\<lambda>x. case x of None \<Rightarrow> calc \<bind> (\<lambda>x. update_st param x \<bind> (\<lambda>_. State_Monad.return x)) | Some x \<Rightarrow> State_Monad.return x)"
g: fixes x :: "complex" assumes "x \<in> S" shows "exp (\<i> * complex_of_real (g x)) = (x - a) /\<^sub>R cmod (x - a) / ((x - b) /\<^sub>R cmod (x - b))"
l_desc_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and X :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf"   and l :: "'a \<Rightarrow> 'a"   and s :: "'a" assumes "Reachability_Problem K S S1 S2" shows "Reachability_Problem.l_desc K S S1 S2 X ct l s = (s \<in> directed_towards S2 (SIGMA s:X. {l s}) \<and> Reachability_Problem.v K S S1 S2 (Finite_Markov_Decision_Process.simple K S ct s) \<le> Reachability_Problem.v K S S1 S2 (Finite_Markov_Decision_Process.simple K S ct (l s)) \<and> l s \<in> maximal (\<lambda>s. Reachability_Problem.v K S S1 S2 (Finite_Markov_Decision_Process.simple K S ct s)) (set_pmf (ct s)))"
simps_2: fixes x :: "'f"   and xs :: "'f list"   and ys :: "'f list" shows "(x # xs) @ ys = x # xs @ ys"
wfP_induct': fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "wfP r"   and "\<And>x. (\<And>y. r y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
typ_of1_decr: fixes Ts :: "typ list"   and T :: "typ"   and Ts' :: "typ list"   and t :: "Core.term"   and ty :: "typ" assumes "typ_of1 (Ts @ [T] @ Ts') t = Some ty"   and "\<not> loose_bvar1 t (length Ts)" shows "typ_of1 (Ts @ Ts') (decr (length Ts) t) = Some ty"
Says_kas_message_form: fixes Kas' :: "agent"   and A :: "agent"   and authK :: "nat"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas' A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "authK \<notin> range shrK \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace> \<or> authTicket \<in> analz (knows Spy evs)"
F2set3_bd': fixes x :: "('e, 'f, 'd) F2"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" shows "|F2set3 x| \<le>o dir_image (r1 +c bd_F2) Abs_bd_type_F"
Farkas'_Lemma: fixes cs :: "rat le_constraint set" assumes "lec_rel ` cs \<subseteq> {Leq_Rel}"   and "finite cs" shows "(\<nexists>v. \<forall>c\<in>cs. v \<Turnstile>\<^sub>l\<^sub>e c) = (\<exists>C const. (\<forall>(r, c)\<in>set C. 0 < r \<and> c \<in> cs) \<and> (\<Sum>(r, c)\<leftarrow>C. Le_Constraint Leq_Rel (r *R lec_poly c) (r *R lec_const c)) = Le_Constraint Leq_Rel 0 const \<and> const < 0)"
admits_diagonal_imp_admits_triangular': assumes "All admits_diagonal_reduction" shows "All admits_triangular_reduction"
pairwise_orthogonal_imp_finite: fixes S :: "'a set" assumes "pairwise orthogonal S" shows "finite S"
src_def: fixes i :: "nat"   and p :: "'p" shows "src i p \<equiv> \<exists>q. channel i = Some (p, q)"
assms_2: shows "\<forall>k<i. P (\<sigma> |\<^sub>s k)"
normalize_stable: fixes q :: "int"   and p :: "int" assumes "0 < q"   and "coprime p q" shows "Rat.normalize (p, q) = (p, q)"
hadd_hmem_cancel: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "(x @+ y \<^bold>\<in> x @+ z) = (y \<^bold>\<in> z)"
some_eq_imp: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "Eps P = a"   and "P b" shows "P a"
distinct_11: fixes x2 :: "'a \<times> nat"   and x4 :: "'a \<times> nat" shows "Formal_in x2 \<noteq> Actual_in x4"
bounds_compare_Some_1: fixes x :: "'a"   and c :: "'a" shows "x \<le>\<^sub>u\<^sub>b Some c = (x \<le> c)"
nat_eventually_critical_path: fixes P :: "nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "P i"   and "\<not> P j"   and "i < j" shows "\<exists>k>i. k \<le> j \<and> \<not> P k \<and> (\<forall>m\<ge>i. m < k \<longrightarrow> P m)"
list_all2_eq: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs = ys) = list_all2 (=) xs ys"
Shift: fixes G :: "('a, 'b) form list" assumes "\<stileturn> rotate1 G" shows "\<stileturn> G"
suff_LNil: fixes A :: "'b set" shows "suff A LNil = A\<^sup>\<infinity>"
path_append_conv: fixes E :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and q :: "'b uprod list"   and v :: "'b" shows "epath E u (p @ q) v = (\<exists>w. epath E u p w \<and> epath E w q v)"
lm008: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "P +< Q = Outside' (Domain Q) P +< Q"
induced_function_nn_integral: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "A \<in> sets M"   and "f \<in> borel_measurable M" shows "integral\<^sup>N M (induced_function A f) = set_nn_integral M (\<Union>n. (T ^^ n) --` A) f"
hd_block: fixes i :: "nat"   and t :: "nat" shows "e_hd (block i t) = i"
comp_cong: fixes f :: "'n \<Rightarrow> 'm"   and g :: "'o \<Rightarrow> 'n"   and x :: "'o"   and f' :: "'p \<Rightarrow> 'm"   and g' :: "'q \<Rightarrow> 'p"   and x' :: "'q" assumes "f (g x) = f' (g' x')" shows "(f \<circ> g) x = (f' \<circ> g') x'"
fresh_index_aforms_def: fixes xs :: "('a \<times> 'a pdevs) list" shows "fresh_index_aforms xs = Max (insert 0 (degree_aform ` set xs))"