hcju/isabelleps huggingface dataset:
ub_arctan: fixes y :: "real"   and x :: "float"   and prec :: "nat" assumes "y \<le> arctan (real_of_float x)" shows "y \<le> real_of_float (ub_arctan prec x)"
abs_abs: fixes abs :: "'d \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (abs a) = abs a"
is_sup_def: fixes x :: "nat"   and y :: "nat"   and s :: "nat" shows "nat_dvd.is_sup x y s = (x dvd s \<and> y dvd s \<and> (\<forall>z. x dvd z \<and> y dvd z \<longrightarrow> s dvd z))"
n_ord_def_alt: fixes x :: "'a"   and y :: "'a" shows "(!x \<le> !y) = (!x \<cdot> !y = !x)"
eigenval_def: fixes k :: "complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "eigenval k H = (\<exists>v. v \<noteq> vec_zero \<and> H *\<^sub>m\<^sub>v v = k *\<^sub>s\<^sub>v v)"
gauss_sum_def: fixes k :: "nat" shows "gauss_sum k = (\<Sum>m = 1..n. \<chi> m * unity_root n (int (m * k)))"
subset_diff: fixes B :: "'b set"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "B \<subseteq> A"   and "finite A" shows "sum g A = sum g (A - B) + sum g B"
nn_integral_measure_spmf': fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf p) f = \<integral>\<^sup>+ x. ennreal (spmf p x) * f x \<partial>count_space (set_spmf p)"
alt_group_carrier: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" shows "(p \<in> carrier (alt_group n)) = (p permutes {1..n} \<and> evenperm p)"
below_sctn_coll_impl_def: fixes ai :: "'b list list"   and sctni :: "real list sctn" shows "below_sctn_coll_impl ai sctni \<equiv> RETURN ai \<bind> (\<lambda>xa. FORWEAK_LIST xa (RETURN True) (\<lambda>xb. RETURN (below_sctn_impl xb sctni)) (\<lambda>xb xc. RETURN (xb \<and> xc)))"
assoc_mult_mat_vec: fixes A :: "'a mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'a mat"   and n\<^sub>3 :: "nat"   and v :: "'a Matrix.vec" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "v \<in> carrier_vec n\<^sub>3" shows "A * B *\<^sub>v v = A *\<^sub>v (B *\<^sub>v v)"
AB_1: shows "(A, B) \<in> \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"
arctan_inverse: fixes x :: "real" assumes "0 < x" shows "arctan (inverse x) = pi / 2 - arctan x"
lprefix_nitpick_simps: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lprefix xs ys = (if lfinite xs then Coinductive_List.finite_lprefix xs ys else xs = ys)"
set_bd_3: fixes x :: "('e, 'f, 'd) F2" shows "|F2set3 x| \<le>o bd_F2"
measure_spmf_bind: fixes p :: "'i spmf"   and f :: "'i \<Rightarrow> 'h spmf" shows "measure_spmf (p \<bind> f) = measure_spmf p \<bind> measure_spmf \<circ> f"
WT_gpv_inline: fixes \<I> :: "('e, 'f) \<I>"   and \<I>' :: "('g, 'h) \<I>"   and callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd, 'g, 'h) gpv"   and gpv :: "('i, 'e, 'f) gpv"   and s :: "'d" assumes "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"   and "\<And>x s. x \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<I>' \<turnstile>g inline callee gpv s \<surd>"
in_homE: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.in_hom f a b" obtains "C.VV.arr f" and "C.VV.dom f = a" and "C.VV.cod f = b"
lm101: fixes x :: "'b"   and X :: "'b set"   and Y :: "'b set" assumes "x \<in> X \<union> Y" shows "X <|| Y ,, x = chi X x"
Un_Int_eq_2: fixes S :: "'c set"   and T :: "'c set" shows "(S \<union> T) \<inter> T = T"
CD_on_idem: fixes ds :: "'d set"   and B :: "'x set" shows "CD_on ds (CD_on ds B) = CD_on ds B"
unit_div_eq_0_iff: 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 b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "(divide a b = zero) = (a = zero)"
fair: shows "\<forall>(t, t')\<in>SIGMA x:UNIV. set_pmf (K x). fair t t' \<omega>"
comp_cong_right: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "x = y" shows "f \<circ> x = f \<circ> y"
ipd_in_pdt: fixes x :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return" shows "(x, ipd x) \<in> pdt"
m_lcomm: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> (y \<oplus>\<^bsub>F\<^esub> z) = y \<oplus>\<^bsub>F\<^esub> (x \<oplus>\<^bsub>F\<^esub> z)"
tensor_mat_add1: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2" shows "tensor_mat (m1 + m2) m3 = tensor_mat m1 m3 + tensor_mat m2 m3"
naturality: fixes f :: "'a" assumes "arr f" shows "\<rho>' (cod f) \<cdot> local.map f = (f \<otimes> \<I>) \<cdot> \<rho>' (local.dom f)"
af_prefix_false: fixes i :: "nat"   and j :: "nat"   and \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "i \<le> j"   and "af \<phi> (prefix i w) \<sim>\<^sub>C false\<^sub>n" shows "af \<phi> (prefix j w) \<sim>\<^sub>C false\<^sub>n"
ttree_restr_possible: fixes x :: "'b"   and S :: "'b set"   and t :: "'b ttree" assumes "x \<in> S"   and "possible t x" shows "possible (ttree_restr S t) x"
a'_def: shows "a' \<equiv> \<lambda>f. \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"
lns_alt_def1: fixes B :: "'b set"   and P :: "('b \<times> 'b) set" shows "local_nonsatiation B P = (\<forall>x\<in>B. \<forall>e>0. \<exists>y\<in>B. dist y x \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P)"
that: fixes c :: "('a, 'b) chart" assumes "c \<in> atlas"   and "x \<in> domain c"   and "apply_chart c x = x0"   and "domain c \<subseteq> U"   and "codomain c = ball x0 e" shows "thesis"
stream_plus: fixes t :: "'a tree"   and t' :: "'a tree" shows "stream (t + t') = stream t + stream t'"
simps_5: fixes dt :: "loc \<Rightarrow> ty option"   and a :: "loc" shows "typeof dt (Addr a) = dt a"
order2_nreduced: fixes a :: "'a"   and A :: "'a set" assumes "a + a = (0::'a)" shows "\<not> reduced_word A [a, a]"
supp_of_multisets: fixes M :: "'a multiset" shows "supp M = \<Union> {supp x |x. x \<in># M}"
a_inv_def: fixes R :: "('e, 'f) ring_scheme" shows "a_inv R = m_inv (add_monoid R)"
extend_1_eval: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e'' :: "expr"   and s'' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e'',s''\<rangle>"   and "P,E \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
takeWhile_tail: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list"   and l :: "'d list" assumes "\<not> P x" shows "takeWhile P (xs @ x # l) = takeWhile P xs"
elims_3: fixes x :: "ereal"   and xa :: "ereal" assumes "\<not> x < xa" obtains xb :: "real" and y :: "real" where "x = ereal xb" and "xa = ereal y" and "\<not> xb < y"   | a :: "ereal" where "x = \<infinity>" and "xa = a__"   | a :: "ereal" where "x = a" and "xa = - \<infinity>"
sets_uniform_measure: fixes M :: "'a measure"   and A :: "'a set" shows "sets (uniform_measure M A) = sets M"
ile_into_Le_enat: fixes m :: "enat"   and n :: "enat" assumes "m \<le> n" shows "(m, n) \<in> Le_enat"
finite_dimensional_vector_space_with_axioms_def: fixes plus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero\<^sub>V\<^sub>S :: "'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_with_axioms plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis \<equiv> finite basis \<and> (with (0::'d) zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>independent\<guillemotright> basis) \<and> (with zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>span\<guillemotright> basis) = UNIV"
AE_T_enabled: fixes s :: "'a" shows "almost_everywhere (K1.T s) (K1.enabled s)"
r_distributive_def: fixes fmul :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "r_distributive fmul fadd \<equiv> \<forall>a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
OUTfromChCorrect_data9: shows "OUTfromChCorrect data9"
vector_scaleR_component: fixes r :: "real"   and x :: "('a, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "(r *\<^sub>R x) $h i = r *\<^sub>R x $h i"
qq3: shows "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> (as \<inter> supp x)"
eval_in_hom: fixes t :: "'c local.term" assumes "Arr t" shows "\<guillemotleft>\<lbrace>t\<rbrace> : \<lbrace>Dom t\<rbrace> \<rightarrow>\<^sub>D \<lbrace>Cod t\<rbrace>\<guillemotright>"
lexext_rev_compat_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "lexext_rev gt ys xs" shows "lexext_rev gt (ys @ [x]) (xs @ [x])"
step_rfunc_usr: fixes p :: "'s_rest spmf"   and q :: "'s_rest' spmf"   and iu :: "'iusr_rest" assumes "S p q"   and "iu \<in> IU" shows "p \<bind> (\<lambda>s1. map_spmf fst (rfunc_usr rest1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (rfunc_usr rest2 s2 iu))"
finter_finsert_left_if1: fixes a :: "'b"   and C :: "'b fset"   and B :: "'b fset" assumes "a |\<in>| C" shows "finsert a B |\<inter>| C = finsert a (B |\<inter>| C)"
ccBinds_def: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallAnalysis.ccBinds ccExp \<Gamma> = (\<Lambda> i. \<Squnion>v\<mapsto>e\<in>map_of \<Gamma>. CoCallAnalysis.ccBind ccExp v e\<cdot>i)"
get_attribute_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and k :: "char list" assumes "h \<turnstile> ok get_attribute element_ptr k" shows "element_ptr |\<in>| element_ptr_kinds h"
LIM_unique: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and a :: "'a"   and M :: "'b" assumes "f \<midarrow>a\<rightarrow> L"   and "f \<midarrow>a\<rightarrow> M" shows "L = M"
emp_def: shows "\<box> \<equiv> \<lambda>s. s = (0::'a)"
BV_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and 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 \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"   and "P,\<Phi> |- \<sigma> [ok]" shows "P,\<Phi> |- \<sigma>' [ok]"
get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "get ptr h = fmlookup (the_heap h) ptr"
AudioPlayer_CTRL_Labels_def: shows "AudioPlayer_CTRL_Labels = {(En ''O'', defaultguard, defaultaction)}"
Inf_from_def: fixes N :: "'a set" shows "Inf_from N = {\<iota> \<in> Inf. set (prems_of \<iota>) \<subseteq> N}"
ground_heads_var_arity: fixes f :: "'s"   and x :: "'v" assumes "f \<in> ground_heads_var x" shows "arity_var x \<le> arity_sym f"
l1_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and gny :: "msg" shows "l1_step3 Ra A B gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xgnx} \<and> dhlvl1.guessed_frame Ra xgny = Some gny \<and> dhlvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> (can_signal s A B \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress s Rb) {xny, xgnx, xgny, xsk} \<and> dhlvl1.guessed_frame Rb xgny = Some gny \<and> dhlvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))))) \<and> (Ra = test \<longrightarrow> Exp gny (NonceF (Ra $ nx)) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp s\<rparr>}"
space_EXP: shows "space EXP = COCOUNT \<rightarrow>\<^sub>M BOOL"
cone_one: fixes U :: "'x set" shows "cone (1, U) = P[U]"
distinct_1: fixes x1 :: "'a"   and x2 :: "'a" shows "Normal x1 \<noteq> Abrupt x2"
entailsI: fixes P :: "assn"   and Q :: "assn" assumes "\<And>h. h \<Turnstile> P \<Longrightarrow> h \<Turnstile> Q" shows "P \<Longrightarrow>\<^sub>A Q"
adv_not_know1: fixes P :: "specID"   and A :: "specID"   and m :: "Keys" assumes "out P \<subseteq> ins A"   and "\<not> know A (kKS m)" shows "\<not> eout P (kE m)"
Conseq: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
finprod_insert: fixes F :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> UNIV"   and "f a \<in> UNIV" shows "prod f (insert a F) = f a * prod f F"
poly_prod_0: fixes ps :: "'c set"   and f :: "'c \<Rightarrow> 'd poly"   and x :: "'d" assumes "finite ps" shows "(poly (prod f ps) x = (0::'d)) = (\<exists>p\<in>ps. poly (f p) x = (0::'d))"
NSCauchy_NSconvergent: fixes X :: "nat \<Rightarrow> 'a" assumes "NSCauchy X" shows "NSconvergent X"
idem: fixes a :: "'c" shows "a \<squnion> a = a"
sel_10: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "getId (RecvMarker x51 x52 x53) = x51"
assms_5: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and J :: "'a set"   and c :: "'a"   and d :: "'a"   and A :: "'a set" assumes "Q I a b"   and "Q J c d"   and "I \<subseteq> A"   and "J \<subseteq> A"   and "A \<in> \<P>"   and "[[b a c]] \<and> a = d" shows "P I J"
uminus_nonpos_Reals_iff: fixes x :: "'a" shows "(- x \<in> \<real>\<^sub>\<le>\<^sub>0) = (x \<in> \<real>\<^sub>\<ge>\<^sub>0)"
assms_8: shows "analytically_valid Dx_pair (\<lambda>p. F p \<bullet> j) i"
mp: shows "\<forall>mm\<in>poly_monoms p. mm \<noteq> m"
inext_cut_greater_conv: fixes t :: "nat"   and n :: "nat"   and I :: "nat set" assumes "t < n" shows "inext n (I \<down>> t) = inext n I"
set_takeWhileD: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "x \<in> set (takeWhile P xs)" shows "x \<in> set xs \<and> P x"
integral_product_swap: fixes f :: "'b \<times> 'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable (M2 \<Otimes>\<^sub>M M1)" shows "LINT (x, y)|(M1 \<Otimes>\<^sub>M M2). f (y, x) = integral\<^sup>L (M2 \<Otimes>\<^sub>M M1) f"
\<Z>_subset: fixes u :: "nat"   and v :: "nat"   and k :: "nat" assumes "u \<le> v" shows "\<Z> k u \<subseteq> \<Z> k v"
increasing_measurable_info: fixes f :: "bool stream \<Rightarrow> 'b"   and s :: "nat"   and N :: "'b measure"   and t :: "nat" assumes "f \<in> G s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> G t \<rightarrow>\<^sub>M N"
bv_smult_length: fixes w1 :: "bit list"   and w2 :: "bit list" shows "length (bv_smult w1 w2) \<le> length w1 + length w2"
step_preserves_le: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and S :: "(char list \<Rightarrow> int) set"   and S' :: "(char list \<Rightarrow> 'b) option"   and c :: "(char list \<Rightarrow> int) set acom"   and c' :: "(char list \<Rightarrow> 'b) option acom" assumes "Abs_Int_Fun num' plus' \<gamma>"   and "S \<subseteq> \<gamma>_option (\<gamma>_fun \<gamma>) S'"   and "c \<le> ACom.map_acom (\<gamma>_option (\<gamma>_fun \<gamma>)) c'" shows "step S c \<le> ACom.map_acom (\<gamma>_option (\<gamma>_fun \<gamma>)) (Abs_Int_Fun.step' num' plus' S' c')"
ex_complex_to_complex2: fixes z :: "complex" shows "\<exists>k. k \<noteq> 0 \<and> k * z = z"
functions_of: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P \<le> card B" shows "functions_of P A B \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B"
prob_space_K: fixes x :: "real \<times> 'a" assumes "x \<in> space S" shows "prob_space (K x)"
rel_stateT_def: fixes S :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "('d, 'f) stateT"   and m' :: "('e, 'g) stateT" shows "rel_stateT S M m m' = (S ===> M) (run_state m) (run_state m')"
rel_Grp: fixes A1 :: "'a set"   and f1 :: "'a \<Rightarrow> 'c"   and A2 :: "'b set"   and f2 :: "'b \<Rightarrow> 'd" shows "rel_blindable\<^sub>m (Grp A1 f1) (Grp A2 f2) = Grp {x. set1_blindable\<^sub>m x \<subseteq> A1 \<and> set2_blindable\<^sub>m x \<subseteq> A2} (map_blindable\<^sub>m f1 f2)"
le_cases: fixes x :: "'g"   and y :: "'g" obtains "x \<le> y"   | "y \<le> x"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_quasi_ordered_set A less_eq" shows "transitive A less_eq"
gen_cong_eval: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "rel (gen_cong R) x y" shows "gen_cong R (eval x) (eval y)"
set_disjoint_witness_axioms: shows "set_disjoint_witness \<alpha>1 invar1 \<alpha>2 invar2 disjoint_witness"
1: shows "search_enum (the (eval (Mn 1 r_search_enum) [b])) b \<down>= 0"
access_matrix_sound: fixes V :: "('a word \<times> 'a wordinterval) list"   and E :: "('a word \<times> 'a word) list"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and s_repr :: "'a word"   and d_repr :: "'a word"   and s_range :: "'a wordinterval"   and s :: "'a word"   and d_range :: "'a wordinterval"   and d :: "'a word" assumes "(V, E) = access_matrix c rs"   and "(s_repr, d_repr) \<in> set E"   and "map_of V s_repr = Some s_range"   and "s \<in> wordinterval_to_set s_range"   and "map_of V d_repr = Some d_range"   and "d \<in> wordinterval_to_set d_range" shows "runFw s d c rs = Decision FinalAllow"
accepts_conv_steps: fixes A :: "'b \<times> ('c \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list" shows "accepts A w = (\<exists>q. (start A, q) \<in> steps A w \<and> fin A q)"
ZFDepthInNat: fixes e :: "(ZF, ZF) Expression" shows "Elem (ZFTFst (Expr2ZF e)) HOLZF.Nat"
closed_others_def: fixes \<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" shows "closed_others \<R> \<equiv> \<forall>c\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C mem\<^sub>C'. (\<langle>c\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>x. mem\<^sub>C x \<noteq> mem\<^sub>C' x \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x) \<longrightarrow> (\<forall>x. dma\<^sub>C mem\<^sub>C x \<noteq> dma\<^sub>C mem\<^sub>C' x \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x) \<longrightarrow> (\<langle>c\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C'\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>"
scale_right_imp_eq: fixes x :: "'b"   and a :: "complex"   and b :: "complex" assumes "x \<noteq> (0::'b)"   and "a *\<^sub>C x = b *\<^sub>C x" shows "a = b"
option_while_rule': fixes C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option"   and s :: "'d"   and ss' :: "'d option"   and M :: "('d \<times> 'd) set"   and I :: "'d option \<Rightarrow> bool" assumes "option_while C B s = ss'"   and "wf M"   and "I (Some s)"   and "\<And>s s'. \<lbrakk>C s; I (Some s); B s = Some s'\<rbrakk> \<Longrightarrow> (s', s) \<in> M"   and "\<And>s s'. \<lbrakk>C s; I (Some s); B s = Some s'\<rbrakk> \<Longrightarrow> I (Some s')"   and "\<And>s. \<lbrakk>C s; I (Some s); B s = None\<rbrakk> \<Longrightarrow> I None" shows "I ss' \<and> (case ss' of None \<Rightarrow> True | Some s' \<Rightarrow> \<not> C s')"
bools_of_nat_length: fixes k :: "nat"   and n :: "nat" shows "k \<le> length (bools_of_nat k n)"
tuple_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and h :: "'a"   and f :: "'a"   and g :: "'a"   and k :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1" shows "elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k \<equiv> if category.commutative_square C f g h k then THE l. C (prj0 f g) l = k \<and> C (prj1 f g) l = h else partial_magma.null C"
chine_assoc_def: shows "chine_assoc \<equiv> \<langle>Prj\<^sub>1\<^sub>1 \<lbrakk>\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1\<rbrakk> \<langle>Prj\<^sub>0\<^sub>1 \<lbrakk>\<nu>.leg0, \<pi>.leg1\<rbrakk> Prj\<^sub>0\<rangle>\<rangle>"
decomps\<^sub>e\<^sub>s\<^sub>t_ik_max_exist: fixes A :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "finite A"   and "finite N" shows "\<exists>D\<in>decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>. \<forall>D'\<in>decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>. ik\<^sub>e\<^sub>s\<^sub>t D' \<subseteq> ik\<^sub>e\<^sub>s\<^sub>t D"
StarLR_Times: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "StarLR (Times r s) t = Times r (StarLR s t)"
open_chart_basis_domain: fixes b :: "'b" shows "open (Collect (chart_basis_domainP b))"
weakStatImpClosed: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "eqvt Rel"   and "\<Psi> \<rhd> P \<lessapprox><Rel> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<lessapprox><Rel> p \<bullet> Q"