hcju/isabelleps huggingface dataset:
eval_def: fixes pred :: "'a Predicate.pred" shows "pred.eval pred = (case pred of pred.Pred x \<Rightarrow> x)"
f1: shows "3 \<le> rk {A, B, C, P}"
sifum_modes_axioms: shows "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"
perm_append_swap: fixes xs :: "'c list"   and ys :: "'c list" shows "xs @ ys <~~> ys @ xs"
bool_aux: fixes a :: "bool"   and b :: "bool" shows "(a \<and> (a \<longrightarrow> b)) = (a \<and> b)"
paths_disjoint: fixes xs :: "'a list"   and ys :: "'a list"   and v :: "'a" assumes "xs \<in> Q"   and "ys \<in> Q"   and "xs \<noteq> ys"   and "v \<in> set xs"   and "v \<in> set ys" shows "v = v0 \<or> v = v1"
e: shows "NTCod \<eta> ## f maps\<^bsub>CatCod (NTCod \<eta>)\<^esub> NTCod \<eta> @@ X to NTCod \<eta> @@ Y"
Inf_empty: shows "\<Sqinter>\<^sub>o\<^sub>w {} = \<top>\<^sub>o\<^sub>w"
inject_3: fixes x2 :: "tyvrs"   and x3 :: "trm"   and x1 :: "ty"   and y2 :: "tyvrs"   and y3 :: "trm"   and y1 :: "ty" shows "((\<lambda>x2<:x1. x3) = (\<lambda>y2<:y1. y3)) = ([x2].x3 = [y2].y3 \<and> x1 = y1)"
assoc: fixes a :: "'t"   and b :: "'t"   and c :: "'t" shows "ord_term_lin.max (ord_term_lin.max a b) c = ord_term_lin.max a (ord_term_lin.max b c)"
simps_1: fixes A :: "'b set" shows "nth_least A 0 = least A"
set_eq_fold: fixes x :: "'a"   and xs :: "'a list" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n set (x # xs) = fold (\<sqinter>) xs x"
oconnect_completeTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and i' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), connect(i, i'), \<sigma>', s') \<in> ocnet_sos S" obtains "((\<sigma>, s), connect(i, i'), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
power_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq a b"   and "less_eq zero a" shows "less_eq (power.power one times a n) (power.power one times b n)"
counit_simps_5: fixes a :: "'a" assumes "C.obj a" shows "D.dom (counit a) = \<tau>\<^sub>0 a \<star>\<^sub>D map\<^sub>0 a"
Target_SAs_Delta_States: fixes SAs :: "'g \<Rightarrow> ('d, 'e, 'f) seqauto set"   and HA :: "'g" shows "Target (\<Union> (Delta ` SAs HA)) \<subseteq> \<Union> (States ` SAs HA)"
null_comp_not_Null: fixes h :: "ref \<Rightarrow> ref"   and q :: "ref"   and g :: "ref \<Rightarrow> ref" assumes "h q \<noteq> Null" shows "(g \<propto> h) q = g (h q)"
\<theta>_def: shows "\<theta> = arccos ((Re z0 - Re z) / r)"
assms_1: fixes x :: "'d"   and n :: "nat" assumes "x \<in> A" shows "norm (f n x) \<le> M n"
ExecSeqOK: fixes s\<^sub>1 :: "stmt"   and \<sigma> :: "char list \<Rightarrow> nat option"   and \<sigma>' :: "char list \<Rightarrow> nat option"   and s\<^sub>2 :: "stmt"   and \<rho> :: "(char list \<Rightarrow> nat option) option" assumes "s\<^sub>1 \<rhd> \<sigma> \<leadsto> Some \<sigma>'"   and "s\<^sub>2 \<rhd> \<sigma>' \<leadsto> \<rho>" shows "Seq s\<^sub>1 s\<^sub>2 \<rhd> \<sigma> \<leadsto> \<rho>"
source_val: assumes "\<exists>v\<in>verts G. enum v \<noteq> \<infinity>" shows "dist s = 0"
Vset_vsubset_mono: fixes i :: "V"   and j :: "V" assumes "i \<subseteq>\<^sub>\<circ> j" shows "Vset i \<subseteq>\<^sub>\<circ> Vset j"
A1n_pos: fixes i :: "'n"   and j :: "'n" shows "0 < A1n $h i $h j"
hcomp_simps_4: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "hseq \<nu> \<mu>" shows "cod (\<nu> \<star> \<mu>) = cod \<nu> \<star> cod \<mu>"
finite_cspan_closed: fixes S :: "'b set" assumes "finite S" shows "closed (cspan S)"
simps_4: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and t :: "('b, 'pred, 'act, 'effect) Tree" shows "FL_valid_Tree P (tAct f \<alpha> t) = (\<exists>\<alpha>' t' P'. tAct f \<alpha> t =\<^sub>\<alpha> tAct f \<alpha>' t' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> FL_valid_Tree P' t')"
c_def: shows "c = \<Prod>\<^sub># (image_mset fract_content (field_poly.prime_factorization (map_poly to_fract p)))"
abs_eq_0: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(abs a = zero) = (a = zero)"
a2_map'_keys': shows "Mapping.keys a2_map' = {tp + 1 - (len + 1)..tp + 1}"
prv_eqv_imp_trans: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (Syntax_with_Connectives.eqv cnj imp \<phi> \<chi>) (imp (Syntax_with_Connectives.eqv cnj imp \<chi> \<psi>) (Syntax_with_Connectives.eqv cnj imp \<phi> \<psi>)))"
pr_sign_key_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and X :: "agent" assumes "(evs, S, A, U) \<in> protocol" shows "Key (priSK X) \<notin> analz (A \<union> spies evs)"
complex_cnj_diff: fixes x :: "complex"   and y :: "complex" shows "cnj (x - y) = cnj x - cnj y"
m'_inverse: fixes x :: "'a"   and l :: "nat" assumes "x \<in> M (V l)" shows "m (m' x) = x"
vwalkI_append_l: fixes p :: "'a list"   and q :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "p \<noteq> []"   and "vwalk (p @ q) G" shows "vwalk p G"
BIND_I: fixes P :: "bool"   and name :: "char list"   and inp :: "nat" assumes "P" shows "Case_Labeling.BIND name inp P"
cases: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" obtains s :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" where "f \<in> borel_measurable M" and "\<And>i. simple_bochner_integrable M (s i)" and "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0" and "(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x"
kat_1_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y \<cdot> !z = y \<cdot> !z) = \<lparr>y\<rparr>!z\<lparr>!x \<cdot> y\<rparr>"
bind_const': fixes M :: "'b measure"   and N :: "'c measure" assumes "prob_space M"   and "subprob_space N" shows "M \<bind> (\<lambda>x. N) = N"
that: fixes A1 :: "'a multiset"   and A2 :: "'a multiset"   and B1 :: "'a multiset"   and B2 :: "'a multiset" assumes "A = A1 + A2"   and "B = B1 + B2"   and "(A1, B1) \<in> multpw ns"   and "\<And>b. b \<in># B2 \<Longrightarrow> \<exists>a. a \<in># A2 \<and> (a, b) \<in> s" shows "thesis"
synth_analz_synth: fixes H :: "msg set" shows "synth (analz (synth H)) = synth (analz H)"
simps_2: fixes ys :: "[:'a:]" assumes "ys \<noteq> \<bottom>" shows "prefix\<cdot>[::]\<cdot>ys = TT"
admissible_leI: fixes luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "mcont luba orda Sup (\<le>) f"   and "mcont luba orda Sup (\<le>) g" shows "ccpo.admissible luba orda (\<lambda>x. f x \<le> g x)"
DSourcesA5_L1: shows "DSources level1 sA5 = {sA42}"
L4_2__aux: fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" assumes "ipath gba.E \<sigma>"   and "\<mu> U\<^sub>r \<eta> \<in> old (\<sigma> 0)"   and "\<forall>j. (\<forall>i<j. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i)) \<longrightarrow> \<eta> \<notin> old (\<sigma> j)" shows "\<forall>i. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i) \<and> \<eta> \<notin> old (\<sigma> i)"
prj_simps_4: shows "Maps.dom SPN_fgh.Prj\<^sub>1\<^sub>1 = Chn ((SPN f \<circ> SPN g) \<circ> SPN h)"
not_numeral_le_zero: 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 "\<not> less_eq (numeral.numeral one plus n) zero"
left_eq: shows "{p. \<exists>y. y \<in> cbox (g2 a) (g1 a) \<and> p = (a, y)} = (\<lambda>y. twoC (0, y)) ` {0..1}"
tm_exhaust_apps_sel: fixes s :: "('a, 'b) tm" obtains "s = apps (Hd (head s)) (args s)"
d_def: shows "d = (\<lambda>v. if v \<in> Vs then c v else (0::'a))"
take_bit_word_beyond_length_eq: fixes n :: "nat"   and w :: "'a word" assumes "LENGTH('a) \<le> n" shows "take_bit n w = w"
SUG: shows "S \<union> \<Union> G \<subseteq> U - K"
Lt_def: fixes a :: "'a aexp"   and b :: "'a aexp" shows "Lt a b \<equiv> Gt b a"
sel_1: fixes x1 :: "name" shows "un_Sconst (Sconst x1) = x1"
content_ball_gt_0_iff: fixes z :: "'c"   and r :: "real" shows "(0 < content (ball z r)) = (0 < r)"
step: shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>"
fin: shows "finite {i. i < length xs \<and> p (xs ! i)}"
power_minus1_odd: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus one) (Suc (2 * n)) = uminus one"
vec_upd_eq: fixes s :: "('c, 'd) vec"   and i :: "'d"   and a :: "'c" shows "HS_VC_Spartan.vec_upd s i a = (\<chi>j. if j = i then a else s $ j)"
digraphI: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<CC>"   and "vsv (\<CC>\<lparr>Dom\<rparr>)"   and "vsv (\<CC>\<lparr>Cod\<rparr>)"   and "vcard \<CC> = 4\<^sub>\<nat>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<And>A B. \<lbrakk>A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>; B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>; A \<in>\<^sub>\<circ> Vset \<alpha>; B \<in>\<^sub>\<circ> Vset \<alpha>\<rbrakk> \<Longrightarrow> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>" shows "digraph \<alpha> \<CC>"
mat_1_fun: fixes a :: "'e"   and b :: "'e" shows "mat (1::'d) $ a $ b = (if a = b then 1::'d else (0::'d))"
G_valid_def: 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" shows "G_valid G = (\<forall>C m l A B I. G\<down>(C, m, l) = Some (A, B, I) \<longrightarrow>  \<Turnstile> \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I)"
Sup_upper: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "x \<le> \<Squnion>A"
sfunc_ind_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "sfunc_ind s i j = inst.funcs (inst s ! i) ! j"
restrictions_inwards: fixes x :: "'a"   and x' :: "'a"   and f :: "'a \<Rightarrow> 'b option"   and y :: "'b" assumes "x \<noteq> x'" shows "(f(x \<mapsto> y))(x' := None) = f(x' := None)(x \<mapsto> y)"
Valid_def: fixes A :: "'c \<Rightarrow> bool" shows "\<turnstile> A \<equiv> \<forall>w. A w"
simps: fixes x :: "'a" shows "sqrt_approx_main x = (if (0::'a) < x then if x * x - n < \<epsilon> then x else sqrt_approx_main ((n / x + x) / (2::'a)) else (0::'a))"
r_inv_ex: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<exists>y\<in>carrier Zp. x \<oplus> y = \<zero>"
thenOrdering_simps_1: fixes y :: "Ordering" shows "thenOrdering\<cdot>LT\<cdot>y = LT"
cross_components_2: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "cross3 x y $ 2 = x $ 3 * y $ 1 - y $ 3 * x $ 1"
gener_bool_list_elem_length: fixes x :: "bool list"   and n :: "nat" assumes "x \<in> gener_bool_list n" shows "length x = n"
polymap_const_inj: fixes p :: "'f poly" assumes "degree p = 0"   and "p \<noteq> 0" shows "inj_on (polymap p) V"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "(a::'b) \<le> sup a (b::'b)"
elementary_divisor_ring_def: shows "class.elementary_divisor_ring TYPE('b) \<equiv> \<forall>A. admits_diagonal_reduction A"
opt: shows "real (T_splay x t) + \<Phi> (splay x t) - \<Phi> t \<le> 3 * \<phi> t + 1"
I_prime: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<otimes> b \<in> I" shows "a \<in> I \<or> b \<in> I"
span_eq_real1: shows "module_on.span (*\<^sub>R) = span"
h4: assumes "Av < 0"   and "Bv \<le> 0" shows "(Av + Bv * sqrt Cv < 0) = (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
nth_append: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < |xs| then xs ! n else ys ! (n - |xs|))"
bisimResComm': fixes x :: "name"   and \<Psi> :: "'b"   and xvec :: "name list"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>"   and "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>P) \<sim> \<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>P"
last_conv_nth: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - 1)"
execute_bind_case: fixes f :: "'c Heap"   and g :: "'c \<Rightarrow> 'b Heap"   and h :: "heap" shows "execute (f \<bind> g) h = (case execute f h of None \<Rightarrow> None | Some (x, h') \<Rightarrow> execute (g x) h')"
simps_2: fixes f :: "'a"   and T :: "('a, 'b) Term.term list" shows "subterms (Fun f T) = {Fun f T} \<union> subterms\<^sub>s\<^sub>e\<^sub>t (set T)"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : t\<^sub>0 \<star> \<pi>.chine \<Rightarrow> s\<^sub>0\<guillemotright>"
a3: shows "\<forall>nP e1 e2. \<not> P (nP e1) (nP e2) \<longrightarrow> \<not> P (nP e1) \<bottom>"
dlist_all_conv_member: fixes P :: "'b \<Rightarrow> bool"   and dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "DList_Set.dlist_all P dxs = (\<forall>x. DList_Set.member dxs x \<longrightarrow> P x)"
one_plus_numeral: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.numeral plus" shows "plus one (numeral.numeral one plus n) = numeral.numeral one plus (num.One + n)"
host_shadow_root_rel_finite: fixes h :: "(_) heap" shows "finite (local.a_host_shadow_root_rel h)"
abc_comp_second_step_eq: fixes s :: "nat"   and lm :: "nat list"   and B :: "abc_inst list"   and sa :: "nat"   and lma :: "nat list"   and A :: "abc_inst list" assumes "abc_step_l (s, lm) (abc_fetch s B) = (sa, lma)" shows "abc_step_l (s + length A, lm) (abc_fetch (s + length A) (A [+] B)) = (sa + length A, lma)"
distinct_13: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list" shows "S \<noteq> Cn x41 x42 x43"
prime_prime_factors: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize p) = {p}"
set_bits_aux_conv: fixes w :: "'b word"   and n :: "nat" shows "local.set_bits_aux w n = (w << n) OR set_bits f AND mask n"
arr_char: fixes f :: "'a \<times> 'a" shows "VV.arr f = (arr (fst f) \<and> arr (snd f) \<and> I.map (fst f) = I.map (snd f))"
R_mult_left_mono: fixes b :: "real"   and a :: "real"   and c :: "real" assumes "b \<le> a"   and "0 \<le> c" shows "c * b \<le> c * a"
subset_conjugation_is_bij: fixes g :: "'a" assumes "g \<in> carrier G" shows "bij_betw (\<lambda>H\<in>{H. H \<subseteq> carrier G}. g <# H #> inv g) {H. H \<subseteq> carrier G} {H. H \<subseteq> carrier G}"
elims_2: fixes x :: "'v"   and xa :: "('v \<times> 'v) list"   and xb :: "'v" assumes "is_path x xa xb" obtains v :: "'v" and v' :: "'v" where "x = va__" and "xa = []" and "xb = v'a__" and "va__ = v'a__ \<and> v'a__ \<in> V"   | v :: "'v" and v1 :: "'v" and v2 :: "'v" and p :: "('v \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, v2) # p" and "xb = v'" and "v = v1 \<and> (v1, v2) \<in> E \<and> is_path v2 p v'"
strict_mono_on_imp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "strict_mono_on f A" shows "inj_on f A"
auxlist_split: shows "auxlist = take (length done) auxlist @ drop (length done) auxlist"
beyond_degree_lookup_zero: fixes f :: "nat \<Rightarrow>\<^sub>0 'b"   and n :: "nat" assumes "degree f \<le> n" shows "lookup f n = (0::'b)"
euler_witness_exists_nat: fixes n :: "nat" assumes "odd n"   and "\<not> prime n"   and "2 < n" shows "\<exists>a. euler_witness (int a) n \<and> coprime a n \<and> 0 < a \<and> a < n"
path_strategies_good: fixes P :: "'a llist"   and n :: "nat" assumes "lset P \<subseteq> winning_region p"   and "enat n < llength P" shows "path_strategies P $ n \<in> good (P $ n)"
OAlist_tc_lex_ord_valI: fixes aux :: "order option"   and k :: "'b"   and xs :: "('b, 'c) oalist_tc"   and ys :: "('b, 'd) oalist_tc"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> order option" assumes "aux \<noteq> Some Eq"   and "k \<in> fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys)"   and "aux = f k (OAlist_tc_lookup xs k) (OAlist_tc_lookup ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys); k' < k\<rbrakk> \<Longrightarrow> f k' (OAlist_tc_lookup xs k') (OAlist_tc_lookup ys k') = Some Eq" shows "OAlist_tc_lex_ord f xs ys = aux"
prems_4: shows "wf_inst \<Theta> \<rho>"
WHILEIT_unfold: fixes I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and x :: "'b" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f x = (if I x then if b x then f x \<bind> WHILE\<^sub>T\<^bsup>I\<^esup> b f else RETURN x else FAIL)"
o_apply: fixes f :: "'ba \<Rightarrow> 'az"   and g :: "'bb \<Rightarrow> 'ba"   and x :: "'bb" shows "(f \<circ> g) x = f (g x)"
IH: fixes fm :: "('a, 'b) fmap" assumes "\<forall>fm'. ListMem fm' fmList \<longrightarrow> agree fm fm'" shows "agree fm (foldr (++) fmList fmempty)"
the_pure_pure: fixes R :: "('a \<times> 'b) set" shows "the_pure (pure R) = R"
closedin_continuous_map_preimage_gen: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'d set" assumes "continuous_map X Y f"   and "closedin X U"   and "closedin Y V" shows "closedin X {x \<in> U. f x \<in> V}"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "CCC.prod f b = null"
Le_enat_eq_ile: fixes m :: "enat"   and n :: "enat" shows "((m, n) \<in> Le_enat) = (m \<le> n)"
mset_eq_permutation: fixes xs :: "'a list"   and ys :: "'a list" assumes "mset xs = mset ys" obtains p :: "nat \<Rightarrow> nat" where "p permutes {..<length ys}" and "permute_list p ys = xs"
homeomorphic_compact_space: fixes X :: "'c topology"   and Y :: "'d topology" assumes "X homeomorphic_space Y" shows "compact_space X = compact_space Y"
ksup_assoc: fixes f :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b set"   and h :: "'a \<Rightarrow> 'b set" shows "f \<squnion> g \<squnion> h = f \<squnion> (g \<squnion> h)"
natural_transformation_axioms: shows "natural_transformation B C F H \<nu>\<tau>.map"
simps: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b" shows "pairself f (a, b) = (f a, f b)"
mset_sorted_list_of_multiset: fixes M :: "'k multiset" shows "mset (sorted_list_of_multiset M) = M"
lta_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LtA Bet Cong C B A D E F"
lemma_NBseq_def2: fixes X :: "'a \<Rightarrow> 'b" shows "(\<exists>K>0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real (Suc N))"
wf_same_length_if_total: fixes n :: "nat" shows "wfP (\<lambda>xs ys. length ys = n \<and> length xs = n \<and> ext gt ys xs)"
concurrent_set_Cons_Snoc: fixes a :: "'a"   and xs :: "'a list"   and x :: "'a" shows "concurrent_set a (xs @ [x]) = concurrent_set a (x # xs)"
word_size_bits_def: shows "word_size_bits = (3::'a)"
w_cmpser_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "w_cmpser G n g = (if n = 0 then d_gchain G n g else d_gchain G n g \<and> (\<forall>l\<le>n - 1. \<natural>\<^bsub>G\<^esub>g l \<triangleright> g (Suc l)))"
p: shows "p = mulexp n [- a, 1::'a] q"
exE: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>x. P x" obtains x :: "'b" where "P x"
convergent_NSconvergent_iff: fixes X :: "nat \<Rightarrow> 'b" shows "convergent X = NSconvergent X"
ipresSwapAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'g, 'b, 'h) model_scheme" shows "ipresSwapAbs hA MOD \<equiv> \<forall>zs z1 z2 us s A. wlsAbs (us, s) A \<longrightarrow> hA (A $[z1 \<and> z2]_zs) = igSwapAbs MOD zs z1 z2 (hA A)"
image_mset_single: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "image_mset f {#x#} = {#f x#}"
18: shows "infs (gaccepting A) (fromN 1 ||| r)"
yx: shows "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b \<le> pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"