hcju/isabelleps huggingface dataset:
assms_2: fixes a :: "'a" assumes "a \<in> carrier R"   and "p \<noteq> []" shows "alg_mult p a \<le> count m a"
degree_mult_right_le: fixes q :: "'b poly"   and p :: "'b poly" assumes "q \<noteq> 0" shows "degree p \<le> degree (p * q)"
sumPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Rel \<subseteq> Rel'"   and "Id \<subseteq> Rel'"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<oplus> U, T) \<in> Rel'" shows "P \<oplus> R \<leadsto>\<^sup>^<Rel'> Q \<oplus> R"
elims_2: fixes x :: "'a \<times> nat" assumes "is_valid_state_id x" obtains i :: "'a" and j :: "nat" where "x = (i, j)" and "j \<le> length (events i)"
m_comm: fixes x :: "int"   and y :: "int" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "x * y = y * x"
cases: fixes x :: "'e list" obtains "x = []"   | y :: "'e" and ys :: "'e list" where "x = y # ys"
space_scale_measure: fixes r :: "ennreal"   and M :: "'b measure" shows "space (scale_measure r M) = space M"
antisym: fixes x :: "'a"   and y :: "'a" assumes "\<not> x \<sqsubset> y"   and "\<not> y \<sqsubset> x"   and "x \<in> A"   and "y \<in> A" shows "x = y"
in_foldr_UnE: fixes x :: "'a"   and xs :: "'a set list" assumes "x \<in> foldr (\<union>) xs {}" obtains A :: "'a set" where "A \<in> set xs" and "x \<in> A"
mult_cancel_le_iff: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "(\<alpha> * \<beta> \<le> \<alpha> * \<gamma>) = (\<beta> \<le> \<gamma> \<or> \<alpha> = 0)"
lexordp_conv_lexordp_eq: fixes xs :: "'a list"   and ys :: "'a list" shows "lexordp xs ys = (lexordp_eq xs ys \<and> \<not> lexordp_eq ys xs)"
igAbsCongSSTR_imp_igAbsCongS: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igAbsCongSSTR MOD" shows "FixSyn.igAbsCongS Delta MOD"
groupQuantifiers_eval: fixes F :: "atom fm"   and L :: "real list" shows "eval F L = eval (groupQuantifiers F) L"
orderI: fixes a :: "'c"   and b :: "'c" assumes "a = a \<sqinter> b" shows "a \<le> b"
tr_Delete_constr_iff: fixes M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and Di :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "ground M" shows "(set Di \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I> \<subseteq> {(t, s) \<cdot>\<^sub>p \<I>} \<and> (t, s) \<cdot>\<^sub>p \<I> \<notin> set D - set Di \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>) = \<lbrakk>M; map (\<lambda>d. \<langle>check: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t) Di @ map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>"
MP_Rel_Pair_def: fixes A :: "int poly \<times> int poly"   and B :: "'a mod_ring poly \<times> 'a mod_ring poly" shows "MP_Rel_Pair A B \<equiv> let (a, b) = A; (c, d) = B in MP_Rel a c \<and> MP_Rel b d"
box_Complex_eq: fixes a :: "real"   and c :: "real"   and b :: "real"   and d :: "real" shows "box (Complex a c) (Complex b d) = (\<lambda>(x, y). Complex x y) ` (box a b \<times> box c d)"
p_fun_compl: fixes x :: "'a"   and y :: "'a" assumes "is_p_fun x" shows "x ; - y \<le> - (x ; y)"
simps_9: fixes f1 :: "'c"   and f2 :: "num \<Rightarrow> 'c"   and f3 :: "num \<Rightarrow> 'c" shows "(case num.One of num.One \<Rightarrow> f1 | num.Bit0 x \<Rightarrow> f2 x | num.Bit1 x \<Rightarrow> f3 x) = f1"
fimage_fminus_fsubset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b fset"   and B :: "'b fset" shows "f |`| A |-| f |`| B |\<subseteq>| f |`| (A |-| B)"
conv_radius_inftyI'': fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>z. summable (\<lambda>n. f n * z ^ n)" shows "conv_radius f = \<infinity>"
mean_divisor_count_asymptotics: shows "(\<lambda>m. real (card {(n, d). n \<in> {0<..m} \<and> d dvd n}) / real (card {0<..m}) - (ln (real m) + 2 * euler_mascheroni - 1)) \<in> O(\<lambda>x. 1 / sqrt (real x))"
wbalanced_bal: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and ys :: "'a list" assumes "n \<le> length xs"   and "bal n xs = (t, ys)" shows "wbalanced t"
set_iterator_rev_linord_rule_P: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S0 :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_rev_linord less_eq it S0"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<And>x'. x' \<in> S0 - S \<Longrightarrow> less_eq x x'; \<And>x'. x' \<in> S \<Longrightarrow> less_eq x' x\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<And>x x'. \<lbrakk>x \<in> S; x' \<in> S0 - S\<rbrakk> \<Longrightarrow> less_eq x x'; \<not> c \<sigma>; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
ass1: shows "\<forall>p<n. Sorted_Less.sorted (rev (ss ! p)) \<and> (ss ! p \<noteq> rev [0..<n] \<longrightarrow> (\<forall>x\<in>set (ss ! p). x < p)) \<and> (ss ! p = rev [0..<n] \<longrightarrow> (\<exists>x\<in>set w. (x, p) \<in> g_E G \<and> x < p)) \<and> (p \<notin> set w \<longrightarrow> stable r step ss p)"
is_Sup_least: fixes A :: "'b set"   and sup :: "'b"   and z :: "'b" assumes "is_Sup A sup"   and "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z" shows "sup \<sqsubseteq> z"
mgenerator1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)"   and "j \<le> n" shows "(\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (mL K P I) (Kb\<^bsub>K n P\<^esub>) n) = (\<nu>\<^bsub>K P j\<^esub>) (Zl_mI K P I j)"
alphaAbs_qAbs_iff_alphaAbs_ex_distinct_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_distinct_qFresh xs x X xs' x' X'"
T_TApp: fixes X :: "tyvrs"   and \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>2 :: "ty"   and T\<^sub>12 :: "ty"   and T\<^sub>11 :: "ty" assumes "X \<sharp> (\<Gamma>, t\<^sub>1, T\<^sub>2)"   and "\<Gamma> \<turnstile> t\<^sub>1 : (\<forall>X<:T\<^sub>11. T\<^sub>12)"   and "\<Gamma>\<turnstile>T\<^sub>2<:T\<^sub>11" shows "\<Gamma> \<turnstile> t\<^sub>1 \<cdot>\<^sub>\<tau> T\<^sub>2 : T\<^sub>12[X \<mapsto> T\<^sub>2]\<^sub>\<tau>"
Cauchy_convergent: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'c" assumes "class.complete_space dist uniformity open"   and "uniform_space.Cauchy uniformity X" shows "topological_space.convergent open X"
power_le_imp_le_exp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "less_eq (power.power one times a m) (power.power one times a n)" shows "m \<le> n"
lexord_trans: fixes x :: "'b list"   and y :: "'b list"   and r :: "('b \<times> 'b) set"   and z :: "'b list" assumes "(x, y) \<in> lexord r"   and "(y, z) \<in> lexord r"   and "trans r" shows "(x, z) \<in> lexord r"
inv_baldL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l + 1 = bheight r"   and "invc2 l"   and "invc r" shows "invh (baldL l a r) \<and> bheight (baldL l a r) = bheight r \<and> invc2 (baldL l a r) \<and> (color r = Black \<longrightarrow> invc (baldL l a r))"
lhd_LCons_ltl: fixes llist :: "'b llist" assumes "\<not> lnull llist" shows "LCons (lhd llist) (ltl llist) = llist"
prob_spaceI: fixes M :: "'c measure" assumes "emeasure M (space M) = 1" shows "prob_space M"
start_in_A: assumes "0 < length is" shows "start \<in> list (length is) A"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Cod t')"
lossless_WT_gpv_induct_strong: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and P :: "('f, 'd, 'e) gpv \<Rightarrow> bool" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>p. \<lbrakk>lossless_spmf p; \<And>out c. IO out c \<in> set_spmf p \<Longrightarrow> out \<in> outs_\<I> \<I>; \<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> lossless_gpv \<I> gpv; \<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> \<I> \<turnstile>g gpv \<surd>; \<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> P gpv\<rbrakk> \<Longrightarrow> P (GPV p)" shows "P gpv"
sup_commute: fixes sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
nth_default_take_eq: fixes dflt :: "'b"   and m :: "nat"   and xs :: "'b list"   and n :: "nat" shows "nth_default dflt (take m xs) n = (if n < m then nth_default dflt xs n else dflt)"
1: shows "\<guillemotleft>cmp\<^sub>U\<^sub>P \<mu>\<nu> : UP (B.dom (fst \<mu>\<nu>)) \<star> UP (B.dom (snd \<mu>\<nu>)) \<Rightarrow> UP (B.cod (fst \<mu>\<nu>) \<star>\<^sub>B B.cod (snd \<mu>\<nu>))\<guillemotright>"
continuous_on_open: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on S f = (\<forall>T. openin (top_of_set (f ` S)) T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
ring_simprules_2: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
img_is_subalgebra: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and K :: "'c set"   and V :: "'c set" assumes "ring_hom_ring R S h"   and "K \<subseteq> carrier R"   and "subalgebra K V R" shows "subalgebra (h ` K) (h ` V) S"
succ: fixes as :: "'a list" shows "set (succ as) = {b. \<exists>a\<in>set as. (a, b) \<in> rel}"
right_unique_conversep: fixes A :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "right_unique A\<inverse>\<inverse> = left_unique A"
WCFG_CondElse: fixes c\<^sub>2 :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and b :: "expr"   and c\<^sub>1 :: "cmd" assumes "c\<^sub>2 \<turnstile> n -et\<rightarrow> n'"   and "n \<noteq> (_Entry_)" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> n \<oplus> #:c\<^sub>1 + 1 -et\<rightarrow> n' \<oplus> #:c\<^sub>1 + 1"
infinity_ne_numeral: fixes k :: "num" shows "\<infinity> \<noteq> numeral k"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "(a \<cong> a') = (\<exists>f. \<guillemotleft>f : a \<Rightarrow> a'\<guillemotright> \<and> local.iso f)"
assms: shows "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
plus_eq_enat_iff: fixes a :: "enat"   and b :: "enat"   and i :: "nat" shows "(a + b = enat i) = (\<exists>j k. a = enat j \<and> b = enat k \<and> j + k = i)"
equivalence_of_categories_axioms: shows "equivalence_of_categories \<F>\<^sub>SC.comp (\<cdot>) D \<F>\<^sub>SC.map \<nu>.map \<mu>.map"
measurable_If: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and M' :: "'d measure"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "{x \<in> space M. P x} \<in> sets M" shows "(\<lambda>x. if P x then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
subformulas_in_subformulas_3: fixes G :: "'a formula"   and H :: "'a formula"   and F :: "'a formula" assumes "G \<^bold>\<rightarrow> H \<in> set (subformulae F)" shows "G \<in> set (subformulae F) \<and> H \<in> set (subformulae F)"
finite_dom_map_of: fixes l :: "('c \<times> 'd) list" shows "finite (dom (map_of l))"
adds_deg_pm_antisym: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "s adds t"   and "deg_pm t \<le> deg_pm s" shows "s = t"
mem_lv_rel_set_rel_iff: fixes y :: "real list set"   and z :: "'a set" shows "((y, z) \<in> \<langle>lv_rel\<rangle>set_rel) = (z = eucl_of_list ` y \<and> (\<forall>x\<in>y. length x = DIM('a)))"
that: assumes "\<not> b" shows "f (0::'b) = (0::'a)"
T_deter_rand: fixes s0 :: "'state"   and qs :: "'request list"   and A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and x :: "'b" shows "real (T s0 qs (off2 A (s0, x) qs)) = T_on_rand' (local.embed A) (return_pmf (s0, x)) qs"
sum_mset_0_left: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c multiset" assumes "\<And>x. (0::'a) \<le> f x"   and "\<Sum>\<^sub># (image_mset f A) = (0::'a)" shows "\<forall>x\<in>#A. f x = (0::'a)"
iprev_predicate_change_exists: fixes a :: "nat"   and b :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P b"   and "P a" shows "\<exists>n\<in>I. a < n \<and> n \<le> b \<and> \<not> P n \<and> P (iprev n I)"
g_def: shows "g \<equiv> \<lambda>s x. complex_of_real x powr - s * complex_of_real (exp (- a * x)) / complex_of_real (1 - exp (- x))"
index_transpose_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_col A"   and "j < dim_row A" shows "A\<^sup>t $$ (i, j) = A $$ (j, i)"
strict_implies_order: fixes b :: "'a list"   and a :: "'a list" assumes "strict_border b a" shows "border b a"
drop_all: fixes xs :: "'k list"   and n :: "nat" assumes "length xs \<le> n" shows "drop n xs = []"
uaE: shows "ua \<in> E \<rightarrow> carrier K"
monadic_filter_rev_rule: fixes l :: "'b list"   and P :: "'b \<Rightarrow> bool nres"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> set l \<Longrightarrow> P x \<le> (spec r. r = Q x)" shows "monadic_filter_rev P l \<le> (spec r. r = filter_rev Q l)"
triangle_right: shows "Cop.H \<epsilon> g \<cdot> \<a>[g, f, g] \<cdot> Cop.H g \<eta> = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
assms: shows "A -\<^sub>\<circ> set {x} \<subseteq>\<^sub>\<circ> B"
sign_changes_neg_inf_def: fixes ps :: "'a poly list" shows "sign_changes_neg_inf ps = length (remdups_adj (list_neq (map poly_neg_inf ps) (0::'a))) - 1"
y_pos: shows "lt_vec 0 (pow_A_1 u)"
merge_finite': fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "finite (local.merge j)" shows "finite (local.merge i)"
sub_ker: shows "local.subspace {x. f x = (0::'c)}"
simps_1: fixes x :: "'b"   and y :: "'b" shows "Some x \<squnion> Some y = Some (x \<squnion> y)"
fin: fixes n :: "nat" shows "finite {N. Nf N \<le> n}"
AnalBinds_delete_bot: fixes ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "ae x = \<bottom>" shows "AnalBinds (delete x \<Gamma>)\<cdot>ae = AnalBinds \<Gamma>\<cdot>ae"
intros: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. borel))" shows "generatorp (emb I J X)"
invariant_arbitraryI: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "\<And>s. s \<in> reachable A I \<Longrightarrow> P s" shows "A \<TTurnstile> (I \<rightarrow>) P"
preserves_ide: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "ide (trg a)"
dependent_inj_imageD: fixes s :: "'b set" assumes "vs2.dependent (f ` s)"   and "inj_on f (vs1.span s)" shows "vs1.dependent s"
apps: fixes \<zeta> :: "('s, 'v) hd"   and ss :: "('s, 'v) tm list" assumes "t = apps (Hd \<zeta>) ss"   and "\<And>sa. sa \<in> set ss \<Longrightarrow> wary sa"   and "enat (length ss) \<le> arity_hd \<zeta>" shows "P"
take_bit_of_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n zero = zero"
l_get_disconnected_nodes_axioms: shows "l_get_disconnected_nodes ShadowRootClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
distinct_mtf2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "distinct (mtf2 n x xs) = distinct xs"
lexmin_SEQ_mem: fixes C :: "(nat \<Rightarrow> 'a) set" assumes "C \<subseteq> SEQ A"   and "C \<noteq> {}" shows "lexmin C \<in> SEQ A"
bin_set_conv_OR: fixes n :: "nat"   and i :: "int" shows "bin_sc n True i = i OR (1 << n)"
sortLemmaL: fixes h :: "'b \<Rightarrow> int"   and t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" assumes "sortedTree h (T t1 x t2)" shows "sortedTree h t1"
simps_4: fixes n :: "nat"   and p :: "pf"   and q :: "pf" shows "dfa_of_pf' n (Or p q) = or_dfa (dfa_of_pf' n p) (dfa_of_pf' n q)"
fract_collapse_2: shows "Fraction_Field.Fract (1::'a) (1::'a) = 1"
ins_height: fixes k :: "nat"   and x :: "'a"   and t :: "'a btree" shows "height_up\<^sub>i (ins k x t) = height t"
inducts: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and prod :: "'c \<times> 'd" assumes "\<And>a b. P (a, b)" shows "P prod"
minus_perm_def: fixes p1 :: "perm"   and p2 :: "perm" shows "p1 - p2 = p1 + - p2"
fC_2: shows "f ` C = insert (f w) (f ` (C \<inter> D))"
arr_Set_arr_ParI: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Par \<alpha> T"   and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>" shows "arr_Set \<alpha> T"
idom_is_ring: fixes R :: "('e, 'f) Ring_scheme" assumes "Idomain R" shows "Ring R"
cong_ex: fixes f1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f1 x = f2 x"   and "\<forall>\<^sub>F x in F. g1 x = g2 x" shows "(f1 \<in> O[F](g1)) = (f2 \<in> O[F](g2))"
chain_subdiv_path_singleton_reverse: fixes \<gamma> :: "real \<Rightarrow> real \<times> real" shows "chain_subdiv_path (reversepath \<gamma>) {(- 1, \<gamma>)}"
FaultProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f" shows "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> Fault f"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "Min (insert x A) = min x (Min A)"
that: fixes A :: "dbfm" assumes "wf_dbfm A"   and "x = \<lbrakk>quot_dbfm A\<rbrakk>e" shows "thesis"
list_of_oalist_of_list_id: fixes xs :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv xs" shows "list_of_oalisty (oalist_of_listy xs) = xs"
continuous_componentwise: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "continuous F f = (\<forall>i\<in>Basis. continuous F (\<lambda>x. f x \<bullet> i))"
fw_int_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and i :: "nat"   and j :: "nat" assumes "dbm_int M n"   and "a \<le> n"   and "b \<le> n"   and "c \<le> n"   and "i \<le> n"   and "j \<le> n"   and "fw M n a b c i j \<noteq> \<infinity>" shows "get_const (fw M n a b c i j) \<in> \<int>"
bind_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c measure" shows "M \<bind> f = (if space M = {} then count_space {} else join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f))"
finite_range_filter_tab: fixes t :: "'b \<Rightarrow> 'a option"   and c :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite (range t)" shows "finite (range (filter_tab c t))"
terminating_path_root_end_converse: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e" shows "terminating_path_root_end e (x\<^sup>T) r"
split_of_bool: fixes one :: "'a"   and zero :: "'a"   and P :: "'a \<Rightarrow> bool"   and p :: "bool" assumes "class.zero_neq_one one zero" shows "P (zero_neq_one.of_bool one zero p) = ((p \<longrightarrow> P one) \<and> (\<not> p \<longrightarrow> P zero))"
ag_pOp_closed: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e"   and y :: "'e" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<in> carrier A"
weight: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set" assumes "BinPacking U w c S L" shows "\<forall>u\<in>U. 0 < w u \<and> w u \<le> real c"
that: fixes t :: "'s" assumes "t \<in> \<S>"   and "viterbi_final s os = viterbi s t os"   and "snd (viterbi s t os) = (MAX t\<in>\<S>. snd (viterbi s t os))" shows "thesis"
division_of_trivial: fixes s :: "'c set set" shows "(s division_of {}) = (s = {})"
gs: shows "gs = (snd (mus_adjuster f n_gs [] (0\<^sub>v n)) + f) # tl gs"
up_ab: shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) f up = up [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> a \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> un [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> b"
inverse_matrix: fixes A :: "(('b, 'c) vec, 'c) vec" shows "inverse_matrix A = (if invertible A then Some (P_Gauss_Jordan A) else None)"
InducedFHModule_axioms: shows "InducedFHModule H G (\<cdot>) V (\<currency>)"
simps_5: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc ({I/x} WHILE b DO C) Q = (\<forall>l s. (I (l(x := Suc (l x))) s \<longrightarrow> pre C I l s) \<and> (0 < l x \<and> I l s \<longrightarrow> bval b s) \<and> (I (l(x := 0)) s \<longrightarrow> \<not> bval b s \<and> Q l s) \<and> vc C I)"
of_rat_neg_one: shows "of_rat (- 1) = - (1::'b)"
assms_3: shows "compl_fam (\<lambda>g. generate G {g}) gs"
BetaNegAnd: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list"   and B :: "('a, 'b) form" assumes "\<stileturn> Neg A # G"   and "\<stileturn> Neg B # G" shows "\<stileturn> Neg (And A B) # G"
bal_list_def: fixes n :: "nat"   and xs :: "'b list" shows "Balance.bal_list n xs = fst (Balance.bal n xs)"
synth_analz_increasing: fixes c :: "msg"   and A :: "msg set" assumes "c \<in> A" shows "c \<in> synth (analz A)"
simps_3: fixes c :: "int"   and s :: "MIR.num"   and t :: "MIR.num" shows "ismaxcoeff (CF c s t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> ismaxcoeff t x)"
g: fixes y :: "complex" assumes "y \<in> V" shows "(g has_derivative g' y) (at y)"