hcju/isabelleps huggingface dataset:
berlekamp_factorization_main: fixes f :: "'a mod_ring poly"   and vs :: "'a mod_ring poly list"   and vs1 :: "'a mod_ring poly list"   and vs2 :: "'a mod_ring poly list"   and n :: "nat"   and us1 :: "'a mod_ring poly list"   and n2 :: "nat"   and us :: "'a mod_ring poly list"   and d :: "nat"   and divs :: "'a mod_ring poly list" assumes "square_free f"   and "vs = vs1 @ vs2"   and "vs = berlekamp_basis f"   and "n = length (berlekamp_basis f)"   and "n = length us1 + n2"   and "us = us1 @ berlekamp_factorization_main d divs vs2 n2"   and "\<And>u. u \<in> set us1 \<Longrightarrow> monic u \<and> irreducible u"   and "\<And>d. d \<in> set divs \<Longrightarrow> monic d \<and> 0 < degree d"   and "\<And>u v i. \<lbrakk>v \<in> set vs1; u \<in> set us1 \<union> set divs; i < CARD('a)\<rbrakk> \<Longrightarrow> gcd u (v - [:of_nat i:]) \<in> {1, u}"   and "f = prod_list (us1 @ divs)"   and "0 < degree f"   and "\<And>g. \<lbrakk>g dvd f; degree g = d\<rbrakk> \<Longrightarrow> irreducible g" shows "f = prod_list us \<and> (\<forall>u\<in>set us. monic u \<and> irreducible u)"
QQ3_a_def: shows "QQ3_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . p \<noteq> nil \<and> gg mrk atom left p \<and> p' = left p \<and> t' = p \<and> mrk' = mrk \<union> {left p} \<and> left' = left(p := t) \<and> right' = right \<and> atom' = atom :]"
simps_29: fixes vb :: "'a"   and vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "({vb} WHILE vc DO vd {ve} \<sqsubseteq> v;; va) = False"
axioms_2: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module_pair s1 s2" shows "module s2"
LeadsTo_weaken: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B \<longmapsto>w B'"
distinct_butlast: fixes xs :: "'a list" assumes "distinct xs" shows "distinct (butlast xs)"
t_stop_def: shows "ego2.t_stop = - v\<^sub>e / a\<^sub>e"
gt_sym_total: fixes f :: "'s"   and g :: "'s" shows "f >\<^sub>s g \<or> g >\<^sub>s f \<or> g = f"
DmdRec2: fixes P :: "state \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<diamond>P"   and "sigma \<Turnstile> \<box>\<not> P$" shows "sigma \<Turnstile> Init P"
thif2: fixes a :: "bool"   and b :: "'b"   and c :: "bool"   and d :: "'b"   and e :: "'b" shows "(if a then b else if c then d else e) = (if c then if a then b else d else if a then b else e)"
rb_aux_domI: fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_inv (fst args)" shows "rb_aux_dom args"
order_tendstoD_2: fixes f :: "'c \<Rightarrow> 'a"   and y :: "'a"   and F :: "'c filter"   and a :: "'a" assumes "(f \<longlongrightarrow> y) F"   and "y < a" shows "\<forall>\<^sub>F x in F. f x < a"
assms_1: assumes "0 \<le> x"   and "0 \<le> y" shows "P (lcm x y)"
pos_divide_less_eq: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "(0::'a) < c" shows "(b / c < a) = (b < a * c)"
eqv_def: 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 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 \<phi> :: "'c"   and \<chi> :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi" shows "Syntax_with_Connectives.eqv cnj imp \<phi> \<chi> = cnj (imp \<phi> \<chi>) (imp \<chi> \<phi>)"
row_echelon_transform: fixes A :: "'c mat"   and b :: "'d" obtains P :: "'c mat" where "P \<in> Units (ring_mat TYPE('c) (dim_row A) b)" and "row_echelon A = P * A"
FORWEAK_def: fixes X :: "'d set"   and d :: "'c nres"   and f :: "'d \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> 'c \<Rightarrow> 'c nres" shows "FORWEAK X d f c = (if X = {} then d else op_set_npick_remove X \<bind> (\<lambda>(a, X). f a \<bind> (\<lambda>b. WHILE (\<lambda>(uu_, X). \<not> op_set_isEmpty X) (\<lambda>(b, X). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(a, X). f a \<bind> (\<lambda>b'. c b b' \<bind> (\<lambda>b. RETURN (b, X)))))) (b, X) \<bind> (\<lambda>(b, uu_). RETURN b))))"
dim_singleton: fixes x :: "('b, 'c) vec" shows "vec.dim {x} = (if x = 0 then 0 else 1)"
unit_simps_5: fixes a :: "'a" assumes "B.obj a" shows "B.cod \<i>\<^sub>B[a] = a"
omega_sum_less_eqlen_iff_cases: fixes ms :: "nat list"   and ns :: "nat list"   and m :: "nat"   and n :: "nat" assumes "length ms = length ns" shows "(omega_sum (m # ms) < omega_sum (n # ns)) = (m < n \<or> m = n \<and> omega_sum ms < omega_sum ns)"
PX5d_linear: fixes d :: "D2" shows "Linear_order (PX5d d)"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>nat. P (TVar nat)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1\<rightarrow>ty2)" shows "P ty"
FM_P_module: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "R module P"
ball_eq_empty: fixes x :: "'b"   and e :: "real" shows "(ball x e = {}) = (e \<le> 0)"
min_def: fixes a :: "'b word"   and b :: "'b word" shows "signed.min a b = (if a \<le>s b then a else b)"
FL_distinguished_bounded_support: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c fs_set"   and x :: "('a, 'e, 'b, 'c) formula"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply"   and "effect_nominal_ts.is_FL_formula L F x"   and "indexed_effect_nominal_ts.is_distinguishing_formula satisfies transition effect_apply x P Q" obtains y :: "('a, 'e, 'b, 'c) formula" where "effect_nominal_ts.is_FL_formula L F y" and "supp y \<subseteq> supp (F, P)" and "indexed_effect_nominal_ts.is_distinguishing_formula satisfies transition effect_apply y P Q"
rel_set_UNION: fixes Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d set"   and B :: "'e set"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f set"   and g :: "'e \<Rightarrow> 'g set" assumes "rel_set Q A B"   and "(Q ===> rel_set R) f g" shows "rel_set R (\<Union> (f ` A)) (\<Union> (g ` B))"
string_unop_type_det: fixes op :: "string_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'b OCL_Types.type"   and \<sigma>\<^sub>2 :: "'b OCL_Types.type" assumes "string_unop_type op \<tau> \<sigma>\<^sub>1"   and "string_unop_type op \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
ide_char: fixes a :: "'b" shows "C'.ide a = (C'.arr a \<and> C'.dom a = a \<and> C'.cod a = a)"
ord_adds: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and t :: "'a" assumes "ordered_powerprod ord ord_strict"   and "s adds t" shows "ord s t"
hom_bot: fixes f :: "'b \<Rightarrow> 'a" shows "hom f \<bottom> = \<bottom>"
that: fixes g :: "real \<Rightarrow> real" assumes "continuous_on {a..b} g"   and "g b = g a"   and "g square_integrable {a..b}"   and "l2norm {a..b} (\<lambda>x. f x - g x) < e" shows "thesis"
to_st_trms_wf: fixes A :: "('fun, 'var) extstrand_step list" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A))" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t (to_st A))"
last_take_nth_conv: fixes n :: "nat"   and l :: "'a list" assumes "n \<le> length l"   and "n \<noteq> 0" shows "last (take n l) = l ! (n - 1)"
abs_eq: fixes x :: "nat" shows "Var x = MPoly (Var\<^sub>0 x)"
additive_left: fixes b :: "'b" shows "additive (\<lambda>a. a ** b)"
Range_Id_on: fixes A :: "'b set" shows "Range (Id_on A) = A"
init_iff_list_eq_upto: fixes f :: "nat \<Rightarrow> nat option"   and vs :: "nat" assumes "f \<in> \<R>"   and "0 < e_length vs" shows "(\<forall>x<e_length vs. f x \<down>= e_nth vs x) = (prefix f (e_length vs - 1) = Partial_Recursive.list_decode vs)"
assms_2: shows "m -as'\<rightarrow>\<^sub>\<surd>* (_Low_)"
simps: fixes f :: "char list \<Rightarrow> 'a"   and xs :: "char list list" shows "Abs_State.dom (FunDom f xs) = xs"
prv_all_inst: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm" shows "prv (imp (all x \<phi>) (subst \<phi> t x))"
len_comp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and a :: "nat"   and c :: "nat"   and xs :: "nat list"   and b :: "nat"   and ys :: "nat list" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.len plus m a c (xs @ b # ys) = plus (linordered_ab_monoid_add.len plus m a b xs) (linordered_ab_monoid_add.len plus m b c ys)"
FvarsT_num: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and num :: "'e set"   and n :: "'e" assumes "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num"   and "n \<in> num" shows "FvarsT n = {}"
BSI_implies_I: assumes "BSI \<V> Tr\<^bsub>(induceES SES)\<^esub>" shows "I \<V> Tr\<^bsub>(induceES SES)\<^esub>"
SPRDetAutoDFS_def: shows "SPRDetAutoDFS \<equiv> \<lambda>agents jkbp envInit envAction envTrans envVal envObsC envObs a. alg_dfs acts_MapOps trans_MapOps (SPRViewDet.spr_simObs envObsC a) (SPRViewDet.spr_simTrans agents jkbp envAction envTrans envVal envObsC envObs a) (SPRViewDet.spr_simAction jkbp envVal envObs a) (map (SPRViewDet.spr_simInit envInit envObsC envObs a \<circ> envObs a) envInit)"
1: shows "\<nu>.cod.leg0 \<cdot> \<nu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>0 = \<pi>.cod.leg1 \<cdot> \<pi>.chine \<cdot> doms.Prj\<^sub>0\<^sub>0"
fdia_arden: fixes x :: "'a"   and p :: "'a"   and q :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)"   and "d p \<le> d q +  |x\<rangle> p" shows "d p \<le>  |x\<^sup>\<star>\<rangle> q"
smult_sum_list_distrib: fixes r :: "'r"   and ms :: "'m list" assumes "r \<in> R"   and "set ms \<subseteq> M" shows "r \<cdot> sum_list ms = sum_list (map ((\<cdot>) r) ms)"
strict_boundedE: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a <s signed.min b c" obtains "a <s b" and "a <s c"
spo: shows "automaton.trans OA = oseqp_sos \<Gamma> i"
from_nat_into: fixes A :: "'c set"   and n :: "nat" assumes "A \<noteq> {}" shows "from_nat_into A n \<in> A"
insertI: fixes A :: "'b set"   and a :: "'b" assumes "finite A" shows "finite (insert a A)"
null_is_valid: fixes bot :: "'a"   and null :: "'a" assumes "class.null bot null" shows "null \<noteq> bot"
singleton_less_list_iff: fixes xs :: "'a list"   and n :: "'a" assumes "sorted xs" shows "([n] < xs) = ({..n} \<inter> list.set xs = {})"
Un_commute: fixes A :: "'h set"   and B :: "'h set" shows "A \<union> B = B \<union> A"
that_2: fixes x :: "'a" assumes "i < x"   and "x < b" shows "g x = f x"
individually_rational_onI: fixes ds :: "D2 set"   and X :: "X3 set" assumes "Theorem_9.CD_on ds X = X"   and "Theorem_9.CH X = X" shows "Theorem_9.individually_rational_on ds X"
listrelp_conj1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and y :: "'b list" assumes "listrelp (\<lambda>x y. R x y \<and> S x y) x y" shows "listrelp R x y"
support_on_if: fixes a :: "'c"   and A :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "a \<noteq> (0::'c)" shows "support_on A (\<lambda>x. if P x then a else (0::'c)) = {x \<in> A. P x}"
rev_nth_bounded: fixes v :: "nat"   and xs :: "'b list" assumes "v < length xs" shows "\<exists>x. xs !. v = Some x"
wf_J_mdecl_def: fixes P :: "'a J_prog"   and C :: "String.literal" shows "wf_J_mdecl P C \<equiv> \<lambda>(M, Ts, T, pns, body). length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> (\<exists>T'. P,[this \<mapsto> Class C, pns [\<mapsto>] Ts] \<turnstile> body :: T' \<and> P \<turnstile> T' \<le> T) \<and> \<D> body \<lfloor>{this} \<union> set pns\<rfloor>"
transitive: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Q \<leadsto>\<guillemotleft>Rel'\<guillemotright> R"   and "eqvt Rel"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''"   and "\<And>S T. (S, T) \<in> Rel \<Longrightarrow> S \<leadsto><Rel> T"   and "(P, Q) \<in> Rel" shows "P \<leadsto>\<guillemotleft>Rel''\<guillemotright> R"
lookup_single_eq: fixes k :: "'e"   and v :: "'d" shows "lookup (monomial v k) k = v"
default_def: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "Mapping.default k v m = (if k \<in> keys m then m else update k v m)"
assms_7: shows "\<not> equivalent_eq_exists t s C (same_values (\<lambda>t. trm_rep t S)) \<sigma> L1"
map_of_def: shows "map_of \<equiv> rec_list Map.empty (\<lambda>p ps psa. psa(fst p \<mapsto> snd p))"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Inv (t \<^bold>\<cdot> u) = Inv u \<^bold>\<cdot> Inv t"
integral_0: fixes S :: "'c set" shows "integral S (\<lambda>x. 0::'b) = (0::'b)"
norm_columns_fst_QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec"   and i :: "'a" assumes "rank A = ncols A" shows "norm (column i (fst (QR_decomposition A))) = 1"
saturated_model: fixes N :: "'a literal multiset set" assumes "saturated N"   and "N \<inter> {{#}} = {}" shows "I_of N \<TTurnstile>e N"
Suc_1: fixes P :: "'a set" assumes "x = card P"   and "finite P" shows "finite {p. distinct p \<and> set p \<subseteq> P}"
set_take_subset_set_take: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "m \<le> n" shows "set (take m xs) \<subseteq> set (take n xs)"
le_strict_2: fixes compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a" assumes "class.Ord_strict compare" shows "Ord.le compare\<cdot>x\<cdot>\<bottom> = \<bottom>"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length fs" shows "d k = (\<Prod>j<k. \<parallel>gso j\<parallel>\<^sup>2)"
is_distinguishing_formula_def: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "('a, 'e, 'b, 'c) formula"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply" shows "indexed_effect_nominal_ts.is_distinguishing_formula satisfies transition effect_apply x P Q \<equiv> effect_nominal_ts.FL_valid satisfies transition effect_apply P x \<and> \<not> effect_nominal_ts.FL_valid satisfies transition effect_apply Q x"
g_inj_on: shows "inj_on g (S \<inter> dom g)"
scf_d_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e"   and d :: "nat" shows "scf_d R S X p d = (SOME f. scf_cond R S X p d f)"
length_pos_if_in_set: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set xs" shows "0 < length xs"
a_set_val_locs_def: fixes character_data_ptr :: "(_) character_data_ptr" shows "a_set_val_locs character_data_ptr \<equiv> all_args (put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr val_update)"
symbSS'_aux: shows "(\<lambda>x. \<forall>u. fst (u 0) = 0 \<longrightarrow> (\<forall>n. (\<forall>i<n. Ex ((case u i of (u, v) \<Rightarrow> \<lambda>(u', v') x z. u = 0 \<and> u' = Suc 0) (u (Suc i)) (x i))) \<longrightarrow> (case u n of (u, v) \<Rightarrow> \<lambda>(u', v') x. 0 < u) (u (Suc n)) (x n))) = \<bottom>"
compactin_subspace: fixes X :: "'b topology"   and S :: "'b set" shows "compactin X S = (S \<subseteq> topspace X \<and> compact_space (subtopology X S))"
Approximation1_5Tr5: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and a :: "'a"   and x :: "'a"   and j :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "a \<in> carrier K"   and "a \<noteq> \<zero>"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x"   and "j \<in> nset (Suc 0) (Suc n)" shows "\<exists>l. \<forall>m>l. 0 < vv j (a \<cdot>\<^sub>r x^\<^bsup>K m\<^esup>)"
invariant3pI: fixes I :: "'a set"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>s tl s'. \<lbrakk>s \<in> I; r s tl s'\<rbrakk> \<Longrightarrow> s' \<in> I" shows "invariant3p r I"
prime_power_cancel: fixes p :: "'a"   and m :: "'a"   and k :: "nat"   and m' :: "'a"   and k' :: "nat" assumes "prime p"   and "m * p ^ k = m' * p ^ k'"   and "\<not> p dvd m"   and "\<not> p dvd m'" shows "k = k'"
sorted_hyp_var: shows "\<forall>q1<length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}). \<forall>q2<length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}). q1 < q2 \<longrightarrow> sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! q1 < sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! q2"
additive_def: fixes f :: "'a \<Rightarrow> 'b" shows "Modules.additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
swap_preserves_fresh_distinct: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y1 :: "'var"   and y2 :: "'var" assumes "good X"   and "xs \<noteq> ys \<or> x \<notin> {y1, y2}" shows "fresh xs x (X #[y1 \<and> y2]_ys) = fresh xs x X"
feq: fixes x :: "'b" assumes "x \<in> t" shows "(f \<circ> h \<circ> k) x = f x"
inf_le1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) x"
fxpnt\<Phi>: shows "\<Inter> (TESL_interpretation_atomic ` set \<Phi>) = \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
add_diff_cancel_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
euler_maclaurin_strong_int': fixes b :: "int" assumes "int a \<le> b" shows "(\<Sum>k = int a..b. f (real_of_int k)) = F (real_of_int b) + C + (1 / 2) *\<^sub>R f (real_of_int b) + (\<Sum>i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\<^sub>R fs (2 * i - 1) (real_of_int b)) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) b"
arity_hd_head: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and s :: "('a, 'b) tm" assumes "arity.wary arity_sym arity_var s" shows "arity.arity_hd arity_sym arity_var (head s) = arity.arity arity_sym arity_var s + enat (num_args s)"
iso_cancel_right: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "VoV.iso g"   and "VoV.comp f g = VoV.comp f' g"   and "VoV.seq f g"   and "VoV.iso g" shows "f = f'"
web_quotient_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "web (quotient_web (quotient_web \<Gamma> f) f)"
assms: shows "s \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) S U"
Sigma_Max_lemma: fixes rf :: "recf"   and xs :: "nat list"   and w :: "nat" assumes "primerec rf (Suc (length xs))" shows "UF.Sigma (rec_exec (Cn (Suc (Suc (length xs))) rec_not [rec_all (recf.id (Suc (Suc (length xs))) (length xs)) (Cn (Suc (Suc (Suc (length xs)))) rec_disj [Cn (Suc (Suc (Suc (length xs)))) rec_le [recf.id (Suc (Suc (Suc (length xs)))) (Suc (Suc (length xs))), recf.id (Suc (Suc (Suc (length xs)))) (Suc (length xs))], Cn (Suc (Suc (Suc (length xs)))) rec_not [Cn (Suc (Suc (Suc (length xs)))) rf (get_fstn_args (Suc (Suc (Suc (length xs)))) (length xs) @ [recf.id (Suc (Suc (Suc (length xs)))) (Suc (Suc (length xs)))])]])])) ((xs @ [w]) @ [w]) = Maxr (\<lambda>args. 0 < rec_exec rf args) xs w"
filterlim_divide_at_infinity: fixes f :: "'b \<Rightarrow> 'b"   and c :: "'b"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g (at (0::'b)) F"   and "c \<noteq> (0::'b)" shows "LIM x F. f x / g x :> at_infinity"
Guard: fixes s :: "'a"   and g :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and f :: "'c"   and c :: "('a, 'b, 'c) com" assumes "s \<in> g" shows "\<Gamma>\<turnstile> (Guard f g c, Normal s) \<rightarrow> (c, Normal s)"
simps_1: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "P \<turnstile> c \<rightarrow>^0 c' = (c' = c)"
inj_on_nth: fixes xs :: "'g list"   and I :: "nat set" assumes "distinct xs"   and "\<forall>i\<in>I. i < length xs" shows "inj_on ((!) xs) I"
gp: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (G i)"
assms: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
emeasure_pmf_UNIV: fixes M :: "'b pmf" shows "emeasure (measure_pmf M) UNIV = 1"
robot_ClockAlg_def: shows "robot_ClockAlg \<equiv> mkClockAuto Agents (\<lambda>_. kbp) envInit envAction envTrans envVal (\<lambda>_. envObs)"
D: shows "evalDdb (C<e2>)\<cdot>env_empty_db = evalDdb (C<e1>)\<cdot>env_empty_db"
simps: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and xs :: "('a \<times> 'b) list"   and ko :: "'c" shows "oalist_raw.lookup_raw rep_key_order (xs, ko) = comparator.lookup_pair (key_compare (rep_key_order ko)) xs"
is_ground_cls_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and C :: "'a literal multiset" shows "substitution_ops.is_ground_cls subst_atm C = (\<forall>L. L \<in># C \<longrightarrow> substitution_ops.is_ground_lit subst_atm L)"
map_gpv_bind_gpv: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and gpv :: "('f, 'e, 'c) gpv"   and h :: "'f \<Rightarrow> ('d, 'e, 'c) gpv" shows "map_gpv f g (bind_gpv gpv h) = bind_gpv (map_gpv id g gpv) (\<lambda>x. map_gpv f g (h x))"
sorted_list_of_set_bij_betw: fixes A :: "'b set" assumes "finite A" shows "bij_betw ((!) (sorted_list_of_set A)) {..<card A} A"
min_height_size_if_incomplete: fixes kdt :: "'a kdt" assumes "\<not> KD_Tree.complete kdt" shows "2 ^ min_height kdt < KD_Tree.size_kdt kdt"
inj_map: fixes f1 :: "'b \<Rightarrow> 'c"   and f2 :: "'d \<Rightarrow> 'e" assumes "inj f1"   and "inj f2" shows "inj (map_prod f1 f2)"
real_divide_code: fixes x :: "rat"   and y :: "rat" shows "Ratreal x / Ratreal y = Ratreal (x / y)"
insertCI: fixes a :: "'q"   and B :: "'q set"   and b :: "'q" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
triangular_to_jnf_vector: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "upper_triangular A" shows "jordan_nf A (triangular_to_jnf_vector A)"