hcju/isabelleps huggingface dataset:
fine_subset: fixes p :: "('a \<times> 'b set) set"   and q :: "('a \<times> 'b set) set"   and d :: "'a \<Rightarrow> 'b set" assumes "p \<subseteq> q"   and "d fine q" shows "d fine p"
subprob_density_distr_real_inverse_aux: fixes M :: "real measure"   and f :: "real \<Rightarrow> ennreal" assumes "has_subprob_density M lborel f" shows "has_subprob_density (distr M borel (\<lambda>x. - inverse x)) lborel (\<lambda>x. f (- inverse x) * ennreal (inverse (x * x)))"
vec_eq_iff: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec" shows "(x = y) = (\<forall>i. x $ i = y $ i)"
case_rews_3: fixes a :: "'r"   and f1 :: "'s"   and f2 :: "'r \<rightarrow> 'r tree \<rightarrow> 'r tree \<rightarrow> 's"   and tree1 :: "'r tree"   and tree2 :: "'r tree" assumes "a \<noteq> \<bottom>" shows "tree_case\<cdot>f1\<cdot>f2\<cdot>(Node\<cdot>a\<cdot>tree1\<cdot>tree2) = f2\<cdot>a\<cdot>tree1\<cdot>tree2"
pow_mono_exp: fixes a :: "'c"   and m :: "nat"   and n :: "nat" assumes "(1::'c) \<le> a"   and "m \<le> n" shows "a ^ m \<le> a ^ n"
entailed_formula_entails_implicates: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set"   and C :: "'at Literal set" assumes "entails_formula S1 S2"   and "entails S2 C" shows "entails S1 C"
disjnt_def: fixes A :: "'b set"   and B :: "'b set" shows "disjnt A B = (A \<inter> B = {})"
IncrEnvironment_axioms: shows "IncrEnvironment jkbp spr_jview envObs spr_jviewInit spr_jviewIncr"
rt_fresher_trans: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt3"
wt: shows "Env\<turnstile>c\<Colon>\<surd>"
hilbert_folgeaxiome_as_strong_as_AX0: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "(AX0 \<union> \<Gamma> \<turnstile>\<^sub>H F) = (AXH \<union> \<Gamma> \<turnstile>\<^sub>H F)"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_on_left_moving_B m rs (l, r) = (\<exists>ln rn. l = Bk \<up> ln @ Oc # Oc \<up> Suc rs @ Bk # Oc \<up> Suc m \<and> r = Bk \<up> rn)"
\<X>_are_terminals: fixes u :: "symbol \<times> character list"   and k :: "nat" assumes "u \<in> \<X> k" shows "is_terminal (terminal_of_token u)"
h1: shows "distinct (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
mfactor_induct: fixes P :: "pol \<Rightarrow> mon \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "mon" assumes "\<And>Pa c. P Pa (Mc c)"   and "\<And>d v va. P (Pc d) (Minj v va)"   and "\<And>d v va. P (Pc d) (MX v va)"   and "\<And>i Pa j M. \<lbrakk>i = j \<Longrightarrow> P Pa M; \<lbrakk>i \<noteq> j; i < j\<rbrakk> \<Longrightarrow> P Pa (Minj (j - i) M)\<rbrakk> \<Longrightarrow> P (Pinj i Pa) (Minj j M)"   and "\<And>i Pa j M. P (Pinj i Pa) (MX j M)"   and "\<And>Pa i Q j M. \<lbrakk>j = 0 \<Longrightarrow> P (PX Pa i Q) M; j \<noteq> 0 \<Longrightarrow> P Pa (Minj j M); \<And>x xa y. \<lbrakk>j \<noteq> 0; x = mfactor Pa (Minj j M); (xa, y) = x\<rbrakk> \<Longrightarrow> P Q (Minj_pred j M)\<rbrakk> \<Longrightarrow> P (PX Pa i Q) (Minj j M)"   and "\<And>Pa i Q j M. \<lbrakk>i = j \<Longrightarrow> P Pa (mkMinj 1 M); \<lbrakk>i \<noteq> j; i < j\<rbrakk> \<Longrightarrow> P Pa (MX (j - i) M); \<lbrakk>i \<noteq> j; \<not> i < j\<rbrakk> \<Longrightarrow> P Pa (mkMinj 1 M)\<rbrakk> \<Longrightarrow> P (PX Pa i Q) (MX j M)" shows "P a0 a1"
symmetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "Q \<approx> P"
finite_comm_group_axioms: shows "finite_comm_group (G\<lparr>carrier := {\<one>}\<rparr>)"
step: fixes Y :: "'a set" assumes "countable Y" shows "countable (F Y)"
algebraic_imp_algebraic_int': fixes p :: "int poly"   and x :: "'a"   and c :: "int" assumes "ipoly p x = (0::'a)"   and "p \<noteq> 0"   and "Polynomial.lead_coeff p dvd c" shows "algebraic_int (of_int c * x)"
simps_2: fixes P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" shows "small_nstep P \<sigma> (Suc n) = {\<sigma>2. \<exists>\<sigma>1. \<sigma>1 \<in> small_nstep P \<sigma> n \<and> \<sigma>2 \<in> small P \<sigma>1}"
Inf_from_def: fixes N :: "'f set" shows "Inf_from N = {\<iota> \<in> Inf. set (prems_of \<iota>) \<subseteq> N}"
pfun_of: fixes x :: "('a, 'b) ffun" shows "pfun_of x \<in> {f. finite (pdom f)}"
Dag_conv_isDag_dag: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" shows "Dag p l r t = (isDag p l r \<and> t = dag p l r)"
assms_3: shows "inj_on f U\<^sub>M\<^sub>_\<^sub>1"
differentiable_image_in_sets_lebesgue: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "S \<in> sets lebesgue"   and "DIM('c) \<le> DIM('d)"   and "f differentiable_on S" shows "f ` S \<in> sets lebesgue"
inverse_unique: fixes f :: "'b"   and g :: "'b" assumes "B.inverse_arrows f g" shows "B.inv f = g"
that: fixes ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> defNode g q-ns\<rightarrow>m"   and "EntryPath g ns"   and "q \<in> phiUses g m"   and "m \<in> set (predecessors g (defNode g p))" shows "thesis"
inverse_composition_commute: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c" assumes "Group_Theory.monoid M composition unit"   and "monoid.invertible M composition unit x"   and "monoid.invertible M composition unit y"   and "x \<in> M"   and "y \<in> M" shows "monoid.inverse M composition unit (composition x y) = composition (monoid.inverse M composition unit y) (monoid.inverse M composition unit x)"
greaterThanAtMost_empty_iff2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and l :: "'b" assumes "class.preorder less_eq less" shows "({} = ord.greaterThanAtMost less_eq less k l) = (\<not> less k l)"
vrat_fsingleton_in_fproduct_vrat: fixes a :: "rat" shows "[a\<^sub>\<rat>]\<^sub>\<circ> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ> ^\<^sub>\<times> 1\<^sub>\<nat>"
inf_sup_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<squnion> y) = x"
A_dim_row_not2: shows "dim_row A \<noteq> 2"
elims: fixes x :: "'a Point"   and xa :: "'a Vector"   and y :: "'a Point" assumes "x \<leadsto> xa = y" obtains p :: "'a Point" and u :: "'a Vector" where "x = p" and "xa = u" and "y = \<lparr>tval = tval p + tdir u, xval = xval p + xdir u, yval = yval p + ydir u, zval = zval p + zdir u\<rparr>"
sWlsInp_def: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and delta :: "'opSym"   and sinp :: "'index \<Rightarrow> 'f option" shows "sWlsInp SEM delta sinp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) sinp \<and> liftAll2 (sWls SEM) (arOf delta) sinp"
eq_onp: shows "eq_onp (\<lambda>x. (\<forall>z\<in>set1_G x. P1 z) \<and> (\<forall>z\<in>set2_G x. P2 z)) x y"
p_bot: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus bot = top"
simps_2: fixes s :: "'b set"   and t :: "'b set"   and ts :: "'b set list" shows "intersection_list s (t # ts) = s \<inter> t # intersection_list s ts"
simps_2: fixes x :: "'b set"   and xs :: "'b set list" shows "disjoint_list_rec (x # xs) = (x \<inter> \<Union> (set xs) = {} \<and> disjoint_list_rec xs)"
coin_def: shows "coin \<equiv> \<lparr>Label = STR ''coin'', Arity = 1, Guards = [], Outputs = [Plus (V (R 2)) (V (I 0))], Updates = [(1, V (R 1)), (2, Plus (V (R 2)) (V (I 0)))]\<rparr>"
not_vend: shows "\<not> (a = STR ''vend'' \<and> b = [])"
hom_sum: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and X :: "'c set" shows "c * sum f X = (\<Sum>x\<in>X. c * f x)"
proj_mult_vec: fixes k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (p ** q) = proj_poly k p * proj_poly k q"
isAbr_simps_2: fixes s :: "'a" shows "isAbr (Abrupt s) = True"
csmall_def: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and combine :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and collect_id :: "'c"   and collect :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c"   and P :: "'a"   and \<sigma> :: "'b" assumes "CollectionSemantics small endset combine collect_id" shows "CollectionSemantics.csmall small collect P \<sigma> \<equiv> {(\<sigma>'', coll). \<sigma>'' \<in> small P \<sigma> \<and> collect P \<sigma> \<sigma>'' = coll}"
sup_dres_addsimps_2: fixes x :: "'a dres" shows "sup x dSUCCEEDi = x"
Seq2: fixes c\<^sub>1 :: "com"   and l\<^sub>1 :: "nat"   and c\<^sub>2 :: "com"   and l\<^sub>2 :: "nat" assumes "\<turnstile> c\<^sub>1 : l\<^sub>1"   and "\<turnstile> c\<^sub>2 : l\<^sub>2" shows "\<turnstile> c\<^sub>1;; c\<^sub>2 : min l\<^sub>1 l\<^sub>2"
f1: shows "U\<^sub>f $$ (i, i - 1) * U\<^sub>f $$ (i - 1, j) = of_nat (f (i div 2)) * U\<^sub>f $$ (i - 1, j)"
F_app: assumes "2\<^sub>\<nat> \<in>\<^sub>\<circ> 3\<^sub>\<nat>" shows "F\<lparr>2\<^sub>\<nat>\<rparr> : A\<lparr>2\<^sub>\<nat>\<rparr> \<mapsto>\<^bsub>if 2\<^sub>\<nat> = []\<^sub>\<circ> \<Rightarrow> \<AA> | 2\<^sub>\<nat> = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>\<^esub> B\<lparr>2\<^sub>\<nat>\<rparr>"
intros_2: fixes l :: "'d tree"   and S :: "'d set"   and r :: "'d tree"   and v :: "'d" assumes "l \<in> trees S"   and "r \<in> trees S"   and "v \<in> S" shows "\<langle>l, v, r\<rangle> \<in> trees S"
lim_Ref_alloc: fixes x :: "'a"   and h :: "heap" shows "lim (snd (Ref.alloc x h)) = Suc (lim h)"
oldest_token_rec: fixes q :: "'b"   and n :: "nat" shows "oldest_token q (Suc n) = (if pre_oldest_tokens q n \<noteq> {} then Some (Min (pre_oldest_tokens q n)) else None)"
fdia_d_simp: fixes x :: "'a"   and y :: "'a" shows " |x\<rangle> d y =  |x\<rangle> y"
zcount_ne_zero_iff: fixes M :: "'c zmultiset"   and x :: "'c" shows "(zcount M x \<noteq> 0) = (x \<in>#\<^sub>z M)"
state_measure_var_type: fixes \<sigma> :: "nat \<Rightarrow> val"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat" assumes "\<sigma> \<in> space (state_measure V \<Gamma>)"   and "x \<in> V" shows "val_type (\<sigma> x) = \<Gamma> x"
nop_onto_axioms_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_onto_axioms A n f \<equiv> n \<in>\<^sub>\<circ> \<omega> \<and> \<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n \<and> \<R>\<^sub>\<circ> f = A"
fcomp: shows "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S"
ub1: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
nat_pow_0: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
FIXME_fiddle1: fixes x :: "'a"   and Y :: "'a set"   and X :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> Y"   and "X \<subseteq> Y"   and "refl_on Y r" shows "insert x (X \<inter> r `` {x}) = insert x X \<inter> r `` {x}"
that: fixes l :: "nat" assumes "k dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> l" shows "thesis"
coset_mult_assoc: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and g :: "'c"   and h :: "'c" assumes "Group.group G"   and "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M #>\<^bsub>G\<^esub> g #>\<^bsub>G\<^esub> h = M #>\<^bsub>G\<^esub> g \<otimes>\<^bsub>G\<^esub> h"
simulation_steps'_map: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and as :: "'a list"   and b :: "'b" assumes "\<And>a b. a \<sim> b \<Longrightarrow> b = f a"   and "A.steps (a # as)"   and "a \<sim> b"   and "PA a"   and "PB b" shows "\<exists>bs. B.steps (b # bs) \<and> bs = map f as \<and> list_all2 (\<sim>) as bs \<and> list_all PA as \<and> list_all PB bs"
sup_compl_top_left2: fixes x :: "'a"   and y :: "'a" shows "x + (- x + y) = 1"
image_empty: fixes f :: "'g \<Rightarrow> 'f" shows "f ` {} = {}"
no_inv_on_left_moving_norm_Bk: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_left_moving_norm (as, lm) (s, l, Bk # r) ires = False"
hEqFG': shows "frechet I ($f f args) \<nu> = FunctionFrechet I f (\<chi>i. sterm_sem I (args i) \<nu>) \<circ> (\<lambda>v. \<chi>i. frechet I (args i) \<nu> v)"
leq: shows "(\<Sum>(a, n)\<leftarrow>C. lec_of_nsc (a *R n)) = Le_Constraint Leq_Rel 0 c"
rec_fact_aux_def: shows "rec_fact_aux = PR (constn 1) (CN rec_mult [CN S [recf.Id 3 0], recf.Id 3 1])"
commute: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" shows "sup_prod a b = sup_prod b a"
split_bintrunc: fixes n :: "nat"   and c :: "int"   and a :: "int"   and b :: "int" assumes "bin_split n c = (a, b)" shows "b = take_bit n c"
measurable_stock_measure_val_type: fixes f :: "'a \<Rightarrow> val"   and M :: "'a measure"   and t :: "pdf_type"   and x :: "'a" assumes "f \<in> M \<rightarrow>\<^sub>M stock_measure t"   and "x \<in> space M" shows "val_type (f x) = t"
that: fixes a :: "'a \<times> bool" assumes "a \<in> set as"   and "a = (v, True)" shows "thesis"
bernoulli_denom_pos: fixes n :: "nat" shows "0 < bernoulli_denom n"
gt_ex: fixes x :: "'a" shows "\<exists>y. x \<sqsubset> y"
KL_div_def: fixes S :: "'b set"   and a :: "'b \<Rightarrow> real"   and d :: "'b \<Rightarrow> real" shows "KL_div S a d = (\<Sum>i\<in>S. a i * log b (a i / d i))"
almost0_in_R1: fixes vs :: "nat list" shows "vs \<odot> 0\<^sup>\<infinity> \<in> \<R>"
smooth_on_add_fun: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (f + g)"
Basic: fixes a :: "'d set"   and A :: "'d set set"   and sp :: "'d set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
n2_is: shows "N2 = N \<union> {(C, active)} \<union> M"
STL5: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<and> G) = (\<box>F \<and> \<box>G)"
simps: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" shows "rb_aux_inv2 (bs, ss, ps) = (rb_aux_inv (bs, ss, ps) \<and> (\<forall>j<length fs. Inr j \<notin> set ps \<longrightarrow> fs ! j \<in> ideal (rep_list ` set (filter (\<lambda>b. component_of_term (lt b) < Suc j) bs)) \<and> (\<forall>b\<in>set bs. component_of_term (lt b) < j \<longrightarrow> (\<exists>s\<in>set ss. s adds\<^sub>t term_of_pair (punit.lt (rep_list b), j)))))"
initSets_def: fixes s :: "('e, 'f, 'g, 'h) SLin_state_scheme" shows "initSets s \<equiv> {ivs. ivs \<noteq> {} \<and> ivs \<subseteq> initVals s}"
map_of_zip_is_None: fixes xs :: "'d list"   and ys :: "'e list"   and x :: "'d" assumes "length xs = length ys" shows "(map_of (zip xs ys) x = None) = (x \<notin> set xs)"
path_split_3: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node"   and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n (as @ a # as') n'" shows "While_CFG.path prog (targetnode a) as' n'"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<rho>\<sigma>_\<tau>.map a = B (\<tau> a) (\<rho>\<sigma>.map a)"
distinct_fst: fixes a :: "('b \<times> 'c) list" assumes "distinct (map fst a)" shows "distinct a"
closed_orbit_periodic: fixes x :: "'a" assumes "closed_orbit x"   and "f x \<noteq> (0::'a)" shows "periodic_orbit x"
removing_offending_flows_makes_invariant_hold: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "\<forall>F\<in>SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP. sinvar (delete_edges G F) nP"
tfr\<^sub>s\<^sub>e\<^sub>t_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and M :: "('a, 'b) Term.term set" assumes "typed_model arity public Ana \<Gamma>" shows "typed_model.tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M \<equiv> \<forall>s\<in>typed_model.SMP arity Ana \<Gamma> M - range Var. \<forall>t\<in>typed_model.SMP arity Ana \<Gamma> M - range Var. (\<exists>\<delta>. Unifier \<delta> s t) \<longrightarrow> \<Gamma> s = \<Gamma> t"
False: shows "\<not> ((Fun f ss, t) \<in> S \<or> (u, Fun f ss) \<in> S \<or> (u, t) \<in> S)"
has_vector_derivative_componentwise_within: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and a :: "real"   and S :: "real set" shows "(f has_vector_derivative f') (at a within S) = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) has_vector_derivative f' \<bullet> i) (at a within S))"
assms: shows "coprime a b \<and> a\<^sup>2 + 3 * b\<^sup>2 = w ^ 3 \<and> odd w"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "linkage \<Gamma> f = (\<forall>x. x \<in> A \<Gamma> \<longrightarrow> d_OUT f x = weight \<Gamma> x)"
reach_nodes: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and i :: "nat" shows "reach A w i \<subseteq> nodes A"
eval\<^sub>G_prop_entailment: fixes S :: "'b ltl set"   and \<phi> :: "'b ltl" shows "S \<Turnstile>\<^sub>P eval\<^sub>G S \<phi> = S \<Turnstile>\<^sub>P \<phi>"
gm_fin: shows "AE x in M. ennreal (f x) * nn_cond_exp M F gm x \<noteq> \<infinity>"
assms: fixes x :: "'a" assumes "x \<in> S" shows "set (coeffs (f x)) \<subseteq> \<real>"
to_vec_def: shows "to_vec \<equiv> map_fun to_fun id (\<lambda>M. \<chi>i j. M i j)"
not_gr_zero: fixes n :: "'a multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
Atm_Wbis: fixes atm :: "'atom" assumes "compatAtm atm" shows "Atm atm \<approx>w Atm atm"
opp_mult_def: fixes x :: "'c"   and y :: "'c" shows "cba_quantale.opp_mult x y \<equiv> y \<sqinter> x"
mset_single_cases: fixes s :: "'c"   and c :: "'c multiset"   and r' :: "'c"   and c' :: "'c multiset" assumes "add_mset s c = add_mset r' c'" obtains "s = r'" and "c = c'"   | "c' = {#s#} + (c' - {#s#})" and "c = {#r'#} + (c - {#r'#})" and "c - {#r'#} = c' - {#s#}"
card_deg_sect: fixes X :: "'c set"   and d :: "nat" assumes "finite X"   and "X \<noteq> {}" shows "card (deg_sect X d) = d + (card X - 1) choose (card X - 1)"
tendsto_within_open: fixes a :: "'d"   and S :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and l :: "'e" assumes "a \<in> S"   and "open S" shows "(f \<longlongrightarrow> l) (at a within S) = f \<midarrow>a\<rightarrow> l"
Y: shows "simple_distributed M Y (\<lambda>x. prob (Y -` {x} \<inter> space M))"
below_ccSquare: fixes G :: "CoCalls"   and S :: "Vars.var set" shows "(G \<sqsubseteq> S\<^sup>2) = (ccField G \<subseteq> S)"
weakly_locally_path_connected_at_def: fixes x :: "'c"   and X :: "'c topology" shows "weakly_locally_path_connected_at x X \<equiv> neighbourhood_base_at x (path_connectedin X) X"
maps_ide_in_hom: fixes a :: "'c" assumes "C.ide a" shows "\<guillemotleft>\<epsilon>o a : GF.map a \<rightarrow>\<^sub>C C.map a\<guillemotright>"
fps_conv_radius_exp: fixes c :: "'a" shows "fps_conv_radius (fps_exp c) = \<infinity>"
l2_inv3_step3: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Nr :: "msg"   and gny :: "msg" shows "{l2_inv3} l2_step3 Ra A B Nr gny {> l2_inv3}"
pol_diff_deg_less: fixes p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "p \<in> carrier R"   and "pol_coeff S c"   and "p = polyn_expr R X (fst c) c"   and "pol_coeff S d"   and "fst c = fst d"   and "snd c (fst c) = snd d (fst d)" shows "p \<plusminus> -\<^sub>a polyn_expr R X (fst d) d = \<zero> \<or> deg_n R S X (p \<plusminus> -\<^sub>a polyn_expr R X (fst d) d) < fst c"
ct_2: fixes s :: "'s" shows "lfp (\<lambda>F s. \<Sqinter>D\<in>K s. \<integral>\<^sup>+ t. g t (F t) \<partial>measure_pmf D) s = \<integral>\<^sup>+ t. g t (lfp (\<lambda>F s. \<Sqinter>D\<in>K s. \<integral>\<^sup>+ t. g t (F t) \<partial>measure_pmf D) t) \<partial>measure_pmf (ct s)"
lead_coeff_comp: fixes q :: "'b poly"   and p :: "'b poly" assumes "0 < degree q" shows "lead_coeff (p \<circ>\<^sub>p q) = lead_coeff p * lead_coeff q ^ degree p"
nterm_to_term_term_to_nterm: fixes t :: "nterm"   and \<Gamma> :: "name list"   and \<Gamma>' :: "name list"   and s :: "name" assumes "frees t |\<subseteq>| fset_of_list \<Gamma>"   and "length \<Gamma> = length \<Gamma>'" shows "alpha_equiv (fmap_of_list (zip \<Gamma> \<Gamma>')) t (fst (run_state (term_to_nterm \<Gamma>' (nterm_to_term \<Gamma> t)) s))"
iff_exI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "(\<exists>x. P x) = (\<exists>x. Q x)"
simps: fixes y :: "'a"   and lx :: "'a"   and ux :: "'a" shows "(y \<in>\<^sub>i Interval lx ux) = (lx \<le> y \<and> y \<le> ux)"
inf_closed: fixes A :: "'b set" assumes "A \<subseteq> carrier \<Y>" shows "\<Sqinter>\<^bsub>\<Y>\<^esub>A \<in> carrier \<Y>"
reflect_poly_smult: fixes c :: "'a"   and p :: "'a poly" shows "reflect_poly (Polynomial.smult c p) = Polynomial.smult c (reflect_poly p)"
simps_3: fixes P :: "'state"   and f :: "'effect"   and \<phi> :: "'pred" shows "FL_valid_Tree P (tPred f \<phi>) = \<langle>f\<rangle>P \<turnstile> \<phi>"
finite_comm_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "finite_comm_group G \<equiv> finite_group G \<and> comm_group G"
assms_1: assumes "cmp u v = Lt" shows "cmp (splus t u) (splus t v) = Lt"
induct: fixes x1 :: "nat"   and x2 :: "com"   and P :: "nat \<Rightarrow> com \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>l. P l SKIP"   and "\<And>x a l. \<lbrakk>sec a \<le> sec x; l \<le> sec x\<rbrakk> \<Longrightarrow> P l (x ::= a)"   and "\<And>l c\<^sub>1 c\<^sub>2. \<lbrakk>l \<turnstile> c\<^sub>1; P l c\<^sub>1; l \<turnstile> c\<^sub>2; P l c\<^sub>2\<rbrakk> \<Longrightarrow> P l (c\<^sub>1;; c\<^sub>2)"   and "\<And>b l c\<^sub>1 c\<^sub>2. \<lbrakk>max (sec b) l \<turnstile> c\<^sub>1; P (max (sec b) l) c\<^sub>1; max (sec b) l \<turnstile> c\<^sub>2; P (max (sec b) l) c\<^sub>2\<rbrakk> \<Longrightarrow> P l (IF b THEN c\<^sub>1 ELSE c\<^sub>2)"   and "\<And>b l c. \<lbrakk>max (sec b) l \<turnstile> c; P (max (sec b) l) c\<rbrakk> \<Longrightarrow> P l (WHILE b DO c)" shows "P x1 x2"
strict_iff_order: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b \<subset># a) = (b \<subseteq># a \<and> a \<noteq> b)"
is_run_def: fixes s :: "nat \<Rightarrow> 'a" shows "is_run s \<equiv> \<exists>l. is_lrun l s"
map_filter_Some: fixes P :: "'d \<Rightarrow> 'c option"   and x :: "'d"   and y :: "'c" assumes "P x = Some y" shows "List.map_filter P [x] = [y]"
cod_in_codomains: fixes f :: "'a" assumes "C.codomains f \<noteq> {}" shows "C.cod f \<in> C.codomains f"
surjective: fixes r :: "'a DiskBlock_scheme" shows "r = \<lparr>mbal = mbal r, bal = bal r, inp = inp r, \<dots> = DiskBlock.more r\<rparr>"
lsorted_LCons': fixes x :: "'a"   and xs :: "'a llist" shows "lsorted (LCons x xs) = (\<not> lnull xs \<longrightarrow> x \<le> lhd xs \<and> lsorted xs)"
permutes_imp_bij: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "bij_betw p S S"
conts'': fixes i :: "'a" shows "\<exists>\<delta>>0. \<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)"