hcju/isabelleps huggingface dataset:
assms_2: fixes \<Gamma> :: "nat \<Rightarrow> nat option"   and x :: "nat"   and C :: "nat"   and CT :: "nat \<Rightarrow> classDef option" assumes "\<Gamma> x = Some C" shows "P CT \<Gamma> (Var x) C"
PivotUpdate_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" shows "PivotUpdate eq_idx_for_lvar pivot update \<equiv> Pivot eq_idx_for_lvar pivot \<and> Update update"
that: fixes as' :: "'edge list" assumes "n -as'\<rightarrow>\<^sub>\<iota>* n'"   and "slice_edges S cs as' = []" shows "thesis"
real_lim_sequentially: fixes f :: "nat \<Rightarrow> complex"   and l :: "complex" assumes "f \<longlonglongrightarrow> l"   and "\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>" shows "l \<in> \<real>"
single_def: fixes x :: "'a" shows "Predicate.single x = pred.Pred ((=) x)"
mult_divide_mult_cancel_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
add_mset_commute: fixes x :: "'b"   and y :: "'b"   and M :: "'b multiset" shows "x, y, M = y, x, M"
INF_inf_distrib: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set"   and g :: "'c::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) ((Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set))) (Inf ((g::'c::type \<Rightarrow> 'b::type) ` A)) = Inf ((\<lambda>a::'c::type. inf (f a) (g a)) ` A)"
coprime_commute: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one b a = algebraic_semidom.coprime times one a b"
simps_7: fixes a :: "Ferrack.num" shows "bound0 (Eq a) = numbound0 a"
lincomb_list_as_mat_mult: fixes ws :: "'b vec list"   and n :: "nat"   and c :: "nat \<Rightarrow> 'b" assumes "\<forall>w\<in>set ws. dim_vec w = n" shows "module.lincomb_list (module_vec TYPE('b) n) c ws = mat_of_cols n ws *\<^sub>v vec (length ws) c"
acc_induct_rule: fixes a :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<And>y. (y, x) \<in> r \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
WhileRedsThrow: fixes b :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and c :: "expr" assumes "P,E \<turnstile> \<langle>b,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>while (b) c,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
simps_6: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e error_result \<Rightarrow> 'c"   and x2 :: "'e error_result" shows "(case Rerr x2 of Rval x \<Rightarrow> f1 x | Rerr x \<Rightarrow> f2 x) = f2 x2"
select_convs_5: fixes types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "memory \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = memory"
count_mset_set_2: fixes A :: "'a set"   and x :: "'a" assumes "infinite A" shows "count (mset_set A) x = 0"
icd_pd_intermediate: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and j :: "nat" assumes "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "k < j"   and "j \<le> i" shows "\<pi> i pd\<rightarrow> \<pi> j"
abst_trans_fm: fixes name :: "name"   and A :: "fm" shows "abst_dbfm name 0 (trans_fm [] A) = trans_fm [name] A"
indRelRp_indRelR_eq: fixes Enc :: "'a \<Rightarrow> 'b" shows "encoding.indRelRp Enc = (\<lambda>x xa. (x, xa) \<in> encoding.indRelR Enc)"
assms_3: shows "z \<notin> circline_set x_axis"
inject_3: fixes x3 :: "'b"   and y3 :: "'b" shows "(Rule x3 = Rule y3) = (x3 = y3)"
t\<^sub>2\<^sub>3: shows "instance_of\<^sub>t t\<^sub>2 t\<^sub>3"
t_3: shows "v \<in> fv (t \<cdot> \<theta>)"
param_nth: fixes i' :: "nat"   and l' :: "'c list"   and i :: "nat"   and l :: "'d list"   and R :: "('d \<times> 'c) set" assumes "i' < length l'"   and "(i, i') \<in> nat_rel"   and "(l, l') \<in> \<langle>R\<rangle>list_rel" shows "(l ! i, l' ! i') \<in> R"
word_size_neg: fixes w :: "32 Word.word" shows "size (- w) = size w"
Cauchy_Schwarz_ineq2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
suf_power: fixes t :: "'b list"   and u :: "'b list"   and k :: "nat" assumes "t \<noteq> \<epsilon>"   and "u \<le>s t \<^sup>@ k" shows "\<exists>m. t \<^sup>@ m \<le>s u \<and> u <s t \<cdot> t \<^sup>@ m"
diagram_axioms: shows "diagram J (\<cdot>\<^sub>C) E"
assms_1: shows "action_needed_asses a s2 \<subseteq>\<^sub>f action_needed_asses a s1"
eq_nodes_def: shows "eq_nodes = (\<lambda>M v p q. \<nexists>n. dist_nodes M n v p q)"
shuffle_to_top_linear: fixes Y :: "'x set"   and d :: "'d" assumes "inj_on Xd Y" shows "Linear_order (shuffle_to_top Y d)"
Hoare_def: fixes p :: "'a"   and S :: "'b \<Rightarrow> 'a"   and q :: "'b" shows "\<Turnstile> p{| S |}q = (p \<sqsubseteq> S q)"
irreducible\<^sub>d_reflect_poly_main: fixes f :: "'a poly" assumes "coeff f 0 \<noteq> (0::'a)"   and "irreducible\<^sub>d (reflect_poly f)" shows "irreducible\<^sub>d f"
strips_state_to_state_inverse_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom s. the (s v) \<in> \<R>\<^sub>+ \<Psi> v" shows "s = \<phi>\<^sub>S\<inverse> \<Psi> \<phi>\<^sub>S \<Psi> s"
algebraic_int_abs_real: fixes x :: "real" shows "algebraic_int \<bar>x\<bar> = algebraic_int x"
False: shows "\<not> poincare_collinear {x, y, z}"
continuous_trivial_limit: fixes net :: "'c filter"   and f :: "'c \<Rightarrow> 'd" assumes "net = bot" shows "continuous net f"
USdB_decomp_head_tail: fixes c :: "'a"   and V :: "'a list"   and c' :: "'a"   and V' :: "'a list"   and SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" assumes "(c # V, c' # V') \<in> Strong_Security.USdB SR DA d" shows "([c], [c']) \<in> Strong_Security.USdB SR DA d \<and> (V, V') \<in> Strong_Security.USdB SR DA d"
One_App: fixes t :: "lam"   and s :: "lam"   and r :: "lam" assumes " App t s \<longrightarrow>\<^sub>1 r" shows "(\<exists>t' s'. r = App t' s' \<and>  t \<longrightarrow>\<^sub>1 t' \<and>  s \<longrightarrow>\<^sub>1 s') \<or> (\<exists>x p p' s'. r = p'[x::=s'] \<and> t = Lam [x].p \<and>  p \<longrightarrow>\<^sub>1 p' \<and>  s \<longrightarrow>\<^sub>1 s' \<and> x \<sharp> (s, s'))"
map_bit_interval_eq: fixes w :: "'a word"   and n :: "nat" shows "map (bit w) [0..<n] = takefill False n (rev (to_bl w))"
le_degree: fixes p :: "'a Polynomial.poly"   and n :: "nat" assumes "poly.coeff p n \<noteq> (0::'a)" shows "n \<le> Polynomial.degree p"
bisim: shows "weak_reduction_bisimulation Rel (STCal Source Target)"
eq': shows "(\<Sum>m\<in>carrier G. \<Phi> m * unity_root d (int m)) = sum (sum (\<lambda>m. \<Phi> m * unity_root d (int m))) (rcosets\<^bsub>G\<^esub> kernel G H f)"
cardinalityOneTheElemIdentity: fixes X :: "'b set" shows "(card X = 1) = (X = {the_elem X})"
drop_0: fixes xs :: "'c list" shows "drop 0 xs = xs"
generalizes_refl: fixes C :: "'a literal multiset" shows "generalizes C C"
select_convs_1: fixes states :: "'a \<Rightarrow> 'b"   and msgs :: "('a, 'c) message \<Rightarrow> nat"   and more :: "'d" shows "states \<lparr>states = states, msgs = msgs, \<dots> = more\<rparr> = states"
round_ivl_correct: fixes A :: "float interval"   and prec :: "nat" shows "set_of A \<subseteq> set_of (round_interval prec A)"
subst_size_emv: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'c) Term.term"   and \<tau> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "s = t \<cdot> \<tau>"   and "num_syms s = num_syms t"   and "num_funs s = num_funs t" shows "emv s t"
hyps: fixes tab :: "'a \<Rightarrow> 'b option"   and tab' :: "'a \<Rightarrow> 'b option"   and ys :: "'b list" assumes "tab z = tab' z" shows "(tab(xs [\<mapsto>] ys)) z = (tab'(xs [\<mapsto>] ys)) z"
first_def: fixes s :: "nat \<Rightarrow> 'b" shows "first s \<equiv> s 0"
countableE_bij: fixes S :: "'c set" assumes "countable S" obtains f :: "nat \<Rightarrow> 'c" and C :: "nat set" where "bij_betw f C S"
not_call_simps_6: fixes l :: "'d"   and fips :: "'a \<Rightarrow> nat set"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}groupcast(fips, fmsg) . p)"
parts_msg_parts: fixes A :: "agent"   and X :: "msg"   and s :: "(agent \<times> msg) set"   and Y :: "msg" assumes "(A, X) \<in> s"   and "Y \<in> parts_msg X" shows "Y \<in> parts (used s)"
qe_eq_repeat_eval': fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (qe_eq_repeat var L F) (xs @ \<Gamma>))"
abs_triangle_ineq3: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus (abs a) (abs b))) (abs (minus a b))"
invar_NodeR2: fixes l :: "('c \<times> nat) tree"   and x :: "'c"   and n :: "nat"   and r :: "('c \<times> nat) tree"   and r' :: "('c \<times> nat) tree" assumes "invar \<langle>l, (x, n), r\<rangle>"   and "sngl r'"   and "n = lvl r + 1"   and "invar r'"   and "lvl r' = n" shows "invar \<langle>l, (x, n), r'\<rangle>"
awalk_hd_in_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "u \<in> pverts (mk_symmetric G)"
mv_boxes_correct: fixes aa :: "nat"   and n :: "nat"   and ba :: "nat"   and lm1 :: "nat list"   and lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "aa + n \<le> ba"   and "aa < ba"   and "length lm1 = aa"   and "length lm2 = n"   and "length lm3 = ba - aa - n" shows "{\<lambda>nl. nl = lm1 @ lm2 @ lm3 @ 0 \<up> n @ lm4} mv_boxes aa ba n {\<lambda>nl. nl = lm1 @ 0 \<up> n @ lm3 @ lm2 @ lm4}"
z_def: shows "f x < ereal z \<and> ereal z < a"
cupcake_alist_to_ns_preserve: fixes env :: "(char list \<times> v) list" assumes "list_all (is_cupcake_value \<circ> snd) env" shows "is_cupcake_ns (alist_to_ns env)"
desargues_config_not_col_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col A A' B'"
not_\<tau>diverge_to_no_\<tau>move: fixes s :: "'s1" assumes "\<not> s -\<tau>1\<rightarrow> \<infinity>" shows "\<exists>s'. s -\<tau>1\<rightarrow>* s' \<and> (\<forall>s''. \<not> s' -\<tau>1\<rightarrow> s'')"
ins_invariant: fixes x :: "nat \<times> nat"   and S :: "nat option list list \<times> (nat \<times> nat) list" assumes "prod_is_node A B x"   and "prod_invariant A B S"   and "\<not> prod_memb x S" shows "prod_invariant A B (prod_ins x S)"
mat_mult_var: fixes f :: "'b \<Rightarrow> 'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c \<Rightarrow> 'd" shows "f \<otimes> g = (\<lambda>i j. \<Sum> {f i k * g k j |k. k \<in> UNIV})"
wp_iff: fixes A :: "'b set"   and act :: "('b \<times> 'b) set"   and B :: "'b set" shows "(A \<subseteq> wp act B) = (act `` A \<subseteq> B)"
monoseq_realpow: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "monoseq ((^) x)"
add: fixes a :: "nat"   and b :: "nat" shows "P a b \<longrightarrow> P a (a + b)"
odd_def: shows "EvenOdd.odd = IF (\<lambda>s. s<Arg> = 0) THEN Loc Res:==(\<lambda>s. 1) ELSE (Loc Res:=CALL Even(\<lambda>s. s<Arg> - 1))"
div_dvd_self: fixes b :: "'b"   and a :: "'b" assumes "b dvd a" shows "a div b dvd a"
preprm_ext_transitive: fixes x :: "('b \<times> 'b) list"   and y :: "('b \<times> 'b) list"   and z :: "('b \<times> 'b) list" assumes "x =p y"   and "y =p z" shows "x =p z"
sums_confine_mult: fixes f :: "nat \<Rightarrow> 'b"   and N :: "nat set"   and a :: "'b"   and c :: "'b" assumes "confine f N sums a" shows "confine (\<lambda>n. c * f n) N sums (c * a)"
assms_4: shows "ipd (\<pi> i) = ipd (\<pi> k)"
assms_2: shows "a \<in>\<^sub>\<circ> \<alpha>"
assoc'_naturality\<^sub>A\<^sub>W\<^sub>C: fixes \<tau> :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "\<tau> \<star> \<mu> \<noteq> null"   and "\<mu> \<star> \<nu> \<noteq> null" shows "local.inv \<a>[cod \<tau>, cod \<mu>, cod \<nu>] \<cdot> (\<tau> \<star> \<mu> \<star> \<nu>) = ((\<tau> \<star> \<mu>) \<star> \<nu>) \<cdot> local.inv \<a>[local.dom \<tau>, local.dom \<mu>, local.dom \<nu>]"
is_interval_simply_connected_1: fixes S :: "real set" shows "is_interval S = simply_connected S"
ContinuousD: fixes H :: "'d upred \<Rightarrow> 'd upred"   and A :: "'d upred set" assumes "Continuous H"   and "\<not> A = {}" shows "H (\<Or>A) = \<Or>(H ` A)"
nonempty_has_size: fixes S :: "'k multiset" shows "(S \<noteq> {#}) = (0 < size S)"
simps_4: fixes n :: "nat"   and a :: "'a atom" shows "local.wf n (Atom a) = wf_atom \<Sigma> n a"
cnj_Ln: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "cnj (Ln z) = Ln (cnj z)"
vec_vec_Tensor_right_empty: fixes x :: "'a list" shows "vec_vec_Tensor x [] = []"
inf_bot_left: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and x :: "'b" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf bot x = bot"
acyclic_subset: fixes s :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set" assumes "acyclic s"   and "r \<subseteq> s" shows "acyclic r"
extends_to_elementary_category_with_binary_products: shows "elementary_category_with_binary_products (\<cdot>) pr0 pr1"
set_iterator_linord_def: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "'b set" shows "set_iterator_linord it S0 \<equiv> set_iterator_genord it S0 (\<le>)"
rev_permutations_of_set: fixes A :: "'a set" shows "rev ` permutations_of_set A = permutations_of_set A"
B: shows "real (sum ((^) m) {..k}) = real (m ^ Suc k - 1) / real (m - 1)"
WHILE_PARALLEL_IND: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and p :: "L"   and G :: "Var \<Rightarrow> L"   and q :: "L"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" assumes "X \<rhd> c : Sec (Q p G) (EQ G q) (EQ G q) \<Phi>"   and "\<forall>s ss. EQ G p s ss \<longrightarrow> evalB b s = evalB b ss"   and "LEQ p q" shows "X \<rhd> While b c : Sec (Q p G) (EQ G q) (EQ G q) (\<lambda>(s, t). (b, EQ G q, \<Phi>, s, t) \<in> var)"
on_line_def: fixes p :: "p_point"   and a :: "p_point"   and b :: "p_point" shows "p_on_line p a b = (p_between p a b \<or> p_between a p b \<or> p_between a b p)"
at_least_at_most_closed: fixes a :: "'a"   and b :: "'a" shows "\<lbrace>a..b\<rbrace> \<subseteq> carrier L"
ag_plus_zero: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier A"   and "y \<in> carrier A" shows "(x = -\<^sub>a y) = (x \<plusminus> y = \<zero>)"
ord_option_map2: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and f :: "'e \<Rightarrow> 'd"   and y :: "'e option" shows "ord_option ord x (map_option f y) = ord_option (\<lambda>x y. ord x (f y)) x y"
not_evaluate_decs_timeout: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and ds :: "dec list" assumes "\<forall>r. \<not> evaluate_decs False mn env s ds r" shows "\<exists>r. evaluate_decs True mn env s ds r \<and> snd r = Rerr (Rabort Rtimeout_error)"
Diagonalize_preserves_Can: fixes t :: "'c local.term" assumes "Can t" shows "Can \<^bold>\<lfloor>t\<^bold>\<rfloor>"
continuous_on_min: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on A f"   and "continuous_on A g" shows "continuous_on A (\<lambda>x. min (f x) (g x))"
is_empty_min_max: fixes l1 :: "int extended"   and h1 :: "int extended"   and l2 :: "int extended"   and h2 :: "int extended" assumes "\<not> is_empty_rep (l1, h1)"   and "\<not> is_empty_rep (l2, h2)" shows "\<not> is_empty_rep (min l1 l2, max h1 h2)"
simps_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Lt x2 = Lt y2) = (x2 = y2)"
13: shows "infs (even \<circ> f) (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p)))"
sel: fixes x2 :: "'d" shows "the \<lfloor>x2\<rfloor> = x2"
index_mat_swapcols_2: fixes k :: "nat"   and l :: "nat"   and A :: "'d mat" shows "dim_row (swapcols k l A) = dim_row A"
uvw_arcs1_1: shows "(u, w) \<in> parcs G"
Minimal_Solutions_gt0: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b" shows "zeroes (length x) <\<^sub>v x"
simps_3: fixes \<rho> :: "(nat \<times> exp) list"   and x :: "nat"   and e :: "exp" shows "psubst \<rho> (ELam x e) = ELam x (psubst ((x, EVar x) # \<rho>) e)"
subdivFace_pres_finals: fixes f :: "face"   and g :: "graph"   and f' :: "face"   and "is" :: "nat option list" assumes "f \<in> set (finals g)"   and "\<not> final f'" shows "f \<in> set (finals (subdivFace g f' is))"
io_targets_reachable: fixes q2 :: "'d"   and M :: "('e, 'f, 'd) FSM"   and q1 :: "'d"   and io :: "('e \<times> 'f) list" assumes "q2 \<in> io_targets M q1 io" shows "q2 \<in> reachable M q1"
simps_2: fixes v :: "int" shows "Num_arc v * MinInfty = MinInfty"
Interleaves_all_nil: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list" shows "xs \<cong> {xs, [], P} = (\<forall>n<length xs. P (xs ! n) (drop (Suc n) xs))"
assms: shows "f k (0::'c) (0::'d) = (0::'a)"
c_def: shows "c \<equiv> lead_coeff g ^ (Suc (degree f) - degree g)"
fmlookup_of_list: fixes m :: "('b \<times> 'c) list" shows "fmlookup (fmap_of_list m) = map_of m"
fU: fixes n :: "nat" shows "f absolutely_integrable_on \<Union> (s ` {..n})"
starfun_star_of: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(*f* f) (star_of x) = star_of (f x)"
norm_pths_2: fixes x :: "'b"   and y :: "'b" shows "(x \<noteq> y) = (\<not> norm (x - y) \<le> 0)"
leading_monomial_tail: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "p = monomial (punit.lc p) (punit.lt p) + punit.tail p"
domain_invol: fixes x :: "'a" shows "ad (ad (ad (ad x))) = ad (ad x)"
mono_bal_i: fixes n :: "nat"   and h :: "nat"   and n' :: "nat"   and h' :: "nat" assumes "bal_i n h"   and "n \<le> n'"   and "h' \<le> h" shows "bal_i n' h'"
hyps: shows "s \<star> rs = s \<star> filter (\<lambda>x. x \<noteq> r) rs"
dnorm_inter: fixes Tr\<^sub>1 :: "'a \<Rightarrow> 'c set"   and Tr\<^sub>2 :: "'b \<Rightarrow> 'c set"   and Up\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and s\<^sub>1 :: "'a"   and Up\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and s\<^sub>2 :: "'b" assumes "\<forall>s\<^sub>1 s\<^sub>2. Tr\<^sub>1 s\<^sub>1 \<inter> Tr\<^sub>2 s\<^sub>2 = {}" shows "((P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>1,Up\<^sub>1\<rbrakk>) s\<^sub>1|||(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>2,Up\<^sub>2\<rbrakk>) s\<^sub>2) = (P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>\<lambda>s. case s of (s\<^sub>1, s\<^sub>2) \<Rightarrow> Tr\<^sub>1 s\<^sub>1 \<union> Tr\<^sub>2 s\<^sub>2,\<lambda>s. case s of (s\<^sub>1, s\<^sub>2) \<Rightarrow> \<lambda>e. if e \<in> Tr\<^sub>1 s\<^sub>1 then (Up\<^sub>1 s\<^sub>1 e, s\<^sub>2) else if e \<in> Tr\<^sub>2 s\<^sub>2 then (s\<^sub>1, Up\<^sub>2 s\<^sub>2 e) else (s\<^sub>1, s\<^sub>2)\<rbrakk>) (s\<^sub>1, s\<^sub>2)"
proots_count_smult: fixes a :: "'a"   and p :: "'a poly"   and s :: "'a set" assumes "a \<noteq> (0::'a)" shows "proots_count (smult a p) s = proots_count p s"
elims_2: fixes x :: "('a, 'b) Term.term" assumes "ground x" obtains f :: "'a" and ts :: "('a, 'b) Term.term list" where "x = Fun f ts" and "Ball (set ts) ground"
conv_radius_geI_ex': fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'a" assumes "\<And>r. \<lbrakk>0 < r; ereal r < R\<rbrakk> \<Longrightarrow> summable (\<lambda>n. f n * of_real r ^ n)" shows "R \<le> conv_radius f"
left_le_right: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "Optimal_BST2 w"   and "Wellfounded.accp (Optimal_BST2.opt_bst2_rel w) (i, j)" shows "(i = j \<longrightarrow> root (Optimal_BST2.opt_bst2 w i j) = i) \<and> (i < j \<longrightarrow> root (Optimal_BST2.opt_bst2 w i j) \<in> {root (Optimal_BST2.opt_bst2 w i (j - 1))..root (Optimal_BST2.opt_bst2 w (i + 1) j)})"
span_alt: fixes B :: "'b set" shows "span B = {\<Sum>x | f x \<noteq> (0::'a). f x *s x |f. {x. f x \<noteq> (0::'a)} \<subseteq> B \<and> finite {x. f x \<noteq> (0::'a)}}"
simps_2: fixes x :: "('a, 'b) Combinators" shows "list2FWpolicy [x] = x"