hcju/isabelleps huggingface dataset:
simps: fixes f :: "natural \<Rightarrow> (bool \<times> term list) option"   and d :: "natural"   and i :: "natural" shows "Quickcheck_Exhaustive.exhaustive_natural' f d i = (if d < i then None else Quickcheck_Exhaustive.orelse (f i) (Quickcheck_Exhaustive.exhaustive_natural' f d (i + 1)))"
ntrancl_on_0: assumes "u = v"   and "n = 0"   and "u \<in> verts G" shows "thesis"
eq_refl: fixes x :: "'b word"   and y :: "'b word" assumes "x = y" shows "x \<le>s y"
inorder_baldL: fixes l :: "('c \<times> color) tree"   and a :: "'c"   and r :: "('c \<times> color) tree" shows "Tree2.inorder (baldL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
UN_extend_simps_7: fixes A :: "'c set"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "A - \<Inter> (B ` C) = (\<Union>x\<in>C. A - B x)"
root_hash_T_def: fixes rha :: "'e \<Rightarrow> 'f" shows "root_hash_T rha = root_hash_T' \<circ> map_T\<^sub>m rha id"
assms: shows "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 \<le> n"
distinct_1: fixes x1 :: "int"   and x2 :: "String.literal" shows "Num x1 \<noteq> Str x2"
of_rat_le_1_iff: fixes r :: "rat" shows "(of_rat r \<le> (1::'a)) = (r \<le> 1)"
norm_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "norm one = 1"
borel_measurable_max: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
arg_min_nat_le: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and m :: "'d \<Rightarrow> nat" assumes "P x" shows "m (arg_min m P) \<le> m x"
summandTransition_4: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "hnf P"   and "a \<noteq> x" shows "P \<longmapsto> a<\<nu>x> \<prec> P' = (<\<nu>x>a{x}.P' \<in> summands P)"
iso_lunit: fixes f :: "'a" assumes "S.ide f" shows "S.iso \<l>[f]"
in_frontier_least: fixes x :: "'b"   and M :: "'b zmultiset" assumes "x \<in>\<^sub>A frontier M" shows "\<forall>y. 0 < zcount M y \<longrightarrow> \<not> y < x"
finite_lin_dep: fixes S :: "'c set" assumes "finite S"   and "lin_dep S"   and "S \<subseteq> carrier M" shows "\<exists>a v. a \<in> S \<rightarrow> carrier R \<and> lincomb a S = \<zero>\<^bsub>M\<^esub> \<and> v \<in> S \<and> a v \<noteq> \<zero>"
E2: shows "E2 = (\<lambda>s. 1 + 3 * (nat (s ''b'') - nat (s ''g'')))"
nat_diff_left_cancel_eq2: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k - m = k - n"   and "n < k" shows "m = n"
inf_unit_def: fixes uu :: "unit"   and uv :: "unit" shows "inf uu uv = ()"
ivl_zero_case: assumes "l1 \<le> u1"   and "l2 \<le> u2" shows "l1 \<le> 0 \<and> 0 \<le> u1 \<and> l2 \<le> 0 \<and> 0 \<le> u2 \<or> l1 \<le> 0 \<and> 0 \<le> u1 \<and> 0 \<le> l2 \<or> l1 \<le> 0 \<and> 0 \<le> u1 \<and> u2 \<le> 0 \<or> 0 \<le> l1 \<and> l2 \<le> 0 \<and> 0 \<le> u2 \<or> u1 \<le> 0 \<and> l2 \<le> 0 \<and> 0 \<le> u2 \<or> u1 \<le> 0 \<and> u2 \<le> 0 \<or> u1 \<le> 0 \<and> 0 \<le> l2 \<or> 0 \<le> l1 \<and> u2 \<le> 0 \<or> 0 \<le> l1 \<and> 0 \<le> l2"
induced_morphism_with_id_is_id: fixes x :: "'a" assumes "x \<in> S" shows "\<phi>\<^bsub>S is_open imS.im_sheaf imS.im_sheaf_morphisms is_open imS.im_sheaf imS.im_sheaf_morphisms (identity S) (\<lambda>U. identity (imS.im_sheaf U)) x\<^esub> = identity (stalk.carrier_stalk is_open imS.im_sheaf imS.im_sheaf_morphisms x)"
frontier: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and v :: "'a" assumes "Prim_Invar2_ctd_loc w g r Q \<pi>"   and "\<pi> u = Some v" shows "v \<in> Prim.S r (A Q \<pi>)"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "0 \<le> f x"
ANDLEFT2_elim: fixes x :: "name"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "x:M \<in> ANDLEFT2 (B AND C) (\<parallel>(C)\<parallel>)" obtains x' :: "name" and M' :: "trm" where "M = AndL2 x'.M' x" and "fin (AndL2 x'.M' x) x" and "x':M' \<in> \<parallel>(C)\<parallel>"
path_component_trans: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "path_component S x y"   and "path_component S y z" shows "path_component S x z"
sum_with_closed: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "comm_monoid_add_ow S pls z"   and "g ` A \<subseteq> S" shows "sum_with pls z g A \<in> S"
true_lit_iff: fixes I :: "'a set"   and L :: "'a literal" shows "(I \<Turnstile>l L) = (\<exists>A. L = Pos A \<and> A \<in> I \<or> L = Neg A \<and> A \<notin> I)"
lookup_fresh: fixes z :: "name"   and \<theta> :: "(name \<times> lam) list"   and x :: "name" assumes "z \<sharp> \<theta>"   and "z \<sharp> x" shows "z \<sharp> lookup \<theta> x"
bnds_cos: fixes l :: "float"   and u :: "float"   and prec :: "nat" shows "\<forall>x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {real_of_float lx..real_of_float ux} \<longrightarrow> real_of_float l \<le> cos x \<and> cos x \<le> real_of_float u"
map_prs_aux: fixes R1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs1 :: "'b \<Rightarrow> 'c"   and rep1 :: "'c \<Rightarrow> 'b"   and R2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs2 :: "'d \<Rightarrow> 'e"   and rep2 :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e"   and l :: "'c list" assumes "Quotient3 R1 abs1 rep1"   and "Quotient3 R2 abs2 rep2" shows "map abs2 (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
bdd_above_Int1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c set" assumes "class.preorder less_eq less"   and "preorder.bdd_above less_eq A" shows "preorder.bdd_above less_eq (A \<inter> B)"
assoc: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "sup (sup a b) c = sup a (sup b c)"
alloc_eq: shows "newOID (Rep_Store s\<lparr>vals := (vals (Rep_Store s))(l := y)\<rparr>) = newOID (Rep_Store s)"
greaterThanLessThan_unfold: fixes a :: "'a"   and b :: "'a" shows "{a<..<b} = {x. a < x \<and> x < b}"
is_expansion_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "is_expansion G basis" shows "is_expansion (times F G) basis"
power_not_zero: fixes a :: "'a"   and n :: "nat" assumes "a \<noteq> (0::'a)" shows "a ^ n \<noteq> (0::'a)"
unat_of_nat_len: fixes x :: "nat" assumes "x < 2 ^ LENGTH('b)" shows "unat (word_of_nat x) = x"
omega_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> '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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "omega one = bot"
eventually_le_abs_nonneg: fixes l :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "\<forall>\<^sub>F x in at_top. 0 \<le> l x"   and "\<forall>\<^sub>F x in at_top. l x \<le> f x" shows "\<forall>\<^sub>F x in at_top. l x \<le> \<bar>f x\<bar>"
sub_num_simps_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one num.One num.One = zero"
axioms_1: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "open_cover_of_open_subset X is_open U I C" shows "open_cover_of_subset X is_open U I C"
init_fin_simulation_silent1: fixes t :: "'t"   and sxm1 :: "(status \<times> 'x1) \<times> 'm1"   and sxm2 :: "(status \<times> 'x2) \<times> 'm2"   and sxm1' :: "(status \<times> 'x1) \<times> 'm1" assumes "t \<turnstile> sxm1 \<approx>i sxm2"   and "\<tau>trsys.silent_move (r1.init_fin t) r1.init_fin_\<tau>move sxm1 sxm1'" shows "\<exists>sxm2'. (\<tau>trsys.silent_move (r2.init_fin t) r2.init_fin_\<tau>move)\<^sup>*\<^sup>* sxm2 sxm2' \<and> t \<turnstile> sxm1' \<approx>i sxm2'"
Diff_partition: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "A \<union> (B - A) = B"
not_zero_iA: shows "\<not> is_zero_row_upt_k i k A"
parallel_converter2_comp2_out: fixes conv'' :: "('h, 'j, 'l, 'n) converter"   and conv :: "('i, 'k, 'p, 'q) converter"   and conv' :: "('p, 'q, 'm, 'o) converter" shows "conv'' |\<^sub>= conv \<odot> conv' = (1\<^sub>C |\<^sub>= conv) \<odot> (conv'' |\<^sub>= conv')"
induct: fixes P :: "proper_base \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "proper_base"   and a1 :: "int" assumes "\<And>b x. (\<not> x < get_base b \<Longrightarrow> P (square_base b) x) \<Longrightarrow> P b x" shows "P a0 a1"
sp_cf_def: fixes S :: "('d, 'e) Ring_scheme"   and a :: "'d"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "sp_cf S a c = (fst c, \<lambda>j. a \<cdot>\<^sub>r\<^bsub>S\<^esub> snd c j)"
mult_assoc: fixes a :: "'c \<Rightarrow> ('d, 'c) Term.term"   and b :: "'c \<Rightarrow> ('d, 'c) Term.term"   and c :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "a \<circ>\<^sub>s b \<circ>\<^sub>s c = a \<circ>\<^sub>s (b \<circ>\<^sub>s c)"
typ_of_eq_ax_7: shows "typ_of eq_abstract_rule_ax = Some propT"
vsv_vimage_iff: fixes b :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" shows "(b \<in>\<^sub>\<circ> VLambda Aa f `\<^sub>\<circ> A) = (\<exists>a. a \<in>\<^sub>\<circ> A \<and> a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f) \<and> VLambda Aa f\<lparr>a\<rparr> = b)"
P_def: shows "P = set (take (Suc n) (prefixes z))"
elim_intro: fixes v :: "'b"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "\<And>t. v \<notin> vars_of (t \<lhd> \<sigma>)" shows "elim \<sigma> v"
distr_density_discrete: fixes A :: "'a set"   and f' :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'a" assumes "countable A"   and "f' \<in> borel_measurable M"   and "g \<in> M \<rightarrow>\<^sub>M count_space A"   and "\<And>x. x \<in> space M \<Longrightarrow> g x \<in> A" shows "density (count_space A) (\<lambda>x. \<integral>\<^sup>+ t. (if g t = x then 1 else 0) * f' t \<partial>M) = distr (density M f') (count_space A) g"
mset_zero_iff_right: fixes x :: "'b list" shows "({#} = mset x) = (x = [])"
adopt_node_def: shows "adopt_node \<equiv> l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node get_owner_document get_parent remove_child get_disconnected_nodes set_disconnected_nodes"
extremum_uniqueI: fixes a :: "'c list" assumes "prefix a []" shows "a = []"
vec_lambda_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> UNIV" shows "($) (vec_lambda y) = y"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_diff a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> head t >\<^sub>h\<^sub>d head s \<and> chkvar t s \<and> chksubs (>\<^sub>t) t s)"
even_Suc_Suc_iff: fixes n :: "nat" shows "even (Suc (Suc n)) = even n"
remove_heap_is_wellformed_preserved_1: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "type_wf h'"
equiv_sym: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.equiv eqv"   and "eqv x y" shows "eqv y x"
fterms_of_hterms_def: fixes ts :: "hterm list" shows "fterms_of_hterms ts \<equiv> map fterm_of_hterm ts"
frontier_complement: fixes S :: "'b set" shows "frontier (- S) = frontier S"
well_founded_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "well_founded R \<equiv> wfP R"
set_filterQuery: fixes QQ :: "(nat set \<times> 'a option list set) set"   and I :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set" assumes "QQ = filterQuery I Q"   and "non_empty_query Q" shows "\<forall>X\<in>Q. (1 \<le> card (fst X \<inter> I)) = (X \<in> QQ)"
g_size_abort_impl: shows "map_size_abort \<alpha> invar g_size_abort"
finite_PiE: fixes S :: "'b set"   and T :: "'b \<Rightarrow> 'c set" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)" shows "finite (Pi\<^sub>E S T)"
eval_ctl_HA_def: fixes a :: "('d, 'e, 'f) hierauto"   and f :: "(('d, 'e, 'f) status, ('d, 'e, 'f) atomar) ctl" shows "a |=H= f = HA2Kripke a,InitStatus a |=c= f"
meet_pres_def: fixes X :: "('c, 'd) gorder_scheme"   and Y :: "('e, 'f) gorder_scheme"   and f :: "'c \<Rightarrow> 'e" shows "meet_pres X Y f \<equiv> lattice X \<and> lattice Y \<and> (\<forall>x\<in>carrier X. \<forall>y\<in>carrier X. f (x \<sqinter>\<^bsub>X\<^esub> y) = f x \<sqinter>\<^bsub>Y\<^esub> f y)"
bij_betwI': fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'d set" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y)"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> Y"   and "\<And>y. y \<in> Y \<Longrightarrow> \<exists>x\<in>X. y = f x" shows "bij_betw f X Y"
RootEx_States_Subset: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" assumes "RootEx F G" shows "States (Root F G) \<subseteq> \<Union> (States ` F)"
1: fixes x2 :: "'a \<times> 'a"   and x :: "'a"   and y :: "'a" assumes "unapp t = Some x2"   and "(x, y) = x2" shows "P x"
antisym_finer_than: fixes S :: "'c topology"   and T :: "'c topology" assumes "S finer_than T"   and "T finer_than S" shows "S = T"
has_disc_negated_alist_and'_append: fixes disc' :: "'b \<Rightarrow> bool"   and neg :: "bool"   and ls1 :: "'b negation_type list"   and ls2 :: "'b negation_type list" shows "has_disc_negated disc' neg (alist_and' (ls1 @ ls2)) = (has_disc_negated disc' neg (alist_and' ls1) \<or> has_disc_negated disc' neg (alist_and' ls2))"
isNSCont_isCont: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "isNSCont f a" shows "isCont f a"
not_empty_M: fixes I :: "'b set"   and P :: "'b set \<Rightarrow> ('b \<Rightarrow> 'c) measure"   and M :: "'b \<Rightarrow> 'c measure"   and i :: "'b" assumes "projective_family I P M"   and "i \<in> I" shows "space (M i) \<noteq> {}"
transversal_segmentE: fixes x :: "'a"   and y :: "'a" assumes "rev.transversal_segment x y" obtains "x \<noteq> y" and "{x--y} \<subseteq> X" and "\<And>z. z \<in> {x--y} \<Longrightarrow> 0 < (- f) z \<bullet> rev.rot (x - y)"   | "x \<noteq> y" and "{x--y} \<subseteq> X" and "\<And>z. z \<in> {x--y} \<Longrightarrow> 0 < (- f) z \<bullet> rev.rot (y - x)"
of_int_floor_le: fixes x :: "'c" shows "of_int \<lfloor>x\<rfloor> \<le> x"
abs_eq: shows "rbt_keys_generator = Generator (rbt_has_next, rbt_keys_next)"
h7: shows "(\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
le_g: fixes x :: "'n" assumes "x \<in> S" shows "norm (f x) \<le> g x"
top_boolean_subset_def: shows "top \<equiv> Abs_boolean_subset top"
eval_ztpoly_nonneg: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and p :: "('v pvar, hmultiset) tpoly" assumes "legal_zpassign A" shows "0 \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p)"
axioms_1: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subring H R" shows "subgroup H (add_monoid R)"
wiring: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and cnv :: "('a, 'b, 'c, 'd) converter"   and f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"   and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "wiring \<I> \<I>' cnv (f, g)"
geom_proc_integrable: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "0 \<le> q"   and "q \<le> 1" shows "integrable N (geom_proc n)"
cases_scheme: fixes r :: "('c, 'd) partial_object_scheme" obtains carrier :: "'c set" and more :: "'d" where "r = \<lparr>carrier = carrier, \<dots> = more\<rparr>"
simps: fixes a1 :: "'e"   and a2 :: "'e"   and r :: "('e \<times> 'e) set" shows "((a1, a2) \<in> r\<^sup>+) = ((\<exists>a b. a1 = a \<and> a2 = b \<and> (a, b) \<in> r) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>+ \<and> (b, c) \<in> r))"
LBINT_I0c_exp_mscale_sin: fixes t :: "real"   and u :: "real" shows "LBINT x=0..ereal t. exp (- (u * x)) * sin x = 1 / (1 + u\<^sup>2) * (1 - exp (- (u * t)) * (u * sin t + cos t))"
pr1_def: fixes a :: "'a"   and b :: "'a" shows "pr1 a b \<equiv> if ide a \<and> ide b then fst (SOME x. has_as_binary_product a b (fst x) (snd x)) else null"
not_Some_eq: fixes x :: "'x option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
\<alpha>_gen_build: fixes As :: "'s \<times> natural \<times> natural"   and bound :: "natural"   and n :: "nat" assumes "invar (fst As)" shows "apfst \<alpha> (local.gen_build bound n As) = Benchmark_Comparison.gen_build bound n (apfst \<alpha> As)"
nsplitTree_invpres: fixes n :: "nat"   and s :: "('b, 'c) FingerTreeStruc"   and p :: "'c \<Rightarrow> bool"   and i :: "'c"   and l :: "('b, 'c) FingerTreeStruc"   and nd :: "('b, 'c) Node"   and r :: "('b, 'c) FingerTreeStruc" assumes "is_leveln_ftree n s"   and "is_measured_ftree s"   and "\<not> p i"   and "p (i + gmft s)"   and "nsplitTree p i s = (l, nd, r)" shows "is_leveln_ftree n l \<and> is_measured_ftree l \<and> is_leveln_ftree n r \<and> is_measured_ftree r \<and> is_leveln_node n nd \<and> is_measured_node nd"
L\<^sub>2_def: fixes X :: "'b ltln set"   and Y :: "'b ltln set" shows "L\<^sub>2 X Y = {w. \<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)}"
simps_1: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "sorted_wrt P [] = True"
ord_eq_le_trans: fixes a :: "'d list"   and b :: "'d list"   and c :: "'d list" assumes "a = b"   and "prefix b c" shows "prefix a c"
natUnion_upperbound: fixes f :: "nat \<Rightarrow> 'b set"   and G :: "'b set" assumes "\<And>n. f n \<subseteq> G" shows "natUnion f \<subseteq> G"
rel_interior_def: fixes S :: "'b set" shows "rel_interior S = {x. \<exists>T. openin (top_of_set (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
elims_3: fixes x :: "Core.term"   and xa :: "nat" assumes "\<not> loose_bvar1 x xa" obtains i :: "nat" and k :: "nat" where "x = Bv i" and "xa = k__" and "i \<noteq> k__"   | t :: "Core.term" and u :: "Core.term" and k :: "nat" where "x = t__ $ u" and "xa = ka__" and "\<not> (loose_bvar1 t__ ka__ \<or> loose_bvar1 u ka__)"   | uu_ :: "typ" and t :: "Core.term" and k :: "nat" where "x = Abs uu_ t" and "xa = k" and "\<not> loose_bvar1 t (k + 1)"   | v :: "String.literal" and va :: "typ" and uw_ :: "nat" where "x = Ct v__ va__" and "xa = uw___"   | v :: "variable" and va :: "typ" and uw_ :: "nat" where "x = Fv v va" and "xa = uw_"
stutter_equiv_def: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<tau> :: "nat \<Rightarrow> 'a" shows "\<sigma> \<approx> \<tau> \<equiv> \<natural>\<sigma> = \<natural>\<tau>"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and tcolor :: "tcolor" shows "P (case tcolor of Red \<Rightarrow> f1 | Black \<Rightarrow> f2) = ((tcolor = Red \<longrightarrow> P f1) \<and> (tcolor = Black \<longrightarrow> P f2))"
inverts_components: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<tau>.B.inverse_arrows (\<tau>'.map a) (\<tau>''.map a)"
descendants_block: fixes k :: "'b"   and i :: "'b"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and ps' :: "('c, 'b) fm list" assumes "descendants k i ((ps, a) # branch) xs" shows "descendants k i ((ps' @ ps, a) # branch) xs"
nonzero_divide_divide_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "d \<in> carrier R"   and "b \<noteq> \<zero>"   and "c \<noteq> \<zero>"   and "d \<noteq> \<zero>" shows "a \<oslash> b \<oslash> (c \<oslash> d) = a \<otimes> d \<oslash> (b \<otimes> c)"
simps_2: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
identity_\<o>_def: shows "(\<^bold>=) \<equiv> (\<^bold>=\<^sub>0)"
Trans: shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<tau> \<prec> R"
\<gamma>_inf_rep: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "\<gamma>_rep (inf_rep p1 p2) = \<gamma>_rep p1 \<inter> \<gamma>_rep p2"
stage_def: fixes p :: "fm"   and d :: "int" shows "stage p d \<equiv> \<lambda>(e, c). evaldjf (\<lambda>j. simpfm (\<sigma> p c (Add e (C j)))) [1..c * d]"
simps_3: fixes x :: "'sz"   and y :: "'sz"   and ODE1 :: "('sf, 'sz) ODE"   and ODE2 :: "('sf, 'sz) ODE" shows "OUrename x y (OProd ODE1 ODE2) = OProd (OUrename x y ODE1) (OUrename x y ODE2)"
preorder_on_axioms: shows "Order_Predicates.preorder_on (carrier \<inter> A) (restrict_relation A le)"
rs_def: shows "rs = rs' @ [defNode g \<phi>\<^sub>r]"