hcju/isabelleps huggingface dataset:
lead_coeff_add_le: fixes p :: "'a poly"   and q :: "'a poly" assumes "degree p < degree q" shows "lead_coeff (p + q) = lead_coeff q"
assms_1: shows "lower_on \<sigma> \<eta> (vars_of t)"
is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) document_ptr" shows "is_document_ptr ptr = (case ptr of document_ptr.Ref x \<Rightarrow> True | document_ptr.Ext document_ptr \<Rightarrow> False)"
range_ex1_eq: fixes f :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "inj f" shows "(b \<in> range f) = (\<exists>!x. b = f x)"
DERIV_f: fixes z :: "real" assumes "a < z"   and "z < b" shows "(f has_real_derivative f' z) (at z)"
finfun_const_False_conv_bot: shows "($) (K$ False) = bot"
AbGroup_nth_subset_inner_dirsum: fixes Gs :: "'b set list"   and n :: "nat" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "n < length Gs" shows "Gs ! n \<subseteq> (\<Oplus>G\<leftarrow>Gs. G)"
pOp_assocTr42: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus> b \<plusminus> c \<plusminus> d = a \<plusminus> (b \<plusminus> c) \<plusminus> d"
simple_distributed_nonneg: fixes M :: "'d measure"   and X :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> real"   and x :: "'e" assumes "simple_distributed M X f" shows "0 \<le> f x"
nn_integral_bij_count_space: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "bij_betw g A B" shows "(\<Sum>\<^sup>+ x\<in>A. f (g x)) = integral\<^sup>N (count_space B) f"
diff_less_0_iff_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) zero = less a b"
NMin_2: shows "\<forall>x\<in>{n. \<exists>msg. (enabled (last exec) msg \<and> (\<exists>p. isReceiverOf p msg)) \<and> n < length exec \<and> enabled (execConf n) msg \<and> (\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> execMsg n')}. nMin \<le> x"
image_constant: fixes x :: "'c"   and A :: "'c set"   and c :: "'d" assumes "x \<in> A" shows "(\<lambda>x. c) ` A = {c}"
fresh_subst_E1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "fresh zs z (X #[Y / y]_ys)"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "fresh zs z X"
induct: fixes P :: "'c rexp \<Rightarrow> 'c rexp \<Rightarrow> bool"   and a0 :: "'c rexp"   and a1 :: "'c rexp" assumes "\<And>r. P Zero r"   and "\<And>r. P One r"   and "\<And>r s t. \<lbrakk>P r t; P s t\<rbrakk> \<Longrightarrow> P (Plus r s) t"   and "\<And>s. P Full s"   and "\<And>v s. P (Atom v) s"   and "\<And>v va s. P (Times v va) s"   and "\<And>v s. P (Star v) s"   and "\<And>v s. P (rexp.Not v) s"   and "\<And>v va s. P (rexp.Inter v va) s"   and "\<And>v s. P (Pr v) s" shows "P a0 a1"
sgn2_nz_eq_arg_pos: fixes x :: "nat" shows "(sgn2 x \<noteq> 0) = (x = 0)"
find_zip_3: fixes cond :: "'a \<Rightarrow> bool"   and p :: "'a"   and v :: "'a list"   and s :: "'a"   and x :: "'a"   and y :: "'a" assumes "find (cond \<circ> snd) (zip (p # v) (v @ [s])) = Some (x, y)"   and "v \<noteq> []" shows "x = p \<or> x \<in> set v \<and> \<not> cond x"
lunit_in_hom_2: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "\<guillemotleft>S.lunit f : S.trg f \<star>\<^sub>S f \<Rightarrow>\<^sub>S f\<guillemotright>"
subspace_cone: fixes hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" shows "phull.subspace (cone hU)"
polyn_minus_m_cf: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "-\<^sub>a polyn_expr R X k c = polyn_expr R X k (m_cf S c)"
fcompose_mult: fixes p1 :: "'b poly"   and p2 :: "'b poly"   and q1 :: "'b poly"   and q2 :: "'b poly" shows "fcompose (p1 * p2) q1 q2 = fcompose p1 q1 q2 * fcompose p2 q1 q2"
Rep_nd_fun_inject: fixes x :: "'b nd_fun"   and y :: "'b nd_fun" shows "(Rep_nd_fun x = Rep_nd_fun y) = (x = y)"
coplanar_perm_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar A B D C"
allE_id: fixes y :: "'var"   and \<phi> :: "'fmla" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "prv (all y \<phi>)" shows "prv \<phi>"
lm076: fixes P :: "('c \<times> 'd) set" assumes "runiq P" shows "card (Domain P) = card P"
lmeasurable_cball: fixes a :: "'c"   and r :: "real" shows "cball a r \<in> lmeasurable"
\<pi>_inverse_exists: fixes m :: "nat" shows "\<exists>n. \<pi> (real n) = real m"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) \<rho>"
cterms_subterms: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "cterms \<Gamma> = {uu_. \<exists>p pn. uu_ = p \<and> p \<in> subterms (\<Gamma> pn) \<and> not_call p \<and> not_choice p}"
mkId2b: fixes h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" shows "Pbij_Rng (mkId h) = Dom h"
vectorize_atomize_poly: fixes p :: "unit \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.vectorize_poly (local.punit.atomize_poly p) = p"
deriv: fixes z :: "real" assumes "z \<in> {0..x} - X" shows "(f has_vector_derivative f' z) (at z)"
keys_swap: fixes b :: "nat"   and i :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 'b" shows "keys (swap0 b i m) = swap b i ` keys m"
Restrp_antisymmetric: shows "antisymmetric UNIV ((\<sqsubseteq>) \<restriction> A \<restriction> UNIV)"
bezout_matrix_row_def: fixes A :: "(('f, 'g) vec, 'e) vec"   and a :: "'e"   and b :: "'e"   and j :: "'g"   and bezout :: "'f \<Rightarrow> 'f \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'h"   and x :: "'e" shows "bezout_matrix_row A a b j bezout x = (let (p, q, u, v, d) = bezout (A $ a $ j) (A $ b $ j) in \<chi>y. if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then 1::'d else (0::'d))"
0: fixes u :: "'a local.term" assumes "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "Diag (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) \<and> Dom (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Dom u \<and> Cod (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Cod t"
simps_2: fixes f :: "'fsym"   and Tl :: "'fsym trm list" shows "Ik.tpOf (Fn f Tl) = resOf f"
nth_w2p: fixes n :: "nat"   and m :: "nat" shows "(2 ^ n) !! m = (m = n \<and> m < LENGTH('b))"
linear_span_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and S :: "('a, 'b) vec set" assumes "linear (*s) (*s) f" shows "vec.span (f ` S) = f ` vec.span S"
mult_strict_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "(0::'a) < c" shows "c * a < c * b"
rbt_def: fixes t :: "('a \<times> color) tree" shows "rbt t = (invc t \<and> invh t \<and> color t = Black)"
crel_vs1_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and v :: "'e"   and f :: "'f Heap" shows "crel_vs1 R v f \<equiv> \<forall>heap. P heap \<longrightarrow> (case execute f heap of None \<Rightarrow> False | Some (v', heap') \<Rightarrow> P heap' \<and> (state_dp_consistency.cmem heap \<longrightarrow> R v v' \<and> state_dp_consistency.cmem heap'))"
sat_plan_to_dimacs: fixes sat_plan_formula :: "sat_plan_variable formula"   and M :: "sat_plan_variable \<Rightarrow> bool" assumes "\<And>sv. sv \<in> atoms sat_plan_formula \<Longrightarrow> valid_state_var sv" shows "M \<Turnstile> sat_plan_formula = M \<circ> dimacs_to_var \<Turnstile> map_formula var_to_dimacs sat_plan_formula"
eval_e\<^sub>1: shows "eval_expr t e\<^sub>1 = (D\<^sub>1, g\<^sub>1)"
gcd_dvdI1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c" shows "dvd.dvd times (gcd a b) c"
ex1I: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "\<exists>!x. P x"
fold_map_idx_nth: fixes j :: "nat"   and xs :: "'a list"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'c \<times> 'b"   and i :: "nat"   and y :: "'c" assumes "j < length xs" shows "snd (fold_map_idx f i y xs) ! j = snd (f (i + j) (fst (fold_map_idx f i y (take j xs))) (xs ! j))"
AE_lborel_singleton: fixes c :: "'d" shows "AE x in lborel. x \<noteq> c"
set_of_appr_ex_append1: fixes xrs :: "real list"   and xs :: "'b list"   and b :: "'b list" assumes "xrs \<in> set_of_appr xs" shows "\<exists>r. r @ xrs \<in> set_of_appr (b @ xs)"
assms_2: shows "\<forall>n. f n \<noteq> (0::'a)"
Pair: fixes a :: "cmsg"   and b :: "cmsg"   and c :: "cmsg"   and d :: "cmsg" assumes "eq a b"   and "eq c d" shows "eq (cPair a c) (cPair b d)"
cond_pass: fixes r :: "'b"   and s :: "'a" assumes "C r s" shows "whileLoop C   B r s = (B r >>= whileLoop C   B) s"
check2_def: fixes S :: "'a set"   and I :: "'a set"   and f :: "'a set" shows "(Kripke S I \<turnstile> f) = (I \<subseteq> S \<inter> f)"
cone_decomp_append: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and S1 :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and S2 :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "direct_decomp T [S1, S2]"   and "cone_decomp S1 ps"   and "cone_decomp S2 qs" shows "cone_decomp T (ps @ qs)"
r_rev_aux_prim: shows "prim_recfn 2 r_rev_aux"
list_ex_simps_1: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "list_ex P (x # xs) = (P x \<or> list_ex P xs)"
aux_3: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "\<not> a \<prec> b"   and "\<not> b \<prec> c"   and "a \<prec> c" shows "c = a"
enumFromTo_def: fixes toEnum :: "Integer \<rightarrow> 'b"   and fromEnum :: "'b \<rightarrow> Integer" shows "Enum.enumFromTo toEnum fromEnum = (\<Lambda> x y. map\<cdot>toEnum\<cdot>(upto\<cdot>(fromEnum\<cdot>x)\<cdot>(fromEnum\<cdot>y)))"
assms_3: shows "f \<equiv> \<lambda>k. k mod m"
equiv_norm_Arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "(monoidal_language.Arr C (free_monoidal_category.norm C t) \<and> monoidal_language.Arr C t \<and> monoidal_language.Dom C (free_monoidal_category.norm C t) = monoidal_language.Dom C t \<and> monoidal_language.Cod C (free_monoidal_category.norm C t) = monoidal_language.Cod C t) \<and> monoidal_language.Diagonalize C (free_monoidal_category.norm C t) = monoidal_language.Diagonalize C t"
assms_2: assumes "\<And>x. m x \<triangleleft>\<triangleright> m' x\<cdot>C\<^sup>\<infinity>" shows "Q"
find_index_eq_some: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ii :: "nat" shows "(find_index P xs = Some ii) = (ii < length xs \<and> P (xs ! ii) \<and> (\<forall>i'<ii. \<not> P (xs ! i')))"
healthy_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "healthy t = (feasible t \<and> mono_trans t \<and> scaling t)"
is_Eulerian_circuit_def: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_Eulerian_circuit v ps v' \<equiv> v = v' \<and> is_Eulerian_trail v ps v'"
subset_class_axioms: fixes M :: "'c measure" shows "subset_class (space M) (sets M)"
hyps: shows "{ys \<in> set (subseqs xs). length ys = length xs} = {xs}"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and t :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "z \<in> A"   and "t \<in> A" shows "dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z) + 2 * delta"
eval_poly_add_aux: fixes p :: "'a list"   and q :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R"   and "length p = length q"   and "a \<in> carrier R" shows "local.eval (poly_add p q) a = local.eval p a \<oplus> local.eval q a"
sqrt_int_ceiling_pos_def: fixes x :: "int" shows "sqrt_int_ceiling_pos x = root_int_ceiling_pos 2 x"
one_sq_matrix_def: shows "1 \<equiv> of_fun (\<lambda>i j. if i = j then 1::'c else (0::'c))"
map_gpv_conv_bind: fixes f :: "'k \<Rightarrow> 'h"   and x :: "('k, 'i, 'j) gpv" shows "map_gpv f (\<lambda>x. x) x = x \<bind> (\<lambda>x. Done (f x))"
P_def: fixes R :: "('a, 'b) ring_scheme" shows "UP R \<equiv> UP R"
conj_equiv: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p" shows "A \<turnstile> (p .\<and>. q) .\<longleftrightarrow>. q"
gd_powerprod_axioms: shows "gd_powerprod (\<preceq>) (\<prec>)"
rtranclp_eq_rtranclp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<lambda>x y. P x y \<or> x = y)\<^sup>*\<^sup>* = P\<^sup>*\<^sup>*"
self_domain_iff_ide: fixes a :: "('obj, 'obj) concrete_category.arr" shows "(a \<in> domains a) = ide a"
adopt_node_preserves_wellformedness: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.adopt_node document_ptr child \<rightarrow>\<^sub>h h'"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "Shadow_DOM.heap_is_wellformed h'"
crel_vs_executeD: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "['f]"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "Q heap"   and "state_dp_consistency.cmem heap" obtains x :: "'f" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "Q heap'" and "state_dp_consistency.cmem heap'" and "R a x"
items_eq_is_filter: fixes k :: "nat"   and I :: "item set" shows "items_eq k I \<subseteq> I"
RedFAssNull: fixes F :: "char list"   and D :: "char list"   and v :: "val"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>null\<bullet>F{D} := Val v,s,b\<rangle> \<rightarrow> \<langle>THROW NullPointer,s,b\<rangle>"
points_index_def: fixes B :: "'c set multiset"   and ps :: "'c set" shows "B index ps \<equiv> size (filter_mset ((\<subseteq>) ps) B)"
fmdrop_fset_fmdrop: fixes S :: "'c fset"   and b :: "'c"   and m :: "('c, 'd) fmap" shows "fmdrop_fset S (fmdrop b m) = fmdrop_fset (finsert b S) m"
Inf_set_def: fixes A :: "'a set set" shows "\<Inter> A = {x. Inf ((\<in>) x ` A)}"
not_in_member: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "member a b"   and "x \<notin> dom (Cp b)" shows "x \<notin> dom (Cp a)"
hs_get_work_locs_def: shows "hs_get_work_locs = prefixed ''hs_get_work_''"
assms_12: shows "r * c = x + u + x * u + r * x + x * c"
res_mult_eq: fixes m :: "int"   and x :: "int"   and y :: "int" assumes "residues m" shows "x \<otimes>\<^bsub>residue_ring m\<^esub> y = x * y mod m"
SigmaE: fixes c :: "'a \<times> 'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "c \<in> Sigma A B" obtains x :: "'a" and y :: "'b" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
fps_X_power_subdegree: fixes n :: "nat" shows "subdegree (fps_X ^ n) = n"
len: shows "i < length (replicate i ze @ on # replicate (n - 1 - i) ze)"
length_dims0_minus_vars2'_is_vars1': shows "{0..<length dims0} - vars2' = {0..<length dims0} \<inter> vars1'"
cidr_split_prefix_single: fixes start :: "'b word"   and "end" :: "'b word" shows "\<Union> (uncurry ipset_from_cidr ` set (cidr_split (iprange_interval (start, end)))) = {start..end}"
order_trans: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "order r A"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
integrable_product_swap_iff: fixes f :: "'b \<times> 'a \<Rightarrow> 'c" shows "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f (y, x)) = integrable (M2 \<Otimes>\<^sub>M M1) f"
le_sup_iff: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b"   and z :: "'a \<times> 'b" shows "less_eq_prod (sup_prod x y) z = (less_eq_prod x z \<and> less_eq_prod y z)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>a b. C a b b a"   and "\<forall>a b p q r s. C a b p q \<and> C a b r s \<longrightarrow> C p q r s"   and "\<forall>a b c. C a b c c \<longrightarrow> a = b" shows "tarski_first3 C"
w_aux: fixes k :: "nat"   and m :: "nat" shows "Max_triangle (triangle k + m) = Max_triangle_aux k m"
linear: fixes x :: "'a word"   and y :: "'a word" shows "x \<le>s y \<or> y \<le>s x"
emeasure_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "\<Omega> \<inter> space M \<in> sets M"   and "A \<subseteq> \<Omega>" shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
psubst_new: fixes c :: "'c"   and p :: "('c, 'd) form"   and x :: "'c" assumes "new c p" shows "psubst (id(c := x)) p = p"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U"   and "a <\<^sub>o\<^sub>w b" shows "a \<le>\<^sub>o\<^sub>w b"
insert_subset: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
self_factor_not_simple: shows "\<not> simple_group (G Mod carrier G)"
items_le_split_via_eq: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k :: "nat"   and J :: "item set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "items_le (Suc k) J = items_le k J \<union> items_eq (Suc k) J"
CompDiag_Diag_Dom: fixes t :: "'a local.term" assumes "Diag t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Dom t = t"
not_eq_order_implies_strict: fixes a :: "'b word"   and b :: "'b word" assumes "a \<noteq> b"   and "a \<le>s b" shows "a <s b"
F_deriv: fixes x :: "real" assumes "x \<in> {real a..} - Y" shows "(F has_vector_derivative f x) (at x)"
regular_fixpoint: fixes f :: "'b set \<Rightarrow> 'b set"   and I :: "'b set" assumes "regular f" shows "f (limit f I) = limit f I"
cmp_combine_def: fixes D1 :: "'b set"   and cmp1 :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res"   and D2 :: "'b set"   and cmp2 :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res"   and a :: "'b"   and b :: "'b" shows "cmp_combine D1 cmp1 D2 cmp2 a b \<equiv> if a \<in> D1 \<and> b \<in> D1 then cmp1 a b else if a \<in> D1 \<and> b \<in> D2 then LESS else if a \<in> D2 \<and> b \<in> D1 then GREATER else cmp2 a b"
ide_is_iso: fixes a :: "('a, 'a set) Maps.arr" assumes "Maps.ide a" shows "Maps.iso a"
sign_simps_37: fixes am :: "'a"   and bm :: "'a"   and n :: "nat" shows "(am * bm) ^ n = am ^ n * bm ^ n"
elims: fixes x :: "'g"   and xa :: "'var"   and xb :: "'node list"   and xc :: "'node"   and y :: "'node set" assumes "phiDefNodes_aux x xa xb xc = y" obtains g :: "'g" and v :: "'var" and unvisited :: "'node list" and n :: "'node" where "x = g" and "xa = v" and "xb = unvisited" and "xc = n" and "y = (if n \<notin> set unvisited \<or> v \<in> defs g n then {} else fold (\<union>) (map (phiDefNodes_aux g v (removeAll n unvisited)) (predecessors g n)) (if length (predecessors g n) \<noteq> 1 then {n} else {}))"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>f x xs y ys. \<lbrakk>f x \<le> f y \<Longrightarrow> P f xs (y # ys); \<not> f x \<le> f y \<Longrightarrow> P f (x # xs) ys\<rbrakk> \<Longrightarrow> P f (x # xs) (y # ys)"   and "\<And>f ys. P f [] ys"   and "\<And>f v va. P f (v # va) []" shows "P a0 a1 a2"
distinct_3: fixes x1 :: "int"   and x3 :: "int" shows "Const' x1 \<noteq> Greater' x3"
cblinfun_image_INF_eq: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and V :: "'c \<Rightarrow> 'a ccsubspace" assumes "isometry U" shows "U *\<^sub>S \<Sqinter> (range V) = (INF i. U *\<^sub>S V i)"
real_average_minus_first: fixes a :: "real"   and b :: "real" shows "(a + b) / 2 - a = (b - a) / 2"
measure_spmf_of_set: fixes S :: "'a set"   and A :: "'a set" shows "Sigma_Algebra.measure (measure_spmf (spmf_of_set S)) A = real (card (S \<inter> A)) / real (card S)"
irred_0: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "\<not> punit.is_red F 0"
simps_2: fixes x :: "'c option"   and y :: "'c" shows "sup_option x (Some y) = Some y"
repeatAmountOfQuantifiers_eval: fixes step :: "atom fm \<Rightarrow> atom fm"   and F :: "atom fm"   and xs :: "real list" assumes "\<And>xs F. eval F xs = eval (step F) xs" shows "eval F xs = eval (repeatAmountOfQuantifiers step F) xs"
assertions_lemma: fixes c :: "'b ann_com" shows "pre c \<in> assertions c"
in_set_replicate: fixes x :: "'c"   and n :: "nat"   and y :: "'c" shows "(x \<in> set (replicate n y)) = (x = y \<and> n \<noteq> 0)"
finfun_Diag_def: fixes f :: "'c \<Rightarrow>f 'd"   and g :: "'c \<Rightarrow>f 'e" shows "($f, g$) = finfun_rec (\<lambda>b. Pair b \<circ>$ g) (\<lambda>a b c. c(a $:= (b, g $ a))) f"
dg_Par_Hom_vifunion_in_Vset: fixes X :: "V"   and Y :: "V" assumes "X \<in>\<^sub>\<circ> Vset \<alpha>"   and "Y \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>X. \<Union>\<^sub>\<circ> (VLambda Y (Hom (dg_Par \<alpha>) a) `\<^sub>\<circ> Y)) `\<^sub>\<circ> X) \<in>\<^sub>\<circ> Vset \<alpha>"
priv_is_guar_priv_def: fixes mdss\<^sub>C :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "priv_is_guar_priv mdss\<^sub>C \<equiv> \<forall>i<length cms. \<forall>j<length cms. i \<noteq> j \<longrightarrow> priv_mem\<^sub>C ! i \<subseteq> (mdss\<^sub>C ! j) GuarNoReadOrWrite"
fgs_generator: fixes H :: "'a set" assumes "H \<subseteq> carrier M" shows "generator R (mdl M (fgs R M H)) H"
assms_3: fixes x :: "'c" shows "h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x"
flow_in_domain: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "flow0 x0 t \<in> X"
scale_right_distrib: fixes x :: "'b"   and y :: "'b"   and a :: "'a" assumes "x \<in> U\<^sub>V\<^sub>S"   and "y \<in> U\<^sub>V\<^sub>S" shows "s a (x + y) = s a x + s a y"
str_del_block_def: fixes b :: "'a set" shows "fin.str_del_block b \<equiv> {#bl - b. bl \<in># {#bl \<in># str_del_block b. bl \<noteq> b#}#}"
config: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) (u @ v) = type1 init x y"