hcju/isabelleps huggingface dataset:
t\<^sub>0_def: shows "t\<^sub>0 = (LEAST t. local.prefixes t j \<down> \<and> x < e_length (the (local.prefixes t j)))"
ANR_imp_absolute_neighbourhood_retract_UNIV: fixes S :: "'a set"   and S' :: "'b set" assumes "ANR S"   and "S homeomorphic S'"   and "closed S'" obtains V :: "'b set" where "open V" and "S' retract_of V"
triangle: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a"   and b :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "C (T (a, monoidal_category.lunit C T \<alpha> \<iota> b)) (\<alpha> (a, monoidal_category.unity C \<iota>, b)) = T (monoidal_category.runit C T \<alpha> \<iota> a, b)"
Collect_ex_eq: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
pp_comp_semi_commute: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> y \<le> - - (x \<sqinter> y)"
fdom_finite: fixes f :: "('a, 'b) ffun" shows "finite (fdom f)"
share_append_Prog\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list"   and S :: "nat \<Rightarrow> bool option" shows "share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis])) S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S"
isometry_on_uniformly_continuous: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "isometry_on X f" shows "uniformly_continuous_on X f"
sorted_snoc_le: fixes xs :: "'c list"   and x :: "'c"   and y :: "'c" assumes "ASSUMPTION (Sorted_Less.sorted (xs @ [x]))"   and "x \<le> y" shows "Sorted_Less.sorted (xs @ [y])"
int_ge_less_than_def: fixes d :: "int" shows "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
strict_uv_refine_lemma: fixes X :: "'b program set"   and F :: "'b program"   and G :: "'b program" assumes "strict_uv_prop X" shows "(\<forall>H. F ok H \<and> G ok H \<and> F \<squnion> H \<in> X \<longrightarrow> G \<squnion> H \<in> X) = (F \<in> X \<longrightarrow> G \<in> X)"
that: fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and x :: "'c" assumes "Regions' (clk_set A) v n x" shows "thesis"
split_min: fixes P :: "'a \<Rightarrow> bool"   and i :: "'a"   and j :: "'a" shows "P (min i j) = ((i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j))"
rec_1: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" shows "rec_nat f1 f2 0 = f1"
residue_is_hom_1: fixes n :: "int"   and m :: "int" assumes "1 < n"   and "1 < m"   and "n dvd m" shows "residue n \<in> ring_hom (residue_ring m) (residue_ring n)"
real_le_abs_sinh: fixes x :: "real" shows "\<bar>x\<bar> \<le> \<bar>(exp x - inverse (exp x)) / 2\<bar>"
covf: shows "ov O f \<subseteq> s \<union> ov \<union> d"
reachableI_2: fixes x :: "'a"   and m :: "'b"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "x \<in> mut_m.tso_store_refs m s" shows "mut_m.reachable m x s"
negligible_iff_measure: fixes S :: "'c set" shows "negligible S = (S \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue S = 0)"
mul_eq_reflexive: fixes M :: "'a multiset"   and r :: "('a \<times> 'a) set" shows "(M, M) \<in> mul_eq r"
freshCompChain_1: fixes xvec :: "name list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" assumes "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>'" shows "xvec \<sharp>* (\<Psi> \<otimes> \<Psi>')"
c_top_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "c_top G H = (\<lambda>X\<in>set_rcs G H. \<lambda>Y\<in>set_rcs G H. {z. \<exists>x\<in>X. \<exists>y\<in>Y. x \<cdot>\<^bsub>G\<^esub> y = z})"
product_run_project_limit_snd: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c" assumes "finite (range (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w))" shows "project_transition_snd ` limit (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w) = limit (run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w)"
permutation_lemma: fixes S :: "'b set"   and p :: "'b \<Rightarrow> 'b" assumes "finite S"   and "bij p"   and "\<forall>x. x \<notin> S \<longrightarrow> p x = x" shows "permutation p"
poly_mapping_plus_induct: fixes P :: "('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow> bool"   and p :: "'d \<Rightarrow>\<^sub>0 'e" assumes "P 0"   and "\<And>p c t. \<lbrakk>c \<noteq> (0::'e); t \<notin> keys p; P p\<rbrakk> \<Longrightarrow> P (monomial c t + p)" shows "P p"
transaction_strand_unlabel_memberD: fixes x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step"   and T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "x \<in> set (unlabel (transaction_strand T))" shows "x \<in> set (unlabel (transaction_receive T)) \<or> x \<in> set (unlabel (transaction_selects T)) \<or> x \<in> set (unlabel (transaction_checks T)) \<or> x \<in> set (unlabel (transaction_updates T)) \<or> x \<in> set (unlabel (transaction_send T))"
simps_22: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm"   and x2 :: "'b" shows "PFn x41 x42 x43 \<noteq> PVar x2"
image_cong: fixes M :: "'b::type set"   and N :: "'b::type set"   and f :: "'b::type \<Rightarrow> 'c::type"   and g :: "'b::type \<Rightarrow> 'c::type" assumes "(M::'b::type set) = (N::'b::type set)"   and "\<And>x::'b::type. x \<in> (N::'b::type set) \<Longrightarrow> (f::'b::type \<Rightarrow> 'c::type) x = (g::'b::type \<Rightarrow> 'c::type) x" shows "(f::'b::type \<Rightarrow> 'c::type) ` (M::'b::type set) = (g::'b::type \<Rightarrow> 'c::type) ` (N::'b::type set)"
no_proots: shows "path_image g \<subseteq> - proots p"
intro: fixes P :: "'c \<Rightarrow> bool" assumes "\<And>t u. P (app t u) = (P t \<and> P u)" shows "simple_syntactic_and P"
hm_size_rule': fixes m :: "'a \<Rightarrow> 'b option"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" shows "<is_hashmap' m l ht> hm_size ht <\<lambda>r. is_hashmap' m l ht * \<up> (r = card (dom m))>"
seq_lexord_def: fixes R :: "('b \<times> 'b) set" shows "seq_lexord R = {(xs, ys). \<exists>i. (sinit i xs, sinit i ys) \<in> lexord R}"
valid_plan_def: shows "valid_plan \<equiv> valid_plan_from I"
Sum_any_when_equal': fixes f :: "'c \<Rightarrow> 'b"   and b :: "'c" shows "(\<Sum>a. f a when b = a) = f b"
simps_5: fixes a :: "'a poly"   and b :: "'a poly" shows "num_params (Mul a b) = max (num_params a) (num_params b)"
typeof_h_eq_NT: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b"   and v :: "'c val" shows "(heap_base.typeof_h typeof_addr h v = \<lfloor>NT\<rfloor>) = (v = Null)"
var\<^sub>1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e" shows "(\<Gamma>, Var x, S) \<Rightarrow> (delete x \<Gamma>, e, Upd x # S)"
fpairs_iff_elts: fixes a :: "V"   and b :: "V"   and r :: "V" shows "([a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> fpairs r) = ([a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
beta_norm_simps_8: fixes xa :: "typ"   and x :: "Core.term"   and xaa :: "Core.term" shows "beta_norm (Abs xa x $ xaa) = beta_norm (subst_bv xaa x)"
update_convs_4: fixes c_glob' :: "('c \<Rightarrow> 'd zmultiset) \<Rightarrow> 'c \<Rightarrow> 'd zmultiset"   and c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_glob_update c_glob' \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob' c_glob, \<dots> = more\<rparr>"
height_split_min: fixes t :: "'b tree234"   and x :: "'b"   and t' :: "'b up\<^sub>d" assumes "split_min t = (x, t')"   and "0 < height t"   and "bal t" shows "height t' = height t"
ide_char': fixes a :: "('o, 'a) arr" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
contour_integrable_continuous_circlepath: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (path_image (circlepath z r)) f" shows "f contour_integrable_on circlepath z r"
m_def: shows "m = (\<lambda>n. Max {B i |i. i \<le> n})"
extend_map_to_freeword_hom': fixes f :: "'b \<Rightarrow> 'a"   and xs :: "('b \<times> bool) list"   and ys :: "('b \<times> bool) list" shows "(\<Sum>(s, b)\<leftarrow>prappend_signed_list xs ys. if b then f s else - f s) = (\<Sum>(s, b)\<leftarrow>xs. if b then f s else - f s) + (\<Sum>(s, b)\<leftarrow>ys. if b then f s else - f s)"
lm007: fixes X :: "'c set"   and Y :: "'d set" shows "injections X Y \<subseteq> injectionsUniverse"
valuate_subst_poly: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "subst_poly \<sigma> p \<lbrace> v \<rbrace> = p \<lbrace> \<lambda>x. \<sigma> x \<lbrace> v \<rbrace> \<rbrace>"
the_eq_least_pos: shows "(THE i. A $ i $ j = (1::'a)) = mod_type_class.from_nat (least_non_zero_position_of_vector (column_iarray (mod_type_class.to_nat j) (matrix_to_iarray A)))"
action_repcs: fixes l :: "'s"   and u :: "'c \<Rightarrow> real"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "action (repcs (l, u) cfg) = rept (l, u) (action cfg)"
positive_pos_imaginary_parts_3: fixes x :: "real_alg_3"   and p :: "int poly" assumes "x \<in> set (pos_imaginary_parts_3 p)" shows "0 < real_of_3 x"
urel32_inj': fixes x :: "uint32"   and z :: "int"   and y :: "uint32" assumes "urel32 x z"   and "urel32 y z" shows "x = y"
partial_order_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
lspace_const: fixes p :: "real"   and S :: "'a set"   and c :: "real" assumes "0 < p"   and "S \<in> lmeasurable" shows "(\<lambda>x. c) \<in> lspace (lebesgue_on S) (ennreal p)"
aext_arestr': fixes a :: "'c \<Longrightarrow> 'd"   and P :: "('e, 'd) uexpr" assumes "a \<natural> P" shows "(P \<restriction>\<^sub>e a) \<oplus>\<^sub>p a = P"
dot_left_diff_distrib: fixes u :: "(real, 'b) vec"   and v :: "(real, 'b) vec"   and x :: "(real, 'b) vec" shows "(u - v) \<bullet> x = u \<bullet> x - v \<bullet> x"
max_word_max: fixes n :: "'a word" shows "n \<le> - 1"
elts_mult_\<omega>E: fixes x :: "V"   and y :: "V" assumes "x \<in> elts (y * \<omega>)" obtains n :: "nat" where "n \<noteq> 0" and "x \<in> elts (y * ord_of_nat n)" and "\<And>m. m < n \<Longrightarrow> x \<notin> elts (y * ord_of_nat m)"
Un_upper1: fixes A :: "'c set"   and B :: "'c set" shows "A \<subseteq> A \<union> B"
map_def: fixes f :: "'a" shows "\<tau>\<sigma>.map f = (if A.arr f then \<tau> (A.cod f) \<cdot>\<^sub>B \<sigma> f else B.null)"
signed_take_bit_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m (take_bit n a) = (if n \<le> m then take_bit n else ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m) a"
convex_rel_interior_inter_two: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "convex T"   and "rel_interior S \<inter> rel_interior T \<noteq> {}" shows "rel_interior (S \<inter> T) = rel_interior S \<inter> rel_interior T"
4: shows "\<forall>i. D (v (Y i)) \<subseteq> D (u (Y i))"
equality: fixes r :: "'c alpha_d_scheme"   and r' :: "'c alpha_d_scheme" assumes "ok r = ok r'"   and "alpha_d.more r = alpha_d.more r'" shows "r = r'"
D: shows "diag_block_mat As \<in> carrier_mat (dim_row (diag_block_mat As)) (dim_col (diag_block_mat As))"
terminates_on_def: shows "terminates_on \<equiv> \<lambda>g. {x. terminates_onp g x}"
NTCompDefined_def: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "\<eta>1 \<approx>>\<bullet> \<eta>2 \<equiv> NatTrans \<eta>1 \<and> NatTrans \<eta>2 \<and> NTCatDom \<eta>2 = NTCatDom \<eta>1 \<and> NTCatCod \<eta>2 = NTCatCod \<eta>1 \<and> NTCod \<eta>1 = NTDom \<eta>2"
dickson_lessD3: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "dickson_less d m s t" shows "s \<prec> t"
induct: fixes P :: "Core.term \<Rightarrow> nat \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "nat"   and a2 :: "Core.term" assumes "\<And>i lev u. P (Bv i) lev u"   and "\<And>T body lev u. P body (lev + 1) u \<Longrightarrow> P (Abs T body) lev u"   and "\<And>f t lev u. \<lbrakk>P f lev u; P t lev u\<rbrakk> \<Longrightarrow> P (f $ t) lev u"   and "\<And>v va uu_ uv_. P (Ct v va) uu_ uv_"   and "\<And>v va uu_ uv_. P (Fv v va) uu_ uv_" shows "P a0 a1 a2"
map_vector_0: fixes v :: "(real, 'c) Finite_Cartesian_Product.vec" shows "(map_vector of_real v = 0) = (v = 0)"
bind_munionM: fixes f :: "'a multiset \<Rightarrow> 'm"   and m :: "'m"   and m' :: "'m" assumes "\<And>A B. f (A + B) = bind (f A) (\<lambda>x. bind (f B) (\<lambda>y. return (x + y)))" shows "bind (munionM m m') f = munionM (bind m f) (bind m' f)"
nodes_connected_ind_trans: fixes F :: "(nat \<times> int \<times> nat) set" assumes "F \<subseteq> E" shows "trans {(x, y) |x y. nodes_connected \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = F\<rparr> x y}"
dghm_dag_Rel_components_2: fixes \<alpha> :: "V" shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr> = VLambda (dg_Rel \<alpha>\<lparr>Arr\<rparr>) converse_Rel"
OclOr_true2: fixes Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(Y or true) = true"
scaleR_2: fixes x :: "'c" shows "2 *\<^sub>R x = x + x"
Re_mult_real: fixes a :: "complex"   and b :: "complex" assumes "is_real a" shows "Re (a * b) = Re a * Re b"
isMaxFlow_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "isMaxFlow f \<equiv> Flow c s t f \<and> (\<forall>f'. Flow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f)"
h1im: shows "h1 ` {0..1} \<subseteq> C"
absolutely_integrable_on_linear_image: fixes g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "linear g" shows "(f absolutely_integrable_on g ` S) = (f \<circ> g absolutely_integrable_on S \<or> det (matrix g) = 0)"
vec_matrix: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" shows "vec_to_iarray (A $ i) = matrix_to_iarray A !! mod_type_class.to_nat i"
simps_2: fixes t :: "('c, 'd) SkewBinomialTree"   and q :: "('c, 'd) SkewBinomialTree list" shows "queue_to_multiset (t # q) = tree_to_multiset t + queue_to_multiset q"
sorted_scaled_inl: fixes X :: "'a \<times> (real \<times> real) pdevs" shows "ccw.sortedP 0 (map ((*\<^sub>R) 2) (ccw.selsort 0 (inl (snd X))))"
inf_bot_right: fixes x :: "'b multiset" shows "x \<inter># \<Empt> = \<Empt>"
assms_4: shows "(b \<rightleftharpoons> c) \<bullet> x = x"
l11_21_b: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and B' :: "'p"   and A' :: "'p"   and C' :: "'p" assumes "B Out A C"   and "B' Out A' C'" shows "A B C CongA A' B' C'"
bd_ctsI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>b M. \<lbrakk>\<And>i. M i \<tturnstile> M (Suc i); \<And>i. sound (M i); \<And>i. bounded_by b (M i)\<rbrakk> \<Longrightarrow> t (Sup_exp (range M)) = Sup_exp (range (t \<circ> M))" shows "bd_cts t"
f_tendsto: fixes x :: "real" assumes "x \<in> {a..b}" shows "(f \<longlongrightarrow> f x) (at x within {a..b})"
Par: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "(P, Q) \<in> Rel" shows "(P \<parallel> R, Q \<parallel> R) \<in> Rel'"
StepCondTrue: fixes b :: "expr"   and s :: "char list \<Rightarrow> val option"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd" assumes "interpret b s = Some true" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> \<langle>if (b) c\<^sub>1 else c\<^sub>2,s,0\<rangle> \<leadsto> \<langle>c\<^sub>1,s,1\<rangle>"
sup_comp: 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"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times (sup x y) z = sup (times x z) (times y z)"
image_sub: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and A1 :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A" shows "f ` A1 \<subseteq> B"
power_strict_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less a b"   and "less_eq zero a" shows "0 < n \<longrightarrow> less (power.power one times a n) (power.power one times b n)"
wf_semiJoin_1: fixes J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQ :: "(nat set \<times> 'a option list set) set"   and st :: "nat set"   and t :: "'a option list" assumes "1 \<le> card J"   and "wf_query n J Q Qn"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = (\<lambda>tab. semiJoin tab (st, t)) ` Q" shows "wf_query n J QQ Qn"
fv_constant0: fixes T :: "Unary_PCF.type" shows "fv (constant0 T) = {|(''bool'', \<B>)|}"
connected_space_closedin: fixes X :: "'c topology" shows "connected_space X = (\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and> topspace X \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
subgame_Digraph: fixes V' :: "'a set" shows "Digraph (G'.subgame V')"
atom_property: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'b"   and L :: "'b Literal" assumes "propositional_atoms atom_ordering"   and "A = atom L" shows "L = Pos A \<or> L = Neg A"
translate_inj_on: fixes a :: "'a"   and A :: "'a set" shows "inj_on ((+) a) A"
out_eq_p: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "a \<in> s"   and "n \<le> j" shows "a j = p"
monom_mult_code: fixes n :: "nat"   and f :: "'a poly" shows "coeffs (monom_mult n f) = (let xs = coeffs f in if xs = [] then xs else replicate n (0::'a) @ xs)"
greaters_Cons: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "greaters R x (y # ys) = (if (x, y) \<in> R \<and> (y, x) \<notin> R then y # greaters R x ys else greaters R x ys)"
has_integral_affinity: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and a :: "'c"   and b :: "'c"   and m :: "real"   and c :: "'c" assumes "(f has_integral i) (cbox a b)"   and "m \<noteq> 0" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (1 / \<bar>m\<bar> ^ DIM('c)) *\<^sub>R i) ((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b)"
is_float_simps_3: fixes x :: "float" shows "is_float (real_of_float x)"
int_or_extra_simps_1: fixes w :: "int" shows "w OR 0 = w"
smallest_prime_betwen_Some: fixes m :: "nat"   and n :: "nat"   and p :: "nat" shows "(smallest_prime_between m n = Some p) = (smallest_prime_beyond m = p \<and> p \<le> n)"
eSubstAbs_simp4: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and ys :: "'varSort"   and eY :: "'d withERR"   and y :: "'var" shows "eSubstAbs MOD ys eY y ERR = ERR"
borel_measurable_P: fixes f :: "nat set set \<Rightarrow> 'a" shows "random_variable borel f"
mismatchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Rel \<subseteq> Rel'" shows "[a\<noteq>b]P \<leadsto>\<guillemotleft>Rel'\<guillemotright> [a\<noteq>b]Q"
\<beta>''_def: shows "\<beta>'' = (\<lambda>q. \<Sum>X\<in>Roots_ms. int (count (prime_factorization (Q X)) q) * \<Prod>\<^sub># (image_mset \<beta>' X))"
trapped_backward_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b"   and K :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.trapped_backward f X x K = (auto_ll_on_open.flow0 f X x ` (auto_ll_on_open.existence_ivl0 f X x \<inter> {..0}) \<subseteq> K)"
linear_continuous_at: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "bounded_linear f" shows "isCont f a"
reachable_extend_eq: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program" assumes "Extend h" shows "reachable (extend h F) = extend_set h (reachable F)"
exhaust: fixes y :: "ipv6addr_syntax" obtains x1 :: "16 word" and x2 :: "16 word" and x3 :: "16 word" and x4 :: "16 word" and x5 :: "16 word" and x6 :: "16 word" and x7 :: "16 word" and x8 :: "16 word" where "y = IPv6AddrPreferred x1 x2 x3 x4 x5 x6 x7 x8"
locally_path_connected_space_open_path_components: fixes X :: "'c topology" shows "locally_path_connected_space X = (\<forall>U c. openin X U \<and> c \<in> path_components_of (subtopology X U) \<longrightarrow> openin X c)"
\<alpha>_s\<beta>': shows "\<alpha> * ZFC_in_HOL.succ \<beta>' = \<alpha> * \<beta>' + \<alpha>"
filterlim_uminus_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" shows "filterlim f at_top F = (LIM x F. - f x :> at_bot)"
leading_coeff_0_iff: fixes p :: "'d poly" shows "(lead_coeff p = (0::'d)) = (p = 0)"
initD_def: shows "initD \<equiv> (\<lambda>_. \<infinity>)(s := 0::'a)"
SNF_divides_diagonal: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and j :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat n m"   and "Smith_normal_form_mat A"   and "j < min n m"   and "i \<le> j" shows "A $$ (i, i) dvd A $$ (j, j)"
card_Diff_singleton_if: fixes A :: "'q set"   and x :: "'q" assumes "finite A" shows "card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
notmatching_notdom: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x (p @ [a]) \<noteq> \<lfloor>a\<rfloor>" shows "x \<notin> dom (C a)"
Id_on_def': fixes A :: "'b \<Rightarrow> bool" shows "s2r {x. A x} = {(x, y). x = y \<and> A x}"