hcju/isabelleps huggingface dataset:
rv_le_iff: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real" assumes "measure_space M" shows "(f \<in> rv M) = (\<forall>a. {w. f w \<le> a} \<in> measurable_sets M)"
pow_sum0: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "deg R ((p \<oplus>\<^bsub>P\<^esub> q) [^]\<^bsub>P\<^esub> n) = deg R p * n"
exhaust: fixes y :: "'b gexp" obtains x1 :: "bool" where "y = Bc x1"   | x21 :: "'b aexp" and x22 :: "'b aexp" where "y = Eq x21 x22"   | x31 :: "'b aexp" and x32 :: "'b aexp" where "y = Gt x31 x32"   | x41 :: "'b" and x42 :: "value list" where "y = In x41 x42"   | x51 :: "'b gexp" and x52 :: "'b gexp" where "y = Nor x51 x52"
idx_dom: fixes xs :: "'a list" assumes "distinct xs" shows "dom (idx xs) = set xs"
mc_SPRAlg_def: shows "mc_SPRAlg \<equiv> mkSPRDetAuto agents jkbp envInit envAction envTrans envVal envObsC envObs"
orthogonal_matrix: fixes Q :: "((real, 'b) vec, 'b) vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = mat 1)"
coeff_0_degree_minus_1: fixes rrr :: "'a poly"   and dr :: "nat" assumes "coeff rrr dr = (0::'a)"   and "degree rrr \<le> dr" shows "degree rrr \<le> dr - 1"
lea456789_lta__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C LtA D E F"   and "D E F LeA G H I" shows "A B C LtA G H I"
Mul_Color_Target: fixes j :: "nat"   and n :: "nat" assumes "0 \<le> j \<and> j < n" shows "\<turnstile> Mul_Color_Target j n  \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts ! j)\<rbrace>"
always_def: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool" shows "\<box>P = (\<lambda>\<sigma>. \<forall>i. P (\<sigma> |\<^sub>s i))"
strong_confluentp_imp_confluentp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "strong_confluentp r" shows "confluentp r"
Min_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq (linorder.Min less_eq A) x"
get_component_to_tree_order_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and c :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c" shows "set nodes \<subseteq> set c"
list_ex_simps_1: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "list_ex P (x # xs) = (P x \<or> list_ex P xs)"
IH: shows "local.coeff p = local.coeff (local.normalize p)"
Y_3: shows "countable (support_flow (\<Squnion> Y))"
C\<^sub>0: fixes x :: "i"   and y :: "i" assumes "SkolemizedAxiomsSetMcL" shows "E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)"
swap_identical_length: fixes i :: "nat"   and j :: "nat"   and t :: "'b list" assumes "i < j"   and "j < length t" shows "length t = length (swap_events i j t)"
measurable_expr_sem: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and V :: "nat set" assumes "\<Gamma> \<turnstile> e : t"   and "free_vars e \<subseteq> V" shows "(\<lambda>\<sigma>. expr_sem \<sigma> e) \<in> state_measure V \<Gamma> \<rightarrow>\<^sub>M subprob_algebra (stock_measure t)"
locally_ringed_space_axioms: shows "locally_ringed_space X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str"
lex_half_segments_lowest_vertex': fixes d :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "d \<in> set (map snd (half_segments_of_aform X))" shows "Counterclockwise_2D_Arbitrary.lex d (lowest_vertex (fst X, nlex_pdevs (snd X)))"
pderiv_power_Suc: fixes p :: "'a poly"   and n :: "nat" shows "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
less_imp_le: fixes x :: "'t"   and y :: "'t" assumes "lt_of_nat_term_order cmp_term x y" shows "le_of_nat_term_order cmp_term x y"
Inr_inject: fixes x :: "'b"   and y :: "'b" assumes "Inr x = Inr y" shows "x = y"
distinct_81: fixes x7 :: "'a fm"   and x91 :: "nat"   and x92 :: "'a fm" shows "ExQ x7 \<noteq> ExN x91 x92"
add_eq_0_iff_both_eq_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
float_defs_6: fixes a :: "('a, 'b) IEEE.float" shows "is_denormal a = (IEEE.exponent a = 0 \<and> fraction a \<noteq> 0)"
add_order0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.add_order plus zero zero = 1"
trivial_ker_imp_transcendental: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R"   and "a_kernel (K [X]) R (\<lambda>p. local.eval p x) = {[]}" shows "(transcendental over K) x"
scaleR_collapse: fixes u :: "real"   and a :: "'a" shows "(1 - u) *\<^sub>R a + u *\<^sub>R a = a"
at_map_def: shows "at_map = Abs_atoms \<circ> atom_map"
fold1_keys_def_alt: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('b, 'c) RBT.rbt" shows "fold1_keys f t = fold f (tl (RBT.keys t)) (hd (RBT.keys t))"
inv_lift_map_eq: fixes i :: "nat" shows "inv (lift_map i) = drop_map i"
under_underS_suc: fixes a :: "'a" assumes "a \<in> Field r"   and "local.aboveS a \<noteq> {}" shows "local.underS (local.suc {a}) = local.under a"
empty_pointermap_def: shows "empty_pointermap \<equiv> \<lparr>entries = [], getentry = Map.empty\<rparr>"
set_disconnected_nodes_get_ancestors_is_l_set_disconnected_nodes_get_ancestors: shows "l_set_disconnected_nodes_get_ancestors set_disconnected_nodes_locs get_ancestors_locs"
oInv_inverse: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "normal F" shows "oInv F (F x) = x"
onorm_inner_left: fixes r :: "'d \<Rightarrow> 'e"   and f :: "'e" assumes "bounded_linear r" shows "onorm (\<lambda>x. r x \<bullet> f) \<le> onorm r * norm f"
last_ConsL: fixes xs :: "'f list"   and x :: "'f" assumes "xs = []" shows "last (x # xs) = x"
rep_nat_pp_inverse: fixes x :: "'a" shows "abs_nat_pp (rep_nat_pp x) = x"
comp1: fixes g :: "'f \<Rightarrow> 'a" assumes "bounded_clinear g" shows "bounded_cbilinear (\<lambda>x. (**) (g x))"
sum: fixes g :: "'d \<Rightarrow> 'a"   and A :: "'d set" shows "sum g A *s x = (\<Sum>xa\<in>A. g xa *s x)"
IH: fixes gpv :: "('a, 'call, 'ret) gpv"   and s :: "'s" assumes "interaction_bound consider' gpv \<le> p" shows "set_spmf (inline1' gpv s) \<subseteq> {Inr (out', out, c', rpv) |out' out c' rpv. if consider' out' then (\<forall>input. (if consider out then eSuc (interaction_bound consider (c' input)) else interaction_bound consider (c' input)) \<le> q) \<and> (\<forall>x. eSuc (interaction_bound consider' (rpv x)) \<le> p) else \<not> consider out \<and> (\<forall>input. interaction_bound consider (c' input) = 0) \<and> (\<forall>x. interaction_bound consider' (rpv x) \<le> p)} \<union> range Inl"
simps_7: fixes \<phi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter (X\<^sub>n \<phi>) \<nu> = \<phi>"
D_PUT_H: fixes f :: "Field"   and e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var" assumes "GAMMA f = high"   and "\<forall>s. noLowDPs s \<longrightarrow> Expr_good e s" shows "(G, Put x f e, HighSec) \<in> Deriv"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>xs. (\<And>x. \<lbrakk>xs \<noteq> []; x \<in> set (remdups xs)\<rbrakk> \<Longrightarrow> P (remove1 x xs)) \<Longrightarrow> P xs" shows "P a0"
cSup_upper: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 x :: "'b"   and X :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "preorder.bdd_above less_eq X" shows "less_eq x (Sup X)"
intros: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "has_bochner_integral M f x" shows "integrable M f"
star_rtc2: fixes x :: "'a" shows "((1::'a) + x \<cdot> x \<le> x) = (x = x\<^sup>\<star>)"
le_measure_iff: fixes M :: "'b measure"   and N :: "'b measure" shows "(M \<le> N) = (if space M = space N then if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"
hoare_while: fixes p :: "'a"   and t :: "'a"   and c :: "'b" assumes "\<lbrace>\<iota> p \<cdot> \<iota> t\<rbrace> c \<lbrace>\<iota> p\<rbrace>" shows "\<lbrace>\<iota> p\<rbrace> While \<iota> t Do c End \<lbrace>! (\<iota> t) \<cdot> \<iota> p\<rbrace>"
iso_char: fixes f :: "'a" shows "S.iso f = (local.iso f \<and> S.arr f \<and> S.arr (local.inv f))"
no_inflow_s: shows "\<forall>e\<in>incoming s. f e = (0::'capacity)"
bounded_increasing_convergent: fixes f :: "nat \<Rightarrow> real" assumes "bounded (range f)"   and "\<And>n. f n \<le> f (Suc n)" shows "\<exists>l. f \<longlonglongrightarrow> l"
delete_def: fixes x :: "'b"   and t :: "'b tree" shows "delete x t = (if t = \<langle>\<rangle> then \<langle>\<rangle> else case splay x t of \<langle>l, a, r\<rangle> \<Rightarrow> if x \<noteq> a then \<langle>l, a, r\<rangle> else if l = \<langle>\<rangle> then r else case splay_max l of \<langle>l', m, r'\<rangle> \<Rightarrow> \<langle>l', m, r\<rangle>)"
Gauss_Jordan_upt_k_iarrays_def: fixes A :: "'b iarray iarray"   and k :: "nat" shows "Gauss_Jordan_upt_k_iarrays A k = snd (foldl Gauss_Jordan_column_k_iarrays (0, A) [0..<Suc k])"
eval_ternary_And_comm: fixes t1 :: "ternaryvalue"   and t2 :: "ternaryvalue" shows "eval_ternary_And t1 t2 = eval_ternary_And t2 t1"
us: shows "us = getFrN (map snd txs) (map fst txs) [dsj \<phi>1 \<phi>2] (length txs)"
le_bot: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.order_bot bot less_eq less"   and "less_eq a bot" shows "a = bot"
less_le_not_le: fixes x :: "'g"   and y :: "'g" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
flpSystem_def: fixes sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat" shows "flpSystem sends \<equiv> (finite UNIV \<and> 2 \<le> card UNIV) \<and> (\<forall>p s m. finite {v. v \<in># sends p s m}) \<and> (\<forall>p s m p2 v. sends p s m <p2, inM v> = 0)"
qAFreshAll_qFreshAll_preserves_alphaAll: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and zs :: "'e"   and z :: "'d"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(FixVars.qGood X \<longrightarrow> qAFresh zs z X \<longrightarrow> (\<forall>X'. X #= X' \<longrightarrow> qFresh zs z X')) \<and> (FixVars.qGoodAbs A \<longrightarrow> qAFreshAbs zs z A \<longrightarrow> (\<forall>A'. A $= A' \<longrightarrow> qFreshAbs zs z A'))"
distrib_inf_le: fixes 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.lattice inf less_eq less sup" shows "less_eq (sup (inf x y) (inf x z)) (inf x (sup y z))"
idempotent: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "Idem f"   and "x \<in> carrier L" shows "f (f x) .= f x"
Knaster_Tarski_top: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "\<top>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> GFP\<^bsub>L\<^esub> f"
m_def: shows "m = Min (poly_deg ` fst ` set (ps\<^sub>+))"
fst_aform_val: shows "fst (aform_val e X) = xl"
assms: shows "\<forall>i. coeff p i \<in> \<rat>"
nths_assoc_three_A: fixes A :: "nat set"   and B :: "nat set"   and C :: "nat set"   and xs :: "'b list" assumes "A \<inter> B = {}"   and "(A \<union> B) \<inter> C = {}" shows "nths (nths xs (ind_in_set (A \<union> B \<union> C) ` (A \<union> B))) (ind_in_set (A \<union> B) ` A) = nths xs (ind_in_set (A \<union> B \<union> C) ` A)"
p_fun_mult_var: fixes x :: "'a"   and y :: "'a" assumes "x\<^sup>T ; x \<le> 1'" shows "(x \<cdot> y)\<^sup>T ; (x \<cdot> y) \<le> 1'"
inj_on_ltake: fixes w :: "'b llist" shows "inj_on (\<lambda>k. ltake k w) {..llength w}"
valuate_def: fixes x :: "(nat \<Rightarrow> real) \<Rightarrow> 'a" shows "valuate x = x ` funcset UNIV {- 1..1}"
Up_To_Technique: fixes d :: "'d"   and R :: "('com list \<times> 'com list) set" assumes "d_Bisimulation_Up_To_USdB d R" shows "R \<subseteq> \<approx>\<^bsub>d\<^esub>"
a2n_def: shows "a2n \<equiv> \<Lambda> a. a\<cdot>1"
disc_compl_inter_ocircline_set: fixes H :: "ocircline" shows "disc_compl H \<inter> ocircline_set H = {}"
sqnf_kno: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:18..PRreq-:20} \<longrightarrow> sqnf (rt \<xi>) (dip \<xi>) = kno)"
log_le_iff: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "Discrete.log m \<le> Discrete.log n"
bounds_consistent_gt_ub: fixes s :: "('a, 'b) state"   and c :: "'b"   and x :: "nat" assumes "\<diamond> s"   and "c <\<^sub>l\<^sub>b \<B>\<^sub>l s x" shows "\<not> c >\<^sub>u\<^sub>b \<B>\<^sub>u s x"
simps_2: fixes m :: "Secrets" shows "KS2Expression (sKS m) = sE m"
op_tdghm_def: fixes \<NN> :: "V" shows "op_tdghm \<NN> = [\<NN>\<lparr>NTMap\<rparr>, op_dghm (\<NN>\<lparr>NTCod\<rparr>), op_dghm (\<NN>\<lparr>NTDom\<rparr>), op_dg (\<NN>\<lparr>NTDGDom\<rparr>), op_dg (\<NN>\<lparr>NTDGCod\<rparr>)]\<^sub>\<circ>"
simps_1: fixes uu :: "'a \<Rightarrow> value option" shows "gval (Bc True) uu = true"
igSubstClsSTR_imp_igSubstCls: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igSubstClsSTR MOD" shows "igSubstCls MOD"
comm_inv_char: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "x \<oplus>\<^bsub>F\<^esub> y = \<zero>\<^bsub>F\<^esub>" shows "inv\<^bsub>add_monoid F\<^esub> x = y"
AGM_Revision_axioms: shows "AGM_Revision Cn \<top> \<bottom> (.\<longrightarrow>.) .\<not> (.\<and>.) (.\<or>.) (.\<longleftrightarrow>.) (\<^bold>*)"
agree_def: fixes s1 :: "('c, 'd) fmap"   and s2 :: "('c, 'd) fmap" shows "agree s1 s2 \<equiv> \<forall>v. v \<in> fmdom' s1 \<and> v \<in> fmdom' s2 \<longrightarrow> fmlookup s1 v = fmlookup s2 v"
the_\<theta>_simps_4: shows "local.dom \<omega>.the_\<theta> = t\<^sub>0 \<star> \<omega>.chine"
h1: shows "\<Inter> {X. s2p (\<phi> X) x} \<subseteq> Y"
finite_cmul_nonzero: fixes c :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int" shows "finite {x. c * lookup a x \<noteq> 0}"
card_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (=) card card"
that_2: fixes l :: "V" assumes "Ord l"   and "ZFC_in_HOL.succ l = k" shows "thesis"
cross_ratio_inf: fixes u :: "complex_homo"   and w :: "complex_homo"   and v :: "complex_homo" assumes "u \<noteq> w"   and "v \<noteq> w" shows "cross_ratio w u v w = \<infinity>\<^sub>h"
ncoplanar_perm_14: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet C B A D"
ll2f_llength: fixes r :: "'b llist"   and A :: "'b set" assumes "r \<in> A\<^sup>\<star>" shows "r !! LList2.llength r = None"
P_def: shows "P \<equiv> \<lambda>k w w\<^sub>1. \<exists>l w\<^sub>2 u. reduced_run q (stake k v) (sdrop k v) l w w\<^sub>1 w\<^sub>2 u"
t_merge_height: fixes l :: "'b tree"   and r :: "'b tree" shows "t_merge l r \<le> max (height l) (height r)"
minim_Under_under: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wo_rel r"   and "A \<noteq> {}"   and "A \<subseteq> Field r" shows "Under r A = under r (minim r A)"
rcos_zfact: fixes k :: "int"   and l :: "int"   and r :: "int" assumes "k \<in> ZMod l r" shows "\<exists>x. k = x * l + r"
assn_def: fixes K :: "'a \<Rightarrow> nat \<Rightarrow> assn"   and V :: "'b \<Rightarrow> 'c \<Rightarrow> assn" shows "iam.assn K V \<equiv> hr_comp is_iam (\<langle>the_pure K, the_pure V\<rangle>map_rel)"
irreducible_connect_int: fixes p :: "int poly" assumes "irreducible\<^sub>d p"   and "content p = 1" shows "irreducible p"
6: shows "bprv (eqv (subst \<chi> \<langle>\<chi>\<rangle> xx) (subst \<phi> \<langle>\<psi>\<rangle> xx))"
saturated_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "calculus.saturated Inf Red_I N = (inference_system.Inf_from Inf N \<subseteq> Red_I N)"
f1: shows "A o\<^sub>C\<^sub>L Proj S = P o\<^sub>C\<^sub>L A"
isos_compose: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and f' :: "'b" assumes "category C"   and "category.iso C f"   and "category.iso C f'"   and "partial_magma.arr C (C f' f)" shows "category.iso C (C f' f)"
routing_ipassmt_wi_def: fixes tbl :: "'b routing_rule list" shows "routing_ipassmt_wi tbl \<equiv> reduce_range_destination (routing_port_ranges tbl wordinterval_UNIV)"
G_conv_def: fixes b_of_a :: "'d \<Rightarrow> 'c"   and G :: "char list \<Rightarrow> 'd list \<Rightarrow> bool" shows "G_conv b_of_a G \<equiv> \<lambda>p bs. G p (map (inv b_of_a) bs)"
open_empty: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.topological_space open" shows "open {}"
perp_perp_in: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A B Perp C A" shows "A PerpAt A B C A "
del_min_height: fixes t :: "'b tree" assumes "braun t" shows "height (del_min t) \<le> height t"
hom_into_subgroup_eq_gen: fixes G :: "('e, 'f) monoid_scheme"   and h :: "'g \<Rightarrow> 'e"   and K :: "('g, 'h) monoid_scheme"   and H :: "'e set" assumes "Group.group G" shows "(h \<in> hom K (subgroup_generated G H)) = (h \<in> hom K G \<and> h ` carrier K \<subseteq> carrier (subgroup_generated G H))"
nstutinv_def: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" shows "NSTUTINV P \<equiv> \<forall>\<sigma> \<tau>. first \<sigma> = first \<tau> \<and> tail \<sigma> \<approx> tail \<tau> \<longrightarrow> P \<sigma> = P \<tau>"
nothingin: fixes s :: "'a" assumes "s \<in> S" shows "s \<notin> set (take (index qs a) qs)"
frechet_derivative_at_real_eq_scaleR: fixes f :: "real \<Rightarrow> 'c"   and x :: "real"   and v :: "real" assumes "f differentiable at x"   and "NO_MATCH (1::'d) v" shows "frechet_derivative f (at x) v = v *\<^sub>R frechet_derivative f (at x) 1"
dom_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.seq g f" shows "CC.dom (CC.comp g f) = CC.dom f"
C2O_inj: fixes n :: "C"   and m :: "C" assumes "C2O n = C2O m" shows "n = m"
orientable_def: fixes x :: "'a" shows "bounded_distrib_allegory.orientable (\<sqinter>) (\<squnion>) bot id x \<equiv> \<exists>y. y \<squnion> id y = x \<and> y \<sqinter> id y = bot"
norm_pn_equiv: fixes x :: "'a itrm"   and f :: "dB" assumes "x \<in> CF" shows "norm_pn f x \<simeq> Pure f \<diamondop> x"
T_on_rand'_as_sum: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and A :: "('b \<Rightarrow> 'e pmf) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) pmf)"   and s0 :: "('b \<times> 'e) pmf"   and rs :: "'c list" shows "On_Off.T_on_rand' step t A s0 rs = (\<Sum>n<length rs. On_Off.T_on_rand' step t A (On_Off.config'_rand step A s0 (take n rs)) [rs ! n])"
rel_Class_iff: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b" shows "x \<sim> y = (y \<in> Sigma I \<FF> \<and> x \<in> rel.Class y)"
grover_state_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" shows "grover_state n f \<equiv> 1 < n \<and> 0 < card {i. i < 2 ^ n \<and> f i} \<and> card {i. i < 2 ^ n \<and> f i} < 2 ^ n"
next_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "next \<phi> \<pi>l \<equiv> \<phi> (map stl \<pi>l)"
weakEqSubset: shows "substClosed congruence \<subseteq> weakBisim"
reduced_normif: fixes env :: "('a, bool) mapping"   and t :: "'a ifex"   and t1 :: "'a ifex"   and t2 :: "'a ifex" shows "reduced (normif env t t1 t2) (Mapping.keys env)"
intro: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')" shows "l_set_child_nodes_get_tag_name_axioms set_child_nodes_locs get_tag_name_locs"
count_prime_factorization_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "count (factorial_semiring.prime_factorization zero times one normalize x) p = factorial_semiring.multiplicity times one p x"
orbit_size: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e" assumes "group_action G \<phi> M"   and "finite (carrier G)"   and "m \<in> M" shows "order G = card (group_action.orbit G \<phi> M m) * card (group_action.stabilizer G \<phi> m)"
add_left: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and a' :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'c" shows "(a + a') *\<^sub>V b = a *\<^sub>V b + a' *\<^sub>V b"
StepRelSem_def: fixes A :: "('d, 'e, 'f) hierauto" shows "StepRelSem A \<equiv> {(ST, ST'). HA ST = A \<and> (HPT ST \<noteq> {} \<longrightarrow> (\<exists>TS\<in>HPT ST. \<exists>U\<in>ResolveRacing TS. ST' = StepStatus ST TS U)) \<and> (HPT ST = {} \<longrightarrow> ST' = StepStatus ST {} DefaultUpdate)}"
closed_Un_complement_component: fixes S :: "'a set"   and c :: "'a set" assumes "closed S"   and "c \<in> components (- S)" shows "closed (S \<union> c)"
subset_one: shows "(A \<inter> B \<subseteq> {\<one>}) = (A \<inter> B = {\<one>})"
split: fixes P :: "'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'c tree list \<Rightarrow> 'b"   and tree :: "'c tree" shows "P (case tree of Node x xa xb \<Rightarrow> f x xa xb) = (\<forall>x1 x2 x3. tree = Node x1 x2 x3 \<longrightarrow> P (f x1 x2 x3))"
wg_def: fixes F :: "'b program"   and Y :: "'b program set" shows "wg F Y \<equiv> \<Union> {X. F \<in> X guarantees Y}"
prems_2: shows "mod_type_class.to_nat a < Suc n"
assms: shows "low_equal s1 s2 \<and> load_word_mem s1 addra 10 = load_word_mem s2 addra 10"
nc: shows "var \<notin> vars (liftPoly 0 0 c)"