hcju/isabelleps huggingface dataset:
closedin_path_component_of_locally_path_connected_space: fixes X :: "'c topology"   and x :: "'c" assumes "locally_path_connected_space X" shows "closedin X (path_component_of_set X x)"
hyps': fixes x :: "real" shows "(real n + 1) * bernpoly n (x + 1) - (real n + 1) * bernpoly n x = real n * x ^ (n - Suc 0) * real (Suc n)"
closure_involutive: fixes Z :: "('c \<Rightarrow> real) set" shows "Closure\<^sub>\<alpha> Closure\<^sub>\<alpha> Z = Closure\<^sub>\<alpha> Z"
rreq_rrep_fresh_2: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and oip :: "nat"   and sip :: "nat" shows "rreq_rrep_fresh crt (Rrep hops dip dsn oip sip) = (sip \<noteq> dip \<longrightarrow> dip \<in> kD crt \<and> sqn crt dip = dsn \<and> the (dhops crt dip) = hops \<and> the (flag crt dip) = val)"
image_sub0: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
A: shows "weak_reduction_simulation (TRel\<^sup>*) Target"
proj2_Col_iff_euclid: fixes a :: "(real, 2) vec"   and b :: "(real, 2) vec"   and c :: "(real, 2) vec" shows "proj2_Col (proj2_pt a) (proj2_pt b) (proj2_pt c) = real_euclid.Col a b c"
HMA_diagonal_to_Smith_i_PQ_aux: fixes P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat"   and P' :: "(('b, 'nr) Finite_Cartesian_Product.vec, 'nr) Finite_Cartesian_Product.vec"   and A' :: "(('b, 'nc) Finite_Cartesian_Product.vec, 'nr) Finite_Cartesian_Product.vec"   and Q' :: "(('b, 'nc) Finite_Cartesian_Product.vec, 'nc) Finite_Cartesian_Product.vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" assumes "HMA_M3 (P, A, Q) (P', A', Q')" shows "HMA_M3 (diagonal_to_Smith_i_PQ_JNF xs i bezout (P, A, Q)) (diagonal_to_Smith_i_PQ xs i bezout (P', A', Q'))"
simps_2: fixes x :: "'a" shows "remdups_adj [x] = [x]"
uminus_galois_var: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(\<partial> \<circ> f = g) = (\<partial> \<circ> g = f)"
sign_simps_2: fixes af :: "'c"   and bf :: "'c" shows "af + bf = bf + af"
HPhase1or2ReadThen_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and p :: "Proc"   and v :: "InputsOrNi" assumes "HPhase1or2ReadThen s s' q d p"   and "valueChosen s v"   and "v \<in> Inputs" shows "valueChosen s' v"
oracle\<^sub>1_def: fixes \<eta> :: "nat" shows "oracle\<^sub>1 \<eta> = vkey_oracle \<eta> \<oplus>\<^sub>O sign_oracle \<eta>"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "max_Var_floatarith (f x) \<le> m"
eq_det: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<simeq> b"   and "determ (wp a)" shows "determ (wp b)"
reachablep_reachable_eq: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool" shows "reachablep A I = (\<lambda>x. x \<in> reachable A I)"
equiv_transD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "equiv_trans t u"   and "sound P" shows "t P = u P"
continuous_map_from_subtopology_mono: fixes X :: "'b topology"   and T :: "'b set"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "continuous_map (subtopology X T) X' f"   and "S \<subseteq> T" shows "continuous_map (subtopology X S) X' f"
homeomorphism_chart_last: shows "homeomorphism (Collect chart_last_domainP) UNIV chart_last chart_last_inv"
simps: fixes G :: "'a set set" shows "vertices G = \<Union> G"
proj_stoch_proc_geom_rng: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q" shows "proj_stoch_proc geom_proc n \<in> N \<rightarrow>\<^sub>M stream_space borel"
selectLiteral_def: fixes Vbl :: "nat set"   and state :: "State" shows "Vbl - vars (elements (getM state)) \<noteq> {} \<longrightarrow> var (selectLiteral state Vbl) \<in> Vbl - vars (elements (getM state))"
op_dg_Hom: fixes \<CC> :: "V"   and a :: "V"   and b :: "V" shows "Hom (op_dg \<CC>) a b = Hom \<CC> b a"
fvi_And: fixes b :: "nat"   and \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "fvi b (And \<phi> \<psi>) = fvi b \<phi> \<union> fvi b \<psi>"
distributed_marginal_eq_joint2: fixes T :: "'e measure"   and S :: "'f measure"   and Y :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> ennreal"   and X :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'f"   and Pxy :: "'f \<times> 'e \<Rightarrow> ennreal" assumes "sigma_finite_measure T"   and "sigma_finite_measure S"   and "distributed (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x))) T Y Py"   and "distributed (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x))) (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "AE y in T. Py y = \<integral>\<^sup>+ x. Pxy (x, y) \<partial>S"
induct: fixes P :: "('b \<times> nat) option \<Rightarrow> 'b \<Rightarrow> lock_action list \<Rightarrow> bool"   and a0 :: "('b \<times> nat) option"   and a1 :: "'b"   and a2 :: "lock_action list" assumes "\<And>l t. P l t []"   and "\<And>l t L Ls. P (upd_lock l t L) t Ls \<Longrightarrow> P l t (L # Ls)" shows "P a0 a1 a2"
lapp_all_invT: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" assumes "r @@ s \<in> A\<^sup>\<infinity>" shows "r \<in> A\<^sup>\<infinity>"
map\<^sub>1_simps_4: fixes f :: "'a" assumes "C.ide f" shows "D.dom (\<tau>\<^sub>1 f) = G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)"
mask_Suc_exp: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask (Suc n) = or (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) (mask n)"
prime_intI: fixes p :: "int" assumes "2 \<le> p"   and "\<And>m n. p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n" shows "prime p"
xX_simp4: shows "backflows (stateful \<union> X) - E - (backflows (X - {x}) - E) = backflows (filternew_flows_state \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> {x}\<rparr>)"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(Atom x1 = Atom y1) = (x1 = y1)"
TameEnum_def: shows "TameEnum \<equiv> \<Union> (TameEnumP ` {..3})"
S'_def: shows "S' \<equiv> {x \<in> U. setdist {x} S \<le> setdist {x} T}"
DERIV_cdivide: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and c :: "'a" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
\<J>_0_0_item_end: 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 x :: "item"   and Doc :: "character list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "x \<in> LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc 0 0" shows "item_end x = 0"
separating_of_bipartite_def: fixes S :: "('v + 'v) set" shows "separating_of_bipartite S = (let A_S = Inl -` S; B_S = Inr -` S in A_S \<inter> B_S \<union> A \<Gamma> \<inter> A_S \<union> B \<Gamma> \<inter> B_S)"
word_not_simps_3: fixes y :: "'b word" shows "\<not> - 1 < y"
map_ident: fixes t :: "'c multiset" shows "{#x. x \<in># t#} = t"
prefix_Nil: fixes xs :: "'c list" shows "prefix xs [] = (xs = [])"
Keys_to_list_map_key_PP: fixes fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.punit.Keys_to_list (map (Poly_Mapping.map_key PP) fs) = map pp.mapping_of (punit.Keys_to_list fs)"
pre_oldest_configuration_element: fixes q' :: "'b"   and n :: "nat"   and ot :: "nat"   and q :: "'b" assumes "oldest_token q' n = Some ot"   and "q = \<delta> q' (w n)" shows "ot \<in> pre_oldest_tokens q n"
restr_restr: fixes A :: "'a set"   and B :: "'a set"   and al :: "('a \<times> 'b) list" shows "restrictA A (restrictA B al) = restrictA (A \<inter> B) al"
numeral_plus_numeral: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m + n)"
ord_def: fixes i :: "'a"   and j :: "'a" shows "ord i j = (if i < j then LESS else if i = j then EQUAL else GREATER)"
negligible_countable_Union: fixes \<F> :: "'b set set" assumes "countable \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> negligible S" shows "negligible (\<Union> \<F>)"
conc_trans_additional_1: fixes A :: "'b nres"   and R :: "('b \<times> 'c) set"   and B :: "'c nres"   and C :: "'c nres" assumes "A \<le> \<Down> R B"   and "B \<le> C" shows "A \<le> \<Down> R C"
order_on_defs_3: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
simps_3: fixes \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'a"   and p :: "'b" shows "Semantics.matches \<gamma> (Match e) p = \<gamma> e p"
eventually_cofinite: fixes P :: "'b \<Rightarrow> bool" shows "Alm_all P = finite {x. \<not> P x}"
Z_dim: shows "Z \<subseteq> carrier_vec (n + 1)"
less_object_ptr_def: fixes x :: "(_) object_ptr"   and y :: "(_) object_ptr" shows "x < y \<equiv> x \<le> y \<and> \<not> y \<le> x"
h_eq_1: fixes t :: "real" assumes "t \<le> 1" shows "h t = 1"
compact_convex_collinear_segment: fixes S :: "'b set" assumes "S \<noteq> {}"   and "compact S"   and "convex S"   and "collinear S" obtains a :: "'b" and b :: "'b" where "S = closed_segment a b"
line_extension_of_combine_set_length_version: fixes u :: "'a"   and Us :: "'a list" assumes "u \<in> carrier R" shows "line_extension K u {combine Ks Us |Ks. length Ks = length Us \<and> set Ks \<subseteq> K} = {combine Ks (u # Us) |Ks. length Ks = length (u # Us) \<and> set Ks \<subseteq> K}"
NewHeapElem_start_heap_obs_start_addrsD: fixes a :: "'addr"   and CTn :: "htype" assumes "NewHeapElem a CTn \<in> set start_heap_obs" shows "a \<in> set start_addrs"
d: fixes p :: "real \<Rightarrow> complex" assumes "path p"   and "pathstart p = pathstart g"   and "pathfinish p = pathfinish g"   and "\<forall>t\<in>{0..1}. cmod (p t - g t) < d" shows "homotopic_paths (- {z}) g p"
set_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = x21 \<triangleright> set x22"
simps_1: fixes \<A> :: "'a \<Rightarrow> bool"   and k :: "'a" shows "\<A> \<Turnstile> Atom k = \<A> k"
dvd_lcm2: fixes b :: "'a"   and a :: "'a" shows "b dvd lcm a b"
nsplitTree_invpres: fixes n :: "nat"   and s :: "('a, 'b) FingerTreeStruc"   and p :: "'b \<Rightarrow> bool"   and i :: "'b"   and l :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node"   and r :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.is_leveln_ftree n s"   and "FingerTreeStruc.is_measured_ftree s"   and "\<not> p i"   and "p (i + FingerTreeStruc.gmft s)"   and "FingerTreeStruc.nsplitTree p i s = (l, nd, r)" shows "FingerTreeStruc.is_leveln_ftree n l \<and> FingerTreeStruc.is_measured_ftree l \<and> FingerTreeStruc.is_leveln_ftree n r \<and> FingerTreeStruc.is_measured_ftree r \<and> FingerTreeStruc.is_leveln_node n nd \<and> FingerTreeStruc.is_measured_node nd"
eq: shows "Ps' = map (extend ( \<Phi> + \<Gamma>' \<Rightarrow>* \<Psi>1 + \<Delta>')) ps"
assms_3: fixes a :: "'a"   and b :: "'a" shows "(A a b \<and> FA b) = A' a b"
intros_2: fixes m :: "'a" shows "lformula0 (Z m)"
Gram_Schmidt_column_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Gram_Schmidt_column_k A k = (\<chi>a b. (if b = mod_type_class.from_nat k then column b A - proj_onto (column b A) {column i A |i. i < b} else column b A) $ a)"
trace_preserves_simple_ownership_distinct: fixes c :: "nat \<Rightarrow> ('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "simple_ownership_distinct (fst (c n))"   and "trace c n k"   and "\<forall>x<k. safe_delayed (c (n + x))"   and "l \<le> k" shows "simple_ownership_distinct (fst (c (n + l)))"
NewC: fixes G :: "prog"   and s0 :: "st"   and C :: "qtname"   and n :: "nat"   and s1 :: "state"   and a :: "loc"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>halloc CInst C\<succ>a\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<midarrow>n\<rightarrow> s2"
return_time_def: fixes P :: "'a set"   and x :: "'a" shows "Y.return_time P x = (if Y.returns_to P x then SOME t. 0 < t \<and> t \<in> Y.existence_ivl0 x \<and> Y.flow0 x t \<in> P \<and> (\<forall>s\<in>{0<..<t}. Y.flow0 x s \<notin> P) else 0)"
subrelI: fixes r :: "('d \<times> 'e) set"   and s :: "('d \<times> 'e) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
minus_add: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "uminus (plus a b) = plus (uminus b) (uminus a)"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'd" shows "dim_row (Matrix.mat nr nc f) = nr"
SubstFormP_graph_aux_def: shows "SubstFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SubstFormP_aux s v i x x' k. x1 = (v, i, x, x') \<and> x2 = VarP v AND TermP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i x x' (Var s) (Var k))) \<and> atom s \<sharp> (v, i, x, x', k) \<and> atom k \<sharp> (v, i, x, x'))"
\<I>_real_def: shows "\<I>_real \<equiv> \<lambda>\<eta>. \<I>_uniform {x. valid_mac_query \<eta> x} (insert None (Some ` (nlists UNIV \<eta> \<times> nlists UNIV \<eta>)))"
A: fixes x :: "nat"   and xi :: "nat" assumes "nofail (icount $ x)" shows "<hn_val nat_rel x xi> icount_impl $ xi <\<lambda>r. hn_val nat_rel x xi * (\<exists>\<^sub>Axa. nat_assn xa r * \<up> (RETURN xa \<le> icount $ x))>\<^sub>t"
ep': shows "epath (insert (Upair u v) E) u p v"
dependent_zero: fixes A :: "'b set" assumes "(0::'b) \<in> A" shows "vs1.dependent A"
oldDefsI: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> defs g n" shows "var g v \<in> oldDefs g n"
PRelM_def: shows "PRelM = {(N1, N2). N1 \<in> PcalM \<and> N2 \<in> PcalM \<and> (\<exists>g\<in>carrier (G\<lparr>carrier := P\<rparr>). N1 = N2 #>\<^bsub>G\<lparr>carrier := P\<rparr>\<^esub> g)}"
refl_reflcl: fixes r :: "('a \<times> 'a) set" shows "refl (r\<^sup>=)"
inverse_arrow_unique: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and g' :: "'a \<times> 'a" assumes "VoV.inverse_arrows f g"   and "VoV.inverse_arrows f g'" shows "g = g'"
ts_per_per_ts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B TS C D"   and "Per B C A"   and "Per B D A" shows "C D TS A B"
betw_events: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "a \<in> \<E> \<and> b \<in> \<E> \<and> c \<in> \<E>"
pconj_lone: fixes b :: "real" assumes "0 \<le> b" shows "1 .& b = b"
quality_increases_def: fixes \<xi> :: "state"   and \<xi>' :: "state" shows "quality_increases \<xi> \<xi>' \<equiv> (\<forall>dip\<in>kD (rt \<xi>). dip \<in> kD (rt \<xi>') \<and> rt \<xi> \<sqsubseteq>\<^bsub>dip\<^esub> rt \<xi>') \<and> (\<forall>dip. sqn (rt \<xi>) dip \<le> sqn (rt \<xi>') dip)"
assms_3: shows "X \<subseteq> {0..<n + 1}"
aha_2: shows "A = (I2, S, E, Es, SS)"
notin_diff: shows "x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i"
valid_cfgI: fixes s :: "'s"   and cfg :: "'s cfg" assumes "s \<in> S"   and "cfg \<in> cfg_on s" shows "cfg \<in> valid_cfg"
S5_excl: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S5 rmhist p \<longrightarrow> S5 rmhist p \<and> \<not> S1 rmhist p \<and> \<not> S2 rmhist p \<and> \<not> S3 rmhist p \<and> \<not> S4 rmhist p"
unitary_intros_1: shows "unitary (\<lambda>s. 0)"
map_bnds: fixes lf :: "float"   and uf :: "float"   and lb :: "nat \<Rightarrow> float \<Rightarrow> float"   and ub :: "nat \<Rightarrow> float \<Rightarrow> float"   and prec :: "nat"   and l :: "float"   and u :: "float"   and f :: "real \<Rightarrow> real"   and x :: "real" assumes "(lf, uf) = map_bnds lb ub prec (l, u)"   and "mono f"   and "x \<in> {real_of_float l..real_of_float u}"   and "real_of_float (lb prec l) \<le> f (real_of_float l)"   and "f (real_of_float u) \<le> real_of_float (ub prec u)" shows "f x \<in> {real_of_float lf..real_of_float uf}"
is_Eq_iff: fixes a :: "atom" shows "(\<forall>i j. a \<noteq> Less i j) = (\<exists>i j. a = Eq i j)"
SUP_spmf_neq_top': fixes x :: "'c"   and Y :: "'c spmf set" shows "(\<Squnion>p\<in>Y. ennreal (spmf p x)) \<noteq> \<top>"
chain_total: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and C :: "'c set"   and x :: "'c"   and y :: "'c" assumes "pred_on.chain A P C"   and "x \<in> C"   and "y \<in> C" shows "P\<^sup>=\<^sup>= x y \<or> P\<^sup>=\<^sup>= y x"
ExistsMsg: fixes cfg1 :: "('p, 'v, 's) configuration"   and m :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration" assumes "cfg1 \<turnstile> m \<mapsto> cfg2" shows "m \<in># msgs cfg1"
BuC_def: shows "BuC = {x \<in> {0<..int p * int q - 1}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0<..(int q - 1) div 2}}"
dghm_HomCod: shows "\<phi> i\<lparr>HomCod\<rparr> = \<AA> i"
connected_iff_interval: fixes U :: "'a set" shows "connected U = (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z\<ge>x. z \<le> y \<longrightarrow> z \<in> U)"
hom: shows "h \<in> Group.iso (sum_group I G) (sum_group I H)"
has_lock_may_lock: fixes l :: "('c \<times> nat) option"   and t :: "'c" assumes "has_lock l t" shows "may_lock l t"
simps_5: fixes x21 :: "guest"   and x22 :: "room"   and x23 :: "key \<times> key"   and x11 :: "guest"   and x12 :: "room"   and x13 :: "key \<times> key" shows "Enter x21 x22 x23 \<noteq> Check_in x11 x12 x13"
delete_def: fixes x :: "'c"   and t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "delete x t = paint Black (del x t)"
nonEmpty_contain: fixes \<Gamma> :: "'b multiset" assumes "\<Gamma> \<noteq> \<Empt>" shows "\<exists>a. a \<in># \<Gamma>"
emeasure_density: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and A :: "'c set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = \<integral>\<^sup>+ x. f x * indicator A x \<partial>M"
convergent_LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'e" shows "convergent X = X \<longlonglongrightarrow> lim X"
hom_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "h (inv x) = inv\<^bsub>H\<^esub> h x"
mono_neutral_left: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'d)" shows "sum g S = sum g T"
ranking_language: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and f :: "nat \<times> 'd \<Rightarrow> nat" assumes "ranking A w f" shows "w \<notin> language A"
SucState_Conf_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "S \<notin> Target TS"   and "A \<in> the (CompFun (HA ST) S)"   and "S \<in> Conf ST"   and "States A \<inter> ChiStar (HA ST) `` Source TS = {}" shows "\<exists>x. x \<in> States A \<and> x \<in> StepConf (HA ST) (Conf ST) TS"
uminus_nonneg_Reals_iff: fixes x :: "'b" shows "(- x \<in> \<real>\<^sub>\<ge>\<^sub>0) = (x \<in> \<real>\<^sub>\<le>\<^sub>0)"
triangular_to_jnf_vector_def: fixes A :: "'b mat" shows "triangular_to_jnf_vector A \<equiv> let B = ??.Jordan_Normal_Form_Existence.step_2 (??.Jordan_Normal_Form_Existence.step_1 A) in concat (map (jnf_vector \<circ> ??.Jordan_Normal_Form_Existence.step_3) (partition_ev_blocks B []))"
left_idem: fixes a :: "'b"   and b :: "'b" shows "a AND a AND b = a AND b"
ennreal_add_left_cancel_less: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a + b < a + c) = (a \<noteq> \<infinity> \<and> b < c)"
assms_1: shows "pseudonaturally_equivalent V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
2: shows "C.inverse_arrows (F (\<eta> b)) (\<epsilon>' (F b))"
row: fixes ia :: "nat" assumes "ia \<le> n" shows "signof (permutation_insert (n - ia) j p) = (- (1::'a)) ^ (n - ia + j) * signof p"
fct_gen_subalgebra_min: fixes M :: "'b measure"   and P :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "subalgebra M P"   and "f \<in> P \<rightarrow>\<^sub>M N" shows "subalgebra P (fct_gen_subalgebra M N f)"
path2_app: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and ms :: "'node list"   and l :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "g \<turnstile> m-ms\<rightarrow>l" shows "g \<turnstile> n-ns @ tl ms\<rightarrow>l"
DERIV_real_root_generic: fixes n :: "nat"   and x :: "real"   and D :: "real" assumes "0 < n"   and "x \<noteq> 0"   and "\<lbrakk>even n; 0 < x\<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"   and "\<lbrakk>even n; x < 0\<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"   and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))" shows "(root n has_real_derivative D) (at x)"
iT_Plus_empty_iff: fixes I :: "nat set"   and k :: "nat" shows "(I \<oplus> k = {}) = (I = {})"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "wbalanced \<langle>l, x, r\<rangle> = (\<bar>int (size l) - int (size r)\<bar> \<le> 1 \<and> wbalanced l \<and> wbalanced r)"
active_minimal: fixes l2 :: "label" assumes "l2 \<noteq> Old" shows "Old \<sqsubset>l l2"
unifiers_insert_ident: fixes t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "unifiers (insert (t, t) E) = unifiers E"
stieltjes_gamma_aux4: fixes k :: "nat" shows "G $ k = (\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)"
sorted_sort_key: fixes f :: "'c \<Rightarrow> 'b word"   and xs :: "'c list" shows "signed.sorted (map f (signed.sort_key f xs))"
det_state_cover_ass_dist: fixes M :: "('d, 'e, 'f) FSM"   and f :: "'f \<Rightarrow> 'd list"   and s1 :: "'f"   and s2 :: "'f" assumes "is_det_state_cover_ass M f"   and "s1 \<in> d_reachable M (initial M)"   and "s2 \<in> d_reachable M (initial M)"   and "s1 \<noteq> s2" shows "\<not> d_reaches M (initial M) (f s2) s1"
solves_ode_singleton: fixes y :: "real \<Rightarrow> 'a"   and t0 :: "real"   and X :: "'a set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a" assumes "y t0 \<in> X" shows "(y solves_ode f) {t0} X"