hcju/isabelleps huggingface dataset:
transpose_mult: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "(A * B)\<^sup>T = B\<^sup>T * A\<^sup>T"
leftmost_Derives1: fixes i :: "nat"   and a :: "symbol list"   and j :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "leftmost i a"   and "Derives1 a j r b" shows "i \<le> j"
star_slide: fixes x :: "'a"   and y :: "'a" shows "(x ; y)\<^sup>\<star> ; x = x ; (y ; x)\<^sup>\<star>"
invar_def: fixes t :: "'b tree" shows "splay.invar t = (True \<and> Sorted_Less.sorted (inorder t))"
rotater_rev': fixes n :: "nat"   and xs :: "'b list" shows "rotater n (rev xs) = rev (rotate n xs)"
vec_eq_iff: fixes x :: "'b vec"   and y :: "'b vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $ i = y $ i))"
nonzero_inverse_scaleC_distrib: fixes a :: "complex"   and x :: "'b" assumes "a \<noteq> 0"   and "x \<noteq> (0::'b)" shows "inverse (a *\<^sub>C x) = inverse x /\<^sub>C a"
l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
Restr_incr: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set"   and A :: "'c set"   and A' :: "'c set" assumes "r \<subseteq> r'"   and "A \<subseteq> A'" shows "Restr r A \<subseteq> Restr r' A'"
reachable_daD: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
c_redu_subst2: fixes M :: "trm"   and M' :: "trm"   and c :: "coname"   and P :: "trm"   and y :: "name" assumes "M \<longrightarrow>\<^sub>c M'"   and "c \<sharp> P"   and "y \<sharp> M" shows "M{c:=y.P} \<longrightarrow>\<^sub>c M'{c:=y.P}"
numeral_less_floor: fixes v :: "num"   and x :: "'b" shows "(numeral v < \<lfloor>x\<rfloor>) = (numeral v + (1::'b) \<le> x)"
exhaust: fixes y :: "pdf_dist" obtains "y = Bernoulli"   | "y = UniformInt"   | "y = UniformReal"   | "y = Poisson"   | "y = Gaussian"
integral_cong_AE: fixes f :: "'e \<Rightarrow> 'f"   and M :: "'e measure"   and g :: "'e \<Rightarrow> 'f" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
floor_less_cancel: fixes x :: "'b"   and y :: "'b" assumes "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor>" shows "x < y"
simps_2: fixes s :: "'c net_state"   and t :: "'c net_state" shows "netmap (SubnetS s t) = netmap s ++ netmap t"
can_lock_preserved: fixes s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t' :: "'thread_id"   and ta' :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list" assumes "s \<in> correct_jvm_state \<Phi>"   and "P \<turnstile> s -t'\<triangleright>ta'\<rightarrow>\<^bsub>jvmd\<^esub> s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "execd_mthr.must_sync P t x (shr s)" shows "execd_mthr.must_sync P t x (shr s')"
untame_def: fixes P :: "graph \<Rightarrow> bool" shows "untame P \<equiv> \<forall>g. final g \<and> P g \<longrightarrow> \<not> tame g"
ex_mset_zip_left: fixes xs :: "'k list"   and ys :: "'l list"   and xs' :: "'k list" assumes "length xs = length ys"   and "mset xs' = mset xs" shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
path_hd: fixes p :: "'c list"   and E :: "('c \<times> 'c) set"   and v :: "'c"   and w :: "'c" assumes "p \<noteq> []"   and "path E v p w" shows "hd p = v"
dense_closed_pp: fixes x :: "'a" assumes "dense x" shows "dense (- - x)"
finite_monoid_imp_card_positive: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "submonoid H G"   and "finite (carrier G)" shows "0 < card H"
orbit_eq: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b" assumes "local_flow f T S \<phi>"   and "s \<in> S" shows "local_flow.orbit f T S s = {\<phi> t s |t. t \<in> T}"
vs_2: shows "set vs = {v'} \<or> set vs = {v, v'}"
wf_J_mdecl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp" shows "wf_J_mdecl P C (M, Ts, T, pns, body) \<equiv> length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> (\<exists>T'. P,[this \<mapsto> Class C, pns [\<mapsto>] Ts] \<turnstile> body :: T' \<and> P \<turnstile> T' \<le> T) \<and> \<D> body \<lfloor>{this} \<union> set pns\<rfloor>"
singleton_in_conc: fixes x :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "([x] \<in> A @@ B) = ([x] \<in> A \<and> [] \<in> B \<or> [] \<in> A \<and> [x] \<in> B)"
ev_shift: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and xl :: "'a list" assumes "ev \<phi> xs" shows "ev \<phi> (xl @- xs)"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q \<alpha> Q'"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* C"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) \<alpha> (P \<parallel> Q')"
simps_6: fixes c :: "'a"   and r :: "'a rexp" shows "pderiv c (Star r) = Timess (pderiv c r) (Star r)"
subst_atm_mset_list_empty: fixes \<sigma> :: "'s" shows "[] \<cdot>aml \<sigma> = []"
member_def: fixes xs :: "'g list"   and x :: "'g" shows "List.member xs x = (x \<in> set xs)"
poincare_distance_zero: fixes x :: "complex_homo" assumes "x \<in> unit_disc" shows "poincare_distance 0\<^sub>h x = (let x' = to_complex x in \<bar>ln (Re (cor ((1 - cmod x') / (1 + cmod x'))))\<bar>)"
apex_def: shows "\<nu>\<pi>.apex \<equiv> C.dom \<nu>\<pi>.leg0"
J: shows "filter (\<lambda>x. key x = k) (nths xs A) = filter (\<lambda>x. key x = k) (nths (rev vs) (A \<inter> {..<maxi xs key})) @ [y]"
im_sheaf_morphisms_def: fixes U :: "'c set"   and V :: "'c set" shows "im_sheaf_morphisms U V \<equiv> \<rho>\<^sub>X (f \<^sup>\<inverse> X U) (f \<^sup>\<inverse> X V)"
matrix_add_0_right: fixes A :: "('c, 'd, 'e) matrix" shows "A + 0 = A"
ads_d_def: fixes x :: "'a" shows "antidomain_semiringl.ads_d antirange_op x = ar ar x"
mycasexy_def: fixes x :: "'a"   and y :: "'a" shows "mycasexy x y = Plus (seq [Atom x, Atom y]) (seq [Atom y, Atom x])"
tn_def: shows "tn = tn' @ [t1, t2, t3]"
image_Un: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and B :: "'f set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
dsg1: fixes x :: "'a" shows "d x \<cdot> x = x"
imply_swap_append: fixes ps :: "('a, 'b) fm list"   and qs :: "('a, 'b) fm list"   and r :: "('a, 'b) fm" assumes "ps @ qs \<turnstile> r" shows "qs @ ps \<turnstile> r"
compare_double_less: fixes x :: "double"   and y :: "double" shows "(compare_double x y < 0) = (Code_Double.is_nan x \<and> \<not> Code_Double.is_nan y \<or> fcompare_double x y = ccode.Lt)"
f_the_inv_into_f: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and y :: "'d" assumes "inj_on f A"   and "y \<in> f ` A" shows "f (the_inv_into A f y) = y"
fresh_star_plus: fixes a :: "atom set"   and p :: "perm"   and q :: "perm" assumes "a \<sharp>* p"   and "a \<sharp>* q" shows "a \<sharp>* (p + q)"
nth_bintr: fixes m :: "nat"   and w :: "int"   and n :: "nat" shows "bit (take_bit m w) n = (n < m \<and> bit w n)"
word_rsplit_len_indep: fixes u :: "'d word"   and v :: "'d word" shows "length (word_rsplit u) = length (word_rsplit v)"
retract_cyclic1: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'d list" assumes "cyclic1 (map f xs) ys" shows "\<exists>zs. cyclic xs zs \<and> ys = map f zs"
rtranclD: fixes a :: "'a"   and b :: "'a"   and R :: "('a \<times> 'a) set" assumes "(a, b) \<in> R\<^sup>*" shows "a = b \<or> a \<noteq> b \<and> (a, b) \<in> R\<^sup>+"
funpower: shows "funpower f (Suc n) = f \<circ> funpower f n"
oguard_def: fixes G :: "'c \<Rightarrow> bool" shows "oguard G \<equiv> \<lambda>s. if G s then Some () else None"
cases: fixes a1 :: "'b set list"   and a2 :: "'b"   and a3 :: "'b" assumes "of_weak_ranking a1 a2 a3" obtains i :: "nat" and j :: "nat" and xs :: "'b set list" and x :: "'b" and y :: "'b" where "a1 = xs" and "a2 = y" and "a3 = x" and "i \<le> j" and "i < length xs" and "j < length xs" and "x \<in> xs ! i" and "y \<in> xs ! j"
get_msgs_def: fixes snd_f :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> 'e"   and cfg :: "'d \<Rightarrow> 'f"   and HO :: "'d \<Rightarrow> 'd set"   and SHO :: "'d \<Rightarrow> 'd set" shows "get_msgs snd_f cfg HO SHO \<equiv> \<lambda>p. {\<mu>. (\<forall>q. (q \<in> HO p) = (\<mu> q \<noteq> None)) \<and> (\<forall>q. q \<in> SHO p \<inter> HO p \<longrightarrow> \<mu> q = Some (snd_f q p (cfg q)))}"
natural_transformation_axioms: shows "natural_transformation C D K M \<mu>\<sigma>.map"
tarski_absolute_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_absolute C B \<equiv> tarski_absolute_space C B \<and> tarski_absolute_axioms C B"
converse_def: shows "converse \<equiv> \<lambda>r. {(xa, x). (\<lambda>x xa. (x, xa) \<in> r)\<inverse>\<inverse> xa x}"
row_factor_def: fixes ris :: "nat list" shows "row_factor ris = a * prod_list (map2 (\<lambda>i B. lookup B [i]) ris (nths Bs I))"
nexts_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<circle>F \<equiv> %s. F (tail s)"
that: fixes f :: "nat \<Rightarrow> real" assumes "aform_val e (aform_of_ivl a b) \<bullet> c = aform_val f (aform_of_ivl (a \<bullet> c) (b \<bullet> c))"   and "f \<in> UNIV \<rightarrow> {- 1..1}" shows "thesis"
sel_1: fixes xs :: "'b list" shows "shd (cycle xs) = hd xs"
Decrypt_def: fixes K :: "nat"   and X :: "msg" shows "Decrypt K X = Abs_Msg (\<Union>U\<in>Rep_Msg X. msgrel `` {DECRYPT K U})"
simps_1: fixes i :: "nat"   and j :: "nat"   and T :: "bool list list" shows "check_eq (Leaf i) (Leaf j) T = (\<not> tr_lookup T i j)"
butlast_elem: fixes n :: "nat"   and ba :: "nat"   and aa :: "nat"   and lm1 :: "'a list"   and lm2 :: "'a list"   and lm3 :: "'a list"   and lm4 :: "'a list" assumes "Suc n \<le> ba - aa"   and "aa < ba"   and "length lm1 = aa"   and "length lm2 = Suc n"   and "length lm3 = ba - Suc (aa + n)" shows "(lm1 @ (0::'a) \<up> n @ last lm2 # lm3 @ butlast lm2 @ (0::'a) # lm4) ! (ba + n) = (0::'a)"
simps_1: fixes xs :: "'c"   and x :: "'d" shows "qSkel (qVar xs x) = Branch Map.empty Map.empty"
fmdom'_drop_set: fixes A :: "'a set"   and m :: "('a, 'b) fmap" shows "fmdom' (fmdrop_set A m) = fmdom' m - A"
tree_sigma_def: fixes M :: "'d measure" shows "tree_sigma M = sigma (trees (space M)) (trees_cyl ` trees (sets M))"
ResB_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x"   and "x \<sharp> P" shows "<\<nu>y>P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> <\<nu>y>P'"
type_unicity_in_context: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "valid \<Gamma>"   and "(x, T\<^sub>1) \<in> set \<Gamma>"   and "(x, T\<^sub>2) \<in> set \<Gamma>" shows "T\<^sub>1 = T\<^sub>2"
g_fin: shows "integral\<^sup>N M g \<noteq> \<infinity>"
leg0_simps_2: shows "arr (tab\<^sub>0 (r \<star> s))"
transpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "transp r"   and "r x y"   and "r y z" obtains "r x z"
simps_4: fixes K :: "nat"   and X :: "freemsg" shows "freediscrim (DECRYPT K X) = freediscrim X - 2"
add_less_cancel_right: fixes a :: "complex"   and c :: "complex"   and b :: "complex" shows "(a + c <\<^sub>\<complex> b + c) = (a <\<^sub>\<complex> b)"
cases: fixes x :: "nat" obtains "x = 0"   | "x = Suc 0"   | n :: "nat" where "x = Suc (Suc n)"
Spy_see_priEK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> ns_public" shows "(Key (priEK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
vars0_def: shows "a.vars0 = vars1 \<union> vars2"
open_enat_iff: fixes A :: "enat set" shows "open A = (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {enat x<..} \<subseteq> A))"
eq: shows "size (queue_to_multiset (xs @ [x])) = size (queue_to_multiset xs) + 2 ^ rank x"
atLeastLessThan_reindex: fixes h :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "bij_betw h {m..<n} {h m..<h n}" shows "sum g {h m..<h n} = sum (g \<circ> h) {m..<n}"
powser_zero: fixes f :: "nat \<Rightarrow> 'a" shows "(\<Sum>n. f n * (0::'a) ^ n) = f 0"
bound_of_sound: fixes P :: "'a \<Rightarrow> real" assumes "sound P" shows "0 \<le> bound_of P"
translation_def: shows "translation = (\<lambda>a\<in>G. \<lambda>x\<in>G. x \<cdot> a)"
upclosed_upset_set: fixes X :: "'a set" shows "upclosed_set (\<Up> X)"
quasi_isometry_betweenD_1: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_between X Y f" shows "lambda C -quasi_isometry_on X f"
pp_star_def: fixes X :: "'a ppath set" shows "pp_star X \<equiv> \<Union> (range (ppath_dioid.power X))"
cexp_mono2: fixes p2 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p2 \<le>o r2"   and "Card_order q"   and "p2 =o czero \<Longrightarrow> r2 =o czero"   and "Card_order p2" shows "q ^c p2 \<le>o q ^c r2"
det_column_add: fixes i :: "'a"   and j :: "'a"   and A :: "(('b, 'a) vec, 'a) vec"   and q :: "'b" assumes "i \<noteq> j" shows "det (column_add A i j q) = det A"
bheight_baliL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "bheight l = bheight r" shows "bheight (baliL l a r) = Suc (bheight l)"
tranclp_eq_tranclp_on: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "r\<^sup>+\<^sup>+ = on (\<lambda>x. True) r\<^sup>+\<^sup>+"
hypnat_of_nat_neq_whn: fixes n :: "nat" shows "hypnat_of_nat n \<noteq> whn"
rref_upt_condition1: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"
inversions_between_permute2: fixes \<pi> :: "nat \<Rightarrow> nat"   and ys :: "'a list"   and xs :: "'a list" assumes "\<pi> permutes {..<length ys}" shows "inversions_between xs (permute_list \<pi> ys) = map_prod id (inv \<pi>) ` inversions_between xs ys"
is_processT: fixes P :: "'a process" shows "([], {}) \<in> F P \<and> (\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<in> F P \<longrightarrow> (s, {}) \<in> F P) \<and> (\<forall>s X Y. (s, Y) \<in> F P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> F P) \<and> (\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P) \<and> (\<forall>s t. s \<in> D P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> D P) \<and> (\<forall>s X. s \<in> D P \<longrightarrow> (s, X) \<in> F P) \<and> (\<forall>s. s @ [tick] \<in> D P \<longrightarrow> s \<in> D P)"
of_int: fixes na :: "nat"   and n :: "int" shows "pbernpoly na (real_of_int n) = pbernpoly na 0"
defAss'_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and g :: "'a"   and m :: "'b"   and v :: "'d" shows "CFG_base.defAss' \<alpha>n invar inEdges' Entry defs g m v = (\<forall>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns m \<longrightarrow> (\<exists>n\<in>set ns. v \<in> defs g n))"
open_segment_eq_empty: fixes a :: "'a"   and b :: "'a" shows "(open_segment a b = {}) = (a = b)"
eucl_less_le_not_le: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
ap_specD3: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) set"   and m :: "nat"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and h :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and g :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "ap_spec ap"   and "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "(set gs \<union> set bs) \<inter> set hs = {}"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "h \<noteq> g"   and "fst h \<noteq> 0"   and "fst g \<noteq> 0"   and "(h, g) \<in>\<^sub>p set ps -\<^sub>p set (ap gs bs ps hs data)"   and "\<And>a b. \<lbrakk>a \<in> set hs; b \<in> set gs \<union> set bs \<union> set hs; (a, b) \<in>\<^sub>p set (ap gs bs ps hs data); fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)" shows "crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)"
cn_arity: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and a :: "abc_inst list"   and b :: "nat"   and c :: "nat" assumes "rec_ci (Cn n f gs) = (a, b, c)" shows "b = n"
IdI: fixes a :: "'d" shows "(a, a) \<in> Id"
bigtheta_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "\<Theta>[F](g) = O[F](g) \<inter> \<Omega>[F](g)"
stepResultsinR_def: fixes p :: "'com option"   and p' :: "'com option"   and R :: "('com list \<times> 'com list) set" shows "stepResultsinR p p' R \<equiv> p = None \<and> p' = None \<or> (\<exists>c c'. p = Some c \<and> p' = Some c' \<and> ([c], [c']) \<in> R)"
shadow_root_eq_h: fixes element_ptr :: "(_) element_ptr"   and shadow_root_opt :: "(_) shadow_root_ptr option" assumes "element_ptr \<noteq> new_element_ptr" shows "h \<turnstile> get_shadow_root element_ptr \<rightarrow>\<^sub>r shadow_root_opt = h2 \<turnstile> get_shadow_root element_ptr \<rightarrow>\<^sub>r shadow_root_opt"
d_zero_ax: shows "d (0::'a) = (0::'a)"
set_zero_plus': fixes C :: "'a set" shows "(0::'a) +o C = C"
assms_3: shows "(i, Store e mo i) \<in> set (node_deliver_messages xs)"
summands_valid: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "A \<subseteq> carrier (md N)"   and "a \<in> A \<rightarrow> carrier R" shows "\<forall>v\<in>A. a v \<odot>\<^bsub>md N\<^esub> v \<in> carrier (md N)"
B_def: shows "B = (\<Union>n. {CHR ''a'' ^^^ n})"
sel_3: fixes x11 :: "nat"   and x12 :: "nat" shows "fork (picks x11 x12) = x12"
beta_subst: fixes M :: "lam"   and M' :: "lam"   and x :: "name"   and N :: "lam" assumes " M \<longrightarrow>\<^sub>\<beta> M'" shows " M[x::=N] \<longrightarrow>\<^sub>\<beta> M'[x::=N]"
lengthClosed: fixes p :: "(name \<times> name) list"   and xs :: "'c list" shows "length (p \<bullet> xs) = length xs"
inf: shows "AlternativeSmallStep.inf \<Gamma> (c # cs) css s"
br_rel_prod: fixes a :: "'a \<Rightarrow> 'd"   and I :: "'a \<Rightarrow> bool"   and b :: "'c \<Rightarrow> 'e"   and J :: "'c \<Rightarrow> bool" shows "br a I \<times>\<^sub>r br b J = br (\<lambda>(x, y). (a x, b y)) (\<lambda>(x, y). I x \<and> J y)"
rim_1: shows "r ` (W \<union> S) \<subseteq> S"
arrow: shows "D.ide (Go (C.cod f)) \<and> \<guillemotleft>\<epsilon>o (C.cod f) : F (Go (C.cod f)) \<rightarrow>\<^sub>C C.cod f\<guillemotright>"
sff_1: shows "p = smult c (\<Prod>(a, i)\<leftarrow>bs. a ^ Suc i)"
simps_8: fixes m :: "nat"   and p :: "('a, 'b) form" shows "closed m (Forall p) = closed (Suc m) p"
min_gallery_pgallery: fixes X :: "'c set set"   and xs :: "'c set list" assumes "ChamberComplex X"   and "SimplicialComplex.min_maxsimpchain X xs" shows "SimplicialComplex.pmaxsimpchain X xs"
CATCH_def: fixes m :: "('f + 'e) nres"   and h :: "'f \<Rightarrow> ('d + 'e) nres" shows "CATCH m h \<equiv> m \<bind> (\<lambda>r. case r of Inl e \<Rightarrow> h e | Inr r \<Rightarrow> RETURN (Inr r))"
vector_eq_ldot: fixes y :: "'a"   and z :: "'a" shows "(\<forall>x. x \<bullet> y = x \<bullet> z) = (y = z)"
poly_cond2_def: fixes p :: "int poly" shows "poly_cond2 p \<equiv> poly_cond p \<and> 1 < degree p"
safe_regex_safe: fixes safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and fv :: "'a \<Rightarrow> 'b set"   and m :: "modality"   and g :: "safety"   and r :: "'a Regex.regex"   and x :: "'a" assumes "\<And>g r. safe g r \<Longrightarrow> safe Lax r"   and "Regex.safe_regex fv safe m g r"   and "x \<in> regex.atms r" shows "safe Lax x"