hcju/isabelleps huggingface dataset:
Step1_4_5b: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCFail crCh rmCh rst p \<and> $S5 rmhist p \<and> S6 rmhist p$ \<and> unchanged (e p, c p, m p) \<longrightarrow> MemFail memCh (resbar rmhist) p"
inv_into_ordermap: fixes \<alpha> :: "V"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "\<alpha> \<in> elts (ordertype A r)" shows "inv_into A (ordermap A r) \<alpha> \<in> A"
separating_networkI: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "'a set" assumes "\<And>p. path \<Delta> (source \<Delta>) p (sink \<Delta>) \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> source \<Delta> \<in> S" shows "separating_network \<Delta> S"
map_of_SomeD: fixes xs :: "('d \<times> 'c) list"   and k :: "'d"   and y :: "'c" assumes "map_of xs k = Some y" shows "(k, y) \<in> set xs"
phi_bot: shows "stone_phi bot = Filters.filter.Abs_filter {top}"
interp_imp_general: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "C \<le> D"   and "D \<le> D'"   and "interp D \<Turnstile> C"   and "interp D' \<subseteq> \<Union> (production ` CC)" shows "\<Union> (production ` CC) \<Turnstile> C"
find_slot_ptr_in_heap: fixes h :: "(_) heap"   and open_flag :: "bool"   and slotable :: "(_) node_ptr"   and slot_opt :: "(_) element_ptr option" assumes "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r slot_opt" shows "slotable |\<in>| node_ptr_kinds h"
simple_syntactic_and_axioms: fixes S :: "name fset" shows "simple_syntactic_and (\<lambda>t. closed_except t S)"
wgraph_by_map_def: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> (nat \<times> 'b) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list option"   and M_invar :: "'a \<Rightarrow> bool" shows "wgraph_by_map M_empty M_update M_delete M_lookup M_invar \<equiv> Map M_empty M_update M_delete M_lookup M_invar"
invar: fixes s :: "('a, 'b) RBT.rbt" shows "rm.invar s"
bl2: shows "bl' \<in># filter_mset ((\<subseteq>) ps) B"
ord_spmf_bind_reflI: fixes p :: "'a spmf"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) (g x)" shows "ord_spmf R (p \<bind> f) (p \<bind> g)"
update_zip: fixes xs :: "'g list"   and ys :: "'h list"   and i :: "nat"   and xy :: "'g \<times> 'h" shows "(zip xs ys)[i := xy] = zip (xs[i := fst xy]) (ys[i := snd xy])"
elist_stmt_inj_term: fixes t :: "expr list"   and w :: "stmt" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
winding_number_from_innerpath: fixes c1 :: "real \<Rightarrow> complex"   and a :: "complex"   and b :: "complex"   and c2 :: "real \<Rightarrow> complex"   and c :: "real \<Rightarrow> complex"   and z :: "complex"   and d :: "complex" assumes "simple_path c1"   and "pathstart c1 = a"   and "pathfinish c1 = b"   and "simple_path c2"   and "pathstart c2 = a"   and "pathfinish c2 = b"   and "simple_path c"   and "pathstart c = a"   and "pathfinish c = b"   and "path_image c1 \<inter> path_image c2 = {a, b}"   and "path_image c1 \<inter> path_image c = {a, b}"   and "path_image c2 \<inter> path_image c = {a, b}"   and "path_image c \<inter> inside (path_image c1 \<union> path_image c2) \<noteq> {}"   and "z \<in> inside (path_image c1 \<union> path_image c)"   and "winding_number (c1 +++ reversepath c) z = d"   and "a \<noteq> b"   and "d \<noteq> 0" obtains "z \<in> inside (path_image c1 \<union> path_image c2)" and "winding_number (c1 +++ reversepath c2) z = d"
prefix_of_appendD: fixes xs :: "'evt list"   and ys :: "'evt list"   and i :: "nat" assumes "xs @ ys prefix of i" shows "xs prefix of i"
knows_composition3: fixes P :: "specID"   and e :: "Expression list"   and PQ :: "specID"   and Q :: "specID" assumes "knows P e"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ"   and "\<forall>m. m mem e \<longrightarrow> (\<exists>z1. m = kE z1) \<or> (\<exists>z2. m = sE z2)"   and "notSpecKeysSecretsExpr P e"   and "notSpecKeysSecretsExpr Q e" shows "knows PQ e"
get_shadow_root_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_shadow_root element_ptr"
under_zero: assumes "Field r \<noteq> {}" shows "local.under local.zero = {local.zero}"
open_proj_space_def: shows "open = openin (map_topology Proj euclidean)"
assms: shows "run_step d i (s, vs, es) = (s', vs', RSNormal es')"
eval_Assoc_Ide: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Ide v"   and "Src t = Trg u"   and "Src u = Trg v" shows "\<lbrace>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<rbrace> = \<alpha> (\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>)"
hyps: fixes y :: "nat" assumes "length Xs \<le> y" shows "flip y Xs = Xs"
insert_preserve_indices': fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and i :: "'c option"   and ys :: "('c \<times> 'd \<times> bool) list" assumes "Ordered_List.insert xs e i = Some ys" shows "fst ` set (the (Ordered_List.insert xs e i)) = fst ` set xs \<union> {fst e}"
disjoint_support: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "disjoint (range (\<lambda>a. set (support p a)))"
esqrt_def: fixes x :: "ereal" shows "esqrt x = enn2ereal (ennsqrt (e2ennreal x))"
suppa2__conga123: 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"   and E :: "'a"   and F :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SuppA Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.SuppA Bet Cong A' B' C' D E F" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"
analz_image_freshK_simps_60: fixes A :: "'a set" shows "{} \<subseteq> A"
conjug_len: fixes u :: "'b list"   and v :: "'b list" assumes "u \<sim> v" shows "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"
inter_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a \<in># A \<inter># B) = (a \<in># A \<and> a \<in># B)"
c3: fixes as :: "'a list"   and a :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "ps = as @ [a]"   and "ls = xs @ [x]"   and "x \<noteq> a" shows "thesis"
llength_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "llength (iterates f x) = \<infinity>"
bisim_waitI: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2"   and x1' :: "'x1"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x1'' :: "'x1"   and m1' :: "'m1"   and x2' :: "'x2"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x2'' :: "'x2"   and m2' :: "'m2"   and w :: "'w" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "r1.silent_moves t (x1, m1) (x1', m1)"   and "r1.r_syntax t x1' m1 ta1 x1'' m1'"   and "\<not> \<tau>move1 (x1', m1) ta1 (x1'', m1')"   and "r2.silent_moves t (x2, m2) (x2', m2)"   and "r2.r_syntax t x2' m2 ta2 x2'' m2'"   and "\<not> \<tau>move2 (x2', m2) ta2 (x2'', m2')"   and "t \<turnstile> (x1'', m1') \<approx> (x2'', m2')"   and "ta1 \<sim>m ta2"   and "Suspend w \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub>"   and "Suspend w \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub>" shows "x1'' \<approx>w x2''"
h_p: shows "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
reduction_P1_interm: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and c0 :: "bool"   and c1 :: "bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf"   and out :: "unit" shows "\<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (S1_14 M out \<bind> D) True\<bar> \<le> 3 * adv_OT12"
induct: fixes P :: "('b, 'c, 'd, 'a) seqp \<Rightarrow> bool"   and a0 :: "('b, 'c, 'd, 'a) seqp" assumes "\<And>l g p. P p \<Longrightarrow> P ({l}\<langle>g\<rangle> p)"   and "\<And>l u p. P p \<Longrightarrow> P ({l}\<lbrakk>u\<rbrakk> p)"   and "\<And>l s\<^sub>i\<^sub>p s\<^sub>m\<^sub>s\<^sub>g p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P ({l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q)"   and "\<And>l s\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>l s\<^sub>i\<^sub>p\<^sub>s s\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>l s\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}send(s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>l s\<^sub>d\<^sub>a\<^sub>t\<^sub>a p. P p \<Longrightarrow> P ({l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p)"   and "\<And>l u\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>p1 p2. \<lbrakk>P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 \<oplus> p2)"   and "\<And>pn. P (call(pn))" shows "P a0"
Range_iff: fixes a :: "'a"   and r :: "('b \<times> 'a) set" shows "(a \<in> Range r) = (\<exists>y. (y, a) \<in> r)"
completion_ex_borel_measurable_real: fixes g :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "g \<in> borel_measurable (completion M)" shows "\<exists>g'\<in>borel_measurable M. AE x in M. g x = g' x"
sub: shows "{x..x + 2 ^ n - 1} \<subseteq> {a..b}"
B_def: shows "B \<equiv> \<Sum>\<^sub>ax\<in>S. norm (f x)"
l12_6: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "A B OS C D"
assms: fixes m :: "nat"   and n :: "nat" shows "u (m + n) \<le> u m + u n"
in_range_scaleR_iff: fixes x :: "'c"   and c :: "real" shows "(x \<in> range ((*\<^sub>R) c)) = (c = 0 \<longrightarrow> x = (0::'c))"
get_elements_by_tag_name_is_strongly_scdom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and results :: "(_) element_ptr list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_elements_by_tag_name ptr' idd \<rightarrow>\<^sub>r results"   and "result \<in> set results" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set sc) = (ptr' \<in> set sc)"
sgn_pos_inf_def: fixes p :: "'a poly" shows "sgn_pos_inf p \<equiv> sgn (lead_coeff p)"
uniform_alt_def_all: shows "\<forall>bl\<in>#\<B>. int (card bl) = \<k>"
ipoly_poly_div: fixes q :: "int poly"   and p :: "int poly"   and x :: "'a"   and y :: "'a" assumes "q \<noteq> 0"   and "ipoly p x = (0::'a)"   and "ipoly q y = (0::'a)"   and "y \<noteq> (0::'a)" shows "ipoly (poly_div p q) (x / y) = (0::'a)"
Minimal_SolutionsI': fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Solutions a b"   and "nonzero x"   and "\<not> (\<exists>(u, v)\<in>Minimal_Solutions a b. u @ v <\<^sub>v x @ y)" shows "(x, y) \<in> Minimal_Solutions a b"
measurable_restrict_space1: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and \<Omega> :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N"
f: shows "finite {a \<in> A. 0 < f a \<and> g a \<noteq> 0}"
asubg_mOp_closed: fixes H :: "'a set"   and x :: "'a" assumes "A +> H"   and "x \<in> H" shows "-\<^sub>a x \<in> H"
poly_assn_alt_def: shows "poly_assn = pure poly_rel"
sparse_update: fixes \<delta> :: "real"   and ps :: "(int \<times> int) list"   and p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "real \<times> real" assumes "sparse \<delta> (set ps)"   and "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) p\<^sub>1 \<le> \<delta>"   and "\<forall>p\<in>set ps. dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) p\<^sub>1 \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))" shows "sparse (dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) p\<^sub>1) (set (p\<^sub>0 # ps))"
S_transform_Tree\<^sub>\<alpha>_preserves_hereditarily_fs: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (S_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
elims_3: fixes x :: "nat"   and xa :: "'b upI" assumes "\<not> full\<^sub>i x xa" obtains n :: "nat" and t :: "'b tree23" where "x = n__" and "xa = TI t" and "\<not> full n__ t"   | n :: "nat" and l :: "'b tree23" and p :: "'b" and r :: "'b tree23" where "x = n" and "xa = OF l p r" and "\<not> (full n l \<and> full n r)"
subdegree_fps_of_ratfps: fixes x :: "'b ratfps" shows "subdegree (fps_of_ratfps x) = ratfps_subdegree x"
inj_on_iff_card_le: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (card A \<le> card B)"
triple_schroeder_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" shows "(x * y * z \<le> - w) = (x\<^sup>T * w * z\<^sup>T \<le> - y)"
Lp_I_3: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)" shows "eNorm (\<LL> (ennreal p) M) f = ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p))"
l9_30: 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 P :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and X :: "'a"   and Y :: "'a"   and Z :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C P"   and "\<not> Tarski_neutral_dimensionless.Col Bet D E F"   and "Tarski_neutral_dimensionless.Coplanar Bet D E F P"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C X"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C Y"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C Z"   and "Tarski_neutral_dimensionless.Coplanar Bet D E F X"   and "Tarski_neutral_dimensionless.Coplanar Bet D E F Y"   and "Tarski_neutral_dimensionless.Coplanar Bet D E F Z" shows "Tarski_neutral_dimensionless.Col Bet X Y Z"
val_pos_mem_Vr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(0 \<le> v x) = (x \<in> carrier (Vr K v))"
cblinfun_compose_zero_left: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "0 o\<^sub>C\<^sub>L U = 0"
onorm_scalarC: fixes f :: "'a \<Rightarrow> 'b"   and r :: "complex" assumes "bounded_clinear f" shows "onorm (\<lambda>x. r *\<^sub>C f x) = cmod r * onorm f"
Rep_zet_zunion: fixes a :: "'b zet"   and b :: "'b zet" shows "Rep_zet (zunion a b) = Rep_zet a \<union> Rep_zet b"
varD_transitive: fixes b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "(b, A, s, t) \<in> varD"   and "transitive A" shows "A s t"
Order: fixes D :: "('c, 'd) Order_scheme" assumes "Worder D" shows "Order D"
Healthy_def': fixes P :: "'d upred"   and H :: "'d upred \<Rightarrow> 'd upred" shows "(P is H) = (H P = P)"
oconf_blank2: fixes P :: "'a prog"   and C :: "char list"   and wt :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and h :: "heap" assumes "is_class P C"   and "wf_prog wt P" shows "P,h \<turnstile> blank P C \<surd>"
nn_integral_nats_reals: fixes f :: "nat \<Rightarrow> real" shows "\<integral>\<^sup>+ i. ennreal (f i) \<partial>count_space UNIV = \<integral>\<^sup>+x\<in>{0..}. ennreal (f (nat \<lfloor>x\<rfloor>))\<partial>lborel"
nemp_pos_len: fixes r :: "'b list" assumes "r \<noteq> \<epsilon>" shows "1 \<le> \<^bold>|r\<^bold>|"
HStartBallot_Inv2a_dblock_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "Inv2a_innermost s q (dblock s q)" shows "Inv2a_innermost s' q (dblock s' q)"
not_Low_High: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and Exit :: "'b"   and H :: "'d set"   and L :: "'d set"   and High :: "'b"   and Low :: "'b"   and V :: "'d" assumes "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice Exit H L High Low"   and "V \<notin> L" shows "V \<in> H"
coeff_mpoly_to_poly: fixes v :: "nat"   and p :: "'b mpoly"   and k :: "nat" shows "poly.coeff (mpoly_to_poly v p) k = MPoly_Type.coeff p (Poly_Mapping.single v k)"
hs_get_roots_locs_subseteq_hp_IdleMarkSweep_locs: shows "hs_get_roots_locs \<subseteq> hp_IdleMarkSweep_locs"
g_bound: fixes x :: "'b" shows "\<alpha> bot \<le> g x"
assms_3: assumes "ps" shows "snd (rep_nat_term u) = snd (rep_nat_term v)"
comm_monoid_add_axioms_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "class.comm_monoid_add_axioms plus zero \<equiv> \<forall>a. plus zero a = a"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
arr_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" shows "A_BxA.arr f = (A_B.arr (fst f) \<and> A.arr (snd f))"
scaleC_add_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "complex"   and b :: "complex"   and x :: "'a" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC (a + b) x = plus (scaleC a x) (scaleC b x)"
cases: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and a1 :: "('n, 'p, 'r, 't, 'v, 'x) T"   and a2 :: "('o, 'q, 's, 'u, 'w, 'x) T" assumes "rel_T L1 Co1 Co2 Contra1 Contra2 a1 a2" obtains x :: "(('n, 'p, 'r, 't, 'v, 'x) T, 'n, 'p, 'r, 't, 'v, 'x) G" and y :: "(('o, 'q, 's, 'u, 'w, 'x) T, 'o, 'q, 's, 'u, 'w, 'x) G" where "a1 = C_T x" and "a2 = C_T y" and "rel_G (rel_T L1 Co1 Co2 Contra1 Contra2) L1 Co1 Co2 Contra1 Contra2 x y"
nn_integral_pair_pmf': fixes A :: "'c pmf"   and B :: "'d pmf"   and f :: "'c \<times> 'd \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf (pair_pmf A B)) f = \<integral>\<^sup>+ a. \<integral>\<^sup>+ b. f (a, b) \<partial>measure_pmf B \<partial>measure_pmf A"
pderiv_power_Suc: fixes p :: "'b poly"   and n :: "nat" shows "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
span_add: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set"   and y :: "'b \<Rightarrow>\<^sub>0 'c" assumes "x \<in> phull S"   and "y \<in> phull S" shows "x + y \<in> phull S"
nondet_left_trel: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and P :: "'b hrel"   and Q :: "'b hrel" shows "(\<sigma>, P \<sqinter> Q) \<rightarrow>\<^sub>u (\<sigma>, P)"
index_not_in_delete_None: fixes i :: "'c"   and xs :: "('c \<times> 'd \<times> bool) list" assumes "i \<notin> fst ` set xs" shows "delete xs i = None"
insert_disjoint_1: fixes a :: "'d"   and A :: "'d set"   and B :: "'d set" shows "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
nb: shows "(\<phi> \<Psi> )\<^sub>G = \<phi>\<^sub>S \<Psi> \<Psi>\<^sub>G\<^sub>+"
inject: fixes x1 :: "char list"   and x2 :: "object list"   and y1 :: "char list"   and y2 :: "object list" shows "(PAction x1 x2 = PAction y1 y2) = (x1 = y1 \<and> x2 = y2)"
one_mod_two_eq_one: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "modulo one (numeral.numeral one plus (num.Bit0 num.One)) = one"
epi_iff_subset: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and G' :: "('d, 'f) monoid_scheme" shows "(f \<in> epi G G') = (f \<in> hom G G' \<and> carrier G' \<subseteq> f ` carrier G)"
power_Suc0_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.monoid_mult one times" shows "power.power one times a (Suc 0) = a"
ring_primeE_2: fixes p :: "'a" assumes "p \<in> carrier R"   and "ring_prime p" shows "prime (Ring_Divisibility.mult_of R) p"
continuous_on_id: fixes s :: "'b set" shows "continuous_on s (\<lambda>x. x)"
splitted_def: fixes p :: "'a list" shows "splitted p = (size (roots p) = degree p)"
invertible_mult_row: fixes q :: "'d"   and k :: "'d"   and a :: "'e" assumes "q * k = (1::'d)"   and "k * q = (1::'d)" shows "invertible (mult_row (mat (1::'d)) a q)"
power_gt1_lemma: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "less one (times a (power.power one times a n))"
intro: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf" shows "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"
Lcm_eq_Lcm_factorial: fixes A :: "'a set" shows "Lcm A = Lcm_factorial A"
Cons_2: shows "set (a # p1) \<subseteq> K"
dlex_pp_alt: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "dlex_pp s t = (deg_pp s < deg_pp t \<or> deg_pp s = deg_pp t \<and> lex_pp s t)"