hcju/isabelleps huggingface dataset:
sup_uminus: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "sup x (mbt_algebra.neg_assert one times inf dual bot x) = one"
tND: fixes p :: "('b set set, 'c) Combinators list"   and x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'c) Combinators"   and a :: "'b set set"   and b :: "'b set set"   and c :: "'b set set"   and d :: "'b set set" assumes "allNetsDistinct p"   and "x \<in> set p"   and "y \<in> set p"   and "a = srcNet x"   and "b = destNet x"   and "c = srcNet y"   and "d = destNet y"   and "a \<noteq> c \<or> b \<noteq> d"   and "singleCombinators [x]"   and "x \<noteq> DenyAll"   and "singleCombinators [y]"   and "y \<noteq> DenyAll" shows "twoNetsDistinct a b c d"
n_st_narrow: fixes X :: "char list set"   and S1 :: "ivl Abs_State.st"   and S2 :: "ivl Abs_State.st" assumes "finite X"   and "set (Abs_State.dom S1) \<subseteq> X"   and "set (Abs_State.dom S2) \<subseteq> X"   and "S2 \<sqsubseteq> S1"   and "\<not> S1 \<sqsubseteq> S1 \<triangle> S2" shows "n_st n_ivl X (S1 \<triangle> S2) < n_st n_ivl X S1"
show_r_zero: fixes x :: "'a"   and a :: "'a" shows "(x + a = x) = (a = (0::'a))"
boundsl_def: fixes s :: "('b, 'a) state" shows "\<B>\<^sub>l s = map_option snd \<circ> Mapping.lookup (\<B>\<^sub>i\<^sub>l s)"
fin_UNT: shows "finite {chi val and\<^sub>p chi val U\<^sub>p (\<psi> and\<^sub>p chi val') |val val'. val \<subseteq> A \<and> val' \<subseteq> A \<and> val' \<noteq> val}"
simps: fixes A :: "'b mat"   and B :: "'b mat" shows "strassen_mat_mult A B = (let nr = dim_row A; n = dim_col A; nc = dim_col B in if strassen_too_small A B then A * B else if strassen_even A B then let nr2 = nr div 2; n2 = n div 2; nc2 = nc div 2; (A1, A2, A3, A4) = split_block A nr2 n2; (B1, B2, B3, B4) = split_block B n2 nc2; M1 = strassen_mat_mult (A1 + A4) (B1 + B4); M2 = strassen_mat_mult (A3 + A4) B1; M3 = strassen_mat_mult A1 (B2 - B4); M4 = strassen_mat_mult A4 (B3 - B1); M5 = strassen_mat_mult (A1 + A2) B4; M6 = strassen_mat_mult (A3 - A1) (B1 + B2); M7 = strassen_mat_mult (A2 - A4) (B3 + B4); C1 = M1 + M4 - M5 + M7; C2 = M3 + M5; C3 = M2 + M4 in Let (M1 - M2 + M3 + M6) (four_block_mat C1 C2 C3) else let nr' = nr div 2 * 2; n' = n div 2 * 2; nc' = nc div 2 * 2; (A1, A2, A3, A4) = split_block A nr' n'; (B1, B2, B3, B4) = split_block B n' nc'; C1 = strassen_mat_mult A1 B1 + A2 * B3; C2 = A1 * B2 + A2 * B4; C3 = A3 * B1 + A4 * B3 in Let (A3 * B2 + A4 * B4) (four_block_mat C1 C2 C3))"
dcache_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and c :: "word20 \<times> word12"   and v :: "word8"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = dcache_mod c v s1 \<and> t2 = dcache_mod c v s2" shows "low_equal t1 t2"
static_mthd_simp: fixes m :: "'b \<times> 'a member_scheme" shows "static (mthd m) = is_static m"
that: fixes a :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
simps_2: fixes P :: "'b \<Rightarrow> bool"   and uv :: "'b" shows "successive P [uv] = True"
STL4_eve: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> \<diamond>F \<longrightarrow> \<diamond>G"
ProofStepInduct_NonTrivial_P_k_pre_axioms: shows "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
sum2: fixes X :: "'a set"   and f :: "'a \<Rightarrow> real \<times> real"   and a :: "real \<times> real" assumes "finite X"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> ccw' 0 (f x) a" shows "ccw' 0 (sum f X) a"
closure_def: shows "closure \<equiv> map_fun Rep_uexpr mk\<^sub>e (\<lambda>P A. \<forall>A'. P A')"
abs_summable_on_normI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "f abs_summable_on A" shows "(\<lambda>x. norm (f x)) abs_summable_on A"
ordermap_mono: fixes w :: "'c"   and x :: "'c"   and r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "(w, x) \<in> r"   and "wf r"   and "w \<in> A"   and "small A" shows "ordermap A r w \<in> elts (ordermap A r x)"
weakPar1: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'" shows "P \<parallel> Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<parallel> Q"
asymp_equivD_const: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "f \<sim>[F] (\<lambda>_. c)" shows "(f \<longlongrightarrow> c) F"
MGT_implies_complete: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>Z. \<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"   and "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
inject: fixes x :: "'b"   and ya :: "'b" shows "(dual x = dual ya) = (x = ya)"
get_disconnected_nodes_locs_impl: shows "get_disconnected_nodes_locs = local.a_get_disconnected_nodes_locs"
adj_dual: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "f \<stileturn> g = \<partial>\<^sub>F g \<stileturn> \<partial>\<^sub>F f"
tail_in_verts: fixes G :: "('a, 'b) pre_digraph"   and e :: "'b" assumes "wf_digraph G"   and "e \<in> arcs G" shows "tail G e \<in> verts G"
mem_existence_ivl_iv_defined_1: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "ll_on_open_it T f X"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0" shows "t0 \<in> T"
Max_in: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A \<in> A"
AAs0'_in_atms_of_CAs0': shows "\<forall>i<n. \<forall>A\<in>#AAs0' ! i. A \<in> atms_of (CAs0' ! i)"
elims: fixes x :: "perm"   and xa :: "('a, 'b) L_action"   and y :: "('a, 'b) L_action" assumes "x \<bullet> xa = y" obtains p :: "perm" and \<alpha> :: "'a" where "x = p__" and "xa = L_action.Act \<alpha>" and "y = L_action.Act (p__ \<bullet> \<alpha>)"   | p :: "perm" and f :: "'b" where "x = p" and "xa = Eff f" and "y = Eff (p \<bullet> f)"
rep_eq: fixes x :: "'a vector"   and xa :: "'a vector" shows "Rep_vector (x \<squnion> xa) = Rep_vector x \<squnion> Rep_vector xa"
cod_simps_2: shows "\<mu>.cod.leg0 = \<mu>.leg0"
Mapping_RBT_def: fixes t :: "('b, 'c) rbt" shows "Mapping_RBT t = Mapping_RBT' (if ord.is_rbt cless t \<or> ID ccompare = None then t else RBT_Impl.fold (ord.rbt_insert cless) t rbt.Empty)"
wary_fun: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and t :: "('a, 'b) tm" assumes "arity.wary arity_sym arity_var t" shows "arity.wary arity_sym arity_var (fun t)"
Iaxiom_def: shows "Iaxiom \<equiv> ([[$\<alpha> vid1**]]Pc pid1 \<rightarrow> [[$\<alpha> vid1]]Pc pid1) \<rightarrow> Pc pid1 \<rightarrow> [[$\<alpha> vid1**]]Pc pid1"
3: shows "(Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q) mod int q = int (1 mod q)"
euclidean_function: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R - {\<zero>}"   and "b \<in> carrier R - {\<zero>}" shows "\<exists>q r. q \<in> carrier R \<and> r \<in> carrier R \<and> a = b \<otimes> q \<oplus> r \<and> (r = \<zero> \<or> \<phi> r < \<phi> b)"
sum_a: shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. n < m \<longrightarrow> sum a {Suc n..m} < e"
find_closest_code_dist_eq: fixes ps :: "(int \<times> int) list"   and \<delta>\<^sub>c :: "int"   and c :: "int \<times> int"   and p :: "int \<times> int"   and \<delta> :: "int" assumes "0 < length ps"   and "(\<delta>\<^sub>c, c) = find_closest_code p \<delta> ps" shows "\<delta>\<^sub>c = dist_code p c"
prems_2: shows "H \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
resTauLeft: fixes Rel :: "(pi \<times> pi) set"   and x :: "name"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "<\<nu>x>\<tau>.P \<leadsto>[Rel] \<tau>.(<\<nu>x>P)"
conjoin_top: fixes c :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (c::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) \<iinter> \<top> = \<top>"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "\<exists>y. execute (f x) h = Some (y, h)"
integer_of_nat_less_0_conv: fixes n :: "nat" shows "\<not> integer_of_nat n < 0"
mask_inner_mask: fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" shows "(a AND mask n) + b AND mask n = a + b AND mask n"
relation_of_Sup: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "action.relation_of (action_of (\<lambda>Aa. Aa \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A})) = (\<lambda>Aa. Aa \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A})"
assms: shows "(\<sigma>, \<zeta>) \<in> oreachable (oclosed (opnet np p)) (\<lambda>_ _ _. True) U"
scaleC_add_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "complex"   and x :: "'b"   and y :: "'b" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (plus x y) = plus (scaleC a x) (scaleC a y)"
zero_space\<^sub>N_def: fixes N :: "'b quasinorm" shows "zero_space\<^sub>N N = {f. eNorm N f = 0}"
ereal_cases: fixes x :: "ereal" obtains r :: "real" where "x = ereal r"   | "x = \<infinity>"   | "x = - \<infinity>"
lapp_allT_iff: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" shows "(r @@ s \<in> A\<^sup>\<infinity>) = (r \<in> A\<^sup>\<star> \<and> s \<in> A\<^sup>\<infinity> \<or> r \<in> A\<^sup>\<omega>)"
exec_elim_cases_4: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>Skip,s\<rangle> \<Rightarrow> t" obtains sa :: "'b" where "s = Normal sa__" and "t = Normal sa__"   | f :: "'c" where "s = Fault f" and "t = Fault f"   | "s = Stuck" and "t = Stuck"   | sa :: "'b" where "s = Abrupt sa" and "t = Abrupt sa"
GF_advice_monotone: fixes X :: "'b ltln set"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "X \<subseteq> Y"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<Pi>\<^sub>1" shows "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<Pi>\<^sub>1"
is_functor: shows "functor DopxC.comp DopxD.comp DopxG.map"
\<Lambda>_pr1'_def: shows "\<Lambda>_pr1' = local.comp (\<Lambda> a b c g) (pr1 a b)"
assms: fixes k :: "'a"   and x :: "'b" shows "f k x (0::'c) = x"
refine: fixes s :: "(integer \<times> edge list) iarray iarray"   and g :: "gState\<^sub>I" shows "RETURN (executable_impl s g) \<le> executable s g"
diagram_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c" shows "diagram J C D \<equiv> category C \<and> category J \<and> functor J C D"
wls_Abs_ainj_ex: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> fresh xs y X \<and> fresh xs y X' \<and> X #[y \<and> x]_xs = X' #[y \<and> x']_xs))"
comm_monoid_def: fixes G :: "('e, 'f) monoid_scheme" shows "Group.comm_monoid G \<equiv> Group.monoid G \<and> Group.comm_monoid_axioms G"
Delta1_subsetof_E1: shows "\<Delta>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub>"
interaction_any_bounded_converter_id: shows "interaction_any_bounded_converter 1\<^sub>C 1"
in_oassert_eq: fixes P :: "bool"   and s :: "'d"   and v :: "unit" shows "(oassert P s = Some v) = P"
relative_homology_group_restrict: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "relative_homology_group p X (topspace X \<inter> S) = relative_homology_group p X S"
ope: fixes v :: "'a set" assumes "v \<in> \<U>" shows "openin X v"
csingleton_iff: fixes b :: "'f"   and a :: "'f" shows "cin b (csingle a) = (b = a)"
basis_list_def: fixes fs :: "'a vec list" shows "basis_list fs = (set fs \<subseteq> carrier_vec n \<and> length fs = n \<and> carrier_vec n \<subseteq> local.span (set fs))"
assms_4: shows "DBM_reset (FW M n) n (v c) d M'"
greatest: fixes r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "MaxR r" shows "set_option (MaxR.MaxR_opt r X) = greatest r (X \<inter> Field r)"
Macaulay_list_lt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and ps :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> phull (set ps)"   and "p \<noteq> 0" obtains g :: "'t \<Rightarrow>\<^sub>0 'c" where "g \<in> set (Macaulay_list ps)" and "g \<noteq> 0" and "lt p = lt g"
state_abscD: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.cfg_on s" shows "state (absc cfg) = abss s"
cond: fixes P :: "'b set"   and b :: "'b set"   and c1 :: "'b com"   and Q :: "'b set"   and c2 :: "'b com" assumes "\<turnstile> (P \<inter> b) c1 Q"   and "\<turnstile> (P \<inter> - b) c2 Q" shows "\<turnstile> P Cond b c1 c2 Q"
fmap_ext: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" assumes "\<And>x. m $$ x = n $$ x" shows "m = n"
fact'_2: fixes x :: "'b" assumes "p \<noteq> 0" shows "(ipoly p x = (0::'b)) = (\<exists>q\<in>set qs. ipoly q x = (0::'b))"
index_a': shows "index (insort a (sorted_list_of_set I)) a = a'"
rep_eq: fixes x :: "'b covector"   and xa :: "'b covector" shows "Rep_covector (x \<squnion> xa) = Rep_covector x \<squnion> Rep_covector xa"
infdist_le2: fixes a :: "'c"   and A :: "'c set"   and x :: "'c"   and d :: "real" assumes "a \<in> A"   and "dist x a \<le> d" shows "infdist x A \<le> d"
fresh_Pair: fixes a :: "atom"   and x :: "'b"   and y :: "'c" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
set_fv_list: fixes e :: "'c" shows "set (fv_list e) = fv e"
INFM_iff_infinite: fixes P :: "'b \<Rightarrow> bool" shows "(\<exists>\<^sub>\<infinity>x. P x) = infinite {x. P x}"
equivI_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<longrightarrow>. q"   and "A \<turnstile> q .\<longrightarrow>. p" shows "A \<turnstile> p .\<longleftrightarrow>. q"
int_pow_0: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
BisT_transC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>T d"   and "mustT c s"   and "mustT d t"   and "s \<approx> t"   and "(c, s) \<rightarrow>c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "c' \<approx>T d'"
Rep_covector: fixes x :: "'c covector" shows "Rep_covector x \<in> covectors"
maybe_or_zero: fixes x :: "trilean" shows "false \<or>? x = x"
rep_eq: fixes x :: "'a mpoly"   and xa :: "'a mpoly" shows "mapping_of (x - xa) = mapping_of x - mapping_of xa"
FinA: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and a :: "abopt"   and c :: "stmt"   and Q :: "'b assn" shows "G,A\<turnstile>{Normal P} .FinA a c. {Q}"
Proj_preservation: fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and l :: "char list"   and T :: "POPLmarkRecord.type"   and v :: "trm" assumes "\<Gamma> \<turnstile> Rcd fs..l : T"   and "fs\<langle>l\<rangle>\<^sub>? = \<lfloor>v\<rfloor>" shows "\<Gamma> \<turnstile> v : T"
simps_1: fixes x :: "Preliminaries.var" shows "GE.wt (Var x) = True"
gen_llength_code_2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b llist" shows "gen_llength n (LCons x xs) = gen_llength (n + 1) xs"
simps_1: fixes x1 :: "'b"   and x2 :: "'d"   and y1 :: "'b"   and y2 :: "'d" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
disj_of_clause_def: fixes c :: "'c literal list" shows "disj_of_clause c \<equiv> \<^bold>\<Or>map form_of_lit c"
run_rel_single_valued: fixes R :: "('h \<times> 'i) set"   and ra :: "nat \<Rightarrow> 'h"   and rb :: "nat \<Rightarrow> 'i" assumes "single_valued R" shows "((ra, rb) \<in> run_rel R) = ((\<forall>i. ra i \<in> Domain R) \<and> rb = the_br_\<alpha> R \<circ> ra)"
fmempty_empty: shows "(uncurry0 (RETURN op_map_empty), uncurry0 (RETURN fmempty)) \<in> unit_rel \<rightarrow>\<^sub>f \<langle>map_fmap_rel\<rangle>nres_rel"
mono_Mprefix1: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and Q :: "'b \<Rightarrow> 'b process" assumes "\<forall>a\<in>A. P a \<sqsubseteq> Q a" shows "D (Mprefix A Q) \<subseteq> D (Mprefix A P)"
lookup_uminus: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "lookup (- f) k = - lookup f k"
beukers_integral3_pos: fixes n :: "nat" shows "0 < beukers_integral3 n"
cgenideal_eq_genideal: fixes i :: "('c multiset \<Rightarrow> 'a) list" assumes "i \<in> carrier (poly_ring L)" shows "PIdl\<^bsub>poly_ring L\<^esub> i = Idl\<^bsub>poly_ring L\<^esub> {i}"
sigma_finite_subalgebra_def: fixes M :: "'c measure"   and F :: "'c measure" shows "sigma_finite_subalgebra M F \<equiv> subalgebra M F \<and> sigma_finite_measure (restr_to_subalg M F)"
abs_eq: shows "unit_circle = abs_circline unit_circle_clmat"
irreducible_imp_maximalideal: fixes p :: "'a" assumes "p \<in> carrier R"   and "ring_irreducible p" shows "maximalideal (PIdl p) R"
eq: shows "q.eq_m C (Polynomial.smult c (\<Prod>\<^sub># fs))"
assms_1: shows "enat p + INV t + 1 \<le> INV s"
con_intro: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<phi> \<pi>l"   and "\<psi> \<pi>l" shows "con \<phi> \<psi> \<pi>l"
\<phi>_invertible: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<exists>!f. \<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright> \<and> \<phi> y f = g"
poly_mult_zero_iff: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" shows "(poly (p * q) x = (0::'b)) = (poly p x = (0::'b) \<or> poly q x = (0::'b))"
fin_succs': fixes G :: "nat graph_rec"   and v :: "nat" assumes "cfg_doms G"   and "v < length (g_V G)" shows "finite (cfg_doms.succs G v)"
augment_flow_value: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and f' :: "nat \<times> nat \<Rightarrow> 'a" assumes "NFlow c s t f"   and "Flow (residualGraph c f) s t f'" shows "Flow.val c s (NFlow.augment c f f') = Flow.val c s f + Flow.val (residualGraph c f) s f'"
linear_surjective_right_inverse: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f"   and "surj f" shows "\<exists>g. linear s2 s1 g \<and> f \<circ> g = id"
same_cs_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and c :: "('d, 'e, 'f) configuration"   and d :: "('d, 'e, 'f) configuration"   and i :: "nat"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and d' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "\<forall>p. (ps c p \<noteq> None) = (ps d p \<noteq> None)"   and "cs c i = cs d i"   and "distributed_system.next channel trans send recv c ev c'"   and "distributed_system.next channel trans send recv d ev d'" shows "cs c' i = cs d' i"
Resolution_taint_assumptions: fixes S :: "'b literal set set"   and T :: "'b literal set set"   and C :: "'b literal set"   and D :: "'b literal set" assumes "S \<union> T \<turnstile> C" shows "\<exists>R\<subseteq>D. (\<union>) D ` S \<union> T \<turnstile> R \<union> C"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> M.events"   and "decseq A" shows "(\<lambda>n. M.prob (A n)) \<longlonglongrightarrow> M.prob (\<Inter> (range A))"
reduce_nested_mpoly_def: fixes pp :: "'b mpoly mpoly" shows "reduce_nested_mpoly pp = insertion (\<lambda>v. monom (Poly_Mapping.single v 1) (1::'b)) pp"
equiv_class_eq_iff: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "equiv A r" shows "((x, y) \<in> r) = (r `` {x} = r `` {y} \<and> x \<in> A \<and> y \<in> A)"
weakRes: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and x :: "name" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "\<lparr>\<nu>x\<rparr>P \<Longrightarrow>\<^sup>^\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
univ_poly_a_inv_length: fixes p :: "'a list" assumes "p \<in> carrier (K [X])" shows "length (\<ominus>\<^bsub>K [X]\<^esub> p) = length p"
poset_to_PosetComplex_OrderingSetMap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ordering less_eq less"   and "\<And>x. x \<in> P \<Longrightarrow> ordering.simplex_like less_eq less {y \<in> P. less_eq y x}" shows "OrderingSetMap less_eq less dual_order.greater_eq dual_order.greater P (ordering.poset_simplex_map less_eq less P)"
induct: fixes P :: "'a\<cdot>'b error \<Rightarrow> bool"   and x :: "'a\<cdot>'b error" assumes "P \<bottom>"   and "\<And>e. P (Err\<cdot>e)"   and "\<And>a. P (Ok\<cdot>a)" shows "P x"
fin_1: shows "finite {R \<in> Rs. (a, b) \<in> R}"
nprv_ballE_aux: fixes F :: "'fmla set"   and x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla"   and t1 :: "'trm" assumes "nprv F (ball x t \<phi>)"   and "nprv F (LLq t1 t)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> atrm"   and "t1 \<in> atrm"   and "x \<in> var"   and "x \<notin> FvarsT t" shows "nprv F (subst \<phi> t1 x)"
Disjunctive_def: fixes Sup_b :: "'d set \<Rightarrow> 'd"   and Sup_c :: "'e set \<Rightarrow> 'e"   and times_abc :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "Disjunctive.Disjunctive Sup_b Sup_c times_abc = {x. \<forall>X. times_abc x (Sup_b X) = Sup_c (times_abc x ` X)}"
psubst_subst_psubst: fixes c :: "name"   and \<theta> :: "(name \<times> trm) list"   and s :: "trm"   and t :: "trm" assumes "c \<sharp> \<theta>" shows "\<theta><t>[c::=s] = ((c, s) # \<theta>)<t>"
innermost_intvl: fixes f :: "atom fm"   and xs :: "'b list"   and l :: "'b"   and x :: "'b"   and u :: "'b"   and y :: "'b" assumes "nqfree f"   and "nolub f xs l x u"   and "l < x"   and "x < u"   and "x \<notin> EQ f xs"   and "DLO.I f (x # xs)"   and "l < y"   and "y < u" shows "DLO.I f (y # xs)"
scope_21: shows "2 \<in> task.scope (1, allI_input)"
tab_in_vhom': shows "\<guillemotleft>\<mu> : u\<^sub>1 \<Rightarrow> u \<star> u\<^sub>0\<guillemotright>"