hcju/isabelleps huggingface dataset:
def_gfp_unfold: fixes A :: "'b::complete_lattice"   and f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "A::'b::complete_lattice \<equiv> gfp (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)"   and "mono (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)" shows "(A::'b::complete_lattice) = (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) A"
oghoare_Call: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and x :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P x Q, A" shows "\<forall>p. x = Call p \<longrightarrow> (\<exists>r n. P = AnnCall r n \<and> (\<exists>as P' f b. \<Theta> p = Some as \<and> as ! n = P' \<and> r \<subseteq> pre P' \<and> \<Gamma> p = Some b \<and> n < length as \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P' b Q, A))"
assms: shows "\<Psi> \<rhd> P \<longmapsto> Rs"
Int_assoc: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
mod_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "Suc (m mod n) mod n = Suc m mod n"
Sink_idle: fixes s :: "('St, 'U, 'C) state" assumes "s \<in> Sink" shows "s \<in> \<delta> s"
axioms_3: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah" assumes "diagram J C D" shows "functor J C D"
enum_set_get_one_Ex_Some: fixes S :: "'b set"   and ss :: "'b list" assumes "S \<subseteq> set ss"   and "S \<noteq> {}" shows "\<exists>x. enum_set_get_one ss S = Some x"
imethds_declI: fixes m :: "qtname \<times> mhead"   and G :: "prog"   and I :: "qtname"   and sig :: "sig" assumes "m \<in> DeclConcepts.imethds G I sig"   and "ws_prog G"   and "is_iface G I" shows "(\<exists>i. iface G (decliface m) = Some i \<and> table_of (imethods i) sig = Some (mthd m)) \<and> G\<turnstile>I\<preceq>I decliface m \<and> m \<in> DeclConcepts.imethds G (decliface m) sig"
FW_cyc_free_preservation: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m i i xs" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len (FW m n) i i xs"
sel_2: fixes x21 :: "'a"   and x22 :: "'a msllist" shows "un_MSLCons2 (MSLCons x21 x22) = x22"
adjoint_equivalence_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" shows "adjoint_equivalence C D F G \<eta> \<epsilon> \<equiv> unit_counit_adjunction C D F G \<eta> \<epsilon> \<and> natural_isomorphism D D (identity_functor.map D) (G \<circ> F) \<eta> \<and> natural_isomorphism C C (F \<circ> G) (identity_functor.map C) \<epsilon>"
finite_UNIV_finfun_def: shows "finite_UNIV = Phantom('a \<Rightarrow>f 'b) (let cb = of_phantom card_UNIV_class.card_UNIV in cb = 1 \<or> of_phantom finite_UNIV \<and> cb \<noteq> 0)"
cont_Cfg: fixes t :: "'a"   and d' :: "'a pmf"   and c' :: "'a \<Rightarrow> 'a cfg"   and s :: "'a" assumes "t \<in> set_pmf d'"   and "state (c' t) = t" shows "cont (Cfg s d' c') t = c' t"
proj_poly_syz_monom_mult: fixes n :: "nat"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly_syz n (monom_mult c t p) = monom_mult c t (proj_poly_syz n p)"
assms: shows "\<Z>_sgrp_basis \<alpha> \<SS>"
assms_4: shows "mat (row_length M2) (length M2) M2"
FVar: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned"   and accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and fn :: "vname" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>{accC,statDeclC,stat}e..fn\<rangle>\<guillemotright> A"
UB: assumes "t1 \<in> U n"   and "t2 \<in> B n" shows "thesis"
asym: fixes a :: "'c Word.word"   and b :: "'c Word.word" assumes "a <s b"   and "b <s a" shows "False"
Cons_in_shuffles_iff: fixes z :: "'g"   and zs :: "'g list"   and xs :: "'g list"   and ys :: "'g list" shows "(z # zs \<in> shuffles xs ys) = (xs \<noteq> [] \<and> hd xs = z \<and> zs \<in> shuffles (tl xs) ys \<or> ys \<noteq> [] \<and> hd ys = z \<and> zs \<in> shuffles xs (tl ys))"
elims_2: fixes x :: "'a"   and xa :: "'b list"   and xb :: "'a" assumes "cas x xa xb" obtains u :: "'a" and v :: "'a" where "x = ua__" and "xa = []" and "xb = va__" and "ua__ = va__"   | u :: "'a" and e :: "'b" and es :: "'b list" and v :: "'a" where "x = u" and "xa = e # es" and "xb = v" and "tail G e = u \<and> cas (head G e) es v"
in_set_butlastD: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set (butlast xs)" shows "x \<in> set xs"
stream_mod: fixes t :: "'a tree"   and t' :: "'a tree" shows "stream (t mod t') = stream t mod stream t'"
scale_one: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "module scale" shows "scale (1::'a) x = x"
PRIO_TAG_def: fixes ma :: "int"   and mi :: "int" shows "PRIO_TAG ma mi \<equiv> True"
has_def: fixes G :: "igraph"   and ac :: "actor \<times> char list" shows "has G ac \<equiv> snd ac \<in> credentials (cgra G (fst ac))"
uminus_sheaf_spec_in_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set" assumes "s \<in> \<O> U"   and "U \<subseteq> Spec" shows "uminus_sheaf_spec U s \<in> \<O> U"
int_pow_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and z :: "int" shows "a [^]\<^bsub>G\<^esub> z = (let p = rec_nat \<one>\<^bsub>G\<^esub> (\<lambda>u b. b \<otimes>\<^bsub>G\<^esub> a) in if z < 0 then inv\<^bsub>G\<^esub> p (nat (- z)) else p (nat z))"
union_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "aggregation.sum_0 g (A \<union> B) = aggregation.sum_0 g A + aggregation.sum_0 g B"
borel_measurable_if_I: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "(\<lambda>x. if x \<in> S then f x else (0::'d)) \<in> borel_measurable lebesgue"
Fun_gt_params: fixes f :: "'c"   and X :: "('c, 'd) Term.term list" shows "Fun f X \<notin> subterms\<^sub>s\<^sub>e\<^sub>t (set X)"
cidrsplit_no_overlaps: fixes x :: "'b prefix_match"   and wi :: "'b wordinterval"   and xa :: "'b prefix_match"   and pt :: "'b word" assumes "x \<in> set (wordinterval_CIDR_split_prefixmatch wi)"   and "xa \<in> set (wordinterval_CIDR_split_prefixmatch wi)"   and "pt && ~~ pfxm_mask x = pfxm_prefix x"   and "pt && ~~ pfxm_mask xa = pfxm_prefix xa" shows "x = xa"
len_of_finite_1_def: fixes x :: "Enum.finite_1 itself" shows "len_of x \<equiv> 1"
adj_surj_inj: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "top_pres ((`) f) = inj g"
\<chi>_is_universal_2: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "D.cones_map (\<psi> (a', a) (\<Psi>.FUN a' (\<iota> \<chi>'))) \<chi> = \<chi>'"
tmneg_nb0: fixes t :: "tm" assumes "tmbound0 t" shows "tmbound0 (tmneg t)"
asymp_equiv_symI: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "g \<sim>[F] f"
U_1: shows "finite (range (\<lambda>i. (A *v pow_A_1 u - sr *s pow_A_1 u) $h i / pow_A_1 u $h i))"
cases: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "r1.not_final_thread s t" obtains x :: "'x1" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x__, ln__)\<rfloor>" and "\<not> final1 x__"   | x :: "'x1" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(xa__, lna__)\<rfloor>" and "lna__ \<noteq> no_wait_locks"   | x :: "'x1" and w :: "'d wait_set_status" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "wset s t = \<lfloor>w\<rfloor>"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "Right.in_hom f a b"   and "Right.in_hom g b c" shows "Right.seq g f"
b: shows "Cn {.\<not> (\<phi> .\<and>. \<psi>) .\<and>. .\<not> \<phi>} = Cn {.\<not> \<phi>}"
fz: shows "(\<lambda>z. f (a + z)) = f \<circ> (+) a"
lift_map: fixes t :: "dB"   and ts :: "dB list"   and i :: "nat" shows "lift (t \<degree>\<degree> ts) i = lift t i \<degree>\<degree> map (\<lambda>t. lift t i) ts"
is_lubD2: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and u :: "'b" assumes "class.po below"   and "po.is_lub below S x"   and "po.is_ub below S u" shows "below x u"
Ints_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<int>" shows "prod f A \<in> \<int>"
hyps_1: fixes t :: "taylor_model" assumes "approx_tm prec ord I a l env = Some t" shows "interpret_floatarith l xs \<in>\<^sub>i range_tm e t"
evalCPS_def: fixes l :: "lambda" shows "\<PR> l = (let ve = Map.empty; \<beta> = Map.empty; f = \<A> (L l) \<beta> ve in \<F>\<cdot>(Discr (f, [Stop], ve, 0)))"
sc_bounded: fixes P :: "'c \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "Expectations.bounded (\<lambda>x. c * P x)"
diff_single_trivial: fixes x :: "'d"   and M :: "'d multiset" assumes "x \<notin># M" shows "M - {#x#} = M"
backward_terminating_path_root_2: fixes r :: "'a"   and x :: "'a" assumes "backward_terminating_path_root r x" shows "backward_terminating x"
eucl_Inf: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" 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 "Inf X = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (INF x\<in>X. inner x i) i) Basis"
scc_of_eq: fixes u :: "'a"   and v :: "'a" assumes "u \<in> H.scc_of v" shows "H.scc_of u = H.scc_of v"
upRuleCharacterise: fixes Ps :: "'a list"   and C :: "'b sequent" assumes "(Ps, C) \<in> upRules" shows "\<exists>F Fs. C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>) \<or> C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* \<Empt>)"
index_zero_vec_1: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "0\<^sub>v n $v i = (0::'d)"
eq_entails: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "P = Q" shows "P \<tturnstile> Q"
in_facesAt_nextVertex: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "f \<in> set (facesAt g (f \<bullet> v))"
eng_add: fixes h1 :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set"   and h2 :: "'a" assumes "h1 \<in> generate_ring R H"   and "h2 \<in> generate_ring R H" shows "h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> generate_ring R H"
strict_mono_on_imp_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "strict_mono_on f A" shows "inj_on f A"
exp_times_has_integral: fixes t0 :: "real"   and t :: "real"   and c :: "real" assumes "t0 \<le> t" shows "((\<lambda>t. exp (c * t)) has_integral (if c = 0 then t else exp (c * t) / c) - (if c = 0 then t0 else exp (c * t0) / c)) {t0..t}"
Group_def: fixes G :: "'a set" shows "Group G \<equiv> G \<noteq> {} \<and> (\<forall>g h. g \<in> G \<longrightarrow> h \<in> G \<longrightarrow> g - h \<in> G)"
in_latticeE: fixes f :: "'a vec"   and fs :: "'a vec list" assumes "f \<in> lattice_of fs" obtains c :: "nat \<Rightarrow> int" where "f = M.sumlist (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])"
that: fixes U :: "'n set"   and T :: "'n set" assumes "convex U"   and "U \<noteq> {}"   and "closedin (top_of_set U) T"   and "S homeomorphic T" shows "thesis"
all_closedin_homeomorphic_image: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and P :: "'b set \<Rightarrow> bool" assumes "homeomorphic_map X Y f" shows "(\<forall>V. closedin Y V \<longrightarrow> P V) = (\<forall>U. closedin X U \<longrightarrow> P (f ` U))"
ImplI: fixes a :: "('a, 'b) form"   and G :: "('a, 'b) form list"   and b :: "('a, 'b) form" assumes "a # G \<turnstile> b" shows "G \<turnstile> Impl a b"
integral_measure_pmf_real: fixes A :: "'a set"   and M :: "'a pmf"   and f :: "'a \<Rightarrow> real" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> 0\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. f a * pmf M a)"
dbl_inc_simps_2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one zero = one"
Un_left_commute: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
assms_2: shows "f \<in> R \<rightarrow>\<^sub>M N"
cardinal_cong: fixes x :: "V"   and y :: "V" assumes "elts x \<approx> elts y" shows "vcard x = vcard y"
simps_5: fixes \<rho> :: "(nat \<times> exp) list"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp" shows "psubst \<rho> (EPrim f e1 e2) = EPrim f (psubst \<rho> e1) (psubst \<rho> e2)"
cases: fixes x :: "tm" obtains "x = Zero"   | k :: "name" where "x = Var k"   | t :: "tm" and u :: "tm" where "x = Eats t u"
G_def: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and p :: "'e \<Rightarrow> 'd"   and I :: "'e set" assumes "comm_monoid_set f z" shows "comm_monoid_set.G f z p I \<equiv> if finite {x \<in> I. p x \<noteq> z} then comm_monoid_set.F f z p {x \<in> I. p x \<noteq> z} else z"
set_tree_diff: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.set_tree (Set2_Join.diff join t1 t2) = Tree2.set_tree t1 - Tree2.set_tree t2"
QuadRes_mod: fixes p :: "int"   and n :: "int"   and a :: "int" assumes "p dvd n" shows "QuadRes p (a mod n) = QuadRes p a"
of_complex_in_o_x_axis_disc: fixes z :: "complex" shows "(of_complex z \<in> disc o_x_axis) = (Im z < 0)"
Var: fixes rs :: "lam list"   and rs' :: "lam list"   and x :: "name" assumes "rs [\<rightarrow>\<^sub>s] rs'" shows "Var x \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> rs'"
reachable_closed_dfs: fixes g :: "(node \<times> node) list"   and xs :: "node list" shows "reachable g xs \<subseteq> set (dfs g xs [])"
Smith_Impl_axioms: shows "Smith_Impl Smith_1x2_JNF Smith_2x2_JNF div_op"
sigma_sets_eq: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "sigma_algebra \<Omega> M" shows "sigma_sets \<Omega> M = M"
deleteAround_empty: fixes g :: "graph"   and a :: "nat" shows "deleteAround g a [] = []"
horizontal_composite: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and H :: "'e \<Rightarrow> 'f"   and K :: "'e \<Rightarrow> 'f"   and \<tau> :: "'e \<Rightarrow> 'f" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation B C H K \<tau>" shows "natural_transformation A C (H \<circ> F) (K \<circ> G) (\<tau> \<circ> \<sigma>)"
SigmaE: fixes c :: "'g \<times> 'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "c \<in> Sigma A B" obtains x :: "'g" and y :: "'h" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
PC_def: shows "PC = \<Union> {Perms, Axioms, Conjs, Disjs, Alls, Exs, Weaks, Contrs, Cuts}"
scaleR_Pair: fixes r :: "real"   and a :: "'d"   and b :: "'e" shows "r *\<^sub>R (a, b) = (r *\<^sub>R a, r *\<^sub>R b)"
induct: fixes P :: "'b hp list \<Rightarrow> bool"   and a0 :: "'b hp list" assumes "P []"   and "\<And>uu_ uv_. P [Hp uu_ uv_]"   and "\<And>uw_ lx ux_ ly. P [Hp uw_ lx, Hp ux_ ly]"   and "\<And>uy_ lx uz_ ly v va. P (v # va) \<Longrightarrow> P (Hp uy_ lx # Hp uz_ ly # v # va)" shows "P a0"
ih_1: assumes "getSATFlag state' = UNDEF" shows "solve_loop_dom (solve_loop_body state' Vbl) Vbl"
fail_bind: fixes f :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind fail f = fail"
exists_vector_in_Berlekamp_basis_dvd_aux: fixes B :: "'a mod_ring poly set"   and P :: "'a mod_ring poly set"   and p_i :: "'a mod_ring poly"   and p_j :: "'a mod_ring poly"   and w :: "'a mod_ring poly" assumes "Berlekamp_subspace.basis B"   and "finite B"   and "finite P"   and "u = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}"   and "p_i \<in> P"   and "p_j \<in> P"   and "p_i \<noteq> p_j"   and "monic u"   and "square_free u"   and "\<not> irreducible w"   and "w dvd u"   and "monic w"   and "p_i dvd w"   and "p_j dvd w" shows "\<exists>v\<in>B. v mod p_i \<noteq> v mod p_j"
N: fixes n :: "nat" assumes "N \<le> n" shows "dist (Sigma_Algebra.measure lebesgue (\<Union> (from_nat_into \<D> ` {..n}))) (Sigma_Algebra.measure lebesgue (\<Union>n. \<Union> (from_nat_into \<D> ` {..n}))) < e"
invT_into: fixes h :: "'n" assumes "h \<in> N" shows "invT h \<in> M"
major: fixes S :: "'a set" assumes "S \<subseteq> A" shows "\<exists>L. isLub S po L"
coinduct: fixes X :: "('h, 'i, 'j, 'k) converter \<Rightarrow> bool"   and x :: "('h, 'i, 'j, 'k) converter"   and \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>conv. x = conv \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> pfinite_gpv \<I>' (run_converter conv x)) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> results_gpv \<I>' (run_converter conv x) \<longrightarrow> X xb \<or> pfinite_converter \<I> \<I>' xb)" shows "pfinite_converter \<I> \<I>' x"
tautology_def: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set" assumes "propositional_atoms atom_ordering" shows "tautology C \<equiv> \<exists>A. Pos A \<in> C \<and> Neg A \<in> C"
flatten_bseq: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" shows "flatten (bseq c1 c2) = flatten c1 @ flatten c2"
inv_4: fixes i :: "nat" assumes "i < m" shows "fs ! i \<in> carrier_vec n"
InvariantImpliedLiteralsWeakerVariant: fixes M :: "(Literal \<times> bool) list"   and F :: "Literal list list" assumes "\<forall>l. l el elements M \<longrightarrow> formulaEntailsLiteral (F @ val2form (decisionsTo l M)) l" shows "\<forall>l. l el elements M \<longrightarrow> formulaEntailsLiteral (F @ val2form (decisions M)) l"
read_only_shared_all_until_volatile_write_subset: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "ownership_distinct ts"   and "weak_sharing_consis ts" shows "read_only (share_all_until_volatile_write ts \<S>) \<subseteq> read_only \<S> \<union> (\<Union>(uu_, uu_, uu_, sb, uu_, \<O>, uu_)\<in>set ts. \<O> \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb))"
VEnd_composite_right: fixes S :: "'v \<Rightarrow> 'v" assumes "VEnd S" shows "VEnd (T \<circ> S)"
RDirProd_hom3: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and R' :: "('b, 'd) ring_scheme"   and g :: "'e \<Rightarrow> 'f"   and S :: "('e, 'g) ring_scheme"   and S' :: "('f, 'h) ring_scheme" assumes "f \<in> ring_hom R R'"   and "g \<in> ring_hom S S'" shows "(\<lambda>(r, s). (f r, g s)) \<in> ring_hom (RDirProd R S) (RDirProd R' S')"
open_cover_of_open_subset_def: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" shows "open_cover_of_open_subset X is_open U I C \<equiv> open_cover_of_subset X is_open U I C \<and> open_cover_of_open_subset_axioms is_open U"
map_to_set_dom: fixes m :: "'d \<Rightarrow> 'e option" shows "dom m = fst ` map_to_set m"
language_Collapse_dfa: shows "dfa.language ST.Collapse_dfa = ST.language"
in_keys_plusI1: fixes t :: "'d"   and p :: "'d \<Rightarrow>\<^sub>0 'e"   and q :: "'d \<Rightarrow>\<^sub>0 'e" assumes "t \<in> keys p"   and "t \<notin> keys q" shows "t \<in> keys (p + q)"
index_le_size: fixes xs :: "'a list"   and x :: "'a" shows "index xs x \<le> length xs"
strict_coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a < c" shows "inf a b < c"
correctness_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool" shows "OT_12.correctness m1 m2 \<equiv> protocol m1 m2 = funct_OT_12 m1 m2"
mul_eq_trans: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset"   and P :: "'a multiset" assumes "trans r"   and "(M, N) \<in> mul_eq r"   and "(N, P) \<in> mul_eq r" shows "(M, P) \<in> mul_eq r"
rt_strictly_fresher_def'': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "(rt1 \<sqsubset>\<^bsub>i\<^esub> rt2) = (rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2 \<and> \<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1)"
wrapS_unwrapS_fib_body: shows "wrapS oo unwrapS oo fib_body = fib_body"
vpairs_ex_vsnd: fixes y :: "V"   and r :: "V" assumes "y \<in>\<^sub>\<circ> vpairs r" shows "\<exists>a. \<langle>a, vsnd y\<rangle> \<in>\<^sub>\<circ> r"
assms: shows "z \<in> interior (convex hull {a, b, c})"
properfactor_divides: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "Group.monoid G"   and "properfactor G a b" shows "a divides\<^bsub>G\<^esub> b"
acyclic_converse: fixes r :: "('b \<times> 'b) set" shows "acyclic (r\<inverse>) = acyclic r"
mat_hom_mult: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "mat\<^sub>h (A * B) = mat\<^sub>h A * mat\<^sub>h B"
inject_5: fixes x51 :: "('a, 'b) com"   and x52 :: "('a, 'b) com"   and y51 :: "('a, 'b) com"   and y52 :: "('a, 'b) com" shows "(x51 | x52 = y51 | y52) = (x51 = y51 \<and> x52 = y52)"
phase_rel_inv_def: shows "phase_rel_inv = (\<lambda>s. (\<forall>x. sys_ghost_hs_in_sync x s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, filter is_mw_Phase (sys_mem_store_buffers gc s)) \<in> phase_rel)"
contains_initially: fixes gamma :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and A :: "formula" assumes "branch subs (pseq gamma) f"   and "A \<in> set gamma" shows "contains f (0, A) 0"
sup_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + (y + z) = y + (x + z)"
poly_shift_id: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f \<ominus>\<^bsub>P\<^esub> monom P (f 0) 0 = X_poly R \<otimes>\<^bsub>P\<^esub> poly_shift f"