hcju/isabelleps huggingface dataset:
cases: fixes a :: "'a"   and r :: "('b \<times> 'a) set" assumes "a \<in> Relation.Range r" obtains aa :: "'b" and b :: "'a" where "a = b" and "(aa, b) \<in> r"
DiffI: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A\<setminus>B"
dvdm_imp_div_mod: fixes m :: "int"   and u :: "int poly"   and g :: "int poly" assumes "poly_mod.dvdm m u g" shows "\<exists>q r. g = q * u + smult m r"
simps_2: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat" shows "up_cl_proc Mkt p (Suc n) = (\<lambda>w. \<Sum>x\<in>support_set p. prices Mkt x (Suc n) w * p x (Suc n) w)"
take_is_prefix: fixes n :: "nat"   and xs :: "'c list" shows "prefix (take n xs) xs"
ACI_norm_eqI: fixes r :: "'a rexp"   and s :: "'a rexp" assumes "ACI r s" shows "\<guillemotleft>r\<guillemotright> = \<guillemotleft>s\<guillemotright>"
basis_det_nonzero: fixes G :: "'a vec list" assumes "basis (set G)"   and "length G = n" shows "det (mat_of_rows n G) \<noteq> (0::'a)"
dist_finmap_lessI: fixes P :: "'c \<Rightarrow>\<^sub>F 'd"   and Q :: "'c \<Rightarrow>\<^sub>F 'd"   and e :: "real" assumes "domain P = domain Q"   and "0 < e"   and "\<And>i. i \<in> domain P \<Longrightarrow> dist ((P)\<^sub>F i) ((Q)\<^sub>F i) < e" shows "dist P Q < e"
count_replicate_mset: fixes n :: "nat"   and x :: "'b"   and y :: "'b" shows "count (replicate_mset n x) y = (if y = x then n else 0)"
add_less_same_cancel1: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b + a \<subset># b) = (a \<subset># {#})"
owner_seskey_other: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner n, SesKey SK) \<in> s" shows "\<exists>A B C D. fst (snd SK) = {A, B} \<and> snd (snd SK) = {C, D} \<and> (Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s \<and> (Owner n, \<lbrace>Num 3, PubKey C\<rbrace>) \<in> s \<and> (Owner n, Crypt (SesK SK) (PubKey D)) \<in> s"
zero_ne_\<omega>: shows "0 \<noteq> \<omega>"
prems_1: shows "w \<le>p r \<cdot> w"
snd_dis: fixes t :: "trm"   and k :: "stack" shows "t \<star> k \<rhd> k"
veval'_shadows: fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and v :: "value" assumes "\<Gamma> \<turnstile>\<^sub>v t \<down> v"   and "not_shadows_vconsts_env \<Gamma>"   and "\<not> shadows_consts t" shows "not_shadows_vconsts v"
cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes object_ptr :: "(_) object_ptr" shows "cast object_ptr = (case object_ptr of object_ptr.Ext (Inl node_ptr) \<Rightarrow> Some node_ptr | object_ptr.Ext (Inr x) \<Rightarrow> Map.empty x)"
less_multiset_ext\<^sub>D\<^sub>M_less: shows "less_multiset_ext\<^sub>D\<^sub>M (<) = (<)"
bisim_loc_id: fixes l :: "'s"   and u :: "'c \<Rightarrow> real"   and l' :: "'s"   and u' :: "'c \<Rightarrow> real" assumes "(l, u) ~ (l', u')" shows "l = l'"
lowest_tops_alt: fixes l :: "'b ifex list" shows "lowest_tops l = lowest_tops_alt l"
assms_2: shows "h - x \<in> orthogonal_complement M"
arrE: fixes f :: "'c local.term set \<times> 'c local.term set" assumes "CC.arr f" obtains "arr (fst f) \<and> arr (snd f)"
pos_in_aug_inf: fixes x :: "ant" assumes "0 \<le> x" shows "x \<in> Z\<^sub>\<infinity>"
inf_sup_aci_2: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "inf_prod (inf_prod x y) z = inf_prod x (inf_prod y z)"
SP: fixes x :: "'b" assumes "x \<in> S" shows "x \<in> P"
fds_zeta_altdef: shows "fds_zeta = fds (\<lambda>n. if n = 0 then 0::'b else (1::'b))"
decode_0_iff: fixes base :: "nat"   and M :: "hmultiset"   and e :: "nat" assumes "2 \<le> base"   and "well_base\<^sub>h base M"   and "aligned\<^sub>d base e M" shows "(decode base e M = 0) = (M = 0)"
g_def: fixes H :: "'a"   and x :: "real"   and i :: "'a"   and j :: "'a" shows "g H x i j = d2 i j (X + x *\<^sub>R H)"
half_segments_of_aform_strict_all: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1<..<1}"   and "length (half_segments_of_aform X) \<noteq> 1" shows "list_all (\<lambda>seg. ccw' (fst seg) (snd seg) (aform_val e X)) (half_segments_of_aform X)"
assms_6: shows "a \<noteq> b \<and> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<and> c \<noteq> d"
ord_equiv_sym: fixes E :: "('g, 'h) Order_scheme" assumes "Order E"   and "ord_equiv D E" shows "ord_equiv E D"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> Rel'" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel'"
"case": fixes f :: "'e \<Rightarrow> 'e stream \<Rightarrow> 'd"   and x1 :: "'e"   and x2 :: "'e stream" shows "(case x1 ## x2 of x ## xa \<Rightarrow> f x xa) = f x1 x2"
emeasure_measure_spmf_conv_measure_pmf: fixes p :: "'c spmf"   and A :: "'c set" shows "emeasure (measure_spmf p) A = emeasure (measure_pmf p) (Some ` A)"
order_def: fixes S :: "('c, 'd) monoid_scheme" shows "order S = card (carrier S)"
isArrLenLoc_simps_3: fixes T :: "Arraytype"   and a :: "nat" shows "isArrLenLoc (arrLenLoc T a) = True"
compseries_is_D_gchain: fixes r :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "compseries G r f" shows "D_gchain G r f"
strict_implies_not_weak: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y"   and "x \<in> A"   and "y \<in> A" shows "\<not> y \<sqsubseteq> x"
onormI: fixes f :: "'b \<Rightarrow> 'c"   and b :: "real"   and x :: "'b" assumes "\<And>x. norm (f x) \<le> b * norm x"   and "x \<noteq> (0::'b)"   and "norm (f x) = b * norm x" shows "onorm f = b"
rel_simps_76: fixes n :: "num" shows "(0::'d) \<noteq> numeral n"
lincomb_append: fixes rs :: "'a list"   and ms :: "'b list"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and ss :: "'a list"   and ns :: "'b list" assumes "length rs = length ms" shows "scalar_mult.lincomb smult (rs @ ss) (ms @ ns) = scalar_mult.lincomb smult rs ms + scalar_mult.lincomb smult ss ns"
z: fixes x :: "'a spmf" assumes "x \<in> Y" shows "ord_spmf (=) x z"
wset_thread_ok_upd_Some: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'d"   and wo :: "'e wait_set_status option"   and xln :: "'f \<times> 'g \<Rightarrow>f nat" assumes "wset_thread_ok ws ts" shows "wset_thread_ok (ws(t := wo)) (ts(t \<mapsto> xln))"
word_not_not: fixes x :: "'b word" shows "NOT (NOT x) = x"
eq: shows "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
isomorphic_group_relative_homology_of_contractible: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "topspace X \<inter> S \<noteq> {}" shows "relative_homology_group p X S \<cong> reduced_homology_group (p - 1) (subtopology X S)"
edges_covered: shows "E \<inter> dom (discovered s) \<times> UNIV - pending s = tree_edges s \<union> cross_edges s \<union> back_edges s"
span_superset: fixes S :: "'a set" shows "S \<subseteq> cspan S"
assms_2: shows "X'' = Ch h (X \<union> X'')"
CondReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>if (e') e\<^sub>1 else e\<^sub>2,s'\<rangle>"
leg1_composite: shows "trg_f.leg1 = trg.leg1 \<cdot> trg_f.prj\<^sub>1"
pts_symbolic_c_preserves_tfr\<^sub>s\<^sub>t\<^sub>p: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "\<forall>S\<in>\<S> \<union> {to_st \<A>}. list_all tfr\<^sub>s\<^sub>t\<^sub>p S" shows "\<forall>S\<in>\<S>' \<union> {to_st \<A>'}. list_all tfr\<^sub>s\<^sub>t\<^sub>p S"
equivp_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "equivp R = (\<forall>x y. R x y = (R x = R y))"
Cons_replicate_eq: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat"   and y :: "'c" shows "(x # xs = replicate n y) = (x = y \<and> 0 < n \<and> xs = replicate (n - 1) x)"
simps_4: fixes i :: "String.literal"   and p :: "fol fm"   and q :: "fol fm" shows "free_in i (Imp p q) = (free_in i p \<or> free_in i q)"
division_segment_nat_def: fixes n :: "nat" shows "division_segment n = 1"
m_inv: fixes h :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set" assumes "h \<in> generate_field R H"   and "h \<noteq> \<zero>\<^bsub>R\<^esub>" shows "inv\<^bsub>R\<^esub> h \<in> generate_field R H"
inducts: 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__, lna__)\<rfloor>" and "\<not> final1 x__"   | x :: "'x1" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(xa__, lnb__)\<rfloor>" and "lnb__ \<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>"
path_While: fixes prog :: "cmd"   and l :: "nat"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and l' :: "nat"   and b :: "expr" assumes "While_CFG.path prog (_ l _) as (_ l' _)" shows "While_CFG.path (while (b) prog) ((_ l _) \<oplus> 2) (as \<oplus>s 2) ((_ l' _) \<oplus> 2)"
weak_sup_carrier: shows "\<Squnion>carrier L .= \<top>"
Union_eq_VUnion: fixes A :: "V" shows "\<Union> (elts ` elts A) = {a. \<exists>x\<in>elts A. a \<in>\<^sub>\<circ> x}"
closest_pair_time: shows "closest_pair_time \<in> O(\<lambda>x. real x * ln (real x))"
simps_2: fixes x :: "char" shows "ODEo x undeft = undefg"
"1_1": shows "wordinterval_CIDR_split1 r = (Some s, u)"
h3: shows "distinct (get_signs_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
uminus_cont: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> bcontfun" shows "(\<lambda>x. - f x) \<in> bcontfun"
sorted_list_of_set_nonempty: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "list_of A = Min A # list_of (A - {Min A})"
gval_fold_gOr: fixes a :: "'a gexp"   and l :: "'a gexp list"   and g :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gOr (a # l) g) s = gval a s \<or>? gval (fold gOr l g) s"
key_compare_Eq: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(key_compare ko x y = Eq) = (x = y)"
extract_rpath_correct: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and dst :: "nat" assumes "valid_PRED c src PRED"   and "dst \<in> dom PRED" shows "Graph.extract_rpath c src dst PRED \<le> (spec p. Graph.isSimplePath c src p dst \<and> length p = Graph.min_dist c src dst)"
op2_def: shows "op2 \<equiv> prod.fst (prod.snd (fix_syn (\<lambda>(root2, op2, rep2, left2, right2). (\<Lambda> pat. rep2\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root2\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op2\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>Null))) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_scons\<cdot>v\<cdot>(\<Lambda> u us. Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>(sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>us))))), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> us v. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>Null)) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> us va. match_scons\<cdot>va\<cdot>(\<Lambda> v vs. Fixrec.succeed\<cdot>(rep2\<cdot>pat\<cdot>(us :@ [:v:], vs)))))))))"
unwrapA_def: shows "unwrapA \<equiv> \<Lambda> f n. f\<cdot>n >>= n2a"
inv_left: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bijection f" shows "inv f (f a) = a"
SUP_absorb: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and k :: "'c"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "k \<in> I" shows "sup (A k) (Sup (A ` I)) = Sup (A ` I)"
that: shows "\<I> \<turnstile>g \<A> \<surd>"
keyh2: shows "\<exists>w. \<forall>y>w. a * y\<^sup>2 < - b * y - c"
ce_set_to_rel_inj: shows "inj ce_set_to_rel"
exec_gpv_invariant: fixes callee :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and I :: "'b \<Rightarrow> bool"   and \<I> :: "('d, 'e) \<I>"   and x :: "'f"   and s' :: "'b"   and gpv :: "('f, 'd, 'e) gpv"   and s :: "'b" assumes "callee_invariant_on callee I \<I>"   and "(x, s') \<in> set_spmf (exec_gpv callee gpv s)"   and "I s"   and "\<I> \<turnstile>g gpv \<surd>" shows "I s'"
\<nu>: shows "\<guillemotleft>\<nu> : R f \<Rightarrow> R f'\<guillemotright>"
succ_chain_valuation_satisfies: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<langle>\<V> (l ! i)\<rangle> \<Turnstile>\<^sub>t \<T> (l ! i) \<longrightarrow> \<langle>\<V> (l ! j)\<rangle> \<Turnstile>\<^sub>t \<T> (l ! j)"
fimage_vsingleton_nin: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "set {[a, b]\<^sub>\<circ>} `\<^sub>\<bullet> A = []\<^sub>\<circ>"
simps_2: fixes literal :: "Literal"   and clause :: "Literal list"   and valuation :: "Literal list" shows "clauseFalse (literal # clause) valuation = (literalFalse literal valuation \<and> clauseFalse clause valuation)"
post_def: shows "post = (\<lambda>S (V0, u0) (V, cyc). gen_dfs_post E U S V0 u0 V (cyc \<noteq> None) \<and> (case cyc of None \<Rightarrow> E `` V \<inter> onstack = {} | Some (p, v) \<Rightarrow> v \<in> onstack \<and> p \<noteq> [] \<and> path E u0 p v))"
step: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g"   and l' :: "'e"   and u' :: "'f \<Rightarrow> 'g"   and l'' :: "'e"   and u'' :: "'f \<Rightarrow> 'g" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"   and "A \<turnstile> \<langle>l', u'\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>"
allzero: fixes L' :: "(real mpoly \<times> nat) list"   and a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list"   and L :: "(real mpoly \<times> nat) list" assumes "\<forall>(p, i)\<in>set L'. p = 0"   and "(a, b) = chophelper L L'" shows "\<forall>(p, i)\<in>set a. p = 0"
inf_concat_mono: fixes n :: "nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and k' :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "\<exists>\<^sub>\<infinity>i. 0 < n i"   and "inf_concat n k = (i, j)"   and "inf_concat n k' = (i', j')"   and "i < i'" shows "k < k'"
y': shows "\<forall>ya\<in>S - {y}. norm (v - y) < norm (v - ya) \<and> v - y \<in> orthogonal_complement S"
TwoRings_axioms_def: fixes R' :: "('c, 'd) Ring_scheme" shows "TwoRings_axioms R' \<equiv> Ring R'"
m_lcomm: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier M1"   and "y \<in> carrier M1"   and "z \<in> carrier M1" shows "x \<oplus>\<^bsub>M1\<^esub> (y \<oplus>\<^bsub>M1\<^esub> z) = y \<oplus>\<^bsub>M1\<^esub> (x \<oplus>\<^bsub>M1\<^esub> z)"
rec_all1_def: fixes f :: "recf" shows "rec_all1 f = CN rec_sign [rec_accum1 f]"
not_sym: fixes t :: "'b"   and s :: "'b" assumes "t \<noteq> s" shows "s \<noteq> t"
sibling_swapLeaves_sibling: fixes t :: "'a tree"   and b :: "'a"   and a :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>s :: "nat" assumes "consistent t"   and "sibling t b \<noteq> b"   and "a \<noteq> b" shows "sibling (swapLeaves t w\<^sub>a a w\<^sub>s (sibling t b)) a = b"
z_onAxis: shows "xval z = (0::'a) \<and> yval z = (0::'a) \<and> zval z = (0::'a)"
simps_4: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
det_rank_iff: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A \<noteq> (0::'a)) = (vec_space.rank n A = n)"
axioms_1: fixes f :: "'d \<Rightarrow> 'e" assumes "antilinear f" shows "additive f"
append_vec_eq: fixes v :: "'a vec"   and n :: "nat"   and v' :: "'a vec"   and w :: "'a vec"   and w' :: "'a vec" assumes "v \<in> carrier_vec n"   and "v' \<in> carrier_vec n" shows "(v @\<^sub>v w = v' @\<^sub>v w') = (v = v' \<and> w = w')"
incr_bv_0: fixes lev :: "nat"   and t :: "Core.term" shows "incr_bv 0 lev t = t"
flowD_outside: fixes g :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and e :: "'v vertex \<times> 'v vertex" assumes "flow \<Delta>'' g"   and "e \<notin> \<^bold>E\<^bsub>\<Delta>''\<^esub>" shows "g e = 0"
inject_1: fixes x1 :: "loc"   and y1 :: "loc" shows "(Loc x1 = Loc y1) = (x1 = y1)"
high_scores: fixes u :: "nat \<Rightarrow> real"   and C :: "real"   and i :: "nat" assumes "\<not> bdd_above (range u)" shows "\<exists>n. (\<forall>l\<le>n. u l \<le> u n) \<and> C \<le> u n \<and> i \<le> n"
dec_def: fixes \<eta> :: "nat"   and k :: "bool list"   and c :: "bool list" shows "dec \<eta> k c \<equiv> Some (k [\<oplus>] c)"
list2: shows "[0..<length fs] = [0..<j] @ [j] @ [Suc j..<length fs]"
delivery_has_a_cause: fixes m :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "Deliver m \<in> set (history i)" shows "\<exists>j. Broadcast m \<in> set (history j)"
Sup_set_def: fixes A :: "'c set set" shows "\<Union> A = {x. Sup ((\<in>) x ` A)}"
subgroupE_4: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H G"   and "a \<in> H"   and "b \<in> H" shows "a \<otimes> b \<in> H"
impl_r_d2: fixes a :: "'a"   and b :: "'a" shows "a r\<rightarrow> b = d2 a (a \<sqinter> b)"
dependent_mono: fixes B :: "'c set"   and A :: "'c set" assumes "vs2.dependent B"   and "B \<subseteq> A" shows "vs2.dependent A"
viterbi_finalE: fixes s :: "'s"   and os :: "'t list" obtains t :: "'s" where "t \<in> \<S>" and "viterbi_final s os = viterbi s t os" and "snd (viterbi s t os) = (MAX t\<in>\<S>. snd (viterbi s t os))"
numeral_Bit0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
merge_empty: fixes f :: "'a \<Rightarrow> 'm" shows "merge empty f = return empty"
EntryPath_triv: fixes g :: "'g"   and n :: "'node" shows "EntryPath g [n]"
simps_8: fixes a :: "MIR.num" shows "bound0 (NEq a) = numbound0 a"
rtrancl_beta_Abs: fixes s :: "dB"   and s' :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" shows "Abs s \<rightarrow>\<^sub>\<beta>\<^sup>* Abs s'"
cf_def: shows "cf \<equiv> map_poly complex_of_real f"
tautology_selection: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A"   and "Tarskian_logic.remainder Cn K \<phi> = {}" shows "selection K \<phi> = {K}"
sets_vimage_algebra: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'d measure" shows "sets (vimage_algebra X f M) = sigma_sets X {f -` A \<inter> X |A. A \<in> sets M}"
value_diff: fixes i :: "nat" assumes "flow_network \<Delta> (f_i i)"   and "value_flow \<Delta> (f_i i) = value_flow \<Delta> (f_aux i)" shows "value_flow (residual_network (f_i i)) (f_aux (Suc i) \<ominus> f_i i) = value_flow \<Delta> (f_aux (Suc i)) - value_flow \<Delta> (f_i i)"
triangle_in_hom_2: shows "\<guillemotleft>(S.UP g \<star>\<^sub>S S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g)) \<cdot>\<^sub>S S.\<a> (S.UP g) (S.UP f) (S.UP g) \<cdot>\<^sub>S (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f) \<star>\<^sub>S S.UP g) : S.trg (S.UP g) \<star>\<^sub>S S.UP g \<Rightarrow>\<^sub>S S.UP g \<star>\<^sub>S S.src (S.UP g)\<guillemotright>"
i: shows "higher_differentiable_on (S \<inter> f -` {0<..} \<union> S \<inter> f -` {..<0}) (\<lambda>x. if 0 < f x then 1 else - 1) n"
assms_8: shows "Domainp C = (\<lambda>x. x \<in> \<UU>C)"
supp_Abs_1: fixes as :: "atom set"   and x :: "'a" shows "supp ([as]set. x) = supp x - as"
that: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" assumes "s \<in> keys f"   and "f \<in> F" shows "thesis"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'b"   and v :: "'d stream" shows "smap g (smap f v) = smap (g \<circ> f) v"
reverse_step_pres_denot: fixes e :: "exp"   and e' :: "exp"   and v :: "val"   and \<rho> :: "(nat \<times> val) list" assumes "e \<longrightarrow> e'"   and "v \<in> E e' \<rho>" shows "v \<in> E e \<rho>"
length_Suc_conv: fixes xs :: "'a list"   and n :: "nat" shows "(|xs| = Suc n) = (\<exists>y ys. xs = y # ys \<and> |ys| = n)"
B: shows "sinks_aux I D U (x # xs) = sinks_aux I D U xs"
k'_def: shows "k' = (if n div 2 \<le> k then k else n - k)"
\<psi>_inner: shows "inner_prod \<psi> \<psi> = 1"
llist_map_unfold: fixes a :: "'b \<rightarrow> 'b" shows "llist_map\<cdot>a = llist_abs oo ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>a)\<cdot>(u_map\<cdot>(llist_map\<cdot>a))) oo llist_rep"
construct_lhs_vector_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" shows "construct_lhs_vector p qs signs \<equiv> vec_of_list (map (\<lambda>w. rat_of_int (int (length (filter (\<lambda>v. v = w) (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)))))) signs)"
map_dds_def: shows "map_dds \<equiv> \<lambda>uu uua uub. corec_dds (\<lambda>(f, g, S) a. map_prod g (\<lambda>uu. Inr (f, g, uu)) (run S (f a))) (uu, uua, uub)"
\<i>_simps_3: fixes a :: "'a" assumes "obj a" shows "S.trg (S.UP \<i>[a] \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (a, a)) = UP.map\<^sub>0 a"
abs_ge_self: fixes abs :: "'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq a (abs a)"