hcju/isabelleps huggingface dataset:
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a interval"   and b :: "'b interval" assumes "rel_interval R a b" obtains x1a :: "'a" and x2a :: "'a" and y1a :: "'b" and y2a :: "'b" where "a = Interval x1a x2a" and "b = Interval y1a y2a" and "R x1a y1a" and "R x2a y2a"
homeomorphic_ANR_iff_ANR: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "ANR S = ANR T"
dickson_less_p_zero: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "gd_term.dickson_less_p pair_of_term ord_term_strict (\<lambda>_. 0) m = ordered_term.ord_strict_p ord_term_strict"
mk_minsky_copy: fixes u :: "nat"   and v :: "nat"   and w :: "nat" assumes "u \<noteq> v"   and "u \<noteq> w"   and "v \<noteq> w" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = v then vs u else if x = w then 0 else vs x))"
diagonal_to_Smith_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith A bezout = diagonal_to_Smith_aux A [0..<min (nrows A) (ncols A) - 1] bezout"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
y1: shows "y [^]\<^bsub>R\<^esub> CARD('a) ^ m' = y"
edge_bipartite_webI2: fixes u :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "u \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub>"   and "u \<notin> A \<Gamma>"   and "u \<notin> B \<Gamma>" shows "edge (bipartite_web_of \<Gamma>) (Inl u) (Inr u)"
add_pow_int_lt: fixes k :: "int"   and R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "k < 0" shows "[k] \<cdot>\<^bsub>R\<^esub> a = \<ominus>\<^bsub>R\<^esub> ([nat (- k)] \<cdot>\<^bsub>R\<^esub> a)"
mergesort_recurrence: shows "mergesort_recurrence \<in> \<Theta>(\<lambda>x. real x * ln (real x))"
b_least2_aux4: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "b_least2 f x y = y"   and "f x y = 0" shows "b_least2 f x (Suc y) = Suc y"
degree_0_content: fixes p :: "int poly" assumes "Polynomial.degree p = 0" shows "content p = \<bar>poly.coeff p 0\<bar>"
t_heap_of_B_bound: fixes xs :: "'b list" shows "t_heap_of_B xs \<le> 6 * length xs + 1"
bounded: fixes K :: "complex set" assumes "compact K"   and "K \<subseteq> S" shows "\<exists>B. \<forall>h\<in>\<H>. \<forall>z\<in>K. cmod (h z) \<le> B"
chain_criterion: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'e"   and q :: "'t \<Rightarrow>\<^sub>0 'e"   and r :: "'t \<Rightarrow>\<^sub>0 'e" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "lp r adds lcs (lp p) (lp q)"   and "component_of_term (lt r) = component_of_term (lt p)"   and "crit_pair_cbelow_on d m F p r"   and "crit_pair_cbelow_on d m F r q" shows "crit_pair_cbelow_on d m F p q"
in_hhomE: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>" obtains "arr \<mu>" and "obj a" and "obj b" and "src \<mu> = a" and "trg \<mu> = b"
subset_T: fixes t1 :: "real" assumes "t1 \<in> cball t et" shows "{t--t1} \<subseteq> cball t et"
intros_35: fixes n :: "i32"   and i :: "nat" assumes "\<not> int_eq n 0" shows "\<lparr>[$C ConstInt32 n, $Br_if i]\<rparr> \<leadsto> \<lparr>[$Br i]\<rparr>"
dbl_simps_4: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (uminus one) = uminus (numeral.numeral one plus (num.Bit0 num.One))"
map_ident: fixes t :: "'b llist" shows "lmap (\<lambda>x. x) t = t"
leftmost_Derives1_propagate: fixes i :: "nat"   and a :: "symbol list"   and j :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "leftmost i a"   and "Derives1 a j r b" shows "is_word b \<and> i = j \<or> (\<exists>k. leftmost k b \<and> i \<le> k)"
secure_mac: fixes q :: "nat" shows "constructive_security MAC.res (\<lambda>_. AUTH.res) (\<lambda>_. CNV Message_Authentication_Code.sim (Inl None)) (\<lambda>\<eta>. \<I>_uniform {x. valid_mac_query \<eta> x} (insert None (Some ` (nlists UNIV \<eta> \<times> nlists UNIV \<eta>)))) (\<lambda>\<eta>. \<I>_uniform UNIV (insert None (Some ` nlists UNIV \<eta>))) (\<lambda>\<eta>. \<I>_uniform (vld \<eta>) UNIV \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` vld \<eta>))) (\<lambda>_. enat q) True (\<lambda>_. (insec_query_of, map_option snd))"
assms_3: assumes "r x y"   and "r y x" shows "thesis"
triangle_left: shows "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^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)) = S.lunit' (S.UP f) \<cdot>\<^sub>S S.runit (S.UP f)"
borel_measurable_if: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "S \<in> sets lebesgue" shows "((\<lambda>x. if x \<in> S then f x else (0::'c)) \<in> borel_measurable lebesgue) = (f \<in> borel_measurable (lebesgue_on S))"
oone_ordIso: fixes x :: "'d" shows "oone =o {(x, x)}"
basis_def: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and A :: "'d set" assumes "vectorspace K V" shows "vectorspace.basis K V A = (\<not> module.lin_dep K V A \<and> LinearCombinations.module.span K V A = carrier V \<and> A \<subseteq> carrier V)"
assms_7: fixes s :: "(nat list \<times> bool list \<times> nat list) pmf" assumes "a \<noteq> b"   and "{a, b} = {x, y}"   and "BIT_inv s a [x, y]"   and "set qs \<subseteq> {a, b}"   and "qs \<in> pattern" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x, y] \<and> T\<^sub>p_on_rand' BIT s qs = bit"
repINT: fixes w :: "32 Word.word" assumes "real_of_int (sint w) < real_of_int (sint POS_INF)"   and "real_of_int (sint NEG_INF) < real_of_int (sint w)" shows "w \<equiv>\<^sub>E real_of_int (sint w)"
ideD_1: fixes a :: "'a" assumes "Right_a.ide a" shows "Right_a.arr a"
shiftl_ge_0: fixes i :: "int"   and n :: "nat" shows "(0 \<le> i << n) = (0 \<le> i)"
map_cong: fixes xs :: "'g list"   and ys :: "'g list"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
subst_1: fixes y :: "name"   and v :: "trm"   and x :: "name" shows "Var x[y::=v] = (if x = y then v else Var x)"
"8_2": shows "f T3 \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t (T1 @ T2) \<union> X"
tangent_field_is_tangent: fixes c :: "real \<Rightarrow> 'a"   and x :: "real" assumes "diff k charts_eucl charts c"   and "0 < k" shows "tangent_field c x \<in> tangent_space (c x)"
splitFace_holds_faces_subset: fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and n :: "nat" assumes "pre_splitFace g' v a f' [countVertices g'..<countVertices g' + n]"   and "minGraphProps g'" shows "faces_subset (snd (snd (splitFace g' v a f' [countVertices g'..<countVertices g' + n])))"
div_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "y \<noteq> \<zero>" shows "x \<oslash> y \<in> carrier R"
timpl_apply_refl: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and a :: "'c set"   and b :: "'c set" shows "t \<in> set \<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle>"
RUV_spawn: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and q :: "'b"   and Ml :: "'d set"   and Me :: "'d set"   and U :: "'a set"   and V :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 1"   and "q \<in># P"   and "(entry fg q, Ml, Me) \<in> RUV_cs fg U V"   and "(mon_n fg u \<union> mon fg p) \<inter> (Ml \<union> Me) = {}" shows "(u, mon fg p \<union> M, Ml \<union> Me) \<in> RUV_cs fg U V"
span_superset: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale" shows "S \<subseteq> module.span scale S"
l_zero: fixes x :: "'a" shows "(0::'a) + x = x"
P': shows "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"
assms: shows "(P \<W> Q \<W> R) \<sigma>"
is_red_union: fixes A :: "('t \<Rightarrow>\<^sub>0 'e) set"   and B :: "('t \<Rightarrow>\<^sub>0 'e) set"   and p :: "'t \<Rightarrow>\<^sub>0 'e" shows "is_red (A \<union> B) p = (is_red A p \<or> is_red B p)"
before_Un: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "set xs = set ys"   and "x \<in> set xs" shows "before x ys = before x xs \<inter> before x ys \<union> after x xs \<inter> before x ys"
ortho: fixes h :: "'a" shows "h - P h \<in> orthogonal_complement (cspan {t})"
diff_strict_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less b a" shows "less (minus c a) (minus c b)"
permutes_prod: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "p permutes S" shows "(\<Prod>s\<in>S. f (p s) s) = (\<Prod>s\<in>S. f s (inv p s))"
monomial_eq_itself: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" shows "monomial (lc p) (lt p) = p"
sep_disj_addD: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj x y \<and> sep_disj x z"
simps_7: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c" shows "(case TernaryTrue of TernaryTrue \<Rightarrow> f1 | TernaryFalse \<Rightarrow> f2 | TernaryUnknown \<Rightarrow> f3) = f1"
dlex_pm_lin: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "dlex_pm s t \<or> dlex_pm t s"
permutes_swap_id: fixes a :: "'d"   and S :: "'d set"   and b :: "'d" assumes "a \<in> S"   and "b \<in> S" shows "Fun.swap a b id permutes S"
integrable_restrict_Int: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'a set" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) integrable_on T) = (f integrable_on S \<inter> T)"
cases: fixes a1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and a2 :: "'d"   and a3 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and a4 :: "nat \<Rightarrow> nat"   and a5 :: "'e \<Rightarrow> nat"   and a6 :: "nat"   and a7 :: "'d"   and a8 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^bsub>a4,a5,a6\<^esub> \<langle>a7, a8\<rangle>" obtains A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" and l :: "'d" and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" and v :: "'e \<Rightarrow> nat" and n :: "nat" and l' :: "'d" and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" and k :: "nat \<Rightarrow> nat" where "a1 = A" and "a2 = l" and "a3 = D" and "a4 = k" and "a5 = v" and "a6 = n" and "a7 = l'" and "a8 = DBM_Normalization.norm (FW D' n) (\<lambda>x. real (k x)) n" and "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"
assms_5: shows "insert_nth j x xs = insert_nth j' x xs'"
prv_exi_imp_gen: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<chi>"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (exi x \<phi>) \<chi>)"
cbody_def: fixes c :: "class" shows "cbody c = \<lparr>access = access c, cfields = cfields c, methods = methods c, init = init c\<rparr>"
rel_pmf_return_pmfI: fixes P :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'b"   and y :: "'d" assumes "P x y" shows "rel_pmf P (return_pmf x) (return_pmf y)"
nmi: shows "\<not> Ifm vs (x # bs) (minusinf p)"
simps: fixes w :: "'a list"   and I :: "(nat + nat set) list" shows "stream_enc (w, I) = smap2 (enc_atom I) nats (w @- sconst any)"
VSLuckyBlocks_def: shows "VSLuckyBlocks = opt_group \<circ> QE_dnf' opt_group (the_real_step_augment luckyFind') \<circ> VSLuckiestBlocks \<circ> opt_group"
is_regular_sequence_snocD_1: fixes fs :: "('c \<Rightarrow>\<^sub>0 'd) list"   and f :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" assumes "is_regular_sequence (fs @ [f])"   and "q * f \<in> ideal (set fs)" shows "q \<in> ideal (set fs)"
coActSimp_2: fixes \<alpha> :: "act" shows "(coAction \<alpha> = \<tau>) = (\<alpha> = \<tau>)"
cont_\<Gamma>: fixes c :: "'a \<Rightarrow> bool"   and b :: "('a \<times> 'a) set" shows "cont (\<Gamma> c b)"
assms_1: shows "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"
List_unique: fixes h :: "'c \<Rightarrow> 'c ref"   and x :: "'c ref"   and as :: "'c list"   and bs :: "'c list" assumes "List h x as"   and "List h x bs" shows "as = bs"
chain_iterates_above: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "a \<in> Field leq" shows "bourbaki_witt_fixpoint.iterates_above lub leq f a \<in> Chains leq"
integral_diff: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on S"   and "g integrable_on S" shows "integral S (\<lambda>x. f x - g x) = integral S f - integral S g"
assms_4: fixes i :: "nat" assumes "i \<in> pdevs_domain d" shows "P i (pdevs_apply b i) = Q i (pdevs_apply d i)"
collinear_trans1: fixes z0 :: "complex"   and z2 :: "complex"   and z1 :: "complex"   and z3 :: "complex" assumes "collinear z0 z2 z1"   and "collinear z0 z3 z1"   and "z0 \<noteq> z1" shows "collinear z0 z2 z3"
symcl_def: fixes Rel :: "('c \<times> 'c) set" shows "symcl Rel = Rel \<union> Rel\<inverse>"
power_minus_mult: fixes n :: "nat"   and a :: "'b list" assumes "0 < n" shows "a \<^sup>@ (n - 1) \<cdot> a = a \<^sup>@ n"
fresh2_eq: fixes x :: "nat"   and xs :: "nat set"   and y :: "nat" assumes "x \<notin> xs" shows "fresh2 xs x y = x"
simps_92: fixes x3 :: "'b formula" shows "atoms (\<^bold>\<not> x3) = atoms x3"
mds_consistent_preds_tyenv_update: fixes P' :: "'BExp set"   and P'' :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and v :: "'Var"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mds :: "Mode \<Rightarrow> 'Var set"   and P :: "'BExp set"   and t :: "'BExp set" assumes "P' = P'' |` {v. stable \<S> v}"   and "v \<in> dom \<Gamma>"   and "mds_consistent mds \<Gamma> \<S> P" shows "mds_consistent mds (\<Gamma>(v \<mapsto> t)) \<S> P'"
mat_add_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<oplus> g \<equiv> \<lambda>i j. f i j + g i j"
group_add_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" shows "class.group_add minus plus zero uminus \<equiv> class.monoid_add plus zero \<and> class.group_add_axioms minus plus zero uminus"
not_final_thread_cases2: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and x :: "'a"   and ln :: "'b \<Rightarrow>f nat" assumes "final_thread.not_final_thread final s t"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "\<not> final x"   | "ln \<noteq> no_wait_locks"   | w :: "'e wait_set_status" where "wset s t = \<lfloor>w\<rfloor>"
boundr_impl_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" shows "boundr_impl a b x y = (\<forall>j<length b. y ! j \<le> max_y_impl a b x j)"
prems: shows "t \<in> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set (a # A))"
kat_4: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and z :: "('b \<times> 'b) set" assumes "rel_dbq.d x O y = (rel_dbq.d x O y) O rel_dbq.d z" shows "rel_dbq.d x O y \<subseteq> y O rel_dbq.d z"
m: shows "\<exists>m\<in>{i..j}. \<forall>n\<in>{i..j}. f m \<le> f n"
fresh_11: fixes a :: "atom"   and "term" :: "Syntax.term" shows "a \<sharp> Prj1 term = a \<sharp> term"
assms: shows "x \<in> pre_oldest_tokens q n - {Suc n}"
root: shows "h \<turnstile> get_root_node_si (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr) \<rightarrow>\<^sub>r root"
pairwise_ortho_scaleR: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and a :: "'b \<Rightarrow> real" assumes "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B" shows "pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
l11_22b: fixes B :: "'p"   and P :: "'p"   and A :: "'p"   and C :: "'p"   and B' :: "'p"   and P' :: "'p"   and A' :: "'p"   and C' :: "'p" assumes "B P OS A C"   and "B' P' OS A' C'"   and "A B P CongA A' B' P'"   and "P B C CongA P' B' C'" shows "A B C CongA A' B' C'"
Suc_4: shows "\<not> ps (S t (Suc n)) p \<noteq> None"
cardOneImageCardOne: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "card A = 1" shows "card (f ` A) = 1"
assms: shows "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')"
Macaulay_red_reducible: fixes f :: "'t \<Rightarrow>\<^sub>0 'd"   and fs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "f \<in> phull (set fs)"   and "F \<subseteq> set fs"   and "lt_set F = lt_set (set fs)" shows "(lin_red (F \<union> set (Macaulay_red (Keys_to_list fs) fs)))\<^sup>*\<^sup>* f 0"
PCall: fixes \<pi> :: "char list \<Rightarrow> com option"   and p :: "char list"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi> p = Some c"   and "\<pi>: (c, s) \<Rightarrow> t" shows "\<pi>: (PCall p, s) \<Rightarrow> t"
Mslp: shows "slp_of_fas' fas Mapping.empty [] = (M, slp)"
Sigma_fm_def: fixes A :: "fm" shows "Sigma_fm A = (\<exists>B. ss_fm B \<and> supp B \<subseteq> supp A \<and> {} \<turnstile> A IFF B)"
Lim_transform_within_open: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and a :: "'b"   and T :: "'b set"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> l) (at a within T)"   and "open s"   and "a \<in> s"   and "\<And>x. \<lbrakk>x \<in> s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x" shows "(g \<longlongrightarrow> l) (at a within T)"
finite_S: shows "finite (\<Union> (funs_term ` T))"
transpose_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "Finite_Cartesian_Product.transpose A = (\<chi>i j. A $ j $ i)"
inclosed_simps_1: fixes \<sigma> :: "'d"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'e" shows "inoclosed \<sigma> (ii\<not>ni:arrive(m)) = eq_newpkt m"
F_LUB: fixes S :: "nat \<Rightarrow> 'c process" assumes "chain S" shows "F (lim_proc (range S)) = \<Inter> (F ` range S)"
ordLess_Field: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r <o r'" shows "|Field r| <o r'"
bQc: shows "bounded_by 1 (\<lambda>s. bound_of Q - Q s)"
cont_While_True: fixes tst :: "'test"   and s :: "'state"   and c :: "('test, 'atom, 'choice) cmd"   and i :: "nat" assumes "tval tst s = True" shows "local.cont (While tst c) s i = c ;; While tst c"
list_sum_mean: fixes xs :: "real list" shows "sum_list xs = mean xs * real (length xs)"
continuous_on_closure_norm_le: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and b :: "real"   and x :: "'a" assumes "continuous_on (closure s) f"   and "\<forall>y\<in>s. norm (f y) \<le> b"   and "x \<in> closure s" shows "norm (f x) \<le> b"
simps_1: fixes d :: "'d" shows "theLe (Le d) = d"
tendsto_at_topE: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and e :: "real" assumes "(f \<longlongrightarrow> l) at_top"   and "0 < e" obtains x0 :: "'b" where "\<And>x. x0 \<le> x \<Longrightarrow> dist (f x) l < e"
right_mat_compose: fixes ws :: "wall" assumes "is_tangle_diagram ws"   and "domain_wall ws = 0" shows "kauff_mat ws = kauff_mat (basic [] \<circ> ws)"
orderE: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
FullNxt_def: shows "FullNxt \<equiv> Enq inp q1 mid \<and> Unchanged (q2, out) \<or> Deq inp q1 mid \<and> Enq mid q2 out \<or> Deq mid q2 out \<and> Unchanged (inp, q1)"
not_conga_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "\<not> A B C CongA D E F" shows "\<not> D E F CongA A B C"
inverse_unique: fixes f :: "'a"   and g :: "'a" assumes "inverse_arrows f g" shows "local.inv f = g"
neutral: fixes \<sigma> :: "'alt \<Rightarrow> 'alt"   and R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R" shows "sds (permute_profile \<sigma> R) = map_pmf \<sigma> (sds R)"
degree_monic_charpoly: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "degree (charpoly A) = CARD('b) \<and> monic (charpoly A)"
Class_eq: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). stk1.rel x y}" shows "e1.Class a = e1.Class b"
ab_3: shows "{a..b} \<subseteq> existence_ivl0 x"
eq\<rho>_refl: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "eq\<rho> ae \<rho> \<rho>"
ln_one_minus_pos_upper_bound: fixes x :: "real" assumes "0 \<le> x"   and "x < 1" shows "ln (1 - x) \<le> - x"
cancel1_set_eq: fixes ys :: "'d list"   and xs :: "'d list" assumes "cancel1 ys xs" shows "set ys = set xs"
indRelRTPO_impl_TRel_weakly_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_respects_barbs (indRelRTPO TRel) (STCalWB SWB TWB)" shows "rel_weakly_respects_barbs TRel TWB"
intros_14: fixes x :: "name"   and a :: "name"   and y :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> y" shows "<\<nu>x>a<y>.P \<equiv>\<^sub>e a<y>.(<\<nu>x>P)"
runit'_simps_4: fixes f :: "'b" assumes "D.ide f" shows "D.dom \<r>\<^sub>D\<^sup>-\<^sup>1[f] = f"
left_neutral: fixes a :: "'b" shows "(1::'b) * a = a"
simps_2: fixes n :: "nat"   and i :: "nat" shows "unit_vecs_first n (Suc i) = unit_vecs_first n i @ [unit_vec n i]"