hcju/isabelleps huggingface dataset:
monom_1_dvd_iff': fixes n :: "nat"   and p :: "'b poly" shows "(monom (1::'b) n dvd p) = (\<forall>k<n. coeff p k = (0::'b))"
special_cosets_lcoset_closed: fixes w :: "'w"   and X :: "'w set" assumes "w \<in> W"   and "X \<in> \<P>" shows "w +o X \<in> \<P>"
trans: fixes a :: "'b::order"   and b :: "'b::order"   and c :: "'b::order" assumes "(a::'b::order) \<le> (b::'b::order)"   and "(b::'b::order) \<le> (c::'b::order)" shows "(a::'b::order) \<le> (c::'b::order)"
f_the_inv_into_f: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and y :: "'c" assumes "inj_on f A"   and "y \<in> f ` A" shows "f (the_inv_into A f y) = y"
bind_right_unit: fixes m :: "'b \<Rightarrow> ('c \<times> 'b) option" shows "bind\<^sub>S\<^sub>E m unit\<^sub>S\<^sub>E = m"
indRelRPOp_indRelRPO_eq: shows "indRelRPOp = indRelRPOinfix"
mlex_leq: fixes f :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" assumes "f x \<le> f y"   and "(x, y) \<in> R" shows "(x, y) \<in> f <*mlex*> R"
trivial_phis_def: fixes phis :: "'c \<Rightarrow> ('a \<times> 'b, 'b list) mapping"   and g :: "'c" shows "CFG_SSA_wf_base_code.trivial_phis phis g = Set.filter (\<lambda>(n, v). CFG_SSA_wf_base_code.trivial_code v (the (Mapping.lookup (phis g) (n, v)))) (Mapping.keys (phis g))"
Decompose: fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S> :: "('fun, 'var) strand_step list set" assumes "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t \<A> \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (\<S>, \<A> @ [Decomp (Fun f T)])"
expl_cond_exp_cong: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> real"   and Z :: "'c \<Rightarrow> real"   and Y :: "'c \<Rightarrow> 'd" assumes "\<forall>w\<in>space M. X w = Z w" shows "\<forall>w\<in>space M. expl_cond_expect M Y X w = expl_cond_expect M Y Z w"
add_ac_2: fixes a :: "'d"   and b :: "'d" shows "a + b = b + a"
fbox_demodalisation3: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(r y \<le> antirange_semiring.bbox (\<cdot>) ar x (r z)) = (x \<cdot> r y \<le> r z \<cdot> x)"
sep_add_commute: fixes x :: "'a"   and y :: "'a" assumes "x ## y" shows "x + y = y + x"
size_1_singleton_mset: fixes M :: "'b multiset" assumes "size M = 1" shows "\<exists>a. M = {#a#}"
pre_oldest_configuration_finite: fixes q :: "'b"   and n :: "nat" shows "finite (pre_oldest_tokens q n)"
assms: shows "\<forall>i. \<exists>j. f (i, j) = top"
phi_one: shows "\<phi> \<one> = \<one>\<^bsub>H\<^esub>"
nipd': fixes l :: "nat" assumes "k < l"   and "l < m" shows "(\<pi> @\<^bsup>n\<^esup> \<pi>') l \<noteq> ipd (\<pi> k)"
Abs_fnat_hom_mult: fixes a :: "nat"   and b :: "nat" shows "word_of_nat a * word_of_nat b = word_of_nat (a * b)"
OneTrans_HPT_SA2: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and T :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and SA :: "('d, 'e, 'f) seqauto"   and U :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" assumes "TS \<in> HPT ST"   and "T \<in> TS"   and "target T \<in> States SA"   and "U \<in> TS"   and "target U \<in> States SA"   and "SA \<in> SAs (HA ST)" shows "T = U"
Keys_mono: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "A \<subseteq> B" shows "Keys A \<subseteq> Keys B"
prv_imp_cnj: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> \<psi>)"   and "prv (imp \<phi> \<chi>)" shows "prv (imp \<phi> (cnj \<psi> \<chi>))"
seq_stop_elim: fixes c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>Stop ;; c, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "c' = c \<and> mds' = mds \<and> mem' = mem"
module_vec_simps_4: fixes ty :: "'a itself"   and n :: "nat" shows "(\<odot>\<^bsub>module_vec ty n\<^esub>) = (\<cdot>\<^sub>v)"
less_asym_TC: fixes x :: "V"   and y :: "V" assumes "x \<sqsubset> y"   and "y \<sqsubset> x" shows "False"
set_less_eq_aux_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq>' B) = (A \<in> infinite_complement_partition \<and> A = B \<or> A \<sqsubset>' B)"
assms_1: shows "insert (y EQ Zero) H \<turnstile> A"
Says_unique_NB: fixes C :: "agent"   and S :: "agent"   and X :: "msg"   and B :: "agent"   and A :: "agent"   and NA :: "nat"   and nb :: "msg"   and evs :: "event list"   and S' :: "agent"   and X' :: "msg"   and B' :: "agent"   and A' :: "agent"   and NA' :: "nat" assumes "Says C S \<lbrace>X, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> \<in> set evs"   and "Gets S' \<lbrace>X', Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace>\<rbrace> \<in> set evs"   and "nb \<notin> analz (knows Spy evs)"   and "evs \<in> yahalom" shows "NA' = NA \<and> A' = A \<and> B' = B"
shiftr_def: fixes w :: "'a word"   and n :: "nat" shows "w >> n = (shiftr1 ^^ n) w"
unit_wfactors: fixes a :: "'a" assumes "a \<in> Units (Ring_Divisibility.mult_of R)" shows "wfactors (Ring_Divisibility.mult_of R) [] a"
double_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(plus a a = zero) = (a = zero)"
lookup_shift_map_keys_plus: fixes t :: "'a"   and c :: "'c"   and xs :: "('t, 'c) oalist_ntm"   and u :: "'t" shows "lookup (MP_oalist (shift_map_keys t ((*) c) xs)) (t \<oplus> u) = c * lookup (MP_oalist xs) u"
rtos_hom7: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "i div s = r - Suc 0" shows "Suc (rtos r s i) mod r = 0"
join_left: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq> x \<squnion> y"
eq_upred_sym: fixes x :: "('c, 'b) uexpr"   and y :: "('c, 'b) uexpr" shows "(x =\<^sub>u y) = (y =\<^sub>u x)"
assms: shows "a \<union>\<^sub>\<circ> b \<in>\<^sub>\<circ> Vset \<alpha>"
tak_pcorrect: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "tak_dom (x, y, z)" shows "tak x y z = (if x \<le> y then y else if y \<le> z then z else x)"
lm088: fixes N :: "'d set"   and G :: "'e list"   and bids :: "'d \<times> 'e set \<Rightarrow> 'f" assumes "N \<noteq> {}"   and "finite N"   and "distinct G"   and "set G \<noteq> {}" shows "takeAll (\<lambda>x. winningAllocationRel N (set G) ((\<in>) x) bids) (allAllocationsAlg N G) \<noteq> []"
LstSeq_imp_Seq_same: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "LstSeq s k y" shows "Seq s k"
Chain_finite_iterates_above_fun_pow_iff: fixes x :: "'a"   and a :: "'a" assumes "\<forall>M\<in>Chains leq. finite M" shows "(x \<in> iterates_above a) = (\<exists>j. x = (f ^^ j) a)"
ss_lt_hd_w: shows "\<forall>x\<in>set (ss ! hd w). x < hd w"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "trimming \<Gamma> f g" obtains "current \<Gamma> g" and "wave \<Gamma> g" and "g \<le> f" and "\<And>x. \<lbrakk>x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f); x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> KIR g x" and "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) - A \<Gamma> = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - A \<Gamma>"
nth_drop: fixes n :: "nat"   and xs :: "'a list"   and i :: "nat" assumes "n \<le> length xs" shows "drop n xs ! i = xs ! (n + i)"
lift_start_obs_def: fixes t :: "'c"   and obs :: "'d list" shows "lift_start_obs t obs = (t, InitialThreadAction) # map (\<lambda>ob. (t, NormalAction ob)) obs"
adjacent_V_1: fixes v :: "'v"   and v' :: "'v" assumes "adjacent v v'" shows "v \<in> V"
sel_1: fixes x1 :: "nat" shows "document_ptr.the_ref (document_ptr.Ref x1) = x1"
sp_append_sub: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and v3 :: "'c \<times> nat" assumes "SubRel.subpath g v1 es v2 subs"   and "(v2, v3) \<in> subs" shows "SubRel.subpath g v1 es v3 subs"
pX: shows "terminals p @ [X] \<in> \<L>\<^sub>P"
Pg_in_Vr: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "Pg K v \<in> carrier (Vr K v)"
exe_const_type_of_ok: fixes cto :: "(String.literal \<times> typ) list"   and ta :: "String.literal \<Rightarrow> nat option"   and sa :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "alist_conds cto" shows "(\<forall>ty\<in>ran (map_of cto). typ_ok_sig (map_of cto, ta, sa) ty) = (\<forall>ty\<in>snd ` set cto. typ_ok_sig (map_of cto, ta, sa) ty)"
orientable_path: fixes x :: "'a"   and y :: "'a" assumes "arc x"   and "x \<le> - - y\<^sup>\<star>" shows "\<exists>z\<le>y. asymmetric z \<and> x \<le> - - z\<^sup>\<star>"
REC_rec: shows "REC = (mid`?`x \<rightarrow> right`!`x \<rightarrow> ack \<rightarrow> REC)"
less_QDelta_def: fixes qd1 :: "QDelta"   and qd2 :: "QDelta" shows "(qd1 < qd2) = (qdfst qd1 < qdfst qd2 \<or> qdfst qd1 = qdfst qd2 \<and> qdsnd qd1 < qdsnd qd2)"
deny_all_def: fixes p :: "'b" shows "D\<^sub>U p = \<lfloor>deny ()\<rfloor>"
mult_map_scale_conv_mult: fixes s :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.map ((*) s) p = monomial s (0::'a) * p"
elims: fixes x :: "'a list"   and xa :: "nat"   and y :: "'a" assumes "nth_el x xa = y" obtains n :: "nat" where "xa__ = []" and "xa = na__" and "y = \<zero>"   | x :: "'a" and xs :: "'a list" where "xa__ = xb__ # xsa__" and "xa = 0" and "y = xb__"   | x :: "'a" and xs :: "'a list" and n :: "nat" where "xa__ = x # xs" and "xa = Suc n" and "y = nth_el xs n"
het_gt_0_imp_noteq_ne: fixes l :: "real list" assumes "0 < het l" shows "list_neq l (mean l) \<noteq> []"
JN_absorb: fixes k :: "'b"   and I :: "'b set"   and F :: "'b \<Rightarrow> 'c program" assumes "k \<in> I" shows "F k \<squnion> JOIN I F = JOIN I F"
set_mset_eq_empty_iff: fixes M :: "'k multiset" shows "(set_mset M = {}) = (M = {#})"
gfp_trans_def: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "gfp_trans T = Sup_trans {t. (\<forall>P. unitary P \<longrightarrow> unitary (t P)) \<and> le_utrans t (T t)}"
exI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P x" shows "\<exists>x. P x"
simps_93: fixes x7 :: "rat"   and x6 :: "nat" shows "Pri_AgrK x7 \<noteq> Passwd x6"
rel_mapping_alt: fixes Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'e \<Rightarrow> 'c option"   and P' :: "'e \<Rightarrow> 'd option" shows "rel_mapping Q P P' = (dom P = dom P' \<and> (\<forall>p\<in>dom P. Q (the (P p)) (the (P' p))))"
wf_matchF_aux_Nil: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and k :: "nat" shows "wf_matchF_aux \<sigma> V n R I r [] 0 k"
cancel_compls1: fixes L :: "'a literal set" shows "L\<^sup>C\<^sup>C = L"
higher_pderiv_of_int_poly: fixes n :: "nat"   and p :: "int poly" shows "(pderiv ^^ n) (of_int_poly p) = of_int_poly ((pderiv ^^ n) p)"
wt_ge_\<delta>\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A" shows "zhmset_of (of_nat \<delta>) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s))"
map_id: fixes t :: "'c llist" shows "lmap id t = t"
h1a: shows "\<forall>i. MPoly_Type.degree (isolate_variable_sparse p var i) var = 0"
subst_apply_fv_union: fixes \<delta> :: "'c \<Rightarrow> ('b, 'a) Term.term"   and V :: "'c set"   and t :: "('b, 'c) Term.term" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V) \<union> fv (t \<cdot> \<delta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` (V \<union> fv t))"
bv_to_int_lower_range: fixes w :: "bit list" shows "- (2 ^ (length w - 1)) \<le> bv_to_int w"
Fvars_num: fixes t :: "trm" assumes "t \<in> num" shows "Fvars t = {}"
DSourcesA42_L1: shows "DSources level1 sA42 = {}"
take_bit_int_eq_self_iff: fixes n :: "nat"   and k :: "int" shows "(take_bit n k = k) = (0 \<le> k \<and> k < 2 ^ n)"
weight_sample_uniform_units: shows "weight_spmf (sample_uniform_units (order \<G>)) = 1"
diff_add_zmset: fixes M :: "'b zmultiset"   and N :: "'b zmultiset"   and Q :: "'b zmultiset" shows "M - (N + Q) = M - N - Q"
ker_def: fixes F :: "('d, 'e) aGroup_scheme"   and G :: "('f, 'g) aGroup_scheme"   and f :: "'d \<Rightarrow> 'f" shows "ker\<^bsub>F,G\<^esub> f = {a \<in> carrier F. f a = \<zero>\<^bsub>G\<^esub>}"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "L'.C.in_hom f a b" obtains "L'.C.arr f" and "L'.C.dom f = a" and "L'.C.cod f = b"
adjuster_wit_small: fixes v :: "'a list"   and a :: "'a vec"   and xs :: "'a vec list"   and x1 :: "'a list"   and x2 :: "'a vec" shows "(adjuster_wit v a xs = (x1, x2)) = (fst (adjuster_wit v a xs) = x1 \<and> x2 = adjuster n a xs)"
retraction_is_epi: fixes g :: "'a" assumes "A.retraction g" shows "A.epi g"
continuous_within_poly: fixes z :: "'b"   and s :: "'b set"   and p :: "'b poly" shows "continuous (at z within s) (poly p)"
class_add_sup_state_opt: fixes P :: "jvm_prog"   and \<tau> :: "(ty list \<times> ty err list) option"   and \<tau>' :: "(ty list \<times> ty err list) option"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> \<tau> \<le>' \<tau>'"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> \<tau> \<le>' \<tau>'"
homotopic_paths_imp_path: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" assumes "homotopic_paths s p q" shows "path p \<and> path q"
injects: fixes a :: "'r"   and slist :: "[:'r:]"   and a' :: "'r"   and slist' :: "[:'r:]" assumes "a \<noteq> \<bottom>"   and "slist \<noteq> \<bottom>" shows "(a :# slist = a' :# slist') = (a = a' \<and> slist = slist')"
project_increasing_I: fixes F :: "'a program"   and G :: "'c program"   and func :: "'a \<Rightarrow> 'e"   and C :: "'c set" assumes "extend h F \<squnion> G \<in> increasing (func \<circ> f)" shows "F \<squnion> project h C G \<in> increasing func"
pure_conj_3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "bool"   and Q' :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (\<lambda>s. P' \<and> Q' s) Q = (\<lambda>s. P' \<and> sep_algebra.sep_conj plus sep_disj Q' Q s)"
filterlim_at_top_iff_smallomega: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" shows "filterlim f at_top F = (f \<in> \<omega>[F](\<lambda>_. 1) \<and> eventually_nonneg F f)"
DERIV_atLeastAtMost_imp_continuous_on: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_field_derivative y) (at x)" shows "continuous_on {a..b} f"
CastRedsAddr: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s',b'\<rangle>"   and "hp s' a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>Cast C e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s',b'\<rangle>"
restricted_idealI: fixes p :: "int mpoly"   and A :: "int mpoly multiset"   and C :: "nat set" assumes "p \<in> pac_ideal (set_mset A)"   and "vars p \<subseteq> C" shows "p \<in> restricted_ideal_to\<^sub>I C A"
select_convs_2: fixes para :: "char list list"   and body :: "expr"   and more :: "'a" shows "body \<lparr>para = para, body = body, \<dots> = more\<rparr> = body"
\<delta>: shows "ball a \<delta> \<subseteq> S"
assms_1: shows "poly (hd ps) x\<^sub>0 \<noteq> 0"
seg_path2: fixes a :: "'a"   and b :: "'a" assumes "segment a b \<noteq> {}" obtains ab :: "'a set" where "path ab a b" and "segment a b \<subseteq> ab"
char_poly_defs_2: fixes A :: "'b mat" shows "char_poly_matrix A = [:0::'b, 1::'b:] \<cdot>\<^sub>m 1\<^sub>m (dim_row A) + map_mat (\<lambda>a. [:- a:]) A"
tendsto_Im5: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im5 (g x)) \<longlongrightarrow> Im5 a) F"
vector_space_ow_axioms: shows "vector_space_ow U\<^sub>M\<^sub>_\<^sub>1 (+) (0::'b) (-) uminus (*s\<^sub>1)"
not_in_inext_fix: fixes n :: "nat"   and I :: "nat set" assumes "n \<notin> I" shows "inext n I = n"
mem_simps_4: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
intro: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x. f x \<circ> f x = f x" shows "comp_fun_idem_axioms f"
the_inv_signed_lconjaction_by_order2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "class.group_add minus plus zero uminus"   and "plus s s = zero" shows "the_inv (signed_funaction (\<lambda>x y. minus (plus x y) x) s) = signed_funaction (\<lambda>x y. minus (plus x y) x) s"
euclidean_size_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size one = 1"
mono_monitor: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<pi>' :: "(char list \<times> 'a list) prefix" assumes "\<pi> \<le> \<pi>'" shows "joiner (\<lambda>k. M (splitter \<pi> k)) \<subseteq> joiner (\<lambda>k. M (splitter \<pi>' k))"
Sup_upper: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 bot :: "'c"   and top :: "'c"   and x :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
enat2_cases: fixes y :: "enat"   and ya :: "enat" obtains nat :: "nat" and nata :: "nat" where "y = enat nat__" and "ya = enat nata"   | nat :: "nat" where "y = enat natb__" and "ya = \<infinity>"   | nat :: "nat" where "y = \<infinity>" and "ya = enat nat"   | "y = \<infinity>" and "ya = \<infinity>"
ord_modulus_prod_coprime: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat"   and x :: "nat" assumes "finite A"   and "\<And>i j. \<lbrakk>i \<in> A; j \<in> A; i \<noteq> j\<rbrakk> \<Longrightarrow> coprime (f i) (f j)" shows "ord (prod f A) x = (LCM i\<in>A. ord (f i) x)"
old_infer_from_def: fixes CC :: "'b literal multiset set"   and \<gamma> :: "'b Inference_System.inference" shows "old_infer_from CC \<gamma> = (set_mset (old_side_prems_of \<gamma> + {#old_main_prem_of \<gamma>#}) \<subseteq> CC)"
A: shows "ipurge_tr_aux I D (U \<union> V) xs = ipurge_tr_aux I D V (ipurge_tr_aux I D U xs)"
chain_map_id_gen: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and f :: "'b \<Rightarrow> 'b" assumes "singular_chain p X c"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = x" shows "chain_map p f c = c"
offs_enum_zero_aux: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and k :: "nat" assumes "index_less index key"   and "0 < n"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "offs_num n xs index key mi ma (n - Suc 0) = 0" shows "offs (enum xs index key n mi ma) k ! (n - Suc 0) = length xs + k"
2: shows "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
lset_lappend_lfinite: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lfinite xs" shows "lset (lappend xs ys) = lset xs \<union> lset ys"
fps_exp_nth: fixes a :: "'a"   and n :: "nat" shows "fps_exp a $ n = a ^ n / of_nat (fact n)"
hindrance_into_hindrance_by: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f" shows "\<exists>\<epsilon>>0. hindrance_by \<Gamma> f \<epsilon>"
product_dvd_irreducibleD: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "irreducible x"   and "a * b dvd x" shows "is_unit a \<or> is_unit b"
ord_recp_functional: fixes T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf"   and x :: "hf"   and y :: "hf"   and y' :: "hf" assumes "ord_recp T G H x y"   and "ord_recp T G H x y'" shows "y' = y"
hyps_2: fixes s :: "char list" shows "(nat_to_string (n div 10) = CHR ''0'' # s) = (n div 10 = 0 \<and> s = [])"
borel_measurable_ereal_iff: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" shows "((\<lambda>x. ereal (f x)) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
blocks_def: fixes R :: "'node set"   and p :: "'node" shows "blocks R p \<equiv> \<forall>Q. quorum_of p Q \<longrightarrow> Q \<inter> R \<noteq> {}"
Liminf_inverse_ereal: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "\<forall>\<^sub>F x in F. 0 \<le> f x" shows "Liminf F (\<lambda>x. inverse (f x)) = inverse (Limsup F f)"
omega_prev: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "[]\<^sub>\<circ> \<in>\<^sub>\<circ> n" obtains k :: "V" where "n = ZFC_in_HOL.succ k"
ordering_axioms: shows "ordering (\<subseteq>#) (\<subset>#)"
interval_eqI: fixes a :: "'a interval"   and b :: "'a interval" assumes "lower a = lower b"   and "upper a = upper b" shows "a = b"