hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "Monitor.args"   and xa :: "nat"   and xb :: "nat"   and xc :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and xd :: "(nat \<times> 'a option list set \<times> 'a option list set) list" assumes "valid_mmuaux' x xa xb xc xd" obtains args :: "Monitor.args" and cur :: "nat" and dt :: "nat" and tp :: "nat" and tss :: "nat queue" and len :: "nat" and maskL :: "bool list" and maskR :: "bool list" and a1_map :: "('a option list, nat) mapping" and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping" and "done" :: "'a option list set list" and done_length :: "nat" and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list" where "x = args" and "xa = cur" and "xb = dt" and "xc = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)" and "xd = auxlist" and "args_L args \<subseteq> args_R args \<and> maskL = join_mask (args_n args) (args_L args) \<and> maskR = join_mask (args_n args) (args_R args) \<and> len \<le> tp \<and> length (linearize tss) = len \<and> sorted (linearize tss) \<and> (\<forall>t\<in>set (linearize tss). t \<le> cur \<and> enat cur \<le> enat t + right (args_ivl args)) \<and> table (args_n args) (args_L args) (Mapping.keys a1_map) \<and> Mapping.keys a2_map = {tp - len..tp} \<and> (\<forall>xs\<in>Mapping.keys a1_map. case Mapping.lookup a1_map xs of Some tp' \<Rightarrow> tp' < tp) \<and> (\<forall>tp'\<in>Mapping.keys a2_map. case Mapping.lookup a2_map tp' of Some m \<Rightarrow> table (args_n args) (args_R args) (Mapping.keys m) \<and> (\<forall>xs\<in>Mapping.keys m. case Mapping.lookup m xs of Some tstp \<Rightarrow> tstp_lt tstp (cur - (left (args_ivl args) - 1)) tp \<and> isl tstp = (0 < left (args_ivl args)))) \<and> length done = done_length \<and> length done + len = length auxlist \<and> rev done = map proj_thd (take (length done) auxlist) \<and> Ball (set (take (length done) auxlist)) (check_before (args_ivl args) dt) \<and> sorted (map fst auxlist) \<and> list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map (args_pos args) tp' a1_map) (\<lambda>ts' tp'. filter_a2_map (args_ivl args) ts' tp' a2_map)) (drop (length done) auxlist) (zip (linearize tss) [tp - len..<tp])"
silent_moves_preds_transfers_path: fixes S :: "'node set"   and f :: "'edge \<Rightarrow> 'state edge_kind"   and n :: "'node"   and s :: "'state"   and as :: "'edge list"   and n' :: "'node"   and s' :: "'state" assumes "S,f \<turnstile> (n,s) =as\<Rightarrow>\<^sub>\<tau> (n',s')"   and "valid_node n" shows "preds (map f as) s \<and> transfers (map f as) s = s' \<and> n -as\<rightarrow>* n'"
NSLIM_minus: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and L :: "'d" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L" shows "(\<lambda>x. - f x) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S - L"
mem_FL_Red_F_because_Prec_FL: fixes Cl :: "'a literal multiset \<times> label"   and N :: "('a literal multiset \<times> label) set" assumes "\<forall>D\<in>\<G>_F (fst Cl). \<exists>El\<in>N. El \<sqsubset> Cl \<and> D \<in> \<G>_F (fst El)" shows "Cl \<in> FL.Red_F_\<G> N"
distinct_35: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stop"
ground_mset_as: shows "is_ground_atm_mset (mset As)"
funpow_neq_less_funpow_dist: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and m :: "nat"   and n :: "nat" assumes "y \<in> orbit f x"   and "m \<le> funpow_dist f x y"   and "n \<le> funpow_dist f x y"   and "m \<noteq> n" shows "(f ^^ m) x \<noteq> (f ^^ n) x"
digitToNlist_def: shows "digitToNlist \<equiv> rec_Digit (\<lambda>a. [a]) (\<lambda>a b. [a, b]) (\<lambda>a b c. [a, b, c]) (\<lambda>a b c d. [a, b, c, d])"
pow_irr: fixes x :: "'a mod_ring poly"   and n :: "nat" shows "x [^]\<^bsub>R\<^esub> n = x ^ n mod f"
a_rcos_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "a_set_inv (H +> x) = H +> \<ominus> x"
of_int_sint: fixes a :: "'c word" shows "of_int (sint a) = signed a"
df: shows "dom\<^bsub>CatDom F\<^esub> f = X"
order_lesseq_imp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(\<forall>z. z \<preceq> x \<longrightarrow> z \<preceq> y) = (x \<preceq> y)"
arity_compatible_length: fixes rs :: "(Term_Class.term list \<times> 'a) fset"   and pats :: "Term_Class.term list"   and rhs :: "'a" assumes "Rewriting_Nterm.arity_compatibles rs"   and "(pats, rhs) |\<in>| rs" shows "length pats = arity rs"
checked_s_def: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) set"   and s :: "('i, 'a) state" shows "checked_s t as s = (invariant_s t as s \<and> \<Turnstile> s)"
card_UNIV: fixes finite_UNIV :: "('b, bool) phantom"   and card_UNIV :: "('b, nat) phantom" assumes "class.card_UNIV finite_UNIV card_UNIV" shows "card_UNIV = Phantom('b) CARD('b)"
simps: fixes \<I> :: "('a, 'b) \<I>"   and callee :: "'a \<Rightarrow> ('b \<times> 'c) spmf" shows "\<I> \<turnstile>c callee \<surd> = (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (callee x) \<longrightarrow> xa \<in> responses_\<I> \<I> x)"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and w :: "'a" shows "G.awalk u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> G.awalk (head G e) es w)"
cases: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and a1 :: "('i, 'j, 'k, 'l) converter"   and a2 :: "('i, 'j, 'k, 'l) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C a1 \<sim> a2" obtains conv :: "('i, 'j, 'k, 'l) converter" and conv' :: "('i, 'j, 'k, 'l) converter" where "a1 = conv" and "a2 = conv'" and "\<And>q. q \<in> outs_\<I> \<I> \<Longrightarrow> eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r. r \<in> responses_\<I> \<I> q)) (eq_\<I>_converter \<I> \<I>')) \<I>' (run_converter conv q) (run_converter conv' q)"
is_pappus_def: shows "is_pappus \<equiv> \<forall>A B C D E F P Q R. is_pappus2 A B C D E F P Q R"
Next_def: fixes S :: "'a set" shows "Next S = (if S = {} then arb else next (Max S))"
\<beta>_def: shows "\<beta> = ZFC_in_HOL.succ \<beta>'"
map_inj_on: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list" assumes "map f xs = map f ys"   and "inj_on f (set xs \<union> set ys)" shows "xs = ys"
measure_pmf_cases: fixes y :: "'b measure" assumes "y \<in> {M. prob_space M \<and> sets M = UNIV \<and> (AE x in M. Sigma_Algebra.measure M {x} \<noteq> 0)}" obtains x :: "'b pmf" where "y = measure_pmf x"
resCasesF': fixes x :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "<\<nu>x>P \<longmapsto> \<alpha> \<prec> P'" obtains P :: "pi" and \<alpha>' :: "freeRes" and P'a :: "pi" and y :: "name" where "<\<nu>x>Pa__ = <\<nu>y>P" and "\<alpha> \<prec> P' = \<alpha>' \<prec> <\<nu>y>P'a" and "P \<longmapsto> \<alpha>' \<prec> P'a" and "y \<sharp> \<alpha>'"
perm_below_to_right: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<pi> :: "perm"   and x :: "'c"   and y :: "'c" assumes "class.cont_pt permute below" shows "(\<pi> \<bullet> x \<sqsubseteq> y) = (x \<sqsubseteq> - \<pi> \<bullet> y)"
types_preserved_set_global_1: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and v :: "v"   and j :: "nat"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [$C v, $Set_global j] : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> [] : ts _> ts'"
\<Delta>: shows "disjAll \<Delta>s s vl s1 vl1"
positions_in_row_def: fixes w :: "('c \<times> bool list) list"   and i :: "nat" shows "positions_in_row w i = Option.these (set (map_index (\<lambda>p a_bs. if snd a_bs ! i then Some p else None) w))"
prj_simps_15: shows "C.dom fgh.Prj\<^sub>0\<^sub>0 = HfHgh.chine"
match_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and h :: "'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C (C h (C g f))"   and "partial_magma.arr C (C g f)" shows "partial_magma.arr C (C h g)"
subspace_on_def: fixes T :: "'a set" shows "subspace T = ((0::'a) \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. x + y \<in> T) \<and> (\<forall>c. \<forall>x\<in>T. c *\<^sub>R x \<in> T))"
card_eq_2_iff_doubleton: fixes A :: "'a set" shows "(card A = 2) = (\<exists>x y. x \<noteq> y \<and> A = {x, y})"
tauChainSum1: fixes P :: "ccs"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P \<noteq> P'" shows "P \<oplus> Q \<Longrightarrow>\<^sub>\<tau> P'"
n_def1: shows "real_of_int q * as * 2 * real_of_int (b (n + 1)) / d (n + 1) powr 2 ^ (n + 1) < 1"
tensor_univ_propTr1: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme"   and Z :: "('h, 'a, 'f) Module_scheme"   and f :: "'e \<times> 'g \<Rightarrow> 'h" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "R module Z"   and "bilinear_map f R M N Z" shows "\<exists>!g. g \<in> mHom R (M \<^bsub>P\<^esub>\<Otimes>\<^bsub>R\<^esub> N) Z \<and> compose (M \<times>\<^sub>c N) g (tnm R P M N) = f"
image_mset_union: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e multiset"   and N :: "'e multiset" shows "image_mset f (M + N) = image_mset f M + image_mset f N"
assms: shows "(y, x) \<in> rtrancl_on F r"
not_eq_newpkt_rreq: fixes hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "\<not> eq_newpkt (Rreq hops dip dsn dsk oip osn sip handled)"
run_state_set: fixes s' :: "'b"   and s :: "'b" shows "run_state (State_Monad.set s') s = ((), s')"
Gmod4_def: shows "Gmod4 = G\<lparr>carrier := K \<inter> H\<rparr> Mod (K1 \<inter> H <#> K \<inter> H1)"
AnnWhile: fixes r :: "'b set"   and i :: "'b set"   and b :: "'b set"   and c :: "'b ann_com"   and q :: "'b set" assumes "r \<subseteq> i"   and "i \<inter> b \<subseteq> pre c"   and "\<turnstile> c  i"   and "i \<inter> - b \<subseteq> q" shows "\<turnstile> AnnWhile r b i c  q"
constant_on_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
overrides_t_def: fixes s :: "'c \<Rightarrow> 'd set"   and t :: "'c \<Rightarrow> 'd set" shows "s \<oplus>\<oplus> t = (\<lambda>k. if t k = {} then s k else t k)"
Units_inv_inv: fixes x :: "'a" assumes "x \<in> Units G" shows "inv (inv x) = x"
l_get_element_by_axioms: shows "l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order"
tensor_lookup_eqI: fixes A :: "'m tensor"   and B :: "'m tensor" assumes "dims A = dims B"   and "\<And>is. is \<lhd> dims A \<Longrightarrow> lookup A is = lookup B is" shows "A = B"
eventually_Inf_base: fixes B :: "'b filter set"   and P :: "'b \<Rightarrow> bool" assumes "B \<noteq> {}"   and "\<And>F G. \<lbrakk>F \<in> B; G \<in> B\<rbrakk> \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G" shows "eventually P (Inf B) = (\<exists>b\<in>B. eventually P b)"
fin: shows "finite (support_set pf1 \<union> support_set pf2)"
diff_fun_order_le: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e"   and l :: "enat" assumes "diff_fun k charts f"   and "l \<le> k" shows "diff_fun l charts f"
star_rtc2: fixes x :: "('a \<times> 'a) set" shows "(Id \<union> x O x \<subseteq> x) = (x = x\<^sup>*)"
Tsub: shows "T \<subseteq> topspace (Euclidean_space n)"
eq: shows "monom 1 1 [^]\<^bsub>R\<^esub> CARD('a) ^ m' = monom 1 1"
domA_not_fresh: fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'c) list" assumes "x \<in> domA \<Gamma>" shows "\<not> atom x \<sharp> \<Gamma>"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
approx_mult_SReal2: fixes a :: "real star"   and x :: "real star" assumes "a \<in> \<real>"   and "x \<approx> 0" shows "a * x \<approx> 0"
l_get_attribute_axioms: shows "l_get_attribute ShadowRootClass.type_wf get_attribute get_attribute_locs"
isDiagonal_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "isDiagonal A = (\<forall>a b. mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b \<longrightarrow> A $ a $ b = (0::'b))"
le_sup_equiv: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<squnion> b = b)"
h1: shows "(u, v) \<in> Partitions (x @ z)"
pth_8: fixes c :: "real"   and x :: "'b"   and d :: "real" shows "c *\<^sub>R x + d *\<^sub>R x \<equiv> (c + d) *\<^sub>R x"
x1: shows "item_rule x = item_rule (inc_dot (length (item_\<alpha> x)) (init_item (item_nonterminal x, item_rhs x) 0))"
assms: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
merge_overlap_helper2: fixes ss :: "('b word \<times> 'b word) list"   and A :: "'b word \<times> 'b word" assumes "\<exists>s'\<in>set ss. \<not> disjoint (interval_of A) (interval_of s')" shows "interval_of A \<union> \<Union> (interval_of ` set ss) = \<Union> (interval_of ` set (merge_overlap A ss))"
DERIV_subset: fixes f :: "'c \<Rightarrow> 'c"   and f' :: "'c"   and x :: "'c"   and s :: "'c set"   and t :: "'c set" assumes "(f has_field_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_field_derivative f') (at x within t)"
hv_zk: fixes h :: "'a"   and x1 :: "nat"   and x2 :: "nat"   and c :: "nat" assumes "h = \<^bold>g [^] x1 \<otimes> g' [^] x2" shows "local.\<Sigma>_protocols_base.R h (x1, x2) c = local.\<Sigma>_protocols_base.S h c"
distinct_3: fixes x1 :: "'g" shows "Node x1 \<noteq> NewExit"
Comm1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and b :: "name"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'"   and "Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'" shows "P \<parallel> Q \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'[x::=b] \<parallel> Q'"
closedt_mono_1: fixes i :: "nat"   and j :: "nat"   and t :: "'b FOL_Fitting.term" assumes "i \<le> j"   and "closedt i t" shows "closedt j t"
set_less_eq_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "ord.set_less_eq less_eq A B = (if A \<in> infinite_complement_partition then ord.set_less_eq_aux'' less_eq A B \<or> B \<notin> infinite_complement_partition else B \<notin> infinite_complement_partition \<and> ord.set_less_eq_aux'' less_eq (- B) (- A))"
col_permutation_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "Col C A B"
LeastI: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
wf_act_adds_are_atoms: fixes effs :: "object ast_effect"   and ae :: "object atom formula" assumes "wf_effect_inst effs"   and "ae \<in> set (adds effs)" shows "is_predAtom ae"
tendsto_uniform_limitI: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and x :: "'c" assumes "uniform_limit S f l F"   and "x \<in> S" shows "((\<lambda>y. f y x) \<longlongrightarrow> l x) F"
simps_11: fixes b :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "assignsE (b ? e1 : e2) = assignsE b \<union> assignsE e1 \<inter> assignsE e2"
SUP_insert: fixes f :: "'e \<Rightarrow> 'c \<times> 'd"   and a :: "'e"   and A :: "'e set" shows "Sup_prod (f ` insert a A) = sup_prod (f a) (Sup_prod (f ` A))"
Cn_same: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and B :: "'b set" assumes "Tarskian_logic Cn" shows "(Cn A = Cn B) = (\<forall>C. (A \<subseteq> Cn C) = (B \<subseteq> Cn C))"
ms_xy: shows "mset (zip xs' ys') = mset (zip xs ys)"
Pr_total: fixes f :: "recf"   and g :: "recf"   and n :: "nat" assumes "Partial_Recursive.total f"   and "Partial_Recursive.total g"   and "recfn (Suc n) (Pr n f g)" shows "Partial_Recursive.total (Pr n f g)"
compat_correct: fixes I :: "('a \<times> 'b) list"   and I' :: "('a \<times> 'b) list" assumes "oneone I"   and "oneone I'" shows "PlaneGraphIso.compat I' I = (let m = map_of I; m' = map_of I' in m \<subseteq>\<^sub>m m ++ m' \<and> inj_on (m ++ m') (dom (m ++ m')))"
flowD_outside: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "flow \<Delta> g"   and "e \<notin> \<^bold>E" shows "g e = 0"
mset_un_single_un_cases: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "add_mset a A = B + C" obtains "a \<in># B" and "A = B - {#a#} + C"   | "a \<in># C" and "A = B + (C - {#a#})"
map_le_graph: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "(f \<subseteq>\<^sub>m g) = (map_graph f \<subseteq> map_graph g)"
induct: fixes x1 :: "'b"   and x2 :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "(x1, x2) \<in> r\<^sup>*"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>(a, b) \<in> r\<^sup>*; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
sup_state_opt_any_Some: fixes P :: "'c prog"   and s :: "ty list \<times> ty err list"   and X :: "(ty list \<times> ty err list) option" shows "P \<turnstile> \<lfloor>s\<rfloor> \<le>' X = (\<exists>s'. X = \<lfloor>s'\<rfloor> \<and> P \<turnstile> s \<le>\<^sub>i s')"
le_minus_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<le> - b) = (b \<le> - a)"
Sigma_mset_plus_distrib1: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a \<Rightarrow> 'b multiset" shows "Sigma_mset (A + B) C = Sigma_mset A C + Sigma_mset B C"
the_dflt_Z_eq: fixes m :: "'a option"   and d :: "'a" shows "(the_default (0::'a) m = d) = (m = None \<and> d = (0::'a) \<or> m = Some d)"
subset: fixes R :: "('e, 'f) ring_scheme"   and N :: "'g set"   and M :: "('e, 'g, 'h) module_scheme" assumes "LinearCombinations.submodule R N M" shows "N \<subseteq> carrier M"
trancl_into_trancl: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and c :: "'d" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
\<mu>_stable_def: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<mu>_stable \<phi> w = (\<G>\<F> \<phi> w = \<F> \<phi> w)"
poly_comp_finsum: fixes n :: "nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and p :: "nat \<Rightarrow> 'a" assumes "\<And>i. i \<le> n \<Longrightarrow> g i \<in> carrier P"   and "q \<in> carrier P"   and "p = finsum P g {..n}" shows "Cring_Poly.compose R p q = (\<Oplus>\<^bsub>P\<^esub>i\<in>{..n}. Cring_Poly.compose R (g i) q)"
are_all_operators_applicable_cons_1: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and ops :: "'a strips_operator list" assumes "are_all_operators_applicable s (op # ops)" shows "is_operator_applicable_in s op"
mat_iarray_def: fixes k :: "'b"   and n :: "nat" shows "mat_iarray k n = tabulate2 n n (\<lambda>i j. if i = j then k else (0::'b))"
Transset_VPow: fixes i :: "V" assumes "Transset i" shows "Transset (VPow i)"
split_option_ex: fixes P :: "'a option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (Some x)))"
deriv_correct: fixes i :: "nat"   and f :: "floatarith"   and xs :: "real list"   and t :: "real"   and n :: "nat"   and S :: "real set" assumes "isDERIV i f (xs[i := t])"   and "i < length xs" shows "((\<lambda>x. interpret_floatarith (local.deriv i f n) (xs[i := x])) has_real_derivative interpret_floatarith (local.deriv i f (Suc n)) (xs[i := t])) (at t within S)"
natUnion_def: fixes f :: "nat \<Rightarrow> 'a set" shows "natUnion f = \<Union> {f n |n. True}"
exhaust: fixes y :: "'b \<times> 'c" obtains a :: "'b" and b :: "'c" where "y = (a, b)"
gs': shows "gs' = filter (\<lambda>gi. \<not> p.dvdm gi h) gs"
surjective: fixes r :: "('a, 'b) pointermap_scheme" shows "r = \<lparr>entries = entries r, getentry = getentry r, \<dots> = more r\<rparr>"
length_mvmult_fa: fixes D :: "nat"   and E :: "nat"   and xs :: "floatarith list"   and ys :: "floatarith list" shows "length (mvmult_fa D E xs ys) = D"
contract_success: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "AGM_Contraction Cn contraction"   and "K = Cn A"   and "\<phi> \<notin> Cn {}" shows "\<phi> \<notin> contraction K \<phi>"
orderE: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<preceq> a" obtains "a = linordered_monoid.max a b"
lm1: fixes x :: "nat" shows "pr_gr (Suc x) = pr_gr x \<or> c_tl (pr_gr (Suc x)) = pr_gr x"
N: shows "N \<in> M \<rightarrow>\<^sub>M subprob_algebra B"
Red_F_of_Red_F_subset_F: fixes N' :: "'f set"   and N :: "'f set" assumes "N' \<subseteq> local.Red_F_\<G> N" shows "local.Red_F_\<G> N \<subseteq> local.Red_F_\<G> (N - N')"
u_c: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) u = type0 init x y"
indspaceD_range': fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set"   and x :: "('a, 'b) aezfun" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V" shows "f x \<in> V"
lincomb_sum: fixes rs :: "'f list"   and ss :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ss \<subseteq> UNIV"   and "set ms \<subseteq> V"   and "length rs \<le> length ss" shows "rs \<bullet>\<cdot> ms + ss \<bullet>\<cdot> ms = (map2 (+) rs ss @ drop (length rs) ss) \<bullet>\<cdot> ms"
sum_divisors_moebius_mu_times_multiplicative: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "multiplicative_function f"   and "0 < n" shows "(\<Sum>d | d dvd n. moebius_mu d * f d) = (\<Prod>p\<in>prime_factors n. (1::'a) - f p)"
Extend_def: fixes S :: "'a fm set"   and C :: "'a fm set set"   and f :: "nat \<Rightarrow> 'a fm" shows "Extend S C f \<equiv> \<Union> (range (extend S C f))"
317: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (- x \<sqinter> y) \<sqinter> (x \<sqinter> z) = x \<sqinter> z"
finite_sysI: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" assumes "incidence_system point_set block_collection"   and "finite point_set" shows "finite_incidence_system point_set block_collection"
same_prefix_prefix: fixes xs :: "[:'r:]"   and ys :: "[:'r:]"   and zs :: "[:'r:]" assumes "xs \<noteq> \<bottom>" shows "prefix\<cdot>(xs :@ ys)\<cdot>(xs :@ zs) = prefix\<cdot>ys\<cdot>zs"
assms_3: fixes i :: "'a"   and e :: "real" assumes "i \<in> Basis"   and "0 < e" shows "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> S"
wellformed_items_\<J>: fixes k :: "nat"   and u :: "nat" shows "wellformed_items (\<J> k u)"
retraction_imp_quotient_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "retraction_map X Y f" shows "quotient_map X Y f"
simulation_def: fixes as\<^sub>0 :: "'e"   and alstep :: "'f \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and cs\<^sub>0 :: "'g"   and clstep :: "'f \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R \<equiv> df_system as\<^sub>0 alstep \<and> df_system cs\<^sub>0 clstep \<and> simulation_axioms as\<^sub>0 alstep cs\<^sub>0 clstep R"
surj_f_inv_f: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "surj f" shows "f (inv f y) = y"
redT_updI_insert_Interrupt: fixes t :: "'b"   and "is" :: "'b set"   and ia :: "'b interrupt_action" assumes "t \<in> redT_updI is ia"   and "t \<notin> is" shows "ia = Interrupt t"
comp_arr_ide: fixes a :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" assumes "S.ide a"   and "S.seq f a" shows "f \<cdot>\<^sub>S a = f"
coeff_monom: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier R" shows "coeff P (monom P a m) n = (if m = n then a else \<zero>)"
hs_is_empty_impl: shows "imp_set_is_empty is_hashset hs_isEmpty"