hcju/isabelleps huggingface dataset:
generalise_output_direct_subsumption_def: fixes t' :: "transition"   and t :: "transition"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and e' :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and s' :: "nat" shows "generalise_output_direct_subsumption t' t e e' s s' = (case stored_reused t' t of None \<Rightarrow> False | Some (r, p) \<Rightarrow> case Outputs t ! p of L v \<Rightarrow> generalise_output_context_check v r s s' e e' | _ \<Rightarrow> False)"
Ident: fixes A :: "fmla" shows "A IMP A \<in> boolean_axioms"
assms_1: shows "((\<lambda>x. f x / g x) \<longlongrightarrow> c) F"
that: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r x y"   and "r x z" shows "\<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>=\<^sup>= z u"
setops\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('c, 'd) stateful_strand_step list"   and N :: "('c, 'd) stateful_strand_step list" assumes "set M \<subseteq> set N" shows "setops\<^sub>s\<^sub>s\<^sub>t M \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t N"
sccs_def: shows "sccs \<equiv> {H. induced_subgraph H G \<and> strongly_connected H \<and> (\<nexists>H'. induced_subgraph H' G \<and> strongly_connected H' \<and> verts H \<subset> verts H')}"
transpose_aux_filter_tail: fixes xss :: "'g list list" shows "concat (map (case_list [] (\<lambda>h t. [t])) xss) = map tl (filter (\<lambda>ys. ys \<noteq> []) xss)"
assms_2: fixes n :: "nat" assumes "k < n" shows "p n = \<zero>"
eq_m_smult_p_removal: fixes k :: "nat"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.Mp (p * p ^ k) (smult p f) = poly_mod.Mp (p * p ^ k) (smult p g)" shows "poly_mod.Mp (p ^ k) f = poly_mod.Mp (p ^ k) g"
simps_27: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat" shows "(case Rrep x21 x22 x23 x24 x25 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25"
sort_oalist_id: fixes ko :: "'a nat_term_order"   and xs :: "('a \<times> 'b) list" assumes "ko_ntm.oalist_inv_raw ko xs" shows "sort_oalist_ko_ntm' ko xs = xs"
simps: fixes a :: "'a"   and s :: "'b + 'a" shows "(a \<in> Basic_BNFs.setr s) = (\<exists>x. a = x \<and> s = Inr x)"
asymp_equiv_sym: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "(f \<sim>[F] g) = (g \<sim>[F] f)"
entt_disjE: fixes A :: "assn"   and M :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>t M"   and "B \<Longrightarrow>\<^sub>t M" shows "A \<or>\<^sub>A B \<Longrightarrow>\<^sub>t M"
marked_deletions_dequeue_Mutate: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and m' :: "'h"   and r :: "'g"   and f :: "'e"   and opt_r' :: "'g option"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers (mutator m') s = mw_Mutate r f opt_r' # ws"   and "mut_m.marked_deletions m s"   and "mut_m.marked_insertions m' s" shows "mut_m.marked_deletions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := ws)\<rparr>))"
respectfulness: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'a set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "af_letter \<phi> \<nu> \<equiv>\<^sub>P af_letter \<psi> \<nu>"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> es' : t2s _> t3s"
sep_dom2_aux: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and z :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp (list2FWpolicy (a \<oplus> y # z)))" shows "x \<in> dom (Cp (a \<oplus> list2FWpolicy (y # z)))"
power_less_one: fixes n :: "nat"   and x :: "'a" assumes "0 < n"   and "(0::'a) \<le> x"   and "x < (1::'a)" shows "x ^ n < (1::'a)"
degree_pCons_eq: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "degree (pCons a p) = Suc (degree p)"
inner_prod_with_itself_Re: fixes u :: "complex vec" shows "0 \<le> Re \<langle>u|u\<rangle>"
RTranCl_elim: fixes h :: "'c"   and h' :: "'c"   and succs :: "'c \<Rightarrow> 'c list" assumes "h [succs]\<rightarrow>* h'" obtains "h' = h"   | g' :: "'c" where "g' \<in> set (succs h)" and "g' [succs]\<rightarrow>* h'"
prod_dvd_prod: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.comm_monoid_mult times one"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times (f a) (g a)" shows "dvd.dvd times (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
fixp_preserves_mcont1: fixes F :: "('c \<Rightarrow> 'd llist) \<Rightarrow> 'c \<Rightarrow> 'd llist"   and f :: "'c \<Rightarrow> 'd llist"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. mono_llist (\<lambda>f. F f x)"   and "f \<equiv> llist.fixp_fun F"   and "\<And>f. mcont lubb ordb lSup (\<sqsubseteq>) f \<Longrightarrow> mcont lubb ordb lSup (\<sqsubseteq>) (F f)" shows "mcont lubb ordb lSup (\<sqsubseteq>) f"
strict_sorted_interact_hd: fixes cs :: "'a list list"   and ds :: "'a list list" assumes "strict_sorted (interact cs ds)"   and "cs \<noteq> []"   and "ds \<noteq> []"   and "hd cs \<noteq> []"   and "hd ds \<noteq> []" shows "hd (hd cs) < hd (hd ds)"
sinks_interference_eq: fixes u :: "'e"   and D :: "'f \<Rightarrow> 'e"   and x :: "'f"   and I :: "('e \<times> 'e) set"   and xs :: "'f list" shows "((u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I)) = (D x \<in> sinks I D u (xs @ [x]))"
independent_aform_def: fixes X :: "'c \<times> 'c pdevs"   and Y :: "'b \<times> 'b pdevs" shows "independent_aform X Y = independent_from (degree_aform X) Y"
det_transpose: fixes A :: "(('c, 'd) vec, 'd) vec" shows "Determinants.det (Finite_Cartesian_Product.transpose A) = Determinants.det A"
simps_2: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e"   and xs :: "'e list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
image_subset_iff: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and B :: "'e set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
map_zip_map: fixes f :: "'c \<times> 'd \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and xs :: "'e list"   and ys :: "'d list" shows "map f (zip (map g xs) ys) = map2 (\<lambda>x y. f (g x, y)) xs ys"
CASRed3: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and v' :: "'addr val" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "extTA,P,t \<turnstile> \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e),s\<rangle> -ta\<rightarrow> \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e'),s'\<rangle>"
hseqE: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.hseq \<nu> \<mu>" obtains "D.arr \<mu>" and "D.arr \<nu>" and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>"
add_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b"   and "less_eq c d" shows "less_eq (plus a c) (plus b d)"
associative_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "associative f \<equiv> \<forall>x y z. f x (f y z) = f (f x y) z"
eq_elem_in: fixes a :: "'c"   and A :: "'c set"   and b :: "'c" assumes "a \<in> A"   and "a = b" shows "b \<in> A"
asymp_equivD: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "((\<lambda>x. if f x = (0::'c) \<and> g x = (0::'c) then 1::'c else f x / g x) \<longlongrightarrow> (1::'c)) F"
diff: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "additive f" shows "f (x - y) = f x - f y"
noDA1sep: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll1 p" shows "noDenyAll1 (separate p)"
map_option_o_empty: fixes f :: "'f \<Rightarrow> 'e" shows "map_option f \<circ> \<emptyset> = \<emptyset>"
pos_line_def: shows "pos_line = linepath (complex_of_real r) (complex_of_real R)"
derangements_skip_one: fixes d :: "nat \<Rightarrow> nat"   and S :: "nat set"   and x :: "nat" assumes "d \<in> derangements S"   and "x \<in> S"   and "d (d x) \<noteq> x" shows "d(x := x, inv d x := d x) \<in> derangements (S - {x})"
borel_closed: fixes A :: "'a set" assumes "closed A" shows "A \<in> sets borel"
select_convs_1: fixes prism_match :: "'a \<Rightarrow> 'b option"   and prism_build :: "'b \<Rightarrow> 'a"   and more :: "'c" shows "match\<^bsub>\<lparr>prism_match = prism_match, prism_build = prism_build, \<dots> = more\<rparr>\<^esub> = prism_match"
rsp: shows "eq_onp (\<lambda>xss. [] \<in> xss \<and> downset xss) {[]} {[]}"
ivl_disj_un_7: fixes l :: "'b"   and u :: "'b" assumes "l < u" shows "{..l} \<union> {l<..<u} = {..<u}"
bnd: shows "AE x in M. \<forall>j. norm (u' x - u j x) \<le> 2 * w x"
prob_space_distr: fixes f :: "'s stream \<Rightarrow> 'a"   and s :: "'s"   and M' :: "'a measure" assumes "T.random_variable s M' f" shows "prob_space (distr (T s) M' f)"
less_tree_def: fixes s :: "'a tree"   and t :: "'a tree" shows "(s < t) = strict (\<le>) s t"
True: shows "[M]\<^bsub>v,n\<^esub> = {}"
disc_6: fixes x3 :: "'b" shows "is_wrong (Goes_wrong x3)"
get_ancestors_wf_is_l_get_ancestors_wf: shows "l_get_ancestors_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.known_ptr ShadowRootClass.known_ptrs ShadowRootClass.type_wf Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_parent"
pmat_extension_def: fixes m :: "'a mat" shows "st4.pmat_extension m = st4.ptensor_mat m (1\<^sub>m st4.d2)"
PO_l3_inv3: shows "reach l3 \<subseteq> l3_inv3"
simps_5: fixes t :: "Core.term"   and u :: "Core.term" shows "tvs (t $ u) = tvs t \<union> tvs u"
exec_blockFault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and f :: "'f"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> \<Rightarrow> Fault f"
kmp_SPEC_def: fixes s :: "'a list"   and t :: "'a list" shows "kmp_SPEC s t = SPEC (\<lambda>x. case x of None \<Rightarrow> \<nexists>i. sublist_at s t i | Some i \<Rightarrow> sublist_at s t i \<and> (\<forall>ii<i. \<not> sublist_at s t ii))"
generate_sincl: fixes A :: "'a set" shows "A \<subseteq> generate G A"
vec_list: fixes v :: "'a Matrix.vec" shows "vec_of_list (list_of_vec v) = v"
Lq_num: shows "let LLq = \<lambda>t1 t2. psubst Lq [(t1, zz), (t2, yy)] in \<forall>\<phi>\<in>fmla. \<forall>q\<in>num. Fvars \<phi> = {zz} \<and> (\<forall>p\<in>num. prv (subst \<phi> p zz)) \<longrightarrow> prv (all zz (imp (LLq (Var zz) q) \<phi>))"
arr_char: fixes f :: "J.Arr" shows "J.arr f = (f \<noteq> J.Null)"
arity_simps_2: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" shows "arity.arity arity_sym arity_var (App s t) = arity.arity arity_sym arity_var s - 1"
S_step_5: shows "snd (cs (S t i) cid) = Done"
fresh_star_def: fixes as :: "atom set"   and x :: "'b" shows "as \<sharp>* x \<equiv> \<forall>a\<in>as. a \<sharp> x"
measure_pmf_eq_density: fixes p :: "'d pmf" shows "measure_pmf p = density (count_space UNIV) (\<lambda>x. ennreal (pmf p x))"
distinct_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "distinct (linorder.sort_key less_eq f xs) = distinct xs"
intv_of_def: fixes k :: "nat"   and c :: "real" shows "intv_of k c \<equiv> if real k < c then Greater k else if \<exists>x. real x = c then Const (nat \<lfloor>c\<rfloor>) else Intv (nat \<lfloor>c\<rfloor>)"
sup_bot_right: fixes x :: "'a multiset" shows "x \<union># \<Empt> = x"
nontrans1: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and u :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "meets u r" shows "\<nexists>t. meets u t \<and> meets t r"
mk_poly_mk_poly2: fixes A :: "'b mat"   and i :: "nat" assumes "0 < dim_col A"   and "i < dim_row A" shows "mk_poly A $$ (i, dim_col A - 1) = mk_poly2 A $ i"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "pp.mapping_of (x + xa) = pp.mapping_of x + pp.mapping_of xa"
try_gpv_def: shows "try_gpv \<equiv> \<lambda>uu uua. corec_gpv (\<lambda>(gpv, gpv'). map_spmf (map_generat id id (\<lambda>uu input. case uu input of Inl gpv \<Rightarrow> Inr (gpv, gpv') | Inr gpv' \<Rightarrow> Inl gpv')) (TRY map_spmf (map_generat id id (id ---> Inl)) (the_gpv gpv) ELSE map_spmf (map_generat id id (id ---> Inr)) (the_gpv gpv'))) (uu, uua)"
order_refl: fixes x :: "'x \<Rightarrow>\<^sub>0 nat" shows "x \<preceq> x"
invar_abs': fixes WAC :: "'a while_algo"   and WAA :: "'b while_algo"   and \<alpha> :: "'a \<Rightarrow> 'b"   and s :: "'a" assumes "wa_refine WAC WAA \<alpha>"   and "s \<in> wa_invar WAC" shows "\<alpha> s \<in> wa_invar WAA"
cot_minus: fixes x :: "'a" shows "cot (- x) = - cot x"
Cont: fixes out :: "'e"   and rpv :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<I> :: "('e, 'f) \<I>"   and x :: "'e" assumes "IO out rpv \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "x \<in> outs_gpv \<I> (rpv input)" shows "x \<in> outs_gpv \<I> gpv"
ordermap_mono_less: fixes w :: "'a"   and x :: "'a"   and r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "(w, x) \<in> r"   and "wf r"   and "trans r"   and "w \<in> A"   and "x \<in> A"   and "small A" shows "ordermap A r w < ordermap A r x"
inv_character_def: fixes \<chi> :: "'a \<Rightarrow> complex" shows "inv_character \<chi> = (\<lambda>a. cnj (\<chi> a))"
Pj_gsurjec: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set" assumes "Group G"   and "G \<triangleright> N" shows "gsurj\<^bsub>G,(G / N)\<^esub> Pj G N"
Sup_image_sup: fixes Y :: "'a set"   and x :: "'a" assumes "Y \<noteq> {}" shows "\<Squnion> ((\<squnion>) x ` Y) = x \<squnion> \<Squnion> Y"
cases: fixes x :: "(nat \<Rightarrow> 'n) \<times> nat" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" where "x = (\<pi>, n)"
map_upds_Cons: fixes m :: "'c \<Rightarrow> 'd option"   and a :: "'c"   and as :: "'c list"   and b :: "'d"   and bs :: "'d list" shows "m(a # as [\<mapsto>] b # bs) = m(a \<mapsto> b, as [\<mapsto>] bs)"
soconf_fupd: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and h :: "heap"   and v :: "val"   and sfs :: "sfields" assumes "P \<turnstile> C has F,Static:T in C"   and "P,h \<turnstile> v :\<le> T"   and "P,h,C \<turnstile>\<^sub>s sfs \<surd>" shows "P,h,C \<turnstile>\<^sub>s sfs(F \<mapsto> v) \<surd>"
intros_41: fixes env :: "v sem_env"   and s :: "'b state"   and p :: "pat"   and v1 :: "v"   and ck :: "bool"   and e :: "exp0"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" assumes "pmatch (c env) (refs s) p v1 [] = Match_type_error" shows "evaluate_match ck env s v1 ((p, e) # pes) err_v (s, Rerr (Rabort Rtype_error))"
cbox: fixes K :: "'a set" assumes "K \<in> \<D>" shows "interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
subreachable_def: fixes A :: "(('e \<Rightarrow> 'f) \<times> 'g, 'h) automaton"   and U :: "('e \<Rightarrow> 'f) \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and J :: "'e set" shows "subreachable A U J \<equiv> \<forall>I. \<forall>s\<in>oreachable A (\<lambda>s s'. I) U. \<exists>\<sigma>. (\<forall>j\<in>J. \<sigma> j = fst s j) \<and> (\<sigma>, snd s) \<in> reachable A I"
smooth_on_fst: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "k-smooth_on S f"   and "open S" shows "k-smooth_on S (\<lambda>x. fst (f x))"
top_eq: shows "\<top> = \<iota> \<squnion> - \<iota>"
minus_equation_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (uminus b = a)"
linear_comb0_2Tr: fixes A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat \<Rightarrow> 'a" assumes "ideal R A" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> m \<in> {j. j \<le> n} \<rightarrow> {\<zero>} \<longrightarrow> l_comb R M n s m = \<zero>"
all_conj_distrib: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
r_universal_def: fixes n :: "nat" shows "r_universal n \<equiv> Cn (Suc n) r_univ [recf.Id (Suc n) 0, r_shift (r_list_encode (n - 1))]"
neq_0_no_wrap: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> x + y"   and "x \<noteq> 0" shows "x + y \<noteq> 0"
check_simplex'_def: shows "check_simplex' = check_cs check_nsc_code"
simps: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat" shows "configuration q n = {x. x \<le> n \<and> token_run x n = q}"
invalidate_rtsf_left: fixes dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt' :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dests dip = None" shows "(invalidate rt dests \<sqsubset>\<^bsub>dip\<^esub> rt') = (rt \<sqsubset>\<^bsub>dip\<^esub> rt')"
can_lunit: fixes a :: "'c local.term set" assumes "ide a" shows "can \<l>[a]"
cobounded2: fixes a :: "'g"   and b :: "'g" shows "min a b \<le> b"
enat_Suc_ltl: fixes n :: "nat"   and xs :: "'c llist" assumes "enat (Suc n) < llength xs" shows "enat n < llength (ltl xs)"
int_floor_fl: fixes x :: "float" shows "real_of_int (int_floor_fl x) \<le> real_of_float x"
tllist_ord_TCons_TNil: fixes x :: "'c"   and xs :: "('c, 'b) tllist"   and b2 :: "'b" shows "local.tllist_ord (TCons x xs) (TNil b2) = False"
rank_eq_dim_col_space': fixes A :: "(('b, 'd) vec, 'c) vec" shows "vec.dim (col_space A) = rank A"
indspace_el_eq': fixes hs :: "'g list"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and f' :: "('f, 'g) aezfun \<Rightarrow> 'v" assumes "set hs \<subseteq> H"   and "H = (\<Union>h\<in>set hs. G + {h})"   and "f \<in> indV"   and "f' \<in> indV"   and "\<forall>i<length hs. f ((1::'f) \<delta>\<delta> hs ! i) = f' ((1::'f) \<delta>\<delta> hs ! i)" shows "f = f'"
fmap_compl: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula" shows "fmap f (- x) = - fmap f x"
h_2: shows "primerec g (Suc (Suc n))"
prod_cancel_nat: fixes B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> nat" assumes "B \<subseteq> A"   and "finite A"   and "\<forall>x\<in>B. f x \<noteq> 0" shows "real (prod f A) / real (prod f B) = real (prod f (A - B))"
x0_in_I: shows "x0 \<in> {- R<..<R}"
v\<sigma>: shows "vars_term_ms (SCF (t \<cdot> \<sigma>)) \<subseteq># vars_term_ms (SCF (Fun f ss \<cdot> \<sigma>))"
action_orderI: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a' :: "nat" assumes "a \<in> actions E"   and "a' \<in> actions E"   and "\<lbrakk>is_new_action (action_obs E a); is_new_action (action_obs E a')\<rbrakk> \<Longrightarrow> a \<le> a'"   and "\<not> is_new_action (action_obs E a) \<Longrightarrow> \<not> is_new_action (action_obs E a') \<and> a \<le> a'" shows "E \<turnstile> a \<le>a a'"
all_roots_real_def: fixes p :: "real poly" shows "all_roots_real p = (\<forall>r\<in>proots (map_poly complex_of_real p). Im r = 0)"
homeomorphic_map_homology_iso: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and p :: "int" assumes "homeomorphic_map X Y f" shows "hom_induced p X {} Y {} f \<in> Group.iso (homology_group p X) (homology_group p Y)"
sd_imp_dictator: fixes A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and u :: "'a"   and v :: "'a"   and j :: "'b" assumes "has 3 A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "hasw [u, v] A"   and "semidecisive swf A Is {j} u v" shows "dictator swf A Is j"
sw_simps3: fixes zs :: "'a"   and xs :: "'a"   and x :: "'b"   and z1 :: "'b"   and z2 :: "'b" assumes "zs \<noteq> xs \<or> x \<notin> {z1, z2}" shows "x @xs[z1 \<and> z2]_zs = x"
bottomD_1: fixes P :: "'a set" assumes "domain.has_bottom P" shows "domain.bottom P \<in> P"
assms_2: shows "orient_lit x1 s1 t1 p1"
assms: shows "\<forall>\<^sub>F x in at_top. f x = 0"
subspace_extensional0: fixes X :: "'c set" shows "subspace (Collect (extensional0 X))"
statImpTauChainDerivative: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" shows "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P') \<Psi>"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "partial_magma.arr C (Leg0 S) \<and> partial_magma.arr C (Leg1 S) \<and> partial_magma.dom C (Leg0 S) = partial_magma.dom C (Leg1 S)" shows "span_in_category_axioms C S"
lin_split_rule: fixes c :: "nat"   and xs :: "('a \<times> 'b) list"   and a :: "('a \<times> 'b) array"   and n :: "nat"   and p :: "'b" shows "<is_pfa c xs (a, n)> lin_split (a, n) p <\<lambda>r. is_pfa c xs (a, n) * \<up> (r \<le> n \<and> (\<forall>j<r. snd (xs ! j) < p) \<and> (r < n \<longrightarrow> p \<le> snd (xs ! r)))>\<^sub>t"
simps_2: fixes x :: "pnat" shows "pnat_min x pnat.PInfty = x"
exponent_uminus: fixes f :: "('a, 'b) IEEE.float" shows "IEEE.exponent (- f) = IEEE.exponent f"
that: fixes v :: "'v" assumes "\<pi> u = Some v"   and "light_edge (S (A Q \<pi>)) v u" shows "thesis"
DERIV_exp: fixes x :: "'b" shows "(exp has_field_derivative exp x) (at x)"
hyps: obtains x :: "'a" where "x \<in> C" and "enat n \<le> esize x"
dualization_rew_1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<not> \<box>F \<equiv> \<diamond>\<not> F"
exE_realizer': fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<times> 'a" assumes "P (snd p) (fst p)" obtains x :: "'b" and y :: "'a" where "P y x"
sign_simps_32: fixes am :: "'b"   and bm :: "'b"   and ck :: "'b" shows "am * bm * ck = am * (bm * ck)"