hcju/isabelleps huggingface dataset:
intros_3: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b"   and e :: "String.literal"   and ts :: "Formula.trm list"   and n :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set" assumes "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"   and "\<forall>x\<in>fv (formula.Pred e ts). x < n"   and "\<forall>t\<in>set ts. trm.is_Var t \<or> trm.is_Const t" shows "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R (MPred e ts) (formula.Pred e ts)"
adds_pp_canc: fixes t :: "'a"   and s :: "'a"   and v :: "'t" shows "(t + s adds\<^sub>p t \<oplus> v) = (s adds\<^sub>p v)"
eulerian_poly': fixes k :: "nat" shows "Abs_fps (\<lambda>n. of_nat (n + 1) ^ k) = fps_of_poly (eulerian_poly k) / (1 - fps_X) ^ (k + 1)"
prems_3: shows "ELam x e = ELam x (subst y v e')"
directly_attracted_V_empty: fixes p :: "Player" shows "directly_attracted p V = {}"
zero_int_def: shows "0 \<equiv> Abs_Integ (0, 0)"
partition_on_mset_subsets: fixes A :: "'b multiset"   and P :: "'b multiset multiset"   and x :: "'b multiset" assumes "partition_on_mset A P"   and "x \<in># P" shows "x \<subseteq># A"
arrI: fixes f1 :: "'b"   and f2 :: "'b" assumes "B.arr f1"   and "B.arr f2" shows "B.VxV.arr (f1, f2)"
ins_height_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Bal t'" shows "height t' \<le> height t + 1"
of_int_minus: fixes R :: "('b, 'c) ring_scheme"   and i :: "int" assumes "ring R" shows "\<guillemotleft>- i\<guillemotright>\<^bsub>R\<^esub> = \<ominus>\<^bsub>R\<^esub> \<guillemotleft>i\<guillemotright>\<^bsub>R\<^esub>"
w_def: fixes i :: "int"   and j :: "int" shows "w i j = sum a {i..j + 1} + sum b {i..j}"
encF: fixes f :: "'trm \<Rightarrow> 'trm" assumes "f \<in> F" shows "encF f \<in> Ops"
pelims: fixes x :: "complex"   and xa :: "complex \<times> complex"   and y :: "complex \<times> complex" assumes "x *\<^sub>s\<^sub>v xa = y"   and "mult_sv_dom (x, xa)" obtains k :: "complex" and x :: "complex" and y :: "complex" where "xa__ = k" and "xa = (x, y)" and "ya__ = (k * x, k * y)" and "mult_sv_dom (k, x, y)"
v11_injective: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>" shows "a = b"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b" assumes "category A"   and "category B"   and "functor A B G"   and "invertible_functor_axioms A B G" shows "invertible_functor A B G"
isomorphic_as_monoids_def: fixes \<M> :: "'c set \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<times> 'c"   and \<M>' :: "'d set \<times> ('d \<Rightarrow> 'd \<Rightarrow> 'd) \<times> 'd" shows "(\<M> \<cong>\<^sub>M \<M>') = (let (M, composition, unit) = \<M>; (M', composition', unit') = \<M>' in \<exists>\<eta>. monoid_isomorphism \<eta> M composition unit M' composition' unit')"
eq: fixes x :: "'b"   and y :: "'b" shows "(c1 x y = Eq) = (x = y)"
simps_10: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "w \<Turnstile> \<phi> U \<psi> = (\<exists>k. suffix k w \<Turnstile> \<psi> \<and> (\<forall>j<k. suffix j w \<Turnstile> \<phi>))"
simps_2: fixes t1 :: "'a bro"   and a1 :: "'a"   and t2 :: "'a bro"   and a2 :: "'a"   and t3 :: "'a bro" shows "n1 (N3 t1 a1 t2 a2 t3) = N2 (N2 t1 a1 t2) a2 (N1 t3)"
exhaust: fixes y :: "Agent" obtains "y = Child0"   | "y = Child1"   | "y = Child2"
minus_eq_not_minus_1: fixes a :: "'a" shows "- a = NOT (a - (1::'a))"
prod_zero_i: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and i :: "'c" assumes "\<forall>k\<in>I. aGroup (A k)"   and "i \<in> I" shows "prod_zero I A i = \<zero>\<^bsub>A i\<^esub>"
select_edge_impl_def_opt: fixes s :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list" shows "select_edge_impl s \<equiv> case s of (S, B, I, P) \<Rightarrow> if P = [] then RETURN (None, S, B, I, P) else let (j, succs) = last P in ASSERT (length B - 1 < length B) \<bind> (\<lambda>_. if B ! (length B - 1) \<le> j then ASSERT (succs \<noteq> {}) \<bind> (\<lambda>_. SPEC (\<lambda>x. x \<in> succs) \<bind> (\<lambda>v. let succs = succs - {v} in ASSERT (P \<noteq> [] \<and> length P - 1 < length P) \<bind> (\<lambda>_. let P = if succs = {} then butlast P else P[length P - 1 := (j, succs)] in RETURN (Some v, S, B, I, P)))) else RETURN (None, S, B, I, P))"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'c \<Rightarrow> 'b"   and U :: "'b set" shows "sinks_aux uu uv U [] = U"
eq_on_trans: fixes r :: "'b \<Rightarrow> 'c"   and s :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and t :: "'b \<Rightarrow> 'c" assumes "r = s on X"   and "s = t on X" shows "r = t on X"
P1: shows "Bet A PO b' \<and> Cong PO b' b Po"
iMOD_inext_nth_diff: fixes r :: "nat"   and m :: "nat"   and a :: "nat"   and b :: "nat" shows "([ r, mod m ] \<rightarrow> a) - ([ r, mod m ] \<rightarrow> b) = (a - b) * m"
fst_nodes: fixes t :: "int tree" shows "fst ` nodes t = set_tree t"
intro: fixes t0 :: "real"   and t :: "nat \<Rightarrow> real" assumes "t0 = t 0" shows "convergent_one_step_axioms t0 t"
prf_domain_finite: fixes prf_key_gen :: "'a spmf"   and prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "bool list set"   and prf_dlen :: "nat"   and prf_clen :: "nat"   and upf_key_gen :: "'b spmf" assumes "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen" shows "finite prf_domain"
cut_less_le_conv_if: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t = (if t \<in> I then I \<down>\<le> t - {t} else I \<down>\<le> t)"
inducts: fixes \<Gamma> :: "('k, 'l) web_scheme"   and \<epsilon> :: "ennreal" assumes "hindered_by \<Gamma> \<epsilon>" obtains f :: "'k \<times> 'k \<Rightarrow> ennreal" where "hindrance_by \<Gamma> f \<epsilon>" and "current \<Gamma> f" and "wave \<Gamma> f"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom (\<mu> \<star> \<nu> \<star> \<pi>))"
equals_suc_AboveS: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> local.AboveS B"   and "\<And>a'. a' \<in> local.AboveS B \<Longrightarrow> (a, a') \<in> r" shows "a = suc B"
less_bool_def: fixes P :: "bool"   and Q :: "bool" shows "(P < Q) = (\<not> P \<and> Q)"
unfold_congs_10: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "Literal \<Rightarrow> nat list"   and f :: "(Literal \<Rightarrow> nat list) \<Rightarrow> Literal \<Rightarrow> nat list"   and f' :: "(Literal \<Rightarrow> nat list) \<Rightarrow> Literal \<Rightarrow> nat list" assumes "r = r'"   and "getWatchList r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "getWatchList_update f r = getWatchList_update f' r'"
trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> b" shows "c \<sqsubseteq> a"
eval_poly_def: fixes h :: "'c \<Rightarrow> 'b"   and p :: "'c poly" shows "eval_poly h p = poly (map_poly h p)"
assms: shows "conv_abscissa f < ereal (s \<bullet> (1::'a))"
simps_2: fixes \<delta> :: "'c"   and uv :: "'c" shows "gt_arctic_delta \<delta> MinInfty_delta (Num_arc_delta uv) = False"
i_shrink_eq_Msg_iUntil_start_event_conv: fixes k :: "nat"   and m :: "'a message_af"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af" assumes "0 < k"   and "m \<noteq> NoMsg"   and "\<And>t. event t = (t mod k = 0)"   and "t0 = t * k" shows "((s \<div> k) t = m) = (s t0 = m \<and> (\<circle> t' t0 [0\<dots>]. s t1 = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t'. event t2) \<or> (\<circle> t' t0 [0\<dots>]. \<not> event t1. t1 \<U> t2 [0\<dots>] \<oplus> t'. s t2 = m \<and> \<not> event t2 \<and> (\<circle> t'' t2 [0\<dots>]. s t3 = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t''. event t4)))"
u_sums_U: fixes x :: "'s"   and y :: "'s" shows "u x y sums U x y"
assms_2: shows "(I, J) = getIJ Q Qn V"
set_iterator_rev_linord_foldli_conv: fixes iti :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'a set" shows "set_iterator_rev_linord iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted (rev l0) \<and> iti = foldli l0)"
l_set_attribute_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_attribute_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_attribute get_attribute_locs set_attribute set_attribute_locs \<equiv> l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_attribute get_attribute_locs \<and> l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs"
mod: shows "\<forall>e f g. list_all (eval e f g) ps \<longrightarrow> eval e f g p"
hyps: shows "g\<cdot>(iterate i\<cdot>f\<cdot>\<bottom>) = iterate i\<cdot>h\<cdot>(g\<cdot>\<bottom>)"
bij_betw_compose: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "bij_betw f A B"   and "bij_betw g B C" shows "bij_betw (compose A g f) A C"
bijective: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and N :: "'c set" assumes "RModuleIso R smult M smult' T N" shows "bij_betw T M N"
ct_list_eq_cons_ct_list: fixes as :: "t list"   and xs :: "ct list"   and ys :: "ct list" assumes "ct_list_eq (to_ct_list as) (xs @ ys)" shows "\<exists>bs bs'. as = bs @ bs' \<and> ct_list_eq (to_ct_list bs) xs \<and> ct_list_eq (to_ct_list bs') ys"
image_uminus_atLeastAtMost: fixes x :: "'a"   and y :: "'a" shows "uminus ` {x..y} = {- y..- x}"
as_needed_asses_submap_exec_vi: fixes v :: "'a"   and a :: "('a, 'b) fmap \<times> 'c"   and s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap" assumes "v \<in> fmdom' (action_needed_asses a s1)"   and "action_needed_asses a s1 \<subseteq>\<^sub>f action_needed_asses a s2" shows "fmlookup (action_needed_asses a s1) v = fmlookup (fst a) v \<and> fmlookup (action_needed_asses a s2) v = fmlookup (fst a) v \<and> fmlookup s1 v = fmlookup (fst a) v \<and> fmlookup s2 v = fmlookup (fst a) v"
linear_frechet_derivative: fixes f :: "'b \<Rightarrow> 'c"   and net :: "'b filter" assumes "f differentiable net" shows "linear (frechet_derivative f net)"
D_lt: fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>x<t. 0 < dist x t \<and> dist x t < d \<longrightarrow> dist ((f x - f t) / (x - t)) D_lt < e"
integrable_normal_moment: fixes \<mu> :: "real"   and k :: "nat" shows "integrable lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ k)"
b: shows "(\<lambda>x. ennreal (1 - 1 / real (x + 2))) \<longlonglongrightarrow> ennreal (1 - 0)"
list_nonempty_induct: fixes xs :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
Kaxiom_def: shows "Kaxiom \<equiv> ([[$\<alpha> vid1]]Pc pid1 \<rightarrow> Pc pid2) \<rightarrow> ([[$\<alpha> vid1]]Pc pid1) \<rightarrow> [[$\<alpha> vid1]]Pc pid2"
LCD_def: fixes B :: "'h set"   and D :: "'i set"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'i" shows "LCD B D f \<equiv> (\<forall>x y z. x \<in> B \<longrightarrow> y \<in> B \<longrightarrow> z \<in> D \<longrightarrow> f x (f y z) = f y (f x z)) \<and> (\<forall>x y. x \<in> B \<longrightarrow> y \<in> D \<longrightarrow> f x y \<in> D)"
mem_scale: fixes x :: "'b \<Rightarrow> real"   and a :: "real" assumes "x \<in> dest.diff_fun_space" shows "a *\<^sub>R x \<in> dest.diff_fun_space"
insertCI: fixes a :: "'x"   and B :: "'x set"   and b :: "'x" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
basis_reduction_cost_def: fixes fs :: "int vec list" shows "basis_reduction_cost fs = (case initial_state_cost fs of (state1, c1) \<Rightarrow> case basis_reduction_main_cost True 0 state1 0 of (state2, c2) \<Rightarrow> (state2, c1 + c2))"
plus_fun: shows "(+) = (\<lambda>f g x. f x + g x)"
bounded_linear_scaleR_right: fixes r :: "real" shows "bounded_linear ((*\<^sub>R) r)"
eSwapInp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'c option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "igWlsInp MOD delta inp" shows "eSwapInp MOD zs z1 z2 (OKI inp) = OKI (igSwapInp MOD zs z1 z2 inp)"
preimage_range: fixes f :: "'c \<Rightarrow> 'd option" shows "preimage f (ran f) = dom f"
pdevs_domain_def: fixes x :: "'b pdevs" shows "pdevs_domain x = {i. pdevs_apply x i \<noteq> (0::'b)}"
behaves_def: fixes step :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and final :: "'d \<Rightarrow> bool"   and load :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "language step final" shows "language.behaves step final load = load OO semantics.sem_behaves step final"
not_Done_def: fixes C :: "char list"   and sh :: "sheap" shows "not_Done C sh \<equiv> sh C = None \<or> (\<exists>sfs i. sh C = \<lfloor>(sfs, i)\<rfloor> \<and> i \<noteq> Done)"
maximal_independent_subset_extend: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set"   and V :: "'e set" assumes "vector_space scale"   and "S \<subseteq> V"   and "\<not> module.dependent scale S" obtains B :: "'e set" where "S \<subseteq> B" and "B \<subseteq> V" and "\<not> module.dependent scale B" and "V \<subseteq> module.span scale B"
simps_26: fixes vb :: "'b"   and v :: "'b acom"   and va :: "'b acom" shows "(SKIP {vb} \<sqsubseteq> v;; va) = False"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
dom_eqI: fixes a :: "'c"   and f :: "'c" assumes "C.ide a"   and "C.seq f a" shows "C.dom f = a"
eval\<^sub>G_subst: fixes S :: "'b ltl set"   and \<phi> :: "'b ltl" shows "eval\<^sub>G S \<phi> = subst \<phi> (\<lambda>\<chi>. Some (eval\<^sub>G S \<chi>))"
HNatInfinite_iff: shows "HNatInfinite = {N. \<forall>n\<in>\<nat>. n < N}"
less_Bits: fixes v :: "int"   and b :: "bool"   and w :: "int"   and c :: "bool" shows "(v BIT b < w BIT c) = (v < w \<or> v \<le> w \<and> \<not> b \<and> c)"
face_of_disjoint_interior: fixes T :: "'b set"   and S :: "'b set" assumes "T face_of S"   and "T \<noteq> S" shows "T \<inter> interior S = {}"
supp_3: fixes x1 :: "name"   and x2 :: "lam" shows "supp (Lam [x1].x2) = supp ([x1].x2)"
that: assumes "a \<in> elts A"   and "b \<in> elts (B a)" shows "thesis"
length_map2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list" shows "length (map2 f xs ys) = min (length xs) (length ys)"
beukers_aux_integrable3: shows "set_integrable lborel D' (\<lambda>(w, x, y). P x * (1 - w) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * w))"
sup_top_left: fixes x :: "'c \<times> 'd" shows "sup_prod top_prod x = top_prod"
dense: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b" assumes "class.dlo less_eq less"   and "less x z" shows "\<exists>y. less x y \<and> less y z"
lub_prod: fixes S :: "nat \<Rightarrow> 'c \<times> 'd" assumes "chain S" shows "(\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
nonzero_inverse_mult_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "inverse (times a b) = times (inverse b) (inverse a)"
derivable_clauses_are_entailed: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set"   and I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool" assumes "derivable C P S \<sigma> k C'"   and "validate_clause_set I (cl_ecl ` P)"   and "fo_interpretation I"   and "\<forall>x\<in>P. finite (cl_ecl x)" shows "validate_clause I (cl_ecl C)"
3: shows "search b = eval r_enumdom [i, the (eval (Mn 1 r_search_enum) [b])]"
subtree_pos: fixes T :: "bool list set"   and ds :: "bool list" shows "subtree T ds \<subseteq> subtree T (ds @ [Left]) \<union> subtree T (ds @ [Right]) \<union> {ds}"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d f n \<in> carrier_mat d d"
nv_5: shows "lin_indpt (insert (normal_vector D) D)"
def_gfp_unfold: fixes A :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "A \<equiv> gfp f"   and "mono f" shows "A = f A"
is_Fun_num_funs_less: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "x \<in> vars_term t"   and "is_Fun t" shows "num_funs (\<sigma> x) < num_funs (t \<cdot> \<sigma>)"
map_of_delete_iff: fixes x :: "'c"   and \<Gamma> :: "('c \<times> 'b) list"   and xa :: "'c"   and e :: "'b" shows "(map_of (delete x \<Gamma>) xa = Some e) = (map_of \<Gamma> xa = Some e \<and> xa \<noteq> x)"
ignores_old_vars_def: fixes P :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool" shows "ignores_old_vars P \<equiv> \<forall>mdss mdss'. length mdss = length mdss' \<and> length mdss' = length cms \<longrightarrow> map (\<lambda>x m. x m \<inter> - range var\<^sub>C_of) mdss = map (\<lambda>x m. x m \<inter> - range var\<^sub>C_of) mdss' \<longrightarrow> P mdss = P mdss'"
combine_code_eq: fixes \<delta>\<^sub>L' :: "int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and \<delta>\<^sub>R' :: "int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and l :: "int"   and \<delta>' :: "int"   and c\<^sub>0' :: "int \<times> int"   and c\<^sub>1' :: "int \<times> int" assumes "\<delta>\<^sub>L' = dist_code p\<^sub>0\<^sub>L p\<^sub>1\<^sub>L"   and "\<delta>\<^sub>R' = dist_code p\<^sub>0\<^sub>R p\<^sub>1\<^sub>R"   and "sorted_snd ps"   and "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"   and "(\<delta>', c\<^sub>0', c\<^sub>1') = combine_code (\<delta>\<^sub>L', p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (\<delta>\<^sub>R', p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps" shows "c\<^sub>0 = c\<^sub>0' \<and> c\<^sub>1 = c\<^sub>1'"
hnorm_mult: fixes x :: "'a star"   and y :: "'a star" shows "hnorm (x * y) = hnorm x * hnorm y"
Agent_LtK: fixes X :: "agent"   and X' :: "ltkey" shows "Messages.Agent X \<noteq> LtK X'"
open_contains_cball_eq: fixes S :: "'c set" assumes "open S" shows "\<forall>x. (x \<in> S) = (\<exists>e>0. cball x e \<subseteq> S)"
complete_hb_in_Runs: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list" assumes "hb_completion s E"   and "\<And>t E ln. ta_hb_consistent P E (llist_of (map (Pair t) (convert_RA ln)))" shows "mthr.Runs s (complete_hb s E)"
higher_pderiv_add: fixes n :: "nat"   and p :: "'a poly"   and q :: "'a poly" shows "(pderiv ^^ n) (p + q) = (pderiv ^^ n) p + (pderiv ^^ n) q"
vector_derivative_within_cbox: fixes a :: "real"   and b :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "'b" assumes "a < b"   and "x \<in> cbox a b"   and "(f has_vector_derivative f') (at x within cbox a b)" shows "vector_derivative f (at x within cbox a b) = f'"
Qg_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "G / H = \<lparr>carrier = set_rcs G H, Group.top = c_top G H, iop = c_iop G H, one = H\<rparr>"
deg_RGB: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> bb 0"
prems: shows "p \<in> setops\<^sub>s\<^sub>s\<^sub>t (a # S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
limA': shows "(sum f' \<longlongrightarrow> limA) (finite_subsets_at_top B)"
local_init_def: fixes init :: "'b set"   and S :: "'b \<Rightarrow> 'a" shows "local_init init S = Inf (S ` init)"
Integer_add_diff_cancel: fixes b :: "Integer"   and a :: "Integer" assumes "b \<noteq> \<bottom>" shows "a + b - b = a"
fmlookup_dom_iff: fixes x :: "'e"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "(x |\<in>| fmdom m) = (\<exists>a. fmlookup m x = Some a)"
u: shows "u = (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R i)"
reachArrLoc: fixes s :: "Store"   and T :: "Arraytype"   and a :: "nat"   and i :: "nat" shows "s\<turnstile> arrLoc T a i reachable_from arrV T a"
aux2: fixes pc :: "nat"   and ins :: "instr array"   and ofs :: "int" assumes "pc < array_length ins"   and "ofs \<in> offsets_is (list_of_array ins)" shows "nat (1 + int pc + ofs) < pc_bound ins"
assms_4: shows "kernel H K g \<subseteq> f ` I"
frame_hoare_r': fixes a :: "'d \<Longrightarrow> 'e"   and r :: "'e upred"   and q :: "'e upred"   and p :: "'e upred"   and S :: "'e hrel" assumes "vwb_lens a"   and "a \<sharp> r"   and "a \<natural> q"   and "\<lbrace>r \<and> p\<rbrace> S \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>r \<and> p\<rbrace> {a}:[S] \<lbrace>r \<and> q\<rbrace>\<^sub>u"
simps_1: shows "trees_cyl \<langle>\<rangle> = {\<langle>\<rangle>}"
d\<^sub>2_def: shows "d\<^sub>2 \<equiv> map_fun eval\<Pi>\<^sub>2 id Some"
ras_singleton_def: fixes x :: "'c" shows "ras_singleton x \<equiv> (array_of_list [x], 1::'d)"