hcju/isabelleps huggingface dataset:
right_neutral: fixes a :: "'a" shows "a * (1::'a) = a"
fun_plus_def: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a" shows "fun_plus f1 f2 \<equiv> \<lambda>v. f1 v + f2 v"
f_def: shows "f \<equiv> \<lambda>x. poly q x / poly p x"
p_def: shows "Rep_nat_int (res ts c) = {p, p + 1}"
simps_1: fixes a :: "'b list" shows "a \<^sup>@ 0 = \<epsilon>"
gram_schmidt_fs_lin_indpt_def: fixes n :: "nat"   and fs :: "'a vec list" shows "gram_schmidt_fs_lin_indpt n fs \<equiv> gram_schmidt_fs_Rn n fs \<and> gram_schmidt_fs_lin_indpt_axioms n fs"
proper_def: fixes S :: "'b set" shows "proper S \<equiv> \<forall>x r. compact (cball x r \<inter> S)"
power2_sum: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "power.power one times (plus x y) 2 = plus (plus (power.power one times x 2) (power.power one times y 2)) (times (times (numeral.numeral one plus (num.Bit0 num.One)) x) y)"
cInf_greatest: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and z :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq z x" shows "less_eq z (Inf X)"
sup_bot_left: fixes x :: "'a" shows "\<bottom> \<squnion> x = x"
atoms_And_append: fixes as1 :: "'a formula list"   and as2 :: "'a formula list" shows "atoms \<^bold>\<And>as1 @ as2 = atoms \<^bold>\<And>as1 \<union> atoms \<^bold>\<And>as2"
concat_lists_listset: fixes xs :: "'a list list" shows "set (concat_lists xs) = listset (map set xs)"
power_le_dvd: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat"   and b :: "'a"   and m :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times (power.power one times a n) b"   and "m \<le> n" shows "dvd.dvd times (power.power one times a m) b"
M1_def: shows "M1 = (\<lambda>i. (f i, f (m + i))) ` {..<m}"
HBL1_rev: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv (PP \<langle>\<phi>\<rangle>)" shows "prv \<phi>"
rep_eq: fixes x :: "'b ugraph" shows "nodes_internal x = fst (Rep_ugraph x)"
lossless_protocol_14_OT: fixes S :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "lossless_spmf (protocol_14_OT S C)"
continuous_on_topological: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
increasing_measurable_info: fixes f :: "bool stream \<Rightarrow> 'a"   and s :: "nat"   and N :: "'a measure"   and t :: "nat" assumes "f \<in> F s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> F t \<rightarrow>\<^sub>M N"
ins_queue_invar: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.tree_invar t"   and "SkewBinomialHeapStruc.queue_invar q" shows "SkewBinomialHeapStruc.queue_invar (SkewBinomialHeapStruc.ins t q)"
collect_clkvt_def: fixes S :: "('e \<times> ('d, 'f) cconstraint \<times> 'g \<times> 'd list \<times> 'e) set" shows "collect_clkvt S = \<Union> {set ((fst \<circ> snd \<circ> snd \<circ> snd) t) |t. t \<in> S}"
at_within_t1_space_avoid_finite: fixes I :: "'a set"   and x :: "'a"   and X :: "'a set" assumes "finite I"   and "x \<notin> I" shows "at x within X - I = at x within X"
homeomorphic_subspaces_eq: fixes S :: "'a set"   and T :: "'b set" assumes "subspace S"   and "subspace T" shows "S homeomorphic T = (dim S = dim T)"
document_ptr_no_shadow_root_ptr_cast: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "\<not> is_document_ptr_kind (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)"
trans_genPrefix: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (genPrefix r)"
interp_rga_def: fixes ops :: "('a \<times> 'a option) list" shows "interp_rga ops \<equiv> foldl insert_rga [] ops"
h4: shows "List.member l2 m \<longrightarrow> (\<exists>n. l2 ! n = m)"
is_fmap_image: fixes M :: "('a \<times> 'b) fset"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "is_fmap M" shows "is_fmap ((\<lambda>(a, b). (a, f a b)) |`| M)"
lcf_in: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> G" shows "i.lcf (focus X g) \<in> P[{x}]"
RSpan_single_closed: fixes m :: "'c" assumes "m \<in> V" shows "GSpan [m] \<subseteq> V"
ParamUses_in_Use: fixes n :: "'node"   and V :: "'var" assumes "valid_node n"   and "V \<in> \<Union> (set (ParamUses n))" shows "V \<in> Use n"
mem_appL: fixes L1 :: "'a list"   and x :: "'a"   and L2 :: "'a list" assumes "List.member L1 x" shows "List.member (L1 @ L2) x"
induct: fixes x1 :: "nat"   and x2 :: "'b tree234"   and P :: "nat \<Rightarrow> 'b tree234 \<Rightarrow> bool" assumes "full x1 x2"   and "P 0 \<langle>\<rangle>"   and "\<And>n l r p. \<lbrakk>full n l; P n l; full n r; P n r\<rbrakk> \<Longrightarrow> P (Suc n) \<langle>l, p, r\<rangle>"   and "\<And>n l m r p q. \<lbrakk>full n l; P n l; full n m; P n m; full n r; P n r\<rbrakk> \<Longrightarrow> P (Suc n) \<langle>l, p, m, q, r\<rangle>"   and "\<And>n l m m' r p q q'. \<lbrakk>full n l; P n l; full n m; P n m; full n m'; P n m'; full n r; P n r\<rbrakk> \<Longrightarrow> P (Suc n) \<langle>l, p, m, q, m', q', r\<rangle>" shows "P x1 x2"
content_eq_zero_iff: fixes p :: "'a poly" shows "(content p = (0::'a)) = (p = 0)"
coeff_cases: fixes y :: "nat \<Rightarrow> 'b" assumes "y \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'b)}" obtains x :: "'b poly" where "y = coeff x"
empty_subsetI: fixes A :: "'c set" shows "{} \<subseteq> A"
cond_action_ok_Join: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and T :: "'f"   and x :: "'x"   and ln :: "'e \<Rightarrow>f nat" assumes "cond_action_ok s t (Join T)"   and "thr s T = \<lfloor>(x, ln)\<rfloor>" shows "final x \<and> ln = no_wait_locks \<and> wset s T = None"
strict_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<prec> b"   and "b \<prec> c" shows "a \<prec> c"
apath_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "S.apath u (e # es) w = (e \<in> parcs sG \<and> fst e = u \<and> S.apath (snd e) es w \<and> fst e \<notin> set (pawalk_verts (snd e) es))"
prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "g (p ^ k) = g_prime_power p k"
WW_eq_UN_WW_seg: shows "WW = \<Union> (range WW_seg)"
dsg2: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
rsa_base_def: fixes prime_set :: "nat set" shows "rsa_base prime_set \<equiv> prime_set \<subseteq> {x. prime x \<and> 2 < x} \<and> finite prime_set \<and> 2 < card prime_set"
wfE_min: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and Q :: "'b set" assumes "wf R"   and "x \<in> Q" obtains z :: "'b" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
orecvmsg_simps_7: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and m :: "'e" shows "orecvmsg P \<sigma> (receive m) = P \<sigma> m"
finals_in_partitions: fixes A :: "'b list set" shows "finals A \<subseteq> UNIV // \<approx>A"
stirling_integral_conv_stirling_integral_Suc: fixes n :: "nat" assumes "0 < n" shows "stirling_integral n s = of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s - complex_of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n)"
SDG_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list" shows "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses \<equiv> CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses \<and> Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"
simps_4: fixes M :: "('fun, 'var) Term.term set"   and uu :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c = (\<lambda>\<I>. Unifier \<I> t t' \<and> \<lbrakk>M; S\<rbrakk>\<^sub>c \<I>)"
aff_dim_2: fixes a :: "'a"   and b :: "'a" shows "aff_dim {a, b} = (if a = b then 0 else 1)"
simps: fixes s1 :: "'e"   and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" shows "P4 (s1, s2) = ((\<forall>p. status s2 p = Sleep) = (initVals s2 = {}))"
top_Err: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Semilat.top (Err.le r) Err"
aff_dim_affine_hull2: fixes S :: "'b set"   and T :: "'b set" assumes "affine hull S = affine hull T" shows "aff_dim S = aff_dim T"
right_unique_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
simps: fixes x :: "'b"   and h :: "'b heap" shows "insert x h = merge (Hp x []) h"
permute_flip_at: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
freshRes: fixes a :: "name"   and P :: "pi" shows "a \<sharp> <\<nu>a>P"
Erdos_Milner_aux: fixes \<alpha> :: "V"   and k :: "nat"   and \<gamma> :: "V"   and \<beta> :: "V" assumes "partn_lst_VWF \<alpha> [ord_of_nat k, \<gamma>] 2"   and "indecomposable \<alpha>"   and "1 < k"   and "Ord \<gamma>"   and "\<beta> \<in> elts \<omega>1" shows "partn_lst_VWF (\<alpha> * \<beta>) [ord_of_nat (2 * k), min \<gamma> (\<omega> * \<beta>)] 2"
smult_closed: fixes R :: "('c, 'd) ring_scheme"   and M :: "('c, 'e, 'f) module_scheme"   and a :: "'c"   and x :: "'e" assumes "Module.module R M"   and "a \<in> carrier R"   and "x \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
subgraph_awalk_imp_awalk: fixes H :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "pre_digraph.awalk H u p v"   and "subgraph H (app_iso hom G)" shows "H.awalk u p v"
ide_in_iso_class: fixes f :: "'a" assumes "B.ide f" shows "f \<in> \<lbrakk>f\<rbrakk>\<^sub>B"
asIMOD_igSubstCls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstCls (asIMOD SEM)"
type_definition_def: fixes Rep :: "'e \<Rightarrow> 'f"   and Abs :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "type_definition Rep Abs A \<equiv> (\<forall>x. Rep x \<in> A) \<and> (\<forall>x. Abs (Rep x) = x) \<and> (\<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y)"
exec_meth_drop_pc: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and n' :: "nat" assumes "exec_meth ci P (ins @ ins') (shift (length ins) xt') t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "length ins \<le> pc"   and "jump_ok ins' 0 n'" shows "length ins \<le> pc'"
dec_clock_def: fixes s :: "'b state" shows "dec_clock s = update_clock (\<lambda>_. clock s - 1) s"
isMaxim_def: fixes A :: "'a set"   and b :: "'a" shows "isMaxim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (a, b) \<in> r)"
less_dual_def_var: fixes x :: "'b dual"   and y :: "'b dual" shows "(x < y) = (\<partial>\<^sup>- y < \<partial>\<^sup>- x)"
pred_Id: shows "(=) = (\<lambda>x y. (x, y) \<in> Id)"
Act_Basic_11: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<A>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<exists>\<alpha>. \<^bold>\<A>\<phi> \<alpha>) in v]"
qGood_iff_good_asTerm: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm" shows "good (asTerm qX) = qGood qX"
monom_span: fixes p :: "nat" shows "poly_vs.span {monom (1::'a) x |x. x \<le> p} = {x. degree x \<le> p}"
p_fusion_hd: fixes ps :: "'a list"   and qs :: "'a list" assumes "ps \<noteq> []"   and "qs \<noteq> []" shows "hd (p_fusion ps qs) = hd ps"
S_at_edge: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "p \<in> S"   and "q \<in> hyp2 \<union> S"   and "r \<in> hyp2 \<union> S"   and "proj2_Col p q r" shows "B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r \<or> B\<^sub>\<real> cart2_pt p cart2_pt r cart2_pt q"
simps_1: fixes lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat"   and n :: "nat" shows "pseudo_divmod_main lc q r d dr (Suc n) = (let rr = smult lc r; qq = coeff r dr; rrr = rr - monom qq n * d; qqq = smult lc q + monom qq n in pseudo_divmod_main lc qqq rrr d (dr - 1) n)"
PO_refines_implies_R_image_trans: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "PO_refines R Ta Tc" shows "{R `` reach Ta} TS.trans Tc {> R `` reach Ta}"
prop_equiv_subset: fixes P :: "'a ltln set" shows "{abs_ltln\<^sub>P \<psi> |\<psi>. nested_prop_atoms \<psi> \<subseteq> P} \<subseteq> {abs_ltln\<^sub>P \<psi> |\<psi>. prop_atoms \<psi> \<subseteq> P}"
right_neutral: fixes a :: "'a upred" shows "(a \<or> false) = a"
f_h_eq: fixes x :: "'a"   and y :: "'b" shows "f (h (x, y)) = x"
l_get_root_node_axioms: shows "l_get_root_node get_root_node get_parent"
sel_1: fixes x1 :: "'b"   and x2 :: "'b stream" shows "shd (x1 ## x2) = x1"
closed_Diff: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "open T" shows "topological_space.closed open (S - T)"
pickBinp_psubstBinp_qPsubstBinp: fixes binp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodBinp binp"   and "goodEnv rho" shows "(pickBinp binp %%[rho], pickBinp binp %%[[pickE rho]]) \<in> alphaBinp"
length_mapi: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list" shows "length (mapi f xs) = length xs"
axioms_1: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "Functor F" shows "FunctorM F"
power_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" shows "power.power one times a 0 = one"
ReachableTrans: fixes cfg1 :: "('p, 'v, 's) configuration"   and cfg2 :: "('p, 'v, 's) configuration"   and cfg3 :: "('p, 'v, 's) configuration" assumes "reachable cfg1 cfg2"   and "reachable cfg2 cfg3" shows "reachable cfg1 cfg3"
eqfun_def: shows "eqfun = (\<lambda>A x. \<approx>A `` {x})"
assms_3: shows "h \<in> hom \<F>\<^bsub>gens\<^esub> G"
indRelRPO_iff_exists_source_target_relation: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<forall>(P, Q)\<in>indRelRPO. Pred (P, Q)) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q)\<in>Rel. Pred (P, Q)) \<and> preorder Rel)"
Cons_eq_map_conv: fixes x :: "'c"   and xs :: "'c list"   and f :: "'d \<Rightarrow> 'c"   and ys :: "'d list" shows "(x # xs = map f ys) = (\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs)"
distinct_27: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm"   and x6 :: "'b ptrm" shows "PApp x31 x32 \<noteq> PFst x6"
termMOD_igSwapAllIPresIGWlsAll: shows "igSwapAllIPresIGWlsAll termMOD"
lc_def: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lcf p = lookup p (lpp p)"
iso_hcomp\<^sub>R\<^sub>W\<^sub>C_2: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "local.iso \<mu>"   and "local.iso \<nu>"   and "sources \<nu> \<inter> targets \<mu> \<noteq> {}" shows "inverse_arrows (\<nu> \<star> \<mu>) (local.inv \<nu> \<star> local.inv \<mu>)"
obj_char: fixes a :: "'c" shows "C'.obj a = (C'.arr a \<and> C.obj a)"
mult_ncols: fixes fmul :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d matrix"   and B :: "'b matrix" assumes "\<forall>a. fmul (0::'d) a = (0::'c)"   and "\<forall>a. fmul a (0::'b) = (0::'c)"   and "fadd (0::'c) (0::'c) = (0::'c)" shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
assms_10: shows "p\<^sub>1 \<in> set ps"
f': shows "\<guillemotleft>f' : src\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright> \<and> C.ide f'"
P3: shows "\<lfloor>\<lambda>w. \<forall>x. ((actualizedAt) x \<^bold>\<rightarrow> Abstract x \<^bold>\<rightarrow> Dependent x) w\<rfloor>"
lim_proc_is_lub2: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "\<forall>u. range S <| u \<longrightarrow> (\<forall>s. s \<notin> D (lim_proc (range S)) \<longrightarrow> Ra (lim_proc (range S)) s = Ra u s)"
remove_var_def: fixes \<sigma> :: "nat \<Rightarrow> val" shows "remove_var \<sigma> = (\<lambda>x. \<sigma> (Suc x))"
n: shows "(derived G ^^ n) (carrier G) = {\<one>}"
abelian_monoidE_4: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "abelian_monoid R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
hd_P_k_v0: shows "hd P_k = v0"
inf_commute: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
left_lang_disjoint: fixes q1 :: "hf"   and q2 :: "hf" assumes "q1 \<noteq> q2" shows "left_lang q1 \<inter> left_lang q2 = {}"
T_mtf_competitive: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0"   and "s0 \<noteq> []"   and "\<forall>i<n. rs ! i \<in> set s0" shows "real_of_int (MTF_Off.T_mtf rs s0 n) \<le> (2 - 1 / real (length s0)) * real_of_int (MTF_Off.T_A as rs s0 n)"
pdec1_le: fixes i :: "nat" shows "pdec1 i \<le> i"
loc: shows "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"
add_edge_def: fixes v :: "'a"   and v' :: "'a"   and G :: "('a, 'b) graph_scheme" shows "add_edge v v' G = \<lparr>nodes = nodes G \<union> {v, v'}, edges = {(v, v')} \<union> edges G\<rparr>"
addpreRT_partly_welldefined: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:18..PRreq-:20} \<union> {PRrep-:2..PRrep-:6} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l \<in> {PRreq-:3..PRreq-:19} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>)))"
nth_mem: fixes n :: "nat"   and xs :: "'g list" assumes "n < length xs" shows "xs ! n \<in> set xs"
simps_10: fixes v :: "Core.term"   and vb :: "String.literal"   and vc :: "typ" shows "strip_all_vars (v $ Ct vb vc) = []"
convex_scaled: fixes S :: "real set"   and c :: "'b" assumes "convex S" shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
monofun_def: fixes f :: "'a \<Rightarrow> 'b" shows "monofun f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
bi_unique_rel_set_lemma: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'c set" assumes "bi_unique R"   and "rel_set R X Y" obtains f :: "'b \<Rightarrow> 'c" where "Y = f ` X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
sumlist_in_span: fixes W :: "'a vec set"   and xs :: "'a vec list" assumes "W \<subseteq> carrier_vec n"   and "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> local.span W" shows "M.sumlist xs \<in> local.span W"
is_median_Cons_odd: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "is_median x xs"   and "odd (length xs)" shows "is_median x (y # xs)"
WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "(WHILE b DO c, s) \<Rightarrow> Suc 0 \<Down> s"
rcsTr2: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> H \<bullet> \<rho> a" shows "b \<cdot> a \<in> H"
coprime_0_left_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one zero a = dvd.dvd times a one"
\<psi>_\<phi>: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<psi> x (\<phi> y f) = f"
OclSelf_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> 'f"   and H :: "'g \<Rightarrow> 'e"   and fst_snd :: "'d state \<times> 'd state \<Rightarrow> ('g, 'h) state_scheme" shows "OclSelf x H fst_snd = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> then if oid_of (x \<tau>) \<in> dom (heap (fst \<tau>)) \<and> oid_of (x \<tau>) \<in> dom (heap (snd \<tau>)) then H \<lceil>heap (fst_snd \<tau>) (oid_of (x \<tau>))\<rceil> else invalid \<tau> else invalid \<tau>)"
pdevs_val_filter_pdevs_eval: fixes e :: "nat \<Rightarrow> real"   and p :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b pdevs" shows "pdevs_val e (filter_pdevs p x) = pdevs_val (\<lambda>i. if p i (pdevs_apply x i) then e i else 0) x"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'f list" assumes "i < n" shows "take n xs ! i = xs ! i"
contour_integrable_eq: fixes f :: "complex \<Rightarrow> complex"   and p :: "real \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "f contour_integrable_on p"   and "\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x" shows "g contour_integrable_on p"
Subst: fixes Ax :: "('a, 'b) Axioms"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and B :: "'a"   and e2 :: "('a, 'b) Expression"   and e3 :: "('a, 'b) Expression"   and C :: "'a" assumes "Axioms Ax"   and "Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e1 : B"   and "Vx : B \<turnstile> e2\<equiv>e3 : C \<in> Axioms.Theory Ax" shows "Vx : A \<turnstile> sub e1 in e2\<equiv>sub e1 in e3 : C \<in> Axioms.Theory Ax"
complete: fixes qs :: "'Q list"   and f :: "'L" assumes "qs \<in> lists Q"   and "A f = Some (length qs)" shows "\<exists>q. q \<rightarrow> f qs \<in> \<delta>"
CARD_vec: shows "CARD(('a, 'b) vec) = CARD('a) ^ CARD('b)"
min_eSuc_eSuc: fixes n :: "enat"   and m :: "enat" shows "min (eSuc n) (eSuc m) = eSuc (min n m)"
k_i'_nonneg: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "0 \<le> k_i' i e"