hcju/isabelleps huggingface dataset:
intro: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" assumes "topological_space S is_open"   and "presheaf_of_rings_axioms is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
assms_1: shows "run (Array.of_list xs) \<sigma> \<sigma>' r"
ContractsWithUnilateralSubstitutesAndIRCAndLAD_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch \<equiv> ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch \<and> ContractsWithLAD Xd Xh Pd Ch"
coeff_zero: fixes R :: "('a, 'b) ring_scheme"   and n :: "nat" assumes "UP_ring R" shows "coeff (UP R) \<zero>\<^bsub>UP R\<^esub> n = \<zero>\<^bsub>R\<^esub>"
Afix_nonrec_def: fixes x :: "Vars.var"   and e :: "exp" shows "Afix_nonrec x e = (\<Lambda> i. \<A>\<^sup>\<bottom>\<^bsub>ABind_nonrec x e\<cdot>i\<^esub> e \<squnion> fst i)"
a_6: fixes x :: "'a"   and y :: "'a" shows "ad (d x \<cdot> y) = ad x + ad y"
sel_2: fixes x1 :: "float poly"   and x2 :: "float interval" shows "tm_bound (TaylorModel x1 x2) = x2"
simps_27: fixes vb :: "char list"   and vc :: "aexp"   and vd :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "(vb ::= vc {vd} \<le> v;; va) = False"
assms_1: shows "\<exists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"
phi_def: fixes g :: "'g"   and v :: "'val" shows "phi g v = phis g (defNode g v, v)"
fold_split_conv_map_fst: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "('c \<times> 'd) list" shows "List.fold (\<lambda>(x, y). f x) xs = List.fold f (map fst xs)"
sum_list_map_eq_sum_count2: fixes xs :: "'b list"   and X :: "'b set"   and f :: "'b \<Rightarrow> nat" assumes "set xs \<subseteq> X"   and "finite X" shows "sum_list (map f xs) = (\<Sum>x\<in>X. count_list xs x * f x)"
msb_pos: fixes w :: "32 Word.word" assumes "msb (- w)"   and "\<not> msb w" shows "uint w \<in> {1..2 ^ (LENGTH(32) - 1) - 1}"
set_map: fixes f :: "'e \<Rightarrow> 'c"   and v :: "'d \<Rightarrow> 'e" shows "range (f \<circ> v) = \<P> f (range v)"
comprehension: fixes x :: "hf"   and P :: "hf \<Rightarrow> bool" shows "\<exists>z. \<forall>u. (u \<^bold>\<in> z) = (u \<^bold>\<in> x \<and> P u)"
is_done_oimpl_def: fixes v :: "'b"   and I :: "'b \<Rightarrow> node_state option" shows "is_done_oimpl v I \<equiv> case I v of None \<Rightarrow> False | Some (STACK nat) \<Rightarrow> False | Some DONE \<Rightarrow> True"
num1_eq_iff: fixes x :: "1"   and y :: "1" shows "(x = y) = True"
wfG': shows "wf_graph \<lparr>nodes = nodes G, edges = edges G\<rparr>"
run_asset_iv_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_asset_iv n s)"
nth_image: fixes l :: "nat"   and xs :: "'a list" assumes "l \<le> length xs" shows "(!) xs ` {0..<l} = set (take l xs)"
poly_nth_root_def: fixes n :: "nat"   and p :: "'a poly" shows "poly_nth_root n p = p \<circ>\<^sub>p monom (1::'a) n"
swapped_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X"   and "(X, Y) \<in> swapped" shows "good Y"
False: shows "\<not> length as \<le> 2 ^ card (prob_dom PROB) - 1"
A: shows "\<exists>xi. xi \<in> Gromov_boundary \<and> Gromov_extension f xi = xi \<and> additive_strength f xi = 0"
f_join_f_expand_iMOD: fixes k :: "nat"   and xs :: "'b message_af list"   and n :: "nat" assumes "0 < k" shows "xs \<odot> k \<Join>\<^sub> [ n * k, mod k ] = xs \<Join>\<^sub> [n\<dots>]"
uniform: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and bl :: "'b set" assumes "block_design point_set block_collection u_block_size"   and "bl \<in># block_collection" shows "int (card bl) = u_block_size"
AE_pair_measure: fixes P :: "'a \<times> 'b \<Rightarrow> bool" assumes "{x \<in> space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"   and "AE x in M1. AE y in M2. P (x, y)" shows "almost_everywhere (M1 \<Otimes>\<^sub>M M2) P"
match_rews_6: fixes a :: "'a"   and slist :: "[:'a:]"   and k :: "'a \<rightarrow> [:'a:] \<rightarrow> 'b match" assumes "a \<noteq> \<bottom>"   and "slist \<noteq> \<bottom>" shows "match_scons\<cdot>(a :# slist)\<cdot>k = k\<cdot>a\<cdot>slist"
simps: fixes ant :: "'b form multiset"   and suc :: "'b form multiset" shows "SRCTransforms.mset ( ant \<Rightarrow>* suc) = ant + suc"
hyps: fixes i :: "nat"   and k :: "nat" assumes "i \<le> k" shows "prefixToLevel_aux M' l i = prefixToLevel_aux M' (l + (k - i)) k"
assms_2: fixes s :: "'a set" assumes "s \<in> f" shows "\<exists>a b. s = cbox a b"
bot_admS_def: shows "bot \<equiv> mklr {\<bottom>}"
fun_upd_idem: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'b" assumes "f x = y" shows "f(x := y) = f"
triangle_paths_2: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "\<exists>Q. Q \<in> \<P> \<and> a \<in> Q \<and> c \<in> Q \<and> a \<noteq> c"
double_dual_iso_def: fixes x :: "'a" shows "double_dual_iso x = (\<lambda>\<chi>. if character G \<chi> then \<chi> x else 0)"
f_range_on_def: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "f_range_on A f = (\<forall>B\<subseteq>A. f B \<subseteq> B)"
finite_Invs: fixes xs :: "'a list"   and ys :: "'a list" shows "finite (Inv xs ys)"
outstanding_refs_subsets_6: fixes sb :: "'b memref list" shows "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> outstanding_refs (Not \<circ> is_volatile) sb"
f_nxt_cong: fixes f :: "'b \<Rightarrow> 'b ttree"   and x' :: "'b"   and f' :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and x :: "'b" assumes "f x' = f' x'" shows "f_nxt f T x x' = f_nxt f' T x x'"
init_fin_bisim_wait_def: fixes bisim_wait :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "FWbisimulation_base.init_fin_bisim_wait bisim_wait = (\<lambda>(status1, x1) (status2, x2). status1 = Running \<and> status2 = Running \<and> bisim_wait x1 x2)"
subgroup_finite: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := P\<rparr>)"   and "finite (carrier (G\<lparr>carrier := P\<rparr>))" shows "finite H"
wf_SN: shows "wfP (\<rightarrow>)\<inverse>\<inverse> = SN {(x, y). x \<rightarrow> y}"
list_assn_aux_len: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l :: "'a list"   and l' :: "'b list" shows "list_assn P l l' = list_assn P l l' * \<up> (length l = length l')"
const_pub_arity_eq_zero: fixes c :: "'fun" assumes "c \<in> {f. public f} \<inter> \<C>" shows "arity c = 0 \<and> public c"
KL_div_pos: fixes S :: "'b set"   and a :: "'b \<Rightarrow> real"   and e :: "'b \<Rightarrow> real" assumes "finite S"   and "S \<noteq> {}"   and "\<And>i. i \<in> S \<Longrightarrow> 0 < a i"   and "\<And>i. i \<in> S \<Longrightarrow> 0 < e i"   and "sum a S = 1"   and "sum e S = 1" shows "0 \<le> KL_div S a e"
arr_inv: fixes f :: "'b" assumes "B.iso f" shows "B.arr (B.inv f)"
pth_4_2: fixes c :: "real" shows "c *\<^sub>R (0::'b) = (0::'b)"
min_def: fixes a :: "'a word"   and b :: "'a word" shows "signed.min a b = (if a \<le>s b then a else b)"
aux79: fixes y :: "('b, 'c) Combinators"   and x :: "('b, 'c) Combinators"   and a :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "y \<in> set (FWNormalisationCore.insort x a l)"   and "y \<noteq> x" shows "y \<in> set a"
fmap_rel_nat_rel_dom_m: fixes A :: "(nat, 'a) fmap"   and B :: "(nat, 'b) fmap"   and R :: "('a \<times> 'b) set" assumes "(A, B) \<in> \<langle>nat_rel, R\<rangle>fmap_rel" shows "dom_m A = dom_m B"
simps_1: fixes A :: "'c set"   and S :: "(('b, 'c) fm list \<times> 'c) set"   and f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c" shows "extend A S f 0 = S"
obj_def': fixes a :: "'a" shows "obj a = (arr a \<and> trg a = a)"
indep_vars_cong: fixes I :: "'d set"   and J :: "'d set"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and Y :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and M' :: "'d \<Rightarrow> 'e measure"   and N' :: "'d \<Rightarrow> 'e measure"   and i :: "'i" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> X i = Y i"   and "\<And>i. i \<in> I \<Longrightarrow> M' i = N' i" shows "Y.indep_vars i M' X I = Y.indep_vars i N' Y J"
space_Sup_eq_UN: fixes M :: "'b measure set" shows "space (Sup M) = \<Union> (space ` M)"
replace_coeff_monom: fixes f :: "'c \<Rightarrow> 'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'c" assumes "f (0::'c) = (0::'b)" shows "replace_coeff f (MPoly_Type.monom m a) = MPoly_Type.monom m (f a)"
refine_init_using_invariants: fixes Tc :: "('a, 'b) TS_scheme"   and R :: "('c \<times> 'a) set"   and Ta :: "('c, 'd) TS_scheme"   and I :: "'c set"   and J :: "'a set" assumes "init Tc \<subseteq> R `` init Ta"   and "init Ta \<subseteq> I"   and "init Tc \<subseteq> J" shows "init Tc \<subseteq> (R \<inter> I \<times> J) `` init Ta"
f_carrier: fixes n :: "nat"   and fs :: "int vec list"   and i :: "nat" assumes "fs_int_indpt n fs"   and "i < length fs" shows "fs ! i \<in> carrier_vec n"
e: shows "[atom ` xs]res. T = [atom ` ys]res. S"
Macaulay_mat_def: fixes ps :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.punit.Macaulay_mat ps = polys_to_mat (pp_pm.punit.Keys_to_list ps) ps"
meet_def: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" shows "meet L x y = Lattice.inf L {x, y}"
IH: shows "limsup (\<lambda>n. u (n + k)) = limsup u"
IH: assumes "list_ops xs" shows "dom (snd (interp_ops xs)) = set (inserted_ids xs) - set (deleted_ids xs)"
\<alpha>\<beta>_sym: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "s \<approx> s'" shows "s' \<approx> s"
inject_1: fixes x31 :: "'b"   and x32 :: "'a FOL_Fitting.term list"   and y31 :: "'b"   and y32 :: "'a FOL_Fitting.term list" shows "(Pred x31 x32 = Pred y31 y32) = (x31 = y31 \<and> x32 = y32)"
remove_rev_def: fixes x :: "'b" shows "remove_rev x \<equiv> filter_rev ((Not \<circ>\<circ> (=)) x)"
assms_2: shows "xs = [] \<or> comp k (fst (hd xs)) = Lt"
max_perm_degree_def: fixes A :: "('b poly, 'c) sq_matrix" shows "max_perm_degree A = (MAX p\<in>{p. p permutes UNIV}. \<Sum>i\<in>UNIV. degree (to_fun A i (p i)))"
reduced_GB_is_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (local.punit.component_of_term ` Keys F)"   and "F \<subseteq> punit.dgrad_p_set d m" shows "punit.is_Groebner_basis (punit.reduced_GB F)"
pa_01q: shows "(path_image (subpath 0 t q) \<union> path_image (subpath 1 t q)) \<inter> closed_segment (a - complex_of_real d) (a + complex_of_real e) = {a - complex_of_real d, a + complex_of_real e}"
A_proj_wf_trms': fixes l :: "'lbl" shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (N1 (proj_unl l A))"
cts_wp_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. bd_cts (wp (a (f s)))" shows "bd_cts (wp (Bind f a))"
ThrowRedsThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>throw e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
\<X>_are_terminals: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and u :: "symbol \<times> character list"   and Doc :: "character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "u \<in> LocalLexing.\<X> \<TT> Lex Doc k" shows "CFG.is_terminal \<TT> (terminal_of_token u)"
dehomogenize_id_iff: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "(dehomogenize x p = p) = (x \<notin> indets p)"
toFunction_def: fixes R :: "('a \<times> 'b) set" shows "toFunction R = (,,) R"
C: shows "Crypt (sesK (d * e)) (Passwd n) \<in> synth (analz (A \<union> spies evsFC5))"
expand: fixes resumption :: "('f, 'g, 'h) Resumption.resumption"   and resumption' :: "('f, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done resumption = Resumption.resumption.is_Done resumption'"   and "\<lbrakk>Resumption.resumption.is_Done resumption; Resumption.resumption.is_Done resumption'\<rbrakk> \<Longrightarrow> Resumption.resumption.result resumption = Resumption.resumption.result resumption'"   and "\<lbrakk>\<not> Resumption.resumption.is_Done resumption; \<not> Resumption.resumption.is_Done resumption'\<rbrakk> \<Longrightarrow> Resumption.resumption.output resumption = Resumption.resumption.output resumption' \<and> Resumption.resumption.resume resumption = Resumption.resumption.resume resumption'" shows "resumption = resumption'"
dom_mult_closed: fixes x :: "'a"   and y :: "'a" shows "d (d x \<cdot> d y) = d x \<cdot> d y"
boolean_is_robbins: shows "class.robbins_algebra uminus (\<sqinter>) (\<squnion>) \<bottom> \<top>"
iteration_denest2: fixes y :: "'a"   and x :: "'a" shows "y\<^sup>\<star> \<cdot> x \<cdot> (x + y)\<^sup>\<infinity> + y\<^sup>\<infinity> = (x + y)\<^sup>\<infinity>"
simps: shows "wf_residual_network = ((\<forall>x xa. (x, xa) \<in> \<^bold>E \<longrightarrow> (xa, x) \<notin> \<^bold>E) \<and> (source \<Delta>, sink \<Delta>) \<notin> \<^bold>E)"
af_abs_equiv: fixes \<psi> :: "'b ltl_prop_equiv_quotient"   and xs :: "'b set list"   and x :: "'b set" shows "foldl \<up>af \<psi> (xs @ [x]) = \<up>step (foldl \<up>af\<^sub>\<UU> (\<up>Unf \<psi>) xs) x"
norm_eff_def: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list" shows "norm_eff i P pc \<tau> = map (\<lambda>pc'. (pc', \<lfloor>eff\<^sub>i (i, P, \<tau>)\<rfloor>)) (succs i \<tau> pc)"
alternating_sum_list_conv_nataction: fixes n :: "nat"   and s :: "'a"   and t :: "'a" shows "sum_list (alternating_list (2 * n) s t) = (s + t) +^ n"
start_points_not_zero_contra: fixes r :: "'a"   and x :: "'a" assumes "connected_root r x"   and "point r"   and "start_points x = 0"   and "x ; r = 0" shows "x = 0"
map_option_o_empty: fixes f :: "'c \<Rightarrow> 'b" shows "map_option f \<circ> Map.empty = Map.empty"
pochhammer_of_real: fixes x :: "real"   and n :: "nat" shows "pochhammer (of_real x) n = of_real (pochhammer x n)"
flush_all_until_volatile_write_append_Prog\<^sub>s\<^sub>b_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and p\<^sub>2 :: "'b"   and mis :: "instr list"   and p\<^sub>1 :: "'b"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p\<^sub>2, is @ mis, \<theta>, sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis], \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
cstrong_operator_topology_def: shows "cstrong_operator_topology = pullback_topology UNIV cblinfun_apply euclidean"
oalist_inv_sort_oalist_raw: fixes xs :: "('a \<times> 'b) list \<times> 'a key_order" shows "ko.oalist_inv (sort_oalist_ko xs)"
info_filtration: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" assumes "trivial_init_filtrated_prob_space M F" shows "init_triv_filt M F"
rep_eq: fixes x :: "'d"   and xa :: "'c"   and xb :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "FL_Formula.formula.Rep_formula (FL_Formula.Act x xa xb) = FL_Formula.Act\<^sub>\<alpha> x xa (FL_Formula.formula.Rep_formula xb)"
minus_apply: fixes A :: "'d \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(A - B) x = A x - B x"
af_letter_atoms: fixes \<phi> :: "'a ltln"   and \<nu> :: "'a set" shows "atoms_ltln (af_letter \<phi> \<nu>) \<subseteq> atoms_ltln \<phi>"
epsilon_def: shows "epsilon = ([], \<lambda>a s. {}, \<lambda>s. s = [])"
hyps: fixes A :: "'a set" assumes "card A < card Aa__"   and "finite A"   and "\<forall>x\<in>A. f x \<in> A \<and> f x \<noteq> x \<and> f (f x) = x" shows "even (card A)"
f7: shows "t' \<otimes> (snd x' \<otimes> r' \<ominus> s' \<otimes> fst x') = \<zero>"
simps_12: fixes cpu_reg' :: "(CPU_register \<Rightarrow> word32) \<Rightarrow> CPU_register \<Rightarrow> word32"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "cpu_reg_update cpu_reg' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg' cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
rel_pmf_measureI': fixes p :: "'c pmf"   and q :: "'d pmf"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>A. A \<subseteq> set_pmf p \<Longrightarrow> measure_pmf.prob p A \<le> measure_pmf.prob q {y \<in> set_pmf q. \<exists>x\<in>A. R x y}" shows "rel_pmf R p q"
rel_new: shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
dim_row_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'b vec" shows "dim_row (mat\<^sub>r nr nc g) = nr"
ord_strict_p_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "ord_strict_p p q = (\<exists>v. lookup p v = (0::'c) \<and> lookup q v \<noteq> (0::'c) \<and> (\<forall>u. lt_of_nat_term_order cmp_term v u \<longrightarrow> lookup p u = lookup q u))"
simps_3: fixes v :: "game" shows "Choiceo undefg (Agame v) = undefg"
abs_summable_on_comparison_test': fixes g :: "'d \<Rightarrow> real"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> g x" shows "f abs_summable_on A"
gr_implies_not_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'b"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
neutr_eq_iff: fixes a :: "nat"   and b :: "nat" shows "(0 = max a b) = (a = 0 \<and> b = 0)"
fps_X_def: shows "fps_X = Abs_fps (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
t_n_closed: fixes x :: "'a" shows "!t x = !x"
HoVH_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "HoVH \<mu> \<equiv> if VVV.arr \<mu> then fst \<mu> \<star> fst (snd \<mu>) \<star> snd (snd \<mu>) else null"
c: shows "(cis (2 * x * pi) = 1) = (x \<in> \<int>)"
set_rel_compp: fixes A :: "('a \<times> 'c) set"   and B :: "('c \<times> 'b) set" shows "\<langle>A O B\<rangle>set_rel = \<langle>A\<rangle>set_rel O \<langle>B\<rangle>set_rel"
in_convex_hull_exchange_unique: fixes S :: "'a set"   and a :: "'a"   and T :: "'a set"   and T' :: "'a set"   and x :: "'a" assumes "\<not> affine_dependent S"   and "a \<in> convex hull S"   and "T \<subseteq> S"   and "T' \<subseteq> S"   and "x \<in> convex hull insert a T"   and "x \<in> convex hull insert a T'" shows "x \<in> convex hull insert a (T \<inter> T')"
prems_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set (x__ # xs)"   and "y \<in> set (x__ # xs)" shows "rel x y \<or> rel y x"
simps_18: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "value" shows "local.less_aexp_aux (Minus e1 e2) (L v) = False"
CDMode_CTRL_States_def: shows "CDMode_CTRL_States = {''Playing'', ''SelectingNextTrack'', ''SelectingPreviousTrack''}"
someI2: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g"   and Q :: "'g \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Inv (t \<^bold>\<star> u) = Inv t \<^bold>\<star> Inv u"
simps: fixes joint_oracle :: "'d \<times> 'e \<Rightarrow> 'f \<Rightarrow> (('g \<times> 'd) \<times> 'g \<times> 'e) spmf"   and oracle1 :: "'d \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'd) spmf"   and bad1 :: "'d \<Rightarrow> bool"   and oracle2 :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and bad2 :: "'e \<Rightarrow> bool"   and gpv :: "('b, 'f, 'g) gpv"   and s1 :: "'d"   and s2 :: "'e" shows "exec_until_bad joint_oracle oracle1 bad1 oracle2 bad2 gpv s1 s2 = (if bad1 s1 \<or> bad2 s2 then pair_spmf (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2) else the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf ((x, s1), x, s2) | IO out f \<Rightarrow> joint_oracle (s1, s2) out \<bind> (\<lambda>((x, s1'), y, s2'). if bad1 s1' \<or> bad2 s2' then pair_spmf (exec_gpv oracle1 (f x) s1') (exec_gpv oracle2 (f y) s2') else exec_until_bad joint_oracle oracle1 bad1 oracle2 bad2 (f x) s1' s2')))"
paths_plus_one_disjoint: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and xs :: "'c list"   and ys :: "'c list"   and v :: "'c" assumes "DisjointPathsPlusOne G v0 v1 paths P_new"   and "xs \<in> DisjointPathsPlusOne.paths_with_new paths P_new"   and "ys \<in> DisjointPathsPlusOne.paths_with_new paths P_new"   and "xs \<noteq> ys"   and "v \<in> set xs"   and "v \<in> set ys" shows "v = v0 \<or> v = v1 \<or> v = DisjointPathsPlusOne.new_last P_new"
loopform: shows "the (snd (usubstappp \<sigma> U \<alpha>**)) = the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))**"
assms_3: fixes l :: "Location" assumes "\<not> s\<turnstile> l reachable_from y" shows "P"
zero_le_double_add_iff_zero_le_single_add: fixes a :: "'a" shows "((0::'a) \<le> a + a) = ((0::'a) \<le> a)"
Dup_touches: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs" shows "only_touches i branch xs"
cap_def: shows "C.cap \<equiv> sum c (outgoing' C)"
lrev_body2_def: shows "lrev_body2 \<equiv> \<mu> lrev_body2. \<Lambda> r v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>ID) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(list2H\<cdot>(wrapH\<cdot>r\<cdot>xs) oo list2H\<cdot>(x :@ lnil))))"