hcju/isabelleps huggingface dataset:
simplePath_empty_conv: fixes s :: "nat"   and t :: "nat" shows "f.cf.isSimplePath s [] t = (s = t)"
dim_ty_sem_Length_def: fixes uu :: "L itself" shows "dim_ty_sem uu = \<^bold>L"
f9a: shows "\<not> (\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
coeff_monom_Suc: fixes a :: "'b"   and d :: "nat"   and p :: "'b poly"   and i :: "nat" shows "coeff (monom a (Suc d) * p) (Suc i) = coeff (monom a d * p) i"
valid_cfg_state_in_S: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg" shows "state cfg \<in> \<S>"
chop_ge_length_eq: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n"   and "xs \<noteq> []"   and "length xs \<le> n" shows "chop n xs = [xs]"
emptyD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Empty]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Undecided"
exhaust: fixes y :: "'b dual" obtains x :: "'b" where "y = \<partial> x"
image_subset_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
linear_surjective_isomorphism: fixes f :: "'b \<Rightarrow> 'b" assumes "linear (*s) (*s) f"   and "surj f"   and "dim UNIV = dim UNIV" shows "\<exists>f'. linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
min_elems3: fixes s :: "'a event list"   and c :: "'a event"   and P :: "'a process" assumes "s @ [c] \<in> D P"   and "s @ [c] \<notin> min_elems (D P)" shows "s \<in> D P"
adv: assumes "\<And>\<eta>. exception_\<I> (\<I>_middle \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g absorb (\<A> \<eta>) (obsf_converter (sim_outer \<eta> |\<^sub>= 1\<^sub>C)) \<surd>" shows "Negligible.negligible (\<lambda>\<eta>. advantage (absorb (\<A> \<eta>) (obsf_converter (sim_outer \<eta> |\<^sub>= 1\<^sub>C))) (obsf_resource (sim_inner \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal \<eta>)) (obsf_resource (middle \<eta>)))"
Achamber_F: shows "SimplicialComplex.maxsimp A F"
congruence_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "congruence x = (equivalence_relation x \<and> compatible_with_structure x)"
recursion: fixes va :: "'a" assumes "va \<in> V\<^bsub>G'\<^esub>" shows "va \<in> G'.winning_region p"
CACH_IF: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and pc :: "int"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and n :: "nat" assumes "ins_is C m l (iftrue pc)"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "Anno\<down>l = None"   and "derivAssum G C m pc (Cachera n)"   and "derivAssum G C m (l + 1) (Cachera n)" shows "deriv G C m l (Cachera n)"
check_ipresIGSwapAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGSwapAll check check (errMOD MOD) MOD"
prob_space_distr: fixes f :: "nat \<Rightarrow>\<^sub>F 'a \<Rightarrow> 'e"   and na :: "nat"   and M' :: "'e measure" assumes "P'.random_variable na M' f" shows "prob_space (distr (P' na) M' f)"
top: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and x :: "'b" assumes "class.SL_top le join Top" shows "le x Top"
assms_8: shows "\<forall>(k, \<gamma>)\<in>one_chain1. valid_path \<gamma>"
isDag_def: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref" shows "isDag p l r = (\<exists>t. Dag p l r t)"
map_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and V :: "'b \<Rightarrow> 'd"   and f :: "'b local.term set" assumes "FreeMonoidalCategory.evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V" shows "FreeMonoidalCategory.evaluation_functor.map C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V f \<equiv> if partial_magma.arr (free_monoidal_category.comp C) f then evaluation_map.eval D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V (free_monoidal_category.rep C f) else partial_magma.null D"
cases: fixes x :: "nat \<times> nat ifex \<times> bdd" obtains bdd :: "bdd" where "x = (0, Falseif, bdd__)"   | bdd :: "bdd" where "x = (Suc 0, Trueif, bdda__)"   | n :: "nat" and v :: "nat" and t :: "nat ifex" and e :: "nat ifex" and bdd :: "bdd" where "x = (Suc (Suc n), IF va__ t e, bdd)"   | va :: "nat" and uw_ :: "bdd" where "x = (Suc (Suc vaa__), Trueif, uw___)"   | va :: "nat" and uw_ :: "bdd" where "x = (Suc (Suc vab__), Falseif, uwa___)"   | v :: "nat" and uw_ :: "bdd" where "x = (Suc vb__, Falseif, uwb___)"   | va :: "nat" and vb :: "nat ifex" and vc :: "nat ifex" and uw_ :: "bdd" where "x = (Suc 0, IF vac__ vba__ vc, uwc___)"   | uw_ :: "bdd" where "x = (0, Trueif, uwd___)"   | va :: "nat" and uw_ :: "bdd" where "x = (Suc (Suc vad__), Trueif, uwe___)"   | v :: "nat" and va :: "nat ifex" and vb :: "nat ifex" and uw_ :: "bdd" where "x = (0, IF vd__ vae__ vbb__, uwf___)"   | v :: "nat" and va :: "nat ifex" and vb :: "nat ifex" and uw_ :: "bdd" where "x = (Suc 0, IF v va vb, uw_)"
new_omit_branch: fixes p :: "('b, 'c) fm"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "new p a branch" shows "new p a (omit_branch xs branch)"
prerun_valid_def: fixes sys :: "('f, 'g, 'h, 'i, 'j, 'k) pre_system_scheme"   and \<phi> :: "('f, 'g, 'h, 'i, 'j) system_state \<Rightarrow> bool" shows "(sys \<Turnstile>\<^bsub>pre\<^esub> \<phi>) = (\<forall>\<sigma>. prerun sys \<sigma> \<longrightarrow> (\<box>\<lceil>\<phi>\<rceil>) \<sigma>)"
vint_struct: shows "\<Z>_vfsequence \<alpha> \<SS>\<^sub>\<int>"
dom_simp: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.dom f = local.dom f"
ord_Sab: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S_a"   and "b \<in> S_b" shows "a < b"
sorted_wrt_nth_less: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and i :: "nat"   and j :: "nat" assumes "sorted_wrt P xs"   and "i < j"   and "j < length xs" shows "P (xs ! i) (xs ! j)"
set_integrable_Gamma_hurwitz_aux2: fixes r :: "real"   and a :: "real"   and s :: "complex" assumes "0 < r"   and "0 < a" shows "complex_set_integrable lborel {r..} (\<lambda>x. complex_of_real x powr - s * complex_of_real (exp (- a * x)) / (1 - exp (- complex_of_real x)))"
dist_triangle_add: fixes x :: "'b"   and y :: "'b"   and x' :: "'b"   and y' :: "'b" shows "dist (x + y) (x' + y') \<le> dist x x' + dist y y'"
power_increasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and N :: "nat"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq one a" shows "less_eq (power.power one times a n) (power.power one times a N)"
assms_2: shows "q \<in> dterms \<Gamma> p"
avl_balR: fixes l :: "('c \<times> nat) tree"   and r :: "('c \<times> nat) tree"   and a :: "'c" assumes "avl l"   and "avl r"   and "height l - 1 \<le> height r \<and> height r \<le> height l + 2" shows "avl (balR l a r)"
comparator_key_compare: fixes ko :: "'a key_order" shows "comparator (key_compare ko)"
sup_red_inf_in_red_liminf: fixes Ns :: "'f set llist" assumes "chain (\<rhd>) Ns" shows "Sup_llist (lmap Red_I Ns) \<subseteq> Red_I (Liminf_llist Ns)"
pCons_2: fixes i :: "nat" shows "coeff (f \<circ>\<^sub>p monom (1::'a) n) (n * i + j) = (if j = 0 then coeff f i else (0::'a))"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind" shows "monad_commute_axioms bind"
partList4: fixes s :: "'b set"   and ts :: "'b set list" shows "partList4 s ts = partList3 s ts"
map_graph_selectors_2: fixes f :: "('d \<times> 'c) set"   and G :: "('b, 'd) labeled_graph" shows "edges (map_graph f G) = on_triple f `` edges G"
swap_cols: fixes A :: "((real, 'n) vec, 'n) vec"   and m :: "'n"   and n :: "'n" assumes "P A"   and "m \<noteq> n" shows "P (\<chi>i j. A $ i $ Fun.swap m n id j)"
simps_1: fixes m :: "nat" shows "bin_rep_aux 0 m = [m]"
simps_6: fixes uw :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and E' :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and b :: "bexp"   and C :: "acom" shows "time ({(uw, E', E)} WHILE b DO C) = E"
find_consistent_signs_at_roots_thm_R: fixes p :: "real poly"   and qs :: "real poly list" shows "find_consistent_signs_at_roots_R p qs = snd (snd (calculate_data_R p qs))"
arr_dom: fixes f :: "'a" assumes "L.arr f" shows "L.arr (L.dom f)"
nmp_eq: fixes x :: "i"   and y :: "i" shows "PP x y = (P x y \<and> \<not> P y x)"
add_mono1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less a b" shows "less (plus a one) (plus b one)"
proj3_invalidate: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat" shows "\<pi>\<^sub>3 (the (invalidate rt dests dip)) = \<pi>\<^sub>3 (the (rt dip))"
DomainI: fixes a :: "'g"   and b :: "'h"   and r :: "('g \<times> 'h) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
attach_c1f22_c1f22_def: fixes cnv1 :: "('t, 'ab, 'af + 'ag, 'ah + 'ai) converter"   and cnv2 :: "('u, 'ac, 'aj + 'ak, 'al + 'am) converter"   and res1 :: "(('p + 'r) + ('af + 'aj) + 'v, ('x + 'z) + ('ah + 'al) + 'ad) resource"   and res2 :: "(('q + 's) + ('ag + 'ak) + 'w, ('y + 'aa) + ('ai + 'am) + 'ae) resource" shows "attach_c1f22_c1f22 cnv1 cnv2 res1 res2 = ((1\<^sub>C |\<^sub>= 1\<^sub>C) |\<^sub>= wiring_c1r22_c1r22 cnv1 cnv2 |\<^sub>= 1\<^sub>C) \<odot> fused_wiring \<rhd> res1 \<parallel> res2"
4: shows "bprv (imp (subst \<phi> \<langle>\<psi>\<rangle> xx) (subst \<chi> \<langle>\<chi>\<rangle> xx))"
cycle_def: fixes xs :: "'a list" shows "cycle xs \<equiv> path xs \<and> 2 < length xs \<and> last xs \<rightarrow> hd xs"
simps_1: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and set_error_code :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s :: "'a"   and d :: "'d" assumes "Controllable_Interruptible_Separation_Kernel kstep output_f s0 current cswitch kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting set_error_code" shows "Controllable_Interruptible_Separation_Kernel.CISK_control aborting waiting set_error_code s d [] = (None, [], s)"
simps_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" assumes "class.wellorder less_eq less" shows "wellorder.enumerate less_eq S 0 = ord.Least less_eq (\<lambda>n. n \<in> S)"
eWls_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and s :: "'sort" shows "eWls MOD s ERR = False"
ext_inject: fixes unit_sys :: "'a"   and more :: "'b"   and unit_sys' :: "'a"   and more' :: "'b" shows "(\<lparr>unit_sys = unit_sys, \<dots> = more\<rparr> = \<lparr>unit_sys = unit_sys', \<dots> = more'\<rparr>) = (unit_sys = unit_sys' \<and> more = more')"
vars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('e, 'd) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
member_bind: fixes x :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b set" shows "(x \<in> A \<bind> f) = (x \<in> \<Union> (f ` A))"
unlabel_nil_only_if_nil: fixes A :: "('b \<times> 'a) list" assumes "unlabel A = []" shows "A = []"
XD_smallest_def: fixes ds :: "'b set"   and X :: "'a set" shows "XD_smallest ds X = - (XH_largest ds X - X)"
set_cond_pmf: fixes p :: "'e pmf"   and s :: "'e set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "set_pmf (cond_pmf p s) = set_pmf p \<inter> s"
sum_Basis_sum_nth_Basis_list: fixes f :: "'b \<Rightarrow> 'a" shows "sum f Basis = (\<Sum>i<DIM('b). f (Basis_list ! i))"
bind_assoc: fixes x :: "'m"   and f :: "unit \<Rightarrow> 'm"   and g :: "unit \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
iprev_eq_if_neq_iMin: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "iprev a I = iprev b I"   and "iMin I \<noteq> a"   and "iMin I \<noteq> b" shows "a = b"
hyps: shows "sqrt (real_of_float x) < real_of_float (sqrt_iteration prec n x)"
remove_key_add: fixes v :: "'b"   and m :: "'b \<Rightarrow>\<^sub>0 'c"   and m' :: "'b \<Rightarrow>\<^sub>0 'c" shows "remove_key v m + remove_key v m' = remove_key v (m + m')"
power_down_le: fixes x :: "real"   and n :: "nat"   and z :: "real"   and p :: "nat" assumes "0 \<le> x"   and "x ^ n \<le> z" shows "power_down p x n \<le> z"
updVal_eqBut_eq: fixes val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and val' :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and ys :: "'varSort"   and y :: "'var"   and sY :: "'b" assumes "eqBut val val' ys y" shows "val (y := sY)_ys = val' (y := sY)_ys"
tautology_monotonous: fixes C :: "'a Literal set"   and D :: "'a Literal set" assumes "tautology C"   and "C \<subseteq> D" shows "tautology D"
assms_3: shows "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"
linear_subspace_kernel: fixes f :: "'d \<Rightarrow> 'e" assumes "clinear f" shows "csubspace {x. f x = (0::'e)}"
simps_2: fixes s :: "'b word"   and e :: "'b word" shows "wi2l (WordInterval s e) = (if e < s then [] else [(s, e)])"
Gmod3_def: shows "Gmod3 = G\<lparr>carrier := H \<inter> K\<rparr> Mod (H1 \<inter> K <#> H \<inter> K1)"
two: shows "(\<lambda>f\<in>Hom A B. set_func (F \<^bsub>\<a>\<^esub> f) (set_func (u A) (Id A))) = (\<lambda>f\<in>Hom A B. set_func (F \<^bsub>\<a>\<^esub> f \<odot> u A) (Id A))"
dom_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "partial_magma.dom C f = (if partial_magma.domains C f \<noteq> {} then SOME a. a \<in> partial_magma.domains C f else partial_magma.null C)"
b2a_def: shows "b2a = b / ((2::'a) * a)"
sum: fixes g :: "'d \<Rightarrow> 'b"   and S :: "'d set" shows "f' y x (sum g S) = (\<Sum>a\<in>S. f' y x (g a))"
sinks_aux_single_event: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and x :: "'b" shows "sinks_aux I D U [x] = (if \<exists>v\<in>U. (v, D x) \<in> I then insert (D x) U else U)"
hdomain_shift_add: fixes f :: "hf"   and delta :: "hf" shows "hdomain (shift f delta) = RepFun (hdomain f) ((@+) delta)"
positively_invariant_barrier_domain: fixes D :: "'a set"   and V :: "'a \<Rightarrow> real"   and V' :: "'a \<Rightarrow> 'a \<Rightarrow> real" assumes "positively_invariant D"   and "\<And>x. (V has_derivative V' x) (at x)"   and "continuous_on D (\<lambda>x. V' x (f x))"   and "\<And>s. \<lbrakk>s \<in> D; V s = 0\<rbrakk> \<Longrightarrow> V' s (f s) < 0" shows "positively_invariant (D \<inter> {x. V x \<le> 0})"
affine_dependent_biggerset: fixes s :: "'c set" assumes "finite s"   and "DIM('c) + 2 \<le> card s" shows "affine_dependent s"
bipartite_web_of_sel_2: fixes \<Gamma> :: "('a, 'b) web_scheme"   and uv :: "'a + 'a"   and u :: "'a" shows "edge (bipartite_web_of \<Gamma>) uv (Inl u) = False"
right_plus_circ: fixes x :: "'a" shows "(x\<^sup>\<star> * x)\<^sup>\<star> = x\<^sup>\<star>"
zdvd_antisym_nonneg: fixes m :: "int"   and n :: "int" assumes "0 \<le> m"   and "0 \<le> n"   and "m dvd n"   and "n dvd m" shows "m = n"
map_disj_None_left: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and x :: "'c" assumes "h\<^sub>0 \<bottom> h\<^sub>1"   and "x \<in> dom h\<^sub>1" shows "h\<^sub>0 x = None"
\<phi>_uniqueness_1: fixes \<mu> :: "'a" assumes "\<guillemotleft>\<mu> : k \<star> p\<^sub>1 \<Rightarrow> h \<star> p\<^sub>0\<guillemotright>" shows "\<mu> = \<phi>"
saturated_def: fixes S :: "(('b, 'c) fm list \<times> 'c) set" shows "saturated S \<equiv> \<forall>p i. (\<^bold>\<diamond> p) at i in' S \<longrightarrow> (\<nexists>a. p = Nom a) \<longrightarrow> (\<exists>j. (\<^bold>@ j p) at i in' S \<and> (\<^bold>\<diamond> Nom j) at i in' S)"
1: shows "\<forall>x. \<exists>k. \<forall>m>k. eint x < val_Zp_dist (s m) a"
max: fixes H' :: "('a, 'b) pre_digraph" assumes "subgraph H' G" shows "verts H' \<noteq> verts H \<or> arcs H' \<subseteq> arcs H"
g_def: shows "g \<equiv> \<lambda>y. b + (4 / ((norm y)\<^sup>2 + 4)) *\<^sub>R (y - 2 *\<^sub>R b)"
nvT_vars: fixes x :: "Preliminaries.var"   and T :: "'a trm" assumes "x \<in> nvT T" shows "x \<in> vars T"
hyps_5: shows "x\<^sub>1 \<noteq> x\<^sub>2"
sorted_list_of_set_append: fixes xs :: "'a set"   and ys :: "'a set" assumes "finite xs"   and "finite ys"   and "\<forall>x\<in>xs. \<forall>y\<in>ys. x < y" shows "sorted_list_of_set (xs \<union> ys) = sorted_list_of_set xs @ sorted_list_of_set ys"
unapp_app: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" shows "unapp (app u\<^sub>1 u\<^sub>2) = Some (u\<^sub>1, u\<^sub>2)"
simps_3: fixes va :: "(val \<times> val) fset"   and v :: "nat" shows "VFun va \<squnion> VNat v = None"
word_of_int_minus: fixes i :: "int" shows "word_of_int (2 ^ LENGTH('b) - i) = word_of_int (- i)"
homotopic_paths_sym: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q" shows "homotopic_paths s q p"
domintros_2: fixes es :: "exp0 list"   and uv :: "(char list, char list) id0 option" assumes "\<And>y. y \<in> set es \<Longrightarrow> is_value_dom y" shows "is_value_dom (Con uv es)"
2: shows "contour_integral (linepath b c) f = contour_integral (linepath b a') f + contour_integral (linepath a' c) f"
even_odd_mirror_ap_tree: fixes even :: "bool"   and f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "even_odd_mirror even (f \<diamondop> x) = even_odd_mirror even f \<diamondop> even_odd_mirror even x"
hom_one: shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (upper n)\<^esub>"
has_fields_fun: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and FDTs' :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C has_fields FDTs'" shows "FDTs' = FDTs"
Cidl: fixes f :: "'b" assumes "f \<in> mor" shows "id (dom f) ;; f = f"
insert_remdups: fixes x :: "'c"   and xs :: "'c list" shows "List.insert x (remdups xs) = remdups (List.insert x xs)"
is_prefix_unsplit: fixes u :: "'a list"   and a :: "'a list" assumes "is_prefix u a" shows "u @ drop (length u) a = a"
from_to_view\<^sub>h: fixes x :: "(view_metadata blindable\<^sub>h \<times> view_data blindable\<^sub>h) rose_tree\<^sub>h" shows "view_merkle.from_view\<^sub>h (view_merkle.to_view\<^sub>h x) = x"
vec_lambda_beta: fixes g :: "'f \<Rightarrow> 'e"   and i :: "'f" shows "vec_lambda g $ i = g i"
sylow_thm: shows "\<exists>H. subgroup H G \<and> card H = q ^ 1"
strict_trans2: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a dvd b \<and> a \<noteq> b"   and "b dvd c" shows "a dvd c \<and> a \<noteq> c"
2: fixes x :: "('b, 'a) trie"   and ks' :: "'b list" shows "lookup_trie (update_trie ks v x) ks' = (if ks = ks' then Some v else lookup_trie x ks')"
finite_imp_less_Inf: fixes X :: "'c set"   and x :: "'c"   and a :: "'c" assumes "finite X"   and "x \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> a < x" shows "a < Inf X"
ideal_prod_simp: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "I = I <+> I \<cdot> J"
Extend_subset: fixes S :: "'b fm set"   and C :: "'b fm set set"   and f :: "nat \<Rightarrow> 'b fm" shows "S \<subseteq> Extend S C f"
encode_initial_state_def: fixes \<Pi> :: "'a strips_problem" shows "\<Phi>\<^sub>I \<Pi> \<equiv> let I = \<Pi>\<^sub>I; vs = \<Pi>\<^sub>\<V> in \<^bold>\<And>map (\<lambda>v. encode_state_variable 0 (index vs v) (I v) \<^bold>\<or> \<bottom>) (filter (\<lambda>v. I v \<noteq> None) vs)"
t_A_def: fixes n :: "nat" shows "t_A n = c_A n + p_A n"
domain_iff_prime: fixes a :: "'a" assumes "a \<in> carrier R - {\<zero>}" shows "domain (R Quot PIdl a) = ring_prime a"