hcju/isabelleps huggingface dataset:
forward: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND" shows "eval nid t t' 0 (\<box>\<^sub>b[\<lambda>nd. pout nd = bc nd]\<^sub>b)"
nrows_not_0: fixes A :: "(('d, 'e) vec, 'f) vec" shows "0 \<noteq> nrows A"
unshared_acquired_or_owned: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "all_unshared sb \<subseteq> all_acquired sb \<union> \<O>"
scaleR_prod_def: fixes r :: "real"   and A :: "'d \<times> 'e" shows "r *\<^sub>R A = (r *\<^sub>R fst A, r *\<^sub>R snd A)"
primroot_is_root: fixes x :: "'b list" assumes "x \<noteq> \<epsilon>" shows "x \<in> \<rho> x*"
diff_less_top_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(a - b < \<top>) = (a < \<top>)"
h2''_def: fixes z :: "complex" shows "h2'' z = h2' (z - of_int \<lfloor>Re z\<rfloor>)"
new_Addr: shows "new_Addr h = \<lfloor>oref\<rfloor>"
map_poly_zero: fixes f :: "'b \<Rightarrow> 'c"   and p :: "'b poly" assumes "\<forall>c. f c = (0::'c) \<longrightarrow> c = (0::'b)" shows "(map_poly f p = 0) = (p = 0)"
decr_grading_p_plus: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat"   and p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" shows "Quasi_PM_Power_Products.punit.decr_grading_p d n (p + q) = Quasi_PM_Power_Products.punit.decr_grading_p d n p + Quasi_PM_Power_Products.punit.decr_grading_p d n q"
filterlim_sequentially_Suc: fixes f :: "nat \<Rightarrow> 'a"   and F :: "'a filter" shows "(LIM x sequentially. f (Suc x) :> F) = filterlim f F sequentially"
c_in: shows "C \<in> {C \<in> N. \<not> I \<Turnstile> C}"
discrete_part: fixes X :: "'a set" assumes "discrete X" shows "part X"
tagged_division: fixes d :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a" assumes "d tagged_division_of cbox a b" shows "(\<Sum>(uu_, y)\<in>d. content y) = content (cbox a b)"
conga_sym_equiv: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" shows "(A B C CongA A' B' C') = (A' B' C' CongA A B C)"
H: shows "F g (\<Union> B) = (F \<circ> F) g B"
cot_def: shows "cot = (\<lambda>x. cos x / sin x)"
of: shows "class.bezout_ring (*) (1::'a) (+) (0::'a) (-) uminus"
fv\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('f, 'e) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
in_mult: shows "(M, N) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> R x y}"
negligible_convex_frontier: fixes S :: "'c set" assumes "convex S" shows "negligible (frontier S)"
eq_fin_indoc: fixes u :: "'item list"   and v :: "'item list"   and c :: "'item" assumes "u =\<^sub>F v"   and "indoc c u" shows "indoc c v"
subcomplex_pgallery: fixes Y :: "'a set set"   and Cs :: "'a set list" assumes "folding_g.ChamberSubcomplex Y"   and "SimplicialComplex.pmaxsimpchain Y Cs" shows "pgallery Cs"
assms_2: shows "a \<noteq> - \<infinity> \<or> b \<noteq> - \<infinity>"
collapse: fixes option :: "'g option" assumes "option \<noteq> None" shows "Some (the option) = option"
comp_id: fixes f :: "'f \<Rightarrow> 'g" shows "f \<circ> id = f"
intros_3: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "P x y"   and "list_emb P xs ys" shows "list_emb P (x # xs) (y # ys)"
effect_to_assignments_ii: fixes as :: "('c \<times> bool) list"   and op :: "'c strips_operator" assumes "as = effect_to_assignments op" obtains as\<^sub>1 :: "('c \<times> bool) list" and as\<^sub>2 :: "('c \<times> bool) list" where "as = as\<^sub>1 @ as\<^sub>2" and "as\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)" and "as\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)"
Sup_atMost: fixes y :: "'a" shows "\<Squnion>{..y} = y"
rbt_unionw_is_rbt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lt :: "('a, 'b) RBT_Impl.rbt"   and rt :: "('a, 'b) RBT_Impl.rbt"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.linorder less_eq less"   and "ord.is_rbt less lt"   and "ord.is_rbt less rt" shows "ord.is_rbt less (ord.rbt_union_with less f lt rt)"
leq_term_minus_Id: shows "leq_term - Id = {(x, y). x < y}"
top_on_Step: fixes X :: "char list set"   and f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a st option \<Rightarrow> 'a st option"   and g :: "bexp \<Rightarrow> 'a st option \<Rightarrow> 'a st option"   and C :: "'a st option acom"   and S :: "'a st option" assumes "\<And>x e S. \<lbrakk>top_on\<^sub>o S X; x \<notin> X; vars e \<subseteq> - X\<rbrakk> \<Longrightarrow> top_on\<^sub>o (f x e S) X"   and "\<And>b S. \<lbrakk>top_on\<^sub>o S X; vars b \<subseteq> - X\<rbrakk> \<Longrightarrow> top_on\<^sub>o (g b S) X"   and "vars C \<subseteq> - X"   and "top_on\<^sub>o S X"   and "top_on\<^sub>c C X" shows "top_on\<^sub>c (Step f g S C) X"
map_gpv_conv_map_gpv': fixes f :: "'q \<Rightarrow> 't"   and g :: "'r \<Rightarrow> 'u" shows "map_gpv f g = map_gpv' f g id"
proj_active_append: fixes n :: "nat"   and i :: "nat"   and n' :: "enat"   and t :: "cnf llist"   and c :: "'id" assumes "n \<le> i"   and "enat i < n'"   and "\<not> lfinite t \<or> n' - 1 < llength t"   and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"   and "\<forall>i'. n \<le> i' \<and> enat i' < n' \<and> enat i' < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i'\<^esub> \<longrightarrow> i' = i" shows "\<pi>\<^bsub>c\<^esub>ltake n' t = \<pi>\<^bsub>c\<^esub>ltake (enat n) t @\<^sub>l (\<sigma>\<^bsub>c\<^esub>lnth t i #\<^sub>l []\<^sub>l)"
mkarr_Lunit': fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.mkarr \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = \<F>C.\<ll>' (\<F>C.mkarr t)"
twos_complement: fixes x :: "'a word" shows "- x = word_succ (NOT x)"
S_eq: shows "S = count_space (X ` space M \<times> Y ` space M \<times> Z ` space M)"
hyps: assumes "abc_notfinal (abc_steps_l (0, lm) A n) A" shows "abc_steps_l (0, lm) (A @ B) n = abc_steps_l (0, lm) A n"
isCont_cdf: fixes x :: "real" shows "isCont M.C x = (M.prob {x} = 0)"
simps: fixes x :: "'a extT" shows "x\<^sup>T = x"
seq_char: fixes g :: "'c local.term set \<times> 'c local.term set"   and f :: "'c local.term set \<times> 'c local.term set" shows "CC.seq g f = (seq (fst g) (fst f) \<and> seq (snd g) (snd f))"
eq_upred_refl: fixes x :: "('e, 'd) uexpr" shows "(x =\<^sub>u x) = true"
View_View': shows "View = (\<lambda>vm vd vs. View' ((vm, vd), vs))"
mat2matofpoly_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "mat2matofpoly A = (\<chi>i j. [:A $ i $ j:])"
has_derivative_add_const: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and net :: "'a filter"   and c :: "'b" assumes "(f has_derivative f') net" shows "((\<lambda>x. f x + c) has_derivative f') net"
imirror_is_empty: fixes I :: "nat set" shows "(imirror I = {}) = (I = {})"
erlang_ith_moment_integrable: fixes l :: "real"   and X :: "'a \<Rightarrow> real"   and k :: "nat"   and i :: "nat" assumes "0 < l"   and "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))" shows "integrable M (\<lambda>x. X x ^ i)"
mult_comp_basic_support_set: fixes Mkt :: "('c, 'd) discrete_market"   and pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'c \<Rightarrow> real" assumes "stock_portfolio Mkt pf1" shows "stock_portfolio Mkt (qty_mult_comp pf1 qty)"
piecewise_differentiable_D1: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "g1 +++ g2 piecewise_differentiable_on {0..1}" shows "g1 piecewise_differentiable_on {0..1}"
max_eSuc_eSuc: fixes n :: "enat"   and m :: "enat" shows "max (eSuc n) (eSuc m) = eSuc (max n m)"
simps_101: fixes x81 :: "expr"   and x82 :: "char list"   and x83 :: "char list list"   and x4 :: "val" shows "x81\<bullet>x82{x83} \<noteq> Val x4"
matrix_transpose_mul: fixes A :: "(('a, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (A ** B) = Finite_Cartesian_Product.transpose B ** Finite_Cartesian_Product.transpose A"
deg_sect_singleton: fixes x :: "'b"   and d :: "nat" shows "deg_sect {x} d = {monomial d x}"
disc_iff_2: fixes n :: "enat"   and xs :: "'c llist" shows "(\<not> lnull (ltake n xs)) = (n \<noteq> 0 \<and> \<not> lnull xs)"
some_geodesic_same_endpoints: fixes x :: "'b"   and S :: "'b set" shows "{x--S--x} = {x}"
wt_gt_\<delta>\<^sub>h_if_superunary: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm" assumes "legal_zpassign A"   and "1 < arity_hd\<^sub>h (head s)" shows "zhmset_of \<delta>\<^sub>h < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
actsUpdate_def: fixes ec :: "'rep"   and A :: "('ma, 'mt) AlgState" shows "actsUpdate ec A \<equiv> update aOps ec (simAction a ec) (aActs A)"
prime_factors_power: fixes n :: "nat"   and x :: "'b" assumes "0 < n" shows "prime_factors (x ^ n) = prime_factors x"
inversion_is_antihomography: shows "is_antihomography inversion"
not_inf: shows "\<not> AlternativeSmallStep.inf \<Gamma> cs css s"
ldrop_eSuc_ltl: fixes n :: "nat"   and xs :: "'b llist" shows "ldropn (Suc n) xs = ldropn n (ltl xs)"
simps_2: fixes las :: "lock_action list" shows "must_acquire_lock (Lock # las) = True"
acc_modi_Protected_le: fixes a :: "acc_modi" assumes "Protected \<le> a" shows "a = Protected \<or> a = Public"
real_tendsto_sandwich: fixes f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real"   and net :: "'b filter"   and h :: "'b \<Rightarrow> real"   and c :: "real" assumes "\<forall>\<^sub>F n in net. f n \<le> g n"   and "\<forall>\<^sub>F n in net. g n \<le> h n"   and "(f \<longlongrightarrow> c) net"   and "(h \<longlongrightarrow> c) net" shows "(g \<longlongrightarrow> c) net"
prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "f (p ^ k) = f_prime_power p k"
cases: fixes a1 :: "('e, 'f, 'g) var0"   and a2 :: "('e, 'f, 'g) var0" assumes "a1 \<sim> a2" obtains x :: "('e, 'f, 'g) var0" where "a1 = xb__" and "a2 = xb__"   | a :: "'e" and x :: "'f" where "a1 = PMF (return_pmf (a__, xc__))" and "a2 = BPS (bsingleton (a__, xc__))"   | a :: "'e" and x :: "'f" where "a1 = BPS (bsingleton (a, x))" and "a2 = PMF (return_pmf (a, x))"
rep_eq: fixes x :: "complex_homo"   and xa :: "complex_homo" shows "dist_fs x xa = dist_fs_hcoords (rep_complex_homo x) (rep_complex_homo xa)"
invE1: fixes G :: "graph"   and S :: "state" assumes "Dijkstra.inv G S" shows "length (est S) = Dijkstra.size G \<and> known_dists G (known_set S) \<and> unknown_set S \<subseteq> verts G"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(f x = f y) = (x = y)"
wls_rawInduct_fresh: fixes vars :: "'varSort \<Rightarrow> 'var set"   and terms :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set set"   and abs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set set"   and envs :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option) set"   and phi :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "\<And>xs us s. (|vars xs| <o |UNIV| \<or> finite (vars xs)) \<and> (|terms s| <o |UNIV| \<or> finite (terms s)) \<and> (\<forall>X\<in>terms s. wls s X) \<and> (|abs (us, s)| <o |UNIV| \<or> finite (abs (us, s))) \<and> (\<forall>A\<in>abs (us, s). wlsAbs (us, s) A) \<and> (|envs| <o |UNIV| \<or> finite envs) \<and> Ball envs wlsEnv"   and "\<And>xs x. phi (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 (\<lambda>(us, s). phiAbs (us, s)) (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)"   and "\<And>s xs x X. \<lbrakk>isInBar (xs, s); wls s X; x \<notin> vars xs; \<And>s' Y. Y \<in> terms s' \<Longrightarrow> fresh xs x Y; \<And>us s' A. A \<in> abs (us, s') \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envs \<Longrightarrow> freshEnv xs x rho; phi s X\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs, s') A \<longrightarrow> phiAbs (xs, s') A)"
grid_part: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and p' :: "(nat \<times> int) list" assumes "d < length b"   and "p \<in> grid b {d}"   and "p' \<in> grid b {d}"   and "lv p d \<le> lv p' d"   and "ix p' d \<le> (ix p d + 1) * 2 ^ (lv p' d - lv p d)"   and "(ix p d - 1) * 2 ^ (lv p' d - lv p d) \<le> ix p' d" shows "p' \<in> grid p {d}"
arr_Par_comp_Par_id_Par_left: fixes \<alpha> :: "V"   and f :: "V"   and A :: "V" assumes "arr_Par \<alpha> f"   and "f\<lparr>ArrCod\<rparr> = A" shows "id_Rel A \<circ>\<^sub>R\<^sub>e\<^sub>l f = f"
set_iterator_union_foldli_conv: fixes as :: "'e list"   and bs :: "'e list" shows "set_iterator_union (foldli as) (foldli bs) = foldli (as @ bs)"
restrict_af_letter: fixes \<phi> :: "'b ltln"   and \<sigma> :: "'b set" shows "restricted_subformulas (af_letter \<phi> \<sigma>) = restricted_subformulas \<phi>"
real_polynomial_function_imp_sum: fixes f :: "real \<Rightarrow> real" assumes "real_polynomial_function f" shows "\<exists>a n. f = (\<lambda>x. \<Sum>i\<le>n. a i * x ^ i)"
digraph_isomorphism_def: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" shows "H.digraph_isomorphism hom \<equiv> wf_digraph (app_iso iso_f G) \<and> inj_on (iso_verts hom) (verts (app_iso iso_f G)) \<and> inj_on (iso_arcs hom) (arcs (app_iso iso_f G)) \<and> (\<forall>a\<in>arcs (app_iso iso_f G). iso_verts hom (tail (app_iso iso_f G) a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head (app_iso iso_f G) a) = iso_head hom (iso_arcs hom a))"
shift_prefix_cases: fixes xl :: "'b list"   and xs :: "'b stream"   and yl :: "'b list"   and ys :: "'b stream" assumes "xl @- xs = yl @- ys" shows "prefix xl yl \<or> prefix yl xl"
g: shows "g \<in> L \<rightarrow>\<^sub>M prob_algebra N"
lF_char_3: fixes f :: "'c" assumes "C.ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> F (trg\<^sub>C f) \<star>\<^sub>D \<mu> = (F \<i>\<^sub>C[trg\<^sub>C f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
minus_mod: fixes q :: "nat"   and b :: "nat"   and a :: "nat" assumes "secure_mult q"   and "b < a" shows "[a - b mod q = a - b] (mod q)"
dims_A'_pow_1: fixes ws :: "nat \<Rightarrow> real" shows "dim_row (A' ws) = last rs ^ N_half"
elims_3: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping"   and x :: "('a \<times> constraint) list"   and xa :: "'a set"   and xb :: "'b ns_constraint list \<times> 'c" assumes "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns"   and "\<not> Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.invariant_cs invariant_nsc to_ns x xa xb" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and ds :: "'b ns_constraint list" and s :: "'c" where "x = cs" and "xa = J" and "xb = (ds, s)" and "\<not> (ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s)"
rep_number_point_group_one: fixes x :: "'a" assumes "x \<in> \<V>" shows "card {g \<in> \<G>. x \<in> g} = 1"
abel_semigroup_set_axioms: shows "abel_semigroup_set (\<^bold>*) \<^bold>1"
exhaust: fixes y :: "('d, 'e) xstate" obtains x1 :: "'d" where "y = Normal x1"   | x2 :: "'d" where "y = Abrupt x2"   | x3 :: "'e" where "y = Fault x3"   | "y = Stuck"
less_eq_cblinfun_def: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "(A \<le> B) = (\<forall>\<psi>. \<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) \<le> \<psi> \<bullet>\<^sub>C (B *\<^sub>V \<psi>))"
assms: shows "r `\<^sub>\<circ> A = s `\<^sub>\<circ> B"
assms_8: shows "u \<circ> h \<circ> t = t \<circ> h \<circ> u"
1: shows "(\<Union>i. a \<inter> A i) = a \<inter> \<Union> (range A)"
UNlift_UNpart: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n" shows "UNlift cl dl P F ` UNpart cl P = UNpart dl (\<lambda>n. F n ` P n)"
A: fixes U :: "'b set" shows "ipurge_ref_aux I D U xs X = ipurge_ref_aux_foldr I D P U xs X"
add_mset_remove_trivial_If: fixes a :: "'k"   and N :: "'k multiset" shows "add_mset a (N - {#a#}) = (if a \<in># N then N else add_mset a N)"
invertible_iff_is_unit: fixes A :: "(('d, 'e) vec, 'e) vec" shows "invertible A = (det A dvd (1::'d))"
assms_1: shows "\<not> Option.is_none (devBC t n nid n')"
log_exp: fixes n :: "nat" shows "Discrete.log (2 ^ n) = n"
prems_1: fixes j :: "nat" assumes "j + 1 < length (x # xs)"   and "remdups_adj (x # xs) ! i = (x # xs) ! j"   and "remdups_adj (x # xs) ! (i + 1) = (x # xs) ! (j + 1)" shows "thesis"
aboveS_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
lgc_invar_lgc: fixes TNs :: "('f inference set \<times> ('f \<times> 'l) set) llist"   and i :: "enat" assumes "chain (\<leadsto>LGC) TNs"   and "active_subset (snd (lhd TNs)) = {}"   and "\<forall>\<iota>\<in>Inf_F. prems_of \<iota> = [] \<longrightarrow> \<iota> \<in> fst (lhd TNs)"   and "i < llength TNs" shows "lgc_invar TNs i"
OclIncludes_including_generic: fixes \<tau> :: "'a state \<times> 'a state"   and a :: "'a state \<times> 'a state \<Rightarrow> 'b"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and S :: "'a state \<times> 'a state \<Rightarrow> Set('b)" assumes "\<And>X x y. X->including\<^sub>S\<^sub>e\<^sub>t(x)->includes\<^sub>S\<^sub>e\<^sub>t(y) = (if \<delta> X then if x \<doteq> y then true else X->includes\<^sub>S\<^sub>e\<^sub>t(y) endif else invalid endif)"   and "\<And>x y. \<delta> (x \<doteq> y) = (\<upsilon> x and \<upsilon> y)"   and "\<tau> \<Turnstile> \<upsilon> a"   and "\<tau> \<Turnstile> \<upsilon> x"   and "\<tau> \<Turnstile> S->includes\<^sub>S\<^sub>e\<^sub>t(x)" shows "\<tau> \<Turnstile> S->including\<^sub>S\<^sub>e\<^sub>t(a)->includes\<^sub>S\<^sub>e\<^sub>t(x)"
Inf_Sec_def: fixes S :: "Sec set" shows "\<Sqinter> S = (if Low \<in> S then Low else High)"
logical_pseudo_reflexivity: fixes \<Gamma> :: "(name \<times> ty) list"   and \<Gamma>' :: "(name \<times> ty) list"   and t :: "(name \<times> trm) list"   and s :: "(name \<times> trm) list" assumes "\<Gamma>' \<turnstile> t is s over \<Gamma>" shows "\<Gamma>' \<turnstile> s is s over \<Gamma>"
one_less_mult: fixes n :: "nat"   and m :: "nat" assumes "Suc 0 < n"   and "Suc 0 < m" shows "Suc 0 < m * n"
inters_comm: fixes l :: "Lines"   and m :: "Lines" shows "inters l m = inters m l"
HFinite_star_of: fixes x :: "'a" shows "star_of x \<in> HFinite"
subsp: fixes a :: "'b" assumes "a \<in> space M" shows "prob_space (K a)"
simps: fixes ps :: "('b, 'c) fm list"   and i :: "'c" shows "nonempty (ps, i) = (ps \<noteq> [])"
ipurge_tr_aux_foldr_sinks_aux: fixes U :: "'e set"   and V :: "'e set"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "U \<subseteq> V" shows "sinks_aux I D U (ipurge_tr_aux_foldr I D P V xs) = U"
le_imp_less_or_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less_eq x y" shows "less x y \<or> x = y"
convex_affine_closure_Int: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "affine T"   and "rel_interior S \<inter> T \<noteq> {}" shows "closure (S \<inter> T) = closure S \<inter> T"
odd_nodes: shows "num_of_odd_nodes (rem_unPath ps G) = 0"
finite_field_mult_group_has_gen: fixes R :: "('a, 'b) ring_scheme" assumes "field R"   and "finite (carrier R)" shows "\<exists>a\<in>carrier (mult_of R). carrier (mult_of R) = {a [^]\<^bsub>R\<^esub> i |i. i \<in> UNIV}"
vdisjnt_vempty1: fixes A :: "V" shows "vdisjnt 0 A"
ASSUME_eq: shows "aASSUME \<equiv> iASSUME areturn"
sin_coeff_Suc: fixes n :: "nat" shows "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
OclValid_def: fixes \<tau> :: "'d state \<times> 'd state"   and P :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> P \<equiv> P \<tau> = true \<tau>"
InvGlobalRecordCount_def: fixes c :: "('c, 'd) configuration" shows "InvGlobalRecordCount c = (\<forall>q. c_records c = GlobalIncomingInfoAt c q + c_glob c q)"
l2_refines_l1: shows "refines (R12s \<inter> reach l1 \<times> (l2_inv1 \<inter> l2_inv2 \<inter> l2_inv3 \<inter> l2_inv4 \<inter> l2_inv4' \<inter> l2_inv5 \<inter> l2_inv6 \<inter> l2_inv7 \<inter> l2_inv8)) med12s l1 l2"
stirling_less: fixes n :: "nat"   and k :: "nat" assumes "n < k" shows "stirling n k = 0"
SV_inv4_inductive_1: shows "init sv_TS \<subseteq> SV_inv4"
closure_aff_dim: fixes S :: "'a set" shows "aff_dim (closure S) = aff_dim S"
elims_2: fixes x :: "'c \<times> 'd"   and xa :: "'c \<Rightarrow> 'd option" assumes "x \<in>\<^sub>m xa" obtains k :: "'c" and v :: "'d" and m :: "'c \<Rightarrow> 'd option" where "x = (k, v)" and "xa = m" and "m k = Some v"
that: fixes ss :: "'a list" assumes "distinct ss"   and "S = set ss" shows "thesis"
mk_disjoint_insert: fixes a :: "'b"   and A :: "'b set" assumes "(a::'b) \<in> (A::'b set)" shows "\<exists>B::'b set. (A::'b set) = (a::'b) \<triangleright> B \<and> a \<notin> B"
v11_vlrestriction: fixes r :: "V"   and A :: "V" assumes "v11 r" shows "v11 (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
coprime_r_nonzero: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list" assumes "factorize_polys ps = (fs, data)"   and "fs \<noteq> []"   and "fss = map real_of_rat_poly fs" shows "coprime_r fss \<noteq> 0"