hcju/isabelleps huggingface dataset:
sup_union: fixes a :: "'a"   and A :: "'a set"   and b :: "'a"   and B :: "'a set" assumes "a \<in> supremum A"   and "b \<in> supremum B" shows "supremum {a, b} = supremum (A \<union> B)"
univ_poly_one: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "\<one>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> = [\<one>\<^bsub>R\<^esub>]"
fps_exp_eq_fps_const_iff: fixes c :: "'a"   and c' :: "'a" shows "(fps_exp c = fps_const c') = (c = (0::'a) \<and> c' = (1::'a))"
eval_val: fixes t :: "db" assumes "val t" shows "t \<Down> t"
mset_set_set: fixes xs :: "'a list" assumes "distinct xs" shows "mset_set (set xs) = mset xs"
body2_def: fixes c_o :: "'c_o"   and c_a :: "'c_a"   and s :: "'s"   and j\<^sub>s :: "nat" shows "body2 c_o c_a s j\<^sub>s = exec_gpv (\<dagger>(oracle c_o) \<oplus>\<^sub>O eval_oracle' c_o c_a) (\<A> c_a) ((0, None), s) \<bind> (\<lambda>(uu_, (id, j\<^sub>0), s'). return_spmf (j\<^sub>0 = Some j\<^sub>s))"
norm_matrix_le_op_norm: fixes x :: "('a, 'b) vec"   and A :: "(('a, 'b) vec, 'c) vec" assumes "\<parallel>x\<parallel> = 1" shows "\<parallel>A *v x\<parallel> \<le> \<parallel>A\<parallel>\<^sub>o\<^sub>p"
sel_2: fixes ws :: "'d list stream" shows "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"
Par1F: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<parallel> Q"
Poincare_recurrence_thm_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "conservative M T"   and "A \<in> sets M" shows "A - qmpt.recurrent_subset_infty T A \<in> null_sets M"
af_subsequence_W: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "af (\<phi> W\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) or\<^sub>n (af \<phi> (prefix (Suc n) w) and\<^sub>n af (\<phi> W\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n]))"
assoc_is_natural_2: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "B.arr \<tau>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<tau>" shows "\<a>\<^sub>B[\<mu>, \<nu>, \<tau>] = \<a>\<^sub>B[B.cod \<mu>, B.cod \<nu>, B.cod \<tau>] \<cdot>\<^sub>B ((\<mu> \<star>\<^sub>B \<nu>) \<star>\<^sub>B \<tau>)"
continuous_closed_preimage_constant: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "continuous_on S f"   and "closed S" shows "closed {x \<in> S. f x = a}"
prems_1: shows "\<mu>_stable (\<phi>1 M\<^sub>n \<phi>2) w"
sum_coefs_def: fixes M :: "hmultiset" shows "sum_coefs M = size (hmsetmset M)"
gim: shows "g__ ` (affine hull T - K__) \<subseteq> rel_frontier U"
dim_times: fixes x :: "('b, 'a, 'c) Quantity_scheme"   and y :: "('b, 'a, 'c) Quantity_scheme" shows "Quantity.dim (x \<cdot> y) = Quantity.dim x \<cdot> Quantity.dim y"
measure_map_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf"   and A :: "'b set" shows "measure (measure_spmf (map_spmf f p)) A = measure (measure_spmf p) (f -` A)"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<t>[f] = null"
find_path1_gen_def: fixes E :: "('b \<times> 'b) set"   and u0 :: "'b"   and P :: "'b \<Rightarrow> bool" shows "find_path1_gen E u0 P \<equiv> find_path E (E `` {u0}) P \<bind> (\<lambda>res. case res of None \<Rightarrow> RETURN None | Some (p, v) \<Rightarrow> RETURN (Some (u0 # p, v)))"
apath_succ_decomp: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and x :: "'a"   and y :: "'a" assumes "sdG.apath u p v"   and "(x, y) \<in> set p"   and "y \<noteq> v" shows "\<exists>p1 z p2. p = p1 @ (x, y) # (y, z) # p2 \<and> x \<noteq> z \<and> y \<noteq> z"
insert_Diff_if: fixes x :: "'k"   and A :: "'k set"   and B :: "'k set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
parts_enc: fixes s' :: "(agent \<times> msg) set"   and K :: "key"   and X :: "msg"   and s :: "(agent \<times> msg) set"   and Y :: "msg" assumes "s' = insert (Spy, Crypt K X) s \<and> (Spy, X) \<in> s \<and> (Spy, Key K) \<in> s"   and "Y \<in> parts_msg X" shows "Y \<in> parts (used s)"
assms_3: fixes x :: "'a" shows "h x (0::'b) (0::'c) = (0::'d)"
mix_pmf_strict_preferred_independence: fixes r :: "'a pmf"   and \<alpha> :: "real"   and p :: "'a pmf"   and q :: "'a pmf" assumes "r \<in> local.\<P>"   and "\<alpha> \<in> {0<..1}"   and "p \<succeq>[\<R>] q \<and> (q, p) \<notin> \<R>" shows "mix_pmf \<alpha> p r \<succeq>[\<R>] mix_pmf \<alpha> q r \<and> (mix_pmf \<alpha> q r, mix_pmf \<alpha> p r) \<notin> \<R>"
ln_ge_zero_imp_ge_one: fixes x :: "real" assumes "0 \<le> ln x"   and "0 < x" shows "1 \<le> x"
orthogonal_comp_subset: fixes U :: "'a set" shows "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
nat_less_omega: fixes n :: "nat" shows "ordinal_of_nat n < \<omega>"
mapl_F_def: fixes l1 :: "'k \<Rightarrow> 'u"   and l2 :: "'l \<Rightarrow> 'v"   and l3 :: "'m \<Rightarrow> 'w" shows "mapl_F l1 l2 l3 = map_F l1 l2 l3 id id id id id id"
deriv: fixes x :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> real" assumes "x \<in> src.diff_fun_space"   and "y \<in> src.diff_fun_space" shows "X (x * y) = x p * X y + y p * X x"
homotopy_eqv_empty2: fixes S :: "'c set" shows "({} homotopy_eqv S) = (S = {})"
L_def: fixes T :: "'a stream \<Rightarrow> enat"   and x :: "'a" shows "L T x = lim_stream x \<bind> (\<lambda>\<omega>. case T \<omega> of enat i \<Rightarrow> distr (lim_stream (\<omega> !! i)) (stream_space M) ((@-) (stake (Suc i) \<omega>)) | \<infinity> \<Rightarrow> return (stream_space M) \<omega>)"
cdib: shows "d\<inverse> O b \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
borel_measurable_iff_halfspace_less: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)"
set_delete_aux: fixes xs :: "('c \<times> 'd) list"   and k :: "'c" assumes "distinct (map fst xs)" shows "set (delete_aux k xs) = set xs - {k} \<times> UNIV"
word_log_esimps_1: fixes x :: "'b word" shows "x && 0 = 0"
ListI: fixes x :: "('b, unit) Old_Datatype.node set" assumes "x \<in> list (range Leaf)" shows "x \<in> List"
discrCf_eff_indis: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and i :: "nat" assumes "discrCf cf"   and "i < brn (fst cf)" shows "snd cf \<approx> snd (cont_eff cf i)"
ins_size_2: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Unbal t'" shows "size t' = size t + 1"
iT_Minus_Plus_cut_assoc: fixes k :: "nat"   and I :: "nat set"   and n :: "nat" shows "k \<ominus> I \<oplus> n = k + n \<ominus> I \<down>\<le> k"
oseq_invariant_ctermI: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "((nat \<Rightarrow> 'f) \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and i :: "nat"   and P :: "(nat \<Rightarrow> 'f) \<times> 'h \<Rightarrow> bool"   and S :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> 'g seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "simple_labels \<Gamma>"   and "automaton.trans A = oseqp_sos \<Gamma> i"   and "\<And>\<sigma> p l. \<lbrakk>(\<sigma>, p) \<in> init A; l \<in> labels \<Gamma> p\<rbrakk> \<Longrightarrow> P (\<sigma>, l)"   and "\<And>\<sigma> \<sigma>' p l. \<lbrakk>(\<sigma>, p) \<in> oreachable A S U; l \<in> labels \<Gamma> p; P (\<sigma>, l); U \<sigma> \<sigma>'\<rbrakk> \<Longrightarrow> P (\<sigma>', l)"   and "\<And>p l \<sigma> a q l' \<sigma>' pp. \<lbrakk>p \<in> cterms \<Gamma>; l \<in> labels \<Gamma> p; P (\<sigma>, l); ((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i; ((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A; l' \<in> labels \<Gamma> q; (\<sigma>, pp) \<in> oreachable A S U; p \<in> sterms \<Gamma> pp; (\<sigma>', q) \<in> oreachable A S U; S \<sigma> \<sigma>' a\<rbrakk> \<Longrightarrow> P (\<sigma>', l')" shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
mat_ext_vars1: fixes A :: "'a mat" shows "mat_extension dims vars1 A = tensor_P A (1\<^sub>m K)"
dfs_match_not_in: fixes xs :: "('c, 'd) bintree option list"   and a :: "'c"   and f :: "('c, 'd) bintree \<Rightarrow> 'e" assumes "\<forall>t. Some t \<in> set xs \<longrightarrow> priority t \<noteq> a" shows "set (dfs f (map (match a) xs)) = set (dfs f xs)"
ovalidNF_K_bind_wp: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'c" assumes "ovalidNF P f Q" shows "ovalidNF P (K_bind f x) Q"
assms_2: shows "x \<cdot> y \<cdot> z \<le> z \<cdot> w"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "(dual xa \<le> dual x) = (x \<le> xa)"
a\<^sub>u: shows "alphabet (splitLeaf t w\<^sub>a a w\<^sub>b b) = alphabet u"
intros_3: fixes xs :: "'b list"   and ys :: "'b list"   and z :: "'b" assumes "xs <~~> ys" shows "z # xs <~~> z # ys"
approx_coarsen: fixes x :: "real"   and a1 :: "real"   and eps1 :: "real"   and a2 :: "real"   and eps2 :: "real" assumes "\<bar>x - a1\<bar> \<le> eps1"   and "\<bar>a1 - a2\<bar> \<le> eps2 - eps1" shows "\<bar>x - a2\<bar> \<le> eps2"
Collect_empty_eq_bot: fixes P :: "'c \<Rightarrow> bool" shows "(Collect P = {}) = (P = bot)"
inv_comp: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c" assumes "C.VV.iso f"   and "C.VV.iso g"   and "C.VV.seq g f" shows "C.VV.inv (C.VV.comp g f) = C.VV.comp (C.VV.inv f) (C.VV.inv g)"
abs_summable_countable: fixes \<mu> :: "'c \<Rightarrow> 'd"   and T :: "'c set" assumes "\<mu> abs_summable_on T" shows "countable {x \<in> T. (0::'d) \<noteq> \<mu> x}"
error_free_Some: fixes x :: "abrupt"   and s :: "st" assumes "\<nexists>err. x = Error err" shows "error_free (Some x, s)"
in_measure: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<Rightarrow> nat" shows "((x, y) \<in> Wellfounded.measure f) = (f x < f y)"
monad_state_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_state return bind get put \<equiv> monad return bind \<and> monad_state_axioms bind get put"
suc: fixes x :: "ordinal" shows "F x < F (oSuc x)"
cyclic_group_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<otimes> b \<otimes> c = a \<otimes> (b \<otimes> c)"
subcls1_wfD: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "P \<turnstile> C \<prec>\<^sup>1 D"   and "wf_prog wf_md P" shows "D \<noteq> C \<and> \<not> (subcls1 P)\<^sup>+\<^sup>+ D C"
sorted_list_of_set_sort_remdups: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq (set xs) = linorder.sort_key less_eq (\<lambda>x. x) (remdups xs)"
one: shows "bv_to_nat xs = bv_to_nat ys"
xor_replicate_bot_right: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs \<le> n"   and "x = \<bottom>" shows "xs [\<oplus>] replicate n x = xs"
g_def: fixes s :: "('St, 'a, 'b) state"   and s' :: "('St, 'c, 'd) state" shows "g s s' \<equiv> \<forall>u1. u1 \<in> GL \<longrightarrow> out (local.getGMState s) u1 = out (local.getGMState s') u1"
ac_Zp_is_Unit: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "ac_Zp x \<in> Units Zp"
trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "sublist a b"   and "sublist b c" shows "sublist a c"
transp_on_map: fixes Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "transp_on Q B"   and "h ` A \<subseteq> B" shows "transp_on (\<lambda>x y. Q (h x) (h y)) A"
distinct_5: fixes x2 :: "label" shows "Cont x2 \<noteq> Ret"
nop_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i = nop_type NOP" shows "snd (nop_instr i s) = False"
resOutputTauTrans: fixes x :: "name"   and a :: "name"   and b :: "name"   and P :: "pi"   and P' :: "pi" assumes "<\<nu>x>a{b}.P \<longmapsto> \<tau> \<prec> P'" shows "False"
coerce_eq_prj: shows "COERCE(udom, 'e) = PRJ('e)"
symp_llist_all2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "symp R" shows "symp (llist_all2 R)"
simps_4: fixes f :: "form"   and g :: "form" shows "fv (FDisj f g) = fv f @ fv g"
that: fixes i :: "'b" assumes "i \<in># K" shows "f i \<le> g i"
has_limits_of_shape_def: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "A.has_limits_of_shape J \<equiv> \<forall>D. diagram J (\<cdot>\<^sub>A) D \<longrightarrow> (\<exists>a \<chi>. limit_cone J (\<cdot>\<^sub>A) D a \<chi>)"
eval_euler_approx_aux_1: shows "euler_approx_aux 0 = 1"
stream_enc_dec: fixes s :: "('a \<times> bool list) stream"   and n :: "nat"   and FO :: "nat set" assumes "\<exists>n. sdrop n (smap fst s) = sconst any"   and "stream_all (\<lambda>x. length (snd x) = n) s"   and "\<forall>i\<in>FO. \<exists>!p. snd (s !! p) ! i" shows "stream_enc (dec_word s, stream_dec n FO s) = s"
adjoint_dim_col: fixes A :: "'b mat" shows "dim_col (adjoint A) = dim_row A"
raw_converter_invariant_interaction_any_bounded_converter: fixes n :: "enat" shows "raw_converter_invariant \<I>_full \<I>_full run_converter (\<lambda>conv. interaction_any_bounded_converter conv n)"
rtrancl_mono: fixes r :: "('c \<times> 'c) set"   and s :: "('c \<times> 'c) set" assumes "r \<subseteq> s" shows "r\<^sup>* \<subseteq> s\<^sup>*"
sum_list_map_cons: fixes y :: "'a"   and ys :: "'a list" shows "sum_list (map weight (y # ys)) = weight y + sum_list (map weight ys)"
fmdrop_fset_fmupd: fixes x :: "'a"   and y :: "'b"   and A :: "'a fset"   and f :: "('a, 'b) fmap" shows "(fmdrop_fset A f)(x $$:= y) = fmdrop_fset (A |-| {|x|}) f(x $$:= y)"
15: shows "(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>"
Atm_transC_invert: fixes atm :: "'atom"   and s :: "'state"   and cf :: "('test, 'atom) com \<times> 'state"   and phi :: "bool" assumes "(Atm atm, s) \<rightarrow>c cf" shows "phi"
mod_mult_self2_is_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a b) b = zero"
AssertionStatImpTrans: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatImp SImp \<Psi> \<Psi>'"   and "assertionAux.AssertionStatImp SImp \<Psi>' \<Psi>''" shows "assertionAux.AssertionStatImp SImp \<Psi> \<Psi>''"
assms_2: shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"
assms_1: shows "rel_stacktraces (Fubx_get F2) xs ys opt"
column_def: fixes j :: "'f"   and A :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "column j A = (\<chi>i. A $h i $h j)"
min_to_max_set: shows "min_set = (`) \<partial> \<circ> max_set \<circ> (`) \<partial>"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "tail G a \<rightarrow>\<^sup>*\<^bsub>bidel_arc.H G M a\<^esub> head G a" shows "bidel_arc_biconnected_axioms G M a"
triple_judgement_def: fixes pre :: "'a \<Rightarrow> bool"   and body :: "'b"   and property :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "triple_judgement pre body property = (\<forall>s. pre s \<longrightarrow> property s body)"
map: shows "map_of ms M = \<lfloor>(Ts, T, m)\<rfloor>"
assms_3: shows "ground_clause (subst_cl (cl_ecl D) \<theta>)"
UP_ring_subring_nat_pow: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "g [^]\<^bsub>UP R\<^esub> n = g [^]\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> n"
scaleR_eq_self_cancel: fixes a :: "real"   and x :: "'a" shows "(a *\<^sub>R x = x) = (a = 1 \<or> x = (0::'a))"
subtopology_eq_discrete_topology_sing: fixes X :: "'b topology"   and a :: "'b" shows "(X = discrete_topology {a}) = (topspace X = {a})"
full_checker_l_full_checker': shows "(uncurry2 full_checker_l, uncurry2 full_checker) \<in> (fully_unsorted_poly_rel O mset_poly_rel \<times>\<^sub>r \<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel\<rangle>fmap_rel) \<times>\<^sub>r \<langle>p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)\<rangle>list_rel \<rightarrow>\<^sub>f \<langle>code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel\<rangle>nres_rel"
oa_facts_5: fixes v :: "i"   and x :: "\<nu>" shows "[\<^bold>\<diamond>\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<box>\<lparr>O!,x\<^sup>P\<rparr> in v]"
step_nonneg: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and i :: "nat"   and x :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "i < k"   and "x\<^sub>1 \<le> x" shows "0 \<le> bs ! i * x + (hs ! i) x"
sel_2: fixes x21 :: "'c"   and x22 :: "'c llist" shows "un_LCons2 (LCons x21 x22) = x22"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "seq g f"
enum_filter_finals_def: fixes succs :: "graph \<Rightarrow> graph list" shows "enum_filter_finals succs = worklist_tree_coll_trie succs final fgraph"
ring_distribs_2: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "(a + b) * c = a * c + b * c"
empty_neq_Literal: fixes b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and s :: "literal" shows "empty_literal \<noteq> Literal b0 b1 b2 b3 b4 b5 b6 s"
regular_UNION: fixes r :: "('d \<times> 'd) set"   and A :: "'e set"   and F :: "'e \<Rightarrow> 'f set" assumes "regular r"   and "|A| <o r"   and "\<And>a. a \<in> A \<Longrightarrow> |F a| <o r" shows "|\<Union> (F ` A)| <o r"
gen_model_rels_tc_worlds: fixes w' :: "'e"   and w'' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and w :: "'e"   and as :: "'f set" assumes "(w', w'') \<in> (\<Union> (relations (gen_model M w) ` as))\<^sup>+" shows "w'' \<in> worlds (gen_model M w)"
length_def: fixes as :: "'c iarray" shows "IArray.length as = length (IArray.list_of as)"
sdrop_sappend: fixes i :: "Integer"   and xs :: "[:'r:]"   and ys :: "[:'r:]" shows "sdrop\<cdot>i\<cdot>(xs :@ ys) = sdrop\<cdot>i\<cdot>xs :@ sdrop\<cdot>(i - slength\<cdot>xs)\<cdot>ys"
lists_Int_eq: fixes A :: "'b set"   and B :: "'b set" shows "lists (A \<inter> B) = lists A \<inter> lists B"
Collect_iMin_le: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "P k" shows "iMin {x. P x} \<le> k"
bisimE_1: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<sim> Q" shows "P \<leadsto>[bisim] Q"
Or\<^sub>n_And\<^sub>n_image_semantics: fixes \<A> :: "'b ltln set set"   and w :: "nat \<Rightarrow> 'b set" assumes "finite \<A>"   and "\<And>\<Phi>. \<Phi> \<in> \<A> \<Longrightarrow> finite \<Phi>" shows "w \<Turnstile>\<^sub>n Or\<^sub>n (And\<^sub>n ` \<A>) = (\<exists>\<Phi>\<in>\<A>. \<forall>\<phi>\<in>\<Phi>. w \<Turnstile>\<^sub>n \<phi>)"
arctan_minus: fixes x :: "real" shows "arctan (- x) = - arctan x"
A: shows "g_add m1 m2 = iterate_add_to_map m1 update (iteratei m2)"
un_TM2_def: fixes tm :: "'a tm" shows "un_TM2 tm = (case tm of ??.Time_Monad.tm.TM x1 x2 \<Rightarrow> x2)"
simps_2: fixes e :: "'c exp"   and es :: "'c exp list" shows "ss_exps (e # es) = (case val_of e of None \<Rightarrow> \<lfloor>ss_exp e\<rfloor> | \<lfloor>v\<rfloor> \<Rightarrow> ss_exps es)"
real_root_pos_pos_le: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "0 \<le> root n x"
lmeasure_integral_UNIV: fixes S :: "'c set" assumes "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue S = integral UNIV (indicat_real S)"
continuous_on_p: fixes A :: "real set"   and s :: "'a"   and s' :: "'a" shows "continuous_on A (p s s')"
inj_imp_inv_eq: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'i \<Rightarrow> 'h" assumes "inj f"   and "\<forall>x. f (g x) = x" shows "inv f = g"
Collect_mem_eq: fixes A :: "'x set" shows "{x. x \<in> A} = A"
preallocatedE: fixes h :: "heap"   and C :: "char list"   and P :: "heap \<Rightarrow> char list \<Rightarrow> bool" assumes "preallocated h"   and "C \<in> sys_xcpts"   and "\<And>fs. h (addr_of_sys_xcpt C) = \<lfloor>(C, fs)\<rfloor> \<Longrightarrow> P h C" shows "P h C"
lenlex_nsets_2_eq: fixes A :: "nat list set" shows "[A]\<^bsup>2\<^esup> = {{x, y} |x y. x \<in> A \<and> y \<in> A \<and> (x, y) \<in> lenlex less_than}"
asig_comp_def: fixes a1 :: "'e set \<times> 'e set \<times> 'e set"   and a2 :: "'e set \<times> 'e set \<times> 'e set" shows "asig_comp a1 a2 \<equiv> (inputs a1 \<union> inputs a2 - (outputs a1 \<union> outputs a2), outputs a1 \<union> outputs a2, internals a1 \<union> internals a2)"
poly_square_subst: fixes f :: "'a poly" shows "poly_square_subst (f \<circ>\<^sub>p Polynomial.monom (1::'a) 2) = f"
reduction: fixes p :: "nat"   and k :: "nat"   and j :: "nat" assumes "prime p"   and "0 < k"   and "k * k = p * (j * j)" shows "k < p * j \<and> 0 < j"
diff_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "minus x y = inf x (uminus y)"
l2polR_eq: fixes N :: "((int \<times> int) set set, int) Combinators list" assumes "N \<noteq> []" shows "C (list2policyR N) = list2policy (map C N)"
sub_block_id: fixes block :: "('b, 'c) fm list \<times> 'c" shows "sub_block id block = block"
bound: fixes z :: "'a"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "bound z n f"   and "n < m" shows "f m = z"
fml: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "(\<nu> \<in> fml_sem I (FUrename x y \<phi>)) = (Radj x y \<nu> \<in> fml_sem I \<phi>)"
exhaust_sel: fixes inference :: "'b inference" obtains "inference = Infer (side_prems_of inference) (main_prem_of inference) (concl_of inference)"
a1: shows "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) y) y = Some yb \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = Some yc"
assms_3: shows "k < length (\<Phi>\<inverse> \<Pi> \<A> t)"