hcju/isabelleps huggingface dataset:
exec_until_bad_fixp_induct: fixes P :: "(('b, 'd, 'e) gpv \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> (('b \<times> 'f) \<times> 'b \<times> 'g) spmf) \<Rightarrow> bool"   and bad1 :: "'f \<Rightarrow> bool"   and bad2 :: "'g \<Rightarrow> bool"   and oracle1 :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and oracle2 :: "'g \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'g) spmf"   and joint_oracle :: "'f \<times> 'g \<Rightarrow> 'd \<Rightarrow> (('e \<times> 'f) \<times> 'e \<times> 'g) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>gpv s1 s2. f ((gpv, s1), s2)))"   and "P (\<lambda>_ _ _. return_pmf None)"   and "\<And>exec_until_bad'. P exec_until_bad' \<Longrightarrow> P (\<lambda>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' (f x) s1' s2')))" shows "P (exec_until_bad joint_oracle oracle1 bad1 oracle2 bad2)"
flow_has_vector_derivative: fixes t :: "real"   and x0 :: "'c" assumes "t \<in> existence_ivl0 x0" shows "(flow0 x0 has_vector_derivative ode (flow0 x0 t)) (at t)"
scheme_def: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f" shows "scheme R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str \<equiv> comm_ring R addition multiplication zero unit \<and> locally_ringed_space X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str \<and> scheme_axioms R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str"
simps: fixes a1 :: "nat"   and a2 :: "'a tree23" shows "full a1 a2 = (a1 = 0 \<and> a2 = Empty \<or> (\<exists>n l r p. a1 = Suc n \<and> a2 = Branch2 l p r \<and> full n l \<and> full n r) \<or> (\<exists>n l m r p q. a1 = Suc n \<and> a2 = Branch3 l p m q r \<and> full n l \<and> full n m \<and> full n r))"
simps: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" shows "add_all_deducible_clauses R S = S \<union> all_deducible_clauses R S"
bind_envT_def: shows "Monad_Overloading.bind = bind_env Monad_Overloading.bind"
frequently_mp: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and F :: "'c filter" assumes "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x"   and "frequently P F" shows "frequently Q F"
Liminf_le_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "F \<noteq> \<bottom>" shows "Liminf F f \<le> Limsup F f"
apply_power_left_mult_order: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a" shows "(f ^ (n * order f a)) \<langle>$\<rangle> a = a"
par_assum_def: shows "par_assum \<equiv> \<lambda>(pre, rely). {c. snd (c ! 0) \<in> pre \<and> (\<forall>i. Suc i < length c \<longrightarrow> c ! i -pe\<rightarrow> c ! Suc i \<longrightarrow> (snd (c ! i), snd (c ! Suc i)) \<in> rely)}"
simps: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list" shows "(Graph V \<Lambda> E)^E = E"
Assoc_List_impl_of: fixes al :: "('b, 'c) assoc_list" shows "Assoc_List (assoc_list.impl_of al) = al"
closure_of_subset_eq: fixes S :: "'b set"   and X :: "'b topology" shows "(S \<subseteq> topspace X \<and> X closure_of S \<subseteq> S) = closedin X S"
compute_is_float_nonneg: fixes m :: "int"   and e :: "int" shows "is_float_nonneg (Float m e) = (0 \<le> m)"
simps_7: shows "[0]\<^sub>1 +\<^sub>1 \<infinity>\<^sub>1 = \<infinity>\<^sub>1"
path_cons_elim: fixes a :: "'transition"   and r :: "'transition list"   and p :: "'state" assumes "path (a # r) p" obtains "enabled a p" and "path r (execute a p)"
has_integral_spike_interior_eq: fixes a :: "'c"   and b :: "'c"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x" shows "(f has_integral y) (cbox a b) = (g has_integral y) (cbox a b)"
U_2: shows "{u \<in> V. real_of_int m \<le> u x} = \<Union> U"
id_1: shows "ias = Atoms (preprocess' (map (Pair ()) nsl) (start_fresh_variable (map (Pair ()) nsl)))"
spectral_radius_1_jnf_max_axioms: shows "spectral_radius_1_jnf_max A n m lam n_as"
surj_def: fixes f :: "'b \<Rightarrow> 'a" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
ring_simprules_20: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> x) = x"
le_post: fixes c :: "'a acom"   and d :: "'a acom" assumes "c \<le> d" shows "post c \<le> post d"
FinDimVectorSpace_axioms: shows "FinDimVectorSpace (\<cdot>) V"
cComm1: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and b :: "name"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "!P \<longmapsto> a[b] \<prec> P''"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a[b] \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> P'[x::=b] \<parallel> P'')"
disjoint_insert_1: fixes B :: "'c set"   and a :: "'c"   and A :: "'c set" shows "(B \<inter> ??.Set.insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
recvmsg_simps_5: fixes P :: "'d \<Rightarrow> bool"   and m :: "'d" shows "recvmsg P (send m) = True"
is_free_simps_3: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c"   and u :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_free unfree (app t u)"
Form_Body_WW: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "Form_Body ka kb xs ys zs" shows "zs \<in> WW"
sig_red_single_regular_lt: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single (\<prec>\<^sub>t) top_tail p q f t" shows "lt q = lt p"
inf_par_product: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "par_distrib (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) ((a::'b::refinement_lattice) \<sqinter> (b::'b::refinement_lattice)) ((c::'b::refinement_lattice) \<sqinter> (d::'b::refinement_lattice)) = par a c \<sqinter> par a d \<sqinter> par b c \<sqinter> par b d"
freshEnv_def: fixes xs :: "'f"   and x :: "'g"   and rho :: "'f \<Rightarrow> 'g \<Rightarrow> ('h, 'i, 'f, 'g, 'j) qTerm set option" shows "freshEnv xs x rho \<equiv> rho xs x = None \<and> (\<forall>ys. liftAll (fresh xs x) (rho ys))"
in_scc_of_self: fixes u :: "'a" assumes "u \<in> verts H" shows "u \<in> H.scc_of u"
ff': shows "\<exists>m m'. incl m \<and> incl m' \<and> seq m f \<and> seq m' f' \<and> m \<cdot> f = m' \<cdot> f'"
assms: shows "(X, Y1 + Y2) \<in> multpw ns"
prj_simps_9: shows "C.cod \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0 = \<rho>.apex"
a: shows "m' \<in> set (normalize_primitive_extract (disc, sel) C f m)"
max_ext_wf: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "wf (max_ext r)"
delete_edge_def: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "delete_edge v e v' g \<equiv> \<lparr>nodes = nodes g, edges = edges g - {(v, e, v')}\<rparr>"
u: shows "u \<in> poincare_circle z1 r \<inter> poincare_circle z2 r"
map_transform_delete: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "map_transform t ae (delete x \<Gamma>) = delete x (map_transform t ae \<Gamma>)"
normalize_left_idem: fixes a :: "'a"   and b :: "'a" shows "gcd (normalize a) b = gcd a b"
perm_inputs: fixes V'' :: "('e \<times> 'f) list set"   and V :: "'e list set"   and M :: "('e, 'f, 'g) FSM"   and vs :: "('e \<times> 'f) list" assumes "V'' \<in> Perm V M"   and "vs \<in> V''" shows "map fst vs \<in> V"
assms_1: shows "uverts H\<^sub>1 \<subseteq> S"
scaleR_nth_eucl: fixes c :: "real"   and x :: "'b"   and i :: "nat" shows "(c *\<^sub>R x) $\<^sub>e i = c *\<^sub>R x $\<^sub>e i"
measurable_expr_sem_eq: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and \<sigma> :: "nat \<Rightarrow> val"   and V :: "nat set" assumes "\<Gamma> \<turnstile> e : t"   and "\<sigma> \<in> space (state_measure V \<Gamma>)"   and "free_vars e \<subseteq> V" shows "(\<rightarrow>\<^sub>M) (expr_sem \<sigma> e) = (\<rightarrow>\<^sub>M) (stock_measure t)"
special_soundness_def: shows "local.\<Sigma>_protocols_base.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub \<longrightarrow> e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> e \<noteq> e' \<longrightarrow> check h a e z \<longrightarrow> check h a e' z' \<longrightarrow> lossless_spmf (ss_adversary h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (ss_adversary h (a, e, z) (a, e', z')). (h, w') \<in> R)"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t1 :: "taylor_model"   and t2 :: "taylor_model" shows "tm_max prec I a t1 t2 = tm_union prec I a t1 t2"
IH: assumes "i < length Vs"   and "Vs ! i = x"   and "distinct Vs" shows "find_indices x Vs = [i]"
assignment: shows "i' \<equiv> i + (j - \<ff> s j + 1)"
mod_double_modulus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "'a"   and x :: "'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step"   and "less zero m"   and "less_eq zero x" shows "modulo x (times (numeral.numeral one plus (num.Bit0 num.One)) m) = modulo x m \<or> modulo x (times (numeral.numeral one plus (num.Bit0 num.One)) m) = plus (modulo x m) m"
\<phi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "inC \<in> C.hom b a \<rightarrow> HomC.set (b, a)"
assms_1: shows "lambda C -quasi_isometry_on {a..b} c"
bind_get: fixes f :: "'s \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (get f) g = get (\<lambda>s. bind (f s) g)"
equivalence_one_closed: shows "matrix_stone_relation_algebra.equivalence mone"
cases: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'d \<times> 'f"   and a2 :: "'e \<times> 'g" assumes "rel_prod R1 R2 a1 a2" obtains a :: "'d" and b :: "'e" and c :: "'f" and d :: "'g" where "a1 = (a, c)" and "a2 = (b, d)" and "R1 a b" and "R2 c d"
ord_less_eq_trans: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a <s b"   and "b = c" shows "a <s c"
set_of_sub_inc_left: fixes A :: "'b interval"   and A' :: "'b interval"   and B :: "'b interval" assumes "set_of A \<subseteq> set_of A'" shows "set_of (A - B) \<subseteq> set_of (A' - B)"
isinP_set_trieP: fixes t :: "trieP"   and xs :: "bool list" shows "isinP t xs = (xs \<in> set_trieP t)"
seqsim_suffix_seqsim: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a" assumes "s \<approx> t" shows "nextsuffix s \<approx> nextsuffix t"
prec_st_simp: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'b" shows "prec_st p r u x = (\<forall>n. (\<forall>i<n. inpt_st r (u i) (u (Suc i)) (x i)) \<longrightarrow> p (u n) (u (Suc n)) (x n))"
add_mset_subseteq_single_iff: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b" shows "(a, M \<subseteq># b, {#}) = (M = {#} \<and> a = b)"
less_not_refl3: fixes s :: "nat"   and t :: "nat" assumes "s < t" shows "s \<noteq> t"
a_star_redu_OrR1_elim: fixes a :: "coname"   and M :: "trm"   and y :: "coname"   and R :: "trm" assumes "OrR1 <a>.M y \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = OrR1 <a>.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
of_int_0_le_iff: fixes z :: "int" shows "((0::'a) \<le> of_int z) = (0 \<le> z)"
ltl_const_equiv_lt_ltl_prop_equiv: shows "(\<sim>\<^sub>C) \<le> (\<sim>\<^sub>P)"
step_preserves_le: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "'av Abs_State.st option"   and c :: "(char list \<Rightarrow> int) set acom"   and c' :: "'av Abs_State.st option acom" assumes "S \<subseteq> \<gamma>\<^sub>o S'"   and "c \<le> \<gamma>\<^sub>c c'" shows "step S c \<le> \<gamma>\<^sub>c (step' S' c')"
vmatch_eq: fixes p :: "Term_Class.term"   and v :: "value" assumes "linear p"   and "vconstructor_value v" shows "rel_option env_eq (vmatch (mk_pat p) v) (match p (value_to_sterm v))"
topo_open: fixes t :: "'a llist set" shows "(t open) = (t \<in> topo B)"
UNIV_I: fixes x :: "'a::type" shows "(x::'a::type) \<in> UNIV"
new_character_data_ptr_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "new_character_data_ptr |\<in>| character_data_ptr_kinds h'"
swap_triple: fixes a :: "atom"   and b :: "atom"   and c :: "atom" assumes "a \<noteq> b"   and "c \<noteq> b"   and "sort_of a = sort_of b"   and "sort_of b = sort_of c" shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
affine_Int: fixes s :: "'b set"   and t :: "'b set" assumes "affine s"   and "affine t" shows "affine (s \<inter> t)"
ODnods_segment_inc_ODord: fixes D :: "'c Order"   and y :: "'c Order set" assumes "Worder D"   and "y \<in> segment ODnods (ordinal_number D)" shows "y \<sqsubset> ordinal_number D"
j_2: shows "\<forall>i\<in>Field r. (i, j) \<in> r"
hom_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id x \<in> carrier (relative_homology_group (int n) (nsphere n) (upper n))"
mat_kernelD_2: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> mat_kernel A" shows "A *\<^sub>v v = 0\<^sub>v nr"
sel: fixes x :: "'a" shows "run_writer (WriterT x) = x"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "loose \<Gamma>" obtains "\<And>f. \<lbrakk>current \<Gamma> f; wave \<Gamma> f\<rbrakk> \<Longrightarrow> f = (\<lambda>_. 0)" and "\<not> hindrance \<Gamma> (\<lambda>_. 0)"
int_pow_diff: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "int"   and m :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> (n - m) = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> m)"
simps_9: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "'a"   and vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a" shows "(v ::= va {vb} \<sqsubseteq> {vc} WHILE vd DO ve {vf}) = False"
InterI: fixes C :: "'b set set"   and A :: "'b" assumes "\<And>X. X \<in> C \<Longrightarrow> A \<in> X" shows "A \<in> \<Inter> C"
elims: fixes x :: "value option"   and xa :: "value option"   and y :: "trilean" assumes "value_eq x xa = y" obtains uu_ :: "value option" where "x = None" and "xa = uu_" and "y = invalid"   | v :: "value" where "x = Some v" and "xa = None" and "y = invalid"   | a :: "value" and b :: "value" where "x = Some a" and "xa = Some b" and "y = (if a = b then true else false)"
type_copy_map_cong0: fixes M :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e"   and N :: "'f \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'f"   and f :: "'c \<Rightarrow> 'g" assumes "M (g x) = N (h x)" shows "(f \<circ> M \<circ> g) x = (f \<circ> N \<circ> h) x"
assms: shows "x \<in> Singleton_for_d.match (gfp_F ds)"
inner_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
dfs_def: fixes E :: "('b \<times> 'b) set"   and src :: "'b"   and tgt :: "'b" shows "dfs E src tgt \<equiv> REC\<^sub>T (\<lambda>dfs (V, v). if v = tgt then RETURN (V, True) else let V = insert v V in FOREACH\<^sub>C (E `` {v}) (\<lambda>(V, brk). \<not> brk) (\<lambda>v' (V, brk). if v' \<notin> V then dfs (V, v') else RETURN (V, False)) (V, False)) ({}, src) \<bind> (\<lambda>(uu_, r). RETURN r)"
maximalD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "real" assumes "maximal t"   and "0 \<le> c" shows "t (\<lambda>_. c) = (\<lambda>_. c)"
hashcode_array_def: fixes a :: "'b array" shows "hashcode a = foldl (\<lambda>h v. h * 33 + hashcode v) 0 (list_of_array a)"
bottom_join: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "weak_bounded_lattice L"   and "x \<in> carrier L" shows "\<bottom>\<^bsub>L\<^esub> \<squnion>\<^bsub>L\<^esub> x .=\<^bsub>L\<^esub> x"
assms: assumes "{x} \<in> \<G>" shows "2 < card \<G>"
strengthen_pre: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>l s. P' l s \<longrightarrow> P l s"   and "\<turnstile>\<^sub>t {P} c {Q}" shows "\<turnstile>\<^sub>t {P'} c {Q}"
disjoint_image_subset: fixes \<A> :: "'b set set"   and f :: "'b set \<Rightarrow> 'b set" assumes "disjoint \<A>"   and "\<And>X. X \<in> \<A> \<Longrightarrow> f X \<subseteq> X" shows "disjoint (f ` \<A>)"
WP1: fixes a :: "('b, 'c) Combinators"   and list :: "('b, 'c) Combinators list" assumes "a \<notin> set list" shows "a \<notin> set (removeShadowRules2 list)"
Sigma_mono: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and D :: "'b \<Rightarrow> 'c set" assumes "A \<subseteq> C"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x" shows "Sigma A B \<subseteq> Sigma C D"
cut_le_Max_all: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I \<le> t" shows "I \<down>\<le> t = I"
alg_closedI_reducible: assumes "\<And>p. \<lbrakk>1 < degree p; monic p; coeff p 0 \<noteq> (0::'b)\<rbrakk> \<Longrightarrow> \<not> irreducible p" shows "OFCLASS('b, alg_closed_field_class)"
gbinomial_int_nonneg: fixes x :: "int"   and k :: "nat" assumes "0 \<le> x" shows "0 \<le> x gchoose k"
functions_of: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" assumes "finite A"   and "finite B"   and "number_partition (card A) N"   and "size N \<le> card B" shows "functions_of A B N \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B"
complement_product_in_sum_complement: fixes x :: "i"   and y :: "i" assumes "x \<oplus> y \<noteq> u" shows "P (\<midarrow> x \<otimes> \<midarrow> y) (\<midarrow> (x \<oplus> y))"
prod_mOp_func: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_mOp I A \<in> carr_prodag I A \<rightarrow> carr_prodag I A"
inner_diff_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (minus y z) = inner x y - inner x z"
open_openin: fixes S :: "'b set" shows "open S = openin euclidean S"
transI: fixes r :: "('d \<times> 'd) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
Expr2ZFType1: fixes e :: "(ZF, ZF) Expression" assumes "ZFType (Expr2ZF e) = 1" shows "\<exists>f e'. e = f E@ e' \<and> Suc (ZFDepth (Expr2ZF e')) = ZFDepth (Expr2ZF e)"
positive_resolvent_is_model_res: shows "less_restrictive ordered_positive_resolvent (ordered_model_resolvent {})"
append_same_eq: fixes ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
dep_vars_Cons_cdep: fixes V :: "'var"   and a :: "'edge"   and as' :: "'edge list"   and n'' :: "'node"   and as'' :: "'edge list"   and n' :: "'node" assumes "V \<in> Use (sourcenode a)"   and "sourcenode a -a # as'\<rightarrow>\<^sub>c\<^sub>d n''"   and "n'' -as''\<rightarrow>\<^sub>d* n'" shows "dependent_live_varsp n' V [] (a # as' @ as'')"
multiplicity_def: fixes p :: "'a"   and x :: "'a" shows "multiplicity p x = (if finite {n. p ^ n dvd x} then Max {n. p ^ n dvd x} else 0)"
Subspace_eqdim_imp_equal: fixes U :: "'b set" assumes "Subspace U"   and "dim_R U = dim_R V" shows "U = V"
Q2: shows "Q2 \<in> carrier_mat (dim_col A) (dim_col A)"
assms: shows "finite (reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0)"
disj_one_left: fixes x :: "'a" shows "- (1::'a) OR x = - (1::'a)"
assms_4: shows "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) (b \<ominus> a)"
Complexs_of_complex: fixes r :: "complex" shows "of_complex r \<in> \<complex>"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_same a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt t = wt s \<and> head t = head s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)))"
\<alpha>_\<beta>: shows "leftderives \<alpha> \<beta>"
prems_6: shows "snd (cs (S t j) cid) = Done"
vec_lambda_eta: fixes g :: "('c, 'd) vec" shows "(\<chi>i. g $ i) = g"