hcju/isabelleps huggingface dataset:
coherent_Assoc_Ide: fixes a :: "'c local.term"   and b :: "'c local.term"   and c :: "'c local.term" assumes "Ide a"   and "Ide b"   and "Ide c"   and "Src a = Trg b"   and "Src b = Trg c" shows "coherent \<^bold>\<a>\<^bold>[a, b, c\<^bold>]"
residue_add_assoc: fixes p :: "int"   and k :: "nat"   and x :: "int"   and y :: "int"   and z :: "int" assumes "padic_integers p" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> z = x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> (y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> z)"
inject_2: fixes x21 :: "nat"   and x22 :: "v list"   and y21 :: "nat"   and y22 :: "v list" shows "(RSBreak x21 x22 = RSBreak y21 y22) = (x21 = y21 \<and> x22 = y22)"
Solve_def: fixes X :: "'b list set"   and ES :: "('b list set \<times> 'b trm set) set" shows "Solve X ES \<equiv> while Cond (Iter X) ES"
lift_character_def: shows "H.lift_character = (\<lambda>(\<chi>, z) xa. if xa \<in> adjoin G {\<one>} x then \<chi> (fst (H.unadjoin xa)) * z ^ snd (H.unadjoin xa) else 0)"
embedS_defs_2: shows "embedS \<equiv> \<lambda>r r' f. embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
is_final_dstate_funpow_imp_deterministic_RP_neq_None: fixes k :: "nat"   and St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" assumes "is_final_dstate ((deterministic_RP_step ^^ k) St)" shows "deterministic_RP St \<noteq> None"
gram_schmidt_sub_result: fixes us :: "'a vec list"   and ws :: "'a vec list"   and us' :: "'a vec list" assumes "gram_schmidt_sub n us ws = us'"   and "set ws \<subseteq> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct (us @ ws)"   and "lin_indpt (set (us @ ws))"   and "corthogonal us" shows "set us' \<subseteq> carrier_vec n \<and> distinct us' \<and> corthogonal us' \<and> local.span (set (us @ ws)) = local.span (set us') \<and> length us' = length us + length ws"
under_Under_singl: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "under r a = Under r {a}"
prefI: fixes p :: "'b list"   and s :: "'b list"   and w :: "'b list" assumes "p \<cdot> s = w" shows "p \<le>p w"
ide_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.ide \<nu>"   and "B.ide \<mu>"   and "src \<nu> = trg \<mu>" shows "B.ide (\<nu> \<star> \<mu>)"
distinct_delete_index: fixes xs :: "'b list"   and n :: "nat" assumes "distinct xs" shows "distinct (delete_index n xs)"
set_of_real_interval_subset: fixes x :: "float interval"   and y :: "float interval" assumes "set_of x \<subseteq> set_of y" shows "set_of (real_interval x) \<subseteq> set_of (real_interval y)"
ldsj: fixes \<phi>s :: "'fmla list" assumes "set \<phi>s \<subseteq> fmla" shows "ldsj \<phi>s \<in> fmla"
exhaust: fixes y :: "('d, 'c) Term.term" obtains x1 :: "'c" where "y = Var x1"   | x21 :: "'d" and x22 :: "('d, 'c) Term.term list" where "y = Fun x21 x22"
map_of_SomeD: fixes xs :: "('b \<times> 'a) list"   and k :: "'b"   and y :: "'a" assumes "map_of xs k = Some y" shows "(k, y) \<in> set xs"
implSpec: shows "Impl = new_configured_list_SecurityInvariant m C description"
INV: shows "cscc_invar v0 D0 (l, p, D, pE)"
translation_invert: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "(+) a ` A = (+) a ` B" shows "A = B"
not_eq_newpkt_rerr: fixes dests :: "nat \<Rightarrow> nat option"   and sip :: "nat" shows "\<not> eq_newpkt (Rerr dests sip)"
A: shows "A = B + \<i> \<cdot>\<^sub>m C"
not_irreducible_one: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.irreducible times one zero one"
Merge_def: fixes A :: "'a\<^sub>m set" shows "Merge A = (if A = {} \<or> infinite A then None else Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (Some (SOME x. x \<in> A)) A)"
irreducible_mult_unit_right: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and p :: "'b" assumes "class.idom times one plus zero minus uminus"   and "dvd.dvd times a one" shows "comm_semiring_1.irreducible times one zero (times p a) = comm_semiring_1.irreducible times one zero p"
Inter_ZeroL: fixes r :: "'a language" shows "Inter Zero r = Zero"
fun_upd_finfun: fixes y :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'d" shows "(y(a := b) \<in> finfun) = (y \<in> finfun)"
comp_arr_dom: fixes f :: "'d"   and a :: "'d" assumes "D.arr f"   and "D.dom f = a" shows "f \<cdot>\<^sub>D a = f"
Pi_mono: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and C :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi A B \<subseteq> Pi A C"
convert_convertDerivative: fixes var :: "nat"   and p :: "real mpoly"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and a :: "real"   and b :: "real"   and c :: "real" assumes "convert_poly var p (xs' @ x # xs) = Some (a, b, c)"   and "length xs' = var" shows "eval (convertDerivative var p) (xs' @ x # xs) = evalUni (convertDerivativeUni (a, b, c)) x"
summable_zero_on_nonneg: fixes f :: "nat \<Rightarrow> 'c"   and X :: "nat set" assumes "summable f"   and "\<And>x. (0::'c) \<le> f x" shows "summable (zero_on f X)"
sub_X: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "p \<in> carrier (UP R)" shows "Cring_Poly.compose R (X_poly R) p = p"
assms_3: fixes s :: "'b" assumes "disc s" shows "ctor (sel s) = s"
o2: shows "t2 = s2 \<or> (t2, s2) \<in> trm_ord"
assms_9: shows "\<forall>j\<le>k + 3. lose (t + j) = [False]"
i_expand_nth_mult: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'a message_af"   and n :: "nat" assumes "0 < k" shows "(f \<odot> k) (n * k) = f n"
insert: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
inC: shows "insertion (nth_default 0 ((xs' @ x # xs)[var := - C / B])) (- c) = - C"
assert_tableau_equiv: fixes s :: "('i, 'a) state"   and v :: "nat \<Rightarrow> 'a"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> (assert a s)"
sum_prod: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Sum>x\<in>A. (f x, g x)) = (sum f A, sum g A)"
f1: assumes "rk {A, a} = 1" shows "rk {A, A', B, P, Q} \<le> 3"
run_fail_env: fixes fail :: "'a"   and r :: "'b" shows "run_env (fail_env fail) r = fail"
to_nat_less_nrows: fixes a :: "'a"   and A :: "(('b, 'c) vec, 'a) vec" shows "to_nat a < nrows A"
subprob_space_axioms_def: fixes M :: "'a measure" shows "subprob_space_axioms M \<equiv> emeasure M (space M) \<le> 1 \<and> space M \<noteq> {}"
pivot_bounds_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<B>\<^sub>i (pivot x\<^sub>i x\<^sub>j s) = \<B>\<^sub>i s"
resolve_action_schema_def: fixes D :: "ast_domain"   and n :: "char list" shows "ast_domain.resolve_action_schema D n = index_by ast_action_schema.name (actions D) n"
assms: shows "f o\<^sub>L g = id_blinfun"
main: fixes x :: "rat" assumes "x \<in> set xs" shows "rat_of_int (case quotient_of x of (n, x) \<Rightarrow> n * dd div x) / rat_of_int dd = x"
select_object_any0\<^sub>S\<^sub>e\<^sub>t_def: fixes f :: "'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and s_set :: "'f list" shows "select_object_any0\<^sub>S\<^sub>e\<^sub>t f s_set = select_object\<^sub>S\<^sub>e\<^sub>t f s_set->any\<^sub>S\<^sub>e\<^sub>t()"
set_eqvt: fixes pi :: "('b \<times> 'b) list"   and xs :: "'a list" shows "pi \<bullet> set xs = set (pi \<bullet> xs)"
eq2: shows "exH_n * pre * exH_n = proj_psi"
w_def: shows "w = (as @ b # bs) \<bullet>\<cdot> (us @ v # vs)"
init_config_def: fixes c :: "('b, 'loc \<times> 't) Exchange.configuration" shows "cri.init_config c \<equiv> (\<forall>p. c_temp c p = {#}\<^sub>z) \<and> (\<forall>p1 p2. c_msg c p1 p2 = []) \<and> (\<forall>p t. 0 \<le> zcount (c_caps c p) t) \<and> (\<forall>p. c_glob c p = cri.records c) \<and> c_data_msg c = {#}"
da_elim_cases_39: fixes Env :: "env"   and B :: "lname set"   and v :: "Term.var"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Acc v\<rangle>\<guillemotright> A" obtains vn :: "lname" where "v = LVar vn" and "vn \<in> B" and "nrm A = B" and "brk A = (\<lambda>k. UNIV)"   | "\<forall>vn. v \<noteq> LVar vn" and "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> A"
rel_with_source_impl_SRel_reflects_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and SRel :: "('procS \<times> 'procS) set" assumes "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b (STCalWB SWB TWB))"   and "\<forall>S1 S2. (S1, S2) \<in> SRel \<longrightarrow> (SourceTerm S1, SourceTerm S2) \<in> Rel" shows "rel_reflects_binary_pred SRel (\<lambda>b. hasBarb b SWB)"
subspace_span: fixes S :: "'c set" shows "ideal.subspace (ideal S)"
nb\<^sub>5: shows "dom (\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) = (\<Union>v\<in>{v |v. v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> (s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) v \<noteq> None}. {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
ipassmt_sanity_disjoint_def: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option" shows "ipassmt_sanity_disjoint ipassmt \<equiv> \<forall>i1\<in>dom ipassmt. \<forall>i2\<in>dom ipassmt. i1 \<noteq> i2 \<longrightarrow> ipcidr_union_set (set (the (ipassmt i1))) \<inter> ipcidr_union_set (set (the (ipassmt i2))) = {}"
matrix_inv_left: fixes M :: "(('a, 'b) vec, 'c) vec" assumes "invertible M" shows "matrix_inv M ** M = mat (1::'a)"
bin_lcp_fst_mismatch_pref: shows "symcode.\<alpha> \<cdot> [c\<^sub>1] \<le>p u\<^sub>1 \<cdot> symcode.\<alpha>"
comm_mult_strict_left_mono: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a < b"   and "(0::'c) < c" shows "c * a < c * b"
assms_1: shows "j \<in>\<^sub>\<circ> Vset \<alpha>"
cod_comp: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq g f" shows "cod (local.comp g f) = cod g"
comp_null_2: fixes f :: "Arr" shows "local.comp f null = null"
not_asym_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> P"   and "y \<in> P" shows "(\<not> x \<sqsubset> y) = (y \<sqsubseteq> x)"
ge_const_equiv: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" assumes "ltl_equivalence eq" shows "(\<sim>\<^sub>C) \<le> eq"
flip_altsublist_adjacent_def: fixes ss :: "'w list"   and ts :: "'w list" shows "flip_altsublist_adjacent ss ts \<equiv> \<exists>s t as bs. ss = as @ pair_relator_halflist s t @ bs \<and> ts = as @ pair_relator_halflist t s @ bs"
CombineStrip': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, bool) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "bool set"   and P :: "'e set"   and c' :: "('e, 'd, bool) com"   and Q :: "'e set"   and A :: "'e set"   and c'' :: "('e, 'd, bool) com"   and c :: "('e, 'd, bool) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c' Q,A"   and "\<Gamma>,{}\<turnstile>\<^bsub>/{} \<^esub>P c'' UNIV,UNIV"   and "c'' = mark_guards False (strip_guards (- F) c')"   and "merge_guards c = merge_guards (mark_guards False c')" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
cball_trans: fixes y :: "'b"   and z :: "'b"   and b :: "real"   and x :: "'b"   and a :: "real" assumes "y \<in> cball z b"   and "x \<in> cball y a" shows "x \<in> cball z (b + a)"
either2: fixes a :: "protocol" shows "(a \<noteq> tcp) = (a = udp)"
partIps_nonempty: fixes ts :: "'a wordinterval list"   and s :: "'a wordinterval" assumes "ts \<noteq> []" shows "partIps s ts \<noteq> []"
sum_set_simps_3: fixes x :: "'b" shows "Basic_BNFs.setr (Inl x) = {}"
rep_eq: fixes x :: "'a scene"   and xa :: "'a scene" shows "(x \<bowtie>\<^sub>S xa) = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. Rep_scene xa (Rep_scene x s\<^sub>1 s\<^sub>2) s\<^sub>3 = Rep_scene x (Rep_scene xa s\<^sub>1 s\<^sub>3) s\<^sub>2)"
group_size_partition_covers_points: shows "\<Union> (\<Union> {{G \<in> \<G>. card G = g} |g. g \<in> group_sizes}) = \<V>"
assms_1: shows "isCont (inverse \<circ> f) a"
additive_content_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>(x, y)\<in>d. content y) = content (cbox a b)"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "D.\<rho>' f = D.null"
head_add_arc: fixes a :: "'b" shows "head (add_arc a) = head G"
Lim_mono: fixes rs :: "'f \<Rightarrow> ('g, 'f) Old_Datatype.node set"   and rs' :: "'f \<Rightarrow> ('g, 'f) Old_Datatype.node set" assumes "\<And>d. rs d \<subseteq> rs' d" shows "Old_Datatype.Lim rs \<subseteq> Old_Datatype.Lim rs'"
lim_f: shows "(sum f \<longlongrightarrow> infsetsum' f A) (finite_subsets_at_top A)"
get_elements_by_class_name_def: shows "Shadow_DOM.get_elements_by_class_name \<equiv> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_class_name Shadow_DOM.to_tree_order get_attribute"
inject_3: fixes x3 :: "val"   and y3 :: "val" shows "(Val x3 = Val y3) = (x3 = y3)"
inverse_design_isomorphism: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "design_isomorphism \<V>' \<B>' \<V> \<B> (inv_into \<V> bij_map)"
umin: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> 1"   and "g t \<in> closure (- S)" shows "u \<le> t"
awalk_appendI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and q :: "'b list"   and w :: "'a" assumes "awalk u p v"   and "awalk v q w" shows "awalk u (p @ q) w"
if_pred_eq: fixes b :: "bool" shows "if_pred b = (if b then single () else \<bottom>)"
simps: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" shows "a1 \<rhd> a2 \<approx> a3 = (\<exists>\<Psi> P Q. a1 = \<Psi> \<and> a2 = P \<and> a3 = Q \<and> \<Psi> \<rhd> P \<lessapprox><weakBisim> Q \<and> \<Psi> \<rhd> P \<leadsto><weakBisim> Q \<and> (\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<approx> Q) \<and> \<Psi> \<rhd> Q \<approx> P)"
polynomial_function_def: fixes p :: "'b \<Rightarrow> 'c" shows "polynomial_function p \<equiv> \<forall>f. bounded_linear f \<longrightarrow> real_polynomial_function (f \<circ> p)"
project_act_extend_act: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and h' :: "'f \<times> 'g \<Rightarrow> 'e"   and act :: "('f \<times> 'f) set" shows "project_act h (extend_act h' act) = {(x, x'). \<exists>s s' y y' z. (s, s') \<in> act \<and> h (x, y) = h' (s, z) \<and> h (x', y') = h' (s', z)}"
jumpF_polyR_0_1: fixes p :: "real poly"   and a :: "real" shows "jumpF_polyR 0 p a = 0"
surj_xor: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "(\<oplus>\<^bsub>L\<^esub>) x ` carrier L = carrier L"
norm_eq_sqrt_inner: fixes x :: "'a" shows "norm x = sqrt (x \<bullet> x)"
Inv_empty2: fixes d :: "'b list"   and c :: "'b list" assumes "length d = 0" shows "Inv c d = {}"
det_one: fixes n :: "nat" shows "det (1\<^sub>m n) = (1::'b)"
leq_imp_less_eq_atm_of: fixes L :: "'a literal"   and M :: "'a literal" assumes "L \<le> M" shows "atm_of L \<le> atm_of M"
initial_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C" shows "category.initial C a \<equiv> partial_magma.ide C a \<and> (\<forall>b. partial_magma.ide C b \<longrightarrow> (\<exists>!f. partial_magma.in_hom C f a b))"
P: shows "P = (\<lambda>x y. S\<^sup>+\<^sup>+ (f x) (f y))"
Suc: fixes q :: "'a poly"   and r :: "'a poly"   and dr :: "nat" shows "fst (pseudo_divmod_main lc q r d dr n) = divide_poly_main lc (smult (lc ^ n) q) (smult (lc ^ n) r) d dr n"
subst_mkv: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and ODE :: "('sf, 'sz) ODE"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<nu>' :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "Oadmit \<sigma> ODE (BVO ODE)"   and "osafe ODE"   and "ssafe \<sigma>" shows "mk_v I (Osubst ODE \<sigma>) \<nu> (fst \<nu>') = mk_v (local.adjoint I \<sigma> \<nu>') ODE \<nu> (fst \<nu>')"
simps_1: fixes y :: "arctic" shows "MinInfty * y = MinInfty"
lookup_default_neutral_combine: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and d :: "'b"   and m1 :: "('c, 'b) mapping"   and m2 :: "('c, 'b) mapping"   and x :: "'c" assumes "\<And>x. f d x = x"   and "\<And>x. f x d = x" shows "Mapping.lookup_default d (Mapping.combine f m1 m2) x = f (Mapping.lookup_default d m1 x) (Mapping.lookup_default d m2 x)"
f_compose_smooth_on: fixes k :: "enat"   and S :: "'b set"   and g :: "'b \<Rightarrow> real" assumes "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f (g x))"
comp_simps_3: fixes i :: "nat"   and j :: "nat" shows "ioa i j \<equiv> \<lparr>ioa.asig = local.asig i j, start = local.start i, trans = local.trans i j\<rparr>"
check_equiv_fold: fixes \<C> :: "t_context"   and es :: "b_e list"   and ts :: "checker_type" shows "check \<C> es ts = foldl (\<lambda>ts e. case ts of Bot \<Rightarrow> Bot | _ \<Rightarrow> check_single \<C> e ts) ts es"
ang_circ_c_simp: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" assumes "E \<noteq> \<mu>1"   and "E \<noteq> \<mu>2" shows "ang_circ_c E \<mu>1 \<mu>2 p1 p2 = \<bar>\<downharpoonright>arg (E - \<mu>2) - arg (E - \<mu>1) + sgn_bool p1 * pi / 2 - sgn_bool p2 * pi / 2\<downharpoonleft>\<bar>"
p_eq_0_iff: fixes s :: "'s" assumes "s \<in> S" shows "(p s = 0) = (\<forall>cfg\<in>cfg_on s. v cfg = 0)"
allow_all_fun_def: fixes f :: "'b \<Rightarrow> 'c" shows "A\<^sub>f f = A\<^sub>p (Some \<circ> f)"
digraph_isomorphism_def: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" shows "digraph_isomorphism hom \<equiv> wf_digraph G \<and> inj_on (iso_verts hom) (verts G) \<and> inj_on (iso_arcs hom) (arcs G) \<and> (\<forall>a\<in>arcs G. iso_verts hom (tail G a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head G a) = iso_head hom (iso_arcs hom a))"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
measurable_restrict_space2_iff: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and \<Omega> :: "'c set" shows "(f \<in> M \<rightarrow>\<^sub>M restrict_space N \<Omega>) = (f \<in> M \<rightarrow>\<^sub>M N \<and> f \<in> space M \<rightarrow> \<Omega>)"
nat_subst\<^sub>E_introduces_no_rids: fixes e :: "('a, 'b, nat) expr"   and x :: "nat"   and e' :: "('a, 'b, nat) expr" shows "RID\<^sub>E (nat_subst\<^sub>E e x e') \<subseteq> RID\<^sub>E e \<union> RID\<^sub>E e'"
enum_distinct: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "distinct enum"
append: fixes xs :: "'c list"   and ys :: "'c list" shows "sum_list (xs @ ys) = sum_list xs + sum_list ys"
ctr: fixes ws :: "'d list stream" shows "flat ws = hd (shd ws) ## flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"
standard_decomp_concat: fixes pss :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list list"   and k :: "nat" assumes "\<And>ps. ps \<in> set pss \<Longrightarrow> standard_decomp k ps" shows "standard_decomp k (concat pss)"
HF_def: shows "Goedel_I.HF \<equiv> hmemrec (\<lambda>f z. if Ord z then W z else Q_HPair (f (hfst z)) (f (hsnd z)))"
almost_full_on_map: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "almost_full_on Q B"   and "h ` A \<subseteq> B" shows "almost_full_on (\<lambda>x y. Q (h x) (h y)) A"
samequot_exec_def: fixes a :: "'a"   and r :: "'b rexp" shows "samequot_exec a r = Times (PLUS (snd (the (while_option fst (rderiv_and_add a) (True, [\<guillemotleft>r\<guillemotright>]))))) (Star (Atom (singleton a)))"
name_swap_bij: fixes a :: "name"   and x :: "'a" shows "[(a, a)] \<bullet> x = x"
disjoint_list_def: fixes ls :: "'a set list" shows "disjoint_list ls \<equiv> distinct ls \<and> disjoint (set ls)"
forall: fixes i :: "nat" assumes "i + n < \<^bold>|w\<^bold>|" shows "w ! i = w ! (i + n)"
lim_1: shows "(f \<longlongrightarrow> c) net"
invar_get_min_rest_1: fixes ts :: "'b tree list"   and t' :: "'b tree"   and ts' :: "'b tree list" assumes "get_min_rest ts = (t', ts')"   and "ts \<noteq> []"   and "invar ts" shows "invar_tree t'"
Cons: fixes exp :: "exp"   and list :: "exp list" assumes "P1 exp"   and "P2 list" shows "P2 (exp # list)"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VxVxV.arr f = (B.arr (fst f) \<and> B.VxV.arr (snd f))"
acyclic_5b_implies_4: fixes x :: "'a" assumes "irreflexive x"   and "acyclic_5b x" shows "acyclic_4 x"
order_subst2: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'e"   and c :: "'e" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
m_inv_def: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" shows "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G \<and> x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)"
Fpow_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "Fpow A \<subseteq> Fpow B"
that: fixes ns :: "'a" assumes "s = NodeS i ns R"   and "ns \<in> init (np i)" shows "thesis"
summable: fixes f :: "'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'b" assumes "bounded_linear f"   and "summable X" shows "summable (\<lambda>n. f (X n))"
distinct_of_prio_hlp: fixes l :: "'b list" assumes "length l < unat (- 1)" shows "distinct (map word_of_nat (map fst (annotate_rlen l)))"
map_disj_None_right: 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>0" shows "h\<^sub>1 x = None"
eval_fds_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b fds"   and s :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> fds_abs_converges (f x) s" shows "eval_fds (prod f A) s = (\<Prod>x\<in>A. eval_fds (f x) s)"
nat_pow_inv: fixes x :: "'a"   and i :: "nat" assumes "x \<in> carrier G" shows "inv x [^] i = inv (x [^] i)"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(cNumber x2 = cNumber y2) = (x2 = y2)"
Der: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and S' :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "\<Psi>' \<rhd> !S \<longmapsto> \<alpha> \<prec> S'"   and "(\<Psi>', S, T) \<in> Rel"   and "bn \<alpha> \<sharp>* \<Psi>'"   and "bn \<alpha> \<sharp>* S"   and "bn \<alpha> \<sharp>* T"   and "guarded T"   and "bn \<alpha> \<sharp>* subject \<alpha>" shows "\<exists>T' U O. \<Psi>' \<rhd> !T \<longmapsto> \<alpha> \<prec> T' \<and> (\<Psi>', S', U \<parallel> !S) \<in> Rel \<and> (\<Psi>', T', O \<parallel> !T) \<in> Rel \<and> (\<Psi>', U, O) \<in> Rel \<and> supp U \<subseteq> supp S' \<and> supp O \<subseteq> supp T'"
rightreg_scalar_multD1: fixes R :: "'a set"   and x :: "'a"   and r :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "Ring1 R"   and "x \<in> R" shows "Ring1.rightreg_scalar_mult R r f x = f (x * r)"
assms_4: shows "\<forall>m. is_zero_row_upt_k m k A"
l12: assumes "\<not> QuadRes (int p) a" shows "[prod Prod local.S2 = a ^ ((p - 1) div 2)] (mod int p)"