hcju/isabelleps huggingface dataset:
finite_t1_space_imp_discrete_topology: fixes X :: "'a topology"   and U :: "'a set" assumes "topspace X = U"   and "finite U"   and "t1_space X" shows "X = discrete_topology U"
extendRuleEmpty: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" shows "extendRule ( \<Empt> \<Rightarrow>* \<Empt>) r = r"
no_prime_divisors_imp_unit: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero"   and "\<And>b. \<lbrakk>dvd.dvd times b a; normalize b = b\<rbrakk> \<Longrightarrow> \<not> comm_semiring_1.prime_elem times one zero b" shows "dvd.dvd times a one"
ladD: fixes f :: "'d set \<Rightarrow> 'd set"   and C :: "'d set"   and B :: "'d set" assumes "lad f"   and "\<And>x. x \<in> C \<Longrightarrow> x \<in> B" shows "card (f C) \<le> card (f B)"
prime_power_cancel2: fixes p :: "'c"   and m :: "'c"   and k :: "nat"   and m' :: "'c"   and k' :: "nat" assumes "prime p"   and "m * p ^ k = m' * p ^ k'"   and "\<not> p dvd m"   and "\<not> p dvd m'" obtains "m = m'" and "k = k'"
c: shows "column i (Gram_Schmidt_matrix A) = column ia (Gram_Schmidt_matrix A)"
assms: shows "ct_list_eq (to_ct_list as) (xs @ to_ct_list ys)"
cindex_pathE_joinpaths: fixes g1 :: "real \<Rightarrow> complex"   and z :: "complex"   and g2 :: "real \<Rightarrow> complex" assumes "finite_ReZ_segments g1 z"   and "finite_ReZ_segments g2 z"   and "path g1"   and "path g2"   and "pathfinish g1 = pathstart g2" shows "cindex_pathE (g1 +++ g2) z = cindex_pathE g1 z + cindex_pathE g2 z"
nn_integral_nonneg_infinite: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "\<not> integrable M f"   and "AE x in M. 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = \<infinity>"
1: shows "line_integral F basis (rec_join l) = (\<Sum>(x, \<gamma>)\<leftarrow>l. real_of_int x * line_integral F basis \<gamma>)"
gaussian_moment_odd_pos: fixes k :: "nat" shows "has_bochner_integral lborel (\<lambda>x. indicat_real {0..} x *\<^sub>R (exp (- x\<^sup>2) * x ^ (2 * k + 1))) (fact k / 2)"
wf_goal: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "wf_ast_problem problem" shows "ast_problem.wf_partial_state problem (ast_problem.astG problem)"
induct: fixes P :: "'c list \<Rightarrow> bool"   and a0 :: "'c list" assumes "P []"   and "\<And>x xs. \<lbrakk>P (filter (\<lambda>y. y < x) xs); P (filter ((\<le>) x) xs)\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P a0"
t'_def: shows "t' = mkTree (stl (trim (sdrop n rs) s')) s''"
Max_by_in: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "A \<noteq> {}" shows "Max_by f A \<in> A"
hseqI: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and a :: "'b"   and b :: "'b"   and \<nu> :: "'b"   and c :: "'b" assumes "horizontal_composition V H src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b"   and "horizontal_homs.in_hhom V src trg \<nu> b c" shows "partial_magma.arr V (H \<nu> \<mu>)"
primerec_getpren: fixes i :: "nat"   and n :: "nat"   and m :: "nat" assumes "i < n"   and "n \<le> m" shows "primerec (get_fstn_args m n ! i) m"
mon_hom_induced_section_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and p :: "int" assumes "section_map X Y f" shows "hom_induced p X {} Y {} f \<in> mon (homology_group p X) (homology_group p Y)"
assms_4: shows "\<gamma> \<sqsubseteq> \<beta>"
id_redu: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "coname"   and \<Delta> :: "(coname \<times> ty) list"   and a :: "name"   and M :: "trm" shows "idn \<Gamma> x,idc \<Delta> a<M> \<longrightarrow>\<^sub>a* M"
Abs_sat_nat_of: fixes n :: "'b sat" shows "Abs_sat (nat_of n) = n"
simps_7: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> X \<phi> = suffix 1 w \<Turnstile> \<phi>"
nLt_le_conv: fixes x :: "'a"   and y :: "'a" shows "(cmp x y \<noteq> Lt) = cmp.le y x"
k: shows "n \<in> {idx k..<idx (Suc k)}"
I: shows "Says n' run' 5 X (Card n') (Crypt (sesK (c * f)) (Passwd n')) \<in> set evsFR5 \<and> Says n' run' 5 (Card n') X (Crypt (sesK (c * f)) (Number 0)) \<in> set evsFR5"
rel_vset_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b set"   and ys :: "'c set" shows "rel_vset R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> (x \<in> xs) = (y \<in> ys)"
nhds_def: fixes a :: "'a" shows "nhds a = Inf (principal ` {S. open S \<and> a \<in> S})"
inv_matching_def: fixes C :: "'b set"   and F :: "'a set set"   and M :: "'a set set" shows "inv_matching C F M = (matching M \<and> M \<subseteq> E \<and> card C \<le> k * card M \<and> (\<forall>e\<in>M. \<forall>f\<in>F. e \<inter> f = {}))"
ball: fixes t :: "complex" assumes "dist z t < d" shows "Im t = 0 \<longrightarrow> 0 < Re t"
Sup_no_Red: shows "Sup_llist Ns - Red_F (Sup_llist Ns) \<Turnstile> {B}"
rbt_map_entry_inv1: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and t :: "('a, 'c) rbt" shows "inv1 (rbt_map_entry k f t) = inv1 t"
simps_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C" shows "monoidal_language.Inv C (t \<^bold>\<otimes> u) = monoidal_language.Inv C t \<^bold>\<otimes> monoidal_language.Inv C u"
msed_rel_invL: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and M :: "'a multiset"   and N :: "'b multiset" assumes "rel_mset R (add_mset a M) N" shows "\<exists>N1 b. N = add_mset b N1 \<and> R a b \<and> rel_mset R M N1"
map_carrier_vec: fixes h :: "'c \<Rightarrow> 'b"   and v :: "'c vec"   and n :: "nat" shows "(map_vec h v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
fsubset_singleton_iff: fixes A :: "'a fset"   and x :: "'a" shows "(A |\<subseteq>| {|x|}) = (A = {||} \<or> A = {|x|})"
F: fixes i :: "nat"   and j :: "nat" assumes "i \<in> I"   and "j \<in> I"   and "i \<le> j" shows "F j \<subseteq> F i"
prob_proj_def: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" shows "prob_proj PROB vs \<equiv> (\<lambda>a. action_proj a vs) ` PROB"
vsv_vinsert: fixes a :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> xs" shows "vsv (vinsert \<langle>a, b\<rangle> xs)"
ASVar: shows "\<forall>xs s. |AS xs s| <o |UNIV| \<or> finite (AS xs s)"
two_comparisons_into_case_order_2: fixes x :: "'a"   and y :: "'a"   and P :: "'c"   and Q :: "'c"   and R :: "'c" shows "(if le x y then if y = x then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
AE_const: fixes P :: "bool" shows "(AE x in M. P) = P"
iless_Suc_eq: fixes m :: "nat"   and n :: "enat" shows "(enat m < eSuc n) = (enat m \<le> n)"
eq: shows "ltake (enat r') E = ltake (enat r') E'"
IH: fixes j :: "nat"   and m' :: "nat" assumes "rank x n = Some i"   and "rank x (n + m) = Some j"   and "m' \<le> m" shows "\<exists>k. rank x (n + m') = Some k"
rsp: shows "eq_onp (\<lambda>x. x \<in> Collect monom_inv) [] []"
lsl_local_eq: fixes x :: "'b"   and y :: "'b" shows "x \<cdot> \<nu> y = \<nu> (x \<cdot> y)"
substEnv_compose1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y1 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y2 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y1"   and "good Y2" shows "rho &[Y1 / y]_ys &[Y2 / y]_ys = rho &[Y1 #[Y2 / y]_ys / y]_ys"
assms_1: shows "insert (Var i IN t) H \<turnstile> Var i IN u"
ferrack_def: fixes p :: "fm" shows "ferrack p = (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p' in if mp = T \<or> pp = T then T else let U = remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (uset p')))) in decr (Ferrack.disj mp (Ferrack.disj pp (evaldjf (simpfm \<circ> usubst p') U))))"
IH: fixes t' :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" assumes "t' |\<in>| cont t"   and "local.wf t'" shows "\<exists>it. local_iwf it (fst (tree.root t'))"
simps_2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
HoHV_eq_ToTC: shows "HoHV = H.ToTC"
simps_2: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and l :: "'d"   and ls :: "'d list"   and s' :: "'b" shows "LTS.path step s (l # ls) s' = (\<exists>sh. step s l sh \<and> LTS.path step sh ls s')"
assoc: fixes a :: "'a upred"   and b :: "'a upred"   and c :: "'a upred" shows "((a \<or> b) \<or> c) = (a \<or> b \<or> c)"
enum_ex_num1_def: fixes P :: "num1 \<Rightarrow> bool" shows "enum_class.enum_ex P = P 1"
append_genPrefix: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs @ ys, zs) \<in> genPrefix r" shows "(xs, zs) \<in> genPrefix r"
map_app_pair_spp_of_vec_of: fixes ps :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "list_all spp_inv_pair ps" shows "map (app_pair spp_of \<circ> app_pair vec_of) ps = ps"
ET_Delta: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<subseteq> ET ST"   and "t \<in> TS"   and "source t \<in> States A"   and "A \<in> SAs (HA ST)" shows "t \<in> Delta A"
completeness': fixes s :: "(nat \<times> form) list"   and f :: "nat \<Rightarrow> nat \<times> (nat \<times> form) list"   and mo :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)" assumes "loc2 s f mo"   and "infinite (deriv s)"   and "init s" shows "\<forall>mA\<in>set s. \<not> FEval mo ntou (snd mA)"
poly_even_odd_def: fixes f :: "'a poly" shows "poly_even_odd f = (case alternate (coeffs f) of (evn, od) \<Rightarrow> (poly_of_list evn, poly_of_list od))"
T_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.arr g" shows "\<F>\<^sub>SC.tensor f g = f \<otimes>\<^sub>S g"
Encode_axioms: shows "Encode var trm fmla Var FvarsT substT Fvars subst num enc"
f_expand_Nil: fixes k :: "nat" shows "[] \<odot> k = []"
2: fixes i :: "nat" assumes "i < length (T \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)" shows "term_variants_pred P ((T \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>) ! i) ((S \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>) ! i)"
comm_semiring_1_cancel_axioms_def: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.comm_semiring_1_cancel_axioms minus times \<equiv> \<forall>a b c. times a (minus b c) = minus (times a b) (times a c)"
length_int_to_bv_upper_limit_gt0: fixes i :: "int"   and k :: "nat" assumes "0 < i"   and "i \<le> 2 ^ (k - 1) - 1" shows "length (int_to_bv i) \<le> k"
mset_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" shows "mset_heap (merge h1 h2) = mset_heap h1 + mset_heap h2"
prems: shows "conv_abscissa f < ereal (s \<bullet> (1::'a))"
ennreal_mono_minus_cancel: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a - b \<le> a - c"   and "a < top"   and "b \<le> a"   and "c \<le> a" shows "c \<le> b"
filterlim_divide_at_infinity: fixes f :: "'a \<Rightarrow> 'a"   and c :: "'a"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'a" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g (at (0::'a)) F"   and "c \<noteq> (0::'a)" shows "LIM x F. f x / g x :> at_infinity"
assms_3: shows "c \<in>\<^sub>\<circ> dg_prod K \<AA>\<lparr>Arr\<rparr>"
sorted_list_of_set_k: fixes k :: "nat" assumes "0 < k" shows "list_of {..<k} = 0 # list_of {0<..<k}"
usolves_ode_on_superset_domain: fixes x :: "real \<Rightarrow> 'a"   and Y :: "'a set" assumes "(x usolves_ode f from t_max) (cball t_max et) (cball (x t_max) ex)"   and "x t_max = x t_max"   and "cball (x t_max) ex \<subseteq> Y" shows "(x usolves_ode f from t_max) (cball t_max et) Y"
converse_tranclp_induct: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r y b \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>+\<^sup>+ z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
Length_Extend: fixes i :: "nat"   and \<AA> :: "interp"   and P :: "nat" shows "Length (Extend () i \<AA> P) = max (Length \<AA>) (len P)"
rs_props_1: shows "g \<turnstile> defNode g r-rs\<rightarrow>defNode g \<phi>\<^sub>r"
star_simulation_right_equal: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqinter> x = y \<sqinter> z" shows "z \<sqinter> top = top \<sqinter> z"
assms: shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
forest_components_wcc: fixes x :: "'a" assumes "injective x" shows "wcc x = forest_components x"
log2_is_max: fixes u :: "nat" assumes "0 < u" shows "log2 u = Max (nat_to_set u)"
bword_neg_one_def: shows "bword_neg_one \<equiv> Abs_bword (- 1)"
fin: fixes Q :: "process set" shows "finite (vote_set (votes s) Q)"
distinct_3: shows "TernaryTrue \<noteq> TernaryUnknown"
proj2_Col_coeff_unique': fixes a :: "(real, 3) vec"   and r :: "(real, 3) vec"   and i :: "real"   and j :: "real" assumes "a \<noteq> 0"   and "r \<noteq> 0"   and "proj2_abs a \<noteq> proj2_abs r"   and "proj2_abs (i *\<^sub>R a + r) = proj2_abs (j *\<^sub>R a + r)" shows "i = j"
sum_case_def: fixes f :: "V \<Rightarrow> 'a"   and g :: "V \<Rightarrow> 'a"   and a :: "V" shows "sum_case f g a \<equiv> THE z. (\<forall>x. a = ZFC_Cardinals.Inl x \<longrightarrow> z = f x) \<and> (\<forall>y. a = ZFC_Cardinals.Inr y \<longrightarrow> z = g y) \<and> (\<not> is_sum a \<longrightarrow> z = undefined)"
p: shows "g \<turnstile> Entry g-ns\<rightarrow>n"
r_lifz_def: fixes f :: "recf"   and g :: "recf" shows "r_lifz f g \<equiv> r_lazyifzero (arity f - 1) (encode f) (encode g)"
supp_2: fixes name :: "name" shows "supp (Var name) = supp name"
in_matched_in_slice2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node" assumes "matched n ns n'"   and "n'' \<in> set ns" shows "n'' \<in> sum_SDG_slice2 n'"
lem2: shows "\<forall>z. z \<noteq> 0 \<longrightarrow> infnorm (sqprojection z) = 1"
abs_fun_def: fixes R :: "('d \<times> 'c) set"   and m :: "'d nres" shows "\<Up> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> if X \<subseteq> Domain R then RES (R `` X) else FAIL"
while_eval_elim': fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>While e c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<and> mem' = mem"
Assign_1: shows "vars a \<union> {x} \<subseteq> dom ps"
assms_2: shows "\<exists>c. hamlet (Rep_run r k c)"
tyenv_eq_sym': fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "dom \<Gamma> \<inter> \<C> = {}"   and "types_wellformed \<Gamma>"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" shows "mem\<^sub>2 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>1"
intersects_x_axis_positive_intersects_x_axis: fixes H :: "circline" assumes "intersects_x_axis_positive H" shows "intersects_x_axis H"
intros_2: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration"   and loc1 :: "'a"   and loc2 :: "'a"   and M :: "'c zmultiset"   and xs :: "('a \<times> 'b \<times> 'a) list"   and s :: "'b"   and loc3 :: "'a" assumes "dataflow_topology summary results_in"   and "dataflow_topology.loc_imps_fw summary results_in c loc1 loc2 M xs"   and "s \<in>\<^sub>A summary loc2 loc3"   and "distinct (xs @ [(loc2, s, loc3)])" shows "dataflow_topology.loc_imps_fw summary results_in c loc1 loc3 ({#results_in t s. t \<in>#\<^sub>z M#} + c_imp c loc3) (xs @ [(loc2, s, loc3)])"
B_Int: shows "basis_in (X \<inter> Y) B_Int"
inv_is_iD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = Aodv_Basic.inv" shows "ip \<in> iD rt"
as': shows "as' = take i as' @ as' ! i # drop (Suc i) as'"
assms: shows "literal_ordering (rename_literal A L1) (rename_literal A L2)"
Abs_poly_mapping_inject: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> {f. finite {x. f x \<noteq> (0::'b)}}"   and "y \<in> {f. finite {x. f x \<noteq> (0::'b)}}" shows "(Abs_poly_mapping x = Abs_poly_mapping y) = (x = y)"
vsv_vsingleton: fixes a :: "V"   and b :: "V" shows "vsv (ZFC_in_HOL.set {\<langle>a, b\<rangle>})"
fmran_restrict_fset: fixes A :: "'f fset"   and m :: "'f \<rightharpoonup>\<^sub>f 'e" shows "fmran (fmrestrict_fset A m) = fmimage m (fmdom m |\<inter>| A)"
corec_transfer: fixes S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and R1 :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_fun S (=)) (rel_fun (rel_fun S R2) (rel_fun (rel_fun S R1) (rel_fun (rel_fun S (=)) (rel_fun (rel_fun S (tllist_all2 R1 R2)) (rel_fun (rel_fun S S) (rel_fun S (tllist_all2 R1 R2))))))) corec_tllist corec_tllist"
residues_n: fixes n :: "nat"   and p :: "int" assumes "n \<noteq> 0"   and "prime p" shows "residues (p ^ n)"
intI: fixes A :: "'c \<Rightarrow> bool" assumes "\<And>w. A w" shows "\<turnstile> A"
reflects_iso: fixes f :: "'c"   and a' :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>"   and "Cop_S.iso (local.map f)" shows "C.iso f"
pp: fixes q :: "nat \<Rightarrow> nat" assumes "q permutes {0..<n}" shows "permutation q"
safe_convert_multiway: fixes \<phi> :: "formula" assumes "safe_formula \<phi>" shows "safe_formula (convert_multiway \<phi>)"
Modules_single_carrier_isom: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme" assumes "R module M"   and "R module N"   and "carrier M = {\<zero>\<^bsub>M\<^esub>}"   and "carrier N = {\<zero>\<^bsub>N\<^esub>}" shows "M \<cong>\<^bsub>R\<^esub> N"
rep_eq: fixes x :: "'b upred" shows "\<lbrakk>[x]\<^sub>u\<rbrakk>\<^sub>e = (\<lambda>A. \<forall>A'. \<lbrakk>x\<rbrakk>\<^sub>e A')"
simps: fixes A :: "'a mat" shows "Smith_2xn A = (if dim_col A = 0 then (1\<^sub>m (dim_row A), A, 1\<^sub>m 0) else if dim_col A = 1 then let (P, S) = Smith_nx1 A in (P, S, 1\<^sub>m (dim_col A)) else if dim_col A = 2 then Smith_2x2 A else let A1 = mat_of_cols (dim_row A) [col A 0]; A2 = mat_of_cols (dim_row A) (map (col A) [1..<dim_col A]); (P1, D1, Q1) = Smith_2xn A2; C = P1 * A1 @\<^sub>c P1 * A2 * Q1; D = mat_of_cols (dim_row A) [col C 0, col C 1]; E = mat_of_cols (dim_row A) (map (col C) [2..<dim_col A]); (P2, D2, Q2) = Smith_2x2 D; H = P2 * D * Q2 @\<^sub>c P2 * E; k = div_op (H $$ (0, 2)) (H $$ (0, 0)); H2 = addcol (- k) 2 0 H; (uu_, uu_, uu_, H2_DR) = split_block H2 1 1; (H_1xn, Q3) = Smith_1xn H2_DR; S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m 1 1) H_1xn; Q1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q1; Q2' = four_block_mat Q2 (0\<^sub>m 2 (dim_col A - 2)) (0\<^sub>m (dim_col A - 2) 2) (1\<^sub>m (dim_col A - 2)); Q_div_k = addrow_mat (dim_col A) (- k) 0 2; Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3 in (P2 * P1, S, Q1' * Q2' * Q_div_k * Q3'))"
implies_e2: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>"
simps_2: fixes dims :: "nat list"   and U :: "complex mat"   and \<rho> :: "complex mat" shows "state_sig.denote dims (Utrans U) \<rho> = U * \<rho> * adjoint U"
put_closure: fixes \<sigma> :: "'b"   and v :: "'a" shows "put \<sigma> v \<in> \<S>"
map_charpair_uniform: fixes ss :: "'a list"   and S :: "'a set" assumes "ss \<in> lists S" shows "map (charpair S) ss = map pairtrue ss"
assms_3: shows "consistent A ({(p # ps, a)} \<union> S)"
zero_in_Field: assumes "Field r \<noteq> {}" shows "r.zero \<in> Field r"
sorted_list_of_fmap_def: fixes m :: "('c, 'd) fmap" shows "sorted_list_of_fmap m = map (\<lambda>k. (k, m $$! k)) (sorted_list_of_fset (fmdom m))"
WTStaticCast: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and D :: "char list"   and C :: "char list" assumes "P,E \<turnstile> e :: Class D"   and "is_class P C"   and "P \<turnstile> Path D to C unique \<or> P \<turnstile> C \<preceq>\<^sup>* D \<and> (\<forall>Cs. P \<turnstile> Path C to D via Cs  \<longrightarrow> Subobjs\<^sub>R P C Cs)" shows "P,E \<turnstile> \<lparr>C\<rparr>e :: Class C"
prime_elem_iff_irreducible: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.ufd times one plus zero minus uminus" shows "comm_semiring_1.prime_elem times one zero x = comm_semiring_1.irreducible times one zero x"
UNIV_I: fixes x :: "'c" shows "s2p UNIV x"
simps_1: fixes e :: "'c exp"   and es :: "'c exp list" shows "map_vals_of (e # es) = (case val_of e of None \<Rightarrow> None | \<lfloor>v\<rfloor> \<Rightarrow> case map_vals_of es of None \<Rightarrow> None | \<lfloor>vs\<rfloor> \<Rightarrow> \<lfloor>v # vs\<rfloor>)"
run_put_writer: fixes s :: "'s"   and m :: "('w, 'a, 'm) writerT" shows "run_writer (local.put_writer s m) = put s (run_writer m)"
bounded_scaling: fixes S :: "'b set"   and c :: "real" assumes "bounded S" shows "bounded ((*\<^sub>R) c ` S)"
l2_step1_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent" shows "l2_step1 Ra A B \<equiv> {(s, s'). Ra \<notin> dom (progress s) \<and> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xgnx}), chan := {Insec A B (Exp Gen (NonceF (Ra $ nx)))} \<union> chan s\<rparr>}"
exhaust: fixes y :: "natpair" obtains x :: "nat \<times> nat" where "y = natpair.Pair x"
IH': fixes i :: "'b" shows "dsafe (Tsubst (args i) \<sigma>)"
the_left_def: fixes j :: "nat" shows "local.the_left j = (GREATEST i. i < j \<and> xs ! i < xs ! j)"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y"   and "x \<in> X"   and "y \<in> X" shows "False"
arcs2: assumes "j # ys @ a # zs = (j # ys) @ a # zs" shows "arcs i i (j # ys @ a # zs) = arcs i a (j # ys) @ arcs a i zs"