hcju/isabelleps huggingface dataset:
gtrE: fixes c :: "'a multiset"   and e :: "'b"   and c' :: "'c multiset"   and T :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'c \<times> 'c multiset) set" assumes "(c, e, c') \<in> gtr T" obtains s :: "'a" and ce :: "'a multiset" and s' :: "'c" and ce' :: "'c multiset" where "c = add_mset s ce" and "c' = add_mset s' ce'" and "((s, ce), e, s', ce') \<in> T"
secure_implies_c_secure: fixes step :: "'d \<Rightarrow> 'a \<Rightarrow> 'd"   and out :: "'d \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'d"   and I :: "('c \<times> 'c) set"   and D :: "'a \<Rightarrow> 'c" assumes "secure (c_process step out s\<^sub>0) I (c_dom D)" shows "c_secure step out s\<^sub>0 I D"
pi_half: shows "pi / 2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
is_const_def: fixes f :: "nat \<Rightarrow> nat" assumes "deutsch f" shows "deutsch.is_const f \<equiv> deutsch.const f 0 \<or> deutsch.const f 1"
hom_pivot_positions_main_gen: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat nr nc" shows "pivot_positions_main (mat\<^sub>h A) nr nc i j = pivot_positions_main A nr nc i j"
cball_eq_empty: fixes x :: "'a"   and e :: "real" shows "(cball x e = {}) = (e < 0)"
assms_2: shows "a \<rightarrow>\<^sup>* b1"
spn_simps_2: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "ide (spn \<mu>)"
antisym: fixes b :: "'q multiset"   and a :: "'q multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
t: shows "t = (p\<^sub>t, is\<^sub>t, \<theta>\<^sub>t, sb\<^sub>t, \<D>\<^sub>t, \<O>\<^sub>t, \<R>\<^sub>t)"
HSem_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "HSem \<Gamma> = (\<Lambda> \<rho>. \<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
eval\<^sub>G_respectfulness_generalized: fixes S :: "'b ltl set"   and y :: "'b ltl"   and \<A> :: "'b ltl set"   and P :: "'b ltl set" assumes "\<And>\<A>. (\<And>x. x \<in> S \<Longrightarrow> \<A> \<Turnstile>\<^sub>P x) \<Longrightarrow> \<A> \<Turnstile>\<^sub>P y"   and "\<And>x. x \<in> S \<Longrightarrow> \<A> \<Turnstile>\<^sub>P eval\<^sub>G P x" shows "\<A> \<Turnstile>\<^sub>P eval\<^sub>G P y"
le_imp_Di_subset: fixes v :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "v \<le>\<^sub>v y" shows "Di i v \<subseteq> Di i y"
plus_subset1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "L F g \<subseteq> L F (\<lambda>x. f x + g x)"
cases: fixes x :: "('b, 'c) Combinators list" obtains "x = []"   | xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs"
supp_set: fixes xs :: "'b list" shows "supp (set xs) = supp xs"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> f). f\<lparr>a\<rparr> = b"
simps_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "vclosed (Vrecabs css name \<Gamma>) = (pred_fmap id (fmmap vclosed \<Gamma>) \<and> pred_fmap (list_all (\<lambda>(pat, t). closed_except t (fmdom \<Gamma> |\<union>| frees pat))) css \<and> name |\<in>| fmdom css \<and> True)"
l3_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "l3_step2 Rb A B Ni gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni \<and> sklvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> implConfid A B Ni \<in> ik s \<and> implInsec A B gnx \<in> ik s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xni, xnr, xgny, xgnx, xsk}), ik := {implConfid B A (NonceF (Rb $ nr))} \<union> ({implInsec B A \<langle>Exp Gen (NonceF (Rb $ ny)), hmac \<langle>Number 0, gnx, Exp Gen (NonceF (Rb $ ny)), Messages.Agent B, Messages.Agent A\<rangle> (hmac Ni (NonceF (Rb $ nr)))\<rangle>} \<union> ik s), signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) else signalsInit s, signalsInit2 := if can_signal s A B then addSignal (signalsInit2 s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit2 s\<rparr>}"
Collect_snth: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and s :: "'b stream" shows "{p. P ((x ## s) !! p)} \<subseteq> {0} \<union> Suc ` {p. P (s !! p)}"
c_f_list_f_is_pr: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "c_f_list f \<in> PrimRec2" shows "f \<in> PrimRec2"
intro: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "maximalideal_axioms I R" shows "maximalideal I R"
vars_mset_empty_iff: fixes s :: "('b, 'a) tm" shows "(vars_mset s = {#}) = ground s"
sublist_ext': fixes lo :: "nat"   and hi :: "nat"   and xs' :: "'c list"   and xs :: "'c list" assumes "\<forall>i. lo \<le> i \<and> i \<le> hi \<longrightarrow> xs' ! i = xs ! i"   and "length xs' = length xs"   and "lo \<le> hi"   and "Suc hi < length xs" shows "WB_Sort.sublist xs' lo hi = WB_Sort.sublist xs lo hi"
eval_abrupt_lemma: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and v :: "vals"   and s' :: "state"   and xc :: "abrupt" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (v, s')" shows "fst s = Some xc \<longrightarrow> s' = s \<and> v = undefined3 t"
d\<mu>: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs'"   and "ii < length fs'" shows "rat_of_int (fs'.d\<mu> ii j) = rat_of_int (fs'.d fs' (Suc j)) * fs'.gs.\<mu> ii j"
prime_factors_primorial: fixes x :: "real" shows "prime_factors (primorial x) = {p. prime p \<and> real p \<le> x}"
L_transform_Pred: fixes f :: "'d"   and \<phi> :: "'b" shows "L_transform (FL_Formula.Pred f \<phi>) = Formula.Act (Eff f) (Formula.Pred \<phi>)"
ESem_simps_as_defined_2: fixes e :: "exp"   and x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> App e x \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> \<down>Fn \<rho> x"
labels_conv_def: fixes c :: "'d \<times> (bool \<times> 'c \<times> 'd) list" shows "labels_conv c = map (\<lambda>q. fst (snd q)) (snd c)"
effective_matrix_match_condn_1: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and A1 :: "'b list list"   and A2 :: "'b list list"   and B1 :: "'b list list"   and B2 :: "'b list list" assumes "plus_mult id f zer g inver"   and "plus_mult.matrix_match A1 A2 B1 B2" shows "\<forall>i j. i < mult.row_length A1 * mult.row_length B1 \<and> j < length A2 * length B2 \<longrightarrow> mult.Tensor f (mat_multI zer g f (mult.row_length A1) A1 A2) (mat_multI zer g f (mult.row_length B1) B1 B2) ! j ! i = f (plus_mult.scalar_product f zer g (row A1 (i div mult.row_length B1)) (col A2 (j div length B2))) (plus_mult.scalar_product f zer g (row B1 (i mod mult.row_length B1)) (col B2 (j mod length B2)))"
fmdom'_filter: fixes P :: "'a \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "fmdom' (fmfilter P m) = Set.filter P (fmdom' m)"
bool_to_ternary_simps_4: fixes X :: "bool" shows "(eval_ternary_Not (bool_to_ternary X) = TernaryFalse) = X"
Sup: shows "(SUP x. ereal (sum a {0..x})) = \<infinity>"
Set_member_code_1: fixes x :: "'b"   and A :: "'b \<Rightarrow> bool" shows "(x \<in> Collect_set A) = A x"
rep_eq: fixes x :: "'k word"   and xa :: "'k word" shows "(x \<le> xa) = (take_bit LENGTH('k) (Word.rep x) \<le> take_bit LENGTH('k) (Word.rep xa))"
TameEnumP_def: fixes p :: "nat" shows "TameEnum\<^bsub>p\<^esub> \<equiv> {g. Seed\<^bsub>p\<^esub> [next_tame\<^bsub>p\<^esub>]\<rightarrow>* g \<and> final g}"
Units_def: shows "transformation.Sym = {u \<in> G \<rightarrow>\<^sub>E G. transformation.invertible u}"
preserves_comp: fixes g :: "'b \<times> 'b"   and f :: "'b \<times> 'b" assumes "B.VV.seq g f" shows "fst (B.VV.comp g f) \<star>\<^sub>B snd (B.VV.comp g f) = (fst g \<star>\<^sub>B snd g) \<cdot>\<^sub>B (fst f \<star>\<^sub>B snd f)"
LoadBalancing_def: fixes m :: "nat" shows "LoadBalancing m \<equiv> 0 < m"
singletonD: fixes b :: "'d"   and a :: "'d" assumes "b \<in> {a}" shows "b = a"
kTS: fixes y :: "'b" assumes "y \<in> T" shows "k y \<in> S"
array_max_def: fixes d :: "'a dynamic_array" shows "array_max d = Array.len (aref d)"
iff_upred_def: shows "(\<Leftrightarrow>) \<equiv> map_fun Rep_uexpr (map_fun Rep_uexpr mk\<^sub>e) (\<lambda>P Q A. P A = Q A)"
exhaust: fixes y :: "'a node" obtains "y = SOURCE"   | x2 :: "'a" where "y = Inner x2"
unit_factor_poly_altdef: fixes p :: "'b poly" shows "unit_factor p = monom (unit_factor (lead_coeff p)) 0"
qfree_qe_eps\<^sub>1: fixes \<phi> :: "atom fm" assumes "nqfree \<phi>" shows "qfree (qe_eps\<^sub>1 \<phi>)"
euler_trail_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "euler_trail u p v \<equiv> trail u p v \<and> set p = arcs G \<and> set (awalk_verts u p) = verts G"
h: fixes w :: "complex"   and y :: "complex" assumes "w \<noteq> z"   and "cmod (y - w) < cmod (w - z)" shows "(h w has_field_derivative 1 / (y - z)) (at y)"
sets_vimage_algebra: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and M :: "'c measure" shows "sets (vimage_algebra X f M) = sigma_sets X {f -` A \<inter> X |A. A \<in> sets M}"
aff_dim_translation_eq: fixes a :: "'c"   and S :: "'c set" shows "aff_dim ((+) a ` S) = aff_dim S"
priv_is_asm_priv_def: fixes mdss\<^sub>C :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "priv_is_asm_priv mdss\<^sub>C \<equiv> \<forall>i<length cms. priv_mem\<^sub>C ! i \<subseteq> (mdss\<^sub>C ! i) AsmNoReadOrWrite"
Domainp_applyI: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and x :: "'e"   and y :: "'f" assumes "(A ===> B) f g"   and "A x y" shows "Domainp B (f x)"
of_int_round_ge: fixes x :: "'a" shows "x - (1::'a) / (2::'a) \<le> of_int (round x)"
subst_typ'_simulates_tsubst_gen: fixes t :: "Core.term"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubst t \<rho> = subst_typ' (map (\<lambda>(x, y). ((x, y), \<rho> x y)) (SOME l. distinct l \<and> tvs t \<subseteq> set l)) t"
tranclp_induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r a y \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r\<^sup>+\<^sup>+ a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
horizontal_boundary_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "horizontal_boundary twoC \<equiv> {(1, \<lambda>x. twoC (x, 0)), (- 1, \<lambda>x. twoC (x, 1))}"
join_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<sqsubseteq> x \<squnion> y"
r_def: fixes x :: "'a" shows "r x = (1::'a) \<sqinter> \<top> \<cdot> x"
intro: fixes set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')" shows "l_set_shadow_root_get_tag_name set_shadow_root_locs get_tag_name_locs"
A_RCOSETS_defs_1: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "a_rcosets\<^bsub>G\<^esub> H = rcosets\<^bsub>add_monoid G\<^esub> H"
oo_assoc: fixes f :: "'t \<rightarrow> 's"   and g :: "'u \<rightarrow> 't"   and h :: "'r \<rightarrow> 'u" shows "f oo g oo h = (f oo g) oo h"
simps: fixes s :: "('b \<times> 'b) set"   and ns :: "('b \<times> 'b) set"   and s2 :: "('c \<times> 'c) set" shows "lex_two s ns s2 = {((a1, b1), a2, b2). (a1, a2) \<in> s \<or> (a1, a2) \<in> ns \<and> (b1, b2) \<in> s2}"
ind_mor_btw_stalks_axioms_def: fixes X :: "'d set"   and x :: "'d" shows "ind_mor_btw_stalks_axioms X x \<equiv> x \<in> X"
general_theorem: fixes P :: "'a \<Rightarrow> bool"   and l :: "nat"   and f :: "'a \<Rightarrow> nat" assumes "\<forall>p. P p \<and> l < f p \<longrightarrow> (\<exists>p'. P p' \<and> f p' < f p)" shows "\<forall>p. P p \<longrightarrow> (\<exists>p'. P p' \<and> f p' \<le> l)"
prefix_fininf_def: fixes u :: "'a list"   and v :: "'a stream" shows "u \<le>\<^sub>F\<^sub>I v \<equiv> \<exists>w. u @- w = v"
frequently_elim1: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "frequently P F"   and "\<And>i. P i \<Longrightarrow> Q i" shows "frequently Q F"
cong: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "cancellative_sep_algebra.precise plus sep_disj \<equiv> cancellative_sep_algebra.precise plus sep_disj"
poly_set_iterateoi_axioms: shows "poly_set_iterateoi \<alpha> invar list_ordered_it"
assms_1: shows "P \<longmapsto> a<\<nu>x> \<prec> P'"
joinpaths_cong: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a" assumes "f = f'"   and "g = g'" shows "f +++ g = f' +++ g'"
that: fixes ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" shows "thesis"
hyps: assumes "(f, g) \<in> set ps" shows "monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f) \<in> set (pdata_pairs_to_list ps)"
induct: fixes x :: "('t, 'f) Language"   and P :: "('t, 'f) Language \<Rightarrow> bool" assumes "x \<in> Theory"   and "\<And>A. A \<in> aAxioms Ax \<Longrightarrow> P A"   and "\<And>A e B. Sig aS \<triangleright> Vx : A \<turnstile> e : B \<Longrightarrow> P Vx : A \<turnstile> e\<equiv>e : B"   and "\<And>A e1 e2 B. \<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory; P Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> e2\<equiv>e1 : B"   and "\<And>A e1 e2 B e3. \<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory; P Vx : A \<turnstile> e1\<equiv>e2 : B; Vx : A \<turnstile> e2\<equiv>e3 : B \<in> Theory; P Vx : A \<turnstile> e2\<equiv>e3 : B\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> e1\<equiv>e3 : B"   and "\<And>A e1 e2 B f C. \<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory; P Vx : A \<turnstile> e1\<equiv>e2 : B; f \<in> Sig aS : B \<rightarrow> C\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> f E@ e1\<equiv>f E@ e2 : C"   and "\<And>A e1 B e2 e3 C. \<lbrakk>Sig aS \<triangleright> Vx : A \<turnstile> e1 : B; Vx : B \<turnstile> e2\<equiv>e3 : C \<in> Theory; P Vx : B \<turnstile> e2\<equiv>e3 : C\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> sub e1 in e2\<equiv>sub e1 in e3 : C" shows "P x"
uint32_sshiftr_def: fixes x :: "uint32"   and n :: "integer" shows "uint32_sshiftr x n = (if n < 0 \<or> 32 \<le> n then undefined (>>>) x n else x >>> nat_of_integer n)"
Annos_code_def: fixes P :: "'b J_prog" shows "Annos_code P = Annos (is_lub_sup P) P"
cases: fixes x :: "instr list \<times> (int \<times> (char list \<Rightarrow> int) \<times> int list) list" obtains P :: "instr list" and cf :: "int \<times> (char list \<Rightarrow> int) \<times> int list" and cf' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" where "x = (Pa__, cf # cf' # cfs)"   | P :: "instr list" where "x = (Pb__, [])"   | P :: "instr list" and v :: "int \<times> (char list \<Rightarrow> int) \<times> int list" where "x = (P, [v])"
finitesum_sub_carrier: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set" assumes "f \<in> I \<rightarrow> {X. submodule R M X}" shows "finitesum M I f \<subseteq> carrier M"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c stream" shows "sset (smap f v) = f ` sset v"
of_complex_power: fixes x :: "complex"   and n :: "nat" shows "of_complex (x ^ n) = of_complex x ^ n"
B_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Simulation_Invariant A B sim PA PB"   and "PB a"   and "B a b" shows "PB b"
adv_P1_def: fixes m1 :: "'msg1"   and m2 :: "'msg2"   and D :: "'view1 \<Rightarrow> bool spmf" shows "adv_P1 m1 m2 D \<equiv> \<bar>spmf (R1 m1 m2 \<bind> D) True - spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S1 m1 o1 \<bind> D)) True\<bar>"
size_list_estimation: fixes x :: "'a"   and xs :: "'a list"   and y :: "nat"   and f :: "'a \<Rightarrow> nat" assumes "x \<in> set xs"   and "y < f x" shows "y < size_list f xs"
finite_jumpFs_linear_pos: fixes c :: "real"   and f :: "real \<Rightarrow> real"   and b :: "real"   and lb :: "real"   and ub :: "real" assumes "0 < c" shows "finite_jumpFs (f \<circ> (\<lambda>x. c * x + b)) lb ub = finite_jumpFs f (c * lb + b) (c * ub + b)"
mk_poly_sub_others: fixes l :: "nat"   and j' :: "nat"   and i :: "nat"   and A :: "'b poly mat"   and j :: "nat" assumes "l \<noteq> j'"   and "i < dim_row A"   and "j' < dim_col A" shows "mk_poly_sub A l j $$ (i, j') = A $$ (i, j')"
c2_def: shows "c2 = \<bar>bernoulli (Suc n)\<bar> / real (Suc n)"
rep_inject: fixes x :: "('c, 'd) mapping"   and y :: "('c, 'd) mapping" shows "(mapping.rep x = mapping.rep y) = (x = y)"
rep_zmultiset_def: shows "rep_zmultiset \<equiv> quot_type.rep Rep_zmultiset"
exec_normalize_iff_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> \<Rightarrow> t = \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
indicator_pos_le: fixes S :: "'d set"   and x :: "'d" shows "(0::'c) \<le> indicator S x"
DSourcesS14: shows "DSources level2 sS14 = {sS11}"
bisimOutputPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<sim> M\<langle>N\<rangle>.Q"
lim: shows "(f \<longlongrightarrow> ereal x) net"
delete_empty: fixes k :: "'b" shows "delete k Mapping.empty = Mapping.empty"
empty: fixes c :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" shows "(c, [], c) \<in> trcl t"
u3_4_3: shows "sum_list u \<le> a ! k"
FAIL_leof: fixes m :: "'b nres" shows "FAIL \<le>\<^sub>n m"
rightUniqueInjectiveOnFirst: fixes P :: "('d \<times> 'e) set" shows "runiq P = inj_on fst P"
simps_1: fixes s :: "'b word"   and e :: "'b word" shows "wordinterval_empty (WordInterval s e) = (e < s)"
weak_complete_lattice_criterion1: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L"   and "\<exists>g. greatest L g (carrier L)"   and "\<And>A. \<lbrakk>A \<subseteq> carrier L; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>i. is_glb L i A" shows "weak_complete_lattice L"
circline_symmetric_inv_homo_disc': fixes a :: "complex_homo"   and a' :: "complex_homo" assumes "circline_symmetric a a' unit_circle" shows "a' = inversion a"
dghm_proj_ObjMap_vrange: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<R>\<^sub>\<circ> (\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>"
WS_observable_move: fixes ms\<^sub>1 :: "'node list"   and s\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms\<^sub>2 :: "'node list"   and s\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set"   and a :: "'edge"   and ms\<^sub>1' :: "'node list"   and s\<^sub>1' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "((ms\<^sub>1, s\<^sub>1), ms\<^sub>2, s\<^sub>2) \<in> WS S"   and "S,kind \<turnstile> (ms\<^sub>1,s\<^sub>1) -a\<rightarrow> (ms\<^sub>1',s\<^sub>1')"   and "s\<^sub>1' \<noteq> []" obtains as :: "'edge list" where "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>1', transfer (slice_kind S a) s\<^sub>2) \<in> WS S" and "S,slice_kind S \<turnstile> (ms\<^sub>2,s\<^sub>2) =as @ [a]\<Rightarrow> (ms\<^sub>1',transfer (slice_kind S a) s\<^sub>2)"
l_inv_ex: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)" shows "\<exists>y\<in>carrier (UP INTEG). y \<oplus>\<^bsub>UP INTEG\<^esub> x = \<zero>\<^bsub>UP INTEG\<^esub>"
Inf_insert: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 a :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (insert a A) = inf a (Inf A)"
diff_iff: fixes f :: "'a \<Rightarrow> 'd" shows "diff_fun k charts f = (\<forall>x\<in>carrier. \<exists>c1\<in>atlas. \<exists>c2\<in>manifold_eucl.atlas k. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1))"
While: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int" shows "(WHILE b DO c, s) \<rightarrow> (IF b THEN c;; WHILE b DO c ELSE SKIP, s)"
llength_ldrop: fixes n :: "enat"   and xs :: "'c llist" shows "llength (ldrop n xs) = (if n = \<infinity> then 0 else llength xs - n)"
ac: shows "index (swaps (snd a) s) (qs ! i) = (\<Sum>e\<in>set s. ALG e qs i (swaps (snd a) s, ()))"
dagger_unfoldl: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<dagger> \<le> x\<^sup>\<dagger>"
trapC_eq: shows "trapC = {p. (fst p)\<^sup>2 + (snd p)\<^sup>2 - 4 \<le> 0} \<inter> {p. 1 / 4 - ((fst p)\<^sup>2 + (snd p)\<^sup>2) \<le> 0}"
gcd_greatest: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "dvd.dvd times c (gcd a b)"
convolution_def: fixes M :: "'c measure"   and N :: "'c measure" shows "(M \<star> N) = distr (M \<Otimes>\<^sub>M N) borel (\<lambda>(x, y). x + y)"
ccloses_id: fixes \<Delta> :: "(coname \<times> ty) list"   and x :: "name" shows "idc \<Delta> x ccloses \<Delta>"
codomains_null: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "partial_magma C" shows "partial_magma.codomains C (partial_magma.null C) = {}"
fhim: shows "f ` h ` {0..1} = {0..1}"
pos_oriented_cmat_mult_negative: fixes H1 :: "complex \<times> complex \<times> complex \<times> complex"   and H2 :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H1 \<and> H1 \<noteq> mat_zero"   and "hermitean H2 \<and> H2 \<noteq> mat_zero"   and "\<exists>k<0. H2 = cor k *\<^sub>s\<^sub>m H1" shows "pos_oriented_cmat H1 = (\<not> pos_oriented_cmat H2)"
plan_is_at_least_singleton_plan_if_trace_has_at_least_two_elements: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" obtains ops :: "'a strips_operator list" and \<pi>' :: "'a strips_operator list list" where "\<pi> = ops # \<pi>'"
foldl_map_append_is_some_if: fixes b :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a"   and ms :: "('b \<Rightarrow> 'a option) list" assumes "b x = Some y \<or> (\<exists>m\<in>set ms. m x = Some y)"   and "\<forall>m'\<in>set ms. m' x = Some y \<or> m' x = None" shows "foldl (++) b ms x = Some y"