hcju/isabelleps huggingface dataset:
true_clss_union: fixes I :: "'b set"   and CC :: "'b literal multiset set"   and DD :: "'b literal multiset set" shows "(I \<Turnstile>s CC \<union> DD) = (I \<Turnstile>s CC \<and> I \<Turnstile>s DD)"
IH2_3: fixes i :: "nat" assumes "i < Suc p" shows "xs''' ! i = xs'' ! i"
continuous_map_eq: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "continuous_map X X' f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "continuous_map X X' g"
arr_char: fixes f :: "'a" shows "S.arr f = right \<omega> f"
vacant2: shows "worklists_vacant_to c (results_in t'' (sum_weights (map (\<lambda>(s, l, t). l) xs)))"
new_element_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'u" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
numsub_nb: fixes t :: "Cooper.num"   and s :: "Cooper.num" assumes "numbound0 t"   and "numbound0 s" shows "numbound0 (numsub t s)"
inversion_involution: fixes z :: "complex_homo" shows "inversion (inversion z) = z"
seq_compact_def: fixes S :: "'c set" shows "seq_compact S = (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
intros_18: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> =n\<Rightarrow> Abrupt s"
nm: shows "\<forall>m. 0 < m \<and> m < n - 1 \<longrightarrow> [a ^ m \<noteq> 1] (mod n)"
function_smult_eval: fixes a :: "nat \<Rightarrow> int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "b \<in> carrier Zp" shows "(a \<odot>\<^bsub>function_ring (carrier Zp) Zp\<^esub> f) b = a \<otimes> f b"
last_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "length xs = length ys" shows "last (zip xs ys) = (last xs, last ys)"
"2_2": shows "u\<^sub>1 \<preceq>\<^sub>F u\<^sub>2"
aab: shows "set {set {a}, set {a, b}} = vinsert (set {a}) (set {set {a, b}})"
index_zero_implies_same_group: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "\<B> index ps = 0" shows "\<exists>G\<in>\<G>. ps \<subseteq> G"
height_bal_tree: fixes n :: "nat"   and t :: "'a tree" assumes "n \<le> size t" shows "height (bal_tree n t) = nat \<lceil>log 2 (real (n + 1))\<rceil>"
simps_1: fixes e :: "(name, nat) finfun" shows "eval_trm e zer = 0"
get_ancestors_locs_def: shows "Shadow_DOM.get_ancestors_locs = Shadow_DOM.get_parent_locs"
mbisim_finite1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2" shows "finite (dom (thr s1))"
zz_def: shows "zz \<equiv> Variable (Suc (Suc (Suc 0)))"
polyfun_subset: fixes N :: "nat set"   and N' :: "nat set"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "N \<subseteq> N'"   and "polyfun N f" shows "polyfun N' f"
relevant_entries_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and i :: "instr"   and pc :: "nat" shows "relevant_entries P i pc = filter (is_relevant_entry P i pc)"
is_processT3_ST: fixes s :: "'b event list"   and t :: "'b event list"   and P :: "'b process" assumes "s @ t \<in> T P" shows "s \<in> T P"
Gmult_distrib_left: fixes g :: "'g"   and v :: "'v"   and v' :: "'v" assumes "g \<in> G"   and "v \<in> V"   and "v' \<in> V" shows "g *\<cdot> (v + v') = g *\<cdot> v + g *\<cdot> v'"
infinite_countable_subset': fixes X :: "'b set" assumes "infinite X" shows "\<exists>C\<subseteq>X. countable C \<and> infinite C"
Joints_def: fixes XS :: "('b \<times> 'b pdevs) list" shows "Joints XS = valuate (\<lambda>e. map (aform_val e) XS)"
rtrancl_fold_product: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set" shows "{((r, s), f a r, f a s) |r s a. a \<in> A}\<^sup>* = {((r, s), fold f w r, fold f w s) |r s w. w \<in> lists A}"
imply_completeness: fixes G :: "'b fm list"   and p :: "'b fm" assumes "\<forall>M s. list_all (semantics M s) G \<longrightarrow> M, s \<Turnstile> p" shows "\<turnstile> imply G p"
simps_2: fixes a :: "'b"   and x :: "'b list" shows "mset (a # x) = add_mset a (mset x)"
strip_unlift_vars: fixes x :: "nat itrm"   and n :: "nat" assumes "opaque x = []" shows "strip_context n (unlift_vars n x) 0 = unlift_vars 0 x"
measure_cond_pmf: fixes p :: "'c pmf"   and A :: "'c set"   and B :: "'c set" assumes "set_pmf p \<inter> A \<noteq> {}" shows "measure_pmf.prob (cond_pmf p A) B = measure_pmf.prob p (A \<inter> B) / measure_pmf.prob p A"
mon_w_empty: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme" shows "mon_w fg [] = {}"
norm_bound_def: fixes A :: "'a mat"   and b :: "real" shows "local.norm_bound A b \<equiv> \<forall>i j. i < dim_row A \<longrightarrow> j < dim_col A \<longrightarrow> norm (A $$ (i, j)) \<le> b"
holomorphic_on_Ln: fixes S :: "complex set" assumes "\<And>z. z \<in> S \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "Ln holomorphic_on S"
inducts: fixes x1 :: "('procS, 'procT) Proc"   and x2 :: "('procS, 'procT) Proc"   and P :: "('procS, 'procT) Proc \<Rightarrow> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "x1 \<R>\<lbrakk>\<cdot>\<rbrakk>L x2"   and "\<And>S. P (TargetTerm (\<lbrakk>S\<rbrakk>)) (SourceTerm S)" shows "P x1 x2"
size_bin_to_bl_aux: fixes n :: "nat"   and w :: "int"   and bs :: "bool list" shows "length (bin_to_bl_aux n w bs) = n + length bs"
inverse_composition_commute: fixes x :: "'a \<Rightarrow> 'a"   and y :: "'a \<Rightarrow> 'a" assumes "transformation.sub.invertible x"   and "transformation.sub.invertible y"   and "x \<in> Translations"   and "y \<in> Translations" shows "transformation.sub.inverse (compose G x y) = compose G (transformation.sub.inverse y) (transformation.sub.inverse x)"
uminus_set_def: fixes A :: "'b set" shows "- A = Collect (- (\<lambda>x. x \<in> A))"
trace_def: fixes sig :: "'c signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
disj_one_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "disj x one = one"
that: fixes k :: "nat"   and l :: "nat" assumes "k \<le> l"   and "\<sigma> (f k) = \<sigma> i"   and "\<sigma> (f l) = \<sigma> j" shows "thesis"
assms_4: fixes w1 :: "'a list"   and w2 :: "'a list" assumes "w1 \<in> \<langle>G\<rangle>"   and "P w1"   and "w2 \<in> \<langle>G\<rangle>"   and "P w2" shows "P (w1 \<cdot> w2)"
strict_implies_order: fixes a :: "'b zmultiset"   and b :: "'b zmultiset" assumes "a \<subset>#\<^sub>z b" shows "a \<subseteq>#\<^sub>z b"
in_roofed_essentialD: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" assumes "x \<in> RF\<^bsub>\<Gamma>\<^esub> S"   and "essential \<Gamma> (B \<Gamma>) S x" shows "x \<in> S"
signed_rconjactionD: fixes s :: "'a"   and x :: "'a \<times> bool" shows "signed_rconjaction s x \<equiv> map_prod (rconjby s) (\<lambda>b. b \<noteq> (fst x = s)) x"
simps_6: fixes v :: "'b" shows "(dFAILi \<le> dRETURN v) = False"
loc1_def: fixes s :: "(nat \<times> form) list"   and f :: "nat \<Rightarrow> nat \<times> (nat \<times> form) list" shows "loc1 s f \<equiv> f = failing_path (deriv s)"
independent_length_le_dimension: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and n :: "nat"   and E :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "ring.dimension R n K E"   and "ring.independent R K Us"   and "set Us \<subseteq> E" shows "length Us \<le> n"
Mk_def: shows "Mk = mat k k (\<lambda>(x, y). \<mu> x y)"
Trans: shows "P \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> P'"
cont_on_top: shows "continuous_on (path_image (\<lambda>x. twoC (x, 1))) F"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>xs gt. \<lbrakk>\<forall>z\<in>set xs. \<forall>y\<in>set xs. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x; \<forall>x\<in>set xs. \<not> gt x x\<rbrakk> \<Longrightarrow> \<not> ext gt xs xs" shows "ext_trans_before_irrefl_axioms ext"
Cons_1: fixes n :: "int"   and m :: "int" assumes "length\<cdot>xs = MkI\<cdot>n"   and "n \<le> m" shows "(xs ++ ys) !! MkI\<cdot>m = ys !! MkI\<cdot>(m - n)"
take_strict: fixes n :: "nat" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>\<bottom> = \<bottom>"
Main: fixes n :: "nat"   and r :: "nat" assumes "n < 2 ^ r" shows "B n \<le> P * 2 ^ (2 * k * r)"
\<theta>_ground_img: shows "ground (subst_range \<theta>)"
nth_weave: fixes i :: "nat"   and A :: "nat set"   and xs :: "'a list"   and ys :: "'a list" assumes "i < length (weave A xs ys)" shows "weave A xs ys ! i = (if i \<in> A then xs ! card {a \<in> A. a < i} else ys ! card {a \<in> - A. a < i})"
nth_eq_iff_index_eq: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < |xs|"   and "j < |xs|" shows "(xs ! i = xs ! j) = (i = j)"
minus_diff_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "uminus (minus a b) = minus b a"
61648: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<squnion> - y \<sqinter> - - x) = bot"
pp_ub2: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c list"   and y :: "'c" assumes "Semilat A r f"   and "set x \<subseteq> A"   and "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> x \<Squnion>\<^bsub>f\<^esub> y"
sample_var_get: fixes x :: "'v"   and f :: "int \<Rightarrow> ('v \<Rightarrow> int option) \<Rightarrow> 'm" shows "bind (sample_var X x) (\<lambda>i. get (f i)) = get (\<lambda>s. bind (sample_var X x) (\<lambda>i. f i s))"
MWLsSteps_det_cases_6: fixes \<iota> :: "nat"   and V :: "('exp, 'id) MWLsCom list"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "\<langle>spawn\<^bsub>\<iota>\<^esub> V,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>" obtains "\<alpha> = V" and "p = None" and "m' = m"
l1_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and gny :: "msg" shows "l1_step3 Ra A B gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xgnx} \<and> guessed_frame Ra xgny = Some gny \<and> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> (can_signal s A B \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress s Rb) {xny, xgnx, xgny, xsk} \<and> guessed_frame Rb xgny = Some gny \<and> guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))))) \<and> (Ra = test \<longrightarrow> Exp gny (NonceF (Ra $ nx)) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp s\<rparr>}"
CARD_1_UNIV: fixes x :: "'d" assumes "class.CARD_1 TYPE('d)" shows "UNIV = {x}"
jordan_nf_via_factored_charpoly: fixes A :: "'b mat"   and n :: "nat"   and es :: "'b list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])" shows "jordan_nf A (jordan_nf_via_factored_charpoly A es)"
m_s_def: fixes S :: "char list \<Rightarrow> 'av"   and X :: "char list set" shows "m\<^sub>s S X = (\<Sum>x\<in>X. m (S x))"
line_integral_around_boundary: shows "one_chain_line_integral F {j} (boundary twoC) = line_integral F {j} (\<lambda>x. twoC (x, 0)) + line_integral F {j} (\<lambda>y. twoC (1, y)) - line_integral F {j} (\<lambda>x. twoC (x, 1)) - line_integral F {j} (\<lambda>y. twoC (0, y))"
linear_cf_def: fixes r :: "('d \<times> 'd) set"   and linear_cf :: "'d set \<Rightarrow> 'd set" shows "linear_cf r linear_cf \<equiv> Linear_order r \<and> (\<forall>X. linear_cf X = set_option (MaxR.MaxR_opt r X))"
comp_in_homE: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr"   and a :: "(bool, bool list) concrete_category.arr"   and c :: "(bool, bool list) concrete_category.arr" assumes "\<guillemotleft>g \<cdot>\<^sub>J f : a \<rightarrow>\<^sub>J c\<guillemotright>" obtains b :: "(bool, bool list) concrete_category.arr" where "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>J c\<guillemotright>"
rawpsubstT: fixes t :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "t \<in> trm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubstT t txs \<in> trm"
split_beta: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and prod :: "'f \<times> 'g" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
diff_numeral_simps_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) (numeral.numeral one plus n) = neg_numeral.sub minus plus one m n"
convex_imp_locally_connected: fixes S :: "'a set" assumes "convex S" shows "locally connected S"
Abs_G_inverse: fixes y :: "'b" assumes "y \<in> G" shows "Rep_G (Abs_G y) = y"
integrable_altD_1: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and a :: "'a"   and b :: "'a" assumes "f integrable_on s" shows "(\<lambda>x. if x \<in> s then f x else (0::'b)) integrable_on cbox a b"
assms_4: fixes x :: "'a" assumes "x \<in> U"   and "P x" shows "Q x"
A_Issues_Tgs: fixes A :: "agent"   and authTicket :: "msg"   and authK :: "nat"   and T2 :: "nat"   and B :: "agent"   and evs :: "event list" assumes "Says A Tgs \<lbrace>authTicket, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "A Issues Tgs with Crypt authK \<lbrace>Agent A, Number T2\<rbrace> on evs"
split_list_first: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
of_nat_mono: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and i :: "nat"   and j :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "i \<le> j" shows "less_eq (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j)"
assms: fixes x :: "V" assumes "x \<in> A" shows "Card x"
pareto_efficient_scf_axioms: shows "pareto_efficient_scf agents' alts' lowered"
interaction_bound_IO_ignore: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and "consider" :: "'e \<Rightarrow> bool"   and input :: "'f" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "\<not> consider out" shows "interaction_bound consider (c input) \<le> interaction_bound consider gpv"
ltl_to_generalized_rabin\<^sub>C_af\<^sub>\<UU>_correct: fixes w :: "nat \<Rightarrow> 'a set"   and \<Sigma> :: "'a set list"   and \<phi> :: "'a ltl" assumes "range w \<subseteq> set \<Sigma>" shows "w \<Turnstile> \<phi> = accept\<^sub>G\<^sub>R_LTS (ltl_to_generalized_rabin\<^sub>C_af\<^sub>\<UU> \<Sigma> \<phi>) w"
scalar_product_def: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec"   and w :: "('b, 'c) Finite_Cartesian_Product.vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $h i * w $h i)"
domintros: fixes B :: "'a set"   and C :: "'a set" assumes "\<And>xa xc. \<lbrakk>f B \<subseteq> C; xa \<in> f B; \<not> local.mk_linear_orders_dom ({}, B - {xa}); xc \<in> f B\<rbrakk> \<Longrightarrow> False"   and "\<And>xc xd. \<lbrakk>\<not> local.mk_linear_orders_dom (C, B - {SOME x. x \<in> f B \<and> x \<notin> C}); xc \<in> f B; xc \<notin> C; xd \<in> f B\<rbrakk> \<Longrightarrow> False" shows "local.mk_linear_orders_dom (C, B)"
assoc: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "semigroup f" shows "f (f a b) c = f a (f b c)"
vsv_vdoubleton: fixes a :: "V"   and c :: "V"   and b :: "V"   and d :: "V" assumes "a \<noteq> c" shows "vsv (set {\<langle>a, b\<rangle>, \<langle>c, d\<rangle>})"
scast_eq_scast_id: fixes a :: "'b signed word"   and b :: "'b signed word" shows "(SCAST('b signed \<rightarrow> 'b) a = SCAST('b signed \<rightarrow> 'b) b) = (a = b)"
not_in_set_insert: fixes x :: "'c"   and xs :: "'c list" assumes "x \<notin> set xs" shows "List.insert x xs = x # xs"
HVZK_def: shows "chaum_ped_sigma.HVZK \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>R. chaum_ped_sigma.R h w e = chaum_ped_sigma.S h e) \<and> (\<forall>h\<in>valid_pub. \<forall>(a, z)\<in>set_spmf (S2 h e). check h a e z)"
ran_CompFun_subseteq: fixes HA :: "('a, 'b, 'c) hierauto" shows "\<Union> (ran (CompFun HA)) \<subseteq> SAs HA"
isRecv_def: fixes event :: "('a, 'b, 'c) event" shows "isRecv event = (\<exists>x31 x32 x33 x34 x35 x36. event = Recv x31 x32 x33 x34 x35 x36)"
inv_FactGroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and X :: "'c set" assumes "H \<lhd> G"   and "X \<in> carrier (G Mod H)" shows "inv\<^bsub>G Mod H\<^esub> X = set_inv\<^bsub>G\<^esub> X"
simps_2: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "ioutput \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = ioutput"
Yb: shows "[[a\<^sub>1 b a\<^sub>n]]"
inj_matrix_vector_mult: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "invertible A" shows "inj ((*v) A)"
sl10: shows "- c * e\<^sup>2 \<in> radical_sqrt"
map_of_append: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "table_of (xs @ ys) = table_of ys ++ table_of xs"
fdia_dom: fixes x :: "('a \<times> 'a) set" shows "relfdia x Id = rdom x"
sameDom_refl: fixes inp :: "'d \<Rightarrow> 'e option" shows "sameDom inp inp"
simps_1: fixes l :: "'a tree"   and v :: "'a"   and r :: "'a tree" shows "right \<langle>l, v, r\<rangle> = r"
adapt_stoch_proc_borel_measurable: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> 'e"   and N :: "'e measure" assumes "filtrated_prob_space M F"   and "adapt_stoch_proc F X N" shows "\<forall>n. X n \<in> M \<rightarrow>\<^sub>M N"
dlist_no_memb_foldr: fixes dl :: "'b dlist"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "\<not> member dl x" shows "DList.foldr f (DList.insert x dl) e = f x (DList.foldr f dl e)"
distinct_35: fixes x31 :: "POPLmarkRecord.type"   and x32 :: "trm"   and x81 :: "pat"   and x82 :: "trm"   and x83 :: "trm" shows "(\<lambda><:x31. x32) \<noteq> (LET x81 = x82 IN x83)"
init_onode_comp: fixes i :: "nat"   and S :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R :: "nat set" shows "init (\<langle>i : S : R\<rangle>\<^sub>o) = {(\<sigma>, NodeS i s R) |\<sigma> s. (\<sigma>, s) \<in> init S}"
inner_eq: fixes m :: "nat" assumes "odd m" shows "(\<Sum>i | \<exists>k. i = 2 ^ k * m \<and> i \<le> n. distinct_of_odd p i * i) = p m * m"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b llist" shows "(case LCons x21 x22 of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
Map_dom: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f" shows "Cop_S.Map (Cop_S.dom f) = Cop_S.Dom f"
0: shows "0 \<le> val_Zp (f' \<bullet> a) \<and> val_Zp (f' \<bullet> a) \<noteq> \<infinity>"
TH: fixes n :: "nat"   and s :: "'s"   and t :: "'s" assumes "prob {\<omega> \<in> space (T' I). \<omega> !! n = s} \<noteq> 0" shows "cond_prob (T' I) (\<lambda>\<omega>. stl \<omega> !! n = t) (\<lambda>\<omega>. \<omega> !! n = s) = pmf (K s) t"
lcm_unique_int: fixes d :: "int"   and a :: "int"   and b :: "int" shows "(0 \<le> d \<and> a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e)) = (d = lcm a b)"
minSetOfComponentsTestL2p1: shows "minSetOfComponents level2 {data10, data13} = {sS1}"
wf_snd_append: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and t :: "('b, 'a) Term.term" shows "wf\<^sub>s\<^sub>t V (S @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t]) = wf\<^sub>s\<^sub>t V S"
pr_ideal_def: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "pr_ideal R I addition multiplication zero unit \<equiv> comm_ring R addition multiplication zero unit \<and> ideal I R addition multiplication zero unit \<and> pr_ideal_axioms R I multiplication"
distinct_2: fixes x21 :: "char list"   and x22 :: "aexp" shows "x21 ::= x22 \<noteq> SKIP"
homotopic_loops_imp_homotopic_paths_null: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and a :: "'b" assumes "homotopic_loops s p (linepath a a)" shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
crdt_ops_unique_last: fixes xs :: "('a \<times> 'b) list"   and oid :: "'a"   and oper :: "'b"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops (xs @ [(oid, oper)]) deps" shows "oid \<notin> set (map fst xs)"
abs_summable_Dirichlet_L: fixes s :: "complex" assumes "1 < Re s" shows "summable (\<lambda>n. cmod (\<chi> n * of_nat n powr - s))"
callcl_host_Some: fixes cl :: "cl"   and t1s :: "t list"   and t2s :: "t list"   and f :: "host"   and ves :: "e list"   and vcs :: "v list"   and n :: "nat"   and m :: "nat"   and s :: "s"   and hs :: "host_state"   and s' :: "s"   and vcs' :: "v list"   and vs :: "v list"   and i :: "nat" assumes "cl = Func_host (t1s _> t2s) f"   and "ves = $$* vcs"   and "length vcs = n"   and "length t1s = n"   and "length t2s = m"   and "host_apply s (t1s _> t2s) f vcs hs = Some (s', vcs')" shows "\<lparr>s;vs;ves @ [Callcl cl]\<rparr> \<leadsto>_ i \<lparr>s';vs;$$* vcs'\<rparr>"
assms: shows "\<forall>i<j. \<tau> \<sigma> i = \<tau> \<sigma>' i"
intro: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" assumes "h \<in> ring_hom R S" shows "ring_hom_cring_axioms R S h"