hcju/isabelleps huggingface dataset:
Nfunc_LeastTr2_2: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "i \<le> n"   and "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "NfuncPair_neq_at f g i" shows "(LEAST k. NfuncPair_neq_at f g k) \<le> i"
cod_tuple: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.dom C f = partial_magma.dom C g" shows "partial_magma.cod C (elementary_category_with_binary_products.tuple C pr0 pr1 f g) = elementary_category_with_binary_products.prod C pr0 pr1 (partial_magma.cod C f) (partial_magma.cod C g)"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes "x1 \<longrightarrow>cbv x2" shows "pi \<bullet> x1 \<longrightarrow>cbv pi \<bullet> x2"
linear_subspace_vimage: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'e set" assumes "clinear f"   and "csubspace S" shows "csubspace (f -` S)"
liminf_bounded_iff: fixes C :: "ereal"   and x :: "nat \<Rightarrow> ereal" shows "(C \<le> liminf x) = (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)"
mker_of_mpj: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "ker\<^bsub>M,M /\<^sub>m H\<^esub> mpj M H = H"
assms: shows "alw (\<lambda>xs. P (shd (stl xs)) (shd xs)) xs"
Reals_add: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>"   and "b \<in> \<real>" shows "a + b \<in> \<real>"
token_fails_alt_def: fixes x :: "nat" shows "token_fails x = (\<forall>\<^sub>\<infinity>n. sink (token_run x n) \<and> token_run x n \<notin> F)"
shift_def: fixes E :: "nat \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" shows "E\<langle>n:x\<rangle> = (\<lambda>m. if m < n then E m else if m = n then x else E (m - 1))"
hyps_2: assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (p * q)) \<subseteq> \<real>"
short_take_append: fixes n :: "nat"   and u :: "'b list"   and v :: "'b list" assumes "n \<le> \<^bold>|u\<^bold>|" shows "take n (u \<cdot> v) = take n u"
chain_def: fixes Y :: "nat \<Rightarrow> 'a" shows "chain Y = (\<forall>i. Y i \<sqsubseteq> Y (Suc i))"
ereal_less_eq_6: fixes r :: "real" shows "(ereal r \<le> 1) = (r \<le> 1)"
objI_src: fixes a :: "'a arrow_of_spans_data" assumes "arr a"   and "src a = a" shows "obj a"
assms: shows "z \<noteq> \<infinity>\<^sub>h \<or> w \<noteq> \<infinity>\<^sub>h"
simps_5: fixes v :: "'b poly"   and va :: "'b poly" shows "degreen (Sub v va) = (\<lambda>m. 0)"
inf_ordering_inv_img: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "infinite X"   and "TernaryOrdering.ordering f ord X" shows "f -` X = {0..}"
IDFT_lower: fixes m :: "nat"   and a :: "nat \<Rightarrow> complex"   and i :: "nat" shows "IDFT (2 * m) a i = IDFT m (\<lambda>i. a (2 * i)) i + (1 / FFT.root (2 * m)) ^ i * IDFT m (\<lambda>i. a (2 * i + 1)) i"
rsd_winners_nonempty: fixes alts' :: "'alt set"   and agents' :: "'agent list" assumes "finite alts"   and "alts' \<noteq> {}"   and "set agents' \<subseteq> agents"   and "alts' \<subseteq> alts" shows "rsd_winners R alts' agents' \<noteq> {}"
asIMOD_igWlsAllDisj: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsNE SEM"   and "sWlsDisj SEM" shows "igWlsAllDisj (asIMOD SEM)"
simps_2: fixes wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat" shows "alphabet.wf wf_atom n Full = True"
sets_1: fixes x :: "'a" assumes "x \<in> A" shows "{x} \<in> sets M"
rel_sum_simps_4: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a2 :: "'d"   and b2 :: "'e" shows "rel_sum R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
valid_reach_src_tgt: fixes tr :: "('state, 'act, 'out) trans list" assumes "valid tr"   and "reach (srcOf (hd tr))" shows "reach (tgtOf (last tr))"
typeVar: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and Vn :: "char list"   and lev :: "secLevel" assumes "\<Gamma> Vn = Some lev" shows "\<Gamma> \<turnstile> Var Vn : lev"
gcd_diff1_nat: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "gcd (m - n) n = gcd m n"
funcsetI: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and B :: "'f set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
res_assoc1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<rhd> (y \<rhd> z) = y \<cdot> x \<rhd> z"
acyclic_6_implies_5a: fixes x :: "'a" assumes "acyclic_6 x" shows "acyclic_5a x"
con_rews_1: shows "StateT\<cdot>\<bottom> = \<bottom>"
fps_deriv_eq_0_iff: fixes f :: "'a fps" shows "(fps_deriv f = 0) = (f = fps_const (fps_nth f 0))"
CompDiag_Inv_Can: fixes t :: "'c local.term" assumes "Can t"   and "Diag t" shows "Inv t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = Dom t"
wens_single_def: fixes act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wens_single act B \<equiv> \<Union>i. (wp act ^^ i) B"
finite_nat_set_iff_bounded_le: fixes N :: "nat set" shows "finite N = (\<exists>m. \<forall>n\<in>N. n \<le> m)"
run_owner_ii_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "case run_owner_ii n s of (s', S, A, x) \<Rightarrow> {(Asset n, Crypt (Auth_ShaKey n) (PriKey S)), (Owner n, \<lbrace>Num 1, PubKey A\<rbrace>)} \<subseteq> s'"
append_prefixD: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "prefix (xs @ ys) zs" shows "prefix xs zs"
no_loops': fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs G" shows "u \<noteq> v"
red_preserves_sconf: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"   and "P,E,hp s,shp s \<turnstile> e : T"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
connectedin_absolute: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin (subtopology X S) S = connectedin X S"
profile_rprD: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set"   and i :: "'d" assumes "profile A Is P"   and "i \<in> Is" shows "rpr A (P i)"
init_dec: fixes \<Delta>0 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and \<Delta>s :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set" assumes "\<Delta>0 \<in> \<Delta>s"   and "\<And>vl vl1. B vl vl1 \<Longrightarrow> \<Delta>0 istate vl istate vl1" shows "\<forall>vl vl1. B vl vl1 \<longrightarrow> disjAll \<Delta>s istate vl istate vl1"
llength_ltake: fixes n :: "enat"   and xs :: "'b llist" shows "llength (ltake n xs) = min n (llength xs)"
normalize_quot_div_unit_right: fixes u :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_unit u" shows "normalize_quot (x, y div u) = (fst (normalize_quot (x, y)) * u, snd (normalize_quot (x, y)))"
x'_def: shows "x' = x - real_of_int (2 * k) * pi"
pseudoinverse: fixes \<omega> :: "real"   and x :: "real" assumes "0 < \<omega>"   and "\<omega> < 1" shows "(\<omega> \<le> M.C x) = (M.I \<omega> \<le> x)"
between_nth_primes_imp_nonprime: fixes k :: "nat"   and n :: "nat" assumes "nth_prime k < n"   and "n < nth_prime (Suc k)" shows "\<not> prime n"
comm_scalar_prod: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> v\<^sub>2 = v\<^sub>2 \<bullet> v\<^sub>1"
B'_wf': shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set B')"
struct_subst_command: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T :: "Syntax.type"   and x :: "nat"   and \<Delta>' :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and U :: "Syntax.type"   and T1 :: "Syntax.type"   and \<alpha> :: "nat"   and \<beta> :: "nat" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T"   and "\<Delta> x = T"   and "\<Gamma> , \<Delta>' \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : U \<Leftarrow> T1"   and "\<Delta> = \<Delta>'\<langle>\<alpha>:T1\<rangle>"   and "\<lbrakk>\<Gamma> , \<Delta>' \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : U \<Leftarrow> T1; \<Delta> = \<Delta>'\<langle>\<alpha>:T1\<rangle>\<rbrakk> \<Longrightarrow> \<Gamma> , \<Delta>'\<langle>\<beta>:U\<rangle> \<turnstile>\<^sub>T t[\<alpha>=\<beta> liftM_ctxt E \<beta>]\<^sup>T : T" shows "\<Gamma> , \<Delta>'\<langle>\<beta>:U\<rangle> \<turnstile>\<^sub>C (<x>t)[\<alpha>=\<beta> liftM_ctxt E \<beta>]\<^sup>C"
env_C_restr_env_C_restr: fixes r :: "C"   and r' :: "C"   and v :: "'a \<Rightarrow> C \<rightarrow> 'b" shows "(v|\<^sup>\<circ>\<^bsub>r'\<^esub>)|\<^sup>\<circ>\<^bsub>r\<^esub> = v|\<^sup>\<circ>\<^bsub>(r' \<sqinter> r)\<^esub>"
mr_iD: fixes p :: "('a, 'b) Combinators list"   and matching_rule :: "'d \<Rightarrow> ('a, 'b) Combinators list \<Rightarrow> 'c"   and x :: "'d" assumes "wellformed_policy1_strong p" shows "matching_rule x p = matching_rule x (insertDeny p)"
expand: fixes code_status :: "'a code_status"   and code_status' :: "'a code_status" assumes "is_cfailed code_status = is_cfailed code_status'"   and "\<lbrakk>is_cfailed code_status; is_cfailed code_status'\<rbrakk> \<Longrightarrow> the_error code_status = the_error code_status'"   and "(code_status = CSUCCESS) = (code_status' = CSUCCESS)" shows "code_status = code_status'"
cycle_free_up_to_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" shows "cycle_free_up_to m k n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (\<forall>j\<le>n. len m i j (rem_cycles i j xs) \<le> len m i j xs) \<and> \<one> \<le> len m i i xs"
denote_while_n_sum_trace: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat"   and n :: "nat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "trace (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n) \<le> trace \<rho>"
Inf_specs_def: fixes d :: "nat"   and X :: "real list set" shows "Inf_specs d X = SPEC (\<lambda>r. length r = d \<and> (\<forall>x\<in>X. list_all2 (\<le>) r x))"
List1Red2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and es' :: "'addr expr1 list"   and s' :: "'heap \<times> 'addr val list"   and v :: "'addr val" assumes "uf,P,t \<turnstile>1 \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val v # es,s\<rangle> [-ta\<rightarrow>] \<langle>Val v # es',s'\<rangle>"
intros: fixes ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and ve' :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and ds :: "'a \<aproc> set list"   and ds' :: "'a \<aproc> set list"   and proc :: "'a \<aproc>"   and b :: "'a" assumes "ve \<lessapprox> ve'"   and "ds \<lessapprox> ds'" shows "(proc, ds, ve, b) \<lessapprox> (proc, ds', ve', b)"
val_nonpos_inv_pos: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "\<not> 0 \<le> v x" shows "0 < v (x\<^bsup>\<hyphen> K\<^esup>)"
project_stable_I: fixes F :: "'c program"   and A :: "'a set"   and C :: "'c set" assumes "F \<in> stable (extend_set h A)" shows "project h C F \<in> stable A"
subst_cls_lists_comp_substs: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and Cs :: "'b literal multiset list"   and \<tau>s :: "'c list"   and \<sigma>s :: "'c list" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_cls_lists subst_atm Cs (substitution_ops.comp_substs comp_subst \<tau>s \<sigma>s) = substitution_ops.subst_cls_lists subst_atm (substitution_ops.subst_cls_lists subst_atm Cs \<tau>s) \<sigma>s"
xivp_1: shows " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)"
distinct_drop: fixes xs :: "'g list"   and i :: "nat" assumes "distinct xs" shows "distinct (drop i xs)"
AboveS_Field: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "x \<in> AboveS r X" shows "x \<in> Field r"
k_is_map: shows "is_left_adjoint r\<^sub>0"
set_rev: fixes xs :: "'a list" shows "set (rev xs) = set xs"
Aaj: shows "A $ a $ j \<noteq> (0::'a)"
simps: fixes b :: "bexp"   and bv :: "bool" shows "lmaps_to_axpr b bv = (\<lambda>ps. vars b \<subseteq> dom ps \<and> pbval b ps = bv)"
finite_comm_group_def: fixes G :: "('d, 'e) monoid_scheme" shows "finite_comm_group G \<equiv> finite_group G \<and> comm_group G"
eqvt_lambda: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'd" shows "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> f (unpermute p x)"
lebesgue_measurable_imp_measurable_on_nnreal: fixes u :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "u \<in> borel_measurable lebesgue"   and "\<And>x. 0 \<le> u x"   and "S \<in> sets lebesgue" shows "u measurable_on S"
a: shows " \<Gamma> \<subseteq> \<Gamma>' "
respects_own_guarantees_def: fixes cm :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)" shows "conc.respects_own_guarantees cm \<equiv> \<forall>x. (x \<in> snd cm GuarNoReadOrWrite \<longrightarrow> conc.doesnt_read_or_modify (fst cm) x) \<and> (x \<in> snd cm GuarNoWrite \<longrightarrow> conc.doesnt_modify (fst cm) x)"
RedCondF: fixes E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>if (false) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e\<^sub>2,s\<rangle>"
mergec_UNION: fixes f :: "'c \<Rightarrow> 'c cset"   and C :: "'c cset"   and g :: "'c \<Rightarrow> 'm" shows "mergec (cUnion (cimage f C)) g = mergec C (\<lambda>x. mergec (f x) g)"
list_ex_rec_list: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "list_ex P xs = rec_list False (\<lambda>x xs. (\<or>) (P x)) xs"
upd_set'_aux1: fixes d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and k :: "'a"   and a :: "'c"   and A :: "('c \<times> 'a) set" shows "upd_set' Mapping.empty d f {b. b = k \<or> (a, b) \<in> A} = Mapping.update k d (upd_set' Mapping.empty d f {b. (a, b) \<in> A})"
K3_imp_K2: 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 "A \<notin> bad"   and "evs \<in> kerbV" shows "\<exists>Ta AT. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta\<rbrace>, AT\<rbrace> \<in> set evs"
i: shows "i < mod_type_class.from_nat (Suc k)"
causal_Ghost\<^sub>s\<^sub>b: shows "causal_program_history (Ghost A L R W # is\<^sub>s\<^sub>b) sb"
subs_Red_I: shows "{\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL} \<subseteq> Red_I_\<G> NL"
assms_5: shows "\<not> M1 \<preceq> M2"
saturated_without_red_F: fixes N :: "'f set" assumes "saturated N" shows "saturated (N - Red_F N)"
dec_X: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "X m \<subseteq> PF.emb {0..<m} {0..<n} (X n)"
simps_8: fixes n :: "nat"   and p :: "pf" shows "dfa_of_pf' n (Neg p) = negate_dfa (dfa_of_pf' n p)"
leg_simps_1: shows "C.arr Dom\<mu>_Dom\<nu>.leg0"
collapse_2: fixes pr_operation' :: "pr_operation'" assumes "is_NONSAT_PUSH' pr_operation'" shows "pr_operation' = NONSAT_PUSH'"
MD_carrier: shows "mat_delete (make_mat n k B) 0 k \<in> carrier_mat (n - 1) (n - 1)"
comp_termFSwImorph: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme"   and h' :: "'g \<Rightarrow> 'j"   and hA' :: "'h \<Rightarrow> 'k"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'j, 'k, 'l) model_scheme" assumes "termFSwImorph h hA MOD"   and "FSwImorph h' hA' MOD MOD'" shows "termFSwImorph (h' \<circ> h) (hA' \<circ> hA) MOD'"
intro: fixes G :: "('d, 'c) pre_digraph"   and arev :: "'c \<Rightarrow> 'c" assumes "\<And>a. (a \<in> arcs G) = (arev a \<noteq> a)"   and "\<And>a. a \<in> arcs G \<Longrightarrow> arev (arev a) = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> tail G (arev a) = head G a" shows "bidirected_digraph_axioms G arev"
V_alt: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "Graph.V c = fst ` Graph.E c \<union> snd ` Graph.E c"
assms_2: shows "D' = D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs (ps -- sps) hs data'))"
inv_def: fixes f :: "'e \<Rightarrow> 'd" shows "inv f = (\<lambda>y. SOME x. f x = y)"
P1: shows "Bet B C C0 \<and> Cong C C0 B' C'"
list_all2_find_element: fixes x :: "'b"   and a :: "'b list"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c list" assumes "x \<in> set a"   and "list_all2 R a b" shows "\<exists>y. y \<in> set b \<and> R x y"
Ne_def: fixes v :: "'a aexp"   and va :: "'a aexp" shows "Ne v va \<equiv> gNot (Eq v va)"
subset_of_face_of: fixes T :: "'b set"   and S :: "'b set"   and u :: "'b set" assumes "T face_of S"   and "u \<subseteq> S"   and "T \<inter> rel_interior u \<noteq> {}" shows "u \<subseteq> T"
subgraph_refl: fixes G :: "('c, 'd) labeled_graph" shows "subgraph G G = graph G"
intros_1: fixes x :: "'b"   and xs :: "'b list" shows "prefixList [] (x # xs)"
getChoices_p_snd: fixes g :: "gState\<^sub>I"   and p :: "pState"   and es :: "edge list" shows "snd ` set (getChoices g p es) \<subseteq> {p}"
tmadd_nb0: fixes t :: "tm"   and s :: "tm" assumes "tmbound0 t"   and "tmbound0 s" shows "tmbound0 (tmadd t s)"
Stone_Weierstrass_polynomial_function: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and e :: "real" assumes "compact S"   and "continuous_on S f"   and "0 < e" shows "\<exists>g. polynomial_function g \<and> (\<forall>x\<in>S. norm (f x - g x) < e)"
pcr_cr_eq: shows "pcr_antichain (=) = cr_antichain"
InstanceOf1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and T :: "ty"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a instanceof T,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
bind_stateT_def: shows "Monad_Overloading.bind = bind_state Monad_Overloading.bind"
pwd_spy_ii_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_ii n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_ii_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
finite_measure_axioms: shows "finite_measure (Pi\<^sub>M (insert i' I) M)"
strict_trans2: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "b \<prec>\<^sub>t a"   and "c \<preceq>\<^sub>t b" shows "c \<prec>\<^sub>t a"
exec_compile: fixes stack :: "'a list" shows "exec (compile e) stack env = eval e env # stack"
a_subid_aux1': fixes y :: "'a"   and x :: "'a" shows "y \<cdot> ar x \<le> y"
select_convs_2: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
CompDiag_Cod_Diag: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.CompDiag C (monoidal_language.Cod C t) t = t"
get_add_neq: fixes x :: "'fun"   and y :: "'fun"   and e :: "'fenv_inca"   and v :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef" assumes "x \<noteq> y" shows "Finca_get (Finca_add e x v) y = Finca_get e y"
single_valued_relcomp: fixes r :: "('c \<times> 'd) set"   and s :: "('d \<times> 'e) set" assumes "single_valued r"   and "single_valued s" shows "single_valued (r O s)"
cond_spmf_fst_try2: fixes x :: "'a"   and p :: "('a \<times> 'b) spmf"   and q :: "('a \<times> 'b) spmf" assumes "x \<notin> fst ` set_spmf p" shows "cond_spmf_fst (TRY p ELSE q) x = (if lossless_spmf p then return_pmf None else cond_spmf_fst q x)"
assms: shows "\<forall>n\<ge>n'. \<gamma> t n"
Sup_aform_def: fixes X :: "'b \<times> 'b pdevs" shows "Sup_aform X = fst X + tdev (snd X)"
Pow_top: fixes A :: "'c set" shows "A \<in> Pow A"
vsv_axioms: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "vsv (VLambda A f)"
it_to_it_fold: fixes it :: "('d list \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'd list \<Rightarrow> 'd list"   and it' :: "('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "proper_it it it'" shows "it_to_it it = it'"
1: shows "\<l>[\<I>] = \<iota> \<and> \<r>[\<I>] = \<iota>"
lm023: fixes X :: "'c set"   and mx :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite X"   and "mx \<in> X"   and "\<forall>x\<in>X. x \<noteq> mx \<longrightarrow> f x < f mx" shows "argmax f X = {mx}"
map_simp: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.map f = f"
wti: shows "wti c pc (\<tau>s ! pc) \<noteq> \<top>"
lexord_append_rightI: fixes y :: "'b list"   and x :: "'b list"   and r :: "('b \<times> 'b) set" assumes "\<exists>b z. y = b # z" shows "(x, x @ y) \<in> lexord r"
set_mult_assoc: fixes M :: "'a set"   and H :: "'a set"   and K :: "'a set" assumes "M \<subseteq> carrier R"   and "H \<subseteq> carrier R"   and "K \<subseteq> carrier R" shows "M <#>\<^bsub>add_monoid R\<^esub> H <#>\<^bsub>add_monoid R\<^esub> K = M <#>\<^bsub>add_monoid R\<^esub> (H <#>\<^bsub>add_monoid R\<^esub> K)"
concat_P1_P1: fixes f :: "nat \<Rightarrow> nat option"   and g :: "nat \<Rightarrow> nat option" assumes "f \<in> \<P>"   and "g \<in> \<P>" shows "(\<lambda>x. if g x \<down> \<and> f (the (g x)) \<down> then Some (the (f (the (g x)))) else None) \<in> \<P>"
not_neutral_obtains_not_neutral: fixes g :: "'c \<Rightarrow> 'b" assumes "Sum_any g \<noteq> (0::'b)" obtains a :: "'c" where "g a \<noteq> (0::'b)"
graph_more_rel_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rw :: "('h \<times> 'k) set" shows "\<langle>Rm, Rv, Rw\<rangle>graph_more_rel \<equiv> {(g, g'). (nodes g, nodes g') \<in> \<langle>Rv\<rangle>set_rel \<and> (edges g, edges g') \<in> \<langle>Rv \<times>\<^sub>r Rw \<times>\<^sub>r Rv\<rangle>set_rel \<and> (graph.more g, graph.more g') \<in> Rm}"
AllI: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<And>x. [\<phi> x in v]" shows "[\<^bold>\<forall>x. \<phi> x in v]"
dom_simp: fixes f :: "'a" assumes "hom.arr f" shows "hom.dom f = local.dom f"