hcju/isabelleps huggingface dataset:
sees_methods_rec: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option"   and Mm' :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D sees_methods Mm"   and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)" shows "P \<turnstile> C sees_methods Mm'"
that: fixes nsx :: "'node list"   and n :: "'node"   and nsx' :: "'node list"   and n' :: "'node" assumes "ns = nsx @ n # nsx'"   and "ns' = n' # nsx'"   and "n' \<in> obs_intra n S"   and "\<forall>nx\<in>set nsx'. \<exists>nx'. call_of_return_node nx nx' \<and> nx' \<in> S"   and "\<forall>xs x xs'. nsx = xs @ x # xs' \<and> obs_intra x S \<noteq> {} \<longrightarrow> (\<exists>x''\<in>set (xs' @ [n]). \<exists>nx. call_of_return_node x'' nx \<and> nx \<notin> S)" shows "thesis"
spec: fixes default_node_properties :: "'a"   and sinvar_spec :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"   and sinvar_impl :: "'b list_graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"   and receiver_violation :: "bool"   and offending_flows_impl :: "'b list_graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<times> 'b) list list"   and node_props_impl :: "('b, 'a) TopoS_Params \<Rightarrow> 'b \<Rightarrow> 'a"   and eval_impl :: "'b list_graph \<Rightarrow> ('b, 'a) TopoS_Params \<Rightarrow> bool" assumes "TopoS_List_Impl default_node_properties sinvar_spec sinvar_impl receiver_violation offending_flows_impl node_props_impl eval_impl" shows "SecurityInvariant sinvar_spec default_node_properties receiver_violation"
bij_betw_comp_iff2: fixes f' :: "'c \<Rightarrow> 'd"   and A' :: "'c set"   and A'' :: "'d set"   and f :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "bij_betw f' A' A''"   and "f ` A \<subseteq> A'" shows "bij_betw f A A' = bij_betw (f' \<circ> f) A A''"
tagged_division_of_trivial: fixes p :: "('a \<times> 'a set) set" shows "(p tagged_division_of {}) = (p = {})"
dense: fixes x :: "'d"   and y :: "'d" assumes "x < y" shows "\<exists>z>x. z < y"
analz_image_freshK_simps_63: fixes c :: "msg"   and G :: "msg set"   and A :: "msg set" assumes "c \<in> analz G" shows "c \<in> analz (A \<union> G)"
i_Exec_Stream_Previous_i_Exec_Stream_Init: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c n = (i_Exec_Comp_Stream trans_fun input c)\<^bsup>\<leftarrow> c\<^esup> n"
CC: fixes r :: "'d rexp"   and r' :: "'d rexp"   and s :: "'d rexp"   and s' :: "'d rexp" assumes "r ~~ r'"   and "s ~~ s'" shows "Conc r s ~~ Conc r' s'"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'a list" shows "(Suc n \<le> |xs|) = (\<exists>x ys. xs = x # ys \<and> n \<le> |ys|)"
assms_4: shows "val_Zp y \<le> val_Zp x"
primrec'_into_primrec: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1'" shows "f \<in> PrimRec1"
wf_lex_prod: fixes ra :: "('c \<times> 'c) set"   and rb :: "('d \<times> 'd) set" assumes "wf ra"   and "wf rb" shows "wf (ra <*lex*> rb)"
le_iff_lt_or_eq: fixes x :: "'a"   and y :: "'a" shows "(le\<cdot>x\<cdot>y = TT) = (lt\<cdot>x\<cdot>y = TT \<or> eq\<cdot>x\<cdot>y = TT)"
t: shows "t \<in>#\<^sub>z Auxiliary.image_zmset f M"
Inst: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and v :: "val"   and s1 :: "state"   and b :: "bool"   and T :: "ref_ty" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1"   and "b = (v \<noteq> Null \<and> G,snd s1\<turnstile>v fits RefT T)" shows "G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<rightarrow> s1"
Pi_gr_1: fixes n :: "nat" shows "Suc 0 < Pi n"
monom_0: fixes a :: "'b" shows "monom a 0 = [:a:]"
fup_eqvt: fixes \<pi> :: "perm" shows "\<pi> \<bullet> fup = fup"
ordertype_WW: shows "ordertype WW (lenlex less_than) = \<omega> \<up> \<omega>"
preserves_comp: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "F (g \<cdot>\<^sub>D f) = F g \<cdot>\<^sub>C F f"
assms: shows "\<forall>i\<le>n. f i = Suc 0"
empt: fixes x :: "'a \<times> 'b \<times> 'a" assumes "x \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0" shows "x \<notin> set []"
dp_consistency_iterator_empty_def: fixes P :: "'a \<Rightarrow> bool"   and lookup :: "'b \<Rightarrow> ('a, 'c option) state"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> ('a, unit) state"   and cnt :: "'b \<Rightarrow> bool"   and nxt :: "'b \<Rightarrow> 'b"   and sizef :: "'b \<Rightarrow> nat"   and empty :: "'a" shows "dp_consistency_iterator_empty P lookup update cnt nxt sizef empty \<equiv> dp_consistency_iterator P lookup update cnt nxt sizef \<and> dp_consistency_empty lookup update P empty"
jcons_def: shows "jcons \<equiv> all xx (neg (cnj (instInp Pinp (Var xx)) (instInpP Pinp 0 (instInp (encF N) (Var xx)))))"
RETURN_def: fixes x :: "'e" shows "RETURN x \<equiv> RES {x}"
dom_char: fixes f :: "'a1 \<times> 'a2" shows "local.dom f = (if C1.arr (fst f) \<and> C2.arr (snd f) then (C1.dom (fst f), C2.dom (snd f)) else (C1.null, C2.null))"
m_anti_mono: fixes h :: "nat"   and m :: "'a \<Rightarrow> nat"   and n :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a" assumes "Measure_wn h m n"   and "x \<le> y" shows "m y \<le> m x"
set_iterator_sng_def: fixes x :: "'e"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma>0 :: "'d" shows "set_iterator_sng x c f \<sigma>0 = (if c \<sigma>0 then f x \<sigma>0 else \<sigma>0)"
equivalence_in_bicategory_axioms: shows "internal_equivalence e e' \<phi> \<psi>'"
xt1_11: fixes b :: "'a"   and a :: "'a" assumes "b \<in> U"   and "a \<in> U"   and "b \<le>\<^sub>o\<^sub>w a"   and "a \<noteq> b" shows "b <\<^sub>o\<^sub>w a"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P mark_guards f c Q,A"
refl_onPD2: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "refl_onP A r"   and "r a b" shows "b \<in> A"
nfa_reach_subset_memb: fixes q0 :: "bool list"   and bss :: "bool list list"   and q :: "bool list" assumes "nfa_reach A q0 bss q"   and "nfa_is_node A q0"   and "list_all (is_alph n) bss" shows "subset_memb q (subset_dfs A q0)"
two_power_odd_mod_12: fixes n :: "nat" assumes "odd n"   and "1 < n" shows "[2 ^ n = 8] (mod 12)"
vmc_path_lnull_ltl_no_deadend: assumes "\<not> lnull (ltl P')" shows "vmc_path_no_deadend G P' (P $ n) p \<sigma>"
sums_minus_ereal: fixes f :: "nat \<Rightarrow> ereal"   and x :: "ereal" assumes "\<not> {- \<infinity>, \<infinity>} \<subseteq> range f"   and "(\<lambda>n. - f n) sums x" shows "f sums - x"
sources_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "sources (\<nu> \<star> \<mu>) = sources \<mu>"
octo_mult_cnj_conv_norm_aux: fixes x :: "octo" shows "octo_of_real ((norm x)\<^sup>2) = x * Octonions.cnj x"
measure_frontier: fixes S :: "'c set" assumes "bounded S" shows "Sigma_Algebra.measure lebesgue (frontier S) = Sigma_Algebra.measure lebesgue (closure S) - Sigma_Algebra.measure lebesgue (interior S)"
eq: shows "Keys (set fs) \<union> (Keys (set gs) \<union> set vs) = Keys (set gs) \<union> Keys (set fs)"
assms_3: shows "ipresIGFreshAll h' hA' MOD' MOD''"
cong_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "base.cong R \<equiv> equivp R \<and> rel_fun (rel R) R eval eval"
gcd_add2: 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 m :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd m (plus m n) = gcd m n"
cond_alt: shows "cond = (\<lambda>s. (V0 \<subseteq> dom (discovered s) \<longrightarrow> stack s \<noteq> []) \<and> \<not> is_break tarjan_params s)"
emeasure_eq_measure: fixes i :: "'i"   and A :: "'a set" shows "emeasure (M i) A = ennreal (M.prob i A)"
boolfun_cases_helper: fixes g :: "bool \<Rightarrow> bool" shows "g = (\<lambda>x. False) \<or> g = (\<lambda>x. x) \<or> g = (\<lambda>x. True) \<or> g = Not"
jensens_inequality: fixes X :: "'a \<Rightarrow> real"   and I :: "real set"   and a :: "real"   and b :: "real"   and q :: "real \<Rightarrow> real" assumes "integrable M X"   and "AE x in M. X x \<in> I"   and "I = {a<..<b} \<or> I = {a<..} \<or> I = {..<b} \<or> I = UNIV"   and "integrable M (\<lambda>x. q (X x))"   and "convex_on I q" shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
0: shows "{1, 2} \<subseteq> Units (residue_ring (p ^ n))"
coprime_imp_coprime: fixes c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "coprime c d"   and "\<And>e. \<lbrakk>\<not> is_unit e; e dvd a; e dvd b\<rbrakk> \<Longrightarrow> e dvd c"   and "\<And>e. \<lbrakk>\<not> is_unit e; e dvd a; e dvd b\<rbrakk> \<Longrightarrow> e dvd d" shows "coprime a b"
Wax_def: shows "Wax \<equiv> {[Pos (Eq (lOfWax \<sigma>) (rOfWax \<sigma>))] |\<sigma>. \<not> isRes \<sigma> \<and> protFw \<sigma>}"
1: shows "local.divide (f \<bullet> ns (k + m)) (f' \<bullet> ns (k + m)) = \<zero>"
p2r_def: fixes P :: "'a \<Rightarrow> bool" shows "\<lceil>P\<rceil> = {(s, s) |s. P s}"
term_equivE: fixes s :: "dB"   and t :: "dB" assumes "s \<leftrightarrow> t" obtains u :: "dB" where "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u" and "t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u"
pequiv_def: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "prog \<simeq> prog' \<equiv> \<forall>P. sound P \<longrightarrow> wp prog P = wp prog' P"
vconn_implies_hpar_3: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "src g = src f"
pow_mono_one: fixes a :: "'b"   and n :: "nat" assumes "(1::'b) \<le> a" shows "(1::'b) \<le> a ^ n"
algo_correct': shows "(all_left, SPEC \<circ> all_left_spec) \<in> \<langle>int_rel\<rangle>list_rel \<rightarrow> \<langle>\<langle>\<langle>nat_rel\<rangle>option_rel\<rangle>list_rel\<rangle>nres_rel"
le_trans_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "le_trans t u \<equiv> \<forall>P. sound P \<longrightarrow> t P \<tturnstile> u P"
split: fixes P :: "'h \<Rightarrow> bool"   and f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h"   and exception :: "'i exception" shows "P (case exception of Fault \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = ((exception = Fault \<longrightarrow> P f1) \<and> (\<forall>x2. exception = OK x2 \<longrightarrow> P (f2 x2)))"
adds_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" shows "plus.adds plus b a = (\<exists>k. a = plus b k)"
map_of_filter_in: fixes xs :: "('d \<times> 'c) list"   and k :: "'d"   and z :: "'c"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "table_of xs k = Some z"   and "P k z" shows "table_of (filter (\<lambda>(x, y). P x y) xs) k = Some z"
bisumTail_is_map_zip: fixes f :: "'b \<times> 'b \<Rightarrow> 'a"   and v :: "'b"   and l1 :: "'b list" shows "sum_list (map f (zip (v # l1) (l1 @ [0::'b]))) = bisumTail (\<lambda>x y. f (x, y)) (v # l1)"
space_measure_spmf: fixes p :: "'c spmf" shows "space (measure_spmf p) = UNIV"
filter_filter_mset: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and M :: "'a multiset" shows "filter_mset P (filter_mset Q M) = {#x \<in># M. Q x \<and> P x#}"
rep_eq: fixes x :: "'a ratfps" shows "local.quot_of_ratfps x = quot_of_fract (Rep_ratfps x)"
pr1_simps_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "arr (pr1 a b)"
np_no_jump: fixes x :: "abopt"   and j :: "jump"   and a' :: "val" assumes "x \<noteq> Some (Jump j)" shows "np a' x \<noteq> Some (Jump j)"
simps_2: fixes t' :: "transition"   and t :: "transition"   and r :: "nat"   and p :: "nat" shows "stored_reused_aux_per_reg t' t (Suc r) p = (if is_generalised_output_of t' t (Suc r) p then Some (Suc r, p) else stored_reused_aux_per_reg t' t r p)"
simps: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a list" shows "linorder_list0.sortedP le a = (a = [] \<or> (\<exists>ys x. a = x # ys \<and> (\<forall>y\<in>set ys. le x y) \<and> linorder_list0.sortedP le ys))"
homotopic_triviality: fixes S :: "'b set"   and T :: "'c set" shows "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and> continuous_on S g \<and> g ` S \<subseteq> T \<longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g) = ((S = {} \<or> path_connected T) \<and> (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c))))"
wf_inst_def: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "wf_inst \<Theta> \<rho> \<equiv> \<forall>v S. \<rho> v S \<noteq> Tv v S \<longrightarrow> has_sort (osig (sig \<Theta>)) (\<rho> v S) S \<and> wf_type (sig \<Theta>) (\<rho> v S)"
dist_le_zero_iff: fixes x :: "'b"   and y :: "'b" shows "(dist x y \<le> 0) = (x = y)"
Call1Params: 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"   and M :: "String.literal" 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\<bullet>M(es),s\<rangle> -ta\<rightarrow> \<langle>Val v\<bullet>M(es'),s'\<rangle>"
succ: fixes t :: "'a"   and u' :: "'c \<Rightarrow> 'a"   and u :: "'c \<Rightarrow> 'a" assumes "(0::'a) \<le> t"   and "u' = u \<oplus> t" shows "local.trans u u'"
rep_list_spair: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list (spair p q) = punit.spoly (rep_list p) (rep_list q)"
take_Nil: fixes n :: "Integer" assumes "n \<noteq> \<bottom>" shows "take\<cdot>n\<cdot>[] = []"
f_join_f_shrink_iT_Div_mod: fixes k :: "nat"   and I :: "nat set"   and xs :: "'b message_af list" assumes "0 < k"   and "\<forall>x\<in>I. x mod k = k - 1" shows "xs \<longmapsto> k \<Join>\<^sub> I = xs \<div> k \<Join>\<^sub> (I \<oslash> k)"
cases: fixes a :: "('a, 'b) ODE" assumes "osafe a" obtains c :: "'b" where "a = OVar c"   | \<theta> :: "('a, 'b) trm" and x :: "'b" where "a = OSing x \<theta>" and "dfree \<theta>"   | ODE1 :: "('a, 'b) ODE" and ODE2 :: "('a, 'b) ODE" where "a = OProd ODE1 ODE2" and "osafe ODE1" and "osafe ODE2" and "ODE_dom ODE1 \<inter> ODE_dom ODE2 = {}"
simps: fixes a1 :: "'t list"   and a2 :: "'t list" shows "ord_term_lin.lexordp_eq a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> x \<prec>\<^sub>t y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> x \<prec>\<^sub>t y \<and> \<not> y \<prec>\<^sub>t x \<and> ord_term_lin.lexordp_eq xs ys))"
supd_gupd: fixes r :: "oref"   and obj :: "obj"   and s :: "state" shows "supd gupd(r\<mapsto>obj) s = (fst s, gupd(r\<mapsto>obj) (snd s))"
gh: fixes x :: "'a" assumes "x \<in> (\<Union> \<F> - C) \<inter> K" shows "g x = h x"
simps: fixes a :: "'e set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
map_\<Gamma>_replace_prefix: fixes f :: "'b set \<Rightarrow> 'b set"   and \<pi> :: "'b Trace.prefix"   and \<sigma> :: "'b Trace.trace" assumes "\<forall>x. f (f x) = f x"   and "prefix_of (pmap_\<Gamma> f \<pi>) \<sigma>" shows "map_\<Gamma> f (replace_prefix \<pi> \<sigma>) = map_\<Gamma> f \<sigma>"
omit_mem: fixes ps :: "('b, 'c) fm list"   and v :: "nat"   and p :: "('b, 'c) fm"   and xs :: "nat set" assumes "ps !. v = Some p"   and "v \<notin> xs" shows "p \<in>. omit xs ps"
root_condI: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" assumes "real_of_rat (rai_lb plr) \<le> x"   and "x \<le> real_of_rat (rai_ub plr)"   and "poly (real_of_int_poly (poly_real_alg_1 plr)) x = 0" shows "root_cond plr x"
simps_1: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "make_tr f 0 i = []"
intro: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "RBTi1 TYPE('b) bal_i" shows "RBTid1 TYPE('b) bal_i"
sin_periodic_nat: fixes x :: "real"   and n :: "nat" shows "sin (x + real n * (2 * pi)) = sin x"
add_diff_cancel_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
blinding_of_same_path: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list"   and t :: "('a, 'b) rose_tree\<^sub>m"   and t' :: "('a, 'b) rose_tree\<^sub>m" assumes "blinding_of_on UNIV h bo" shows "blinding_of_tree h bo (tree_of_zipper\<^sub>m (p, t)) (tree_of_zipper\<^sub>m (p, t')) = blinding_of_tree h bo t t'"
run_owner_v_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "let (s', S, A, B, C, D) = run_owner_v n s; SK = (Some S, {A, B}, {C, D}) in {(Asset n, SesKey SK), (Owner n, Crypt (SesK SK) (Pwd n))} \<subseteq> s'"
ACI_refl: fixes r :: "'a rexp" shows "ACI r r"
valid_def: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and c :: "'f list multiset" shows "valid fg c \<equiv> \<forall>s s'. {#s, s'#} \<subseteq># c \<longrightarrow> mon_s fg s \<inter> mon_s fg s' = {}"
simps_1: fixes uu :: "'c"   and h :: "'a" shows "|Inr (uu, h)|\<^sub>h = h"
sep_conj_ac_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P Q = sep_algebra.sep_conj plus sep_disj Q P"
leftderives_induct: fixes a :: "symbol list"   and b :: "symbol list"   and P :: "symbol list \<Rightarrow> bool" assumes "leftderives a b"   and "P a"   and "\<And>y z. \<lbrakk>leftderives a y; leftderives1 y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
set_map2_ex: fixes t :: "'b list"   and s :: "'c list"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd" assumes "length t = length s" shows "set (Map2.map2 f s t) = {x. \<exists>i<length t. x = f (s ! i) (t ! i)}"
inject_2: fixes b :: "'d"   and b' :: "'d" shows "(Inr b = Inr b') = (b = b')"
cnj_mono: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and C :: "fmla"   and D :: "fmla" assumes "insert A H \<turnstile> B"   and "insert C H \<turnstile> D" shows "insert (A AND C) H \<turnstile> B AND D"
simple_functionD_1: fixes M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "simple_function M g" shows "finite (g ` space M)"
lconf_def: fixes P :: "'b prog"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and E :: "String.literal \<Rightarrow> ty option" shows "P,h \<turnstile> l (:\<le>) E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
left: fixes t1 :: "('a, 'b) tm"   and t2 :: "('a, 'b) tm" shows "App t1 t2 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b t1"
zero_in_succ: fixes i :: "V" assumes "Ord i" shows "0 \<in>\<^sub>\<circ> ZFC_in_HOL.succ i"
path0E: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a" assumes "graph weights"   and "graph.path weights l1 l2 []" obtains "l2 = l1"
program_invI: fixes prog :: "'a program_scheme" assumes "0 < IArray.length (states prog)"   and "IArray.length (states prog) = IArray.length (processes prog)"   and "\<And>s. s \<in> set (IArray.list_of (states prog)) \<Longrightarrow> 0 < IArray.length s"   and "\<And>sidx. sidx \<in> ran (lm.\<alpha> (proc_data prog)) \<Longrightarrow> sidx < IArray.length (processes prog) \<and> fst (processes prog !! sidx) = sidx"   and "\<And>sidx start procArgs args. (sidx, start, procArgs, args) \<in> set (IArray.list_of (processes prog)) \<Longrightarrow> \<exists>s. start = Index s \<and> s < IArray.length (states prog !! sidx)" shows "program_inv prog"
INT_iff: fixes b :: "'e"   and B :: "'f \<Rightarrow> 'e set"   and A :: "'f set" shows "(b \<in> \<Inter> (B ` A)) = (\<forall>x\<in>A. b \<in> B x)"
inner1: shows "mk_v I (OUrename x y ODE) (a, b) c = (\<chi>i. (if i \<in> ODE_vars I (OUrename x y ODE) then c else a) $ i, \<chi>i. (if i \<in> ODE_vars I (OUrename x y ODE) then ODE_sem I (OUrename x y ODE) c else b) $ i)"
get_match_def: fixes rule :: "'c rule" shows "get_match rule = (case rule of Rule x1 x2 \<Rightarrow> x1)"
simps_2: fixes uu :: "'b extended" shows "(uu \<le> \<infinity>) = True"
finite_measure_subalgebra_axioms: shows "finite_measure_subalgebra M Invariants"
assms_4: shows "aff_dim V \<le> aff_dim U"
lt_row_space_pivot_fun: fixes S :: "'t set"   and A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "card S = dim_col A"   and "pivot_fun A f (dim_col A)"   and "p \<in> row_to_poly (pps_to_list S) ` row_space A"   and "p \<noteq> 0" shows "lt p \<in> lt_set (set (mat_to_polys (pps_to_list S) A))"
open_translation: fixes S :: "'a set"   and a :: "'a" assumes "open S" shows "open ((+) a ` S)"
diff_diff_add: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a - b - c = a - (b + c)"
gamma2_def: shows "gamma2 = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
concat_bit_eq: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "concat_bit n k l = take_bit n k + push_bit n l"
C_def: shows "C = fpxs_X_power (- (rat_of_nat N * r))"
infsetsum_altdef: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "infsetsum f A = set_lebesgue_integral (count_space UNIV) A f"
Omega_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" shows "Omega_fun f g = (\<sqinter>) g \<circ> (\<circ>) f"
left_right_simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and l :: "'c"   and r :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.right unapp (app l r) = r"
bound': fixes \<eta> :: "nat" shows "interaction_bounded_by' (Not \<circ> isl) (\<A> \<eta>) (q' \<eta>)"
retraction_is_epi: fixes g :: "'a" assumes "B.retraction g" shows "B.epi g"