hcju/isabelleps huggingface dataset:
pfp_wn_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "pfp_wn f x = (case iter_widen f x of None \<Rightarrow> None | Some p \<Rightarrow> iter_narrow f p)"
strategy_attractsE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts p \<sigma> A W"   and "v0 \<in> A" shows "visits_via P A W"
B: shows "next_dim.cone (Y \<union> Z) = next_dim.polyhedral_cone B"
MonotProblem_axioms: shows "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi>"
size_simps_1: fixes t :: "'a"   and ts :: "'a list" assumes "t \<in> set ts" shows "size t < Suc (size_list size ts)"
dtor1_ctor1: fixes x :: "('b, 'b JF1, 'b JF2) F1" shows "dtor1 (ctor1 x) = x"
fset_of_list_surj: shows "surj fset_of_list"
strictly_preferred_by_a_man_def: fixes \<mu> :: "('m \<times> 'w) set"   and \<mu>' :: "('m \<times> 'w) set" shows "strictly_preferred_by_a_man \<mu> \<mu>' = (\<exists>m. \<exists>w\<in>{w. (m, w) \<in> \<mu>'}. w \<in> m_strictly_prefers m \<mu>)"
is_lbI: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and l :: "'c" assumes "class.po below"   and "\<And>x. x \<in> S \<Longrightarrow> below l x" shows "po.is_lb below S l"
rtrancl_idemp_self_comp: fixes R :: "('c \<times> 'c) set" shows "R\<^sup>* O R\<^sup>* = R\<^sup>*"
spmf_rel_map2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a spmf"   and g :: "'c \<Rightarrow> 'b"   and y :: "'c spmf" shows "rel_spmf R x (map_spmf g y) = rel_spmf (\<lambda>x y. R x (g y)) x y"
Compl_eq_Compl_iff: fixes A :: "'c set"   and B :: "'c set" shows "(- A = - B) = (A = B)"
is_fmapD: fixes M :: "('a \<times> 'b) fset"   and a :: "'a"   and b\<^sub>1 :: "'b"   and b\<^sub>2 :: "'b" assumes "is_fmap M"   and "(a, b\<^sub>1) |\<in>| M"   and "(a, b\<^sub>2) |\<in>| M" shows "b\<^sub>1 = b\<^sub>2"
assms_1: shows "w \<in> set_shadow_root_locs element_ptr"
expand_fps_eq: fixes p :: "'a fps"   and q :: "'a fps" shows "(p = q) = (\<forall>n. p $ n = q $ n)"
UN_extend_simps_3: fixes A :: "'d set"   and B :: "'e \<Rightarrow> 'd set"   and C :: "'e set" shows "A \<union> \<Union> (B ` C) = (if C = {} then A else \<Union>x\<in>C. A \<union> B x)"
fairInfiniteExecution_def: fixes fe :: "nat \<Rightarrow> ('p, 'v, 's) configuration list"   and ft :: "nat \<Rightarrow> ('p, 'v) message list" shows "fairInfiniteExecution fe ft \<equiv> infiniteExecution fe ft \<and> (\<forall>n n0. n0 < length (fe n) \<longrightarrow> (\<forall>p msg. enabled (fe n ! n0) msg \<and> isReceiverOf p msg \<and> correctInfinite fe ft p \<longrightarrow> (\<exists>n'\<ge>n. \<exists>n0'\<ge>n0. n0' < length (ft n') \<and> msg = ft n' ! n0')))"
test_import_3: fixes x :: "'a"   and y :: "'a" assumes "test x" shows "(! x \<squnion> y) \<sqinter> x = y \<sqinter> x"
IH: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "deg R ((p \<oplus>\<^bsub>P\<^esub> q) [^]\<^bsub>P\<^esub> na__) = deg R p * na__"
chop_empty_right: shows "\<forall>i. N_Chop(i,i,bot)"
dual_space_restrict: fixes E :: "'c \<Rightarrow> real"   and S :: "'c set" assumes "E \<in> dual_space S" shows "extensional0 S E"
assms: shows "\<forall>l\<le>n. M1 l l \<le> \<one>"
substitute_cong: fixes f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set"   and t :: "'a ttree"   and f' :: "'a \<Rightarrow> 'a ttree" assumes "\<And>x. x \<in> carrier (substitute f T t) \<Longrightarrow> f x = f' x" shows "substitute f T t = substitute f' T t"
coeff_prod_const: fixes xs :: "'b set"   and y :: "'b"   and f :: "'b \<Rightarrow> 'c poly"   and i :: "nat" assumes "finite xs"   and "y \<notin> xs"   and "\<And>x. x \<in> xs \<Longrightarrow> degree (f x) = 0" shows "coeff (prod f (insert y xs)) i = (\<Prod>x\<in>xs. coeff (f x) 0) * coeff (f y) i"
get_exp_def: fixes g :: "'a" shows "get_exp g = (\<lambda>a. SOME k. a = g [^] k)"
qAFresh_qFresh_preserves_alpha: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAFresh zs z X"   and "X #= X'" shows "qFresh zs z X'"
JF2map: fixes u :: "'d \<Rightarrow> 'e" shows "dtor2 \<circ> JF2map u = F2map u (JF1map u) (JF2map u) \<circ> dtor2"
P_reach: fixes n :: "nat" shows "P n \<subseteq> reach A w n"
l_def_real: shows "real_of_int l = real b powr real_of_int (\<lfloor>log (real b) (real a)\<rfloor> + 1)"
defNode_2: fixes v :: "'val"   and g :: "'g" assumes "v \<in> allVars g" shows "v \<in> allDefs g (defNode g v)"
xcpt_app_def: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list" shows "xcpt_app i P pc mxs xt \<tau> = (\<forall>(f, t, C, h, d)\<in>set (relevant_entries P i pc xt). is_class P C \<and> d \<le> length (fst \<tau>) \<and> d < mxs)"
simps_5: fixes m :: "'node"   and x :: "nat" shows "valid_SDG_node (Actual_out (m, x)) = (\<exists>a Q p f ins outs. valid_edge a \<and> kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f \<and> targetnode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length outs)"
fixing_ide_gives_functor_2: fixes a2 :: "'a" assumes "ide a2" shows "functor (\<cdot>) (\<cdot>) (\<lambda>f1. T (f1, a2))"
map_vals_of_throw: fixes vs :: "val list"   and e :: "'c exp"   and es' :: "'c exp list" shows "map_vals_of (map Val vs @ throw e # es') = None"
assms_6: shows "g \<in> A \<rightarrow> carrier G"
homeomorphism_imp_open_map: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and U :: "'a set" assumes "homeomorphism S T f g"   and "openin (top_of_set S) U" shows "openin (top_of_set T) (f ` U)"
wf_relcomp_compatible: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "wf R"   and "R O S \<subseteq> S O R" shows "wf (S O R)"
concl_compact_consequence_relation_axioms: shows "concl_compact_consequence_relation Bot (\<Turnstile>)"
ShiftP_Mem_D: fixes H :: "fm set"   and f :: "tm"   and k :: "tm"   and del :: "tm"   and g :: "tm"   and a :: "tm"   and x :: "name"   and x' :: "name"   and y :: "name" assumes "H \<turnstile> ShiftP f k del g"   and "H \<turnstile> a IN g"   and "atom x \<sharp> (x', y, a, f, del, k)"   and "atom x' \<sharp> (y, a, f, del, k)"   and "atom y \<sharp> (a, f, del, k)" shows "H \<turnstile> SyntaxN.Ex x (SyntaxN.Ex x' (SyntaxN.Ex y (a EQ HPair (Var x') (Var y) AND HaddP del (Var x) (Var x') AND HPair (Var x) (Var y) IN f AND Var x IN k)))"
nb_1: fixes k :: "nat" assumes "k < length (\<Phi>\<inverse> \<Pi> \<A> t)" shows "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) (\<Phi>\<inverse> \<Pi> \<A> t)) - 1"
WT: fixes q :: "'b"   and s :: "'a" assumes "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g callee s q \<surd>"
esize_infinite_enat: fixes A :: "'a set"   and k :: "nat" assumes "infinite A" shows "enat k < esize A"
zip_eq_butlast_last: fixes xs :: "'a list"   and ys :: "'b list" assumes "0 < length xs"   and "length xs = length ys" shows "zip xs ys = zip (butlast xs) (butlast ys) @ [(last xs, last ys)]"
a_translate_functions: fixes c :: "'a" assumes "c \<in> carrier R" shows "a_translate c \<in> carrier (function_ring (carrier R) R)"
isomorphic_to_identity_is_equivalence: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'd" assumes "endofunctor A F"   and "natural_isomorphism A A F (identity_functor.map A) \<phi>" shows "equivalence_functor A A F"
restr_conv': fixes A :: "'c set"   and al :: "('c \<times> 'd) list" shows "map_of (restrict A al) = map_of al |` A"
cclfp_def: 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 f :: "'c \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "countable_complete_lattice.cclfp Sup bot f = Sup (range (\<lambda>i. (f ^^ i) bot))"
obj_simps_5: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "cod a = a"
bot': fixes f :: "'a \<Rightarrow> 'b" shows "L bot f = UNIV"
Domain_converse: fixes r :: "('d \<times> 'c) set" shows "Domain (r\<inverse>) = Range r"
freshnat_def: fixes S :: "nat set" shows "freshnat S \<equiv> Suc (Max (0 \<triangleright> S))"
swap_vsubst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X" shows "X #[y1 // y]_ys #[z1 \<and> z2]_zs = X #[z1 \<and> z2]_zs #[y1 @ys[z1 \<and> z2]_zs // y @ys[z1 \<and> z2]_zs]_ys"
Od_rel: fixes R :: "('a \<times> 'a) set"   and S :: "'a set" shows "rel (D\<lparr>carrier := S, rel := R\<rparr>) = R"
alphaAbs_all_equal_or_qAFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
z_neq_new_last: shows "z \<noteq> new_last"
nat_filtration_space: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "space (infinite_coin_toss_space.nat_filtration M n) = UNIV"
orbit_subset_eq_affected: fixes a :: "'a"   and f :: "'a perm" assumes "a \<in> affected f" shows "orbit f a \<subseteq> affected f"
read_state_reg_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {sreg_type RDY, sreg_type RDPSR, sreg_type RDWIM, sreg_type RDTBR}" shows "snd (read_state_reg_instr i s) = False"
Basis_prod_def: shows "Basis = (\<lambda>u. (u, 0::'c)) ` Basis \<union> Pair (0::'b) ` Basis"
oghoare_seq_sound: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P' :: "'b set"   and P :: "('b, 'a, 'c) ann"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P' P c Q, A" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P' c Q, A"
fresh_ineq_at_base: fixes a :: "atom"   and b :: "'a" assumes "a \<noteq> atom b" shows "a \<sharp> b"
cut_less_empty_iff: fixes I :: "'b set"   and t :: "'b" shows "(I \<down>< t = {}) = (\<forall>x\<in>I. t \<le> x)"
simps: fixes ly :: "nat list"   and n :: "nat"   and e :: "nat"   and as :: "nat"   and am :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_inv_2 ly n e (as, am) (s, l, r) ires = (let ss = start_of ly as; am' = abc_lm_s am n (abc_lm_v am n - Suc 0); am'' = abc_lm_s am n (abc_lm_v am n) in if s = 0 then False else if s = ss + 2 * n then inv_locate_a (as, am) (n, l, r) ires else if s = ss + 2 * n + 1 then inv_locate_n_b (as, am) (n, l, r) ires else if s = ss + 2 * n + 2 then dec_first_on_right_moving n (as, am'') (s, l, r) ires else if s = ss + 2 * n + 3 then dec_after_clear (as, am') (s, l, r) ires else if s = ss + 2 * n + 4 then dec_right_move (as, am') (s, l, r) ires else if s = ss + 2 * n + 5 then dec_check_right_move (as, am') (s, l, r) ires else if s = ss + 2 * n + 6 then dec_left_move (as, am') (s, l, r) ires else if s = ss + 2 * n + 7 then dec_after_write (as, am') (s, l, r) ires else if s = ss + 2 * n + 8 then dec_on_right_moving (as, am') (s, l, r) ires else if s = ss + 2 * n + 9 then dec_after_clear (as, am') (s, l, r) ires else if s = ss + 2 * n + 10 then inv_on_left_moving (as, am') (s, l, r) ires else if s = ss + 2 * n + 11 then inv_check_left_moving (as, am') (s, l, r) ires else if s = ss + 2 * n + 12 then inv_after_left_moving (as, am') (s, l, r) ires else if s = ss + 2 * n + 16 then inv_stop (as, am') (s, l, r) ires else False)"
iso_is_arr: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "category.iso C f" shows "partial_magma.arr C f"
assms_2: assumes "\<Phi>" shows "(m', m) \<in> \<langle>R\<rangle>nres_rel"
a_subset_N: fixes k :: "nat" shows "a k \<subseteq> N"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_after_write (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml + mr = Suc m \<and> ml = Suc m \<and> lm2 \<noteq> [] \<and> (if lm1 = [] then l = Bk # Oc \<up> ml @ Bk # Bk # ires else l = Bk # Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> tl r = <lm2> @ Bk \<up> rn)"
dropD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Drop]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalDeny"
assms_3: shows "q\<^sub>0 \<notin> F"
return_sets_cong: fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N" shows "return M = return N"
theI_unique: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<exists>!x. P x" shows "P x = (x = The P)"
num_params_tm_abs_le: fixes t :: "taylor_model"   and X :: "nat"   and prec :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t) \<le> X" shows "num_params (tm_poly (tm_abs prec I a t)) \<le> X"
GatewayReq_def: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" shows "GatewayReq req dt a stop lose d ack i vc \<equiv> msg 1 req \<and> msg 1 a \<and> msg 1 stop \<and> ts lose \<longrightarrow> (\<forall>t. (ack t = [init_state] \<and> req (Suc t) = [init] \<and> lose (t + 1) = [False] \<and> lose (t + 2) = [False] \<longrightarrow> ack (t + 2) = [connection_ok]) \<and> (ack t = [connection_ok] \<and> req (Suc t) = [send] \<and> (\<forall>k\<le>d + 1. lose (t + k) = [False]) \<longrightarrow> i (Suc t + d) = inf_last_ti dt t \<and> ack (Suc t) = [sending_data]) \<and> (ack (t + d) = [sending_data] \<and> a (Suc t) = [sc_ack] \<and> (\<forall>k\<le>d + 1. lose (t + k) = [False]) \<longrightarrow> vc (Suc t + d) = [vc_com]))"
gdpr_actors_def: shows "gdpr_actors \<equiv> {''Patient'', ''Doctor''}"
rel_fun_def_butlast: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and T :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h \<Rightarrow> 'i" shows "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> rel_fun S T (f x) (g y))"
valid_sop_def: fixes sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" shows "valid_sop sop \<equiv> \<forall>D f \<theta>. sop = (D, f) \<longrightarrow> D \<subseteq> dom \<theta> \<longrightarrow> f \<theta> = f (\<theta> |` D)"
assms_1: shows "f \<approx>>\<^bsub>CatDom F\<^esub> g"
cs_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "relation.cs r a b \<equiv> \<exists>s. r\<^sup>*\<^sup>* a s \<and> r\<^sup>*\<^sup>* b s"
inj_Suc: fixes N :: "nat set" shows "inj_on Suc N"
equivI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "refl_on A r"   and "sym r"   and "trans r" shows "equiv A r"
power3_eq_cube: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.monoid_mult one times" shows "power.power one times a 3 = times (times a a) a"
sum_simp_setup_1: shows "\<forall>v\<in>V. sum f (outgoing v) = (\<Sum>u\<in>V. f (v, u))"
ext_inject: fixes basepoint :: "'b Point"   and direction :: "'b Vector"   and more :: "'c"   and basepoint' :: "'b Point"   and direction' :: "'b Vector"   and more' :: "'c" shows "(\<lparr>basepoint = basepoint, direction = direction, \<dots> = more\<rparr> = \<lparr>basepoint = basepoint', direction = direction', \<dots> = more'\<rparr>) = (basepoint = basepoint' \<and> direction = direction' \<and> more = more')"
float_defs_9: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "(a < b) = flt a b"
iterate_iterate: fixes m :: "nat"   and F :: "'c \<rightarrow> 'c"   and n :: "nat"   and x :: "'c" shows "iterate m\<cdot>F\<cdot>(iterate n\<cdot>F\<cdot>x) = iterate (m + n)\<cdot>F\<cdot>x"
sl_cs_2: fixes cs :: "'a list" assumes "cs \<in> set css" shows "set (some_list cs) = set cs"
clock_simAbs_def: fixes X :: "'b odlist \<times> 'b odlist" shows "clock_simAbs X \<equiv> rel_ext (\<lambda>uu_. \<exists>s. uu_ = (toSet (fst X), s) \<and> s \<in> toSet (snd X))"
AbsI: fixes x :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 y. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<alpha>\<nu> y" shows "[\<lparr>A!,x\<rparr> in v]"
assms: shows "orientable_8 (x \<sqinter> - (1::'a))"
abs_le_D1: fixes abs :: "'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq a b"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H (add_monoid G)" shows "H \<subseteq> carrier G"
quasi_inverses_symmetric: fixes f :: "'a"   and g :: "'a" assumes "B.quasi_inverses f g" shows "B.quasi_inverses g f"
jordan_nf_exists: fixes A :: "'c mat"   and n :: "nat"   and as :: "'c list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'c:])" shows "\<exists>n_as. jordan_nf A n_as"
pdevs_apply_Pdevs_eq_0: fixes xs :: "(nat, 'd) slist"   and i :: "nat" shows "(pdevs_apply (Pdevs xs) i = (0::'d)) = (\<forall>x. (i, x) \<in> set_slist xs \<longrightarrow> x = (0::'d))"
AG_def: fixes f :: "'a set" shows "AG f \<equiv> gfp (\<lambda>Z. f \<inter> AX Z)"
Union_path_components_of: fixes X :: "'a topology" shows "\<Union> (path_components_of X) = topspace X"
clock_sim_range: shows "sim_range Clock.MC clock_simMC clock_sim"
nneg_o: fixes P :: "'c \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c" assumes "nneg P" shows "nneg (P \<circ> f)"
induct: fixes P :: "Syntax.term \<Rightarrow> bool"   and "term" :: "Syntax.term" assumes "P Unit"   and "\<And>x. P (Var x)"   and "\<And>x1 x2a. P x2a \<Longrightarrow> P (Lam x1 x2a)"   and "\<And>x1 x2a. P x2a \<Longrightarrow> P (Rec x1 x2a)"   and "\<And>x. P x \<Longrightarrow> P (Inj1 x)"   and "\<And>x. P x \<Longrightarrow> P (Inj2 x)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (Syntax.Pair x1 x2a)"   and "\<And>x1 x2a x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P (Syntax.Let x1 x2a x3)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (App x1 x2a)"   and "\<And>x1 x2a x3. \<lbrakk>P x1; P x2a; P x3\<rbrakk> \<Longrightarrow> P (Case x1 x2a x3)"   and "\<And>x. P x \<Longrightarrow> P (Prj1 x)"   and "\<And>x. P x \<Longrightarrow> P (Prj2 x)"   and "\<And>x. P x \<Longrightarrow> P (Roll x)"   and "\<And>x. P x \<Longrightarrow> P (Unroll x)"   and "\<And>x. P x \<Longrightarrow> P (Auth x)"   and "\<And>x. P x \<Longrightarrow> P (Unauth x)"   and "\<And>x. P (Hash x)"   and "\<And>x1 x2a. P x2a \<Longrightarrow> P (Hashed x1 x2a)" shows "P term"
comp_permute: fixes f :: "'c"   and g :: "'c"   and k :: "'c"   and l :: "'c"   and h :: "'c" assumes "f \<cdot>\<^sub>C g = k \<cdot>\<^sub>C l"   and "C.seq f g"   and "C.seq g h" shows "f \<cdot>\<^sub>C g \<cdot>\<^sub>C h = k \<cdot>\<^sub>C l \<cdot>\<^sub>C h"
chain_total: fixes S :: "nat \<Rightarrow> 'a set"   and i :: "nat"   and j :: "nat" assumes "chain S" shows "S i \<subseteq> S j \<or> S j \<subseteq> S i"
right_unit: fixes a :: "'a" assumes "a \<in> A" shows "addA a zeroA = a"
i_def_2: shows "zip (p # v) (v @ [s]) ! i = (x, y)"
max_add_distrib_left: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "subset_mset.max x y + z = subset_mset.max (x + z) (y + z)"
cart_basis_def: shows "cart_basis = {axis i (1::'c) |i. i \<in> UNIV}"
irreflp_less: shows "irreflp (\<lambda>a b. cmp a b = Less)"
simps_1: fixes x :: "name" shows "[] maps x to Var x"
additive_freeword_funlift: fixes f :: "'a \<Rightarrow> 'b" shows "UGroupHom (freeword_funlift f)"
has_all_wt_instances_of_def: fixes \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> 'c"   and N :: "('a, 'b) Term.term set"   and M :: "('a, 'b) Term.term set" shows "has_all_wt_instances_of \<Gamma> N M \<equiv> \<forall>t\<in>N. \<exists>s\<in>M. is_wt_instance_of_cond \<Gamma> t s"
induct: fixes P :: "('e, 'f, 'g) Tree \<Rightarrow> ('e, 'f, 'g) Tree \<Rightarrow> bool"   and a0 :: "('e, 'f, 'g) Tree"   and a1 :: "('e, 'f, 'g) Tree" assumes "\<And>tset1 tset2. (\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. P t1 t2 \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>\<phi>1 \<phi>2. P (tPred \<phi>1) (tPred \<phi>2)"   and "\<And>\<alpha>1 t1 \<alpha>2 t2. \<lbrakk>\<And>x. P (x \<bullet> t1) t2; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t2) t2\<rbrakk> \<Longrightarrow> P (tAct \<alpha>1 t1) (tAct \<alpha>2 t2)"   and "\<And>v va. P (tNot v) (tConj va)"   and "\<And>v va. P (tNot v) (tPred va)"   and "\<And>v va vb. P (tNot v) (tAct va vb)"   and "\<And>v va. P (tPred v) (tConj va)"   and "\<And>v va. P (tPred v) (tNot va)"   and "\<And>v va vb. P (tPred v) (tAct va vb)"   and "\<And>v va vb. P (tAct v va) (tConj vb)"   and "\<And>v va vb. P (tAct v va) (tNot vb)"   and "\<And>v va vb. P (tAct v va) (tPred vb)"   and "\<And>va v. P (tConj va) (tNot v)"   and "\<And>va v. P (tPred va) (tNot v)"   and "\<And>va vb v. P (tAct va vb) (tNot v)"   and "\<And>va v. P (tConj va) (tPred v)"   and "\<And>va v. P (tNot va) (tPred v)"   and "\<And>va vb v. P (tAct va vb) (tPred v)"   and "\<And>vb v va. P (tConj vb) (tAct v va)"   and "\<And>vb v va. P (tNot vb) (tAct v va)"   and "\<And>vb v va. P (tPred vb) (tAct v va)" shows "P a0 a1"
resultant_impl_main_def: fixes div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b"   and G1 :: "'b poly"   and G2 :: "'b poly" shows "resultant_impl_main div_exp G1 G2 = (if G2 = 0 then if degree G1 = 0 then 1::'b else (0::'b) else case subresultant_prs div_exp G1 G2 of (Gk, hk) \<Rightarrow> if degree Gk = 0 then hk else (0::'b))"
is_filter_def: fixes F :: "('d \<Rightarrow> bool) \<Rightarrow> bool" shows "is_filter F \<equiv> F (\<lambda>x. True) \<and> (\<forall>P Q. F P \<longrightarrow> F Q \<longrightarrow> F (\<lambda>x. P x \<and> Q x)) \<and> (\<forall>P Q. (\<forall>x. P x \<longrightarrow> Q x) \<longrightarrow> F P \<longrightarrow> F Q)"
has_codomain_iff_arr: fixes f :: "'b" shows "(\<phi>.B.codomains f \<noteq> {}) = \<phi>.B.arr f"
INF_id_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and A :: "'b set" shows "Inf (id ` A) = Inf A"
diff_eq_diff_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less a b = less c d"