hcju/isabelleps huggingface dataset:
Par2B: fixes Q :: "pi"   and a :: "subject"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> Q'"
ToCT_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "ToCT f \<equiv> if CCC.arr f then T (fst f, T (fst (snd f), snd (snd f))) else C.null"
permutations_of_multiset_empty: shows "permutations_of_multiset {#} = {[]}"
cycle_rotate: fixes xs :: "nat list"   and i :: "nat"   and j :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and a :: "nat" assumes "1 < length xs"   and "(i, j) \<in> arcs' xs" shows "\<exists>ys zs. len M a a xs = len M i i (j # ys @ a # zs) \<and> xs = zs @ i # j # ys"
mset_Cons: fixes A :: "'a formula"   and S :: "'a formula list" shows "mset (A # S) = A, mset S"
pointwise_max_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "pointwise_max f g = (\<lambda>x. max (f x) (g x))"
def_hashmap_size_ta_rule_def: shows "def_hashmap_size \<equiv> \<lambda>_. 32"
Card_order_Plus_empty1: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r =o |Field r <+> {}|"
nat_of_Abs_sat': fixes n :: "nat" shows "nat_of (Abs_sat' n) = min LENGTH('b) n"
simps_6: fixes z :: "octo" shows "Im5 (Octonions.cnj z) = - Im5 z"
some_lub_def: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" shows "some_lub r x y = (SOME z. is_lub r x y z)"
eq_idx_for_lvar: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "EqForLVar eq_idx_for_lvar"   and "x \<in> lvars t" shows "eq_idx_for_lvar t x < length t \<and> lhs (t ! eq_idx_for_lvar t x) = x"
assms: shows "evaluate_decs False mn env (update_clock (\<lambda>_. count1) s) ds (update_clock (\<lambda>_. count1) s', r)"
cop_F_def: fixes ds :: "'b set"   and XH :: "'a set" shows "cop_F ds XH = XH \<union> CD_on ds (- Rf CH XH)"
ex_def: fixes P :: "'b \<Rightarrow> (nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "local.ex P \<equiv> \<lambda>t n. \<exists>y. P y t n"
subst_comp: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "y \<notin> FvarsT t" shows "subst (subst \<phi> (Var x) y) t x = subst (subst \<phi> t x) t y"
wtTE_intT: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" assumes "Ik.wtE \<xi>" shows "intT (TE.tpOfV x) (\<xi> x)"
transpose_mult: fixes A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "Square_Matrix.transpose (A * B) = Square_Matrix.transpose B * Square_Matrix.transpose A"
correct_state_impl_Some_method: fixes P :: "jvm_prog"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]" shows "\<exists>b m Ts T. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C"
a2: shows "x = (s |\<^bsub>rel\<^esub> \<one>)"
ode_fa_nth: fixes i :: "nat"   and xs :: "floatarith list" assumes "i < length ode_e" shows "ode_fa xs ! i = ode_fa_nth xs i"
abs_in: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and \<alpha> :: "nat \<times> bool \<Rightarrow> 'c set" assumes "t \<in> M" shows "t \<cdot>\<^sub>\<alpha> \<alpha> \<in> M \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t \<alpha>"
a1_def: shows "a1 = distinct6 A B C A' B' C'"
that: shows "(0::'a) \<in>\<^sub>i A \<or> (0::'a) \<in>\<^sub>i B"
disjoint_iff: fixes A :: "'d set"   and B :: "'d set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
monom_mult: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and n :: "nat"   and m :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "monom (UP Zp) (a \<otimes> b) (n + m) = monom (UP Zp) a n \<otimes>\<^bsub>UP Zp\<^esub> monom (UP Zp) b m"
S6MClkReply_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> ($S6 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> <MClkReply memCh crCh cst p>_(c p) \<longrightarrow> S1 rmhist p$"
mod_ex_dist: fixes h :: "pheap"   and P :: "'b \<Rightarrow> assn" shows "(h \<Turnstile> (\<exists>\<^sub>Ax. P x)) = (\<exists>x. h \<Turnstile> P x)"
f_minus: fixes v :: "'c"   and w :: "'c" assumes "v \<in> carrier V"   and "w \<in> carrier V" shows "T (v \<ominus>\<^bsub>V\<^esub> w) = T v \<ominus>\<^bsub>W\<^esub> T w"
elims: fixes x :: "'a list"   and xa :: "'a list"   and y :: "'a list" assumes "splice x xa = y" obtains ys :: "'a list" where "x = []" and "xa = ys__" and "y = ys__"   | xb :: "'a" and xs :: "'a list" and ys :: "'a list" where "x = xb # xs" and "xa = ys" and "y = xb # splice ys xs"
rotateR_def: fixes t :: "('c, 'd) tree" shows "rotateR t = (if t = Tip then t else if lsub t = Tip then t else let lt = lsub t in Node (lsub lt) (key lt) (nval lt) (Node (rsub lt) (key t) (nval t) (rsub t)))"
right_idem: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> b \<sqinter> b = a \<sqinter> b"
has_fps_expansion_const: fixes c :: "'a" shows "(\<lambda>_. c) has_fps_expansion fps_const c"
igSwapIGVar_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapGVar MOD" shows "igSwapIGVar (fromMOD MOD)"
diff: fixes x :: "real"   and y :: "real"   and xa :: "'b" shows "(x - y) *\<^sub>R xa = x *\<^sub>R xa - y *\<^sub>R xa"
simps: fixes G :: "('c, 'd) pre_digraph"   and p :: "'d list"   and q :: "'d list"   and r :: "'d list"   and s :: "'d list" assumes "wf_digraph G" shows "wf_digraph.is_awalk_cyc_decomp G p (q, r, s) = (p = q @ r @ s \<and> (\<exists>u v w. pre_digraph.awalk G u q v \<and> pre_digraph.awalk G v r v \<and> pre_digraph.awalk G v s w) \<and> 0 < length r \<and> (\<exists>u. distinct (pre_digraph.awalk_verts G u q)))"
star_Diff_Nil_fold: fixes A :: "'b list set" shows "(A - {[]}) @@ star A = star A - {[]}"
card_of_Func_option_Func: fixes A :: "'a set"   and B :: "'b set" shows "|Func_option A B| =o |Func A B|"
has_integral_cmul_iff: fixes c :: "real"   and f :: "'a \<Rightarrow> 'b"   and I :: "'b"   and A :: "'a set" assumes "c \<noteq> 0" shows "((\<lambda>x. c *\<^sub>R f x) has_integral c *\<^sub>R I) A = (f has_integral I) A"
sig_regular_reduced_unique: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_GB_upt d G (lt q)"   and "p \<in> dgrad_sig_set' (length fs) d"   and "q \<in> dgrad_sig_set' (length fs) d"   and "lt p = lt q"   and "lc p = lc q"   and "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G p"   and "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G q" shows "rep_list p = rep_list q"
access_matrix: fixes V :: "('a word \<times> 'a wordinterval) list"   and E :: "('a word \<times> 'a word) list"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and s :: "'a word"   and d :: "'a word" assumes "(V, E) = access_matrix c rs" shows "(\<exists>s_repr d_repr s_range d_range. (s_repr, d_repr) \<in> set E \<and> map_of V s_repr = Some s_range \<and> s \<in> wordinterval_to_set s_range \<and> map_of V d_repr = Some d_range \<and> d \<in> wordinterval_to_set d_range) = (runFw s d c rs = Decision FinalAllow)"
globEANow: fixes n :: "nat"   and i :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and \<gamma> :: "(nat \<Rightarrow> 'ND) \<Rightarrow> nat \<Rightarrow> bool" assumes "n \<le> i"   and "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<box>\<^sub>b\<gamma>)" shows "eval c t t' i \<gamma>"
icd: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
urel64_minus: fixes x :: "uint64"   and y :: "int"   and x' :: "uint64"   and y' :: "int" assumes "urel64 x y"   and "urel64 x' y'" shows "urel64 (minus_p64 pp x x') (minus_p p y y')"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and s1 :: "'c stream"   and s2 :: "'d stream" shows "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)"
cut_greater_finite: fixes I :: "'b set"   and t :: "'b" assumes "finite I" shows "finite (I \<down>> t)"
smult_minus_1_back: fixes v :: "'c" assumes "v \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> v = \<ominus> \<one> \<odot>\<^bsub>M\<^esub> v"
a_ac_3: fixes x :: "'a vec"   and y :: "'a vec"   and z :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + (y + z) = y + (x + z)"
Sigma_Un_distrib2: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'd set" shows "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
iso_qiter_fun: fixes x :: "'b"   and y :: "'b" shows "mono (qiter_fun x y)"
simps_1: fixes p :: "assertion" shows "conc_asn (Ribbon_conc p) = p"
axioms_1: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_vfsequence \<alpha> \<SS>"
OTNoTN: fixes p :: "('b, 'c) Combinators list"   and x :: "('b, 'c) Combinators" assumes "OnlyTwoNets p"   and "x \<noteq> DenyAll"   and "x \<in> set p" shows "onlyTwoNets x"
assms: shows "low_equal s1 s2 \<and> t1 = add_data_cache s1 address w bm \<and> t2 = add_data_cache s2 address w bm"
strong_operator_topology_continuous_evaluation: fixes x :: "'d" shows "continuous_map strong_operator_topology euclidean (\<lambda>f. blinfun_apply f x)"
social_choice_function_axioms: shows "social_choice_function agents alts scf"
ConjI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<and> [\<psi> in v]" shows "[\<phi> \<^bold>& \<psi> in v]"
graph_finite: shows "finite (nodeAbs ` {x. isNode x})"
inf_prefix_length_append: fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length (zs @ xs) (zs @ ys) = length zs + inf_prefix_length xs ys"
maxsimpchain_in_subcomplex: fixes Y :: "'a set set"   and ys :: "'a set list" assumes "order.greater_eq X Y \<and> SimplicialComplex Y"   and "order.greater_eq Y (set ys)"   and "gallery ys" shows "SimplicialComplex.maxsimpchain Y ys"
closed_segment_translation_eq: fixes d :: "'a"   and x :: "'a"   and a :: "'a"   and b :: "'a" shows "(d + x \<in> closed_segment (d + a) (d + b)) = (x \<in> closed_segment a b)"
runStateT_zeroU: fixes s :: "'a" shows "runStateT\<cdot>zeroU\<cdot>s = mzero"
edge_None1E: fixes y :: "'v option" assumes "edge' None y" obtains z :: "'v" where "y = \<langle>z\<rangle>" and "z \<in> \<^bold>V" and "z \<notin> A \<Gamma>"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Cod (fst Fg'))"
A_2: shows "\<Prod>\<^sub># A = normalize (primitive_part p)"
union: fixes M :: "'b set set"   and B :: "'b set set" assumes "\<And>t. t \<in> M \<Longrightarrow> t open\<^bsub>topo B\<^esub>" shows "\<Union> M open\<^bsub>topo B\<^esub>"
onorm: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "bounded_linear f" shows "norm (f x) \<le> onorm f * norm x"
inner_Basis_zero: fixes a :: "'a" assumes "finite B"   and "a \<in> S" shows "(local.inner_Basis a a = 0) = (a = (0::'a))"
simps_7: fixes v :: "Formula.formula"   and va :: "Formula.formula" shows "remove_neg (formula.Or v va) = formula.Or v va"
entail_set_all_formulas: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile> N2) = (\<forall>C\<in>N2. N1 \<Turnstile> {C})"
effect_heapI: fixes h' :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap"   and r :: "'a" assumes "h' = snd (f h)"   and "r = fst (f h)" shows "effect (Heap_Monad.heap f) h h' r"
map_id: fixes t :: "('a, 'b, 'c) gpv" shows "map_gpv id id t = t"
flat_ordI: fixes x :: "'g"   and a :: "'g"   and y :: "'g" assumes "x \<noteq> a \<Longrightarrow> x = y" shows "flat_ord a x y"
minus_Box_def: fixes x :: "'c Box"   and y :: "'c Box" shows "x - y \<equiv> bliftM2 (\<Lambda> a b. a - b)\<cdot>x\<cdot>y"
ipathD: fixes E :: "('h \<times> 'h) set"   and r :: "nat \<Rightarrow> 'h"   and i :: "nat" assumes "ipath E r" shows "(r i, r (Suc i)) \<in> E"
chain': fixes x :: "'a" shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>f y. f (x, y)) ` Y)"
repPOS_INF: fixes r :: "real" assumes "real_of_int (sint POS_INF) \<le> r" shows "POS_INF \<equiv>\<^sub>E r"
not_suffix_cases: fixes ps :: "'c list"   and ls :: "'c list" assumes "\<not> suffix ps ls" obtains "ps \<noteq> []" and "ls = []"   | a :: "'c" and as :: "'c list" and x :: "'c" and xs :: "'c list" where "ps = as__ @ [a__]" and "ls = xs__ @ [x__]" and "x__ = a__" and "\<not> suffix as__ xs__"   | a :: "'c" and as :: "'c list" and x :: "'c" and xs :: "'c list" where "ps = as @ [a]" and "ls = xs @ [x]" and "x \<noteq> a"
abs_eq: fixes xa :: "'b ltl"   and x :: "'b set list" shows "af_abs.f_foldl_abs (Abs xa) x = Abs (af xa x)"
M_Notes_PG: fixes M :: "agent"   and LID_M :: "nat"   and P :: "agent"   and C :: "agent"   and etc :: "msg"   and evs :: "event list" assumes "Notes M \<lbrace>Number LID_M, Agent P, Agent M, Agent C, etc\<rbrace> \<in> set evs"   and "evs \<in> set_pur" shows "\<exists>j. P = PG j"
intro: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real" assumes "normed_vectorspace V norm" shows "normed_vectorspace_with_fn_norm V norm"
simps_1: fixes "is" :: "'b set" shows "interrupt_actions_ok is [] = True"
def_Least_le: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "a \<equiv> LEAST x. P x"   and "P k" shows "a \<le> k"
"0_2": shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (N1 (unlabel A))"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "local.eval (v # va) = (\<lambda>x. lead_coeff (v # va) \<otimes> x [^] degree (v # va) \<oplus> local.eval (tl (v # va)) x)"
init_fin_bisim_simps_4: fixes final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and bisim :: "'c \<Rightarrow> 'a \<times> 'd \<Rightarrow> 'b \<times> 'e \<Rightarrow> bool"   and t :: "'c"   and s1 :: "status"   and x1 :: "'a"   and m1 :: "'d"   and x2 :: "'b"   and m2 :: "'e" shows "FWbisimulation_base.init_fin_bisim final1 final2 bisim t ((s1, x1), m1) ((PreStart, x2), m2) = (s1 = PreStart \<and> bisim t (x1, m1) (x2, m2))"
i: shows "x = a [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> i"
T\<^sub>p_spliting: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and R :: "'a list" assumes "x \<noteq> y"   and "set xs \<subseteq> {x, y}"   and "set ys \<subseteq> {x, y}"   and "R \<in> {[x, y], [y, x]}" shows "T\<^sub>p R (xs @ [x, x]) (OPT2 (xs @ [x, x]) R) + T\<^sub>p [x, y] ys (OPT2 ys [x, y]) = T\<^sub>p R (xs @ [x, x] @ ys) (OPT2 (xs @ [x, x] @ ys) R)"
sets: shows "{x \<in> space M. u x \<noteq> 0} \<in> sets M"
that: assumes "bij p"   and "\<forall>x. x \<notin> S \<longrightarrow> p x = x" shows "thesis"
to_poly_nat_pow: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "to_polynomial R a [^]\<^bsub>P\<^esub> n = to_polynomial R (a [^] n)"
lm051: fixes a :: "('a \<times> 'b set) set" shows "pseudoAllocation a = {(fst pair, {y}) |y pair. y \<in> snd pair \<and> pair \<in> a}"
r_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier G"   and "a \<in> carrier G" shows "(a \<oplus> x = x) = (a = \<zero>)"
covering_space_power_punctured_plane: fixes n :: "nat" assumes "0 < n" shows "covering_space (- {0}) (\<lambda>z. z ^ n) (- {0})"
OUT_plus_flow: fixes \<Delta> :: "('k, 'l) graph_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> f e = 0"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> g (y, x) \<le> f (x, y)"   and "d_IN g x \<noteq> \<top>" shows "d_OUT (f \<oplus>\<^bsub>\<Delta>\<^esub> g) x = d_OUT f x + (\<Sum>\<^sup>+ y. g (x, y) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (x, y)) - (\<Sum>\<^sup>+ y. g (y, x) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (x, y))"
ma_1: shows "1 = real_of (ma_of_rat 1)"
simps_2: fixes x :: "('b, 'c) Combinators"   and l :: "'b set list" shows "FWNormalisationCore.sorted [x] l = True"
dgr_def: fixes D :: "'a \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and U :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" shows "dgr D U = (step D \<circ> U) \<sqinter> [. - grd (step D) .]"
simps_2: fixes \<sigma> :: "'tp" shows "TE_arOf (Tag \<sigma>) = [\<sigma>]"
gval_take: fixes g :: "vname gexp"   and a :: "nat"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "GExp.max_input g < Some a" shows "gval g (join_ir i r) = gval g (join_ir (take a i) r)"
Ord_IN_Ord0: fixes H :: "fm set"   and l :: "tm"   and k :: "tm" assumes "H \<turnstile> l IN k" shows "insert (OrdP k) H \<turnstile> OrdP l"
Inf_if_Inf_between: fixes \<iota> :: "'f inference"   and N :: "'f set"   and M :: "'f set" assumes "\<iota> \<in> Inf_between N M" shows "\<iota> \<in> Inf"
A'_def: shows "A' = row_add A (mod_type_class.from_nat n) i ((res (A $ i $ j) (A $ mod_type_class.from_nat n $ j) - A $ mod_type_class.from_nat n $ j) div A $ i $ j)"
ii: fixes L :: "(nat \<times> int \<times> nat) list"   and a :: "nat"   and aa :: "int"   and b :: "nat" assumes "fromlist L"   and "(a, aa, b) \<in> set L" shows "uGraph_from_list_\<alpha>_weight L (Upair a b) = aa"
cong: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysis.Afix Aexp \<equiv> ArityAnalysis.Afix Aexp"
trail_simps_1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G" shows "pre_digraph.trail G u [] v = (u = v \<and> u \<in> verts G)"
semiconnex_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "semiconnex B (\<sqsubset>)"
covdi: shows "ov O d\<inverse> \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
ball_translation: fixes a :: "'a"   and c :: "'a"   and r :: "real" shows "ball (a + c) r = (+) a ` ball c r"
vector_smult_component: fixes c :: "'a"   and y :: "('a, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "(c *s y) $h i = c * y $h i"
tendsto_iff_Liminf_eq_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and f0 :: "'b" assumes "F \<noteq> \<bottom>" shows "(f \<longlongrightarrow> f0) F = (Liminf F f = f0 \<and> Limsup F f = f0)"
p2r_def: fixes P :: "'b \<Rightarrow> bool" shows "p2r P = {(s, s) |s. P s}"
update_simps_9: fixes up :: "(nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and s :: "('a, 'b) state" shows "\<T> (\<B>\<^sub>i\<^sub>l_update up s) = \<T> s"
wiring_id_converter: fixes \<I> :: "('i, 'j) \<I>" shows "wiring \<I> \<I> 1\<^sub>C (id, id)"
rel_spmf_OO_trans: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'c spmf" assumes "rel_spmf R p q"   and "rel_spmf S q r" shows "rel_spmf (R OO S) p r"
P'_def: shows "P' = mult_row (interchange_rows P i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)"
INF_cong_simp: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and Inf :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
assms_3: fixes b :: "'al" assumes "b \<in> B" shows "\<exists>x\<in>A. x \<le>\<^sub>o\<^sub>w b"
finite_Cs: shows "\<forall>C\<in>Cs. finite C"
indecomposable_iff_\<omega>_power: fixes \<mu> :: "V" shows "indecomposable \<mu> = (\<mu> = 0 \<or> (\<exists>\<delta>. \<mu> = \<omega> \<up> \<delta> \<and> Ord \<delta>))"
Limit_vifunion_def: fixes \<alpha> :: "V" assumes "Limit \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>\<alpha>. a) `\<^sub>\<circ> \<alpha>) = \<alpha>"
snapshot_state_unchanged: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "ps c p \<noteq> None" shows "ps c p = ps c' p"
llist_set_induct: fixes x :: "'c"   and xs :: "'c llist"   and P :: "'c \<Rightarrow> 'c llist \<Rightarrow> bool" assumes "x \<in> lset xs"   and "\<And>xs. \<not> lnull xs \<Longrightarrow> P (lhd xs) xs"   and "\<And>xs y. \<lbrakk>\<not> lnull xs; y \<in> lset (ltl xs); P y (ltl xs)\<rbrakk> \<Longrightarrow> P y xs" shows "P x xs"
fmrel_on_fsetI: fixes S :: "'a fset"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'a \<rightharpoonup>\<^sub>f 'b"   and n :: "'a \<rightharpoonup>\<^sub>f 'c" assumes "\<And>x. x |\<in>| S \<Longrightarrow> rel_option P (fmlookup m x) (fmlookup n x)" shows "fmrel_on_fset S P m n"
pdevs_val_zip: fixes e :: "nat \<Rightarrow> real"   and xs :: "'b list" shows "pdevs_val e (pdevs_of_list xs) = sum_list (map2 (\<lambda>i. (*\<^sub>R) (e i)) [0..<length xs] xs)"
rdvd01_cs: fixes u :: "real"   and n :: "int"   and i :: "int"   and s :: "real" assumes "0 \<le> u"   and "u < 1"   and "0 < real_of_int n" shows "(real_of_int i rdvd real_of_int n * u - s) = (\<exists>j\<in>{0..n - 1}. real_of_int n * u = s - real_of_int \<lfloor>s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>s\<rfloor>))"
arbitrage_process_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "disc_filtr_prob_space M F" shows "disc_filtr_prob_space.arbitrage_process M F Mkt p = (\<exists>m. self_financing Mkt p \<and> disc_filtr_prob_space.trading_strategy F p \<and> (\<forall>w\<in>space M. val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < \<P>(w in M. 0 < cls_val_process Mkt p m w))"
fresh_psubst_tyS: fixes X :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list"   and S :: "tyS" assumes "X \<sharp> \<theta>"   and "X \<sharp> S" shows "X \<sharp> \<theta><S>"
update_by_fun_gr_pair_eq_update_by_fun_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv_raw x xs" shows "update_by_fun_gr_pair x k f xs = update_by_fun_pair x k f xs"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'b" shows "replicate (n + m) x = replicate n x @ replicate m x"
mapping_empty_def: shows "mapping_empty = (\<lambda>_. Mapping.empty)"
cases: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" assumes "a1 is-a2\<rightarrow>\<^sub>d* a3" obtains n :: "'node SDG_node" where "a1 = nb__" and "a2 = []" and "a3 = nb__" and "valid_SDG_node nb__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a1 = nc__" and "a2 = nsa__ @ [n''__]" and "a3 = n'a__" and "nc__ is-nsa__\<rightarrow>\<^sub>d* n''__" and "n''__ s\<longrightarrow>\<^bsub>cd\<^esub> n'a__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a1 = nd__" and "a2 = nsb__ @ [n''a__]" and "a3 = n'b__" and "nd__ is-nsb__\<rightarrow>\<^sub>d* n''a__" and "n''a__ s-V\<rightarrow>\<^sub>d\<^sub>d n'b__" and "n''a__ \<noteq> n'b__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a1 = n" and "a2 = ns @ [n'']" and "a3 = n'" and "n is-ns\<rightarrow>\<^sub>d* n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"