hcju/isabelleps huggingface dataset:
cases: fixes x :: "('b \<times> nat) tree" obtains l :: "('b \<times> nat) tree" and a :: "'b" and lv :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (a, lv), r\<rangle>"   | "x = \<langle>\<rangle>"
assms: shows "locks_ticket (map_ticket (\<lambda>t. t mod N) (ts t)) tki"
path_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and k :: "nat" shows "IFC_def.path prob \<sigma> k = fst ((IFC_def.step prob ^^ k) (IFC_def.entry prob, \<sigma>))"
subset_atMost_Max_le_conv: fixes A :: "'a set"   and n :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(A \<subseteq> {..n}) = (Max A \<le> n)"
hull_unique: fixes s :: "'c set"   and t :: "'c set"   and S :: "'c set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t"   and "\<And>t'. \<lbrakk>s \<subseteq> t'; S t'\<rbrakk> \<Longrightarrow> t \<subseteq> t'" shows "S hull s = t"
Diff_eq: fixes A :: "'b set"   and B :: "'b set" shows "A - B = A \<inter> - B"
exp_eq_polar: fixes z :: "complex" shows "exp z = complex_of_real (exp (Re z)) * cis (Im z)"
action_proj_def: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs :: "'d set" shows "action_proj a vs \<equiv> (fmrestrict_set vs (fst a), fmrestrict_set vs (snd a))"
fold_mset_rewr: fixes a :: "'b"   and l :: "'a list" shows "fold_mset f a (mset l) = fold f l a"
mulexp_Suc: fixes n :: "nat"   and p :: "'a list"   and q :: "'a list" shows "mulexp (Suc n) p q = p *** mulexp n p q"
hS: fixes x :: "'b"   and y :: "'b"   and c :: "'a" assumes "x \<in> S"   and "h y" shows "h (c *s x + y)"
CESem_simps_4: fixes b :: "bool"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Bool b \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> (C\<cdot>r). CB\<cdot>(Discr b))"
cmod_square_less_1_plus: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
chine_simps_1: shows "C.arr \<nu>'.chine"
segment_in_aform_val: fixes e :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "f \<in> UNIV \<rightarrow> {- 1..1}" shows "closed_segment (aform_val e X) (aform_val f X) \<subseteq> Affine X"
l2norm_triangle: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "l2norm S (\<lambda>x. f x + g x) \<le> l2norm S f + l2norm S g"
eval_code: fixes f :: "unit \<Rightarrow> 'a seq" shows "eval (Seq f) = member (f ())"
osubst_preserves_BVO: fixes ODE :: "('c + 'd, 'b) ODE"   and \<sigma> :: "'d \<Rightarrow> ('c, 'b) trm" shows "BVO (OsubstFO ODE \<sigma>) = BVO ODE"
antisym_def: fixes r :: "('b \<times> 'b) set" shows "antisym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
zero_r_def: shows "(0::'a) = -r (1::'a)"
simps_2: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and n :: "nat" shows "iterated_serial_on A r x (Suc n) = (SOME y. y \<in> A \<and> (iterated_serial_on A r x n, y) \<in> r)"
equality: fixes r :: "'a a0n_state_scheme"   and r' :: "'a a0n_state_scheme" assumes "signals r = signals r'"   and "a0n_state.more r = a0n_state.more r'" shows "r = r'"
emeasure_subprob_space_less_top: fixes A :: "bool stream set" shows "emeasure M A \<noteq> \<top>"
r_result_prim: shows "prim_recfn 3 r_result"
qtmset_append_union: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and q' :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.queue_to_multiset (q @ q') = BinomialHeapStruc.queue_to_multiset q + BinomialHeapStruc.queue_to_multiset q'"
tendsto_sgn: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<noteq> (0::'b)" shows "((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
add_def: shows "add \<equiv> map_fun id (map_fun Rep_\<I> Abs_\<I>) (\<lambda>n (a, b). (a, b + enat n))"
Stirling_closed_form: fixes n :: "nat"   and k :: "nat" shows "real (Stirling n k) = (\<Sum>j\<le>k. (- 1) ^ (k - j) * real (k choose j) * real j ^ n) / fact k"
ssap: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'c" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {Generic_Syntax.Variable var 10}" shows "ssap \<phi> \<in> F"
amin_generateTr: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<forall>w\<in>carrier K - {\<zero>}. \<exists>z. v w = z *\<^sub>a AMin {x \<in> v ` carrier K. 0 < x}"
Im_complex_div_eq_0: fixes a :: "complex"   and b :: "complex" shows "is_real (a / b) = is_real (a * cnj b)"
lookup_tl': fixes xs :: "('c, 'b) oalist_ntm"   and k :: "'c" shows "OAlist_lookup_ntm (OAlist_tl_ntm xs) k = (if k = fst (OAlist_hd_ntm xs) then 0::'b else OAlist_lookup_ntm xs k)"
s1': shows "redT_upd s1 t ta1 x1' m1' s1'"
hcomp_arr_obj: fixes f :: "'b"   and a :: "'b" assumes "D.arr f"   and "D.obj a"   and "src\<^sub>D f = a" shows "f \<star>\<^sub>D a = f"
ih: fixes ua :: "('s, 'v) tm"   and ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#hsize ua, hsize ta, hsize sa#} < {#hsize u, hsize t, hsize s#}"   and "ua >\<^sub>t ta"   and "ta >\<^sub>t sa" shows "ua >\<^sub>t sa"
surj_fun_eq: fixes f :: "'e \<Rightarrow> 'd"   and X :: "'e set"   and g1 :: "'d \<Rightarrow> 'f"   and g2 :: "'d \<Rightarrow> 'f" assumes "f ` X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
DAcc_DSourcesNOT: fixes X :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" shows "(X \<notin> DAcc i C) = (C \<notin> DSources i X)"
IH1: shows "continuous_on UNIV (\<lambda>x. Blinfun (extendf_deriv I i \<theta>\<^sub>1 \<nu> x))"
simps_2: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "minus_poly_rev_list_i ops xs [] = xs"
pair_measure_countable: fixes S1 :: "'b set"   and S2 :: "'c set" assumes "countable S1"   and "countable S2" shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
finite_size_bounded_msets: fixes S :: "'a set"   and len :: "nat" assumes "finite S" shows "finite {m. size m \<le> len \<and> set_mset m \<subseteq> S}"
order_antisym_conv: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "(x \<le> y) = (x = y)"
eq_strict_2: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and y :: "'b" assumes "class.Eq_strict eq" shows "eq\<cdot>\<bottom>\<cdot>y = \<bottom>"
equiv_listrel: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "equiv A r" shows "equiv (lists A) (listrel r)"
Theorem3derivProp_def: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and q :: "L" shows "Theorem3derivProp X p G c H q = (\<exists>\<Phi>. X \<rhd> c : Sec (Q p H) (EQ G q) (EQ H q) \<Phi>)"
norm_vector_to_cblinfun: fixes x :: "'b" shows "norm (vector_to_cblinfun x) = norm x"
card_of_lists_infinite: fixes A :: "'b set" assumes "infinite A" shows "|lists A| =o |A|"
closedin_def: fixes U :: "'c topology"   and S :: "'c set" shows "closedin U S = (S \<subseteq> topspace U \<and> openin U (topspace U - S))"
unit_rcs_in_set_rcs: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "H \<in> set_rcs G H"
weakRepl: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<parallel> !P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "P' \<noteq> P \<parallel> !P" shows "!P \<Longrightarrow>\<alpha> \<prec> P'"
phi_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and v :: "'val" shows "uninst.phi p g v = p g (uninst.defNode p g v, v)"
finmap_of_eq_iff: fixes i :: "'b set"   and j :: "'b set"   and m :: "'b \<Rightarrow> 'c"   and n :: "'b \<Rightarrow> 'c" assumes "finite i"   and "finite j" shows "(Fin_Map.finmap_of i m = Fin_Map.finmap_of j n) = (i = j \<and> (\<forall>k\<in>i. m k = n k))"
chamber_adj: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "D \<in> X"   and "C \<sim> D" shows "SimplicialComplex.maxsimp X D"
poly_add_zero_1: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "polynomial K p" shows "poly_add p [] = p"
onlD: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and P :: "'e \<times> 'd \<Rightarrow> bool"   and \<xi> :: "'e"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "onl \<Gamma> P (\<xi>, p)" shows "\<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"
valid_plan_valid_tail: fixes h :: "'a"   and as :: "'a list"   and PROB :: "'a set" assumes "h # as \<in> valid_plans PROB" shows "as \<in> valid_plans PROB"
cod_simp: fixes f :: "'a" assumes "L.arr f" shows "L.cod f = cod f"
v0a: shows "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {0..a}"
naturality: fixes f :: "'a" assumes "Adj.C.arr f" shows "Adj.\<epsilon> (Adj.C.cod f) \<cdot>\<^sub>C Adj.FG.map f = Adj.C.map f \<cdot>\<^sub>C Adj.\<epsilon> (Adj.C.dom f)"
con_comp_sinks_no_fake: fixes p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e"   and u :: "'h option"   and I :: "('h \<times> 'h) set"   and D :: "'f \<Rightarrow> 'h"   and E :: "'g \<Rightarrow> 'h"   and xs :: "'e list" assumes "range p \<union> range q = UNIV"   and "u \<in> range Some" shows "sinks I (the \<circ> con_comp_map D E p q) (the u) xs = the ` sinks (con_comp_pol I) (con_comp_map D E p q) u xs"
measure_Un2: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M (B - A)"
inject_2: fixes x21 :: "nat"   and x22 :: "int poly \<times> rat \<times> rat"   and y21 :: "nat"   and y22 :: "int poly \<times> rat \<times> rat" shows "(Irrational x21 x22 = Irrational y21 y22) = (x21 = y21 \<and> x22 = y22)"
simps_5: fixes C :: "String.literal" shows "default_val (Class C) = Null"
sym_digraphI_app_iso: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "sym_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "sym_digraph (app_iso hom G)"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "B.ide a"   and "B.seq f a" shows "B.dom f = a"
neq_iff: fixes x :: "'t"   and y :: "'t" shows "(x \<noteq> y) = (x \<prec>\<^sub>t y \<or> y \<prec>\<^sub>t x)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Right.in_hom f a b" shows "Right.in_hom (Right.R f) (Right.R a) (Right.R b)"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'f"   and u :: "'f"   and y :: "'f"   and v :: "'f" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
vcg_all_lift: fixes P :: "'d \<Rightarrow> ('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com"   and Q :: "'d \<Rightarrow> ('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<And>x. \<lbrace>P x\<rbrace> c \<lbrace>Q x\<rbrace>" shows "\<lbrace>\<lambda>s. \<forall>x. P x s\<rbrace> c \<lbrace>\<lambda>s. \<forall>x. Q x s\<rbrace>"
exhaust_sel: fixes option :: "'a option" obtains "option = None"   | "option = Some (the option)"
\<beta>s_rev: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s (Abs_fv x \<tau>' s $ Fv x \<tau>')"
Target_def: fixes T :: "('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" shows "Target T = target ` T"
refl_multpw: fixes ns :: "('a \<times> 'a) set" assumes "refl ns" shows "refl (multpw ns)"
symvar_var_imp_var2: fixes sv :: "'c \<times> nat"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "'c \<Rightarrow> nat" assumes "sv \<in> Bexp.vars (adapt_bexp e s)" shows "fst sv \<in> Bexp.vars e"
set_of_appr_of_ivl_point: fixes xs :: "real list" shows "set_of_appr (appr_of_ivl ops xs xs) = {xs}"
a_rcos_zero: fixes I :: "'a list set"   and i :: "'a list" assumes "ideal I (K [X])"   and "i \<in> I" shows "I +>\<^bsub>K [X]\<^esub> i = I"
image_Un: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'e set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
fPowD: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<in>| fPow B" shows "A |\<subseteq>| B"
that: fixes g :: "'a" assumes "g \<in> carrier G"   and "Q = g <# (P #> inv g)" shows "thesis"
return_def: shows "return = COERCE(udom \<rightarrow> udom\<cdot>'d, 'c \<rightarrow> 'c\<cdot>'d)\<cdot>returnU"
FP: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "t \<in> timpl_closure t TI"
steps: shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* (cs', css', t)"
Pi_power_coprime: fixes i :: "nat"   and j :: "nat"   and m :: "nat"   and n :: "nat" assumes "i \<noteq> j" shows "coprime (Pi i ^ m) (Pi j ^ n)"
take_bit_rec: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n a = (if n = 0 then zero else plus (times (take_bit (n - 1) (divide a (numeral.numeral one plus (num.Bit0 num.One)))) (numeral.numeral one plus (num.Bit0 num.One))) (modulo a (numeral.numeral one plus (num.Bit0 num.One))))"
ev: shows "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P"
sigma_finite_measures: fixes i :: "'i" shows "sigma_finite_measure (M i)"
mmq: shows "m = 2 ^ s1 * word_of_nat mq"
H_iso_cond2: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'b"   and z :: "'b"   and x :: "'c" assumes "mono f"   and "y \<le> z"   and "x \<le> f y" shows "x \<le> f z"
r2: shows "r2 = 0 \<or> degree r2 < degree g"
card_of_Plus_infinite1: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|A <+> B| =o |A|"
suffix_second: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "second (s |\<^sub>s n) = s (Suc n)"
determ_maximalD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "determ t" shows "maximal t"
to_vec_diag: fixes x :: "'b" shows "to_vec (diag x) = mat x"
alw_True: fixes \<omega> :: "'a stream" shows "alw (\<lambda>x. True) \<omega>"
iso_imp_img_comm_group: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> Group.iso G H" shows "comm_group (H\<lparr>one := h \<one>\<rparr>)"
diff: fixes x :: "complex"   and y :: "complex"   and xa :: "'d" shows "(x - y) *\<^sub>C xa = x *\<^sub>C xa - y *\<^sub>C xa"
pref_comp_not_spref: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie> v"   and "\<not> u <p v" shows "v \<le>p u"
YM3_auth_B_to_A_lemma: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and NA :: "nat"   and nb :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "B \<notin> bad \<longrightarrow> Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> \<in> set evs"
PiE_eq: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set"   and T :: "'a \<Rightarrow> 'b set" shows "(Pi\<^sub>E I S = Pi\<^sub>E I T) = (Pi\<^sub>E I S = {} \<and> Pi\<^sub>E I T = {} \<or> (\<forall>i\<in>I. S i = T i))"
triangulize_divisor: fixes A :: "'a mat"   and d :: "nat"   and r' :: "'a"   and A' :: "'a mat" assumes "mute_fun mf"   and "det_selection_fun sel_fun"   and "A \<in> carrier_mat d d"   and "local.triangulize A = (r', A')" shows "r' \<noteq> (0::'a)"
assms: shows "\<forall>i. P (\<omega> !! i) (\<omega>' !! i)"
cinner_mult_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and a :: "'c"   and m :: "complex"   and b :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "\<langle>a, of_complex m * b\<rangle> = m * \<langle>a, b\<rangle>"
comp_quasi_inverse_2: fixes f :: "'a" assumes "equivalence_map f" shows "f \<star> f\<^sup>~ \<cong> trg f"
rbt_lookup_in_tree: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt"   and k :: "'a"   and v :: "'b" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "(ord.rbt_lookup less t k = Some v) = ((k, v) \<in> set (RBT_Impl.entries t))"
left_ideal_explicit: fixes S :: "'a set" shows "left_ideal_generated S = {y. \<exists>f U. finite U \<and> U \<subseteq> S \<and> (\<Sum>i\<in>U. f i * i) = y}"
diff: fixes x :: "'a"   and y :: "'a" shows "f'' (x - y) = f'' x - f'' y"
elims: fixes x :: "'b rule list"   and y :: "'b rule list" assumes "normalize_rules_dnf x = y" obtains "x = []" and "y = []"   | m :: "'b match_expr" and a :: "action" and rs :: "'b rule list" where "x = Rule m a # rs" and "y = map (\<lambda>m. Rule m a) (normalize_match m) @ normalize_rules_dnf rs"
cd_split: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob i \<pi> k"   and "\<not> IFC_def.is_icdi prob i \<pi> k" obtains m :: "nat" where "IFC_def.is_icdi prob i \<pi> m" and "IFC_def.is_cdi prob m \<pi> k"
elims_2: fixes x :: "'c arctic_delta"   and xa :: "'c arctic_delta" assumes "x \<le> xa" obtains xb :: "'c arctic_delta" where "x = MinInfty_delta" and "xa = xb__"   | y :: "'c" and xb :: "'c" where "x = Num_arc_delta y" and "xa = Num_arc_delta xb" and "y \<le> xb"
max_input_def: fixes g :: "vname gexp" shows "GExp.max_input g = (let inputs = enumerate_gexp_inputs g in if inputs = {} then None else Some (Max inputs))"
preserves_comp_2: fixes f' :: "'d \<times> 'c"   and f :: "'d \<times> 'c" assumes "DopxC.seq f' f" shows "\<Phi> (DopxC.comp f' f) = S.comp (\<Phi> f') (Hom_FopxC.map f)"
split_option_ex: fixes P :: "'m option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (Some x)))"
map_poly_pCons_hom: fixes a :: "'a"   and p :: "'a poly" shows "map_poly hom (pCons a p) = pCons (hom a) (map_poly hom p)"
swap_def: fixes a :: "'e"   and b :: "'e"   and f :: "'e \<Rightarrow> 'f" shows "Fun.swap a b f = f(a := f b, b := f a)"
null_sets_outer: fixes S :: "'a set" shows "(S \<in> null_sets M) = (\<forall>e>0. \<exists>T\<in>fmeasurable M. S \<subseteq> T \<and> Sigma_Algebra.measure M T < e)"
Bseq_NSBseq_iff: fixes X :: "nat \<Rightarrow> 'a" shows "Bseq X = NSBseq X"
neq_if_length_neq: fixes xs :: "'c list"   and ys :: "'c list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
bounded_f': shows "bounded ((\<lambda>xa. f' (h * xa + t, x (h * xa + t))) ` {0..1})"
leq_lit_def: fixes L :: "('a, 'b) Term.term literal"   and K :: "('a, 'b) Term.term literal" shows "leq_lit L K = (case (K, L) of (Pos x, Pos a) \<Rightarrow> leq_head (atm_of L) (atm_of K) | (Pos x, Neg xa) \<Rightarrow> False | (Neg x, Pos xa) \<Rightarrow> True | (Neg x, Neg a) \<Rightarrow> leq_head (atm_of L) (atm_of K))"
braun2_of_replicate: fixes x :: "'b"   and n :: "nat"   and s :: "'b tree"   and t :: "'b tree" assumes "braun2_of x n = (s, t)" shows "list s = replicate n x \<and> list t = replicate (n + 1) x"
simps_2: fixes \<gamma> :: "'b constr"   and \<Gamma> :: "'b constr list" shows "\<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk> \<gamma> \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
scale_zero_right: fixes a :: "'d" shows "a * (0::'d) = (0::'d)"
P3_def: shows "P3 = {p. p \<in># prime_factorization n \<and> [p = 3] (mod 4)}"
pmf_abs_summable: fixes p :: "'b pmf"   and A :: "'b set" shows "pmf p abs_summable_on A"
Int_assoc: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
axioms_2: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "semiring_of_sets \<Omega> M" shows "semiring_of_sets_axioms M"
induct: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and prod :: "'b \<times> 'c" assumes "\<And>a b. P (a, b)" shows "P prod"
image_mset_filter_swap: fixes f :: "'c \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and M :: "'c multiset" shows "image_mset f {#x \<in># M. P (f x)#} = filter_mset P (image_mset f M)"
VSagree_supset: fixes B :: "'a set"   and A :: "'a set"   and \<nu> :: "(real, 'a) vec"   and \<nu>' :: "(real, 'a) vec" assumes "B \<subseteq> A"   and "VSagree \<nu> \<nu>' A" shows "VSagree \<nu> \<nu>' B"
in_set_butlastD: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set (butlast xs)" shows "x \<in> set xs"
Diff_insert0: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "A - Set.insert x B = A - B"
approximation_preproc_int_6: fixes i :: "int"   and j :: "int" shows "real_of_int (i * j) = real_of_int i * real_of_int j"
ba_iso_inv_lifted_pair: fixes xf :: "('c, 'd) lifted_boolean_algebra"   and f :: "('c, 'd) phi" shows "(Rep_lifted_boolean_algebra xf f, Rep_phi f (- Rep_lifted_boolean_algebra xf f)) \<in> triple.pairs (Rep_phi f)"
lformula_nFEx: fixes b :: "'k"   and \<phi> :: "('a, 'k) aformula" assumes "lformula (FEx b \<phi>)" shows "lformula (nFEx b \<phi>)"
compl_closed: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<^bold>- x \<in> carrier L"
guard_def: fixes b :: "bool" shows "guard b = (if b then Some () else None)"
select_convs_2: fixes g_V :: "'d set"   and g_E :: "('d \<times> 'd) set"   and g_V0 :: "'d set"   and sa_L :: "'d \<Rightarrow> 'e"   and more :: "'b" shows "sa_rec.more \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, sa_L = sa_L, \<dots> = more\<rparr> = more"
WriterT_runWriterT: fixes m :: "'e\<cdot>('f, 'g) writerT" shows "WriterT\<cdot>(runWriterT\<cdot>m) = m"
supset_lbound_ofI: fixes x :: "'b set"   and b :: "'b set"   and y :: "'b set" assumes "order.greater_eq b x"   and "order.greater_eq b y" shows "dual_order.lbound_of x y b"
rep_eq: fixes x :: "'b ratfps"   and xa :: "'b ratfps" shows "Rep_ratfps (x - xa) = Rep_ratfps x - Rep_ratfps xa"