hcju/isabelleps huggingface dataset:
gcd_right_idem: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (gcd a b) b = gcd a b"
emb_cases_reverse: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" obtains t' :: "('c, 'd) tm" where "t = t'__" and "s = t'__"   | t' :: "('c, 'd) tm" and u :: "('c, 'd) tm" and s' :: "('c, 'd) tm" where "t = t'" and "s = s'" and "t' \<rightarrow>\<^sub>e\<^sub>m\<^sub>b u" and "u \<unrhd>\<^sub>e\<^sub>m\<^sub>b s'"
content_real_if: fixes a :: "real"   and b :: "real" shows "content {a..b} = (if a \<le> b then b - a else 0)"
Q: shows "Q \<in> sets (N \<Otimes>\<^sub>M M)"
prj_simps_6: shows "C.cod f_src.prj\<^sub>0 = src.apex"
remdups_clss_Nil_iff: fixes Cs :: "('a literal list \<times> nat) list" shows "(remdups_clss Cs = []) = (Cs = [])"
path_image_semicircle_Re_ge: fixes r :: "real"   and c :: "complex" assumes "0 \<le> r" shows "path_image (part_circlepath c r (- pi / 2) (pi / 2)) = sphere c r \<inter> {s. Re c \<le> Re s}"
simps_2: fixes v :: "int" shows "Num_arc v + MinInfty = Num_arc v"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "D.arr f"   and "D.cod f = b" shows "b \<cdot>\<^sub>D f = f"
neq_commute: fixes a :: "'a"   and b :: "'a" shows "(a \<noteq> b) = (b \<noteq> a)"
monom_mult_dist_left: fixes c :: "'c"   and d :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult (c + d) t p = monom_mult c t p + monom_mult d t p"
Smith_normal_form_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "Smith_normal_form A = ((\<forall>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < nrows A \<and> mod_type_class.to_nat b + 1 < ncols A \<longrightarrow> A $ a $ b dvd A $ (a + (1::'f)) $ (b + (1::'e))) \<and> isDiagonal A)"
assms_5: shows "io_targets M2 q2 io = {q2t}"
contV': shows "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) (\<lambda>x. V' x (f x))"
test2: fixes p :: "'a"   and q :: "'a" shows "\<iota> p \<cdot> \<iota> q \<cdot> \<iota> p = \<iota> p \<cdot> \<iota> q"
fls_deriv_eq_iff: fixes f :: "'a fls"   and g :: "'a fls" shows "(fls_deriv f = fls_deriv g) = (f = fls_const (fls_nth f 0 - fls_nth g 0) + g)"
assms_2: shows "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t"
fps_of_poly_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "fps_of_poly (p * q) = fps_of_poly p * fps_of_poly q"
emptiness_preservance: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle>"   and "R = {}" shows "R' = {}"
sd_chain_None_propagate: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m"   and "sd_chain n = None" shows "sd_chain m = None"
divergence_preservation_cond: shows "enc_preserves_divergence = (\<forall>S. S \<longmapsto>Source\<omega> \<longrightarrow> \<lbrakk>S\<rbrakk> \<longmapsto>Target\<omega>)"
limseq: shows "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
hensel_factor_def: shows "t = val_Zp (f \<bullet> a) - eint 2 * val_Zp (f' \<bullet> a)"
ext_surjective: fixes x :: "'a point_scheme" shows "x = \<lparr>xpos = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst point_ext_Tuple_Iso \<circ> id)) x, ypos = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst point_ext_Tuple_Iso \<circ> id)) x, \<dots> = (Record.iso_tuple_snd point_ext_Tuple_Iso \<circ> id) x\<rparr>"
hyps_2: shows "proj_poly_syz (length bs) p = sum_list (map2 (\<odot>) (cofactor_list_syz (length bs) p) bs)"
deriv_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "deriv f x \<equiv> SOME D. (f has_field_derivative D) (at x)"
cut3: fixes A :: "fm"   and B :: "fm"   and C :: "fm"   and D :: "fm"   and H :: "fm set" assumes "{A, B, C} \<turnstile> D"   and "H \<turnstile> A"   and "H \<turnstile> B"   and "H \<turnstile> C" shows "H \<turnstile> D"
simps: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" shows "local.root_1 (p, l, r) = (if n = 0 \<or> r = 0 then Rational 0 else if 0 < r then local.root_pos_1 (p, l, r) else uminus_2 (local.root_pos_1 (uminus_1 (p, l, r))))"
opar_comp_def': fixes s :: "((nat \<Rightarrow> 'd) \<times> 'e, 'g seq_action) automaton"   and i :: "nat"   and t :: "('f, 'g seq_action) automaton" shows "s \<langle>\<langle>\<^bsub>i\<^esub> t = \<lparr>init = {(\<sigma>, s\<^sub>l, t\<^sub>l) |\<sigma> s\<^sub>l t\<^sub>l. (\<sigma>, s\<^sub>l) \<in> init s \<and> t\<^sub>l \<in> init t}, trans = oparp_sos i (automaton.trans s) (automaton.trans t)\<rparr>"
gsp_reach: fixes q :: "'q"   and s :: "'s"   and q' :: "'q"   and s' :: "'s" assumes "((q, s), q', s') \<in> prod.E\<^sup>*" shows "(q, q') \<in> igba.E\<^sup>* \<and> (s, s') \<in> sa.E\<^sup>*"
thread_oks_redT_updTs_conv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'g, 'k) new_thread_action list"   and tas :: "('f, 'i, 'j) new_thread_action list" shows "thread_oks (redT_updTs' ts ta) tas = thread_oks (redT_updTs ts ta) tas"
ennexp_eln: fixes x :: "ennreal" shows "ennexp (eln x) = x"
dense_top_closed: shows "top \<le> top \<sqinter> top"
fresh_fact: fixes x :: "name"   and v :: "trm"   and t :: "trm"   and y :: "name" assumes "x \<sharp> v"   and "x \<sharp> t" shows "x \<sharp> t[y::=v]"
holds: shows "\<exists>x. Domainp A x \<and> f x"
fair_stl: fixes rs :: "'rule stream" assumes "fair rs" shows "fair (stl rs)"
2: fixes A :: "'b set"   and B :: "'c set" assumes "A \<in> sets M1"   and "B \<in> sets M2" shows "f -` (A \<times> B) \<inter> space M \<in> sets M"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('v \<Rightarrow> 'v) \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme"   and v :: "'v" assumes "CharacterDataClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
receiver_violation_def: shows "SINVAR_BLPtrusted.receiver_violation \<equiv> True"
h1: shows "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
some_eq_imp: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "Eps P = a"   and "P b" shows "P a"
benv_in_ve_def: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "benv_in_ve ve = \<Union> {benv_in_d d |d. d \<in> ran ve}"
add_cong: fixes x :: "int"   and y :: "int" shows "x mod m \<oplus> y mod m = (x + y) mod m"
div_mod_int_poly_then_pdivmod_1: fixes p :: "int poly"   and q :: "int poly"   and r :: "int poly"   and m :: "int poly" assumes "div_mod_int_poly p q = Some (r, m)" shows "(of_int_poly p div of_int_poly q, of_int_poly p mod of_int_poly q) = (of_int_poly r, of_int_poly m)"
step: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and l :: "'d"   and Z :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and k :: "nat \<Rightarrow> nat"   and v :: "'e \<Rightarrow> nat"   and n :: "nat"   and l' :: "'d"   and Z' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l'' :: "'d"   and Z'' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub>* \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub> \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub>* \<langle>l'', Z''\<rangle>"
coboundedI1: fixes c :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" assumes "c \<preceq> a" shows "c \<preceq> a \<oplus> b"
hd: shows "e_hd b = hd (Partial_Recursive.list_decode b)"
gval_fold_gOr_foldr: fixes l :: "'a gexp list"   and g :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gOr l g) s = gval (foldr gOr l g) s"
x_def: fixes n :: "nat" shows "x n = root n (real n) - 1"
is_run_def: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" shows "graph_defs.is_run (degeneralize_ext m) r \<equiv> r 0 \<in> g_V0 (degeneralize_ext m) \<and> ipath (g_E (degeneralize_ext m)) r"
cring_simprules_27: fixes x :: "int" shows "x * 0 = 0"
ln_complex_def: shows "Ln \<equiv> \<lambda>z. THE w. exp w = z \<and> - pi < Im w \<and> Im w \<le> pi"
direct_sum_def: fixes M1 :: "('g, 'e, 'h) module_scheme"   and M2 :: "('g, 'f, 'i) module_scheme" shows "direct_sum M1 M2 = \<lparr>carrier = carrier M1 \<times> carrier M2, mult = \<lambda>v w. (\<zero>\<^bsub>M1\<^esub>, \<zero>\<^bsub>M2\<^esub>), one = (\<zero>\<^bsub>M1\<^esub>, \<zero>\<^bsub>M2\<^esub>), zero = (\<zero>\<^bsub>M1\<^esub>, \<zero>\<^bsub>M2\<^esub>), add = \<lambda>v w. (fst v \<oplus>\<^bsub>M1\<^esub> fst w, snd v \<oplus>\<^bsub>M2\<^esub> snd w), smult = \<lambda>c v. (c \<odot>\<^bsub>M1\<^esub> fst v, c \<odot>\<^bsub>M2\<^esub> snd v)\<rparr>"
kernel_dim: shows "kernel_dim (char_matrix (jordan_matrix n_as) ev ^\<^sub>m k) = K.dim"
pos_divideR_le_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and b :: "'a"   and a :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq (scaleR (inverse c) b) a = less_eq b (scaleR c a)"
e'_def: shows "e' = e / (2 * sqrt (real DIM('a)))"
Max_imp_iSince: fixes I :: "nat set"   and Q :: "nat \<Rightarrow> bool"   and P :: "nat \<Rightarrow> bool" assumes "finite I"   and "I \<noteq> {}"   and "Q (Max I)" shows "P t'. t' \<S> t I. Q t"
neval_Suc_simp: fixes x :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)"   and y :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)" shows "A.neval x (Suc 0) y = eval_abv\<^sub>A x y"
list_order_def: fixes ops :: "('c \<times> 'c option) list"   and x :: "'c"   and y :: "'c" shows "Insert_Spec.list_order ops x y \<equiv> \<exists>xs ys zs. interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
is_equivalent_to_strict_subcategory: shows "equivalence_of_categories (\<cdot>\<^sub>S) (\<cdot>) \<F>C.D local.map \<F>C.\<nu> \<F>C.\<mu>"
semi_complete_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "semi_complete_on r A = (WN_on r A \<and> CR_on r A)"
mk_poly2_row_col: fixes pv :: "'b poly vec"   and n :: "nat"   and A :: "'b mat"   and j :: "nat"   and d :: "nat" assumes "dim_vec pv = n"   and "dim_row A = n"   and "j < dim_col A" shows "mk_poly2_row A d j pv n = pv + map_vec (\<lambda>a. monom a d) (col A j)"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<sim>\<^sup>s P"
ofilter_ordLess: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "Well_order r"   and "ofilter r A" shows "(A \<subset> Field r) = (Restr r A <o r)"
i_drop_the_conv: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Up> k = (THE g. \<exists>xs. length xs = k \<and> xs \<frown> g = f)"
indep_imp_subset_basis: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep X" shows "\<exists>B. indep_system.basis carrier indep B \<and> X \<subseteq> B"
dom_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.dom (trg\<^sub>B \<mu>) = trg\<^sub>B \<mu>"
unat_eq_of_nat: fixes n :: "nat"   and x :: "'c word" assumes "n < 2 ^ LENGTH('c)" shows "(unat x = n) = (x = word_of_nat n)"
real_scalar_mult2: fixes x :: "'b"   and A :: "'a" shows "real_of (inner x x) *\<^sub>R A = inner x x * A"
iSince_True_left: fixes I :: "nat set"   and Q :: "nat \<Rightarrow> bool" shows "(True. t' \<S> t I. Q t) = (\<diamond> t I. Q t)"
insort_insert_triv: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "insort_insert x xs = xs"
inEdges_def: fixes g :: "'a"   and n :: "'node" shows "gp2.inEdges g n \<equiv> map (\<lambda>(f, d). (f, d, n)) (inEdges2 g n)"
fold_mset_rewr: fixes x :: "'b \<Rightarrow> int"   and a :: "'a \<Rightarrow> int"   and l :: "(('b \<times> 'a) \<times> int) list" shows "fold_mset (\<lambda>((r, c), v) y. y(c := y c + x r * v)) a (mset l) = fold (\<lambda>((r, c), v) y. y(c := y c + x r * v)) l a"
llist_of_inf_llist_prefix: fixes u :: "'a list"   and v :: "'a stream" assumes "u \<le>\<^sub>F\<^sub>I v" shows "llist_of u \<le> llist_of_stream v"
axioms_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "fmpt M T" shows "mpt M T"
Abs_fls_inverse: fixes y :: "int \<Rightarrow> 'a" assumes "y \<in> {f. \<forall>\<^sub>\<infinity>n. f (- int n) = (0::'a)}" shows "fls_nth (Abs_fls y) = y"
omega_sum_less_iff: fixes ms :: "nat list"   and ns :: "nat list" shows "(((length ms, omega_sum ms), length ns, omega_sum ns) \<in> less_than <*lex*> VWF) = ((ms, ns) \<in> lenlex less_than)"
prems_6: shows "x \<in> dom (Cp (AllowPortFromTo c d po))"
false_sup_1: fixes P :: "'b upred" shows "false \<sqinter> P = P"
invocation_methd: fixes G :: "prog"   and statT :: "ref_ty"   and mode :: "inv_mode"   and s :: "st"   and a' :: "val"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "wf_prog G"   and "statT \<noteq> NullT"   and "\<forall>statC. statT = ClassT statC \<longrightarrow> is_class G statC"   and "\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM"   and "\<forall>T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM"   and "G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"   and "dynlookup G statT (invocation_class mode s a' statT) sig = Some m" shows "methd G (invocation_declclass G mode s a' statT sig) sig = Some m"
LNil_eq_lmap: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "(LNil = lmap f xs) = (xs = LNil)"
\<iota>_def: shows "\<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<equiv> \<I>"
inline1_in_sub_gpvs_callee: fixes out :: "'call'"   and callee' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('c, 'call, 'ret) rpv"   and gpv :: "('c, 'call, 'ret) gpv"   and s :: "'s" assumes "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "\<exists>call\<in>outs_\<I> \<I>. \<exists>s. I s \<and> (\<forall>x\<in>responses_\<I> \<I>' out. callee' x \<in> sub_gpvs \<I>' (callee s call))"
is_group: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "Group.group G"
simps_1: fixes a :: "'c" shows "lsumr (Inl a) = Inl (Inl a)"
Quotient3_rep_abs: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and r :: "'c" assumes "Quotient3 R Abs Rep"   and "R r r" shows "R (Rep (Abs r)) r"
unfold_enumFromToStep: fixes y :: "int"   and n :: "int\<^sub>\<bottom>" shows "unfold\<cdot>(enumFromToStep\<cdot>(up\<cdot>y))\<cdot>(up\<cdot>n) = enumFromToL\<cdot>n\<cdot>(up\<cdot>y)"
simps_3: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and e1 :: "('d, 'c) expr"   and e2 :: "('d, 'c) expr"   and env :: "'d \<Rightarrow> 'c" shows "eval (Binop f e1 e2) env = f (eval e1 env) (eval e2 env)"
IH: fixes \<phi> :: "'a ltl" assumes "\<^bold>G \<phi> = {}" shows "af \<phi> xs = af\<^sub>G \<phi> xs"
Card_Aleph: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "Card (\<aleph>\<alpha>)"
t_fusion_def: fixes x :: "'b \<times> ('c \<times> 'b) list"   and y :: "'b \<times> ('c \<times> 'b) list" shows "t_fusion x y \<equiv> if Dioid_Models.last x = first y then (fst x, snd x @ snd y) else undefined"
suffix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "Sublist.suffix xs ys = (\<exists>zs. ys = zs @ xs)"
qSwap_preserves_qGood: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" shows "qGood (X #[[x \<and> y]]_zs) = qGood X"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<squnion> b) * c = a * c \<squnion> b * c"
not_principal_aux2: fixes c :: "('a, 'b) sequent"   and T :: "'b"   and Ts :: "('a, 'b) form list"   and M :: "'b"   and \<Gamma> :: "('a, 'b) form multiset"   and N :: "'b"   and \<Delta> :: "('a, 'b) form multiset"   and F :: "'a"   and Fs :: "('a, 'b) form list" assumes "ModalSequents.mset c = \<LM> Modal T Ts  \<RM>"   and "M \<cdot> \<Gamma> + antec c = N \<cdot> \<Delta> \<oplus> Compound F Fs" shows "False"
nn_def: shows "nn = 1 / (complex_of_real (2 * pi) * \<i>) * contour_integral h (\<lambda>w. 1 / (w - z))"
amor_Unbal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree"   and n' :: "nat"   and h' :: "nat" assumes "ins3 n d x t = Unbal2 t' n' h'"   and "bal_i n (height t + d)" shows "real (T_ins3 n d x t) + \<Phi> t' - \<Phi> t \<le> real (2 * size1 t) + (6 * e + 1) * real (height t)"
chine_assoc_props_2: shows "Prj\<^sub>1 \<cdot> chine_assoc = Prj\<^sub>1\<^sub>1"
integrable_normal_moment: fixes \<sigma> :: "real"   and \<mu> :: "real"   and k :: "nat" assumes "0 < \<sigma>" shows "integrable lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ k)"
select_convs_3: fixes E_ES :: "'c set"   and I_ES :: "'c set"   and O_ES :: "'c set"   and Tr_ES :: "'c list set"   and more :: "'d" shows "O_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = O_ES"
hyps_3: shows "\<forall>t\<in>{s<..<b}. Re (g t - z) \<noteq> 0"
msubset_mset_set_iff: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(mset_set A \<subseteq># mset_set B) = (A \<subseteq> B)"
wt_method_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and Ts :: "ty list"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "instr list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "wt_method P C Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s \<equiv> 0 < length is \<and> length \<tau>s = length is \<and> check_types P mxs (1 + length Ts + mxl\<^sub>0) (map OK \<tau>s) \<and> wt_start P C Ts mxl\<^sub>0 \<tau>s \<and> (\<forall>pc<length is. P,T\<^sub>r,mxs,length is,xt \<turnstile> is ! pc,pc :: \<tau>s)"
subgroup_sum_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "subgroup {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}} (product_group I G)"
indep_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "\<E>.indep_in \<E> X = (X \<subseteq> \<E> \<and> indep_in carrier X)"
gen_subalgebra_def: fixes M :: "'b measure"   and G :: "'b set set" shows "gen_subalgebra M G = sigma (space M) G"
zhmset_of_plus: fixes A :: "hmultiset"   and B :: "hmultiset" shows "zhmset_of (A + B) = zhmset_of A + zhmset_of B"
prv_imp_imp_neg_rev: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (imp \<phi> \<chi>) (imp (neg \<chi>) (neg \<phi>)))"
img_is_subgroup: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h" shows "subgroup (h ` carrier G) H"
nat_pow_Suc: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "nat" assumes "Group.monoid G" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
card_of_set_of_all_possible_states: fixes X :: "'a set" assumes "finite X" shows "card {s. fmdom' s = X} = 2 ^ card X"
update_def: fixes get :: "('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and put :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'e"   and m :: "'d" shows "monad_state_base.update get put f m = get (\<lambda>s. put (f s) m)"
sifum_modes_axioms: shows "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"
accI: fixes x :: "'b"   and r :: "('b \<times> 'b) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
linear_hensel_main: fixes p :: "int"   and D1 :: "int poly"   and S :: "int poly"   and H1 :: "int poly"   and T :: "int poly"   and C :: "int poly"   and n :: "nat"   and D :: "int poly"   and H :: "int poly"   and D' :: "int poly"   and H' :: "int poly" assumes "poly_mod.Mp p (D1 * S + H1 * T) = poly_mod.Mp p 1"   and "poly_mod.Mp p (D1 * H1) = poly_mod.Mp p C"   and "monic D1"   and "poly_mod.Mp p D1 = D1"   and "poly_mod.Mp p H1 = H1"   and "linear_hensel_main C p S T D1 H1 n = (D, H)"   and "n \<noteq> 0"   and "prime p"   and "poly_mod.coprime_m p D1 H1" shows "poly_mod.Mp (p ^ n) (D * H) = poly_mod.Mp (p ^ n) C \<and> monic D \<and> poly_mod.Mp p D = poly_mod.Mp p D1 \<and> poly_mod.Mp p H = poly_mod.Mp p H1 \<and> poly_mod.Mp (p ^ n) D = D \<and> poly_mod.Mp (p ^ n) H = H \<and> (poly_mod.Mp (p ^ n) (D' * H') = poly_mod.Mp (p ^ n) C \<longrightarrow> poly_mod.Mp p D' = poly_mod.Mp p D1 \<longrightarrow> poly_mod.Mp p H' = poly_mod.Mp p H1 \<longrightarrow> poly_mod.Mp (p ^ n) D' = D' \<longrightarrow> poly_mod.Mp (p ^ n) H' = H' \<longrightarrow> monic D' \<longrightarrow> D' = D \<and> H' = H)"
wMaxU: fixes \<theta>\<^sub>1 :: "trm"   and \<theta>\<^sub>2 :: "trm"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([trm.Max \<theta>\<^sub>1 \<theta>\<^sub>2]<>\<nu>) = (let (l1, u1) = [\<theta>\<^sub>1]<>\<nu>; (l2, u2) = [\<theta>\<^sub>2]<>\<nu> in (wmax l1 l2, wmax u1 u2))"
sum_nonneg: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "\<And>x. x \<in> A \<Longrightarrow> (0::'a) \<le> f x" shows "(0::'a) \<le> sum_0 f A"
dist_eq_0_iff: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open" shows "(dist x y = 0) = (x = y)"
power_commuting_commutes: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = times y x" shows "times (power.power one times x n) y = times y (power.power one times x n)"
norm_28: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "((\<lambda>s. False) \<R> P) = \<box>P"
prems_1: shows "emb_step_at (a # p) d t = u"
UP_a_closed: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
weak_bisimilarity_implies_weak_equivalence: fixes P :: "'state"   and Q :: "'state" assumes "P \<approx>\<cdot>\<^sub>S Q" shows "S_transform.weakly_logically_equivalent P Q"
le_ubound_def: fixes c :: "'a"   and b :: "'a option" shows "c \<le>\<^sub>u\<^sub>b b = \<unlhd>\<^sub>u\<^sub>b (<) c b"
asym: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subset># a"   and "a \<subset># b" shows "False"