hcju/isabelleps huggingface dataset:
steps'_distinct: fixes qs :: "'b list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'b list" assumes "length qs = length as"   and "length as = n" shows "distinct (steps' s qs as n) = distinct s"
order_iff: fixes b :: "'e"   and a :: "'e" shows "(b \<le> a) = (a = sup a b)"
VSEqualityBlocks_def: shows "VSEqualityBlocks = opt_group \<circ> QE_dnf' opt_group (the_real_step_augment qe_eq_repeat) \<circ> VSLuckiestBlocks \<circ> opt_group"
limit_ordinal_not_oSuc: fixes p :: "ordinal" shows "\<not> limit_ordinal (oSuc p)"
a: shows "<a>:M \<in> ORRIGHT1 (B OR C) (\<parallel><B>\<parallel>)"
assms_6: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"
rep_eq: fixes x :: "'b zmultiset" shows "count (mset_neg x) = (case rep_zmultiset x of (Mp, Mn) \<Rightarrow> count (Mn - Mp))"
guardT: fixes \<xi>' :: "'e"   and g :: "'e \<Rightarrow> 'e set"   and \<xi> :: "'e"   and l :: "'h"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<xi>' \<in> g \<xi>" shows "((\<xi>, {l}\<langle>g\<rangle> p), \<tau>, \<xi>', p) \<in> seqp_sos \<Gamma>"
Enq_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "Enq ic q oc \<equiv> ic$ \<noteq> $ic \<and> q$ = $q @ [ic$] \<and> oc$ = $oc"
set_zip: fixes xs :: "'g list"   and ys :: "'h list" shows "set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
assms: shows "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>"
card_set_berlekamp_basis: shows "card (set (berlekamp_basis u)) = length (berlekamp_basis u)"
elims_1: fixes x :: "('e, 'f, 'g) Tree"   and xa :: "('e, 'f, 'g) Tree"   and y :: "bool" assumes "(x =\<^sub>\<alpha> xa) = y" obtains tset1 :: "('e, 'f, 'g) Tree set['e]" and tset2 :: "('e, 'f, 'g) Tree set['e]" where "x = tConj tset1" and "xa = tConj tset2" and "y = rel_bset (=\<^sub>\<alpha>) tset1 tset2"   | t1 :: "('e, 'f, 'g) Tree" and t2 :: "('e, 'f, 'g) Tree" where "x = tNot t1__" and "xa = tNot t2__" and "y = (t1__ =\<^sub>\<alpha> t2__)"   | \<phi>1 :: "'f" and \<phi>2 :: "'f" where "x = tPred \<phi>1" and "xa = tPred \<phi>2" and "y = (\<phi>1 = \<phi>2)"   | \<alpha>1 :: "'g" and t1 :: "('e, 'f, 'g) Tree" and \<alpha>2 :: "'g" and t2 :: "('e, 'f, 'g) Tree" where "x = tAct \<alpha>1 t1" and "xa = tAct \<alpha>2 t2" and "y = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"   | v :: "('e, 'f, 'g) Tree" and va :: "('e, 'f, 'g) Tree set['e]" where "x = tNot v__" and "xa = tConj va__" and "\<not> y"   | v :: "('e, 'f, 'g) Tree" and va :: "'f" where "x = tNot vb__" and "xa = tPred vaa__" and "\<not> y"   | v :: "('e, 'f, 'g) Tree" and va :: "'g" and vb :: "('e, 'f, 'g) Tree" where "x = tNot vc__" and "xa = tAct vab__ vba__" and "\<not> y"   | v :: "'f" and va :: "('e, 'f, 'g) Tree set['e]" where "x = tPred vd__" and "xa = tConj vac__" and "\<not> y"   | v :: "'f" and va :: "('e, 'f, 'g) Tree" where "x = tPred ve__" and "xa = tNot vad__" and "\<not> y"   | v :: "'f" and va :: "'g" and vb :: "('e, 'f, 'g) Tree" where "x = tPred vf__" and "xa = tAct vae__ vbb__" and "\<not> y"   | v :: "'g" and va :: "('e, 'f, 'g) Tree" and vb :: "('e, 'f, 'g) Tree set['e]" where "x = tAct vg__ vaf__" and "xa = tConj vbc__" and "\<not> y"   | v :: "'g" and va :: "('e, 'f, 'g) Tree" and vb :: "('e, 'f, 'g) Tree" where "x = tAct vh__ vag__" and "xa = tNot vbd__" and "\<not> y"   | v :: "'g" and va :: "('e, 'f, 'g) Tree" and vb :: "'f" where "x = tAct vi__ vah__" and "xa = tPred vbe__" and "\<not> y"   | va :: "('e, 'f, 'g) Tree set['e]" and v :: "('e, 'f, 'g) Tree" where "x = tConj vai__" and "xa = tNot vj__" and "\<not> y"   | va :: "('e, 'f, 'g) Tree set['e]" and v :: "'f" where "x = tConj vaj__" and "xa = tPred vk__" and "\<not> y"   | vb :: "('e, 'f, 'g) Tree set['e]" and v :: "'g" and va :: "('e, 'f, 'g) Tree" where "x = tConj vb" and "xa = tAct v va" and "\<not> y"
simps: fixes a :: "'g list"   and A :: "'g set" shows "(a \<in> lists A) = (a = [] \<or> (\<exists>aa l. a = aa # l \<and> aa \<in> A \<and> l \<in> lists A))"
OrdP_fresh_iff: fixes a :: "atom"   and x :: "tm" shows "a \<sharp> OrdP x = a \<sharp> x"
SubstFormP_Eq: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstFormP v i (Q_Eq x y) (Q_Eq x' y')"
collapse_1: fixes tllist :: "('a, 'b) tllist" assumes "is_TNil tllist" shows "TNil (terminal tllist) = tllist"
Max_le_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Max less_eq A) x = (\<forall>a\<in>A. less_eq a x)"
put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def: fixes character_data_ptr :: "(_) character_data_ptr"   and character_data :: "'t option RCharacterData_scheme" shows "put character_data_ptr character_data = put (cast character_data_ptr) (cast character_data)"
tl_lens_def: shows "tl\<^sub>L = \<lparr>lens_get = tl, lens_put = \<lambda>xs. (#) (hd xs)\<rparr>"
cod_src_eq_dom_src: shows "HfHgh.cod.src = HfHgh.dsrc"
hyps: fixes as :: "nat list list"   and bs :: "nat list list" assumes "length as + length bs < length asa__ + length bsa__"   and "strict_sorted (concat as)"   and "strict_sorted (concat bs)"   and "as \<in> lists (- {[]})"   and "bs \<in> lists (- {[]})"   and "hd as < hd bs"   and "as \<noteq> []"   and "bs \<noteq> []"   and "\<And>a b. \<lbrakk>a \<in> list.set as; b \<in> list.set bs\<rbrakk> \<Longrightarrow> a < b \<or> b < a" shows "\<exists>us. Ex (merge as bs us)"
bij_betw_plane_ball: shows "bij_betw (\<lambda>x. (\<i> - x) / (\<i> + x)) {x. 0 < Im x} (ball 0 1)"
enum_P: fixes P :: "'a \<Rightarrow> bool"   and i :: "nat" assumes "infinitely_many1 P" shows "P (infinitely_many1.enum P i)"
ENR_def: fixes S :: "'a set" shows "ENR S \<equiv> \<exists>U. open U \<and> S retract_of U"
\<phi>_2: fixes i :: "nat"   and x :: "'a" assumes "i \<in> {0, 1}"   and "x \<in> carrier" shows "0 \<le> \<phi> i x"
f4: fixes n :: "nat"   and ss :: "symbol list"   and ssa :: "symbol list" shows "take (length (take n ss)) ssa = take (length ss) (take n ssa)"
iso_char: fixes f :: "'d" shows "D'.iso f = (D.iso f \<and> D'.arr f \<and> D'.arr (D.inv f))"
induct: fixes P :: "'d list \<Rightarrow> ('c option \<Rightarrow> 'c) \<Rightarrow> ('d, 'c) trie \<Rightarrow> bool"   and a0 :: "'d list"   and a1 :: "'c option \<Rightarrow> 'c"   and a2 :: "('d, 'c) trie" assumes "\<And>f v ps. P [] f (Trie v ps)"   and "\<And>k ks f v ps. (\<And>x. P ks f x) \<Longrightarrow> P (k # ks) f (Trie v ps)" shows "P a0 a1 a2"
mod_poly_code: fixes f :: "'b poly"   and g :: "'b poly" shows "f mod g = (let cg = coeffs g in if cg = [] then f else let cf = coeffs f; ilc = inverse_class.inverse (last cg); ch = map ((*) ilc) cg; r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg) in poly_of_list (rev r))"
polyn_mem: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "polyn_expr R X k c \<in> carrier R"
simps_1: fixes g :: "'b ugraph"   and u :: "'b"   and v :: "'b" shows "path g u [] v = (u = v)"
b1: fixes x :: "'a set" assumes "closed_csubspace x" shows "x +\<^sub>M orthogonal_complement x = UNIV"
cis_rcis_eq: fixes a :: "real" shows "cis a = rcis 1 a"
uniform_limit_theorem: fixes A :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'a filter"   and l :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F n in F. continuous_on A (f n)"   and "uniform_limit A f l F"   and "F \<noteq> bot" shows "continuous_on A l"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "g x \<noteq> 0"
set_case_simps_1: fixes c_empty :: "'a"   and c_sing :: "'b \<Rightarrow> 'a"   and c_else :: "'a" shows "set_case {} c_empty c_sing c_else = c_empty"
get_ancestors_di_locs_def: shows "get_ancestors_di_locs \<equiv> l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di_locs Shadow_DOM.get_parent_locs get_host_locs get_disconnected_document_locs"
chromatic_number_def: fixes G :: "nat set \<times> nat set set" shows "chromatic_number G \<equiv> INF c\<in>vertex_colorings G. enat (card c)"
star_right_induct: fixes z :: "'b"   and x :: "'b"   and y :: "'b" shows "z \<squnion> x * y \<le> x \<longrightarrow> z * y\<^sup>\<star> \<le> x"
comb_planar_def: fixes G :: "('c, 'd) pre_digraph" shows "comb_planar G \<equiv> \<exists>M. digraph_map G M \<and> pre_digraph_map.euler_genus G M = 0"
homogeneous_set_Polys: fixes X :: "'c set" shows "homogeneous_set P[X]"
assms_2: shows "AE x in M. (\<lambda>xa. ereal \<bar>birkhoff_sum f xa x\<bar>) \<longlonglongrightarrow> \<infinity>"
update_convs_2: fixes edges' :: "'c edge set \<Rightarrow> 'c edge set"   and root :: "'c"   and edges :: "'c edge set"   and more :: "'d" shows "edges_update edges' \<lparr>root = root, edges = edges, \<dots> = more\<rparr> = \<lparr>root = root, edges = edges' edges, \<dots> = more\<rparr>"
filter_filter: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
sw_simps1: fixes xs :: "'b"   and x :: "'a"   and y :: "'a" shows "x @xs[x \<and> y]_xs = y"
inject: fixes x1 :: "'b form multiset"   and x2 :: "'b form multiset"   and y1 :: "'b form multiset"   and y2 :: "'b form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
pre: fixes s :: "char list \<Rightarrow> int" shows "pre C Q s \<le> P s"
single_uminus: fixes k :: "'b"   and a :: "'c" shows "single k (- a) = - single k a"
SucNat_def: fixes x :: "ZF" shows "SucNat x \<equiv> HOLZF.union x (Singleton x)"
eval_elim_cases_19: fixes G :: "prog"   and s :: "st"   and e0 :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (e0 ? e1 : e2)\<succ>\<rightarrow> (v, s')" obtains b :: "val" and s1 :: "state" and v :: "val" where "va__ = In1 v" and "G\<turnstile>Norm s \<midarrow>e0-\<succ>b\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<rightarrow> s'"
generates_max_idealD_2: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and F' :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "generates_max_ideal X F"   and "F' \<subseteq> P[X]"   and "ideal F \<subset> ideal F'" shows "ideal F' = UNIV"
ustar_refines_nu: fixes P :: "'b hrel" shows "(\<nu> X \<bullet> (P ;; X) \<sqinter> II) \<sqsubseteq> P\<^sup>\<star>"
sel_transfer: fixes R1 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "(rel_nondetT' R1 R2 ===> R2) run_nondet run_nondet"
lconf_hext: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P,h \<turnstile> l (:\<le>) E"   and "h \<unlhd> h'" shows "P,h' \<turnstile> l (:\<le>) E"
half_gt_zero_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One))) = less zero a"
the_inv_into_f_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'d" assumes "inj_on f A"   and "f x = y"   and "x \<in> A" shows "the_inv_into A f y = x"
that_3: shows "(A ===> A ===> (=)) (\<bullet>) (\<bullet>)"
image_subsetI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
box_real_1: fixes a :: "real"   and b :: "real" shows "box a b = {a<..<b}"
delta_def: fixes A :: "'c \<times> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> bool)" shows "delta A = foldl2 (next A)"
neutral_fixed_point_def: fixes f :: "'b \<Rightarrow> 'b" shows "neutral_fixed_point f = (SOME xi. xi \<in> Gromov_boundary \<and> Gromov_extension f xi = xi \<and> additive_strength f xi = 0)"
tendsto_arctan_at_top: shows "(arctan \<longlongrightarrow> pi / 2) at_top"
DSTEP_comp: fixes l :: "('c + 'd) list"   and r :: "('c + 'd) list"   and t :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list" assumes "l \<cdot> r -G\<rightarrow> t" shows "\<exists>l' r'. l -G\<rightarrow>\<^sup>= l' \<and> r -G\<rightarrow>\<^sup>= r' \<and> t = l' \<cdot> r'"
lfilter_eq_LNil: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "(lfilter P xs = LNil) = (\<forall>x\<in>lset xs. \<not> P x)"
uop_ueval: fixes f :: "'c \<Rightarrow> 'a"   and x :: "('c, 'b) uexpr"   and b :: "'b" shows "\<lbrakk>uop f x\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b)"
is_propositional: fixes G :: "\<Pi>\<^sub>1"   and \<phi> :: "\<nu> \<Rightarrow> \<kappa> \<Rightarrow> \<o>" shows "IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)"
subtopology_def: fixes U :: "'b topology"   and V :: "'b set" shows "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
gcd_list_sub: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> set ys" shows "gcd_list ys dvd gcd_list xs"
vfsequence_vlrestriction_vcard: shows "xs\<^sub>1 \<restriction>\<^sup>l\<^sub>\<circ> vcard xs\<^sub>1 = xs\<^sub>1"
succs_Return: fixes \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" shows "succs Return \<tau> pc = []"
u: shows "u = (punit.lt (rep_list p) - punit.lt (rep_list g1)) \<oplus> lt g1"
small_refl_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f \<in> o[F](f)) = (\<forall>\<^sub>F x in F. f x = (0::'b))"
minus_equality: fixes y :: "'a"   and x :: "'a" assumes "y \<oplus> x = \<zero>"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x = y"
list_all2_append: fixes xs :: "'g list"   and ys :: "'h list"   and P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and us :: "'g list"   and vs :: "'h list" assumes "length xs = length ys" shows "list_all2 P (xs @ us) (ys @ vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
chain_map_compose: fixes p :: "nat"   and h :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "chain_map p (h \<circ> g) = chain_map p h \<circ> chain_map p g"
nonempty_card_pos: fixes A :: "'e set" assumes "finite A"   and "A \<noteq> {}" shows "0 < card A"
drop_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(drop n xs = []) = order.greater_eq n (length xs)"
less_le_not_le: fixes x :: "'a"   and y :: "'a" assumes "x \<in> U"   and "y \<in> U" shows "(x <\<^sub>o\<^sub>w y) = (x \<le>\<^sub>o\<^sub>w y \<and> \<not> y \<le>\<^sub>o\<^sub>w x)"
fold_mset_fun_left_comm: fixes x :: "'m"   and s :: "'m"   and M :: "'m multiset" shows "munionM x (fold_mset munionM s M) = fold_mset munionM (munionM x s) M"
chordal_circles_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "chordal_circles_cmat H = (let (A, B, C, D) = H; dsc = sqrt (Re ((D - A)\<^sup>2 + 4 * (B * cnj B))); a1 = (A - D + cor dsc) / (2 * C); r1 = sqrt ((4 - Re (- 4 * a1 / B * A)) / (1 + Re (a1 * cnj a1))); a2 = (A - D - cor dsc) / (2 * C); r2 = sqrt ((4 - Re (- 4 * a2 / B * A)) / (1 + Re (a2 * cnj a2))) in ((a1, r1), a2, r2))"
uint8_divmod_def: fixes x :: "uint8"   and y :: "uint8" shows "uint8_divmod x y = (if y = 0 then (undefined (div) x 0, undefined (mod) x 0) else (x div y, x mod y))"
induct: fixes x1 :: "((Class \<times> Method) \<times> nat) list"   and x2 :: "Prim"   and x3 :: "nat"   and P :: "((Class \<times> Method) \<times> nat) list \<Rightarrow> Prim \<Rightarrow> nat \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> TP_prim"   and "\<And>\<Sigma> i. P \<Sigma> (IntPrim i) 0"   and "\<And>\<Sigma> f x. P \<Sigma> (UnPrim f x) 0"   and "\<And>\<Sigma> f x y. P \<Sigma> (BinPrim f x y) 0"   and "\<And>\<Sigma>. P \<Sigma> NilPrim 0"   and "\<And>\<Sigma> x y. P \<Sigma> (ConsPrim x y) 1"   and "\<And>\<Sigma> f n args. \<Sigma>\<down>f = Some n \<Longrightarrow> P \<Sigma> (CallPrim f args) n" shows "P x1 x2 x3"
last_append: fixes p :: "'a"   and xs :: "('b \<times> 'a) list"   and ys :: "('b \<times> 'a) list" shows "Dioid_Models_New.last (p, xs @ ys) = Dioid_Models_New.last (Dioid_Models_New.last (p, xs), ys)"
RPREM_def: fixes P :: "bool" shows "RPREM P = P"
INJ: fixes x :: "'a"   and y :: "'b"   and y' :: "'b" assumes "(x, y) \<in> R"   and "(x, y') \<in> R" shows "y = y'"
image_image: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "\<P> f (\<P> g A) = \<P> (\<lambda>x. f (g x)) A"
leg1_uniquely_isomorphic_1: shows "r\<^sub>1 \<cong> t\<^sub>1 \<star> \<omega>.chine"
simps: fixes A :: "'a \<Rightarrow> bool"   and a :: "'a list" shows "listsp A a = (a = [] \<or> (\<exists>aa l. a = aa # l \<and> A aa \<and> listsp A l))"
closure_difference: fixes A :: "'a list set"   and B :: "'a list set" assumes "regular A"   and "regular B" shows "regular (A - B)"
dirichlet_prod_prime_power: fixes p :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "prime p" shows "dirichlet_prod f g (p ^ k) = (\<Sum>i\<le>k. f (p ^ i) * g (p ^ (k - i)))"
update_def: fixes f :: "'b \<Rightarrow> 'b" shows "update f = get \<bind> local.set \<circ> f"
vars_def: fixes g :: "'a" shows "CFG_base2.vars g \<equiv> fold (\<union>) (map (uses2 g) (\<alpha>n2 g)) {}"
subgraph_trans: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph"   and G\<^sub>3 :: "('e, 'f) labeled_graph" assumes "subgraph G\<^sub>1 G\<^sub>2"   and "subgraph G\<^sub>2 G\<^sub>3" shows "subgraph G\<^sub>1 G\<^sub>3"
intro: fixes len :: "nat"   and f :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list" assumes "\<And>xs ys. \<lbrakk>length xs = len; length ys = len\<rbrakk> \<Longrightarrow> length (f xs ys) = len" shows "otp f len"
coef_\<zeta>_eq: fixes i :: "nat" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta> i)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta> i))"
elts_of_set: fixes X :: "V set" shows "elts (ZFC_in_HOL.set X) = (if small X \<Rightarrow> X | otherwise \<Rightarrow> {})"
predecessors_def: fixes g :: "'a"   and n :: "'node" shows "i.predecessors g n \<equiv> map getFrom (i.inEdges g n)"
Buridan: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) in v]"
upd_None_map_le: fixes f :: "'g \<Rightarrow> 'h option"   and x :: "'g" shows "f(x := None) \<subseteq>\<^sub>m f"
Gcd_eq_1_I: fixes a :: "'a"   and A :: "'a set" assumes "is_unit a"   and "a \<in> A" shows "Gcd A = (1::'a)"
inv_comp_right_2: fixes g :: "'a1"   and f :: "'a1" assumes "C1.iso (g \<cdot>\<^sub>1 f)"   and "C1.iso f" shows "C1.iso g"
product_def: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM" shows "product A B \<equiv> \<lparr>succ = \<lambda>a (p\<^sub>1, p\<^sub>2). succ A a p\<^sub>1 \<times> succ B a p\<^sub>2, inputs = inputs A \<union> inputs B, outputs = outputs A \<union> outputs B, initial = (initial A, initial B)\<rparr>"
that: fixes k :: "'a strand_label"   and b :: "'b" assumes "a = (k, b)"   and "k = ln l \<or> k = \<star>" shows "thesis"
mod_less_dividend: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m \<le> n" shows "n mod m < n"
inner_scaleR_left: fixes r :: "real"   and x :: "'a"   and y :: "'a" shows "inner (r *\<^sub>R x) y = r * inner x y"
nat_pow_Suc: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "x [^]\<^bsub>P\<^esub> Suc n = x [^]\<^bsub>P\<^esub> n \<otimes>\<^bsub>P\<^esub> x"
inj_int_decode: fixes A :: "nat set" shows "inj_on ni_bij A"
wf_alpha_Tree_rel_hull_rel_Tree_wf: shows "wf (Validity.alpha_Tree_rel O hull_rel O Tree_wf)"
simps_4: fixes e\<^sub>1 :: "'a exp"   and bop :: "bop"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
intros: fixes P :: "'a prog"   and U :: "ty"   and T :: "ty"   and V :: "ty" assumes "P \<turnstile> U \<le> T"   and "P \<turnstile> V \<le> T"   and "\<And>T'. \<lbrakk>P \<turnstile> U \<le> T'; P \<turnstile> V \<le> T'\<rbrakk> \<Longrightarrow> P \<turnstile> T \<le> T'" shows "P \<turnstile> lub(U, V) = T"
order_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "class.order less_eq less \<equiv> class.preorder less_eq less \<and> class.order_axioms less_eq"
mHom_lin_nsumTr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'j) Module_scheme"   and t :: "'g \<Rightarrow> 'i"   and f :: "nat \<Rightarrow> 'g"   and n :: "nat" assumes "R module M"   and "R module N"   and "t \<in> mHom R M N" shows "f \<in> {j. j \<le> n} \<rightarrow> carrier M \<longrightarrow> t (\<Sigma>\<^sub>e M f n) = \<Sigma>\<^sub>e N cmp t f n"
Bounded_vec_mono: fixes b :: "'b"   and B :: "'b" assumes "b \<le> B" shows "Bounded_vec b \<subseteq> Bounded_vec B"
simps_2: fixes p :: "linear_poly"   and a :: "'a" shows "ns_constraint_const (GEQ_ns p a) = a"
splus_mono_strict: fixes v :: "'a"   and w :: "'a"   and t :: "'a" assumes "v \<prec> w" shows "local.punit.splus t v \<prec> local.punit.splus t w"
rPar2: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> !P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "Prop \<Psi> (!P) M N P'" shows "Prop \<Psi> (P \<parallel> !P) M N (P \<parallel> P')"
success_makeI: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and h :: "heap" shows "success (Array.make n f) h"
cP: shows "\<forall>a b. a \<in> carrier G \<and> b \<in> carrier G \<and> a \<sim> b \<longrightarrow> f a \<in> carrier G \<and> f b \<in> carrier G \<and> f a \<sim> f b"
wfI: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and B :: "'c set" assumes "r \<subseteq> A \<times> B"   and "\<And>x P. \<lbrakk>\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x; x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> P x" shows "wf r"
is_hashmap'_def: fixes m :: "'e \<Rightarrow> 'f option"   and l :: "('e \<times> 'f) list list"   and ht :: "('e, 'f) hashtable" shows "is_hashmap' m l ht = is_hashtable l ht * \<up> (map_of (concat l) = m)"