hcju/isabelleps huggingface dataset:
hext_typeof_mono: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty" assumes "h \<unlhd> h'"   and "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "typeof\<^bsub>h'\<^esub> v = \<lfloor>T\<rfloor>"
tendsto_mult_left_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "(f \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>x. f x * c) \<longlongrightarrow> (0::'b)) F"
least_upper_bound: fixes F :: "i \<Rightarrow> bool"   and x :: "i" assumes "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))" shows "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>z. (\<forall>y. F y \<longrightarrow> P y z) \<longrightarrow> P x z)"
monic_power: fixes p :: "'b poly"   and n :: "nat" assumes "monic p" shows "monic (p ^ n)"
card_order_csum: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 +c r2)"
map_cong: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
eval_sees_field_i_i_i_o_o_o_conv: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal" shows "pred.eval (sees_field_i_i_i_o_o_o P C F) = (\<lambda>(T, fm, D). P \<turnstile> C sees F:T (fm) in D)"
fork: fixes c :: "('exp, 'id) MWLfCom"   and V :: "('exp, 'id) MWLfCom list"   and m :: "'id \<Rightarrow> 'val" shows "SRabbr (fork c V) m (c # V) m"
ntrp_unsplit: fixes w :: "('p, 'ba) label list el_step list"   and wa :: "('p, 'ba) label list el_step list"   and wb :: "('p, 'ba) label list list"   and s :: "'n list"   and ca :: "'n list multiset"   and s' :: "'n list"   and ca' :: "'n list multiset"   and cb :: "'n list multiset"   and cb' :: "'n list multiset" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>nl fg\<^esub> map ENV wb"   and "((s, ca), wa, s', ca') \<in> trcl (ntrp fg)"   and "(cb, wb, cb') \<in> trcl (ntr fg)"   and "mon_c fg ({#s#} + ca) \<inter> (mon_c fg cb \<union> mon_ww fg wb) = {}"   and "mon_c fg cb \<inter> (mon_c fg ({#s#} + ca) \<union> mon_ww fg (map le_rem_s wa)) = {}" shows "((s, ca + cb), w, s', ca' + cb') \<in> trcl (ntrp fg)"
assms_2: shows "rk {A, B, C, A', B', C'} = 3"
assms: shows "inorder \<langle>l, k, r\<rangle> = [i..j]"
linear_order_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> Relation.total_on A r"
finite_weight: fixes x :: "'a" shows "\<integral>\<^sup>+ x. ennreal (R x x) \<partial>count_space UNIV < \<infinity>"
that_1: assumes "xs <\<^sub>v vs"   and "ys \<le>\<^sub>v ws" shows "thesis"
poincare_between_poincare_collinear: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc"   and "poincare_between u v w" shows "poincare_collinear {u, v, w}"
convex_hull_carrier: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n" shows "convex_hull Vs \<subseteq> carrier_vec n"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "B.par f f'"   and "B.R f = B.R f'" shows "f = f'"
image_is_empty: fixes f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set" shows "((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set) = {}) = (A = {})"
locally_faithful_iff_faithful: shows "locally_faithful = faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F"
fresh_star_list_nil_1: fixes xvec :: "name list" shows "xvec \<sharp>* []"
polychain_of_singleton_iff: fixes p :: "'a"   and xs :: "'a list"   and a :: "'a \<times> 'a" shows "(polychain_of p xs = [a]) = (fst a = p \<and> xs = [snd a - p])"
dim_col_tensor_mat: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" shows "dim_col (A \<Otimes> B) = dim_col A * dim_col B"
ntrs_c_no_mon: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> trcl (ntrs fg)" shows "mon_c fg c' = mon_c fg c"
max_min_distrib2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linorder less_eq less" shows "ord.max less_eq a (ord.min less_eq b c) = ord.min less_eq (ord.max less_eq a b) (ord.max less_eq a c)"
wf_interp_for_formula_FOr: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi>1 :: "'a formula"   and \<phi>2 :: "'a formula" shows "wf_interp_for_formula (w, I) (FOr \<phi>1 \<phi>2) = (wf_interp_for_formula (w, I) \<phi>1 \<and> wf_interp_for_formula (w, I) \<phi>2)"
the_\<nu>_simps_4: shows "local.dom \<chi>.the_\<nu> = s\<^sub>1"
G_def: fixes f :: "'c" shows "G f = (if C.arr f then terminal_arrow_from_functor.the_coext (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go (C.cod f)) (\<epsilon>o (C.cod f)) (Go (C.dom f)) (f \<cdot>\<^sub>C \<epsilon>o (C.dom f)) else D.null)"
has_key_alist_def: fixes xs :: "('b \<times> 'c) list"   and a :: "'b" shows "has_key_alist xs a = (\<exists>p\<in>set xs. fst p = a)"
mapmeasure_PiF: fixes M :: "('i \<Rightarrow> 'e) measure"   and n :: "nat"   and N :: "'e measure"   and X :: "(nat \<Rightarrow>\<^sub>F 'e) set" assumes "space M = space (Pi\<^sub>M (J n) (\<lambda>_. N))"   and "sets M = sets (Pi\<^sub>M (J n) (\<lambda>_. N))"   and "space N = UNIV"   and "X \<in> sets (Pi\<^sub>F (Collect finite) (\<lambda>_. N))" shows "emeasure (mapmeasure n M (\<lambda>_. N)) X = emeasure M (fm n -` X \<inter> extensional (J n))"
assms_1: shows "a \<in> circline_set (poincare_line a1 a2)"
smult_one: fixes x :: "'b" shows "x *\<^sub>S 1 = diag x"
SD_inefficientI: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b pmf"   and p :: "'b pmf"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "q \<in> lotteries_on alts"   and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q"   and "i \<in> agents"   and "p \<prec>[SD (R i)] q" shows "\<not> SD_efficient R p"
distinguish_attach: fixes d :: "('y, 'z) distinguisher"   and conv :: "('y, 'z, 'aa, 'ab) converter"   and res :: "('aa, 'ab) resource" shows "connect d (conv \<rhd> res) = connect (absorb d conv) res"
inj_fun_not_hypnat_in_SHNat: fixes f :: "nat \<Rightarrow> nat" assumes "inj f" shows "(*f* f) whn \<notin> \<nat>"
dvd_proper_imp_size_less: fixes a :: "'a"   and b :: "'a" assumes "a dvd b"   and "\<not> b dvd a"   and "b \<noteq> (0::'a)" shows "euclidean_size a < euclidean_size b"
ptrm_infer_type_swp_types: fixes a :: "'b"   and b :: "'b"   and \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and T :: "PreSimplyTyped.type"   and S :: "PreSimplyTyped.type"   and X :: "'b ptrm" assumes "a \<noteq> b" shows "ptrm_infer_type (\<Gamma>(a \<mapsto> T, b \<mapsto> S)) X = ptrm_infer_type (\<Gamma>(a \<mapsto> S, b \<mapsto> T)) ([a \<leftrightarrow> b] \<bullet> X)"
polar_form_eq: fixes p :: "(real, 2) vec" shows "norm p *\<^sub>R of_radiant (radiant_of p) = p"
local_trap: fixes n :: "nat"   and i :: "nat"   and vs :: "v list" shows "\<lparr>[Local n i vs [Trap]]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
numeral_ne_infinity: fixes k :: "num" shows "numeral k \<noteq> \<infinity>"
v11D_2: fixes r :: "V" assumes "v11 r" shows "vsv (r\<inverse>\<^sub>\<circ>)"
Min_ge_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq x (linorder.Min less_eq A) = (\<forall>a\<in>A. less_eq x a)"
rbt_insertw_rbt_sorted: fixes t :: "('a, 'c) rbt"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted t" shows "rbt_sorted (rbt_insert_with f k v t)"
simps_4: shows "cospan_shape.Dom cospan_shape.AT = cospan_shape.AA"
lookup_update_cases: fixes k :: "'b"   and v :: "'a"   and m :: "('b, 'a) mapping"   and k' :: "'b" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
with_proj_simps_5: fixes G :: "'c pair_pre_digraph" shows "head (with_proj G) = snd"
simE_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto><Rel> Q"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
elims: fixes x :: "variable"   and xa :: "variable set"   and y :: "variable \<times> variable set" assumes "variant_variable x xa = y" obtains n :: "String.literal" and S :: "variable set" where "x = Free na__" and "xa = S__" and "y = (let s' = fresh_name' (name ` S__) in (Free s', insert (Free s') S__))"   | n :: "String.literal" and uu_ :: "int" and S :: "variable set" where "x = Var (n, uu_)" and "xa = S" and "y = (let s' = fresh_name' (name ` S) in (Var (s', 0), insert (Var (s', 0)) S))"
evalC_app: fixes b :: "nat"   and ns :: "C list"   and ms :: "C list" shows "evalC b (C (ns @ ms)) = evalC b (C ns) + evalC b (C ms)"
mod_div_mult_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times (divide a b) b) = a"
CSP3_def: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP3 P \<equiv> action.relation_of Skip ;; P"
s2r_surj: shows "\<forall>R\<subseteq>Id. \<exists>A. R = s2r A"
intros_1: fixes V :: "'var"   and n' :: "'node" assumes "V \<in> Use n'" shows "(V, [], []) \<in> dependent_live_vars n'"
ik_snd_empty: fixes X :: "('a, 'b) Term.term list" shows "ik\<^sub>s\<^sub>t (map Send X) = {}"
Ms_def: shows "Ms = map (\<lambda>w. w `\<rightarrow> C0) (map ((+) s) (sums ts))"
of_ocircline_opposite_ocircline: fixes H :: "ocircline" shows "of_ocircline (opposite_ocircline H) = of_ocircline H"
ENR_ANR: fixes S :: "'b set" shows "ENR S = (ANR S \<and> locally compact S)"
k_frontier_def: fixes a :: "'a" shows "SPRdet.k_frontier a \<equiv> map (local.spr_simInit a \<circ> envObs a) envInit"
fs_var_inst: shows "fs TYPE('a) TYPE(NominalSequents.var)"
extremum_unique: fixes a :: "'b list" shows "prefix a [] = (a = [])"
one_less_ennreal: fixes x :: "real" shows "(1 < ennreal x) = (1 < x)"
mcontI: fixes orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i"   and luba :: "'h set \<Rightarrow> 'h"   and lubb :: "'i set \<Rightarrow> 'i" assumes "monotone orda ordb f"   and "cont luba orda lubb ordb f" shows "mcont luba orda lubb ordb f"
sum_insert: fixes x :: "'a"   and F :: "'a set"   and P :: "'a \<Rightarrow> 'b" assumes "x \<notin> F"   and "finite F" shows "sum P (insert x F) = sum P F + P x"
neg_equal_iff_equal: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "(uminus a = uminus b) = (a = b)"
infinite: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "freeultrafilter F"   and "eventually P F" shows "infinite {x. P x}"
has_greatest_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "has_greatest_fixpoint f \<equiv> \<exists>x. is_greatest_fixpoint f x"
degree_def: fixes p :: "'b poly" shows "degree p = (LEAST n. \<forall>i>n. coeff p i = (0::'b))"
add_mset_le_right_le: fixes A :: "'b multiset"   and B :: "'b multiset"   and a :: "'b" assumes "A \<le> B" shows "add_mset a A \<le> add_mset a B"
resCasesB: fixes y :: "name"   and P :: "pi"   and a :: "name"   and x :: "name"   and RP' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "<\<nu>y>P \<longmapsto> a<\<nu>x> \<prec> RP'"   and "x \<noteq> y"   and "\<And>P'. \<lbrakk>P \<longmapsto> a[y] \<prec> P'; a \<noteq> y\<rbrakk> \<Longrightarrow> F ([(x, y)] \<bullet> P')"   and "\<And>P'. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; y \<noteq> a\<rbrakk> \<Longrightarrow> F (<\<nu>y>P')" shows "F RP'"
assms_5: shows "formulaFalse (getF state) (elements (getM state))"
simps_8: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c" shows "(case Lt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f2"
complex_rotations_def: shows "complex_rotations \<equiv> {x. complex_rotationsp x}"
cases: fixes x :: "'b val \<times> 'b val" obtains i1 :: "32 word" and i2 :: "32 word" where "x = (Intg i1, Intg i2)"   | v2 :: "'b val" where "x = (Unit, v2__)"   | v2 :: "'b val" where "x = (Null, v2a__)"   | v :: "bool" and v2 :: "'b val" where "x = (Bool v__, v2b__)"   | v :: "'b" and v2 :: "'b val" where "x = (Addr va__, v2)"   | v1 :: "'b val" where "x = (v1__, Unit)"   | v1 :: "'b val" where "x = (v1a__, Null)"   | v1 :: "'b val" and v :: "bool" where "x = (v1b__, Bool vb__)"   | v1 :: "'b val" and v :: "'b" where "x = (v1, Addr v)"
pt_set_bij2: fixes x :: "'a"   and X :: "'a set"   and pi :: "('b \<times> 'b) list" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "x \<in> X" shows "pi \<bullet> x \<in> pi \<bullet> X"
lcm_dvdD2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "lcm a b dvd c" shows "b dvd c"
distinct_sorted_wrt_irrefl: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "irreflp rel"   and "transp rel"   and "sorted_wrt rel xs" shows "distinct xs"
map\<^sub>0_def: fixes a :: "'b" shows "map\<^sub>0 a \<equiv> src\<^sub>B (local.map a)"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "'b"   and a2 :: "('b \<times> color) tree" assumes "\<And>l x r. \<lbrakk>\<And>x21 x22 x23 xa y. \<lbrakk>\<not> bheight r \<le> bheight l; r = \<langle>x21, x22, x23\<rangle>; (xa, y) = x22; y = Red\<rbrakk> \<Longrightarrow> P l x x21; \<And>x21 x22 x23 xa y. \<lbrakk>\<not> bheight r \<le> bheight l; r = \<langle>x21, x22, x23\<rangle>; (xa, y) = x22; y = Black\<rbrakk> \<Longrightarrow> P l x x21\<rbrakk> \<Longrightarrow> P l x r" shows "P a0 a1 a2"
ang_vec_a_scale1: fixes k :: "real"   and z1 :: "complex"   and z2 :: "complex" assumes "k \<noteq> 0" shows "\<angle>a (cor k * z1) z2 = \<angle>a z1 z2"
is_prefix_add: fixes ls :: "'a list"   and vs :: "'a list"   and as :: "'a list" assumes "is_prefix ls vs" shows "is_prefix (as @ ls) (as @ vs)"
f_Exec_Equal_State: fixes localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c1 :: "'d"   and c2 :: "'d"   and xs :: "'f message_af list" assumes "Exec_Equal_State localState trans_fun"   and "localState c1 = localState c2"   and "xs \<noteq> []" shows "f_Exec_Comp trans_fun xs c1 = f_Exec_Comp trans_fun xs c2"
oneideal: shows "ideal (carrier (R Quot I)) (R Quot I)"
elims_2: fixes x :: "nat"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "wadjust_on_left_moving x xa xb" obtains m :: "nat" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = rs" and "xb = (l, r)" and "wadjust_on_left_moving_O m rs (l, r) \<or> wadjust_on_left_moving_B m rs (l, r)"
frontierE: fixes x :: "'a"   and s :: "'a set" assumes "x \<in> frontier s" obtains "x \<in> closure s" and "x \<in> core s \<Longrightarrow> False"
assms: fixes C :: "'a" assumes "C \<in># M"   and "P C" shows "f C = g C"
weakenSimI: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<Rel> Q"
qtable_l\<delta>: fixes \<phi>s :: "Formula.formula list"   and mr :: "mregex"   and A :: "nat set"   and n :: "nat"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and rels :: "event_data option list set list"   and Q :: "event_data list \<Rightarrow> Formula.formula Regex.regex \<Rightarrow> bool"   and rel :: "(mregex, event_data option list set) mapping" assumes "ok (length \<phi>s) mr"   and "fv_regex (from_mregex mr \<phi>s) \<subseteq> A"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) j \<phi>)) \<phi>s rels"   and "\<forall>ms\<in>LPD mr. qtable n A (mem_restr R) (\<lambda>v. Q (map the v) (from_mregex ms \<phi>s)) (lookup rel ms)" shows "qtable n A (mem_restr R) (\<lambda>v. \<exists>s\<in>Regex.lpd (Formula.sat \<sigma> V (map the v)) j (from_mregex mr \<phi>s). Q (map the v) s) (l\<delta> (\<lambda>x. x) rel rels mr)"
cont: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> M"   and "decseq A"   and "\<Inter> (range A) = {}" shows "(\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
one_le_of_rat_iff: fixes r :: "rat" shows "((1::'a) \<le> of_rat r) = (1 \<le> r)"
r_neg2: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<oplus> (\<ominus> x \<oplus> y) = y"
subgraph_euler_genus_le: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph"   and GM :: "'d pre_map" assumes "subgraph H G"   and "digraph_map G GM"   and "\<exists>rev. bidirected_digraph H rev" obtains HM :: "'d pre_map" where "digraph_map H HM" and "pre_digraph_map.euler_genus H HM \<le> pre_digraph_map.euler_genus G GM"
fds_norm_def: fixes f :: "'a fds" shows "fds_norm f = fds (\<lambda>n. of_real (norm (fds_nth f n)))"
least_non_zero_position_of_vector_def: fixes A :: "'d iarray" shows "least_non_zero_position_of_vector A = least_non_zero_position_of_vector_from_index A 0"
unsatReport: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list" assumes "InvariantImpliedLiterals F M"   and "InvariantEquivalent F0 F"   and "decisions M = []"   and "formulaFalse F (elements M)" shows "\<not> satisfiable F0"
Abs_eq_iff2_1: fixes bs :: "atom set"   and x :: "'b"   and bs' :: "atom set"   and y :: "'b" shows "([bs]set. x = [bs']set. y) = (\<exists>p. (bs, x) \<approx>set (=) supp p (bs', y) \<and> supp p \<subseteq> bs \<union> bs')"
map_upds_apply_nontin: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b option"   and ys :: "'b list" assumes "x \<notin> set xs" shows "(f(xs [\<mapsto>] ys)) x = f x"
convergent_bounded_family: fixes A :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> bounded (range (\<lambda>n. f n x))"   and "countable A" obtains k :: "nat \<Rightarrow> nat" where "strict_mono k" and "\<And>x. x \<in> A \<Longrightarrow> convergent (\<lambda>n. f (k n) x)"
nth_map_conv: fixes xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "length xs = length ys"   and "\<forall>i<length xs. f (xs ! i) = g (ys ! i)" shows "map f xs = map g ys"
unitary_embed: fixes P :: "'b \<Rightarrow> bool" shows "unitary \<guillemotleft> P \<guillemotright>"
union_disjoint: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "prod g (A \<union> B) = prod g A * prod g B"
phi'_nonzero: fixes m :: "nat" assumes "0 < m" shows "0 < phi' m"
axioms_2: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "rational_preference carrier relation" shows "rational_preference_axioms carrier relation"
SkipRule: fixes P :: "'d set"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> Q" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnExpr P) Skip Q, A"
K_S_None: fixes x :: "'s"   and n :: "nat" shows "K2.p (Some x) None n = 0"
intro: fixes cl :: "'b potype" assumes "cl \<in> PartialOrder" shows "PO cl"
pdevs_val_perm_ex: fixes xs :: "'b list"   and ys :: "'b list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "xs <~~> ys"   and "e \<in> UNIV \<rightarrow> I" shows "\<exists>e'. e' \<in> UNIV \<rightarrow> I \<and> pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
add_closed: fixes A :: "'q set"   and x :: "'q"   and y :: "'q" assumes "ring_closed A"   and "x \<in> A"   and "y \<in> A" shows "x + y \<in> A"
finite_ran: fixes p :: "'g \<Rightarrow> 'h option" assumes "finite (dom p)" shows "finite (ran p)"
mult_sign_intros_7: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "(0::'a) < b" shows "a * b < (0::'a)"
sgn_greater: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (sgn a) = less zero a"
sorted_tl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (tl xs)"
oalist_inv_def: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a" shows "oalist_raw.oalist_inv rep_key_order xs = ko.oalist_inv_raw (rep_key_order (snd xs)) (fst xs)"
order_0_monom: fixes c :: "'a"   and n :: "nat" assumes "c \<noteq> (0::'a)" shows "Polynomial.order (0::'a) (Polynomial.monom c n) = n"
mtx_row_R_def: fixes sign_list :: "rat list list"   and index_list :: "nat list \<times> nat list" shows "mtx_row_R sign_list index_list \<equiv> map (z_R index_list) sign_list"
simps_1: fixes m :: "nat \<Rightarrow> nat" shows "flush [] m = m"
enat_numeral: fixes k :: "num" shows "enat (numeral k) = numeral k"
action_axioms_def: fixes G :: "('a, 'b) monoid_scheme"   and act :: "'c \<Rightarrow> 'a \<Rightarrow> 'c" shows "action_axioms G act \<equiv> (\<forall>b. act b \<one>\<^bsub>G\<^esub> = b) \<and> (\<forall>g h b. g \<in> carrier G \<and> h \<in> carrier G \<longrightarrow> act (act b g) h = act b (g \<otimes>\<^bsub>G\<^esub> h))"
gt_subst: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "t >\<^sub>t s" shows "subst \<rho> t >\<^sub>t subst \<rho> s"
\<delta>_def: shows "\<delta> = (s \<bullet> (1::'a) - s2) / 2"
silent_moves_into_\<tau>rtrancl3p: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and s' :: "'g" assumes "(\<tau>trsys.silent_move trsys \<tau>move)\<^sup>*\<^sup>* s s'" shows "\<tau>trsys.\<tau>rtrancl3p trsys \<tau>move s [] s'"
trans: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"