hcju/isabelleps huggingface dataset:
Lim_within: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and a :: "'b"   and S :: "'b set" shows "(f \<longlongrightarrow> l) (at a within S) = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
homotopic_paths_rinv: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set" assumes "path p"   and "path_image p \<subseteq> s" shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
add_order: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.monoid_add plus zero" shows "power.power zero plus a (monoid_add.add_order plus zero a) = zero"
polynomialI: fixes p :: "'a list"   and K :: "'a set" assumes "set p \<subseteq> K"   and "lead_coeff p \<noteq> \<zero>" shows "polynomial K p"
extended_predist_def: fixes x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "extended_predist x y = real_of_ereal (min (esqrt (extended_Gromov_distance x y)) (eexp (ereal (- epsilonG TYPE('b)) * extended_Gromov_product_at basepoint x y)))"
length_takeWhile_less: fixes a :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> ran\<^sub>l xs"   and "\<not> P a" shows "length (takeWhile P xs) < length xs"
hseqE: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.hseq \<nu> \<mu>" obtains "D.arr \<mu>" and "D.arr \<nu>" and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>"
de_morgan1: fixes X :: "'a set" shows "- \<Squnion> X = \<Sqinter> (uminus ` X)"
changes_scale_const: fixes s :: "'a"   and xs :: "'a list" assumes "s \<noteq> (0::'a)" shows "changes (map ((*) s) xs) = changes xs"
invh_paint: fixes t :: "('b \<times> color) tree"   and c :: "color" assumes "invh t" shows "invh (paint c t)"
hom_induced_eq: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and p :: "int"   and S :: "'a set"   and Y :: "'b topology"   and T :: "'b set" assumes "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "hom_induced p X S Y T f = hom_induced p X S Y T g"
sum_mat_def: fixes A :: "'b mat" shows "sum_mat A = sum (($$) A) ({0..<dim_row A} \<times> {0..<dim_col A})"
Diff_Compl: fixes A :: "'k set"   and B :: "'k set" shows "A - - B = A \<inter> B"
hyps: fixes cs :: "('a, 'b, 'c) com list" shows "\<Gamma>\<turnstile>cs,css \<Down> Stuck"
fourier_primitive_2: shows "cmod (gauss_sum 1 / complex_of_real (sqrt (real n))) = 1"
quadratic_linear_at_most_2_intersections_help': fixes qf :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and a11 :: "complex"   and a12 :: "complex"   and a22 :: "complex"   and b1 :: "complex"   and b2 :: "complex"   and c :: "complex"   and x :: "complex"   and n :: "complex"   and k1 :: "complex"   and y :: "complex"   and yf :: "complex \<Rightarrow> complex" assumes "qf = (\<lambda>x y. a11 * x\<^sup>2 + 2 * a12 * x * y + a22 * y\<^sup>2 + b1 * x + b2 * y + c)"   and "x = - n / k1"   and "k1 \<noteq> 0"   and "qf x y = 0"   and "yf = (\<lambda>y. k1\<^sup>2 * a22 * y\<^sup>2 + (- 2 * a12 * n * k1 + b2 * k1\<^sup>2) * y + a11 * n\<^sup>2 - b1 * n * k1 + c * k1\<^sup>2)" shows "yf y = 0"
int_cases4: fixes m :: "int" obtains n :: "nat" where "m = int na__"   | n :: "nat" where "0 < n" and "m = - int n"
aux_integral: shows "((\<lambda>t. exp (u - t *\<^sub>R u) * u) has_integral exp u - 1) {0..1}"
rtf_init_invar: fixes L_left :: "nat list" assumes "distinct L_left"   and "set L_left = V - {s, t}" shows "rtf_invar c s t pp_init_f pp_init_l rtf_init_n [] L_left"
four: fixes x :: "'a" shows "4 \<times> x = x \<squnion> x \<squnion> x \<squnion> x"
distinct_1: fixes x2 :: "'m" shows "None \<noteq> Some x2"
b: shows "\<lbrakk>X\<rbrakk>\<^sub>\<sim> = \<lbrakk>Y\<rbrakk>\<^sub>\<sim>"
infinite_contains_3_elems: fixes A :: "'a set" assumes "infinite A" shows "\<exists>x y z. x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A"
fm_def: fixes n :: "nat" shows "fm n = finmap_of (Utn ` J n) \<circ> (\<lambda>g. compose (Utn ` J n) g (from_nat_into (\<Union> (range J))))"
eventually_finite_subsets_at_top: fixes P :: "'c set \<Rightarrow> bool"   and A :: "'c set" shows "eventually P (finite_subsets_at_top A) = (\<exists>X. finite X \<and> X \<subseteq> A \<and> (\<forall>Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A \<longrightarrow> P Y))"
numberofIsbeschr: fixes xs :: "'b list"   and ys :: "'b list"   and y :: "'b" assumes "dist_perm xs ys"   and "y \<in> set xs" shows "index xs y \<le> index ys y + card {(x, y) |x. x < y in xs \<and> y < x in ys}"
fps_uminus_def: shows "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - fps_nth f n))"
N: fixes n :: "nat" assumes "N \<le> n" shows "norm (x ^ n) < e"
right_def: fixes t :: "'a" shows "right t = (case unapp t of Some (x, r) \<Rightarrow> r)"
path_def: fixes g :: "real \<Rightarrow> 'c" shows "path g = continuous_on {0..1} g"
fun_relI: fixes A :: "('d \<times> 'e) set"   and f :: "'d \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'g"   and B :: "('f \<times> 'g) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
map_of_domAD: fixes \<Gamma> :: "('b \<times> 'a) list"   and x :: "'b"   and e :: "'a" assumes "map_of \<Gamma> x = Some e" shows "x \<in> domA \<Gamma>"
less_eq_BEState_ext_def: fixes x :: "('a, 'b, 'c, 'd) BEState_scheme"   and y :: "('a, 'b, 'c, 'd) BEState_scheme" shows "x \<le> y \<equiv> es x < es y \<or> es x = es y \<and> (ps x < ps y \<or> ps x = ps y \<and> BEState.more x \<le> BEState.more y)"
cod_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "cod (src \<mu>) = src \<mu>"
twiddle_def: fixes s :: "Var \<Rightarrow> Val"   and ss :: "Var \<Rightarrow> Val" shows " s \<approx> ss  = (\<forall>x. CONTEXT x = low \<longrightarrow> s x = ss x)"
Inf_many_def: fixes P :: "'a \<Rightarrow> bool" shows "Inf_many P = infinite {x. P x}"
hasRead_def: fixes s :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" shows "hasRead s p d q = (\<exists>br\<in>blocksRead s p d. proc br = q)"
ffun_member_plus: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun" shows "((x, y) \<in>\<^sub>f f + g) = (x \<notin> fdom g \<and> (x, y) \<in>\<^sub>f f \<or> (x, y) \<in>\<^sub>f g)"
simps_2: fixes f :: "atom \<Rightarrow> nat \<Rightarrow> atom"   and n :: "nat" shows "map_fm_binders f FalseF n = FalseF"
len_of_numeral_defs_3: fixes uu :: "'a bit0 itself" shows "len_of uu = 2 * LENGTH('a)"
g_cond: shows "\<forall>g\<in>set (take n gs). terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
nxt_single: fixes x :: "'a"   and x' :: "'a" shows "nxt (single x) x' = TTree.empty"
two_events: shows "\<exists>a\<in>\<E>. \<exists>b\<in>\<E>. a \<noteq> b"
rtrancl_into_rtrancl: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
f2r_surj: shows "top_pres (\<P> \<R>)"
C_def: shows "C = (\<Union>x. \<Union>y\<in>- {x}. B x \<inter> B y)"
set_merge_wrt: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" shows "set (merge_wrt rel xs ys) = set xs \<union> set ys"
NS_staticSecret_parts_Spy: fixes m :: "msg"   and evs :: "event list"   and A :: "agent" assumes "m \<in> parts (knows Spy evs)"   and "m \<in> staticSecret A"   and "evs \<in> ns_public" shows "A \<in> bad"
Subset_HFun_Sigma: fixes s :: "tm"   and s' :: "tm" shows "{HFun_Sigma s, s' SUBS s} \<turnstile> HFun_Sigma s'"
perm_rows_def: shows "perm_rows \<equiv> map_fun to_fun (map_fun id of_fun) (\<lambda>M p i. M (p i))"
assign: fixes E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and e :: "'b"   and m :: "'c \<Rightarrow> 'a"   and v :: "'a"   and x :: "'c"   and BMap :: "'a \<Rightarrow> bool" assumes "E e m = v" shows "((x:=e, m), [], m(x := v)) \<in> MWLf_semantics.MWLfSteps_det E BMap"
split_pairs: fixes A :: "'b"   and B :: "'c"   and X :: "'b \<times> 'c" shows "((A, B) = X) = (fst X = A \<and> snd X = B)"
monoE: fixes g :: "'a"   and f :: "'a"   and f' :: "'a" assumes "local.mono g"   and "seq g f"   and "seq g f'"   and "g \<cdot> f = g \<cdot> f'" shows "f' = f"
wf_converse_trancl: fixes r :: "('b \<times> 'b) set" assumes "wf (r\<inverse>)" shows "wf ((r\<^sup>+)\<inverse>)"
list_emb_Cons_cases: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and y :: "'d"   and ys :: "'d list" assumes "list_emb P xs (y # ys)" obtains "xsa__ = []"   | "list_emb P xsa__ ys"   | x :: "'d" and xs :: "'d list" where "xsa__ = x # xs" and "P x y" and "list_emb P xs ys"
is_poincare_line_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "is_poincare_line_cmat H = (let (A, B, C, D) = H in hermitean (A, B, C, D) \<and> A = D \<and> (cmod A)\<^sup>2 < (cmod B)\<^sup>2)"
pgwt_ground: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "public_ground_wf_term t" shows "fv t = {}"
Sup_admS_def: fixes A :: "'a admS set" shows "Sup A = Inf {y. \<forall>x\<in>A. x \<le> y}"
nrcons_invlevel: fixes n :: "nat"   and t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_leveln_ftree n t"   and "is_leveln_node n nd" shows "is_leveln_ftree n (nrcons t nd)"
add_inf_distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + inf b c = inf (a + b) (a + c)"
hyps_2: fixes \<rho> :: "nat \<rightarrow> ValD" shows "evalDdb (P<DBFix P/0>)\<cdot>\<rho> = evalDdb V\<cdot>\<rho>"
that: fixes C :: "'a set"   and T :: "'a set" assumes "gdelta C"   and "T \<in> null_sets lebesgue"   and "S \<union> T = C"   and "disjnt S T" shows "thesis"
minus_mat_limit: fixes B :: "complex mat"   and m :: "nat"   and X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" assumes "B \<in> carrier_mat m m"   and "limit_mat X A m" shows "limit_mat (mat_seq_minus X B) (A - B) m"
ltlr_expand_Until: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r \<phi> U\<^sub>r \<psi> = \<xi> \<Turnstile>\<^sub>r \<psi> or\<^sub>r (\<phi> and\<^sub>r X\<^sub>r (\<phi> U\<^sub>r \<psi>))"
sum_list_addf: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.comm_monoid_add plus zero" shows "monoid_add.sum_list plus zero (map (\<lambda>x. plus (f x) (g x)) xs) = plus (monoid_add.sum_list plus zero (map f xs)) (monoid_add.sum_list plus zero (map g xs))"
inv_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "inv (x \<otimes> y) = inv x \<otimes> inv y"
simps_1: fixes prec :: "nat"   and t :: "nat"   and f :: "floatarith"   and ivl :: "float interval"   and cmp :: "float interval \<Rightarrow> bool" shows "approx_tse_form' prec t f 0 ivl cmp = (case approx_tse prec 0 t (mid ivl) 1 f [Some ivl] of None \<Rightarrow> False | Some x \<Rightarrow> cmp x)"
Inter_lower: fixes B :: "'a set"   and A :: "'a set set" assumes "B \<in> A" shows "\<Inter> A \<subseteq> B"
assms_12: shows "set vs \<inter> FvarsT r = {}"
empty_ran_restrict: fixes f :: "'b \<Rightarrow> 'a option"   and A :: "'b set" shows "(ran (f |` A) = {}) = (\<forall>x\<in>A. f x = None)"
invariant_def: fixes pm :: "'a extended"   and sm :: "'a extended" shows "invariant pm sm = (pm \<in> set (ext_ids s) \<and> sm \<in> set (ext_ids s) \<and> Psi.subset (a pm, a sm) (a (P m), a (S m)) \<and> elem (a \<lbrakk>I m\<rbrakk>) (a pm, a sm))"
weakInputPres: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and M :: "'a"   and N :: "'a" assumes "\<And>Tvec \<Psi>''. length xvec = length Tvec \<Longrightarrow> (\<Psi> \<otimes> \<Psi>'', P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<leadsto><Rel> M\<lparr>\<lambda>*xvec N\<rparr>.Q"
les_trans: fixes s1 :: "'s"   and s2 :: "'s"   and s3 :: "'s" assumes "les s1 s2"   and "les s2 s3" shows "les s1 s3"
lookup_next: shows "Mapping.lookup (phis g) next = Some next_args"
simps_2: fixes x :: "'sz"   and y :: "'sz"   and z :: "'sz"   and \<theta> :: "('sf, 'sz) trm" shows "PUrename x y (z := \<theta>) = (swap x y z := TUrename x y \<theta>)"
finite_subset_induct': fixes F :: "'a set"   and A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite F"   and "F \<subseteq> A"   and "P {}"   and "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F"
S1: shows "c_assoc_have_key (pr_gr upb1) (c_pair (c_snd (c_snd n)) x) = 0"
bind_assoc: fixes x :: "'e option"   and f :: "'e \<Rightarrow> 'd option"   and g :: "'d \<Rightarrow> 'c option" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
norm_of_real: fixes r :: "real" shows "norm (of_real r) = \<bar>r\<bar>"
points_sep_groups_ne: fixes G1 :: "'a set"   and G2 :: "'a set"   and x :: "'a"   and y :: "'a" assumes "G1 \<in> \<G>"   and "G2 \<in> \<G>"   and "G1 \<noteq> G2"   and "x \<in> G1"   and "y \<in> G2" shows "x \<noteq> y"
rel_fundefs_Some1: fixes f :: "'l \<Rightarrow> ('b, 'm, 'n, 'o, 'e) Inca.instr fundef option"   and g :: "'l \<Rightarrow> ('b, 'm, 'n, 'o, 'e, 'g, 'h, 'i) Ubx.instr fundef option"   and x :: "'l"   and y :: "('b, 'm, 'n, 'o, 'e) Inca.instr fundef" assumes "rel_fundefs f g"   and "f x = Some y" shows "\<exists>z. g x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) y z"
full: shows "full_chain R\<^sup>+\<^sup>+ xs"
supp_Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and \<alpha> :: "'g" assumes "finite (supp t\<^sub>\<alpha>)" shows "supp (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>) = supp \<alpha> \<union> supp t\<^sub>\<alpha> - bn \<alpha>"
that: fixes r :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b" assumes "keys r \<subseteq> B"   and "Poly_Mapping.range r \<subseteq> punit.dgrad_p_set d m"   and "p = ideal.rep r" shows "thesis"
two: fixes A :: "'a set"   and B :: "'a set"   and e :: "real" assumes "closed A"   and "A \<subseteq> S"   and "closed B"   and "B \<subseteq> S"   and "A \<inter> B = {}"   and "0 < e"   and "e < 1" shows "\<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>x\<in>A. f x < e) \<and> (\<forall>x\<in>B. 1 - e < f x)"
p1mod22k': fixes b :: "int"   and n :: "nat" shows "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
first_iff_u_greater_strict_preff: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> local.\<P>"   and "q \<in> local.\<P>"   and "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>" shows "(local.my_U q < local.my_U p) = (mix_pmf (local.my_U p) local.some_best local.some_worst \<succeq>[\<R>] mix_pmf (local.my_U q) local.some_best local.some_worst \<and> (mix_pmf (local.my_U q) local.some_best local.some_worst, mix_pmf (local.my_U p) local.some_best local.some_worst) \<notin> \<R>)"
assms_1: shows "u \<le>p r\<^sup>\<omega>"
assms_1: shows "qGoodAbs A \<or> qGoodAbs A'"
add_mset_replicate_mset_safe: fixes M :: "'d multiset"   and a :: "'d" assumes "NO_MATCH {#} M" shows "add_mset a M = {#a#} + M"
im_submodule: fixes N :: "'v set" assumes "RModule FG (\<cdot>) N \<and> N \<subseteq> V" shows "RModule FG (\<star>) (T ` N) \<and> T ` N \<subseteq> ImG"
trivial_VectorSpaceHom: fixes smult' :: "'f \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<And>a. smult' a (0::'a) = (0::'a)" shows "VectorSpaceHom (\<cdot>) V smult' 0"
lnull_ldropn: fixes n :: "nat"   and xs :: "'a llist" shows "lnull (ldropn n xs) = (llength xs \<le> enat n)"
eval_polyexp_def: fixes xs :: "('b \<times> nat \<times> 'b) list" shows "eval_polyexp xs = (\<lambda>n. \<Sum>(a, k, b)\<leftarrow>xs. a * of_nat n ^ k * b ^ n)"
stake_shift: fixes i :: "nat"   and w :: "'b list"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "stake i (w @- \<sigma>) = take i w @ stake (i - length w) \<sigma>"
cons: fixes c :: "'c"   and a :: "'d"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set"   and w :: "'d list"   and c'' :: "'c" assumes "(c, a, c') \<in> t"   and "(c', w, c'') \<in> trcl t" shows "(c, a # w, c'') \<in> trcl t"
is_Arg_2pi_iff: fixes z :: "complex"   and r :: "real"   and k :: "int" shows "is_Arg z (r + real_of_int k * (2 * pi)) = is_Arg z r"
indicator_def: fixes S :: "'l set"   and x :: "'l" shows "indicator S x = (if x \<in> S then 1::'k else (0::'k))"
mod_uexpr_def: fixes u :: "('e, 'f) uexpr"   and v :: "('e, 'f) uexpr" shows "u mod v = bop (mod) u v"
cr_real_def: shows "cr_real \<equiv> \<lambda>x y. realrel x x \<and> Real x = y"
map_component_def: fixes f :: "nat \<Rightarrow> nat"   and v :: "('a, 'b) pp \<times> nat" shows "map_component_pprod f v = (fst v, f (snd v))"
union_list'_nodes_card: fixes AA :: "('c, 'd) dca list" assumes "list_all (finite \<circ> DCA.nodes) AA" shows "card (DGCA.nodes (union_list' AA)) \<le> prod_list (map (card \<circ> DCA.nodes) AA)"
norm_prod_le: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a))"
vlrestriction_vsingleton_nin: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "set {\<langle>a, b\<rangle>} \<restriction>\<^sup>l\<^sub>\<circ> A = 0"
sup_compl_top_left2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (sup (uminus x) y) = top"
strict_prefI: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set" assumes "x \<^bsub>r\<^esub>\<preceq> y"   and "(y, x) \<notin> r" shows "x \<^bsub>r\<^esub>\<prec> y"
is_processT6_S1: fixes X :: "'a event set"   and s :: "'a event list"   and P :: "'a process" assumes "tick \<notin> X"   and "(s @ [tick], {}) \<in> F P" shows "(s, X) \<in> F P"
admissible_imp: fixes Q :: "bool"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "Q \<Longrightarrow> ccpo.admissible lub ord P" shows "ccpo.admissible lub ord (\<lambda>x. Q \<longrightarrow> P x)"
I_subst\<^sub>22: fixes f :: "atom fm"   and xs :: "'a list"   and l :: "nat"   and x :: "'a"   and u :: "nat" assumes "nqfree f"   and "xs ! l < x \<and> x < xs ! u"   and "nolub f xs (xs ! l) x (xs ! u)"   and "\<forall>x\<in>{xs ! l<..<xs ! u}. DLO.I f (x # xs) \<and> x \<notin> EQ f xs" shows "DLO.I (subst\<^sub>2 l u f) xs"
N2: shows "of_nat K * emeasure P (space M) / epsilon \<le> of_nat N2"
add_diff_cancel: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus (plus a b) b = a"
ssubst_Some: fixes z :: "char list"   and u :: "sterm"   and t :: "sterm" shows "ssubst_option z u (Some t) = Some ([z \<rightarrow> u] t)"
aux7: fixes a :: "'a set set"   and b :: "'a set set"   and p :: "('a set set, 'b) Combinators list"   and c :: "'a set set"   and d :: "'a set set" assumes "DenyAllFromTo a b \<in> set p"   and "allNetsDistinct (DenyAllFromTo c d # p)"   and "a \<noteq> c \<or> b \<noteq> d" shows "twoNetsDistinct a b c d"
nle__lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> A B Le C D" shows "C D Lt A B"
var_rel_def: shows "var_rel = br \<phi> (\<lambda>_. True)"
type_definition_assn: shows "type_definition Rep_assn Abs_assn (Collect proper)"
simps_10: fixes f1 :: "ct list \<Rightarrow> 'a"   and f2 :: "t list \<Rightarrow> 'a"   and f3 :: "'a"   and x2 :: "t list" shows "(case Type x2 of TopType x \<Rightarrow> f1 x | Type x \<Rightarrow> f2 x | Bot \<Rightarrow> f3) = f2 x2"
CONSTRAINT_def: fixes f :: "'a"   and R :: "('b \<times> 'a) set" shows "CONSTRAINT f R \<equiv> True"
termination_0_1_invar: fixes guard :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and p :: "real"   and body :: "'a \<Rightarrow> 'a spmf"   and s :: "'a" assumes "\<And>s. \<lbrakk>guard s; I s\<rbrakk> \<Longrightarrow> p \<le> weight_spmf (loop_spmf.while guard body s)"   and "0 < p"   and "\<And>s. \<lbrakk>guard s; I s\<rbrakk> \<Longrightarrow> lossless_spmf (body s)"   and "\<And>s s'. \<lbrakk>s' \<in> set_spmf (body s); I s; guard s\<rbrakk> \<Longrightarrow> I s'"   and "I s" shows "lossless_spmf (loop_spmf.while guard body s)"
rev_flow_eq: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> ll_on_open.existence_ivl (range (preflect t0)) (\<lambda>t. - f) X t0 x0" shows "ll_on_open.flow (range (preflect t0)) (\<lambda>t. - f) X t0 x0 t = flow0 x0 (preflect t0 t - t0)"
AlphaNegOr: fixes A :: "('a, 'b) form"   and B :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<stileturn> Neg A # Neg B # G" shows "\<stileturn> Neg (Or A B) # G"
discI_4: fixes rexp :: "'a rexp"   and x41 :: "'a rexp"   and x42 :: "'a rexp" assumes "rexp = Plus x41 x42" shows "is_Plus rexp"
execn_callUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and init :: "'d \<Rightarrow> 'd"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and s :: "'d"   and n :: "nat" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =Suc n\<Rightarrow> Stuck"
match_length_2: fixes fps :: "(char list \<times> pat) list"   and ft :: "(char list \<times> trm) list"   and ts :: "trm list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Delta> :: "binding list" assumes "\<turnstile> fps [\<rhd>] ft \<Rightarrow> ts"   and "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>" shows "\<parallel>ts\<parallel> = \<parallel>\<Delta>\<parallel>"
set_iterator_image_foldli_conv: fixes g :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "set_iterator_image g (foldli xs) = foldli (map g xs)"
assms_6: shows "valid_unMultigraph.is_Eulerian_trail G2 v2 ps2 v2'"
simps_26: fixes f1 :: "b_e \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "cl \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> v list \<Rightarrow> e list \<Rightarrow> 'a" shows "(case Trap of $x \<Rightarrow> f1 x | Trap \<Rightarrow> f2 | Callcl x \<Rightarrow> f3 x | Label x xa xb \<Rightarrow> f4 x xa xb | Local x xa xb xc \<Rightarrow> f5 x xa xb xc) = f2"
ffmember_filter: fixes x :: "'x"   and P :: "'x \<Rightarrow> bool"   and A :: "'x fset" shows "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
update_convs_6: fixes sid' :: "nat \<Rightarrow> nat"   and id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "sid_update sid' \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid' sid, \<dots> = more\<rparr>"
path_component_of_nonoverlap: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "(path_component_of_set X x \<inter> path_component_of_set X y = {}) = (x \<notin> topspace X \<or> y \<notin> topspace X \<or> path_component_of X x \<noteq> path_component_of X y)"
alpha_Tree_supp_rel: fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree" assumes "t1 =\<^sub>\<alpha> t2" shows "supp_rel (=\<^sub>\<alpha>) t1 = supp_rel (=\<^sub>\<alpha>) t2"