hcju/isabelleps huggingface dataset:
affine_parallel_commut: fixes A :: "'a set"   and B :: "'a set" assumes "affine_parallel A B" shows "affine_parallel B A"
fcompose_const: fixes a :: "'b"   and q :: "'b poly"   and r :: "'b poly" shows "fcompose [:a:] q r = [:a:]"
subcls1I: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and rest :: "fdecl list \<times> 'a option mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "P \<turnstile> C \<prec>\<^sup>1 D"
A: shows "c_sources I D u ys \<subseteq> c_sources I D u zs"
msubst_nb: fixes p :: "fm"   and t :: "tm"   and s :: "tm"   and c :: "poly"   and d :: "poly" assumes "islin p"   and "tmbound0 t"   and "tmbound0 s" shows "bound0 (msubst p ((c, t), d, s))"
keys_of_monomial: fixes c :: "'c"   and t :: "'d" assumes "c \<noteq> (0::'c)" shows "keys (monomial c t) = {t}"
plus_subgs: fixes H1 :: "'a set"   and H2 :: "'a set" assumes "M +> H1"   and "M +> H2" shows "M +> H1 \<minusplus> H2"
simps_1: shows "\<J> 0 0 = \<pi> 0 {} Init"
image_subset_iff_subset_vimage: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" shows "(f ` A \<subseteq> B) = (A \<subseteq> f -` B)"
univalentD: fixes R :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c"   and z :: "'c" assumes "univalent R"   and "(x, y) \<in> R"   and "(x, z) \<in> R" shows "z = y"
deriv_bodies: shows "\<forall>p\<in>Procs. \<forall>\<sigma> Z. \<Gamma>,\<Theta> \<union> (\<Union>q\<in>Procs. \<Union>Z. {(P q Z \<inter> {s. ((s, q), \<sigma>, p) \<in> r}, q, Q q Z, A q Z)})\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"
simps_2: fixes uw :: "'b \<times> 'b"   and ux :: "'b"   and e :: "'b \<times> 'b" shows "co_path uw ux [e] = [e]"
arr_char: fixes f :: "'d \<times> 'd" shows "D.VV.arr f = (D.arr (fst f) \<and> D.arr (snd f) \<and> src\<^sub>D (fst f) = trg\<^sub>D (snd f))"
assms: shows "- \<infinity> < liminf (\<lambda>x. ereal (u x))"
arrE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr"   and "partial_magma.arr (subcategory.comp C Arr) f" shows "Arr f"
linear_construct: fixes B :: "('a, 'b) vec set"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "vec.independent B" shows "linear (*s) (*s) (vec.construct B f)"
b1_def: shows "b1 \<equiv> compute_bound_tm prec I a t1"
right_v2: shows "right (len v2 ts c) = right (ext v2)"
nodes_delete_edge: fixes v :: "'c"   and e :: "'d"   and v' :: "'c"   and G :: "('c, 'd, 'e) graph_scheme" shows "nodes (delete_edge v e v' G) = nodes G"
meissel_mertens_bounds: shows "\<bar>meissel_mertens - 1 + ln (ln 2)\<bar> \<le> 2 / ln 2"
max_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "max x y = x"
ex_label_map: shows "\<exists>\<phi>. label_wrt C0 \<phi>"
list_update1: fixes t :: "'b tree"   and n :: "nat"   and x :: "'b" assumes "braun t"   and "n \<in> {1..size t}" shows "list (update1 n x t) = (list t)[n - 1 := x]"
successive_split: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "xs = ys @ zs"   and "successive P xs" shows "successive P ys \<and> successive P zs"
strongly_connected_iso: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "pre_digraph.digraph_isomorphism G hom" shows "strongly_connected (app_iso hom G) = strongly_connected G"
arr_cod_iff_arr: fixes f :: "'a2" shows "A2.arr (A2.cod f) = A2.arr f"
single_uminus: fixes k :: "'a"   and a :: "'b" shows "Poly_Mapping.single k (- a) = - Poly_Mapping.single k a"
Exit_Def_empty: fixes V :: "'d" assumes "V \<in> Def Exit" shows "False"
subsum_step: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat"   and prb' :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb"   and "(subsumee sub \<noteq> subsumer sub \<and> fst (subsumee sub) = fst (subsumer sub) \<and> subsumee sub \<in> red_vertices prb \<and> subsumee sub \<notin> subsumers (subs prb) \<and> subsumee sub \<notin> subsumees (subs prb) \<and> subsumer sub \<in> red_vertices prb \<and> subsumer sub \<notin> subsumees (subs prb) \<and> out_edges (red prb) (subsumee sub) = {} \<and> subs prb' = subs prb \<union> {sub}) \<and> \<not> marked prb (subsumer sub) \<and> \<not> marked prb (subsumee sub) \<and> confs prb (subsumee sub) \<sqsubseteq> confs prb (subsumer sub) \<and> prb' = \<lparr>red = red prb, black = black prb, subs = insert sub (subs prb), init_conf = init_conf prb, confs = confs prb, marked = marked prb, strengthenings = strengthenings prb, \<dots> = pre_RedBlack.more prb\<rparr>" shows "RedBlack prb'"
n_mult_left_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "!z \<cdot> x \<le> !z \<cdot> y"
cos_diff_limit_1: fixes \<theta> :: "nat \<Rightarrow> real"   and \<Theta> :: "real" assumes "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1" obtains k :: "nat \<Rightarrow> int" where "(\<lambda>j. \<theta> j - real_of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>"
map\<^sub>0_def: fixes a :: "'a" shows "map\<^sub>0 a \<equiv> src\<^sub>D (F a)"
set_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "set (map f xs) = f ` set xs"
primepow_decompose: fixes n :: "'a" assumes "primepow n" shows "aprimedivisor n ^ multiplicity (aprimedivisor n) n = n"
ab_diff_conv_add_uminus: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a -\<^sub>o\<^sub>w b = a +\<^sub>o\<^sub>w -\<^sub>o\<^sub>w b"
closedE: fixes T :: "'b set set"   and s :: "'b set" assumes "s closed\<^bsub>T\<^esub>" obtains "carrier\<^bsub>T\<^esub> - s open\<^bsub>T\<^esub>"
sf_finite: fixes \<phi> :: "'b ltln" shows "finite (sf \<phi>)"
det_wf_nfa: shows "wf_dfa (det_nfa A) n"
uint16_shiftr_def: fixes x :: "uint16"   and n :: "integer" shows "uint16_shiftr x n = (if n < 0 \<or> 16 \<le> n then undefined drop_bit x n else drop_bit (nat_of_integer n) x)"
tendsto_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'b \<Rightarrow> 'd"   and F :: "'c filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F" shows "((\<lambda>x. \<Sum>i\<in>I. f i x) \<longlongrightarrow> sum a I) F"
Un: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> null_sets M"   and "b \<in> null_sets M" shows "a \<union> b \<in> null_sets M"
gets_def: fixes f :: "'b \<Rightarrow> 'c" shows "gets f \<equiv> do s \<leftarrow> get; return (f s) od"
infsetsum_Un_disjoint: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "f abs_summable_on A"   and "f abs_summable_on B"   and "A \<inter> B = {}" shows "infsetsum f (A \<union> B) = infsetsum f A + infsetsum f B"
fixedaction_Log: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and ms :: "'c match_expr list" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m Log) ms) Undecided = Undecided"
SNF_2x2_works: shows "\<forall>A\<in>carrier_mat 2 2. is_SNF A (Smith_2x2 A)"
sinks_aux_sinks_aux_less: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list" shows "sinks_aux I D U xs = U \<union> sinks_aux_less I D U xs"
nt_mult_mp: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "m < 0"   and "0 \<le> n" shows "m\<triangleright>n\<triangleright>a\<^bsub>A\<^esub>\<^bsub>A\<^esub> = (m * n)\<triangleright>a\<^bsub>A\<^esub>"
simps_1: fixes uu :: "nat \<Rightarrow> 'a"   and uv :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and uw :: "'c \<Rightarrow> 'a list \<Rightarrow> bool" shows "\<lbrakk>uu, uv, uw\<rbrakk> \<^bold>\<bottom> = False"
exhaust_sel: fixes id :: "'d Monomorphic_Monad.id" obtains "id = return_id (extract id)"
len_ts: shows "length ts = dim_col (row_echelon (polys_to_mat ts ps))"
continuous_on: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
echelon_foldl_condition1: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and ma :: "'f"   and k :: "nat" assumes "is_bezout_ext bezout"   and "A $ ma $ mod_type_class.from_nat (Suc k) \<noteq> (0::'d)"   and "k < ncols A" shows "\<exists>m. \<not> is_zero_row_upt_k m (Suc (Suc k)) (bezout_iterate (interchange_rows A (0::'f) (LEAST n. A $ n $ mod_type_class.from_nat (Suc k) \<noteq> (0::'d))) (nrows A - Suc 0) (0::'f) (mod_type_class.from_nat (Suc k)) bezout)"
functor_axioms: shows "functor VV.comp (\<cdot>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu>)"
While_1: fixes P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "\<forall>\<rho>\<in>density_states. trace (wlp S P * \<rho>) = trace (P * denote S \<rho>) + trace \<rho> - trace (denote S \<rho>)"
rtrancl_insert: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" shows "(insert (a, b) r)\<^sup>* = r\<^sup>* \<union> {(x, y). (x, a) \<in> r\<^sup>* \<and> (b, y) \<in> r\<^sup>*}"
suffix: fixes i :: "nat" assumes "i < Suc (Suc n)" shows "suffix k w \<Turnstile>\<^sub>n af (if i \<le> n then \<phi> else \<psi>) (w [min i n \<rightarrow> k])[X]\<^sub>\<nu>"
padic_mult_in_padic_set: fixes f :: "nat \<Rightarrow> int"   and p :: "int"   and g :: "nat \<Rightarrow> int" assumes "f \<in> padic_set p"   and "g \<in> padic_set p"   and "prime p" shows "padic_mult p f g \<in> padic_set p"
that: fixes i :: "nat" assumes "i < num_params p" shows "e i = f i"
dvd_imp_mod_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times a b" shows "modulo b a = zero"
ConseqMGT: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and P' :: "'d \<Rightarrow> 'd set"   and c :: "('d, 'e, 'f) com"   and Q' :: "'d \<Rightarrow> 'd set"   and A' :: "'d \<Rightarrow> 'd set"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c (Q' Z),(A' Z)"   and "\<And>s. s \<in> P \<Longrightarrow> s \<in> P' s \<and> (\<forall>t. t \<in> Q' s \<longrightarrow> t \<in> Q) \<and> (\<forall>t. t \<in> A' s \<longrightarrow> t \<in> A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
godel_code'_butlast_last_id: fixes xs :: "nat list"   and j :: "nat" assumes "xs \<noteq> []" shows "godel_code' xs (Suc j) = godel_code' (butlast xs) (Suc j) * Pi (length xs + j) ^ last xs"
some: fixes b :: "'b"   and \<sigma>' :: "'a" assumes "ioprog a \<sigma> = Some (b, \<sigma>')"   and "\<sigma>' \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p S ioprog) (P (b # s))" shows "Q"
row_i_A_not_zero: shows "\<not> is_zero_row i A"
fgh: shows "f = g \<circ>\<^sub>p Polynomial.monom (1::'a) 2 + Polynomial.monom (1::'a) 1 * h \<circ>\<^sub>p Polynomial.monom (1::'a) 2"
smooth_compat_def: fixes k :: "enat"   and c1 :: "('b, 'c) chart"   and c2 :: "('b, 'c) chart" shows "k-smooth_compat c1 c2 = (k-smooth_on (apply_chart c1 ` (domain c1 \<inter> domain c2)) (apply_chart c2 \<circ> inv_chart c1) \<and> k-smooth_on (apply_chart c2 ` (domain c1 \<inter> domain c2)) (apply_chart c1 \<circ> inv_chart c2))"
wt_elim_cases_8: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and x :: "val"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Lit x)\<Colon>T" obtains T :: "ty" where "Ta__ = Inl T" and "typeof dt x = Some T"
distinct_drop: fixes xs :: "'b list"   and i :: "nat" assumes "distinct xs" shows "distinct (drop i xs)"
drop_eq_nths: fixes n :: "nat"   and xs :: "'b list" shows "drop n xs = nths xs {i. n \<le> i}"
In0_inject: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" assumes "In0 M = In0 N" shows "M = N"
hyp: fixes x :: "trm"   and z :: "trm" assumes "\<And>y. x \<mapsto> y \<Longrightarrow> SN y"   and "\<And>y. x \<mapsto> y \<Longrightarrow> P y z"   and "\<And>u. z \<mapsto> u \<Longrightarrow> SN u"   and "\<And>u. z \<mapsto> u \<Longrightarrow> P x u" shows "P x z"
sel_rews_3: fixes app :: "'e\<cdot>'f writer\<cdot>'g" shows "runWriterT\<cdot>(WriterT\<cdot>app) = app"
ip: shows "is_path (\<pi> @\<^bsup>l\<^esup> \<pi>l)"
aka_cond_def: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "if p then x else y = d p \<cdot> x + ad p \<cdot> y"
simps: fixes x :: "'b"   and uu :: "'b hp list" shows "get_min (Some (Hp x uu)) = x"
assms_2: shows "f absolutely_integrable_on g ` S \<or> f \<circ> g absolutely_integrable_on S"
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xb :: "'a + 'b"   and xc :: "'a + 'b" assumes "\<not> sum_le x xa xb xc" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xd :: "'a" and y :: "'a" where "x = P__" and "xa = Q__" and "xb = Inl xd__" and "xc = Inl y__" and "\<not> P__ xd__ y__"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xd :: "'b" and y :: "'b" where "x = Pa__" and "xa = Qa__" and "xb = Inr xd" and "xc = Inr y" and "\<not> Qa__ xd y"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and v :: "'b" and va :: "'a" where "x = Pb__" and "xa = Qb__" and "xb = Inr v__" and "xc = Inl va__"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and va :: "'a" and v :: "'b" where "x = P" and "xa = Q" and "xb = Inl va" and "xc = Inr v"
approx_indep_ge: fixes x :: "'a pmf"   and y :: "'a pmf"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and \<alpha> :: "real"   and outcomes :: "'a set" assumes "x \<approx>[\<R>] y"   and "\<alpha> \<in> {0..1}"   and "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>" shows "\<forall>r\<in>\<P> outcomes. mix_pmf \<alpha> y r \<succeq>[\<R>] mix_pmf \<alpha> x r"
IN_g_B: fixes x :: "'v" assumes "x \<in> B \<Gamma>" shows "d_IN (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>)) x = d_OUT g \<langle>x\<rangle>"
approximate_sets_axioms: shows "approximate_sets ops set_of_appr appr_rell optns"
splitAt_in_snd: fixes v :: "'a"   and ram :: "'a"   and vs :: "'a list" assumes "v \<in> set (snd (splitAt ram vs))" shows "v \<in> set vs"
d'_neq: shows "d' \<noteq> a' \<and> d' \<noteq> b' \<and> d' \<noteq> c'"
assms_1: shows "set (coeffs (p * q)) \<subseteq> \<real>"
sqrtu_2_eq: fixes x :: "real" shows "sqrtu x 2 = 1 / 8 * (x ^ 4 + 28 * x ^ 3 + 70 * x\<^sup>2 + 28 * x + 1) / ((x + 1) * (x\<^sup>2 + 6 * x + 1))"
that: fixes d :: "real" assumes "0 < d"   and "\<forall>x\<in>s. d \<le> norm (f x)" shows "thesis"
set_remdups: fixes xs :: "'d list" shows "set (remdups xs) = set xs"
pos_less_minus_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a < - (b /\<^sub>R c)) = (c *\<^sub>R a < - b)"
4: shows "a \<odot>\<^bsub>UP R\<^esub> g [^]\<^bsub>UP R\<^esub> n = a \<odot>\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> g [^]\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> n"
lfinite_inf_llist: fixes f :: "nat \<Rightarrow> 'a" shows "linfinite (inf_llist f)"
fst_sel_fun: shows "det_selection_fun fst_sel_fun"
\<rho>: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> w) \<rho> (src w) (g \<star> w)"
Rep3: fixes x :: "'f" shows "T3 (Rep3 x) x"
snd_plus_commute: fixes x0 :: "'a" shows "snd \<circ> (+) (x0, x0) = (+) x0 \<circ> snd"
take_map: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map f xs \<down> n = map f (xs \<down> n)"
exprel_imp_eq_freefun: fixes U :: "freeExp"   and V :: "freeExp" assumes "U \<sim> V" shows "freefun U = freefun V"
"1_2": shows "g \<turnstile> n-ns\<rightarrow>m"
simps_2: fixes \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "run \<delta> q\<^sub>0 w (Suc i) = \<delta> (run \<delta> q\<^sub>0 w i) (w i)"
disjunctiveI: fixes a :: "'b \<Rightarrow> 'c" assumes "\<And>b c. a (b \<squnion> c) = a b \<squnion> a c" shows "a \<in> Apply.disjunctive"
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_def: fixes n :: "nat" shows "PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n = P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>P n,Up\<^sub>P\<rbrakk>"
assms: shows "domA \<Gamma> \<inter> edom \<rho> = {}"
mat1_mult_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nr :: "nat"   and nc :: "nat"   and m :: "'a list list" assumes "class.semiring_1 one times plus zero"   and "mat nr nc m" shows "mat_multI zero plus times nr m (mat1I zero one nc) = m"
nonneg_euler_genus: shows "0 \<le> G.euler_genus"
to_fun_X_minus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_fun (X_poly_minus R a) b = b \<ominus> a"
integrable_MIN: fixes I :: "'b set"   and M :: "'c measure"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> real" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "integrable M (\<lambda>x. MIN i\<in>I. f i x)"
\<phi>J: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {xx}" shows "\<phi>J \<psi> \<in> fmla"
that: fixes as' :: "'edge list"   and a' :: "'edge" assumes "as = as' @ [a']"   and "n -as'\<rightarrow>* sourcenode a'"   and "valid_edge a'"   and "n' = targetnode a'" shows "thesis"
funion_fempty_left: fixes B :: "'a fset" shows "{||} |\<union>| B = B"
extf_ext_trans_before_irrefl: fixes f :: "'s" shows "ext_trans_before_irrefl (extf f)"
GF_advice_\<nu>LTL_1: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "\<phi>[X]\<^sub>\<nu> \<in> \<nu>LTL"
swap_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" shows "X #[x \<and> y]_zs = X #[y \<and> x]_zs"
prodag_sameTr4: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and B :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_zero I A = prod_zero I B"
lossless: fixes \<eta> :: "nat" assumes "lossless" shows "plossless_converter (\<I>_common' \<eta>) (\<I>_common \<eta>) (conv \<eta>)"
simps_2: fixes v :: "nat" shows "T_chop (Suc v) [] = 1"
cond_cases_1: fixes h :: "heap" assumes "success p h"   and "cond p h" shows "execute p h = Some (True, h)"
gr_zeroI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
frontier_of_components_subset: fixes C :: "'a set"   and S :: "'a set" assumes "C \<in> components S" shows "frontier C \<subseteq> frontier S"
represents_plus: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and r :: "'d \<Rightarrow>\<^sub>0 'c"   and x :: "'d"   and B :: "'d set"   and s :: "'d \<Rightarrow>\<^sub>0 'c"   and y :: "'d" assumes "module scale"   and "module.represents scale A r x"   and "module.represents scale B s y" shows "module.represents scale (A \<union> B) (r + s) (x + y)"
smult_assoc1: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M1" shows "a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M1\<^esub> x = a \<odot>\<^bsub>M1\<^esub> (b \<odot>\<^bsub>M1\<^esub> x)"
DERIV_LIM_iff: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and D :: "'a" shows "(\<lambda>h. (f (a + h) - f a) / h) \<midarrow>0::'a\<rightarrow> D = (\<lambda>x. (f x - f a) / (x - a)) \<midarrow>a\<rightarrow> D"
min_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
k_def: shows "k = (LEAST k. x < f (Suc k))"
inclusion_of_generators_def: shows "inclusion_of_generators \<equiv> \<lambda>f. if C.arr f then mkarr \<^bold>\<langle>f\<^bold>\<rangle> else null"
max_Var_floatarith_DERIV_floatarith: fixes x :: "nat"   and fa :: "floatarith" shows "max_Var_floatarith (DERIV_floatarith x fa) \<le> max_Var_floatarith fa"
simps: fixes a :: "'a array"   and n :: "nat" shows "assoc_list_of_array_code a n = (if array_length a \<le> n then [] else (n, array_get a n) # assoc_list_of_array_code a (n + 1))"
False: shows "\<not> (a = 0 \<or> b = 0)"
Neg_atm_of_iff: fixes L :: "'a literal" shows "(Neg (atm_of L) = L) = is_neg L"
fib_plus_2: fixes n :: "nat" shows "fib (n + 2) = fib (n + 1) + fib n"
keys_map_key: fixes f :: "'c \<Rightarrow> 'd"   and p :: "'d \<Rightarrow>\<^sub>0 'e" assumes "inj f" shows "keys (Poly_Mapping.map_key f p) = f -` keys p"
summable_pre_arctan: fixes x :: "real" assumes "norm x < 1" shows "summable (\<lambda>n. mssnth arctan_series_stream n * x ^ n)"
inject_1: fixes x11 :: "'a trm"   and x12 :: "'a trm"   and y11 :: "'a trm"   and y12 :: "'a trm" shows "(Eq x11 x12 = Eq y11 y12) = (x11 = y11 \<and> x12 = y12)"
zmult_zless_mono2_lemma: fixes i :: "Quotient_Int.int"   and j :: "Quotient_Int.int"   and k :: "nat" assumes "i < j"   and "0 < k" shows "of_nat k * i < of_nat k * j"
Class_unit_is_normal: shows "normal_subgroup Normal G (\<cdot>) \<one>"
digraph_op: shows "digraph \<alpha> (op_dg \<AA>)"
igSubstBinp_def: fixes MOD :: "('m, 'n, 'o, 'p, 'q, 'r, 's, 'l, 't) model_scheme"   and ys :: "'o"   and Y :: "'s"   and y :: "'r"   and binp :: "'k \<Rightarrow> 'l option" shows "igSubstBinp MOD ys Y y binp \<equiv> lift (igSubstAbs MOD ys Y y) binp"
rcos_const: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and h :: "'c" assumes "subgroup H G"   and "Group.group G"   and "h \<in> H" shows "H #>\<^bsub>G\<^esub> h = H"
\<theta>_simps_4: shows "local.dom \<theta> = (s\<^sub>0 \<star> p\<^sub>0) \<star> w"
linear_add: fixes f :: "'a \<Rightarrow> 'b"   and b1 :: "'a"   and b2 :: "'a" assumes "linear f" shows "f (b1 + b2) = f b1 + f b2"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list"   and y :: "'d"   and ys :: "'d list" shows "zipwith0 f (x # xs) (y # ys) = f x y # zipwith0 f xs ys"
a71: shows "\<Union> (Acc level1 ` DAcc level1 sA71) = Acc level1 sA81"
on_new_root_refine: fixes v0 :: "'a"   and s :: "('a, 'c) state_scheme"   and v0i :: "'a"   and si :: "('a, 'd) simple_state_scheme"   and si' :: "('a, 'd) simple_state_scheme"   and s' :: "('a, 'c) state_scheme" assumes "a.a.gen_dfs.pre_new_root v0 s"   and "(v0i, v0) \<in> Id"   and "(si, s) \<in> \<langle>ES\<rangle>simple_state_rel"   and "(si', s') \<in> \<langle>ES\<rangle>simple_state_rel"   and "nf_inres (gbs_new_root a.a.gbs v0 s) s'" shows "on_new_root parami v0i si' \<le> \<Down> ES (on_new_root param v0 s')"
minusinf_qfree: fixes p :: "fm" assumes "qfree p" shows "qfree (minusinf p)"
set_delete: fixes x :: "'b list"   and s :: "'b trie" assumes "True" shows "Trie_Fun.set (Trie_Fun.delete x s) = Trie_Fun.set s - {x}"
fu1: fixes x :: "real" assumes "x \<in> {u<..1}" shows "f x \<noteq> f 0"
wrap_iso_arcs_permutes: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'f" assumes "pre_digraph.digraph_isomorphism G hom"   and "A \<subseteq> arcs G"   and "f permutes A" shows "pre_digraph.wrap_iso_arcs G hom f permutes iso_arcs hom ` A"
nonneg_Reals_inverse_I: fixes a :: "'a" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "inverse a \<in> \<real>\<^sub>\<ge>\<^sub>0"