hcju/isabelleps huggingface dataset:
termdiffs_aux: fixes c :: "nat \<Rightarrow> 'a"   and K :: "'a"   and x :: "'a" assumes "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"   and "norm x < norm K" shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0::'a\<rightarrow> (0::'a)"
weakSimE_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "\<alpha> \<noteq> \<tau>" shows "\<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>', P', Q') \<in> Rel)"
vec_simps_21: fixes ac :: "'a"   and bc :: "'a"   and cb :: "'a" shows "ac - (bc - cb) = ac + cb - bc"
sup_id_star1: fixes x :: "'a" assumes "(1::'a) \<le> x" shows "x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
inv_mult_group: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H" shows "\<ominus>\<^bsub>H\<^esub> (x \<oplus>\<^bsub>H\<^esub> y) = \<ominus>\<^bsub>H\<^esub> y \<oplus>\<^bsub>H\<^esub> \<ominus>\<^bsub>H\<^esub> x"
Rep_Abs_LUB: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "Rep_process (Abs_process (\<Inter>a. F (S a), \<Inter>a. D (S a))) = (\<Inter>a. F (S a), \<Inter>a. D (S a))"
gen_def: fixes x :: "String.literal"   and s :: "thm" shows "gen x s \<equiv> Thm (Forall x (concl s))"
inv_solve_right': fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c = a) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
antisym_conv3: fixes y :: "'a"   and x :: "'a" assumes "\<not> lt y x" shows "(\<not> lt x y) = (x = y)"
set_minus_imp_plus: fixes a :: "'c"   and b :: "'c"   and C :: "'c set" assumes "a - b \<in> C" shows "a \<in> b +o C"
proj_proj_poly_syz: fixes k :: "nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (proj_poly_syz n p) = proj_poly (k + n) p"
integrable_restrict_UNIV: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "((\<lambda>x. if x \<in> s then f x else (0::'c)) integrable_on UNIV) = (f integrable_on s)"
f_borel_Suc_real_cond_exp: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and f :: "bool stream \<Rightarrow> real"   and n :: "nat"   and g :: "bool stream \<Rightarrow> 'b"   and N :: "'b measure" assumes "infinite_cts_filtration p M F"   and "f \<in> borel_measurable (F (Suc n))"   and "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N"   and "\<forall>w. g -` {g w} \<in> sets (F n)"   and "\<forall>r\<in>range g \<inter> space N. \<exists>A\<in>sets N. range g \<inter> A = {r}"   and "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z"   and "0 < p"   and "p < 1" shows "AEeq M (real_cond_exp M (fct_gen_subalgebra M N g) f) (\<lambda>w. p * f (pseudo_proj_True n w) + (1 - p) * f (pseudo_proj_False n w))"
univ_asAbs_alphaAbs: fixes fAbs :: "('c, 'd, 'varSort, 'var, 'e) qAbs \<Rightarrow> 'f"   and A :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "fAbs respectsP ($=)"   and "qGoodAbs A" shows "Equiv_Relation2.univ fAbs (asAbs A) = fAbs A"
R: shows "\<forall>S T. P S \<and> P T \<longrightarrow> (finite union_of P) (S \<inter> T)"
chambersubcomplex_fold_A_im: shows "ChamberComplex.ChamberSubcomplex A (fold_A \<turnstile> A)"
show10: fixes d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and s :: "nat" assumes "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "0 < s"   and "\<forall>m\<ge>s. \<exists>n\<ge>m. d (n + 1) * (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j))) \<le> real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1)))" shows "\<forall>m\<ge>s. \<exists>n\<ge>m. d (n + 1) powr 2 ^ (n + 1) * (\<Prod>j = 1..n. real_of_int (a j)) * (1 / (\<Prod>j = 1..s - 1. real_of_int (a j))) \<le> real_of_int (a (n + 1))"
preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>D (local.map \<mu>) = F.map\<^sub>0 (src\<^sub>C \<mu>)"
star: fixes c :: "real"   and x :: "'a" assumes "0 \<le> c"   and "c < 1"   and "a + x \<in> S" shows "a + c *\<^sub>R x \<notin> S"
nonpos_Reals_of_nat_iff: fixes n :: "nat" shows "(of_nat n \<in> \<real>\<^sub>\<le>\<^sub>0) = (n = 0)"
det_row_mul: fixes k :: "'d"   and c :: "'c"   and a :: "'d \<Rightarrow> ('c, 'd) vec"   and b :: "'d \<Rightarrow> ('c, 'd) vec" shows "Determinants.det (\<chi>i. if i = k then c *s a i else b i) = c * Determinants.det (\<chi>i. if i = k then a i else b i)"
ordering_top_axioms: shows "ordering_top (\<le>) (<) 1"
measure\<^sub>s\<^sub>t_trans: shows "trans measure\<^sub>s\<^sub>t"
GB_replace_red_stable_pmdl_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis G"   and "p \<in> G"   and "red (G - {p}) p q" shows "pmdl (insert q (G - {p})) = pmdl G"
inv_image_def: fixes r :: "('h \<times> 'h) set"   and f :: "'g \<Rightarrow> 'h" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"
Sbis_finished_discr_L: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(c, d) \<in> PL_Indis.Sbis aval tval cval indis"   and "proper d"   and "finished d" shows "PL_Indis.discr aval tval cval indis c"
ordLeq_finite_Field: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" assumes "r \<le>o s"   and "finite (Field s)" shows "finite (Field r)"
erf_remainder_integral_aux_bound_1: fixes x :: "real"   and m :: "nat"   and n :: "nat" assumes "0 < x" shows "norm (integral {x..x + real m} (\<lambda>t. exp (- t\<^sup>2) / t ^ (2 * n))) \<le> exp (- x\<^sup>2) / x ^ (2 * n + 1)"
Fake_insert_dy_fake_msg: fixes M :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set"   and X :: "msg set" assumes "M \<in> dy_fake_msg bad IK CH"   and "extr bad IK CH \<subseteq> synth (analz X)" shows "synth (analz (insert M IK)) \<subseteq> synth (analz X)"
that_1: fixes i :: "'a"   and j :: "'a" assumes "i \<in> A"   and "j \<in> A"   and "i \<noteq> j" shows "coprime (m i) (m j)"
aforms_err_def: fixes e :: "nat \<Rightarrow> real"   and xs :: "((real \<times> real pdevs) \<times> real) list" shows "aforms_err e xs = listset (map (aform_err e) xs)"
geometric_pmf_prob_lessThan: fixes p :: "real"   and n :: "nat" assumes "p \<in> {0<..1}" shows "measure_pmf.prob (geometric_pmf p) {..<n} = 1 - (1 - p) ^ n"
simps_6: fixes a :: "Cooper.num"   and b :: "Cooper.num" shows "zsplit0 (Sub a b) = (let (ia, a') = zsplit0 a; (ib, b') = zsplit0 b in (ia - ib, Sub a' b'))"
distinct_7: fixes x5 :: "char list" shows "Accept \<noteq> Call x5"
is_element_ptr_kind_obtains: fixes node_ptr :: "(_) node_ptr" assumes "is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" obtains element_ptr :: "(_) element_ptr" where "node_ptr = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr"
exesort_leq_def: fixes sub :: "('a \<times> 'b) list"   and s1 :: "'a set"   and s2 :: "'b set" shows "exesort_leq sub s1 s2 = (\<forall>c\<^sub>2\<in>s2. \<exists>c\<^sub>1\<in>s1. execlass_leq sub c\<^sub>1 c\<^sub>2)"
nn_integral_split: fixes f :: "(nat \<Rightarrow> 'a) \<Rightarrow> ennreal"   and i :: "nat" assumes "P.random_variable borel f" shows "integral\<^sup>N (Pi\<^sub>M UNIV (\<lambda>i. M)) f = \<integral>\<^sup>+ \<omega>. \<integral>\<^sup>+ \<omega>'. f (comb_seq i \<omega> \<omega>') \<partial>Pi\<^sub>M UNIV (\<lambda>i. M) \<partial>Pi\<^sub>M UNIV (\<lambda>i. M)"
pairwiseI: fixes S :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; x \<noteq> y\<rbrakk> \<Longrightarrow> R x y" shows "pairwise R S"
sf_subset_subfrmlsn: fixes \<phi> :: "'b ltln" shows "sf \<phi> \<subseteq> subfrmlsn \<phi>"
VDMComp: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and d :: "OBJ"   and B :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> c : A"   and "G \<rhd> d : B" shows "G \<rhd> Comp c d : \<lambda>s t. \<exists>r. A s r \<and> B r t"
hom_in_carrier: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme"   and x :: "'e" assumes "h \<in> hom G H"   and "x \<in> carrier G" shows "h x \<in> carrier H"
distinct_19: fixes x41 :: "'d regex"   and x42 :: "'d regex"   and x5 :: "'d regex" shows "Times x41 x42 \<noteq> Star x5"
induct: fixes P :: "'state \<Rightarrow> ('b, 'pred, 'act) Tree \<Rightarrow> bool"   and a0 :: "'state"   and a1 :: "('b, 'pred, 'act) Tree" assumes "\<And>P tset. (\<And>x. x \<in> set_bset tset \<Longrightarrow> P P x) \<Longrightarrow> P P (tConj tset)"   and "\<And>P t. P P t \<Longrightarrow> P P (tNot t)"   and "\<And>P \<phi>. P P (tPred \<phi>)"   and "\<And>P \<alpha> t. (\<And>x xa xb. \<lbrakk>tAct \<alpha> t =\<^sub>\<alpha> tAct x xa; P \<rightarrow> \<langle>x,xb\<rangle>\<rbrakk> \<Longrightarrow> P xb xa) \<Longrightarrow> P P (tAct \<alpha> t)" shows "P a0 a1"
less_eq_pratt1_def: fixes x :: "pratt1"   and y :: "pratt1" shows "x \<le> y \<equiv> x + y = y"
reward_until_def: fixes \<iota> :: "'c \<Rightarrow> 'c \<Rightarrow> ennreal"   and \<rho> :: "'c \<Rightarrow> ennreal"   and X :: "'c set" assumes "MC_with_rewards \<iota> \<rho>" shows "MC_with_rewards.reward_until \<iota> \<rho> X = lfp (\<lambda>F s \<omega>. if s \<in> X then 0 else \<rho> s + \<iota> s (shd \<omega>) + F (shd \<omega>) (stl \<omega>))"
defensive_strategy_Odd_def: fixes p :: "'a list" shows "defensive_strategy_Odd p \<equiv> SOME m. move_defensive_by_Odd m p"
locally_trans_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset" shows "locally_trans R A B C \<equiv> \<forall>t u v. t \<in># A \<longrightarrow> u \<in># B \<longrightarrow> v \<in># C \<longrightarrow> (t, u) \<in> R \<longrightarrow> (u, v) \<in> R \<longrightarrow> (t, v) \<in> R"
vec_index_conjugate: fixes i :: "nat"   and v :: "'b vec" assumes "i < dim_vec v" shows "conjugate v $ i = conjugate (v $ i)"
polyfun_add: fixes N :: "nat set"   and f :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'c"   and g :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "polyfun N f"   and "polyfun N g" shows "polyfun N (\<lambda>x. f x + g x)"
hull_concat_lists: fixes G :: "'b list set" shows "\<langle>G\<rangle> = concat ` lists G"
zero_neq_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "zero \<noteq> numeral.numeral one plus n"
rtrancl_Un_rtrancl: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" shows "(R\<^sup>* \<union> S\<^sup>*)\<^sup>* = (R \<union> S)\<^sup>*"
demonic_assert_comp: fixes r :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d \<Rightarrow> bool" shows "[: r :] \<circ> {. p .} = {.x. \<forall>y. r x y \<longrightarrow> p y.} \<circ> [: r :]"
integrable_spike_finite: fixes S :: "'c set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "f integrable_on T" shows "g integrable_on T"
nonzero_memI: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "domain R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>" shows "a \<in> nonzero R"
INF_const: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and f :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Inf ((\<lambda>i. f) ` A) = f"
linear_directional_derivative: fixes k :: "enat"   and a :: "'c" assumes "k \<noteq> 0" shows "linear (directional_derivative k a)"
CFG_call_determ: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r' :: "'e"   and p' :: "'f"   and fs' :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p'\<^esub>fs'"   and "sourcenode a = sourcenode a'"   and "local.CFG.pred (kind a) s"   and "local.CFG.pred (kind a') s" shows "a = a'"
set_intros_1: fixes x1 :: "'b"   and x2 :: "'b hp list" shows "x1 \<in> set_hp (Hp x1 x2)"
two_comparisons_into_case_order_11: fixes x :: "'a"   and y :: "'a"   and P :: "'c"   and R :: "'c"   and Q :: "'c" shows "(if x = y then P else if le y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
assms_1: shows "H \<turnstile> ShiftP f k del g"
chordal_circle_cmat_hermitean_nonzero: fixes a :: "complex \<times> complex"   and r :: "real" assumes "a \<noteq> vec_zero" shows "chordal_circle_cvec_cmat a r \<in> hermitean_nonzero"
simps_2: fixes A :: "'b mat"   and i :: "nat" shows "??.Jordan_Normal_Form_Existence.identify_block A (Suc i) = (if A $$ (i, Suc i) = (1::'b) then ??.Jordan_Normal_Form_Existence.identify_block A i else Suc i)"
simps_1: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "bezout_iterate A 0 i j bezout = A"
frac_le: 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 y :: "'a"   and x :: "'a"   and w :: "'a"   and z :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero y"   and "less_eq x y"   and "less zero w"   and "less_eq w z" shows "less_eq (divide x z) (divide y w)"
scnj_def: fixes F :: "'fmla set" shows "scnj F = lcnj (asList F)"
project_set_is_UN_slice: fixes A :: "'c set" shows "project_set h A = \<Union> (range (slice A))"
deformation_retract_imp_isomorphic_relative_homology_groups: fixes X :: "'a topology"   and X' :: "'a topology"   and r :: "'a \<Rightarrow> 'a"   and V :: "'a set"   and U :: "'a set"   and p :: "int" assumes "retraction_maps X X' r id"   and "V \<subseteq> U"   and "r ` U \<subseteq> V"   and "homotopic_with (\<lambda>h. h ` U \<subseteq> U) X X r id" shows "relative_homology_group p X U \<cong> relative_homology_group p X' V"
openI: fixes m :: "'b set"   and T :: "'b set set" assumes "m \<in> T" shows "m open\<^bsub>T\<^esub>"
iff: fixes n :: "nat"   and f :: "(real, 'm) vec \<Rightarrow> (real, 'a) vec"   and b :: "(real, 'a) vec" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on \<Union> (F ` {..n}) \<and> integral (\<Union> (F ` {..n})) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` \<Union> (F ` {..n}) \<and> integral (g ` \<Union> (F ` {..n})) f = b)"
PiM_eq: fixes M' :: "('i \<Rightarrow> 'a) measure" assumes "sets M' = P.events"   and "\<And>J F. \<lbrakk>finite J; J \<subseteq> I; \<And>j. j \<in> J \<Longrightarrow> F j \<in> M.events\<rbrakk> \<Longrightarrow> emeasure M' (emb I J (Pi\<^sub>E J F)) = (\<Prod>j\<in>J. emeasure borel (F j))" shows "M' = Pi\<^sub>M I (\<lambda>_. borel)"
q: shows "((\<gamma>1'', \<gamma>2), (s, [\<beta>_step] @ \<upsilon>_step), s, [\<alpha>_step] @ \<rho>_step) \<in> pex r"
lt_ord_p: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt_of_nat_term_order cmp_term (lt p) (lt q)" shows "ord_strict_p p q"
fold_union_elemI: fixes x :: "'a"   and ys :: "'a set"   and xss :: "'a set list"   and xs :: "'a set" assumes "x \<in> ys"   and "ys \<in> set xss \<union> {xs}" shows "x \<in> fold (\<union>) xss xs"
vlrestriction_atE1: fixes A :: "V"   and a :: "V"   and b :: "V" assumes "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A" obtains "r\<lparr>a\<rparr> = b"
min_diff_distrib_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "minus (ord.min less_eq x y) z = ord.min less_eq (minus x z) (minus y z)"
xsx2_last: shows "(xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2)\<lparr>vcard xs\<^sub>2\<rparr> = x\<^sub>2"
ball_eq_ball: fixes m :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>m. P x = Q x" shows "(\<forall>x\<in>m. P x) = (\<forall>x\<in>m. Q x)"
is_arg_min_eq: fixes m :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a" assumes "is_arg_min m P x"   and "P z"   and "m z = m x" shows "is_arg_min m P z"
listset_finite: fixes XS :: "'b set list" assumes "list_all (\<lambda>A. A \<noteq> {}) XS" shows "finite (listset XS) = list_all finite XS"
DVD_mono: fixes x :: "real"   and bs :: "real list"   and i :: "int" assumes "0 \<le> x"   and "x < 1" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (DVD i n s) = Ifm (x # bs) (Dvd i a)"
eqvtI: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and p :: "(name \<times> name) list" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'" shows "p \<bullet> P \<Longrightarrow>\<^sup>^p \<bullet> \<alpha> \<prec> p \<bullet> P'"
max_0_1_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq one zero = one"
list_vec: fixes v :: "complex Matrix.vec" assumes "v \<in> state_qbit 1" shows "list_of_vec v = [v $ 0, v $ 1]"
R_def: shows "R = B \<union> (*\<^sub>C) \<i> ` B"
seq: shows "(s, t) \<in> (R \<union> S)\<^sup>* O S"
sbelow_sctns_coll_def: fixes XS :: "'c set"   and sctns :: "'c sctn set" shows "sbelow_sctns_coll XS sctns = (sets_of_coll XS ::: \<langle>\<langle>appr_rel\<rangle>list_wset_rel\<rangle>nres_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN True) (\<lambda>X. sbelow_sctns X sctns) (\<lambda>a b. RETURN (a \<and> b)))"
set_trieP_deleteP: fixes xs :: "bool list"   and t :: "trieP" shows "set_trieP (deleteP xs t) = set_trieP t - {xs}"
abs_1: shows "\<bar>1::'a\<bar> = (1::'a)"
simps_2: fixes v :: "('c, 'd) SkewBinomialTree"   and va :: "('c, 'd) SkewBinomialTree list" shows "meldUniq (v # va) [] = v # va"
inext_imirror_iprev_conv: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<le> iMin I + Max I" shows "inext (mirror_elem n I) (imirror I) = mirror_elem (iprev n I) I"
shadow_root_ptr_casts_commute: fixes document_ptr :: "(_) document_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" shows "(cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr = Some shadow_root_ptr) = (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = document_ptr)"
laz_len: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a list"   and b :: "'b list" assumes "list_all_zip P a b" shows "length a = length b"
ide_mkIde: fixes A :: "'s set" assumes "A \<subseteq> S.Univ" shows "S.ide (S.mkIde A)"
return_def: fixes a :: "'e" shows "return a = State (Pair a)"
Rice_2: fixes A :: "nat set" assumes "index_set A"   and "A \<noteq> {}"   and "A \<noteq> UNIV" shows "\<not> computable A"
int_of_integer_minus: fixes x :: "integer"   and y :: "integer" shows "int_of_integer (x - y) = int_of_integer x - int_of_integer y"
coprimeTr: fixes A :: "'a set"   and B :: "'a set" assumes "ideal R A"   and "ideal R B" shows "coprime_ideals R A B = (\<exists>a\<in>A. \<exists>b\<in>B. a \<plusminus> b = 1\<^sub>r)"
rep_eq: fixes x :: "'a prefix" shows "plen x = length (Rep_prefix x)"
keys_monom_mult_subset: fixes c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (MPoly_Type_Class.punit.monom_mult c t p) \<subseteq> MPoly_Type_Class.punit.splus t ` keys p"
sym: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "sympartp r x y" shows "sympartp r y x"
\<Z>_srng_\<Z>_csgrp_vplus: shows "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"
onllIl: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'h, 'f, 'e) seqp"   and p :: "('g, 'h, 'f, 'e) seqp"   and p' :: "('g, 'h, 'f, 'e) seqp"   and P :: "('i \<times> 'e) \<times> 'j \<times> 'i \<times> 'e \<Rightarrow> bool"   and \<xi> :: "'i"   and a :: "'j"   and \<xi>' :: "'i" assumes "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')" shows "onll \<Gamma> P ((\<xi>, p), a, \<xi>', p')"
future_reach_And: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "future_reach (And \<phi> \<psi>) = max (future_reach \<phi>) (future_reach \<psi>)"
diff_diff_add: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - b - c = a - (b + c)"