hcju/isabelleps huggingface dataset:
min_range_antimono_def: fixes p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and DF :: "floatarith \<Rightarrow> floatarith"   and l :: "float"   and u :: "float"   and X :: "(real \<times> real pdevs) \<times> real" shows "min_range_antimono p F DF l u X = (let L = Num l; U = Num u in approx p (Max (DF L) (DF U)) [] \<bind> (\<lambda>aivl. let a = upper aivl; A = Num a in approx p (Half (F L + F U - A * (L + U))) [] \<bind> (\<lambda>bivl. let (b, be) = mid_err bivl; (B, Be) = (Num (float_of b), Num (float_of be)) in approx p (Add (Half (F L - F U + A * (U - L))) Be) [] \<bind> (\<lambda>divl. Some (affine_unop p (real_of_float a) b (real_of_float (upper divl)) X)))))"
le_lt_convs_2: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y \<noteq> Gt) = le_of_comp comp x y"
rel_S_pos_distr_cond_def: fixes Co1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Co1' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and Contra1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra2 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and uu :: "('n \<times> 'o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't) itself" shows "rel_S_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1' L2 L2'. rel_S L1 L2 Co1 Co2 Contra1 Contra2 OO rel_S L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_S (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2'))"
grounding_set_def: fixes S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list" shows "grounding_set S \<sigma> = (\<forall>x. x \<in> S \<longrightarrow> ground_clause (subst_cl (cl_ecl x) \<sigma>))"
algorithmic_weak_head_closure: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and s' :: "trm"   and t' :: "trm" assumes "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T"   and "s' \<leadsto> s"   and "t' \<leadsto> t" shows "\<Gamma> \<turnstile> s' \<Leftrightarrow> t' : T"
LeastI2_wellorder: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
AE_T_alw: fixes P :: "'s stream \<Rightarrow> bool"   and x :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P"   and "\<And>s. (x, s) \<in> acc \<Longrightarrow> almost_everywhere (T s) P" shows "almost_everywhere (T x) (alw P)"
total: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlo le" shows "le x y \<or> le y x"
vconstructor_value_def: fixes C_info :: "(name, dt_def) fmap" shows "constructors.vconstructor_value C_info \<equiv> value_pred.pred (\<lambda>_ _. True) (\<lambda>name. name |\<in>| constructors.C C_info) (\<lambda>_. True)"
init_fin_bisim_wait_def: shows "init_fin_bisim_wait = (\<lambda>(status1, x1) (status2, x2). status1 = Running \<and> status2 = Running \<and> x1 \<approx>w x2)"
untime_length_def: fixes s :: "'b message_af list" shows "untime_length s \<equiv> length (untime s)"
dghm_up_ObjMap_vdomain: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" shows "\<D>\<^sub>\<circ> (dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>) = \<CC>\<lparr>Obj\<rparr>"
a: shows "adjoint (2 \<cdot>\<^sub>m proj_psi) = 2 \<cdot>\<^sub>m proj_psi"
le_Err: fixes e :: "'b err"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "e \<sqsubseteq>\<^bsub>Err.le r\<^esub> Err"
q2_sat_props: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p (drop (length qs div 2) qs) (get_subsets_R (calculate_data_R p (drop (length qs div 2) qs))) (get_signs_R (calculate_data_R p (drop (length qs div 2) qs))) (get_matrix_R (calculate_data_R p (drop (length qs div 2) qs)))"
union_lcomm: fixes M :: "'d multiset"   and N :: "'d multiset"   and K :: "'d multiset" shows "M + (N + K) = N + (M + K)"
sup_ge1: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) x (y::'b::type))"
mult_zero_left: fixes x :: "'a" assumes "x \<in> V" shows "0 \<cdot> x = (0::'a)"
supp_eqvt_1: fixes pi :: "(tyvrs \<times> tyvrs) list"   and x :: "'a" shows "pi \<bullet> supp x = supp (pi \<bullet> x)"
dvd_content_dvd: fixes f :: "'a poly"   and g :: "'a poly" assumes "content f dvd content g"   and "primitive_part f dvd primitive_part g" shows "f dvd g"
le_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "le r = list_all2 (\<lambda>x. lesub x r)"
int_euclid_pyth_triples: fixes a :: "int"   and b :: "int"   and c :: "int" assumes "coprime a b"   and "odd a"   and "a\<^sup>2 + b\<^sup>2 = c\<^sup>2" shows "\<exists>p q. a = p\<^sup>2 - q\<^sup>2 \<and> b = 2 * p * q \<and> \<bar>c\<bar> = p\<^sup>2 + q\<^sup>2 \<and> coprime p q"
cases: fixes a :: "'a set" assumes "finite a" obtains "a = {}"   | A :: "'a set" and aa :: "'a" where "a = insert aa A" and "finite A"
is_universal: fixes a' :: "'b"   and \<chi>' :: "'j \<Rightarrow> 'b" assumes "\<chi>_cod_a.D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow>\<^sub>B l (A.cod a)\<guillemotright> \<and> \<chi>_cod_a.D.cones_map f (\<chi> (A.cod a)) = \<chi>'"
to_int_mod_ring_plus: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" shows "to_int_mod_ring (x + y) = M (to_int_mod_ring x + to_int_mod_ring y)"
v_subset_carrier: shows "v \<subseteq> carrier\<^bsub>T\<^esub>"
characteristic_iso_2: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>\<iota> \<star> f : (b \<star> b) \<star> f \<Rightarrow>\<^sub>S b \<star> f\<guillemotright>"
vars_u_s: shows "vars_mset s \<subseteq># vars_mset u"
\<gamma>_aboveI: fixes i :: "int"   and iv :: "ivl"   and j :: "int" assumes "i \<in> \<gamma>_ivl iv"   and "i \<le> j" shows "j \<in> \<gamma>_ivl (Abs_Int2_ivl.above iv)"
ggb: fixes x :: "'b" assumes "x \<in> C" shows "g x = gb x"
and_not_mask: fixes w :: "'b word"   and n :: "nat" shows "w AND NOT (mask n) = w >> n << n"
IH: fixes i :: "nat" assumes "(a, b) \<in> set (arcs i j xs)"   and "a \<noteq> i" shows "\<exists>c. xs = [a] \<or> (\<exists>ys. xs = a # b # ys) \<or> (\<exists>ys. xs = ys @ [c, a] \<and> b = j) \<or> (\<exists>ys zs. xs = ys @ c # a # b # zs)"
lideal: fixes r :: "'a"   and a :: "'a" assumes "r \<in> R"   and "a \<in> I" shows "r \<cdot> a \<in> I"
floor_log_add_eqI: fixes b :: "nat"   and a :: "nat"   and r :: "real" assumes "1 < b"   and "1 \<le> a"   and "0 \<le> r"   and "r < 1" shows "\<lfloor>log (real b) (real a + r)\<rfloor> = \<lfloor>log (real b) (real a)\<rfloor>"
pcp_alt: fixes C :: "'b formula set set" shows "pcp (C::'b formula set set) = (\<forall>S::'b formula set\<in>C. \<bottom> \<notin> S \<and> (\<forall>k::'b. Atom k \<in> S \<longrightarrow> \<^bold>\<not> (Atom k) \<in> S \<longrightarrow> False) \<and> (\<forall>(F::'b formula) (G::'b formula) H::'b formula. Con F G H \<longrightarrow> F \<in> S \<longrightarrow> G \<triangleright> H \<triangleright> S \<in> C) \<and> (\<forall>(F::'b formula) (G::'b formula) H::'b formula. Dis F G H \<longrightarrow> F \<in> S \<longrightarrow> G \<triangleright> S \<in> C \<or> H \<triangleright> S \<in> C))"
Bad0_meas: fixes n :: "nat" shows "Bad0 n \<in> sets M"
leg0_simps_3: shows "trg\<^sub>C f = src\<^sub>C r"
hyps_3: fixes f :: "'a \<Rightarrow> 'b" assumes "S = edom f" shows "fv f \<subseteq> edom f"
p: shows "g \<turnstile> Entry g-ns\<rightarrow>n''"
dvd_div_mult: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b" shows "times (divide b c) a = divide (times b a) c"
that: fixes g :: "'a \<Rightarrow> complex" assumes "continuous_on S g"   and "\<And>x. x \<in> S \<Longrightarrow> f x = (g x)\<^sup>2" shows "thesis"
\<pi>: shows "\<pi> \<in> (I \<rightarrow> Univ) \<inter> extensional' I \<rightarrow> Univ \<and> inj_on \<pi> ((I \<rightarrow> Univ) \<inter> extensional' I)"
times_u_def: fixes x :: "'a\<^sub>\<bottom>"   and y :: "'a\<^sub>\<bottom>" shows "x * y = (case x of up\<cdot>a \<Rightarrow> \<Lambda> (up\<cdot>b). up\<cdot>(a * b))\<cdot>y"
inf_def: fixes cs :: "com list"   and s :: "state" shows "PsHoareTotal.inf cs s = (\<exists>f. f 0 = (cs, s) \<and> (\<forall>i. f i \<rightarrow> f (Suc i)))"
min_max_for_sol: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "c \<bullet> x = b \<bullet> y"
IH_2: assumes "is_cupcake_all_env (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env)"   and "is_cupcake_exp e'" shows "s2 = fst bv"
pt_list_prm_eq: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and xs :: "'a list" assumes "pt TYPE('a) TYPE('b)"   and " pi1 \<triangleq> pi2 " shows "pi1 \<bullet> xs = pi2 \<bullet> xs"
vec_of_basis_enum_inverse: fixes w :: "'a" shows "basis_enum_of_vec (vec_of_basis_enum w) = w"
I_def: fixes insert_message :: "('c, 'd) insert_message" shows "I insert_message = (case insert_message of InsertMessage x1 x2 x3 x4 \<Rightarrow> x2)"
subset: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme" assumes "subgroup H G" shows "H \<subseteq> carrier G"
pointwise_simp: fixes f :: "'a set \<Rightarrow> 'b set"   and X :: "'a set" assumes "pointwise f" shows "f X = \<Union> {f {x} |x. x \<in> X}"
that: fixes p' :: "'a list" assumes "set p' \<subseteq> carrier R"   and "p = p' @ [a]" shows "thesis"
abelian_monoidE_1: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
Hintikka_def: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set" shows "Hintikka A H \<equiv> (((\<forall>i j x. Nom j at i in' H \<longrightarrow> Pro x at j in' H \<longrightarrow> \<not> (\<^bold>\<not> Pro x) at i in' H) \<and> (\<forall>i a. Nom a at i in' H \<longrightarrow> \<not> (\<^bold>\<not> Nom a) at i in' H)) \<and> (\<forall>i p. (\<^bold>\<not> \<^bold>\<not> p) at i in' H \<longrightarrow> p at i in' H) \<and> (\<forall>i p q. (p \<^bold>\<or> q) at i in' H \<longrightarrow> p at i in' H \<or> q at i in' H) \<and> (\<forall>i p q. (\<^bold>\<not> (p \<^bold>\<or> q)) at i in' H \<longrightarrow> (\<^bold>\<not> p) at i in' H \<and> (\<^bold>\<not> q) at i in' H)) \<and> ((\<forall>p i. (\<nexists>a. p = Nom a) \<longrightarrow> (\<^bold>\<diamond> p) at i in' H \<longrightarrow> (\<exists>j. (\<^bold>\<diamond> Nom j) at i in' H \<and> (\<^bold>@ j p) at i in' H)) \<and> (\<forall>i p j. (\<^bold>\<not> (\<^bold>\<diamond> p)) at i in' H \<longrightarrow> (\<^bold>\<diamond> Nom j) at i in' H \<longrightarrow> (\<^bold>\<not> (\<^bold>@ j p)) at i in' H) \<and> (\<forall>a i p. (\<^bold>@ i p) at a in' H \<longrightarrow> p at i in' H)) \<and> (\<forall>a i p. (\<^bold>\<not> (\<^bold>@ i p)) at a in' H \<longrightarrow> (\<^bold>\<not> p) at i in' H) \<and> (\<forall>i p. i \<in> nominals p \<longrightarrow> (\<exists>a. p at a in' H) \<longrightarrow> (\<exists>ps. (ps, i) \<in> H)) \<and> (\<forall>p i j. (\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A) \<longrightarrow> p at i in' H \<longrightarrow> Nom j at i in' H \<longrightarrow> p at j in' H)"
canonical_factorization: fixes t :: "'c local.term" assumes "Arr t" shows "coherent t = (\<lbrace>t\<rbrace> = D.inv \<lbrace>Cod t\<^bold>\<down>\<rbrace> \<cdot>\<^sub>D \<lbrace>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<rbrace> \<cdot>\<^sub>D \<lbrace>Dom t\<^bold>\<down>\<rbrace>)"
simps_1: fixes w :: "nat"   and a :: "'a"   and b :: "'a" shows "freq (Leaf w a) b = (if b = a then w else 0)"
simps_3: fixes uv :: "int" shows "(Any \<sqsubseteq> Const uv) = False"
trms_subsumes_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'c eclause"   and D :: "'c eclause"   and \<sigma> :: "('c \<times> 'c trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "trms_subsumes C D \<sigma> = (subst_cl (cl_ecl C) \<sigma> = cl_ecl D \<and> subst_set (trms_ecl C) \<sigma> \<subseteq> trms_ecl D)"
typing_inversion_7: fixes \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes "\<Gamma> \<turnstile> TRUE : T" obtains "T = tBOOL"
diamond_cube_valid_two_cube: shows "valid_two_cube diamond_cube"
Infinitesimal_add_hrabs_hypreal_of_real_less: fixes x :: "real star"   and r :: "real"   and y :: "real" assumes "x \<in> Infinitesimal"   and "\<bar>hypreal_of_real r\<bar> < hypreal_of_real y" shows "\<bar>hypreal_of_real r + x\<bar> < hypreal_of_real y"
mirror_ap_tree: fixes f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "mirror (f \<diamondop> x) = mirror f \<diamondop> mirror x"
indRelLTPO_impl_TRel_reflects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelLTPO TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
nat_pow_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "x [^] n \<in> carrier R"
quotient_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "A // r = (\<Union>x\<in>A. {r `` {x}})"
hom_0_iff: fixes x :: "'d" shows "([:x:] = 0) = (x = (0::'d))"
avl_mkt: fixes l :: "'b tree"   and r :: "'b tree"   and x :: "'b" assumes "avl l"   and "avl r"   and "height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1" shows "avl (mkt x l r)"
inject_1: fixes x3 :: "name"   and x2 :: "name"   and x1 :: "pi"   and y3 :: "name"   and y2 :: "name"   and y1 :: "pi" shows "(x3{x2}.x1 = y3{y2}.y1) = (x3 = y3 \<and> x2 = y2 \<and> x1 = y1)"
Un: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a \<union> b \<in> fmeasurable M"
ide_in_iso_class: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C f" shows "f \<in> category.iso_class C f"
simps_1: fixes t :: "('sequent \<times> 'rule) tree"   and a :: "'a" shows "shd (konigDtree t a) = (tree.root t, a)"
right_lang_def: fixes q :: "hf" shows "Power.right_lang q \<equiv> {u. Power.nextl q u \<in> dfa.final Power_dfa}"
subset_empty: fixes A :: "'e set" shows "(A \<subseteq> {}) = (A = {})"
aperiodic_def: fixes C :: "('s \<times> 's option) set" shows "KN.aperiodic C = (C \<in> UNIV // KN.communicating \<and> KN.period C = 1)"
transpose_of_jozsa_transform: shows "U\<^sub>f\<^sup>t = U\<^sub>f"
clauseTrueAndClauseFalseImpliesInconsistent: fixes clause :: "Literal list"   and valuation :: "Literal list" assumes "clauseTrue clause valuation"   and "clauseFalse clause valuation" shows "inconsistent valuation"
iso_is_arr: fixes f :: "'a1" assumes "C1.iso f" shows "C1.arr f"
m: shows "n \<in> {idx m..<idx (Suc m)}"
FactRing_iso_set: assumes "h ` carrier R = carrier S" shows "(\<lambda>X. the_elem (h ` X)) \<in> ring_iso (R Quot a_kernel R S h) S"
traces_dom_events: fixes u :: "'a"   and D :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and P :: "'b process" assumes "u \<in> range D" shows "(xs \<in> traces P) = (next_dom_events P D u xs \<union> ref_dom_events P D u xs \<noteq> {})"
MapsToE: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and X :: "'g"   and Y :: "'g" assumes "f maps\<^bsub>C\<^esub> X to Y" obtains "f \<in> mor\<^bsub>C\<^esub>" and "dom\<^bsub>C\<^esub> f = X" and "cod\<^bsub>C\<^esub> f = Y"
subgroup_adjoin_induct: fixes H0 :: "'a set"   and P :: "('a, 'b) monoid_scheme \<Rightarrow> bool" assumes "subgroup H0 (G\<lparr>carrier := H'\<rparr>)"   and "P (G\<lparr>carrier := H', carrier := H0\<rparr>)"   and "\<And>H a. \<lbrakk>subgroup H (G\<lparr>carrier := H'\<rparr>); H0 \<subseteq> H; a \<in> carrier (G\<lparr>carrier := H'\<rparr>) - H; P (G\<lparr>carrier := H', carrier := H\<rparr>)\<rbrakk> \<Longrightarrow> P (G\<lparr>carrier := H', carrier := adjoin (G\<lparr>carrier := H'\<rparr>) H a\<rparr>)" shows "P (G\<lparr>carrier := H'\<rparr>)"
naturality: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "A.map (J.cod f) \<cdot> A.map f = A.map f \<cdot> A.map (J.dom f)"
quasi_measure_preserving_def: fixes M :: "'b measure"   and N :: "'c measure" shows "quasi_measure_preserving M N = {f \<in> M \<rightarrow>\<^sub>M N. \<forall>A\<in>sets N. (f -` A \<inter> space M \<in> null_sets M) = (A \<in> null_sets N)}"
FL_valid_Tree\<^sub>\<alpha>_def: shows "FL_valid_Tree\<^sub>\<alpha> \<equiv> (id ---> rep_Tree\<^sub>\<alpha> ---> id) FL_valid_Tree"
swapWatchesEffect_1: fixes clause :: "nat"   and state :: "State"   and clause' :: "nat" shows "getWatch1 (swapWatches clause state) clause' = (if clause = clause' then getWatch2 state clause' else getWatch1 state clause')"
distinct_257: fixes x11a :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Sqrt x11a \<noteq> x151 ^\<^sub>e x152"
cases: fixes x :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set" obtains N :: "'a literal multiset set" and P :: "'a literal multiset set" and Q :: "'a literal multiset set" where "x = (N, P, Q)"
mcont_sup2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<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 y :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "mcont Sup less_eq Sup less_eq (\<lambda>x. sup x y)"
state_successors_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "state_successors PROB s \<equiv> state_succ s ` PROB - {s}"
bij_lift_map: fixes i :: "nat" shows "bij (lift_map i)"
add_diff_cancel_left: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and c :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
bernoulli_Not: fixes r :: "real" shows "map_pmf Not (bernoulli_pmf r) = bernoulli_pmf (1 - r)"
simps_2: fixes p :: "fm"   and q :: "fm" shows "islin (Or p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
cupcake_nsBind_preserve: fixes ns :: "(char list, char list, v) namespace"   and v0 :: "v"   and k :: "char list" assumes "is_cupcake_ns ns"   and "is_cupcake_value v0" shows "is_cupcake_ns (nsBind k v0 ns)"
T_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "\<F>C.tensor f g = f \<otimes> g"
eSum_in_SubmoduleTr: fixes H :: "'a set"   and K :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "H \<subseteq> carrier M"   and "K \<subseteq> H" shows "f \<in> {j. j \<le> n} \<rightarrow> K \<and> s \<in> {j. j \<le> n} \<rightarrow> carrier R \<longrightarrow> l_comb R (mdl M (fgs R M K)) n s f = l_comb R M n s f"
vdomain_atE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains b :: "V" where "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" and "r\<lparr>a\<rparr> = b"
power2_eq_square: fixes x :: "'a" assumes "x \<in> carrier R" shows "x [^] 2 = x \<otimes> x"
l4_ports_negate_one_not_has_disc_negated_generic: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and ports :: "ipt_l4_ports" assumes "\<forall>a. \<not> disc (Prot a)" shows "\<not> has_disc_negated disc False (l4_ports_negate_one C ports)"
add_vec_span: fixes us :: "'a vec list"   and n :: "nat"   and j :: "nat"   and i :: "nat"   and c :: "'a" assumes "set us \<subseteq> carrier_vec n"   and "j < length us"   and "i < length us"   and "i \<noteq> j" shows "LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (set us) = LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (set (us[i := us ! i + c \<cdot>\<^sub>v us ! j]))"
subseq_filter: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "subseq xs ys" shows "subseq (filter P xs) (filter P ys)"
fsmultD': fixes a :: "'f"   and v :: "'w" shows "a \<sharp>\<star> v = (a \<delta>\<delta> (0::'g)) \<star> v"
cong_m: shows "[a ^ (m * c) = 1] (mod b)"
induct: fixes P :: "instr array \<Rightarrow> integer \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "instr array"   and a1 :: "integer"   and a2 :: "nat" assumes "\<And>ins s pc. \<lbrakk>\<And>x21 x22. \<lbrakk>pc < array_length ins; array_get ins pc = TestI x21 x22; bval x21 s\<rbrakk> \<Longrightarrow> P ins s (pc + 1); \<And>x21 x22 x. \<lbrakk>pc < array_length ins; array_get ins pc = TestI x21 x22; \<not> bval x21 s; x = nat (int (pc + 1) + x22); pc < x\<rbrakk> \<Longrightarrow> P ins s x; \<And>x3 x xa. \<lbrakk>pc < array_length ins; array_get ins pc = ChoiceI x3; x = concat (map (\<lambda>(b, i). if bval b s then [nat (int (pc + 1) + i)] else []) x3); x \<noteq> []; xa \<in> set x; pc < xa\<rbrakk> \<Longrightarrow> P ins s xa; \<And>x4 x. \<lbrakk>pc < array_length ins; array_get ins pc = GotoI x4; x = nat (int (pc + 1) + x4); pc < x\<rbrakk> \<Longrightarrow> P ins s x\<rbrakk> \<Longrightarrow> P ins s pc" shows "P a0 a1 a2"
comp_null_2: fixes f :: "'a" shows "R.comp f R.null = R.null"
span_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "ideal A \<subseteq> ideal B"
unaffected_domains_single_dom: fixes X :: "'a set"   and D :: "'a \<Rightarrow> 'b"   and I :: "('b \<times> 'b) set"   and u :: "'b"   and xs :: "'a list" shows "{x \<in> X. D x \<in> unaffected_domains I D {u} xs} = ipurge_ref I D u xs X"
vector_covector: fixes v :: "'a"   and w :: "'a" assumes "vector v"   and "vector w" shows "v \<sqinter> w\<^sup>T = v * w\<^sup>T"
ctr_simps_subset_pow_iff'': fixes S :: "'a set set"   and U :: "'a set" shows "(S \<subseteq> {S. S \<subseteq> U}) = (S \<subseteq> Pow U)"
finite_hfset: fixes a :: "hf" shows "finite (hfset a)"
lM: shows "\<forall>t\<in>U. linorder.Min (\<sqsubseteq>) U \<sqsubseteq> t"
path_connected_Times: fixes s :: "'a set"   and t :: "'b set" assumes "path_connected s"   and "path_connected t" shows "path_connected (s \<times> t)"
bw_diff_gbinomial_pow: fixes l :: "nat"   and n :: "int"   and k :: "nat" shows "(bw_diff ^^ l) (\<lambda>x. x + n gchoose k) = (if l \<le> k then \<lambda>x. x + n - int l gchoose (k - l) else (\<lambda>_. 0))"