hcju/isabelleps huggingface dataset:
DSource_Sources: fixes S :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" assumes "S \<in> DSources i C" shows "Sources i S \<subseteq> Sources i C"
polyfun_eq_coeffs: fixes c :: "nat \<Rightarrow> 'a"   and n :: "nat"   and d :: "nat \<Rightarrow> 'a" shows "(\<forall>x. (\<Sum>i\<le>n. c i * x ^ i) = (\<Sum>i\<le>n. d i * x ^ i)) = (\<forall>i\<le>n. c i = d i)"
z_append: fixes xs :: "nat list"   and as :: "rat list"   and ys :: "nat list"   and bs :: "rat list" assumes "\<And>i. i \<in> set xs \<Longrightarrow> i < length as" shows "z (xs @ map ((+) (length as)) ys) (as @ bs) = z xs as * z ys bs"
AE_iff_measurable: fixes N :: "'b set"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "N \<in> sets M"   and "{x \<in> space M. \<not> P x} = N" shows "almost_everywhere M P = (emeasure M N = 0)"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> P"   and "valid P" shows "valid P'"
distinct_3: fixes x31 :: "com"   and x32 :: "com" shows "Skip \<noteq> x31;; x32"
linear: shows "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1::'a:])"
assm: shows "\<exists>va u. v=va--u \<and> (\<exists>v u. va=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)) \<and> (\<exists>v ua. u=v--ua \<and> (\<exists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)) \<and> (\<exists>v u. ua=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)))"
that: fixes steps :: "nat \<Rightarrow> nat"   and start :: "int" assumes "\<And>i. i < m \<Longrightarrow> 0 < steps i"   and "is_multi_arith_prog_on (l + 1) m start steps a b"   and "let f = multi_arith_prog m start steps in \<forall>c\<in>{0..<m} \<rightarrow> {0..l}. \<forall>s<m. (\<forall>j\<le>s. c j < l) \<longrightarrow> col (f c) = col (f (\<lambda>i. if i \<le> s then 0 else c i))" shows "thesis"
Arcsin_sin: fixes z :: "complex" assumes "\<bar>Re z\<bar> < pi / 2 \<or> \<bar>Re z\<bar> = pi / 2 \<and> Im z = 0" shows "Arcsin (sin z) = z"
isolated_verts_def: fixes G :: "('e, 'f) pre_digraph" shows "pre_digraph.isolated_verts G \<equiv> {v \<in> verts G. out_arcs G v = {}}"
set_spmf_I_e_d_1: fixes N :: "nat"   and e :: "nat"   and d :: "nat" assumes "((N, e), d) \<in> set_spmf I" shows "1 < e"
unowned_shared': fixes a :: "nat" assumes "\<forall>i<length ts. a \<notin> owned (ts ! i)" shows "a \<in> dom \<S>"
locally_faithful_def: shows "UP.locally_faithful \<equiv> \<forall>f g \<mu> \<mu>'. \<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright> \<and> \<guillemotleft>\<mu>' : f \<rightarrow>\<^sub>B g\<guillemotright> \<and> UP \<mu> = UP \<mu>' \<longrightarrow> \<mu> = \<mu>'"
add_independentS_doubleD: fixes A :: "'b set"   and B :: "'b set"   and b :: "'b"   and a :: "'b" assumes "add_independentS [A, B]"   and "b \<in> B"   and "a \<in> A"   and "a + b = (0::'b)" shows "a = (0::'b)"
lookup_focus_in_Polys: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" shows "lookup (focus X p) t \<in> P[- X]"
rett_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> \<not> snd (rett_instr instr s1) \<and> \<not> snd (rett_instr instr s2) \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (rett_instr instr s1)) \<and> t2 = snd (fst (rett_instr instr s2))" shows "low_equal t1 t2"
preserves_inv: fixes f :: "'b" assumes "D.iso f" shows "src\<^sub>D (D.inv f) = D.inv (src\<^sub>D f)"
ls_insls_map_of': fixes ls :: "('a \<times> 'b) list list"   and ld :: "('a \<times> 'b) list list"   and n :: "nat" assumes "ht_hash ls"   and "ht_distinct ls"   and "ht_hash ld"   and "ht_distinct ld"   and "1 < length ld"   and "n < length ls" shows "map_of (concat (ls_insls (ls ! n) ld)) ++ map_of (concat (take n ls)) = map_of (concat ld) ++ map_of (concat (take (Suc n) ls))"
that: fixes d :: "real" assumes "0 < d"   and "a + d *\<^sub>R l \<in> frontier S"   and "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> interior S" shows "thesis"
Mem_not_sym: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (x IN y) (insert (y IN x) H) \<turnstile> A"
cont2cont_lambda: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "\<And>y. cont (\<lambda>x. f x y)" shows "cont f"
scaleR_right_distrib: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>R (x + y) = a *\<^sub>R x + a *\<^sub>R y"
subset_UnE: fixes C :: "'d set"   and A :: "'d set"   and B :: "'d set" assumes "C \<subseteq> A \<union> B" obtains A' :: "'d set" and B' :: "'d set" where "A' \<subseteq> A" and "B' \<subseteq> B" and "C = A' \<union> B'"
tendsto_add_ereal_general1: fixes y :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
in_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> O[F](h)) = (g \<in> O[F](h))"
VectorSpaceHomSet_def: fixes fsmult :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set"   and fsmult' :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and W :: "'b set" shows "VectorSpaceHomSet fsmult V fsmult' W \<equiv> {T. VectorSpaceHom fsmult V fsmult' T} \<inter> {T. T ` V \<subseteq> W}"
cong_exp_iff_simps_6: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus num.One) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 q))) = False"
order_trans_rules_25: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
injectivity_subexpr2: fixes a :: "'b"   and A :: "'b set"   and P :: "'b set set"   and P' :: "'b set set"   and k' :: "nat" assumes "a \<notin> A"   and "insert {a} P = insert {a} P'"   and "(partition_on A P \<and> card P = k') \<and> partition_on A P' \<and> card P' = k'" shows "P = P'"
flow_initial_time: fixes x0 :: "'a" assumes "0 \<in> UNIV"   and "x0 \<in> X" shows "rev.flow0 x0 0 = x0"
dmu: shows "d\<mu>_impl_cost fs_init = (st, c)"
whatineed: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) v = length v - 2"
psiFreshSet_2: fixes X :: "name set"   and M :: "'a"   and I :: "('a, 'b, 'c) input" shows "X \<sharp>* M\<lparr>I = (X \<sharp>* M \<and> X \<sharp>* I)"
arcosh_eq_0_iff: fixes x :: "real" assumes "1 \<le> x" shows "(arcosh x = 0) = (x = 1)"
check_mult_def: fixes A :: "(nat, int mpoly) fmap"   and \<V> :: "nat set"   and p :: "nat"   and q :: "int mpoly"   and i :: "nat"   and r :: "int mpoly" shows "check_mult A \<V> p q i r = SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars q \<subseteq> \<V> \<and> vars r \<subseteq> \<V> \<and> the (fmlookup A p) * q - r \<in> More_Modules.ideal polynomial_bool)"
bisims1List1: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and es :: "'addr expr1 list" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e # es,h \<turnstile> (e' # es, xs) [\<leftrightarrow>] (stk, loc, pc, xcp)"
list_all_list_of_pdevsI: fixes X :: "'a pdevs"   and P :: "'a \<Rightarrow> bool" assumes "\<And>i. i \<in> pdevs_domain X \<Longrightarrow> P (pdevs_apply X i)" shows "list_all P (map snd (list_of_pdevs X))"
le_less_trans: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
astDom_def: shows "astDom \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> D"
fps_divide_unit: fixes g :: "'b fps"   and f :: "'b fps" assumes "g $ 0 \<noteq> (0::'b)" shows "f / g = f * inverse g"
statement53_existence: fixes p :: "((proj2, 4) vec, 2) vec" assumes "\<forall>i. proj2_no_3_Col (range (($) (p $ i)))" shows "\<exists>C. \<forall>j. apply_cltn2 (p $ 0 $ j) C = p $ 1 $ j"
complementTA_def: shows "complementTA \<equiv> \<lparr>ta_initial = Q - Qi, ta_rules = \<delta>\<rparr>"
simps_2: fixes v :: "'b"   and \<mu> :: "'b" shows "\<^bold>\<langle>v\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>v\<^bold>\<rangle>"
cases: fixes x :: "rat \<times> constraint" obtains r :: "rat" and cc :: "constraint" where "x = (r, cc)"
bezout_ring_class_def: shows "OFCLASS('b, bezout_ring_class) \<equiv> (OFCLASS('b, comm_ring_1_class) &&& class.bezout_ring_axioms (*) (+))"
that: fixes l :: "nat \<Rightarrow> real"   and u :: "nat \<Rightarrow> real" assumes "einterval a b = (\<Union>i. {l i..u i})"   and "incseq u"   and "decseq l"   and "\<And>i. l i < u i"   and "\<And>i. a < ereal (l i)"   and "\<And>i. ereal (u i) < b"   and "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a"   and "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b" shows "thesis"
rep_eq: fixes x :: "complex"   and xa :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "(*\<^sub>V) (x *\<^sub>C xa) = (\<lambda>xb. x *\<^sub>C (xa *\<^sub>V xb))"
distinct_7: fixes x1 :: "'e" shows "prot_atom.Atom x1 \<noteq> Bottom"
nqfree_nnf: fixes \<phi> :: "'a fm" assumes "qfree \<phi>" shows "nqfree (nnf \<phi>)"
set_intros_3: fixes x1 :: "'d" shows "x1 \<in> vars_term (Var x1)"
has_int': shows "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral I) {l..r}"
cone_decompD: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "cone_decomp T ps" shows "direct_decomp T (map cone ps)"
loop_orientation_symmetric: fixes y :: "'a"   and x :: "'a" assumes "loop_orientation x y" shows "symmetric x"
prems_2: fixes k :: "nat"   and h :: "heap" assumes "I k h"   and "i \<le> k"   and "k < j" shows "success (f k) h"
q_def: shows "q \<equiv> [:z0, complex_of_real r:]"
cut_less_cut_ge_ident: fixes t2 :: "'b"   and t1 :: "'b"   and I :: "'b set" assumes "t2 \<le> t1" shows "I \<down>< t1 \<union> I \<down>\<ge> t2 = I"
rGamma_reflection_complex: fixes z :: "complex" shows "rGamma z * rGamma (1 - z) = sin (complex_of_real pi * z) / complex_of_real pi"
assert_fun_dual: fixes p :: "'b" shows "(assert_fun p \<circ> \<top>) \<sqinter> dual_fun (assert_fun p) = assert_fun p"
Dom_Inv: fixes t :: "'a local.term" assumes "Can t" shows "Dom (Inv t) = Cod t"
simps_4: fixes v :: "'b poly"   and va :: "'b poly" shows "degreen (Add v va) = (\<lambda>m. 0)"
algebraic_imp_represents_irreducible_cf_pos: fixes x :: "'a" assumes "algebraic x" shows "\<exists>p. p represents x \<and> irreducible p \<and> 0 < lead_coeff p \<and> primitive p"
a_known_ptrs_def: fixes h :: "(_) heap" shows "local.a_known_ptrs h = (\<forall>ptr\<in>fset (object_ptr_kinds h). known_ptr ptr)"
ci_app: fixes x :: "('a, 'b) check_instr" shows "ci_app x \<in> Collect wf_ci"
gcd_0_left: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd zero a = normalize a"
approx_scaleR1: fixes a :: "real star"   and b :: "real"   and c :: "'a star" assumes "a \<approx> hypreal_of_real b"   and "c \<in> HFinite" shows "scaleHR a c \<approx> b *\<^sub>R c"
assms: fixes l :: "('a \<times> 'b) list" assumes "distinct (map fst l)"   and "sorted_wrt (key_rel R) l" shows "foldli l c f \<sigma> = result (map_of l)"
SeqStTermP_Zero: fixes s :: "name"   and k :: "name"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, v, i)"   and "atom k \<sharp> (v, i)" shows "{VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i Zero Zero (Var s) (Var k)))"
ipurge_tr_rev_aux_append_nil: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and xs :: "'c list"   and ys :: "'c list" shows "(ipurge_tr_rev_aux I D U (xs @ ys) = ipurge_tr_rev_aux I D U xs) = (ipurge_tr_rev_aux I D U ys = [])"
2: assumes "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs" shows "thesis"
arbitrary_intersection_of_inc: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "P S" shows "(arbitrary intersection_of P) S"
create_element_known_ptr: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r new_element_ptr" shows "known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M (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>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr)"
\<phi>1s: shows "set \<phi>1s \<subseteq> fmla"
6: shows "\<forall>x y. f (x \<sqinter> - g y) \<le> - - f x \<sqinter> - y \<and> g (y \<sqinter> - f x) \<le> - - g y \<sqinter> - x"
sconst_alt: fixes s :: "'b stream"   and x :: "'b" shows "(s = sconst x) = (sset s = {x})"
inner_elements_not_triv: fixes i :: "nat" assumes "i + 1 < length \<GG>"   and "0 < i" shows "\<GG> ! i \<noteq> {\<one>}"
continuous_on_id: fixes s :: "'a set" shows "continuous_on s (\<lambda>x. x)"
tauSupportDerivative: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'" shows "supp P' \<subseteq> supp P"
insert: fixes A :: "'d set"   and x :: "'d"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
simps_17: fixes v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "'b"   and ve :: "bexp"   and vf :: "'b acom"   and vg :: "'b" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> {vd} WHILE ve DO vf {vg}) = False"
trancl_def: shows "trancl \<equiv> \<lambda>r. {(xa, x). (\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ xa x}"
nsqn_def: shows "nsqn \<equiv> \<lambda>rt dip. case rt dip of None \<Rightarrow> 0 | Some r \<Rightarrow> nsqn\<^sub>r r"
lower_character_lift_character_2: fixes \<chi> :: "'a \<Rightarrow> complex"   and z :: "complex" assumes "\<chi> \<in> characters (G\<lparr>carrier := H\<rparr>)" shows "lift_character (\<chi>, z) a = z"
liminf_finite_then_bounded_below: fixes u :: "nat \<Rightarrow> real" assumes "- \<infinity> < liminf (\<lambda>x. ereal (u x))" shows "\<exists>C. \<forall>n. C \<le> u n"
prems: shows "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (msx,sx)"
lookupDef_to_\<alpha>n': fixes g :: "'g"   and n :: "'node"   and v :: "'var"   and val :: "'var \<times> 'node \<times> Def" assumes "lookupDef g n v = val"   and "n \<in> set (\<alpha>n g)" shows "defNode val \<in> set (\<alpha>n g)"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "fin_digraph_axioms G" shows "fin_digraph G"
singleton_sub_trivial_uniq: fixes x :: "'c"   and X :: "'c set" assumes "{x} \<subseteq> X"   and "trivial X" shows "x = the_elem X"
assms_12: shows "x0' + (s2' * h' * rkp') *\<^sub>R ode x0' \<in> Csafe"
lub_least: fixes S :: "'a set"   and L :: "'a" assumes "S \<subseteq> A"   and "L \<in> A"   and "\<forall>x\<in>S. (x, L) \<in> r" shows "(lub S cl, L) \<in> r"
R15_1: shows "pmf (sds R15) a = 1 / 2"
hyp2_S_z_non_zero: fixes p :: "proj2" assumes "p \<in> hyp2 \<union> S" shows "z_non_zero p"
assms_7: shows "coreflexive (q\<^sup>T * q \<sqinter> f\<^sup>T * f)"
cinner_sum_left_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and b :: "'c"   and f :: "'c \<Rightarrow> complex" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "b \<in> CBasis" shows "cinner (comm_monoid_add.sum plus zero (\<lambda>i. scaleC (f i) i) CBasis) b = cnj (f b)"
R'_def: shows "R' = {(x, y). (x, y) \<in> set_pmf p \<times> set_pmf q \<and> R x y}"
to_nat_is_inv: shows "the_inv_into {0..<CARD('a)} mod_type_class.from_nat = mod_type_class.to_nat"
coerce_def: shows "COERCE('c, 'd) \<equiv> PRJ('d) oo EMB('c)"
mig_lv_def: fixes a :: "'b list"   and b :: "'b list" shows "mig_lv a b = map2 mig_real a b"
countable_image: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "countable A" shows "countable (f ` A)"
Jacobi_mod: fixes n :: "int"   and m :: "int"   and a :: "int" assumes "n dvd m" shows "Jacobi (a mod m) n = Jacobi a n"
yun_monic_factorization_def: fixes Gcd :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"   and p :: "'a poly" shows "yun_gcd.yun_monic_factorization Gcd p = (let pp = pderiv p; u = Gcd p pp; b0 = p div u; c0 = pp div u in filter (\<lambda>(a, i). a \<noteq> 1) (yun_gcd.yun_factorization_main Gcd b0 c0 0 []))"
cases: fixes x :: "'b tree" obtains "x = \<langle>\<rangle>"   | xa :: "'b" and r :: "'b tree" where "x = \<langle>\<langle>\<rangle>, xa__, r__\<rangle>"   | v :: "'b tree" and va :: "'b" and vb :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>\<langle>v, va, vb\<rangle>, xa, r\<rangle>"
ND0aux2: fixes x :: "'b"   and y :: "'b"   and b :: "('b, 'c) Combinators list" assumes "DenyAllFromTo x y \<in> set b" shows "y \<in> set (net_list_aux b)"
set_less_aux_Compl_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and ao :: "'a option"   and s1 :: "'d"   and s2 :: "'e" shows "set_less_aux_Compl_fusion g1 g2 ao s1 s2 = set_less_aux_Compl ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
sat_models_finite: fixes P :: "'a ltln set" assumes "finite P" shows "finite {sat_models (abs_ltln\<^sub>P \<phi>) \<inter> Pow P |\<phi>. prop_atoms \<phi> \<subseteq> P}"
tdghm_NTMap_vrange: shows "\<R>\<^sub>\<circ> (\<NN>\<lparr>NTMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
not_eq_order_implies_strict: fixes a :: "'b word"   and b :: "'b word" assumes "a \<noteq> b"   and "b \<le>s a" shows "b <s a"
GreenThm_typeII_twoCube_1: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "green_typeII_cube i j twoC F" shows "integral (cubeImage twoC) (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} (boundary twoC)"
inv_equality: fixes y :: "'a \<Rightarrow> complex"   and x :: "'a \<Rightarrow> complex" assumes "y \<otimes>\<^bsub>Characters G\<^esub> x = \<one>\<^bsub>Characters G\<^esub>"   and "x \<in> carrier (Characters G)"   and "y \<in> carrier (Characters G)" shows "inv\<^bsub>Characters G\<^esub> x = y"
cupcake_nsLookup_preserve: fixes ns :: "(char list, char list, v) namespace"   and n :: "(char list, char list) id0"   and v0 :: "v" assumes "is_cupcake_ns ns"   and "nsLookup ns n = Some v0" shows "is_cupcake_value v0"
\<mu>_def: shows "\<mu> \<equiv> LEAST \<mu>. Ord \<mu> \<and> \<not> \<mu> \<le> h \<mu> \<and> \<mu> \<in> D"
Restrp_transitive: shows "transitive UNIV ((\<sqsubset>) \<restriction> A \<restriction> UNIV)"
IF1rel_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "IF1rel R = (BNF_Def.Grp (IF1in {(x, y). R x y}) (IF1map fst))\<inverse>\<inverse> OO BNF_Def.Grp (IF1in {(x, y). R x y}) (IF1map snd)"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c n p. (\<And>x. n = x \<Longrightarrow> P p) \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (v)\<^sub>p"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
counterex_reducing_inference_system_axioms: shows "counterex_reducing_inference_system Bot (\<Turnstile>) Inf I_of"
span_def: fixes S :: "'b set" shows "cspan S = csubspace hull S"
quotient_is_empty: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "(A // r = {}) = (A = {})"
abs_conv_abscissa_integral: fixes a :: "'a"   and f :: "'a fds" shows "abs_conv_abscissa (fds_integral a f) = abs_conv_abscissa f"
cases: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b" assumes "Domainp r a" obtains aa :: "'b" and b :: "'c" where "a = aa" and "r aa b"
diff_add_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.comm_monoid_diff plus minus zero" shows "minus a (plus a b) = zero"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(ENat x2 = ENat y2) = (x2 = y2)"
check_extension_l_def: fixes spec :: "'a"   and A :: "(nat, 'b) fmap"   and \<V> :: "char list set"   and i :: "nat"   and v :: "char list"   and p :: "(char list list \<times> int) list" shows "check_extension_l spec A \<V> i v p = (let b = i \<notin># dom_m A \<and> v \<notin> \<V> \<and> ([v], - 1) \<in> set p in if \<not> b then check_extension_l_dom_err i \<bind> (\<lambda>c. RETURN (error_msg i c)) else let p' = remove1 ([v], - 1) p; b = vars_llist p' \<subseteq> \<V> in if \<not> b then check_extension_l_new_var_multiple_err v p' \<bind> (\<lambda>c. RETURN (error_msg i c)) else mult_poly_full p' p' \<bind> (\<lambda>p2. let p' = map (\<lambda>(a, b). (a, - b)) p' in add_poly_l (p2, p') \<bind> (\<lambda>q. weak_equality_l q [] \<bind> (\<lambda>eq. if eq then RETURN CSUCCESS else check_extension_l_side_cond_err v p p' q \<bind> (\<lambda>c. RETURN (error_msg i c))))))"