hcju/isabelleps huggingface dataset:
p3p2: fixes Q :: "'c \<Rightarrow> real" assumes "unitary Q" shows "\<guillemotleft> I \<guillemotright> && wp B Q = wp B (\<guillemotleft> H \<guillemotright> && Q)"
row_to_poly_vec_plus: fixes ts :: "'c list"   and n :: "nat"   and f1 :: "nat \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'd" assumes "distinct ts"   and "length ts = n" shows "row_to_poly ts (vec n (f1 + f2)) = row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)"
pathstart_linepath: fixes a :: "'b"   and b :: "'b" shows "pathstart (linepath a b) = a"
deq: shows "|~ Deq ic q oc \<longrightarrow> BInv ic q oc \<longrightarrow> \<circle>BInv ic q oc"
False: shows "\<not> (arg z1 \<noteq> pi \<and> arg z2 \<noteq> pi)"
reds_hext_incr: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and es :: "'addr expr list"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and es' :: "'addr expr list"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA,P,t \<turnstile> \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>" shows "hp s \<unlhd> hp s'"
matchNil: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "[a\<frown>b]P \<sim> \<zero>"
commute_left_comp: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and g :: "'a \<Rightarrow> ('b, 'c set) mapping" shows "insert_mapping y \<circ> (insert_mapping x \<circ> g) = insert_mapping x \<circ> (insert_mapping y \<circ> g)"
fmrel_cases: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'f \<rightharpoonup>\<^sub>f 'd"   and n :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" assumes "fmrel P m n" obtains "fmlookup m x = None" and "fmlookup n x = None"   | a :: "'d" and b :: "'e" where "fmlookup m x = Some a" and "fmlookup n x = Some b" and "P a b"
mult_pos_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less zero b" shows "less zero (times a b)"
cf_h_special_cf: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g" assumes "PolynRg A B Y"   and "h \<in> rHom S B" shows "polyn_expr A Y (Suc 0) (cf_h h (ext_cf S (Suc 0) (C\<^sub>0 S))) = polyn_expr A Y (Suc 0) (ext_cf B (Suc 0) (C\<^sub>0 B))"
nat_ceiling_le_eq: fixes x :: "real"   and a :: "nat" shows "(nat \<lceil>x\<rceil> \<le> a) = (x \<le> real a)"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "append (Cons x xs) ys = Cons x (append xs ys)"
function_smult_r_distr: fixes a :: "'a"   and x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "x \<in> carrier F"   and "y \<in> carrier F" shows "a \<odot>\<^bsub>F\<^esub> (x \<oplus>\<^bsub>F\<^esub> y) = a \<odot>\<^bsub>F\<^esub> x \<oplus>\<^bsub>F\<^esub> a \<odot>\<^bsub>F\<^esub> y"
distinct_7: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and x51 :: "floatarith"   and x52 :: "floatarith"   and x53 :: "floatarith" shows "Bound x11 x12 x13 x14 \<noteq> AtLeastAtMost x51 x52 x53"
same_fw_behaviour_one_equi_3: fixes x :: "'a word"   and y :: "'a word"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list"   and z :: "'a word" assumes "same_fw_behaviour_one x y c rs \<and> same_fw_behaviour_one y z c rs" shows "same_fw_behaviour_one x z c rs"
intros_3: fixes G :: "'b formula"   and F :: "'b formula" shows "G \<^bold>\<rightarrow> F \<^bold>\<or> G \<in> AX10"
finfun_fst_const: fixes bc :: "'b \<times> 'c" shows "finfun_fst (finfun_const bc) = finfun_const (fst bc)"
swap: fixes g :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and B :: "'d set"   and A :: "'c set" shows "(\<Prod>i\<in>A. prod (g i) B) = (\<Prod>j\<in>B. \<Prod>i\<in>A. g i j)"
union_subseteq_elim: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<union> B \<subseteq> C" obtains "A \<subseteq> C" and "B \<subseteq> C"
dpd1: fixes x :: "'a" shows "x \<le> d x \<cdot> x"
finite_vimage_IntI: fixes F :: "'a set"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "finite F"   and "inj_on h A" shows "finite (h -` F \<inter> A)"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat list"   and a5 :: "nat" assumes "mtype(a1,a2,a3) = a4 \<rightarrow> a5" obtains CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and m :: "nat" and mDef :: "methodDef" and Bs :: "nat list" and B :: "nat" where "a1 = CT__" and "a2 = m__" and "a3 = C__" and "a4 = Bs__" and "a5 = B__" and "CT__ C__ = Some CDef__" and "lookup (cMethods CDef__) (\<lambda>md. mName md = m__) = Some mDef" and "varDefs_types (mParams mDef) = Bs__" and "mReturn mDef = B__"   | CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and m :: "nat" and D :: "nat" and Bs :: "nat list" and B :: "nat" where "a1 = CT" and "a2 = m" and "a3 = C" and "a4 = Bs" and "a5 = B" and "CT C = Some CDef" and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = None" and "cSuper CDef = D" and "mtype(CT,m,D) = Bs \<rightarrow> B"
odd_iff_lsb: fixes x :: "'b word" shows "odd (unat x) = x !! 0"
is_monomial_set_def: fixes A :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "is_monomial_set A = (\<forall>p\<in>A. is_monomial p)"
interrupt_actions_ok'_collect_interrupts_imp_interrupt_actions_ok: fixes "is" :: "'a set"   and ias :: "'a interrupt_action list"   and is' :: "'a set" assumes "interrupt_actions_ok' is ias"   and "collect_interrupts ias \<subseteq> is"   and "interrupt_actions_ok is' ias" shows "interrupt_actions_ok is ias"
abs_eq: fixes x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "\<^bold>\<box>make\<o> x = make\<o> (\<lambda>s w. \<forall>v. x s v)"
trimmed_imp_eventually_nz: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "trimmed F" shows "\<forall>\<^sub>F x in at_top. eval F x \<noteq> 0"
igWlsAbsIsInBar_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igWlsAbsIsInBar MOD \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> isInBar (us, s)"
Immediate: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G\<turnstile> mbr m declared_in C"   and "declclass m = C" shows "G \<turnstile> m member_of C"
jumpF_polyL_mult_cancel: fixes p' :: "real poly"   and q :: "real poly"   and p :: "real poly"   and a :: "real" assumes "p' \<noteq> 0" shows "jumpF_polyL (p' * q) (p' * p) a = jumpF_polyL q p a"
fun_type: fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "0 < arity f" shows "\<Gamma> (Fun f T) = Fun f (map \<Gamma> T)"
subgroup_int_pow_closed: fixes H :: "'a set"   and h :: "'a"   and k :: "int" assumes "subgroup H G"   and "h \<in> H" shows "h [^] k \<in> H"
permute_plus: fixes p :: "perm"   and q :: "perm"   and x :: "'a" shows "(p + q) \<bullet> x = p \<bullet> q \<bullet> x"
monoid_incl_imp_submonoid: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.monoid G"   and "H \<subseteq> carrier G"   and "Group.monoid (G\<lparr>carrier := H\<rparr>)" shows "submonoid H G"
monad_fail_writerT: assumes "monad_fail return bind fail" shows "monad_fail local.return_writer local.bind_writer local.fail_writer"
assms_2: fixes n :: "nat" shows "p n \<in> carrier R"
mem_imp_less_HMSet: fixes k :: "hmultiset"   and L :: "hmultiset multiset" assumes "k \<in># L" shows "k < HMSet L"
mid_in_real_interval: fixes i :: "float interval" shows "real_of_float (mid i) \<in>\<^sub>r i"
negatively_invariant_complI: fixes M :: "'a set" assumes "rev.positively_invariant M" shows "rev.negatively_invariant (X - M)"
assms_5: shows "S\<lparr>ArrCod\<rparr> = T\<lparr>ArrCod\<rparr>"
natural_isomorphism_axioms: shows "natural_isomorphism CCC.comp (\<cdot>) T.ToTC T.ToCT \<alpha>"
proots_count_emtpy: fixes p :: "'a poly" shows "proots_count p {} = 0"
sup_assoc: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (sup (x::'b) (y::'b)) (z::'b) = sup x (sup y z)"
wf_rexp_of'': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "WS1S_Equivalence_Checking.wf enum_class.enum n (rexp_of'' n \<phi>)"
E\<^sub>2rv: shows "Edges (r # between (vertices f\<^sub>2) r v @ [v]) = (if r = u then {} else Edges (r # between (vertices f) r u @ [u])) \<union> Edges (u # vs @ [v])"
RModuleHom_axioms: shows "RModuleHom FG (\<cdot>) V (\<star>) T"
sum_upto_mangoldt_le: fixes x :: "real" assumes "0 \<le> x" shows "sum_upto mangoldt x \<le> 3 / 2 * x"
path_image_join_subset: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" shows "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
frontier_change_zmset_frontier: fixes t :: "'a"   and M1 :: "'a zmultiset"   and M0 :: "'a zmultiset" assumes "t \<in>\<^sub>A frontier M1 - frontier M0" shows "zcount (zmset_of (mset_set (set_antichain (frontier M1)))) t = 1 \<and> zcount (zmset_of (mset_set (set_antichain (frontier M0)))) t = 0"
zero_in_PPs: fixes X :: "'c set" shows "0 \<in> .[X]"
continuous_within_Ln: fixes z :: "complex"   and S :: "complex set" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "continuous (at z within S) Ln"
unhindered_bipartite_web_of: assumes "loose (bipartite_web_of \<Gamma>)" shows "\<not> hindered (bipartite_web_of (bipartite_web_of \<Gamma>))"
fst_changed_by_recv_recording_trace: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "i < j"   and "j \<le> length t"   and "trace init t final"   and "fst (cs (S t i) cid) \<noteq> fst (cs (S t j) cid)"   and "channel cid = Some (p, q)" shows "\<exists>k\<ge>i. k < j \<and> (\<exists>p q u u' m. t ! k = Recv cid q p u u' m) \<and> snd (cs (S t k) cid) = Recording"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) trg trg trg"
assms: fixes x :: "V" assumes "x \<in> X" shows "Transset x"
uniform_limit_cong: fixes X :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'a filter"   and h :: "'b \<Rightarrow> 'c"   and i :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F y in F. \<forall>x\<in>X. f y x = g y x"   and "\<And>x. x \<in> X \<Longrightarrow> h x = i x" shows "uniform_limit X f h F = uniform_limit X g i F"
index_smult_vec_1: fixes i :: "nat"   and v :: "'a Matrix.vec"   and a :: "'a" assumes "i < dim_vec v" shows "(a \<cdot>\<^sub>v v) $ i = a * v $ i"
eout_knows_nonKS_s: fixes m :: "Secrets"   and A :: "specID" assumes "m \<notin> specSecrets A"   and "eout A (sE m)"   and "eoutKnowsECorrect A (sE m)" shows "knows A [sE m]"
circ_separate_5: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> x * x\<^sup>\<circ> * (x \<squnion> y)" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * y\<^sup>\<circ>"
is_interp_def: fixes I :: "('a, 'b, 'c) interp" shows "is_interp I \<equiv> \<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x) \<and> continuous_on UNIV (\<lambda>x. Blinfun (FunctionFrechet I i x))"
wfE_replace: fixes \<Delta> :: "binding list"   and B :: "binding"   and \<Gamma> :: "binding list"   and B' :: "binding" assumes "\<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B'"   and "is_TVarB B' = is_TVarB B" shows "\<Delta> @ B' \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
trm_ord_reduction_right: shows "\<forall>x1 x2 y. (x1, x2) \<in> trm_ord \<longrightarrow> (y \<cdot> x1, y \<cdot> x2) \<in> trm_ord"
Fin: fixes f :: "'a \<Rightarrow> nat" assumes "finite A"   and "inj_on f A" shows "thesis"
decode_plan_step_element_then_2: fixes k :: "nat"   and t :: "nat"   and op :: "'b strips_operator"   and \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "k < t"   and "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)" shows "\<A> (Operator k (index (\<Pi>\<^sub>\<O>) op))"
nth_coeffs_coeff: fixes i :: "nat"   and f :: "'b poly" assumes "i < length (coeffs f)" shows "coeffs f ! i = poly.coeff f i"
pcr_fpxs_def: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_fpxs T \<equiv> rel_fun (=) T OO cr_fpxs"
R0_def: shows "R0 = {(V, V'). V = [skip] \<and> V' = [skip]}"
mode_le: shows "mds\<^sub>2 \<sqsubseteq> mds\<^sub>1"
rComm1: fixes M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and P'' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "\<Psi> \<rhd> !P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''"   and "\<And>C. Prop C \<Psi> (!P) (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'')"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop C \<Psi> (P \<parallel> !P) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"
prems_6: shows "\<forall>c. (c \<in> set newWl \<or> c \<in> set (clause # Wl')) = (Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c)"
ct_prefix_nil_2: fixes x :: "ct"   and xs :: "ct list" shows "\<not> ct_prefix (x # xs) []"
test_restrictr: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "x \<cdot> p \<le> x"
preserves_reflects_arr: fixes f :: "'a" shows "L'.C.arr (H\<^sub>L a' f) = L'.C.arr f"
nb\<^sub>1\<^sub>5: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> dom ((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op'))"   and "(v, a) \<notin> dom (map_of (effect_to_assignments op'))" shows "(\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) (v, a) = Some (the (s v) = a)"
map_sel: fixes f1 :: "'f \<Rightarrow> 'b"   and f2 :: "'g \<Rightarrow> 'd"   and a :: "('f, 'g, 'e) gpv" shows "the_gpv (map_gpv f1 f2 a) = map_spmf (map_generat f1 f2 ((\<circ>) (map_gpv f1 f2))) (the_gpv a)"
l_set_shadow_root_get_shadow_root_axioms: shows "l_set_shadow_root_get_shadow_root type_wf set_shadow_root set_shadow_root_locs get_shadow_root get_shadow_root_locs"
hyps: fixes acc :: "(bool \<times> 'b \<times> 'c) list" shows "set acc \<subseteq> set (fold (\<lambda>a. case a of (ic, a) \<Rightarrow> case a of (q_in_bs, a) \<Rightarrow> case a of (p, q) \<Rightarrow> \<lambda>ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc)"
non_npD: fixes v :: "val"   and P :: "'a prog"   and h :: "heap"   and C :: "char list" assumes "v \<noteq> Null"   and "P,h \<turnstile> v :\<le> Class C" shows "\<exists>a C' fs. v = Addr a \<and> h a = \<lfloor>(C', fs)\<rfloor> \<and> P \<turnstile> C' \<preceq>\<^sup>* C"
step_sb: shows "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"
actionCases: fixes \<alpha> :: "'d action" obtains M :: "'d" and N :: "'d" where "\<alpha> = M__\<lparr>N__\<rparr>"   | M :: "'d" and xvec :: "name list" and N :: "'d" where "\<alpha> = M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>"   | "\<alpha> = \<tau>"
Hilbert_fun_eq_Hilbert_poly_plus_card: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and k :: "nat"   and d :: "nat" assumes "X \<noteq> {}"   and "valid_decomp X ps"   and "hom_decomp ps"   and "cone_decomp T ps"   and "standard_decomp k ps"   and "exact_decomp X 0 ps"   and "\<b> ps (Suc 0) \<le> d" shows "int (Hilbert_fun T d) = int (card {h. (h, {}) \<in> set ps \<and> poly_deg h = d}) + Hilbert_poly (\<b> ps) (int d)"
smult_unique_scalars: fixes vs :: "'v list"   and v :: "'v"   and a :: "'f" assumes "set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs"   and "v \<in> V" shows "(THE cs. length cs = length vs \<and> a \<cdot> v = cs \<bullet>\<cdot> vs) = map ((*) a) (THE cs. length cs = length vs \<and> v = cs \<bullet>\<cdot> vs)"
open_countable_basis_ex: fixes p :: "'d set \<Rightarrow> bool"   and B :: "'d set set"   and X :: "'d set" assumes "countable_basis p B"   and "p X" shows "\<exists>B'\<subseteq>B. X = \<Union> B'"
that: fixes C :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite C"   and "disjnt C (\<Union> \<G>)"   and "card C \<le> card \<F>"   and "continuous_on (\<Union> \<F> - C) g"   and "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T"   and "\<And>x. x \<in> \<Union> \<G> \<Longrightarrow> g x = f x" shows "thesis"
pathstart_flow_to_path: fixes x :: "'a"   and t :: "real"   and t' :: "real" shows "pathstart (flow_to_path x t t') = flow0 x t"
\<epsilon>'_def: shows "\<epsilon>' = \<epsilon> / 2"
FAILURES_def: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" shows "FAILURES P = fst P"
fmpred_add: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" assumes "fmpred P m"   and "fmpred P n" shows "fmpred P (m ++\<^sub>f n)"
inner_sum_left_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and b :: "'a"   and f :: "'a \<Rightarrow> real" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "b \<in> Basis" shows "inner (comm_monoid_add.sum plus zero (\<lambda>i. scaleR (f i) i) Basis) b = f b"
rep_eq: fixes x :: "('a, 'b) pp" shows "varnum_pp x = varnum {} (pp.mapping_of x)"
Units_inv_closed: fixes x :: "'a" assumes "x \<in> Units R" shows "inv x \<in> carrier R"
equivalence_is_left_adjoint: fixes f :: "'a" assumes "equivalence_map f" shows "is_left_adjoint f"
finprod_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a vec" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier_vec n"   and "f a \<in> carrier_vec n" shows "finsum V f (insert a F) = f a + finsum V f F"
nat_to_string_neq_nil: fixes n :: "nat" shows "nat_to_string n \<noteq> []"
Reception: fixes evsr :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" assumes "evsr \<in> yahalom"   and "Says A B X \<in> set evsr" shows "Gets B X # evsr \<in> yahalom"
unl_Rall: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and w :: "'c" shows "(\<forall>x. A x) w \<equiv> \<forall>x. A x w"
inv_comp_left_2: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.iso (CC.comp g f)"   and "CC.iso g" shows "CC.iso f"
cancel_separate: fixes x :: "'a"   and y :: "'a" assumes "coreflexive (x * y)" shows "x\<^sup>\<star> * y\<^sup>\<star> \<le> x\<^sup>\<star> \<squnion> y\<^sup>\<star>"
V01_learn_cons: shows "learn_cons \<phi> V\<^sub>0\<^sub>1 s\<^bsub>01\<^esub>"
continuous_on_eq_continuous_at: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open s" shows "continuous_on s f = (\<forall>x\<in>s. isCont f x)"
ide_char': fixes a :: "'a \<times> 'a" shows "VV.ide a = (VV.arr a \<and> (VV.dom a = a \<or> VV.cod a = a))"
extended_distance_set_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool" assumes "class.metric_space dist uniformity open" shows "metric_space.extended_distance_set dist = {z. metric_space.extended_distance dist z z = 0}"
map_id: fixes t :: "'b pmf" shows "map_pmf id t = t"
permutes_inverses_2: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "p permutes S" shows "inv p (p x) = x"
apply_guards_empty: fixes s :: "vname \<Rightarrow> value option" shows "apply_guards [] s"
e: shows "change_respecting (cms\<^sub>2 ! k, subst h (snd (mems ! k))) \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), mem\<^sub>h\<rangle> (differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k) g2"
simps_2: fixes x :: "'c"   and xs :: "'c llist" shows "lstream (LCons x xs) = Yield x xs"
try_swap_def: fixes p :: "'person"   and c\<^sub>1 :: "'color"   and c\<^sub>2 :: "'color"   and w :: "'person \<Rightarrow> 'color"   and x :: "'person" shows "local.try_swap p c\<^sub>1 c\<^sub>2 w x = (if c\<^sub>1 = blue \<or> c\<^sub>2 = blue \<or> x \<noteq> p then w x else Fun.swap c\<^sub>1 c\<^sub>2 id (w x))"
d_OUT_h_plus_not_infty: fixes i :: "nat"   and x :: "'v" shows "d_OUT (h_plus i) x \<noteq> \<top>"
RelS_def: shows "RelS = _Collect (S1, S2) ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> RelT)"
seqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "H.VV.arr f"   and "H.VV.arr g"   and "H.VV.dom g = H.VV.cod f" shows "H.VV.seq g f"
VVp: fixes v :: "'a"   and p :: "Player"   and W :: "'a set" assumes "v \<in> VV p"   and "\<exists>w. v \<rightarrow> w \<and> w \<in> attractor_inductive p W" shows "v \<in> attractor_inductive p W"
Sup_least: fixes Inf :: "'k set \<Rightarrow> 'k"   and Sup :: "'k set \<Rightarrow> 'k"   and inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bot :: "'k"   and top :: "'k"   and A :: "'k set"   and z :: "'k" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
plus_flow_simps: fixes G :: "('k, 'l) graph_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k"   and y :: "'k" shows "(f \<oplus>\<^bsub>G\<^esub> g) (x, y) = (if edge G x y then f (x, y) + g (x, y) - g (y, x) else 0)"