hcju/isabelleps huggingface dataset:
vec_set_def: fixes v :: "'b vec" shows "set\<^sub>v v = ($) v ` {..<dim_vec v}"
complex_inverse: fixes a :: "real"   and b :: "real" shows "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
wt_ParT_notWtFT_finished: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and i :: "nat"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "properL cl"   and "n < length cl"   and "i < PL.brn (cl ! n)"   and "WtFT cl \<noteq> 1"   and "finished (cl ! n)" shows "PL.wt aval tval cval (ParT cl) s (PL.brnL cl n + i) = 0"
is_run_gbg: fixes m :: "'b" shows "graph_defs.is_run (to_gbg_ext m) = is_run"
list_order_append: fixes pre :: "('a \<times> 'a option) list"   and suf :: "('a \<times> 'a option) list"   and x :: "'a"   and y :: "'a" assumes "insert_ops (pre @ suf)"   and "list_order pre x y" shows "list_order (pre @ suf) x y"
C_3: shows "rep_Bonk_Schramm_extension z \<in> extended_distance_set"
4: shows "\<turnstile> DBSpec \<longrightarrow> WF(Deq mid q2 out)_vars"
adverse_Suc_not_hyp_change: fixes x :: "nat"   and z :: "nat"   and i :: "nat"   and j :: "nat" assumes "0 < x"   and "\<not> (s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x))" shows "local.adverse z i j (Suc x) \<down>= z"
compose_inv_into_id: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "compose A (restrict (inv_into A f) B) f = (\<lambda>x\<in>A. x)"
vderiv: shows " D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0<--<t + r / 2}"
dvd_power_same: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b"   and y :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times x y" shows "dvd.dvd times (power.power one times x n) (power.power one times y n)"
igSwapAllIPresIGWlsAllSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igSwapAllIPresIGWlsAllSTR MOD \<equiv> igSwapIPresIGWlsSTR MOD \<and> igSwapAbsIPresIGWlsAbsSTR MOD"
uedge_degen: fixes a :: "'b" shows "uedge (a, a) = {a}"
del_list_sorted2: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted (xs @ a # ys)"   and "x < a" shows "del_list x (xs @ a # ys) = del_list x xs @ a # ys"
yun_wrel_def: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" shows "yun_wrel F c f = (of_int_poly F = Polynomial.smult c f)"
update_mat_def: fixes A :: "'b mat"   and ij :: "nat \<times> nat"   and a :: "'b" shows "A |\<^sub>m ij \<mapsto> a = mat (dim_row A) (dim_col A) (\<lambda>ij'. if ij' = ij then a else A $$ ij')"
cmp_def: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "cmp g f = (\<lambda>x. g (f x))"
vec_mat_Tensor_length: fixes xs :: "int poly fract list"   and ys :: "int poly fract list list" shows "length (rat_poly.vec_mat_Tensor xs ys) = length ys"
dvd_iff_poly_eq_0: fixes c :: "'a"   and p :: "'a poly" shows "([:c, 1::'a:] dvd p) = (poly p (- c) = (0::'a))"
simps_5: fixes r :: "'d set"   and e\<^sub>1 :: "('d, 'e, 'f) ann"   and e\<^sub>2 :: "('d, 'e, 'f) ann" shows "pre (AnnBin r e\<^sub>1 e\<^sub>2) = r"
f: shows "f = (\<lambda>s. 3 * (nat (s ''a'') + 2))"
inv_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y \<otimes> x = \<one>"   and "x \<otimes> y' = \<one>"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "y' \<in> carrier R" shows "y = y'"
integrableD_3: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>"
f_2: shows "\<forall>x\<in>s. f x = (0::'b) \<longrightarrow> x = (0::'a)"
x2: shows "(\<Union>x\<in>R `` {x}. {R `` {x}}) = {R `` {x}}"
elims_2: fixes x :: "Formula.formula" assumes "is_constraint x" obtains t1 :: "Formula.trm" and t2 :: "Formula.trm" where "x = formula.Eq t1__ t2__"   | t1 :: "Formula.trm" and t2 :: "Formula.trm" where "x = formula.Less t1a__ t2a__"   | t1 :: "Formula.trm" and t2 :: "Formula.trm" where "x = formula.LessEq t1b__ t2b__"   | t1 :: "Formula.trm" and t2 :: "Formula.trm" where "x = formula.Neg (formula.Eq t1c__ t2c__)"   | t1 :: "Formula.trm" and t2 :: "Formula.trm" where "x = formula.Neg (formula.Less t1d__ t2d__)"   | t1 :: "Formula.trm" and t2 :: "Formula.trm" where "x = formula.Neg (formula.LessEq t1 t2)"
rel_refl: fixes Ra :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'e \<Rightarrow> 'd" assumes "\<And>x. Ra x x" shows "rel_fun (=) Ra x x"
w1: fixes b1 :: "'a"   and b2 :: "'a"   and y :: "'b" shows "h (b1 + b2) y = h b1 y + h b2 y"
cp_dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<S>\<A>\<L>\<A>\<R>\<Y>_at_pre: fixes X :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(X.salary@pre) \<tau> = (\<lambda>_. X \<tau>.salary@pre) \<tau>"
sup_idem: fixes x :: "'a" shows "x \<squnion> x = x"
null_rec_1: fixes x :: "'d"   and xs :: "'d list" shows "List.null (x # xs) = False"
holds_if_pred: fixes b :: "bool" shows "holds (if_pred b) = b"
that: fixes n :: "nat" assumes "m \<le> \<^bold>|r \<^sup>@ n\<^bold>|"   and "n \<noteq> 0" shows "thesis"
open_vimage: fixes s :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "open s"   and "continuous_on UNIV f" shows "open (f -` s)"
lax: shows "\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)"
fst_dominant_term_uminus: 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 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" shows "fst (dominant_term (uminus F)) = - fst (dominant_term F)"
foldr_conv_fold: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list" shows "foldr f xs = List.fold f (rev xs)"
stc_Infinitesimal_add_SComplex: fixes x :: "complex star"   and e :: "complex star" assumes "x \<in> SComplex"   and "e \<in> Infinitesimal" shows "stc (x + e) = x"
\<omega>_limit_set_in_compact_\<omega>_limit_set_contained: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "\<forall>y\<in>rev.\<omega>_limit_set x. rev.\<omega>_limit_set y \<subseteq> rev.\<omega>_limit_set x"
adds_term_refl: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_term pair_of_term v v"
mod_false: fixes h :: "heap \<times> nat set" shows "\<not> h \<Turnstile> false"
monotone_lconcat: shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) lconcat"
Fin: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and C1 :: "assigned"   and c2 :: "stmt"   and C2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1"   and "Env\<turnstile> B \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2"   and "nrm A = nrm C1 \<union> nrm C2"   and "brk A = (brk C1 \<Rightarrow>\<union>\<^sub>\<forall> nrm C2) \<Rightarrow>\<inter>  brk C2" shows "Env\<turnstile> B \<guillemotright>\<langle>c1 Finally c2\<rangle>\<guillemotright> A"
simps: fixes ap :: "abc_inst list" shows "tpairs_of ap = zip (map (start_of (layout_of ap)) [0..<length ap]) ap"
simps_4: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
l_get_dom_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
rel_interior_closed_segment: fixes a :: "'a"   and b :: "'a" shows "rel_interior (closed_segment a b) = (if a = b then {a} else open_segment a b)"
nc_c: shows "nc \<cdot> 1\<^sub>\<pi> = 1\<^sub>\<pi>"
left_def: fixes \<tau> :: "'a" shows "left \<tau> \<equiv> \<lambda>\<mu>. \<tau> \<star> \<mu> \<noteq> null"
sbintr_ge: fixes n :: "nat"   and w :: "int" shows "- (2 ^ n) \<le> signed_take_bit n w"
assms: shows "unifier\<^sub>t\<^sub>s \<sigma> ts"
rbt_lookup_rbt_insert: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt"   and k :: "'c"   and v :: "'d" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (ord.rbt_insert less k v t) = ord.rbt_lookup less t(k \<mapsto> v)"
0: shows "h (inv a) \<otimes>\<^bsub>S\<^esub> h a = \<one>\<^bsub>S\<^esub>"
eq_minus_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(a = uminus (divide b c)) = (if c \<noteq> zero then times a c = uminus b else a = zero)"
fold_remove1_split: fixes xs :: "'k list"   and f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and x :: "'k" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"   and "x \<in> set xs" shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
l_get_ancestors_def: fixes get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_get_ancestors get_ancestors \<equiv> (\<forall>node_ptr h. pure (get_ancestors node_ptr) h) \<and> (\<forall>h ptr. h \<turnstile> ok get_ancestors ptr \<longrightarrow> ptr |\<in>| object_ptr_kinds h) \<and> (\<forall>h ptr ancestors. h \<turnstile> get_ancestors ptr \<rightarrow>\<^sub>r ancestors \<longrightarrow> ptr \<in> set ancestors)"
awalk_simps_1: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and v :: "'b" assumes "wf_digraph G" shows "pre_digraph.awalk G u [] v = (u = v \<and> u \<in> verts G)"
eventually_subadditive_def: fixes u :: "nat \<Rightarrow> real"   and N0 :: "nat" shows "eventually_subadditive u N0 = (\<forall>m>N0. \<forall>n>N0. u (m + n) \<le> u m + u n)"
fds_const_zero: shows "fds_const (0::'a) = 0"
mon_sI: fixes n :: "'f"   and s :: "'f list"   and m :: "'g"   and fg :: "('f, 'h, 'i, 'g, 'j) flowgraph_rec_scheme" assumes "n \<in> set s"   and "m \<in> mon_n fg n" shows "m \<in> mon_s fg s"
lookup_proj_poly: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and k :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and t :: "'b" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.proj_poly term_of_pair k p) t = lookup p (term_of_pair (t, k))"
init1: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and w :: "'d list"   and k :: "nat"   and T :: "nat \<times> nat \<Rightarrow> 'c list" assumes "init G w k = Some T" shows "init' G w k = T"
doesnt_modify_def: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var" shows "doesnt_modify c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> mem x = mem' x \<and> dma mem x = dma mem' x)"
class_of_def: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "dl.class_of U s \<equiv> eq.Class (U, s)"
intro: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "CL cl"   and "CLF_axioms cl f" shows "CLF cl f"
"case": fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(case NondetT x of NondetT x \<Rightarrow> f x) = f x"
taut_test: fixes ifex_of :: "'a \<Rightarrow> 'b ifex"   and val :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'a" assumes "reduced_bdt_checkers ifex_of val" shows "taut_test ifex_of b = (\<forall>s. val b s)"
to_fun_monic_monom_diff: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "\<exists>c. c \<in> carrier R \<and> to_fun (monom P \<one> n) a \<ominus> to_fun (monom P \<one> n) b = (a \<ominus> b) \<otimes> c"
of_real_diff: fixes x :: "real"   and y :: "real" shows "of_real (x - y) = of_real x - of_real y"
signed_scast_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and w :: "'d word" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.signed one plus zero minus (scast w) = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (LENGTH('c) - Suc 0) (ring_1.signed one plus zero minus w)"
distinct_3: fixes pi' :: "pi" shows "\<zero> \<noteq> \<tau>.pi'"
eventually_prod_same: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and F :: "'b filter" shows "eventually P (F \<times>\<^sub>F F) = (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
even_add: fixes a :: "'a"   and b :: "'a" shows "even (a + b) = (even a = even b)"
xt1_3: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "order_ow U le ls"   and "a = b"   and "le c b" shows "le c a"
ssorted_monoD: fixes s :: "'b stream"   and i :: "nat"   and j :: "nat" assumes "ssorted s"   and "i \<le> j" shows "s !! i \<le> s !! j"
sel: fixes x :: "'b \<Rightarrow> bool" shows "pred.eval (pred.Pred x) = x"
simps_2: fixes "is" :: "'b set"   and t :: "'b" shows "interrupt_action_ok is (ClearInterrupt t) = True"
column_add_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and n :: "'c"   and m :: "'c"   and q :: "'b" shows "column_add A n m q = (\<chi>i j. if j = n then A $ i $ n + A $ i $ m * q else A $ i $ j)"
impr_ded: fixes a :: "'b N"   and b :: "'b N"   and c :: "'b N" shows "a * b r\<rightarrow> c = b r\<rightarrow> a r\<rightarrow> c"
isTable_ExcessTable: fixes g :: "graph"   and vs :: "nat list" shows "isTable (ExcessAt g) vs (ExcessTable g vs)"
sub_fun: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and u :: "('a, 'b) tm" assumes "sub s t" shows "sub s (App u t)"
swap_cols_rows_carrier_3: fixes A :: "'a mat"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n n" shows "swap_cols_rows k l A \<in> carrier_mat n n"
vector_smult_lneg: fixes c :: "'b"   and x :: "('b, 'c) vec" shows "- c *s x = - (c *s x)"
set_mult_comm: fixes H :: "'a set"   and J :: "'a set" assumes "H \<subseteq> carrier G"   and "J \<subseteq> carrier G" shows "H <#> J = J <#> H"
simps_2: fixes r :: "'form" shows "outPorts (Assumption r) = {|Reg r|}"
timeslu: shows "wtimes l\<^sub>1 u\<^sub>2 \<equiv>\<^sub>E rl1 * ru2"
grdLit_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and x :: "Preliminaries.var" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.grdLit x \<equiv> Neg (Pr (Guard (Signature.tpOfV x)) [Var x])"
F'_3: shows "is_expansion_aux F' (\<lambda>x. f x - eval C x * b x powr e) (b # basis')"
abs_minus_commute: fixes abs :: "'a \<Rightarrow> '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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (minus a b) = abs (minus b a)"
lQuot_inter: fixes a :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "lQuot a (A \<inter> B) = lQuot a A \<inter> lQuot a B"
cross_no_root: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "\<forall>x. a < x \<and> x < b \<longrightarrow> poly p x \<noteq> 0" shows "cross p a b = 0"
simps_7: fixes \<theta> :: "('d + 'f, 'e) trm"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) trm" shows "TsubstFO (Differential \<theta>) \<sigma> = Differential (TsubstFO \<theta> \<sigma>)"
induct: fixes P :: "'b tree \<Rightarrow> 'b \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b"   and a2 :: "'b tree" assumes "\<And>a. P \<langle>\<rangle> a \<langle>\<rangle>"   and "\<And>x a. P \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> a \<langle>\<rangle>"   and "\<And>l1 x1 r1 a l2 x2 r2. \<lbrakk>\<lbrakk>\<not> (a \<le> x1 \<and> a \<le> x2); x1 \<le> x2\<rbrakk> \<Longrightarrow> P l1 a r1; \<lbrakk>\<not> (a \<le> x1 \<and> a \<le> x2); \<not> x1 \<le> x2\<rbrakk> \<Longrightarrow> P l2 a r2\<rbrakk> \<Longrightarrow> P \<langle>l1, x1, r1\<rangle> a \<langle>l2, x2, r2\<rangle>"   and "\<And>vc vd ve va vb b. P \<langle>\<langle>vc, vd, ve\<rangle>, va, vb\<rangle> b \<langle>\<rangle>"   and "\<And>v va vc vd ve b. P \<langle>v, va, \<langle>vc, vd, ve\<rangle>\<rangle> b \<langle>\<rangle>"   and "\<And>b v va vb. P \<langle>\<rangle> b \<langle>v, va, vb\<rangle>" shows "P a0 a1 a2"
in_set_CollectI: fixes P :: "V \<Rightarrow> bool"   and x :: "V" assumes "P x"   and "small {x. P x}" shows "x \<in>\<^sub>\<circ> set {x. P x}"
trcl_cons_conv: fixes u :: "'a"   and a :: "'b"   and xs :: "'b list"   and v :: "'a"   and R :: "('a \<times> 'b \<times> 'a) set" shows "((u, a # xs, v) \<in> trcl R) = (\<exists>uh. (u, a, uh) \<in> R \<and> (uh, xs, v) \<in> trcl R)"
insertion_sym_mpoly: fixes X :: "nat set"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "finite X" shows "insertion f (sym_mpoly X k) = (\<Sum>Y | Y \<subseteq> X \<and> card Y = k. prod f Y)"
swap_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "X #[x \<and> y]_zs #[x' \<and> y']_zs' = X #[x' \<and> y']_zs' #[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']_zs"
harmonic_aux: shows "\<forall>m>0. 1 / 2 \<le> (\<Sum>n = 2 ^ (m - 1) + 1..2 ^ m. 1 / real n)"
inv: fixes t :: "int poly factor_tree"   and f :: "int poly" assumes "t \<in> sub_trees (Factor_Node f__ l r)"   and "factor_node_info t = f" shows "f = Mp (\<Prod>\<^sub># (factors_of_factor_tree t))"
\<sigma>_vnaccpt: shows "\<forall>k \<mu> \<eta>. \<mu> U\<^sub>r \<eta> \<in> old (\<sigma> k) \<longrightarrow> \<not> (\<forall>i. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> (k + i)) \<and> \<eta> \<notin> old (\<sigma> (k + i)))"
execute_guard_2: fixes P :: "heap \<Rightarrow> bool"   and h :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap" assumes "P h" shows "execute (guard P f) h = Some (f h)"
rcosets_subset_PowG: fixes H :: "'a set" assumes "subgroup H G" shows "rcosets H \<subseteq> Pow (carrier G)"
MP_setI: fixes x :: "('a, 'b) strand_step"   and S :: "('a, 'b) strand_step list" assumes "x \<in> set S" shows "trms\<^sub>s\<^sub>t\<^sub>p x \<subseteq> trms\<^sub>s\<^sub>t S"
tr_wf': fixes D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and A :: "('fun, 'var) stateful_strand_step list"   and X :: "'var set"   and A' :: "('fun, 'var) strand_step list" assumes "\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "\<forall>(t, t')\<in>set D. fv t \<union> fv t' \<subseteq> X"   and "wf'\<^sub>s\<^sub>s\<^sub>t X A"   and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "A' \<in> set (tr A D)" shows "wf\<^sub>s\<^sub>t X A'"
vcard: shows "vcard xs\<^sub>1 = vcard xs\<^sub>2"
atLeast0_lessThan_Suc: fixes n :: "nat" shows "{0..<Suc n} = insert n {0..<n}"
apply_rsp': fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'c"   and y :: "'d" assumes "(R1 ===> R2) f g"   and "R1 x y" shows "R2 (f x) (g y)"
obs_def: shows "obs \<equiv> SOME x. x \<in> \<O>\<^sub>s"
HPow_def: fixes x :: "hf" shows "HPow x = (THE z. \<forall>u. (u \<^bold>\<in> z) = (u \<le> x))"
greatest_Lower_cong_l: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and x' :: "'c"   and A :: "'c set" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> x'"   and "x \<in> carrier L"   and "x' \<in> carrier L" shows "is_glb L x A = is_glb L x' A"
orthogonal_currentD_A: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set"   and x :: "'a" assumes "orthogonal_current \<Gamma> f S"   and "x \<in> S"   and "x \<in> A \<Gamma>"   and "x \<notin> B \<Gamma>" shows "d_OUT f x = weight \<Gamma> x"
derivableE: fixes x :: "'a" assumes "derivable x" obtains P :: "'a set" where "derivation P" and "x \<in> P"
lipschitz: shows "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
abs_eq: fixes y :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "eq_onp overrider (\<lambda>ya x. y x ya) (\<lambda>ya x. y x ya)" shows "- Abs_scene (\<lambda>ya x. y x ya) = Abs_scene y"
delete_first_eq_some: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list"   and l' :: "'a list" shows "(delete_first P l = Some l') = (\<exists>l1 x l2. P x \<and> (\<forall>x\<in>set l1. \<not> P x) \<and> l = l1 @ x # l2 \<and> l' = l1 @ l2)"