hcju/isabelleps huggingface dataset:
MtransC_MtransT: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and cf'' :: "'state" assumes "cf \<rightarrow>*c cf'"   and "cf' \<rightarrow>*t cf''" shows "cf \<rightarrow>*t cf''"
tighten_poly_bounds_def: fixes l :: "rat"   and r :: "rat"   and sr :: "rat" shows "local.tighten_poly_bounds l r sr = (let m = (l + r) / 2; sm = sgn (ipoly p m) in if sm = sr then (l, m, sm) else (m, r, sr))"
Null: fixes as :: "nat list list" assumes "as \<noteq> []" shows "merge as [] [concat as] []"
maintain_prim_invar_impl_1: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and d :: "nat"   and Qi' :: "'q"   and \<pi>i' :: "'m" assumes "prim_invar_impl Qi \<pi>i"   and "\<not> Q_is_empty Qi"   and "Q_getmin Qi = (u, d)"   and "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')" shows "prim_invar_impl (Q_delete u Qi') \<pi>i'"
add2_self: fixes p :: "'a"   and q :: "'a" shows "ccw (0::'a) (p + q) p = ccw (0::'a) q p"
assms: shows "\<turnstile>\<^sub>3\<^sub>' {P} c { Q}"
subst_idem_def: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_idem s \<equiv> s \<circ>\<^sub>s s = s"
deg_pm_mono: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "s adds t" shows "deg_pm s \<le> deg_pm t"
\<C>_vars\<^sub>C_def: fixes x :: "var\<^sub>C" shows "\<C>_vars\<^sub>C x \<equiv> if x = buffer\<^sub>C then {control_var\<^sub>C} else {}"
error_msg_def: fixes i :: "'a"   and msg :: "char list" shows "error_msg i msg = CFAILED (''s CHECKING failed at line '' @ show i @ '' with error '' @ msg)"
hcomp_in_hhomE: fixes \<nu> :: "'d"   and \<mu> :: "'d"   and a :: "'d"   and c :: "'d" assumes "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D c\<guillemotright>" obtains "D.arr \<mu>" and "D.arr \<nu>" and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" and "src\<^sub>D \<mu> = a" and "trg\<^sub>D \<nu> = c"
subst_elim_rm_vars_dest: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and v :: "'c"   and vs :: "'c set" assumes "subst_elim \<sigma> v"   and "v \<notin> vs" shows "subst_elim (rm_vars vs \<sigma>) v"
inj_imp_image_group_iso: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme" assumes "inj_on f (carrier G)" shows "f \<in> Group.iso G (image_group f G)"
perm_vars_nth_lt: fixes n :: "nat"   and vs :: "nat list"   and i :: "nat" assumes "perm_vars n vs"   and "i < n" shows "vs ! i < n"
exec_Call_body_aux: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and bdy :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and t :: "('a, 'c) xstate" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> \<langle>Call p,s\<rangle> \<Rightarrow> t = \<Gamma>\<turnstile> \<langle>bdy,s\<rangle> \<Rightarrow> t"
simps: fixes a1 :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and a2 :: "typ"   and a3 :: "String.literal set" shows "has_sort a1 a2 a3 = ((\<exists>sub S S' tcs a. a1 = (sub, tcs) \<and> a2 = Tv a S \<and> a3 = S' \<and> sort_leq sub S S') \<or> (\<exists>tcs \<kappa> dm S sub Ts. a1 = (sub, tcs) \<and> a2 = Ty \<kappa> Ts \<and> a3 = S \<and> tcs \<kappa> = Some dm \<and> (\<forall>c\<in>S. \<exists>Ss. dm c = Some Ss \<and> list_all2 (has_sort (sub, tcs)) Ts Ss)))"
returnOk_bindE: fixes x :: "'g"   and f :: "'g \<Rightarrow> 'd \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool" shows "returnOk x >>=E f = f x"
cancel_div_mod_rules_1: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "plus (plus (times (divide a b) b) (modulo a b)) c = plus a c"
sorted_list_of_set_empty: shows "sorted_list_of_set {} = []"
lunit_simps_3: fixes f :: "'d" assumes "D.ide f" shows "trg\<^sub>D \<l>\<^sub>D[f] = trg\<^sub>D f"
as_ms: shows "primitive_extractor (disc1, sel1) m = (as, ms)"
supp_finite_atom_set: fixes S :: "atom set" assumes "finite S" shows "supp S = S"
inject_1: fixes x1 :: "'c"   and y1 :: "'c" shows "(Le x1 = Le y1) = (x1 = y1)"
str_max: shows "\<forall>i<n. ground_resolution_with_selection.strictly_maximal_wrt (As ! i) (Cs ! i)"
simps_2: fixes xs :: "'d list"   and y :: "'e"   and ys :: "'e list" shows "zip xs (y # ys) = (case xs of [] \<Rightarrow> [] | z # zs \<Rightarrow> (z, y) # zip zs ys)"
size_braun_adds: fixes t :: "'b tree"   and n :: "nat"   and xs :: "'b list" assumes "braun t"   and "size t = n" shows "size (adds xs n t) = size t + length xs \<and> braun (adds xs n t)"
symclp_conversep: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "symclp r\<inverse>\<inverse> = symclp r"
correct_fold: fixes ts :: "((nat \<times> nat) \<times> int) list"   and x :: "nat \<Rightarrow> int" assumes "distinct (map fst ts)"   and "\<forall>((r, c), uu_)\<in>set ts. r < n" shows "fold (\<lambda>((r, c), v) y. y(c := y c + x r * v)) ts (\<lambda>_. 0) = local.pr (\<alpha> ts) x"
assms_1: fixes u :: "real" assumes "x0 \<in> X" shows "(u \<in> existence_ivl t0 x0) = (g u \<in> existence_ivl s0 x0)"
remdups_append1: fixes xs :: "'a list"   and ys :: "'a list" shows "remdups (remdups xs @ ys) = remdups (xs @ ys)"
Borsukian_alt: fixes S :: "'a set" shows "Borsukian S = (\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> - {0} \<and> continuous_on S g \<and> g ` S \<subseteq> - {0} \<longrightarrow> homotopic_with_canon (\<lambda>h. True) S (- {0}) f g)"
minus_divide_left: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "b \<noteq> \<zero>" shows "\<ominus> (a \<oslash> b) = \<ominus> a \<oslash> b"
plus_Suc_gbinomial: fixes x :: "int"   and k :: "nat" shows "x + (1 + int k) gchoose Suc k = x + int k gchoose k + (x + int k gchoose Suc k)"
vcarr: shows "lincomb l2 W \<in> carrier_vec n"
that: fixes a :: "'b" assumes "g a \<noteq> \<^bold>1" shows "thesis"
Union_iff: fixes A :: "'c"   and C :: "'c set set" shows "s2p (\<mu> C) A = (\<exists>X\<in>C. s2p X A)"
singular_boundary_set_eq_cycle_singleton: fixes X :: "'b topology"   and a :: "'b"   and p :: "nat" assumes "topspace X = {a}"   and "1 \<le> p" shows "singular_relboundary_set p X {} = singular_relcycle_set p X {}"
upd_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and x :: "nat"   and y :: "nat"   and v :: "'c" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.upd m x y v = m(x := (m x)(y := v))"
num_mod_den'2_unique: fixes x :: "int tree" assumes "x = Node 0 (2 * num') x" shows "x = 2 * num_mod_den'"
distinct_1: fixes x11 :: "'a trm"   and x12 :: "'a trm"   and x21 :: "'b"   and x22 :: "'a trm list" shows "Eq x11 x12 \<noteq> Pr x21 x22"
extends_to_equivalence_of_bicategories: shows "equivalence_of_bicategories (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi> F.right_map F.right_cmp F.unit\<^sub>0 F.unit\<^sub>1 F.counit\<^sub>0 F.counit\<^sub>1"
non_empty_cycle_root_no_start_end_points: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "no_start_end_points x"
obtain_sorted_carrier''_def: shows "obtain_sorted_carrier'' = SPEC (\<lambda>L. \<alpha> ` set L = E \<and> (\<forall>(a, wv, b)\<in>set L. weight (\<alpha> (a, wv, b)) = wv) \<and> set L \<subseteq> superE) \<bind> (\<lambda>l. SPEC (\<lambda>L. sorted_wrt edges_less_eq L \<and> set L = set l))"
forest_components_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "forest_components x \<le> forest_components y"
elims_2: fixes x :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and xa :: "char list \<Rightarrow> ty option"   and xb :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and xc :: "expr list"   and xd :: "ty list" assumes "x,xa,xb \<turnstile> xc [:]\<^bsub>NT\<^esub> xd" obtains P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" and E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" where "x = Pa__" and "xa = Ea__" and "xb = ha__" and "xc = []" and "xd = []"   | P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" and E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and e :: "expr" and es :: "expr list" and T :: "ty" and Ts :: "ty list" where "x = P" and "xa = E" and "xb = h" and "xc = e # es" and "xd = T # Ts" and "P,E,h \<turnstile> e :\<^bsub>NT\<^esub> T \<and> P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts"
measurable_PiF: fixes N :: "'d measure"   and f :: "'d \<Rightarrow> 'e \<Rightarrow>\<^sub>F 'f"   and I :: "'e set set"   and M :: "'e \<Rightarrow> 'f measure" assumes "\<And>x. x \<in> space N \<Longrightarrow> domain (f x) \<in> I \<and> (\<forall>i\<in>domain (f x). (f x)\<^sub>F i \<in> space (M i))"   and "\<And>J S. \<lbrakk>J \<in> I; \<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)\<rbrakk> \<Longrightarrow> f -` Pi' J S \<inter> space N \<in> sets N" shows "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>F I M"
to_unique_collection_det: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "to_unique_collection \<tau> \<sigma>\<^sub>1"   and "to_unique_collection \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
assms_1: shows "p\<^sub>0 \<in> ps\<^sub>L"
dot_matrix_vector_mul: fixes A :: "((real, 'a) vec, 'a) vec"   and x :: "(real, 'a) vec"   and B :: "((real, 'a) vec, 'a) vec"   and y :: "(real, 'a) vec" shows "(A *v x) \<bullet> (B *v y) = (rowvector x ** (Finite_Cartesian_Product.transpose A ** B ** columnvector y)) $ 1 $ 1"
Int_Union: fixes A :: "'c set"   and B :: "'c set set" shows "A \<inter> \<Union> B = \<Union> ((\<inter>) A ` B)"
min_eq: shows "min (nrows A) (ncols A) - 1 = Suc (min (nrows A) (ncols A) - 2)"
is_lub_def: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b" assumes "class.po below" shows "po.is_lub below S x = (po.is_ub below S x \<and> (\<forall>u. po.is_ub below S u \<longrightarrow> below x u))"
erH_mem: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and p :: "'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "p \<in> carrier R" shows "erH R S X A B Y h p \<in> carrier A"
ord_suc_nonzero: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>"   and "ord_Zp x = int n" shows "x (Suc n) \<noteq> 0"
vreal_impl_in_Vset_ss_omega: shows "vreal_impl \<in>\<^sub>\<circ> Vset (\<omega> + \<omega>)"
mat_assoc_test_4: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A - B + (B - C) = A + - B + B + - C"
if_0_minus_poly_rev_list: fixes a :: "'b"   and x :: "'b list"   and y :: "'b list" shows "(if a = (0::'b) then x else minus_poly_rev_list x (map ((*) a) y)) = minus_poly_rev_list x (map ((*) a) y)"
u_2: shows "u \<longlonglongrightarrow> c"
adjacent_nodes_def: fixes u :: "nat" shows "cf.adjacent_nodes u \<equiv> cf.E `` {u} \<union> cf.E\<inverse> `` {u}"
inject_3: fixes x3 :: "'b"   and y3 :: "'b" shows "(mid x3 = mid y3) = (x3 = y3)"
cong: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "f = g"   and "x = y" shows "f x = g y"
ex1_implies_ex: fixes P :: "'c \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "\<exists>x. P x"
strict_border_step: fixes xs :: "'a list"   and ys :: "'a list" shows "strict_border xs ys = strict_border (xs @ [ys ! length xs]) (ys @ [ys ! length xs])"
mkId4b: fixes l :: "Location"   and ll :: "Location"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "(l, ll) \<in> mkId h" shows "l \<in> Dom h \<and> l = ll"
EncS: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<lbrace>x,F\<rbrace> in v] = (\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.en r)"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Arr (t \<^bold>\<cdot> u) = VSeq t u"
take_zip: fixes n :: "nat"   and xs :: "'c list"   and ys :: "'d list" shows "take n (xs || ys) = take n xs || take n ys"
igSwapIGVar_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) Iteration.model_scheme" shows "igSwapIGVar MOD \<equiv> \<forall>zs z1 z2 xs x. igSwap MOD zs z1 z2 (igVar MOD xs x) = igVar MOD xs (x @xs[z1 \<and> z2]_zs)"
vc_foreach_finite: fixes S :: "'v set"   and V :: "'v set"   and v :: "'v" assumes "rpre S (V, v)" shows "finite (E `` {v})"
cong_0_iff: 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b zero a = dvd.dvd times a b"
max_face_ex: fixes p :: "nat"   and g :: "graph" assumes "Seed\<^bsub>p\<^esub> [next_plane0\<^bsub>p\<^esub>]\<rightarrow>* g" shows "\<exists>f\<in>set (finals g). |vertices f| = maxGon p"
HEndPhase0_HInv4a_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv2a s"   and "Inv2c s"   and "HInv4d s p"   and "Inv1 s"   and "HInv4a s p" shows "HInv4a s' p"
transTerminationLessState1I: fixes x :: "State"   and y :: "State"   and Vbl :: "nat set"   and z :: "State" assumes "(x, y) \<in> terminationLessState1 Vbl"   and "(y, z) \<in> terminationLessState1 Vbl" shows "(x, z) \<in> terminationLessState1 Vbl"
Ssegment_inc: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b = (a \<in> Ssegment D b)"
mon_loc_ileq: fixes w :: "('a, 'b) label list el_step list"   and w' :: "('a, 'b) label list el_step list"   and fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme" assumes "w \<preceq> w'" shows "mon_loc fg w \<subseteq> mon_loc fg w'"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>r x. f (r *\<^sub>C x) = cnj r *\<^sub>C f x" shows "antilinear_axioms f"
Red1AAssBounds: fixes s :: "'heap \<times> 'addr val list"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr val" assumes "typeof_addr (hp s) a = \<lfloor>Array_type T n\<rfloor>"   and "i <s 0 \<or> int n \<le> sint i" shows "uf,P,t \<turnstile>1 \<langle>addr a\<lfloor>Val (Intg i)\<rceil> := Val e,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ArrayIndexOutOfBounds),s\<rangle>"
none_remains_none: fixes SES :: "('c, 'd) SES_rec"   and s :: "'c"   and t :: "'d list"   and e :: "'d" assumes "path SES s t = None" shows "path SES s (t @ [e]) = None"
prod_as_plus: fixes X :: "'c \<Longrightarrow> 'e"   and Y :: "'d \<Longrightarrow> 'f" shows "X \<times> Y = X ;\<^sub>L fst\<^sub>L +\<^sub>L Y ;\<^sub>L snd\<^sub>L"
pre_hensel_1: fixes n :: "nat" shows "val_Zp (f' \<bullet> a) < val_Zp (a \<ominus> ns n)"
hd_b_nin_tl_b: shows "hd b \<notin> set (tl b)"
set_relE1: fixes s :: "'a set"   and s' :: "'b set"   and R :: "('a \<times> 'b) set"   and x :: "'a" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x \<in> s" obtains x' :: "'b" where "x' \<in> s'" and "(x, x') \<in> R"
distinct_105: fixes x7 :: "logic_type"   and x9 :: "ctrl_type" shows "logic_type x7 \<noteq> ctrl_type x9"
trivial_group_alt: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "trivial_group G = (\<exists>a. carrier G \<subseteq> {a})"
vec_carrier_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and m :: "nat" shows "(vec n f \<in> carrier_vec m) = (n = m)"
fac_dvd2: fixes q :: "nat"   and n :: "nat" assumes "Suc 0 < q"   and "q dvd n!"   and "q \<le> n" shows "\<not> q dvd Suc (n!)"
assms: shows "xs @ [Broadcast (i, Store e mo i)] prefix of j"
sgnx_top: shows "sgnx (poly p) at_top = 1"
eq: shows "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
concat_suc: fixes as :: "nat"   and xs :: "'a list list" assumes "Suc as \<le> length xs" shows "concat (take (Suc as) xs) = concat (take as xs) @ xs ! as"
uint_shiftr_eq: fixes w :: "'b word"   and n :: "nat" shows "uint (w >> n) = uint w div 2 ^ n"
unfold: shows "sconcatMap = (\<Lambda> f. sconcat oo smap\<cdot>f)"
simps: fixes p :: "rat poly"   and q :: "rat poly" shows "sturm_aux_rat p q = (if degree q = 0 then [p, q] else p # sturm_aux_rat q (- (p mod q)))"
assms_2: shows "f \<otimes> \<I> = (a \<otimes> \<iota>) \<cdot> \<a>[a, \<I>, \<I>]"
Op: fixes delta :: "'opSym"   and inp' :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp' :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and inp :: "'index \<Rightarrow> 'a option"   and binp :: "'bindex \<Rightarrow> 'b option" assumes "wlsInp delta inp'"   and "wlsBinp delta binp'"   and "gWlsInp MOD delta inp"   and "gWlsBinp MOD delta binp"   and "liftAll2 phi (arOf delta) inp"   and "liftAll2 phiAbs (barOf delta) binp" shows "phi (stOf delta) (gOp MOD delta inp' inp binp' binp)"
wfactors_def: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c" shows "wfactors G fs a = ((\<forall>x\<in>set fs. irreducible G x) \<and> foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<sim>\<^bsub>G\<^esub> a)"
filter_insort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)"   and "P x" shows "filter P (linorder.insort_key less_eq f x xs) = linorder.insort_key less_eq f x (filter P xs)"
freshStarPair_2: fixes xvec :: "name list"   and x :: "'e"   and y :: "'f" shows "xvec \<sharp>* (x, y) = (xvec \<sharp>* x \<and> xvec \<sharp>* y)"
mset_partition: fixes t :: "'b tree"   and p :: "'b"   and l' :: "'b tree"   and r' :: "'b tree" assumes "bst_wrt (\<le>) t"   and "Splay_Heap.partition p t = (l', r')" shows "mset_tree t = mset_tree l' + mset_tree r'"
inv1: shows "access root path user\<^sub>1 {} = Some (Env att dir)"
cases: fixes x :: "fterm" obtains p :: "char list" and ts :: "fterm list" where "x = Fun p ts"   | v :: "char list" where "x = Var v"
preserves_hom: fixes f :: "(bool, unit) X.arr"   and a :: "(bool, unit) X.arr"   and b :: "(bool, unit) X.arr" assumes "X.in_hom f a b" shows "\<guillemotleft>a0xa1.mkCone p0 p1 f : p0p1.A.map a \<rightarrow> a0xa1.map b\<guillemotright>"
induct: fixes P1 :: "'b exp \<Rightarrow> bool"   and P2 :: "'b exp list \<Rightarrow> bool"   and exp :: "'b exp" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3. P1 (x1\<bullet>\<^sub>sx2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (INIT x1 (x2,x3) \<leftarrow> x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x2; P1 x4\<rbrakk> \<Longrightarrow> P1 (RI (x1,x2) ; x3 \<leftarrow> x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 exp"
sum_indicator_mult: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "(\<Sum>x\<in>A. indicator B x * f x) = sum f (A \<inter> B)"
simps_70: fixes x5 :: "'d regex" shows "atms (Star x5) = atms x5"
AE_equal_preimage_then_null_or_conull: fixes A :: "'a set" assumes "A \<in> sets M"   and "A \<Delta> T --` A \<in> null_sets M" shows "A \<in> null_sets M \<or> space M - A \<in> null_sets M"
urel32_uminus: fixes x :: "uint32"   and y :: "int" assumes "urel32 x y" shows "urel32 (uminus_p32 pp x) (uminus_p p y)"
RedBinOp: fixes bop :: "bop"   and v\<^sub>1 :: "val"   and v\<^sub>2 :: "val"   and v :: "val"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "redp P E (Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> Val v\<^sub>2) s (Val v) s"
t_del_left_bound: fixes t :: "'b tree" assumes "t \<noteq> \<langle>\<rangle>" shows "t_del_left t \<le> 2 * height t"
Limsup_MInfty: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> ereal" assumes "net \<noteq> bot" shows "(f \<longlongrightarrow> - \<infinity>) net = (Limsup net f = - \<infinity>)"
subsumed_by_two_imp: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and v :: "'c \<times> nat"   and v1 :: "'c \<times> nat"   and v2 :: "'c \<times> nat" assumes "wf_sub_rel subs"   and "(v, v1) \<in> subs"   and "(v, v2) \<in> subs" shows "v1 = v2"
rtrancl_into_rtrancl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
dg_Set_Arr_iff: fixes x :: "V"   and \<alpha> :: "V" shows "(x \<in>\<^sub>\<circ> dg_Set \<alpha>\<lparr>Arr\<rparr>) = arr_Set \<alpha> x"
are_complementary_symmetric: fixes x :: "'a"   and y :: "'a" shows "are_complementary x y = are_complementary y x"