hcju/isabelleps huggingface dataset:
psubstt_image_1: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b FOL_Fitting.term" shows "paramst (psubstt f t) = f ` paramst t"
ordLeq_csum2: fixes p2 :: "('d \<times> 'd) set"   and p1 :: "('e \<times> 'e) set" assumes "Card_order p2" shows "p2 \<le>o p1 +c p2"
UN_iff: fixes b :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
HD': shows "class.Hermite_ring TYPE('a) \<and> (\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a}))"
branch_contains_initially: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and x :: "formula" assumes "branch subs (pseq fs) f"   and "x \<in> set fs" shows "contains f (0, x) 0"
simps_1: fixes w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "splitLeaf\<^sub>F [] w\<^sub>a a w\<^sub>b b = []"
SUP_const: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and f :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Sup ((\<lambda>i. f) ` A) = f"
coeff_zero: fixes n :: "nat" shows "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
assms_2: shows "f \<in> P \<rightarrow>\<^sub>M N"
notDom: fixes x :: "'b"   and A :: "'b \<Rightarrow> 'c option" assumes "x \<in> dom A" shows "A x \<noteq> \<bottom>"
mset_heap_def: fixes ho :: "'b hp option" shows "mset_heap ho = (case ho of None \<Rightarrow> {#} | Some h \<Rightarrow> mset_hp h)"
compute_rapprox_posrat: fixes prec :: "nat"   and x :: "nat"   and y :: "nat" shows "rapprox_posrat prec x y = (let l = rat_precision prec (int x) (int y); (r, s) = if 0 \<le> l then (int x * 2 ^ nat l, int y) else (int x, int y * 2 ^ nat (- l)); d = r div s; m = r mod s in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
mult_diff_mult: fixes x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "'a" shows "x * y - a * b = x * (y - b) + (x - a) * b"
P2: shows "Bet X' A' B' \<and> Cong A' B' A B"
assms_2: fixes q :: "real poly" assumes "q \<in> set qs" shows "coprime p q"
accepts_conc: fixes L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "accepts (RegExp2NA.conc L R) w = (\<exists>u v. w = u @ v \<and> accepts L u \<and> accepts R v)"
ide_in_iso_class: fixes f :: "'a" assumes "ide f" shows "f \<in> \<lbrakk>f\<rbrakk>"
join_compl2_top: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> \<^bold>-\<^bsub>L\<^esub> x = \<top>\<^bsub>L\<^esub>"
commute: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "linordered_monoid.min a b = linordered_monoid.min b a"
vunionE: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A \<union>\<^sub>\<circ> B" obtains "x \<in>\<^sub>\<circ> A"   | "x \<in>\<^sub>\<circ> B"
factor_preserving_hom_axioms: shows "factor_preserving_hom poly_lift"
MakeCatMapsTo: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "f maps\<^bsub>C\<^esub> X to Y" shows "f maps\<^bsub>MakeCat C\<^esub> X to Y"
hom: fixes f :: "complex \<Rightarrow> 'a"   and g :: "complex \<Rightarrow> 'a" assumes "continuous_on (sphere 0 1) f"   and "f ` sphere 0 1 \<subseteq> S"   and "continuous_on (sphere 0 1) g"   and "g ` sphere 0 1 \<subseteq> S" shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g"
mat_kernelD_1: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> mat_kernel A" shows "v \<in> carrier_vec nc"
simps: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a set['c]"   and a2 :: "'b set['c]" shows "rel_bset R a1 a2 = (\<exists>x. a1 = map_bset fst x \<and> a2 = map_bset snd x \<and> set_bset x \<subseteq> {(x, y). R x y})"
AlwaysI: fixes F :: "'b program"   and A :: "'b set" assumes "Init F \<subseteq> A"   and "F \<in> Stable A" shows "F \<in> Always A"
class_to_linorder: shows "linorder (dflt_cmp (\<le>) (<))"
cons_canceled: fixes a :: "bool \<times> 'd"   and x :: "(bool \<times> 'd) list" assumes "canceled (a # x)" shows "canceled x"
polypow: fixes bs :: "'a list"   and n :: "nat"   and p :: "poly" shows "\<lparr>p ^\<^sub>p n\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ^ n"
simps_41: fixes l :: "Formula.formula list" shows "mmonitorable_exec (formula.Ands l) = (let (pos, neg) = partition mmonitorable_exec l in pos \<noteq> [] \<and> list_all mmonitorable_exec (map remove_neg neg) \<and> \<Union> (set (map fv neg)) \<subseteq> \<Union> (set (map fv pos)))"
edges_split_face1: fixes f :: "face"   and u :: "nat"   and v :: "nat"   and vs :: "nat list" assumes "pre_split_face f u v vs" shows "\<E> (fst (split_face f u v vs)) = Edges (v # rev vs @ [u]) \<union> Edges (u # between (vertices f) u v @ [v])"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> r" shows "\<exists>a b. x = \<langle>a, b\<rangle>"
that: fixes x :: "'a" assumes "(x, y) \<in> set (zip xs ys)" shows "thesis"
polymul_properties_1: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
pmat_extension_def: fixes m :: "'a mat" shows "st2.pmat_extension m = st2.ptensor_mat m (1\<^sub>m st2.d2)"
redT_updTs_new_thread_ts: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and tas :: "('f, 'g, 'i) new_thread_action list"   and t :: "'f"   and x :: "'g"   and m :: "'i" assumes "thread_oks ts tas"   and "NewThread t x m \<in> set tas" shows "redT_updTs ts tas t = \<lfloor>(x, no_wait_locks)\<rfloor>"
that: fixes p :: "'a list" assumes "p \<in> carrier (K [X])"   and "p \<noteq> []"   and "local.eval p x = \<zero>" shows "thesis"
upRules_not_right_principal_for_modal: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and M :: "'b"   and Ms :: "('a, 'b) form list"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "r \<in> upRules" shows "\<not> rightPrincipal r (Modal M Ms) R"
simps_4: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (And \<phi> \<psi>) \<Gamma> = (eval \<phi> \<Gamma> \<and> eval \<psi> \<Gamma>)"
rquot_steps: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and x :: "nat"   and w :: "bool list list" shows "dfa_steps (rquot M n) x w = dfa_steps M x w"
cases: fixes a :: "('state \<times> 'rule) tree" assumes "local.wf a" obtains t :: "('state \<times> 'rule) tree" where "a = t" and "snd (tree.root t) \<in> R" and "effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t)" and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> local.wf t'"
\<psi>_def: shows "\<psi> \<equiv> \<psi>.map"
DomainPI: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "r a b" shows "Domainp r a"
pred_inject_2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and aa :: "'a tree list" shows "pred_tree P (Node a aa) = (P a \<and> list_all (pred_tree P) aa)"
FmToFo: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and X :: "'a"   and Y :: "'b" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>"   and "Y \<in> obj\<^bsub>CatCod F\<^esub>"   and "F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y" shows "F @@ X = Y"
Suc_1: fixes j :: "nat"   and r :: "mregex" assumes "j \<le> i" shows "LPDi j (MStar r) \<subseteq> {MStar r} \<union> MTimesR (\<Union>j\<le>i. LPDi j r) (MStar r)"
preserve_deadlocked_axioms: shows "preserve_deadlocked final r convert_RA wf_state"
single_eq_single: fixes a :: "'b"   and b :: "'b" shows "({#a#} = {#b#}) = (a = b)"
Kelly_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and A :: "'b set" shows "Kelly R B A = (\<forall>a\<in>A. \<forall>b\<in>B. R b a)"
assms_3: shows "g2 \<in> stabilizer G \<phi> x"
zip_Cons_Cons: fixes x :: "'c"   and xs :: "'c list"   and y :: "'d"   and ys :: "'d list" shows "x # xs || y # ys = (x, y) # (xs || ys)"
surjective: fixes r :: "('b, 'c) fsm_scheme" shows "r = \<lparr>states = states r, init = init r, final = final r, next = next r, \<dots> = more r\<rparr>"
lm015: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" shows "P || (X \<inter> Y) \<subseteq> P || X \<and> Outside' (X \<union> Y) P \<subseteq> Outside' X P"
cod_lunit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.cod \<l>[a] = a"
is_sublist_simp: fixes a :: "'a"   and v :: "'a"   and as :: "'a list"   and vs :: "'a list" assumes "a \<noteq> v" shows "is_sublist (a # as) (v # vs) = is_sublist (a # as) vs"
v0_VV: fixes p :: "Player" shows "v0 \<in> VV p \<or> v0 \<in> VV p**"
assms: shows "theta \<subseteq> ZOretr (theta \<union> ZObis)"
module_hom_iff: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "module_hom s1 s2 f = (module s1 \<and> module s2 \<and> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x)))"
alphaAbs_all_equal_or_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_all_equal_or_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
iso_iff2: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> (\<forall>a\<in>Field r. \<forall>b\<in>Field r. ((a, b) \<in> r) = ((f a, f b) \<in> r')))"
alist_map_corr: fixes al :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "distinct (map fst al)" shows "((k, v) \<in> set al) = (map_of al k = Some v)"
sym: fixes x :: "'a"   and y :: "'a" shows "invert_order (comp x y) = comp y x"
leof_FAIL: fixes m :: "'b nres" shows "m \<le>\<^sub>n FAIL"
P_inf_False: fixes t :: "'s" shows "P_inf t (\<lambda>\<omega>. False) = 0"
find_mu_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" shows "fx0.find_mu f x0 \<equiv> (\<lambda>s. s\<lparr>m := 0, tortoise := x0, hare := fx0.seq' f x0 (l s)\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>tortoise := f (tortoise s), hare := f (hare s), m := m s + 1\<rparr>)"
restrict0_lift_f_Ker_Q: shows "order.greater_eq (ker (res_freeword_funlift f S) \<inter> FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>"
finite_two_chain_vertical_boundary: fixes two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set" assumes "finite two_chain" shows "finite (two_chain_vertical_boundary two_chain)"
B_def: shows "B \<equiv> liminf (\<lambda>x. ereal (aa (x + 1) / aa x * bb x / bb (x + 1)))"
ssubst_forget: fixes x :: "name"   and k :: "stack"   and v :: "trm" assumes "x \<sharp> k" shows "ssubst x v k = k"
\<Z>_\<alpha>_\<alpha>\<omega>: shows "\<alpha> \<in>\<^sub>\<circ> \<alpha> + \<omega>"
tendsto_mult_left_iff: fixes c :: "'e"   and f :: "'f \<Rightarrow> 'e"   and l :: "'e"   and F :: "'f filter" assumes "c \<noteq> (0::'e)" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F = (f \<longlongrightarrow> l) F"
steps_mono: fixes M :: "('a \<times> ('a, 'b) Op) set"   and M' :: "('a \<times> ('a, 'b) Op) set" assumes "M \<subseteq> M'" shows "(step M)\<^sup>* \<subseteq> (step M')\<^sup>*"
CLB: shows "carrier_vec n = local.span (set (Ls @ Bs))"
sff_4: assumes "p = 0" shows "c = (0::'a) \<and> bs = []"
starfunC_approx_Re_Im_iff: fixes f :: "'a \<Rightarrow> complex"   and x :: "'a star"   and z :: "complex star" shows "((*f* f) x \<approx> z) = ((*f* (\<lambda>x. Re (f x))) x \<approx> hRe z \<and> (*f* (\<lambda>x. Im (f x))) x \<approx> hIm z)"
step_invariantD: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool"   and s :: "'c"   and a :: "'d"   and s' :: "'c" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "P (s, a, s')"
split_minNoneN1: fixes t :: "'b bro"   and h :: "nat" assumes "t \<in> U h"   and "split_min t = None" shows "t = N1 N0"
order_smult: fixes c :: "'b"   and x :: "'b"   and p :: "'b poly" assumes "c \<noteq> (0::'b)" shows "order x (Polynomial.smult c p) = order x p"
pds_finite: fixes x :: "'n" shows "finite {y. (x, y) \<in> pdt}"
mkKripke_def: fixes ws :: "'d set"   and rels :: "'b \<Rightarrow> ('d \<times> 'd) set"   and val :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "mkKripke ws rels val \<equiv> \<lparr>worlds = ws, relations = \<lambda>a. Restr (rels a) ws, valuation = val\<rparr>"
override_on_upd: fixes x :: "'d"   and S :: "'d set"   and \<rho> :: "'d \<Rightarrow> 'e"   and \<rho>' :: "'d \<Rightarrow> 'e"   and z :: "'e" assumes "x \<in> S" shows "\<rho> ++\<^bsub>S\<^esub> \<rho>'(x := z) = (\<rho> ++\<^bsub>S - {x}\<^esub> \<rho>')(x := z)"
endofunctor_axioms: shows "endofunctor (\<cdot>) L"
assms: shows "a<x>.P \<longmapsto> b<\<nu>y> \<prec> P'"
RModuleHomI: fixes T :: "'m \<Rightarrow> 'a"   and smult' :: "'r \<Rightarrow> 'a \<Rightarrow> 'a" assumes "GroupHom M T"   and "\<And>r m. \<lbrakk>r \<in> R; m \<in> M\<rbrakk> \<Longrightarrow> T (r \<cdot> m) = smult' r (T m)" shows "RModuleHom R (\<cdot>) M smult' T"
indep_setD_ev2: fixes A :: "'a set set"   and B :: "'a set set" assumes "indep_set A B" shows "B \<subseteq> events"
trans_def: fixes r :: "('c \<times> 'c) set" shows "trans r = (\<forall>x y z. x \<^bsub>r\<^esub>\<preceq> y \<longrightarrow> y \<^bsub>r\<^esub>\<preceq> z \<longrightarrow> x \<^bsub>r\<^esub>\<preceq> z)"
Cons: fixes x :: "'a"   and y :: "'b"   and r :: "('a \<times> 'b) set"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> r"   and "(xs, ys) \<in> listrel r" shows "(x # xs, y # ys) \<in> listrel r"
simps: fixes H :: "chmsg \<Rightarrow> bool"   and a :: "atom" shows "atomsp H a = ((\<exists>c A B M At. a = At \<and> H (StatCh c A B (Msg M)) \<and> At \<in> set M) \<or> (\<exists>c K M At. a = At \<and> H (DynCh c K (Msg M)) \<and> At \<in> set M))"
basis_finmap_def: shows "local.basis_finmap = {Pi' I S |I S. finite I \<and> (\<forall>i\<in>I. S i \<in> local.basis_proj)}"
assms_5: fixes a :: "complex_homo"   and b :: "complex_homo"   and c :: "complex_homo"   and M :: "moebius" assumes "P a b c" shows "P (moebius_pt M a) (moebius_pt M b) (moebius_pt M c)"
dist_midpoint_4: fixes a :: "'a"   and b :: "'a" shows "dist (midpoint a b) b = dist a b / 2"
add: fixes b1 :: "'n"   and b2 :: "'n" shows "f (b1 + b2) = f b1 + f b2"
Rep_boolean_subset: fixes x :: "'b boolean_subset" shows "Rep_boolean_subset x \<in> {x. \<exists>y. x = - y}"
bounded_component_cart: fixes s :: "('a, 'b) vec set"   and i :: "'b" assumes "bounded s" shows "bounded ((\<lambda>x. x $ i) ` s)"
witness: shows "\<bar>e (degree x - 1)\<bar> *\<^sub>R \<bar>pdevs_apply x (degree x - 1)\<bar> < \<bar>pdevs_apply x (degree x - 1)\<bar>"
run_asset_v_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_asset_v n s))) \<notin> used s"
assms_2: shows "{x, y} \<in> arc_to_block ` E"
IH2: shows "continuous_on UNIV (\<lambda>x. Blinfun (extendf_deriv I i \<theta>\<^sub>2 \<nu> x))"
cong_add_rcancel: 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 uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and x :: "'a"   and a :: "'a"   and y :: "'a"   and n :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (plus x a) (plus y a) n = unique_euclidean_semiring.cong modulo x y n"
encompasses_convex_polygon3: fixes p :: "real \<times> real"   and x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and z :: "(real \<times> real) \<times> real \<times> real"   and zs :: "((real \<times> real) \<times> real \<times> real) list" assumes "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) (x # y # z # zs)"   and "convex_polygon (x # y # z # zs)"   and "ccw'.sortedP (fst x) (map snd (butlast (x # y # z # zs)))" shows "p \<in> convex hull set (map fst (x # y # z # zs))"
linf_norm_vec_ge_0: fixes v :: "'a vec" shows "(0::'a) \<le> \<parallel>v\<parallel>\<^sub>\<infinity>"
inject_3: fixes x5 :: "nat"   and y5 :: "nat" shows "(ShaK x5 = ShaK y5) = (x5 = y5)"
finite_path_iff: fixes x :: "'a" shows "finite_path x = (backward_finite_path x \<and> forward_finite_path x)"
map_of_delete_aux': fixes xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst xs)" shows "map_of (AList.delete_aux k xs) = (map_of xs)(k := None)"
trivial_group: shows "trivial_group G = (\<exists>a. carrier G = {a})"
Sum_any_when_independent: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and g :: "'d \<Rightarrow> 'c"   and P :: "bool" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero (\<lambda>a. zero.when zero (g a) P) = zero.when zero (comm_monoid_add.Sum_any plus zero g) P"
normalized_gauss_int_2: fixes z :: "gauss_int" assumes "normalize z = z" shows "0 \<le> ImZ z"
lNact_least: fixes n :: "nat"   and n' :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n = n' \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
methdMembr_def: fixes m :: "qtname \<times> mdecl" shows "methdMembr m = (fst m, mdecl (snd m))"
s_neq_1: fixes m :: "nat"   and m' :: "nat" shows "s1 m \<noteq> s2 m'"
strongSimEarly_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto>[Rel] Q \<equiv> (\<forall>a y Q'. Q \<longmapsto> a<\<nu>y> \<prec> Q' \<longrightarrow> y \<sharp> P \<longrightarrow> (\<exists>P'. P \<longmapsto> a<\<nu>y> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
find_nu_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" shows "fx0.find_nu f x0 \<equiv> (\<lambda>s. s\<lparr>nu := 1, tortoise := f x0, hare := f (f x0)\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>nu := nu s + 1, tortoise := f (tortoise s), hare := f (f (hare s))\<rparr>)"
scale_zero_right: fixes a :: "'a" shows "a *s (0::'b) = (0::'b)"
J_def: shows "J \<equiv> \<lambda>(A, A'). (ok A \<longrightarrow> ok A') \<and> alpha_d.more A = alpha_d.more A'"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
complete_digraph_def: fixes n :: "nat"   and G :: "('b, 'c) pre_digraph" shows "K\<^bsub>n\<^esub> G \<equiv> graph G \<and> card (verts G) = n \<and> arcs_ends G = {(u, v). (u, v) \<in> verts G \<times> verts G \<and> u \<noteq> v}"
bij_pointE: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "bij f" obtains x :: "'c" where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
equivalent_objects_def: fixes a :: "'c"   and b :: "'c" shows "C.equivalent_objects a b \<equiv> \<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.equivalence_map f"
geom_proc_stoch_proc_filt: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "stoch_proc_filt N geom_proc borel n = fct_gen_subalgebra N (stream_space borel) (proj_stoch_proc geom_proc n)"
rsp_fold_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "rsp_fold f = (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
entt_trans: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>t B"   and "B \<Longrightarrow>\<^sub>t C" shows "A \<Longrightarrow>\<^sub>t C"
monoidal_category_with_alternate_unit_axioms: shows "monoidal_category_with_alternate_unit (\<cdot>) T \<alpha> \<iota> \<iota>\<^sub>1"
pause_option_def: fixes out :: "'o"   and c :: "'i \<Rightarrow> ('a, 'm) optionT" shows "local.pause_option out c = OptionT (pause out (\<lambda>i. run_option (c i)))"
SD_inefficient_support_subset: fixes p' :: "'alt pmf"   and p :: "'alt pmf" assumes "\<not> SD_efficient R p'"   and "set_pmf p' \<subseteq> set_pmf p"   and "p \<in> lotteries" shows "\<not> SD_efficient R p"
135: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- - x \<sqinter> y) = - (- - x \<sqinter> y)"
r: shows "int (size (filter_mset ((\<in>) x) \<B>)) = r"
algebraic_times_i_2: fixes x :: "complex" assumes "algebraic x" shows "algebraic (- \<i> * x)"
IH_2: fixes u :: "('a \<times> bool) list"   and ass :: "'a \<Rightarrow> bool"   and ass' :: "'a \<Rightarrow> bool" assumes "ifex_no_twice e"   and "ifex_sat_list e = Some u"   and "ass = update_assignment u ass'" shows "val_ifex e ass = True"
fempty_iff: fixes c :: "'x" shows "(c |\<in>| {||}) = False"
Cons_1: fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and s' :: "('a, 'b) fmap"   and ss :: "('a, 'b) fmap list" assumes "vars_change as vs s = s' # ss" shows "\<exists>as1 act as2. as = as1 @ act # as2 \<and> vars_change as1 vs s = [] \<and> state_succ (exec_plan s as1) act = s' \<and> vars_change as2 vs (state_succ (exec_plan s as1) act) = ss"
document_ptr_casts_commute: fixes ptr :: "(_) object_ptr"   and document_ptr :: "(_) document_ptr" shows "(cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some document_ptr) = (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^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 document_ptr = ptr)"
pathCard_Cons_other: fixes x' :: "Vars.var"   and x :: "Vars.var"   and xs :: "Vars.var list" assumes "x' \<noteq> x" shows "pathCard (x # xs) x' = pathCard xs x'"
simps_4: fixes G :: "('d, 'c) labeled_graph"   and l :: "'d" shows ":G:\<lbrakk>A_Lbl l\<rbrakk> = getRel l G"
assms: shows "f $$ fls_subdegree f * y = (1::'a)"
compare_real_alg: fixes x :: "real_alg"   and y :: "real_alg" shows "compare (real_of x) (real_of y) = compare x y"
hyps_3: shows "count (image_mset f (mset_set F)) b = card {a \<in> F. f a = b}"
ExistsP_graph_aux_def: shows "ExistsP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>ExistsP_aux x p q v y x'. x1 = (p, q) \<and> x2 = SyntaxN.Ex x (SyntaxN.Ex x' (SyntaxN.Ex y (SyntaxN.Ex v (FormP (Var x) AND VarNonOccFormP (Var v) (Var y) AND AbstFormP (Var v) Zero (Var x) (Var x') AND p EQ Q_Imp (Var x) (Var y) AND q EQ Q_Imp (Q_Ex (Var x')) (Var y))))) \<and> atom x \<sharp> (p, q, v, y, x') \<and> atom x' \<sharp> (p, q, v, y) \<and> atom y \<sharp> (p, q, v) \<and> atom v \<sharp> (p, q))"
hom_uminus: fixes x :: "'a poly"   and a :: "'a" shows "poly (- x) a = - poly x a"
mt: shows "mtype(CT,m,C) = Cs \<rightarrow> C0"