hcju/isabelleps huggingface dataset:
assert_bound'_def: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and dir :: "('b, 'a) Direction"   and i :: "'b"   and x :: "nat"   and c :: "'a"   and s :: "('b, 'a) state" assumes "Update update" shows "Update.assert_bound' update dir i x c s \<equiv> if \<unrhd>\<^sub>u\<^sub>b (Direction.lt dir) c (Direction.UB dir s x) then s else let s' = update\<B>\<I> (Direction.UBI_upd dir) i x c s in if \<lhd>\<^sub>l\<^sub>b (Direction.lt dir) c (Direction.LB dir s x) then set_unsat [i, Direction.LI dir s x] s' else if x \<notin> lvars (\<T> s') \<and> Direction.lt dir c (\<langle>\<V> s\<rangle> x) then update x c s' else s'"
mult_inverse_of_nat_commute: 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 xa :: "nat"   and x :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times (inverse (semiring_1.of_nat one plus zero xa)) x = times x (inverse (semiring_1.of_nat one plus zero xa))"
R2: shows "RETURN l' \<le> map_to_sorted_list R s'"
inducts: fixes f :: "nat \<Rightarrow> bool" assumes "wf_set_bits_int f" obtains n :: "nat" where "\<forall>n'\<ge>n__. \<not> f n'"   | n :: "nat" where "\<forall>n'\<ge>n. f n'"
map_def: shows "\<Psi>.map = transformation_by_components.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) Cones.map (\<lambda>a. S.inv (\<Phi> a))"
simps_2: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "black_depth (Node l R k v r) = black_depth l"
cfcomp1: fixes f :: "'e \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'e" shows "f oo g = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
prj_simps_1: shows "Maps.seq SPN_fgh.\<mu>\<nu>.prj\<^sub>1 (Maps.PRJ\<^sub>1 (SPN_fgh.\<nu>.leg0 \<odot> SPN_fgh.\<mu>\<nu>.prj\<^sub>0) SPN_fgh.\<pi>.leg1)"
ccw_scale23: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and r :: "real" assumes "ccw 0 a b"   and "0 < r" shows "ccw 0 (r *\<^sub>R a) (r *\<^sub>R b)"
inverse_aformE: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "real \<times> real pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "0 < Inf_aform' p X \<or> Sup_aform' p X < 0" obtains Y :: "(real \<times> real pdevs) \<times> real" where "inverse_aform p X = Some Y" and "inverse (aform_val e X) \<in> aform_err e Y"
euler_imp_connected: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "euler_trail u p v" shows "Digraph_Component.connected G"
f: shows "\<forall>\<^sub>F x in at_top. f x \<le> ln (b * x) powr - (e / 2 + 1)"
iso_char: fixes f :: "'c \<times> 'c" shows "C.VxV.iso f = (C.iso (fst f) \<and> C.iso (snd f))"
rel_fundef_body_length: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and fd1 :: "'a fundef"   and fd2 :: "'b fundef" assumes "rel_fundef r fd1 fd2" shows "length (body fd1) = length (body fd2)"
normalize_mult_normalize_left: fixes a :: "'a"   and b :: "'a" shows "normalize (normalize a * b) = normalize (a * b)"
proots_def: fixes p :: "'b poly" shows "proots p = {x. poly p x = (0::'b)}"
5: shows "set_func (Id\<^bsub>Set\<^esub> F \<^bsub>\<o>\<^esub> A) a = a"
natLeq_on_ordLeq_less: fixes m :: "nat"   and n :: "nat" shows "(natLeq_on m <o natLeq_on n) = (m < n)"
image_mset_cong: fixes M :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x" shows "image_mset f M = image_mset g M"
invar_link: fixes t\<^sub>1 :: "'b tree"   and t\<^sub>2 :: "'b tree" assumes "invar_tree t\<^sub>1"   and "invar_tree t\<^sub>2"   and "rank t\<^sub>1 = rank t\<^sub>2" shows "invar_tree (link t\<^sub>1 t\<^sub>2)"
gcd_rat_to_gcd_int: fixes f :: "int poly"   and g :: "int poly" shows "gcd (of_int_poly f) (of_int_poly g) = smult (inverse (rat_of_int (lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
1: shows "e \<star> f \<star> src f \<cong> e \<star> f' \<star> src f"
eq_refl: fixes x :: "'b"   and y :: "'b" assumes "x = y" shows "x \<le> y"
ex_r: shows "\<exists>r. take_cols R (map snd (pivot_positions R)) \<in> carrier_mat n r \<and> take_rows R [0..<length (pivot_positions R)] \<in> carrier_mat r nc"
l1_iagreement_def: shows "l1_iagreement \<equiv> {s. \<forall>A B N. l1_state.signals s (Commit A B N) \<le> l1_state.signals s (Running A B N)}"
sum_unfold_sum_mset: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f A = comm_monoid_add.sum_mset plus zero (image_mset f (mset_set A))"
ZFfun_def: fixes d :: "ZF"   and r :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "ZFfun d r f \<equiv> Opair (Opair d r) (Lambda d f)"
M_def: fixes m :: "int"   and x :: "int" shows "poly_mod.M m x = x mod m"
M_inv: shows "\<lbrace>\<lambda>\<sigma>. \<not> exec_stop \<sigma> \<and> cond \<sigma> \<and> P \<sigma>\<rbrace> M \<lbrace>\<lambda>_. P\<rbrace>"
tranclp_trans_induct: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x y"   and "\<And>x y. r x y \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>r\<^sup>+\<^sup>+ x y; P x y; r\<^sup>+\<^sup>+ y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
homom2: fixes H :: "'a set"   and N :: "'a set" assumes "G \<triangleright> H"   and "G \<triangleright> N"   and "H \<subseteq> N" shows "gbij\<^bsub>(G / H / carrier (\<natural>N / H)),(G / N)\<^esub> (Qm\<^bsub>G H,N\<^esub>\<dieresis>\<^bsub>(G / H),(G / N)\<^esub>)"
induct: fixes x1 :: "'b val"   and x2 :: "'b rexp"   and P :: "'b val \<Rightarrow> 'b rexp \<Rightarrow> bool" assumes "\<turnstile> x1 : x2"   and "\<And>v1 r1 v2 r2. \<lbrakk>\<turnstile> v1 : r1; P v1 r1; \<turnstile> v2 : r2; P v2 r2\<rbrakk> \<Longrightarrow> P (Seq v1 v2) (Times r1 r2)"   and "\<And>v1 r1 r2. \<lbrakk>\<turnstile> v1 : r1; P v1 r1\<rbrakk> \<Longrightarrow> P (val.Left v1) (Plus r1 r2)"   and "\<And>v2 r2 r1. \<lbrakk>\<turnstile> v2 : r2; P v2 r2\<rbrakk> \<Longrightarrow> P (val.Right v2) (Plus r1 r2)"   and "P Void One"   and "\<And>c. P (Atm c) (Atom c)"   and "\<And>r. P (Stars []) (Star r)"   and "\<And>v r vs. \<lbrakk>\<turnstile> v : r; P v r; \<turnstile> Stars vs : Star r; P (Stars vs) (Star r)\<rbrakk> \<Longrightarrow> P (Stars (v # vs)) (Star r)" shows "P x1 x2"
ecl_ord_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.ecl_ord trm_ord = {(x, y). (basic_superposition.mset_ecl x, basic_superposition.mset_ecl y) \<in> mult (mult trm_ord)}"
item_rhs_split: fixes x :: "item" shows "item_rhs x = item_\<alpha> x @ item_\<beta> x"
mset_insert: fixes a :: "'b"   and h :: "'b hp option" shows "mset_heap (insert a h) = {#a#} + mset_heap h"
fg_eq: fixes w :: "complex" assumes "w \<in> s"   and "w \<noteq> z" shows "f w = g w * (w - z) powr of_int n"
oprod_def: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "r *o r' = {((x1, y1), x2, y2). (y1, y2) \<in> r' - Id \<and> x1 \<in> Field r \<and> x2 \<in> Field r \<or> (y1, y2) \<in> Restr Id (Field r') \<and> (x1, x2) \<in> r}"
Ri_subset_\<Gamma>: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set" shows "standard_redundancy_criterion.Ri \<Gamma> N \<subseteq> \<Gamma>"
simps_2: fixes x2 :: "'a" shows "None \<noteq> Some x2"
distributed_count_space: fixes M :: "'c measure"   and A :: "'d set"   and X :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> ennreal"   and a :: "'d" assumes "distributed M (count_space A) X P"   and "a \<in> A"   and "finite A" shows "P a = emeasure M (X -` {a} \<inter> space M)"
simps_3: fixes x2 :: "'d" shows "Some x2 \<noteq> None"
map_project_conv_fold: fixes f :: "'c \<Rightarrow> 'b option"   and xs :: "'c list" shows "List.map_project f (set xs) = fold (\<lambda>x A. case f x of None \<Rightarrow> A | Some y \<Rightarrow> insert y A) xs {}"
Monotheism_normalModel: shows "\<lfloor>(\<lambda>w. \<exists>x. \<forall>xa. (\<forall>x. (\<P> x \<^bold>\<rightarrow> x xa) w) = (x = xa))\<rfloor>"
get_shadow_root_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "h \<turnstile> ok get_shadow_root element_ptr" shows "element_ptr |\<in>| element_ptr_kinds h"
write_cpu_npc_privilege: fixes s' :: "'b sparc_state"   and w :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_cpu w nPC s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
poly_sum: fixes p :: "'b \<Rightarrow> 'a poly"   and A :: "'b set"   and x :: "'a" shows "poly (sum p A) x = (\<Sum>k\<in>A. poly (p k) x)"
simps: fixes x :: "nat"   and y :: "nat" shows "gcd x y = (if y = 0 then x else gcd y (x mod y))"
inner_gt_zero_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(0 < inner x x) = (x \<noteq> zero)"
dfs_match_distinct: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and xs :: "('d, 'e) bintree option list"   and a :: "'d" assumes "distinct (dfs f xs)" shows "distinct (dfs f (map (match a) xs))"
measure_nonneg: fixes M :: "'a measure"   and A :: "'a set" shows "0 \<le> Sigma_Algebra.measure M A"
sees_method_decl_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "'a prog"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and b' :: "staticb"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'a"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M, b' :  Ts'\<rightarrow>T' = m' in D'" shows "P \<turnstile> D' \<preceq>\<^sup>* D"
indets_monomial: fixes c :: "'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'c)" shows "indets (monomial c t) = keys t"
step_k: shows "S t k \<turnstile> t ! k \<mapsto> S t (k + 1)"
dvd_mult_cancel_left: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.idom times one plus zero minus uminus" shows "dvd.dvd times (times c a) (times c b) = (c = zero \<or> dvd.dvd times a b)"
assms_1: shows "prices Mkt asset (Suc n) w \<noteq> 0"
assms_4: fixes x :: "'b" assumes "x \<in> N - M" shows "0 \<le> f x"
assms: shows "grun A w V (fromN (Suc k) ||| r) (k, p)"
group_by_nonempty_inner: fixes b :: "'a"   and cs :: "'b set"   and f :: "'c \<Rightarrow> 'a \<times> 'b"   and as :: "'c set" assumes "(b, cs) \<in> group_by f as" obtains c :: "'b" where "c \<in> cs"
subspace_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale" shows "module.subspace scale S = ((0::'d) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. scale c x \<in> S))"
one_tensor_of_H_is_H: shows "Matrix.mat (2 ^ 1) (2 ^ 1) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>1\<^esub>  j / sqrt 2 ^ 1)) = H"
Existso_undef: fixes x :: "variable"   and \<phi> :: "fml option" shows "(Existso x \<phi> = undeff) = (\<phi> = undeff)"
equinumerous_def: fixes A :: "'b set"   and B :: "'c set" shows "(A \<cong> B) = (\<exists>f. bij_betw f A B)"
l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"
acyclic_4c_def: fixes x :: "'a" shows "acyclic_4c x \<equiv> \<forall>y\<le>x. y \<sqinter> (x \<sqinter> - y)\<^sup>\<star> = bot"
dres_Inf_chain_cases: fixes M :: "'a dres set" assumes "is_chain M" obtains "M \<subseteq> {dFAIL}" and "Inf M = dFAIL"   | r :: "'a" where "M \<subseteq> {dFAIL, dRETURN r}" and "dRETURN r \<in> M" and "Inf M = dRETURN r"   | "dSUCCEED \<in> M" and "Inf M = dSUCCEED"
coinduct: fixes X :: "'b llist \<Rightarrow> bool"   and x :: "'b llist"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> (\<exists>xa. x = LCons xa LNil) \<or> (\<exists>xs xa. x = LCons xa xs \<and> (X xs \<or> chain R xs) \<and> R xa (lhd xs))" shows "chain R x"
cases: fixes 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 a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "ty list" assumes "P,a1 \<turnstile> a2 [::] a3" obtains E :: "char list \<Rightarrow> ty option" where "a1 = Eb__" and "a2 = []" and "a3 = []"   | E :: "char list \<Rightarrow> ty option" and e :: "expr" and T :: "ty" and es :: "expr list" and Ts :: "ty list" where "a1 = E" and "a2 = e # es" and "a3 = T # Ts" and "P,E \<turnstile> e :: T" and "P,E \<turnstile> es [::] Ts"
finite_gpv_rel''D2: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and gpv :: "('e, 'g, 'i) gpv"   and gpv' :: "('f, 'h, 'j) gpv"   and \<I> :: "('h, 'j) \<I>"   and \<I>' :: "('g, 'i) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "finite_gpv \<I> gpv'"   and "rel_\<I> C R \<I>' \<I>" shows "finite_gpv \<I>' gpv"
complete_def: fixes S :: "'a set" shows "complete S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l))"
multiplicity_power_power: fixes n :: "nat"   and p :: "'a"   and x :: "'a" assumes "0 < n" shows "multiplicity (p ^ n) (x ^ n) = multiplicity p x"
subst_self_idemp: fixes x :: "'a"   and t :: "('b, 'a) Term.term" assumes "x \<notin> fv t" shows "subst x t \<circ>\<^sub>s subst x t = subst x t"
returns_to_congI2: fixes x :: "'a set"   and A :: "'a" assumes "auto_ll_on_open.returns_to g Y x A" shows "returns_to x A"
mono_meet: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and meet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and x :: "'a"   and x' :: "'a"   and y :: "'a"   and y' :: "'a" assumes "class.L_top_bot le join Top meet bot"   and "le x x'"   and "le y y'" shows "le (meet x y) (meet x' y')"
indep_sets_sigma_sets_iff: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'a set set" assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I = indep_sets F I"
brk_rel_def: fixes R :: "('b \<times> 'c) set" shows "brk_rel R \<equiv> {((False, x), False, y) |x y. (x, y) \<in> R} \<union> {((True, x), False, y) |x y. True}"
regular_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "regular x"   and "regular y" shows "regular (x * y)"
Imp_2: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "(F2, F2, \<Gamma> \<Rightarrow> \<Delta> \<longrightarrow> F2, \<Gamma> \<Rightarrow> \<Delta>) \<and> (\<Gamma> \<Rightarrow> F2, F2, \<Delta> \<longrightarrow> \<Gamma> \<Rightarrow> F2, \<Delta>)"
product_coneI: fixes a :: "'c"   and \<pi> :: "nat \<Rightarrow> 'c" assumes "D.limit_cone a \<pi>" shows "product_cone J.comp C (\<lambda>j. null) a \<pi>"
nextVertex_prevVertex: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f" shows "f \<bullet> (f\<^bsup>-1\<^esup> \<bullet> v) = v"
rtranclp_tranclp_tranclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
divide_minus1: fixes x :: "'a" shows "x / - (1::'a) = - x"
nor_equiv: fixes a :: "'a gexp"   and b :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (gNot (gOr a b)) s = gval (Nor a b) s"
exhaust_sel: fixes prot_atom :: "'e prot_atom" obtains "prot_atom = prot_atom.Atom (prot_atom.un_Atom prot_atom)"   | "prot_atom = Value"   | "prot_atom = SetType"   | "prot_atom = AttackType"   | "prot_atom = Bottom"   | "prot_atom = OccursSecType"
starfun_Re: fixes f :: "'a \<Rightarrow> complex" shows "*f* (\<lambda>x. Re (f x)) = (\<lambda>x. hRe ((*f* f) x))"
inverse_power_summable: fixes s :: "nat" assumes "2 \<le> s" shows "summable (\<lambda>n. inverse (of_nat n ^ s))"
cases: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d list"   and a2 :: "'d list" assumes "ord.lexordp_eq less a1 a2" obtains ys :: "'d list" where "a1 = []" and "a2 = ys__"   | x :: "'d" and y :: "'d" and xs :: "'d list" and ys :: "'d list" where "a1 = x__ # xs__" and "a2 = y__ # ysa__" and "less x__ y__"   | x :: "'d" and y :: "'d" and xs :: "'d list" and ys :: "'d list" where "a1 = x # xs" and "a2 = y # ys" and "\<not> less x y" and "\<not> less y x" and "ord.lexordp_eq less xs ys"
integrable_suminf: fixes M :: "'a measure"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>i. integrable M (f i)"   and "AE x in M. summable (\<lambda>i. norm (f i x))"   and "summable (\<lambda>i. LINT x|M. norm (f i x))" shows "integrable M (\<lambda>x. \<Sum>i. f i x)"
ne: shows "real_of_int (a mod b) / real_of_int b \<noteq> 0"
arctan_upper_31_def: shows "arctan_upper_31 \<equiv> \<lambda>x. - (pi / 2) - (64 + 735 * x\<^sup>2 + 945 * x ^ 4) / (15 * x * (15 + 70 * x\<^sup>2 + 63 * x ^ 4))"
lsemimodule2: fixes s :: "'b"   and t :: "'b"   and a :: "'a" shows "(s \<oplus> t) \<circ> a = s \<circ> a + t \<circ> a"
simps: fixes a :: "'f set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
right_impl_times: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) * a = (b l\<rightarrow> a) * b"
counterModel_def: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "counterModel f = Abs_model (counterM f, counterEvalP f)"
NSTD_lemma2': fixes c :: "nat" assumes "rec_exec rec_NSTD [c] = Suc 0" shows "NSTD c"
cospan_\<mu>\<nu>: shows "C.cospan \<mu>.dom.leg0 \<nu>.dom.leg1"
segment_a'_a_ne: shows "segment G.face_cycle_succ a' a \<noteq> {}"
\<T>_is_injection_2: fixes e :: "'s" assumes "e \<in> Ya'.SET a" shows "\<E> (\<T> e) = e"
GF_advice_congruent_axioms_def: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and normalise :: "'b ltln \<Rightarrow> 'b ltln" shows "GF_advice_congruent_axioms eq normalise \<equiv> ((\<forall>\<phi>. eq \<phi> (normalise \<phi>)) \<and> (\<forall>w \<phi> X. w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu> \<longrightarrow> w \<Turnstile>\<^sub>n normalise \<phi>[X]\<^sub>\<nu>)) \<and> (\<forall>w \<phi> X. w \<Turnstile>\<^sub>n normalise \<phi>[X]\<^sub>\<nu> \<longrightarrow> (\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu>)) \<and> (\<forall>\<phi> \<psi> X. eq \<phi> \<psi> \<longrightarrow> eq (normalise \<phi>[X]\<^sub>\<nu>) (normalise \<psi>[X]\<^sub>\<nu>))"
assms_3: shows "Sigma_Algebra.measure M A = Sigma_Algebra.measure M (space M)"
Max_gr_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Max less_eq A) = (\<exists>a\<in>A. less x a)"
is_extensional: fixes f :: "'a" assumes "\<not> B.arr f" shows "B.\<ll> f = B.null"
liftAll_lift_ext: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c"   and inp :: "'b \<Rightarrow> 'a option" shows "liftAll (\<lambda>x. f x = g x) inp = (lift f inp = lift g inp)"
normal_subgroup_in_kernel_axioms_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and unit' :: "'d"   and L :: "'c set" shows "normal_subgroup_in_kernel_axioms \<eta> G unit' L \<equiv> L \<subseteq> group_homomorphism.Ker \<eta> G unit'"
map_of_delete: fixes k' :: "'a"   and k :: "'a"   and al :: "('a \<times> 'b) list" assumes "k' \<noteq> k" shows "map_of (delete k al) k' = map_of al k'"
add_le_cancel_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus c a) (plus c b) = less_eq a b"
ray_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.ray \<P> betw a b \<equiv> insert b (MinkowskiBetweenness.segment \<P> betw a b \<union> MinkowskiBetweenness.prolongation \<P> betw a b)"
monotone_applyI: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'c"   and x :: "'d" assumes "monotone orda ordb F" shows "monotone (fun_ord orda) ordb (\<lambda>f. F (f x))"
integrable_count_space_nat_iff: fixes f :: "nat \<Rightarrow> 'b" shows "integrable (count_space UNIV) f = summable (\<lambda>x. norm (f x))"
valid_maximal_conforming_path_0: fixes P :: "'a llist"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "\<not> lnull P"   and "valid_path P"   and "maximal_path P"   and "path_conforms_with_strategy p P \<sigma>" shows "vmc_path G P (P $ 0) p \<sigma>"
fst_apsnd: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'c \<times> 'e" shows "fst (apsnd f x) = fst x"
simple_fw_alt: fixes r :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme" shows "simple_fw r p = simple_fw_alt r p"
bal_tree_def: fixes n :: "nat"   and t :: "'a tree" shows "bal_tree n t = bal_list n (inorder t)"
Lower_is_closed: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L" shows "is_closed\<^bsub>L\<^esub> (Lower L A)"
eq: fixes i :: "'b"   and j :: "'c" assumes "I i j" shows "\<mu>' (G i j) = \<mu> i j"
interfree_swap_def: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set" shows "interfree_swap \<Gamma> \<Theta> F \<equiv> \<lambda>(x, a, xs, as). \<forall>y<length xs. interfree_aux \<Gamma> \<Theta> F (x, a, xs ! y, fst (as ! y)) \<and> interfree_aux \<Gamma> \<Theta> F (xs ! y, as ! y, x, fst a)"
distributed_imp_emeasure_nonzero: fixes MX :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> ennreal" assumes "distributed M MX X Px" shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0"