hcju/isabelleps huggingface dataset:
abs_eq: shows "topfloat = Abs_float (0, - 1 - 1, 2 ^ LENGTH('b) - 1)"
emptyE: fixes a :: "'d"   and P :: "bool" assumes "a \<in> {}" shows "P"
open_mapping_thm: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and U :: "complex set" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "open U"   and "U \<subseteq> S"   and "\<not> f constant_on S" shows "open (f ` U)"
nxtActLe: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
fds_liouville_lambda: shows "fds liouville_lambda = fds_ind is_square * fds moebius_mu"
denote_while_scale: fixes \<rho> :: "complex mat"   and c :: "real"   and x1 :: "nat \<Rightarrow> complex mat"   and x2 :: "com" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1"   and "0 \<le> c"   and "measurement d 2 x1"   and "well_com x2"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>; trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1\<rbrakk> \<Longrightarrow> denote x2 (complex_of_real c \<cdot>\<^sub>m \<rho>) = complex_of_real c \<cdot>\<^sub>m denote x2 \<rho>" shows "denote_while (x1 0) (x1 1) (denote x2) (complex_of_real c \<cdot>\<^sub>m \<rho>) = complex_of_real c \<cdot>\<^sub>m denote_while (x1 0) (x1 1) (denote x2) \<rho>"
clo: fixes S :: "'a set" assumes "S \<in> \<A>" shows "closedin X S"
poly_X_def: shows "poly_X = Polynomial.monom (1::'b) 1"
pmc_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div>\<^bsub>\<gamma>\<^esub> \<phi> \<subseteq> K"
lconst_LNil: fixes a :: "'b" shows "lconst a \<noteq> LNil"
gexp_max_input_Nor: fixes g1 :: "vname gexp"   and g2 :: "vname gexp" shows "GExp.max_input (Nor g1 g2) = max (GExp.max_input g1) (GExp.max_input g2)"
distinct_fst_foldr_Cons: fixes f :: "'c \<Rightarrow> 'b"   and as :: "'c list"   and P :: "'c \<Rightarrow> 'c list \<Rightarrow> bool" assumes "distinct (map f as)" shows "distinct (map f (foldr (\<lambda>x xs. if P x xs then x # xs else xs) as []))"
GFP_greatest_fixed_point: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L"   and "x \<in> fps L f" shows "x \<sqsubseteq>\<^bsub>L\<^esub> GFP\<^bsub>L\<^esub> f"
DiffD1: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A - B" shows "c \<in> A"
M_car1: shows "mat_of_rows n G \<in> carrier_mat n n"
ltake_ltake: fixes n :: "enat"   and m :: "enat"   and xs :: "'b llist" shows "ltake n (ltake m xs) = ltake (min n m) xs"
Nmlize_in_Hom_2: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> \<in> VHom \<^bold>\<lfloor>Dom t\<^bold>\<rfloor> \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
seq_filter_increasing: fixes P :: "'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "\<forall>n. \<exists>m>n. P (s m)" shows "is_increasing (seq_filter s P)"
comp_in_homI': fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and c :: "'a" assumes "C.arr f"   and "C.arr g"   and "C.dom f = a"   and "C.cod g = c"   and "C.dom g = C.cod f" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow>\<^sub>C c\<guillemotright>"
h3: assumes "odd i \<and> i mod 2 = 1" shows "SQ ^ (i div 2) * real (i mod 2) * sqrt SQ + SQ ^ (i div 2) * (1 - real (i mod 2)) = sqrt SQ ^ i"
intrinsic_borderI': fixes ys :: "'a list" assumes "ys \<noteq> []" shows "strict_border (intrinsic_border ys) ys"
t_twice_def: shows "t_twice = adjust0 t_twice_compile"
right_adjoint_to_equivalence_is_equivalence: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.equivalence_map f"   and "Span.adjoint_pair f g" shows "Span.equivalence_map g"
com_validity_def: fixes P :: "'b com"   and pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and post :: "'b set" shows "\<Turnstile> P sat [pre, rely, guar, post] \<equiv> \<forall>s. cp (Some P) s \<inter> assum (pre, rely) \<subseteq> comm (guar, post)"
cr_nonzero_def: shows "cr_nonzero \<equiv> \<lambda>x y. x = Rep_nonzero y"
cobounded1: fixes a :: "'g"   and b :: "'g" shows "a \<le> sup a b"
put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'm, 'n, 'v option) RShadowRoot_scheme" shows "put shadow_root_ptr shadow_root = put (cast shadow_root_ptr) (cast shadow_root)"
bangUnfold: fixes P :: "ccs" shows "!P \<sim> P \<parallel> !P"
ETP_base_def: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "ETP_base I domain range F F\<^sub>i\<^sub>n\<^sub>v \<equiv> etp I domain range F F\<^sub>i\<^sub>n\<^sub>v"
truncate_down_def: fixes prec :: "nat"   and x :: "real" shows "truncate_down prec x = round_down (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>) x"
intros_2: fixes h :: "'c"   and H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
split_comb_seq: fixes P :: "'b \<Rightarrow> bool"   and i :: "nat"   and \<omega> :: "nat \<Rightarrow> 'b"   and \<omega>' :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "P (comb_seq i \<omega> \<omega>' j) = ((j < i \<longrightarrow> P (\<omega> j)) \<and> (\<forall>k. j = i + k \<longrightarrow> P (\<omega>' k)))"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'c list"   and xs\<^sub>2 :: "'c list"   and ys\<^sub>1 :: "'c list"   and ys\<^sub>2 :: "'c list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if length xs\<^sub>1 \<le> length ys\<^sub>1 then xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2 else take (length ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
mult_le_cancel_iff1: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "(0::'c) < z" shows "(x * z \<le> y * z) = (x \<le> y)"
nsetTr1: fixes j :: "nat"   and a :: "nat"   and b :: "nat" assumes "j \<in> nset a b"   and "j \<noteq> a" shows "j \<in> nset (Suc a) b"
proj_psi_l_dim: fixes l :: "nat" shows "proj_psi_l l \<in> carrier_mat N N"
tighten_bounds_2_1: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "real_of_2 (tighten_bounds_2 x) = real_of_2 x"
d: fixes x :: "real" assumes "x \<in> norm ` comb' ` sphere 0 1" shows "d \<le> dist 0 x"
conj_comm: fixes P :: "'b upred"   and Q :: "'b upred" shows "(P \<and> Q) = (Q \<and> P)"
compat_low_eq: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and x :: "'Var" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "dma x = Low"   and "\<forall>i<length cms\<^sub>1. x \<notin> snd (cms\<^sub>1 ! i) AsmNoRead" shows "mem\<^sub>1 x = mem\<^sub>2 x"
proots_ball_plane_eq: fixes p :: "complex poly" assumes "p \<noteq> 0" shows "proots_count p (ball 0 1) = proots_count (fcompose p [:\<i>, - 1:] [:\<i>, 1:]) {x. 0 < Im x}"
distinct_31: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x17 :: "nat" shows "Add x11 x12 \<noteq> Var x17"
CHAOS_all_tickfree_traces1: fixes A :: "'b set" shows "T (CHAOS A) = {s. set s \<subseteq> ev ` A}"
irred_idom_nz: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "irred x = irreducible x"
alw_iff_sdrop: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "alw P \<omega> = (\<forall>m. P (sdrop m \<omega>))"
ex1_apartment_iso: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'" shows "\<exists>!f. ChamberComplexIsomorphism A A' f \<and> fixespointwise f (\<Union> (A \<inter> A')) \<and> fixespointwise f (- \<Union> A)"
isCont_blinfunD: fixes a :: "'b"   and f' :: "'b \<Rightarrow> 'c \<Rightarrow>\<^sub>L 'd"   and e :: "real" assumes "isCont f' a"   and "0 < e" shows "\<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. blinfun_apply (f' x) v - blinfun_apply (f' a) v) < e"
assms_2: shows "Q \<Longrightarrow>\<^sub>\<tau> Q'"
map_def: fixes f :: "'a1" shows "A1.map f = (if A1.arr f then f else A1.null)"
simps_4: fixes x21 :: "'a"   and x22 :: "'a" shows "DenyAll \<noteq> DenyAllFromTo x21 x22"
Seq2: fixes c1 :: "com"   and s :: "char list \<Rightarrow> val"   and c1' :: "com"   and s' :: "char list \<Rightarrow> val"   and c2 :: "com" assumes "(c1, s) \<rightarrow> (c1', s')" shows "(c1;; c2, s) \<rightarrow> (c1';; c2, s')"
part_is_subset: fixes S :: "'a set"   and P :: "'a set set"   and p :: "'a set" assumes "part S P"   and "p \<in> P" shows "p \<subseteq> S"
finite_ex_and1: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite {b. \<exists>a. P a b}" shows "finite {b. \<exists>a. P a b \<and> Q a b}"
injective_map_axioms: shows "injective_map \<eta> R R'"
non_trivial_coprime_neq: fixes a :: "int"   and b :: "int" assumes "1 < a"   and "1 < b"   and "coprime a b" shows "a \<noteq> b"
code: fixes a :: "'a"   and b :: "'a" shows "mssalternate a b = MSSCons a (mssalternate b a)"
poly_vars_list: fixes p :: "('c monom \<times> 'd) list" shows "set (poly_vars_list p) = poly_vars p"
rank_tower: fixes i :: "nat"   and j :: "nat"   and x :: "nat"   and n :: "nat"   and m :: "nat"   and y :: "nat" assumes "i \<le> j"   and "rank x n = Some j"   and "rank x (n + m) = Some j"   and "rank y n = Some i" shows "rank y (n + m) = Some i"
hensel_4: shows "\<forall>n>0. degree (hensel_fpxs1 $ n) < degree g"
at_exists_fresh: fixes x :: "'b" assumes "at TYPE('a)"   and "finite (supp x)" obtains c :: "'a" where "c \<sharp> x"
implies_curry: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z - y - x = z - (x \<squnion> y)"
primepow_prime: fixes n :: "'a" assumes "prime n" shows "primepow n"
liouville_lambda_divisors_sum: fixes n :: "nat" shows "(\<Sum>d | d dvd n. liouville_lambda d) = ind is_square n"
prems_2: shows "set (x # xs) \<subseteq> s"
stable_no_blocking_onI: fixes X :: "'x set"   and ds :: "'d set" assumes "\<And>h X''. \<lbrakk>X'' = Ch h (X \<union> X''); X'' \<noteq> Ch h X; X'' \<subseteq> CD_on ds (X \<union> X'')\<rbrakk> \<Longrightarrow> False" shows "stable_no_blocking_on ds X"
assms_8: shows "(i_begin, i_end) \<in> set (identify_blocks A k)"
encode_problem_parallel_complete: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list" assumes "is_valid_problem_strips \<Pi>"   and "is_parallel_solution_for_problem \<Pi> \<pi>" shows "valuation_for_plan \<Pi> \<pi> \<Turnstile> \<Phi> \<Pi> length \<pi>"
multi_member_split: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "\<exists>A. M = add_mset x A"
reach9: fixes s1 :: "Store"   and x :: "Value"   and s2 :: "Store"   and l :: "Location" assumes "\<forall>l. s1\<turnstile> l reachable_from x \<longrightarrow> s1@@l = s2@@l" shows "s1\<turnstile> l reachable_from x = s2\<turnstile> l reachable_from x"
ConsCons1: fixes e1 :: "'a \<times> 'a"   and e2 :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list" assumes "e1 = (fst e, w) \<and> e2 = (w, snd e)"   and "P e w es" shows "P e w (e1 # e2 # es)"
tree_decomposition_exists: shows "\<exists>T bag. TreeDecomposition G T bag"
idinfo_crypt: fixes K :: "enc_key"   and n :: "nat"   and X :: "msg"   and s :: "(agent \<times> msg) set" assumes "(Spy, Crypt K \<langle>n, X\<rangle>) \<in> s"   and "s\<^sub>0 \<Turnstile> s" shows "\<langle>n, X\<rangle> \<in> spied s"
tl_def: fixes list :: "'a list" shows "tl list = (case list of [] \<Rightarrow> [] | x21 # x22 \<Rightarrow> x22)"
\<dagger>: fixes y :: "real" shows "(\<lambda>x. if y \<le> g x then f x else 0) \<in> {\<lambda>x. 0, f} \<union> (\<Union>z. {\<lambda>x. if z < x then f x else 0}) \<union> (\<Union>z. {\<lambda>x. if z \<le> x then f x else 0})"
interpretation_grounds_all: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and v :: "'c" assumes "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "fv (\<theta> v) = {}"
rrsmult: fixes H :: "'a set"   and G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and rrsmult :: "('b, 'a) aezfun \<Rightarrow> (('b, 'a) aezfun \<Rightarrow> 'c) \<Rightarrow> ('b, 'a) aezfun \<Rightarrow> 'c" assumes "InducedFinGroupRepresentation H G smult V rrsmult" shows "rrsmult = Ring1.rightreg_scalar_mult (aezfun_setspan H)"
sets_Sup_in_sets: fixes M :: "'b measure set"   and N :: "'b measure" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> space m = space N"   and "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N" shows "sets (Sup M) \<subseteq> sets N"
fresh_string: fixes a :: "'v"   and s :: "char list" shows "a \<sharp> s"
assms_3: fixes c :: "'b"   and t :: "'a" shows "f (monom_mult c t p) = monom_mult c t (f p)"
AX0p_AX0_eq: shows "AX0p = (\<lambda>x. x \<in> AX0)"
valid_weak_action_modality: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" shows "P \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> P \<Rightarrow>\<langle>\<alpha>''\<rangle> P' \<and> P' \<Turnstile> x')"
intros_5: fixes a1 :: "aexp"   and s :: "char list \<Rightarrow> val"   and r1 :: "real"   and a2 :: "aexp"   and r2 :: "real" assumes "taval a1 s (Rv r1)"   and "taval a2 s (Rv r2)" shows "taval (Plus a1 a2) s (Rv (r1 + r2))"
intro_elim_1: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]"   and "[\<psi> in v]" shows "[\<phi> \<^bold>& \<psi> in v]"
proj_fst_k_le_one: fixes k :: "nat" shows "proj_fst_k k \<le>\<^sub>L 1\<^sub>m K"
induct: fixes P :: "'a mod_ring poly \<Rightarrow> 'a mod_ring poly \<Rightarrow> nat \<Rightarrow> (nat \<times> 'a mod_ring poly) list \<Rightarrow> bool"   and a0 :: "'a mod_ring poly"   and a1 :: "'a mod_ring poly"   and a2 :: "nat"   and a3 :: "(nat \<times> 'a mod_ring poly) list" assumes "\<And>v w d res. \<lbrakk>\<And>x xa xb. \<lbrakk>v \<noteq> 1; \<not> degree v < d + d; x = w ^ CARD('a) mod v; xa = Suc d; xb = gcd (x - monom 1 1) v; xb = 1\<rbrakk> \<Longrightarrow> P v x xa res; \<And>x xa xb xc. \<lbrakk>v \<noteq> 1; \<not> degree v < d + d; x = w ^ CARD('a) mod v; xa = Suc d; xb = gcd (x - monom 1 1) v; xb \<noteq> 1; xc = v div xb\<rbrakk> \<Longrightarrow> P xc (x mod xc) xa ((xa, xb) # res)\<rbrakk> \<Longrightarrow> P v w d res" shows "P a0 a1 a2 a3"
card_eq_1_iff: fixes A :: "'a set" shows "(card A = 1) = (A \<noteq> {} \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x = y))"
eval_ring_hom: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and s :: "'e" assumes "UP_pre_univ_prop R S h"   and "s \<in> carrier S" shows "eval R S h s \<in> ring_hom (UP R) S"
b2: shows "b = q * a + smult p (smult (p ^ (l - 1)) r)"
Let_1: fixes x :: "Vars.var"   and ea :: "exp"   and aa :: "Arity"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "(x, ea) \<in> set \<Gamma>" shows "eq aa (\<lbrakk> \<T>\<^bsub>aa\<^esub> ea \<rbrakk>\<^bsub>\<rho>\<^esub>) (\<lbrakk> ea \<rbrakk>\<^bsub>\<rho>\<^esub>)"
power_Suc2: fixes a :: "('b \<times> 'b) set"   and n :: "nat" shows "rel_dioid.power a (Suc n) = rel_dioid.power a n O a"
dghm_singleton_ObjMap_vrange: fixes j :: "V"   and \<CC> :: "V" shows "\<R>\<^sub>\<circ> (dghm_singleton j \<CC>\<lparr>ObjMap\<rparr>) = (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)\<lparr>Obj\<rparr>"
fact: shows "{V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))} = {V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V)}"
elims: fixes x :: "'b trm"   and y :: "Preliminaries.var set" assumes "nvT x = y" obtains x :: "Preliminaries.var" where "xa__ = Var x" and "y = {x}"   | f :: "'b" and Tl :: "'b trm list" where "xa__ = Fn f Tl" and "y = {}"
star_invol: fixes x :: "('a \<times> 'a) set" shows "(x\<^sup>*)\<^sup>* = x\<^sup>*"
rev_permutations_of_multiset: fixes A :: "'a multiset" shows "rev ` permutations_of_multiset A = permutations_of_multiset A"
less_eqE: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># b" obtains c :: "'a multiset" where "b = a + c"
pt_open_on_good: fixes Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "pt_open_on (good Q) (SEQ A)"
mset_size_le1_cases: fixes M :: "'c multiset" assumes "size M \<le> Suc 0" obtains "M = {#}"   | m :: "'c" where "M = {#m#}"
frameResChainFreshSet: fixes Xs :: "name set"   and xvec :: "name list"   and F :: "'d frame" shows "Xs \<sharp>* (\<lparr>\<nu>*xvec\<rparr>F) = (\<forall>x\<in>Xs. x mem xvec \<or> x \<sharp> F)"
edges_return: fixes x :: "'n" assumes "(local.return, x) \<in> local.edges" shows "x = local.return"
match_list_append: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m1 :: "'b match_expr list"   and m2 :: "'b match_expr list"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> (m1 @ m2) a p = (\<not> match_list \<gamma> m1 a p \<longrightarrow> match_list \<gamma> m2 a p)"
mk_linear_orders_MaxR_opt_f: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set"   and D :: "'a set"   and x :: "'a" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C A"   and "MaxR.MaxR_opt (linord_of_list r) D = Some x"   and "iia_on A f"   and "D \<subseteq> A" shows "x \<in> f D"
k: shows "(\<natural>\<sigma>) (Suc k) = (\<natural>\<sigma>) k"
assms_1: shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs @ A_y # A_ys) @ A_neg) ((zs @ x # xs @ y # ys) @ L_neg)"
corthogonal_def: fixes vs :: "'a vec list" shows "corthogonal vs \<equiv> \<forall>i<length vs. \<forall>j<length vs. (vs ! i \<bullet>c vs ! j = (0::'a)) = (i \<noteq> j)"
trancl_induct2: fixes ax :: "'a"   and ay :: "'b"   and bx :: "'a"   and "by" :: "'b"   and r :: "(('a \<times> 'b) \<times> 'a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>+"   and "\<And>a b. ((ax, ay), a, b) \<in> r \<Longrightarrow> P a b"   and "\<And>a b aa ba. \<lbrakk>((ax, ay), a, b) \<in> r\<^sup>+; ((a, b), aa, ba) \<in> r; P a b\<rbrakk> \<Longrightarrow> P aa ba" shows "P bx by"
Nodes_def: fixes i :: "nat"   and levellist :: "ref list list" shows "Nodes i levellist = (\<Union>k\<in>{k. k < i}. set (levellist ! k))"
domD: fixes a :: "'c"   and m :: "'c \<Rightarrow> 'd option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
strong_ordertype_eq: fixes D :: "V set"   and \<beta> :: "V" assumes "D \<subseteq> elts \<beta>"   and "Ord \<beta>" obtains L :: "V set list" where "\<Union> (list.set L) = D" and "\<And>X. X \<in> list.set L \<Longrightarrow> indecomposable (tp X)" and "\<And>M. \<lbrakk>M \<subseteq> D; \<And>X. X \<in> list.set L \<Longrightarrow> tp X \<le> tp (M \<inter> X)\<rbrakk> \<Longrightarrow> tp M = tp D"
primroot_rotate_comm: fixes w :: "'b list"   and n :: "nat" assumes "w \<noteq> \<epsilon>" shows "\<rho> (rotate n w) = rotate n (\<rho> w)"
combine_t_wise_balance_index: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = \<t>" shows "int (\<B>\<^sup>+ index ps) = \<Lambda>\<^sub>t + \<Lambda>\<^sub>t'"
comp_fun_idem_merge: shows "comp_fun_idem (\<lambda>x yo. yo \<bind> m x)"
lcos_mult_one: fixes G :: "('d, 'e) monoid_scheme"   and M :: "'d set" assumes "Group.group G"   and "M \<subseteq> carrier G" shows "\<one>\<^bsub>G\<^esub> <#\<^bsub>G\<^esub> M = M"
mono_Span: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and u :: "'a" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R"   and "u \<in> carrier R" shows "ring.Span R K Us \<subseteq> ring.Span R K (u # Us)"
not_unprot_inImage_ntsem: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "\<not> unprot \<sigma>"   and "\<not> protFw \<sigma>"   and "a \<notin> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}" shows "ntsem \<sigma> a = eintF (Tag \<sigma>) [a]"
mutual_information_distr': fixes S :: "'b measure"   and T :: "'f measure"   and X :: "'a \<Rightarrow> 'b"   and Px :: "'b \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and Pxy :: "'b \<times> 'f \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "finite_entropy S X Px"   and "finite_entropy T Y Py"   and "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "mutual_information b S T X Y = LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
PRE_D1: fixes Q :: "'a \<Rightarrow> bool"   and x :: "'a"   and P :: "'a \<Rightarrow> bool"   and S1 :: "('b \<times> 'a) set"   and S :: "'a \<Rightarrow> bool" shows "Q x \<and> P x \<longrightarrow> comp_PRE S1 Q (\<lambda>x _. P x) S x"
image_empty: fixes f :: "'h \<Rightarrow> 'g" shows "f ` {} = {}"
remdups_map_remdups: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "remdups (list.map f (remdups xs)) = remdups (list.map f xs)"
cases: fixes x :: "('b \<times> nat) tree" obtains t1 :: "('b \<times> nat) tree" and a :: "'b" and lva :: "nat" and t2 :: "('b \<times> nat) tree" and b :: "'b" and lvb :: "nat" and t3 :: "('b \<times> nat) tree" and c :: "'b" and lvc :: "nat" and t4 :: "('b \<times> nat) tree" where "x = \<langle>t1, (a, lva), \<langle>t2, (b, lvb), \<langle>t3, (c, lvc), t4\<rangle>\<rangle>\<rangle>"   | "x = \<langle>\<rangle>"   | v :: "('b \<times> nat) tree" and va :: "'b \<times> nat" where "x = \<langle>v__, va__, \<langle>\<rangle>\<rangle>"   | v :: "('b \<times> nat) tree" and va :: "'b \<times> nat" and vc :: "('b \<times> nat) tree" and vd :: "'b \<times> nat" where "x = \<langle>v, va, \<langle>vc, vd, \<langle>\<rangle>\<rangle>\<rangle>"
rev_curr_FinFunc: assumes "Field r \<noteq> {}" shows "local.rev_curr ` FinFunc r (s *o t) = FinFunc (r ^o s) t"
cld_for_path: fixes P :: "cld list"   and ctx :: "ctx"   and fqn :: "fqn"   and path :: "(ctx \<times> cld) list" assumes "find_path_f P ctx (cl_fqn fqn) = Some path" shows "\<exists>ctxcld. find_cld_f P ctx fqn = Some ctxcld"
ub_rangeI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'b"   and x :: "'b" assumes "class.po below"   and "\<And>i. below (S i) x" shows "po.is_ub below (range S) x"
WCFG_SeqFirst: fixes c\<^sub>1 :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and c\<^sub>2 :: "cmd" assumes "c\<^sub>1 \<turnstile> n -et\<rightarrow> n'"   and "n' \<noteq> (_Exit_)" shows "c\<^sub>1;; c\<^sub>2 \<turnstile> n -et\<rightarrow> n'"
distinct_5: fixes x1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree" shows "tConj x1 \<noteq> tAct x41 x42 x43"
oalist_inv_sort_oalist_aux: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "oalist_inv (sort_oalist_aux ko xs, ko)"
csl: shows "cs\<^bsup>\<pi>\<^sub>1\<^esup> l = cs\<^bsup>\<pi>\<^sub>1'\<^esup> l'"
a_mul_d': fixes x :: "'a" shows "ad x \<cdot> apd_d x = (zero_class.zero::'a)"
map2_val_compatD2: fixes f :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'f) list"   and zs :: "('a \<times> 'e) list" assumes "map2_val_compat f"   and "oalist_inv_raw zs" shows "fst ` set (f zs) \<subseteq> fst ` set zs"