hcju/isabelleps huggingface dataset:
Equality_E: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (t SUBS u) (insert (u SUBS t) H) \<turnstile> A" shows "insert (t EQ u) H \<turnstile> A"
boundedI: fixes a :: "'h"   and b :: "'h"   and c :: "'h" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> min b c"
rroots3_def: shows "rroots3 = roots3_generic solve_depressed_cubic_real"
suffix_ConsD2: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "suffix (x # xs) (y # ys)" shows "suffix xs ys"
CARD_1: assumes "class.CARD_1 TYPE('b)" shows "CARD('b) = 1"
image_restrict_eq: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "restrict f A ` A = f ` A"
trcl_unconcat: fixes c :: "'a"   and w1 :: "'b list"   and w2 :: "'b list"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" assumes "(c, w1 @ w2, c') \<in> trcl t" shows "\<exists>ch. (c, w1, ch) \<in> trcl t \<and> (ch, w2, c') \<in> trcl t"
get_shadow_root_locs_def: shows "get_shadow_root_locs \<equiv> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_locs"
existN_eval: fixes n :: "nat"   and \<phi> :: "atom fm" shows "\<forall>xs. eval (ExN n \<phi>) xs = (\<exists>L. length L = n \<and> eval \<phi> (L @ xs))"
star_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c" assumes "class.left_near_kleene_algebra plus times one less_eq less star" shows "star one = one"
Sup_set_fold: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and xs :: "'c list" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (set xs) = fold sup xs bot"
reify_monom1: shows "\<Theta>(\<lambda>x. x) = \<Theta>(eval_primfun (LnChain 0, 1))"
nro_i: shows "no_write_to_read_only_memory \<S> sb\<^sub>i"
f1: shows "i < dim_row U\<^sub>f \<and> j < dim_col U\<^sub>f"
set_eq_iff_mset_eq_distinct: fixes x :: "'b list"   and y :: "'b list" assumes "distinct x"   and "distinct y" shows "(set x = set y) = (mset x = mset y)"
IH: fixes e\<^sub>1 :: "expr" assumes "e = \<lparr>C\<rparr>e\<^sub>0"   and "e'' = \<lparr>C\<rparr>e\<^sub>1" shows "P,E \<turnstile> \<langle>e\<^sub>0,s\<rangle> \<rightarrow>* \<langle>e\<^sub>1,s''\<rangle>"
cases: fixes x :: "nat list" obtains "xa__ = []"   | x :: "nat" where "xa__ = [xb__]"   | x :: "nat" and v :: "nat" and va :: "nat list" where "xa__ = x # v # va"
r_dummy: fixes n :: "nat"   and f :: "recf" shows "r_dummy n f = Cn (arity f + n) f (map (recf.Id (arity f + n)) [0..<arity f])"
conjugate_zero: shows "conjugate (0::'a) = (0::'a)"
ord_le_eq_trans: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f"   and b :: "'f"   and c :: "'f" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
eigenvalue_imp_nonzero_dim: fixes A :: "'a mat"   and n :: "nat"   and ev :: "'a" assumes "A \<in> carrier_mat n n"   and "eigenvalue A ev" shows "0 < n"
eval_monom_list: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'd"   and m :: "('c \<times> nat) list"   and n :: "('c \<times> nat) list" assumes "class.linorder less_eq less" shows "linorder.eval_monom_list \<alpha> (m @ n) = linorder.eval_monom_list \<alpha> m * linorder.eval_monom_list \<alpha> n"
minus_unique: fixes y :: "nat \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'a"   and y' :: "nat \<Rightarrow> 'a" assumes "y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"   and "x \<oplus>\<^bsub>P\<^esub> y' = \<zero>\<^bsub>P\<^esub>"   and "x \<in> carrier P"   and "y \<in> carrier P"   and "y' \<in> carrier P" shows "y = y'"
fds_abs_converges_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b fds"   and s :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> fds_abs_converges (f x) s" shows "fds_abs_converges (prod f A) s"
preorder_idempotent: fixes x :: "'a" assumes "preorder x" shows "idempotent x"
pq_sum: fixes k :: "nat" shows "real_of_int (p k) / real_of_int (q k) = (\<Sum>i = 0..k. real_of_int (b i) / real_of_int (a i))"
ih: fixes C :: "'a literal list" assumes "reduce Ds C C' \<noteq> C'" shows "\<exists>L\<in>set C'. is_reducible_lit Ds (C @ remove1 L C') L"
inter_guards_execn_Normal_Fault: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "'d"   and n :: "nat"   and f :: "'f" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> Fault f \<or> \<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> =n\<Rightarrow> Fault f"
hcomp_simps_1: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.hseq \<nu> \<mu>" shows "src\<^sub>D (\<nu> \<star>\<^sub>D \<mu>) = src\<^sub>D \<mu>"
attractive_order_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.attractive_order less_eq less \<equiv> attractive_ordering UNIV less_eq less"
expand: fixes interval :: "'a interval"   and interval' :: "'a interval" assumes "low interval = low interval' \<and> high interval = high interval'" shows "interval = interval'"
trms\<^sub>s\<^sub>s\<^sub>t_in: fixes t :: "('e, 'f) Term.term"   and S :: "('e, 'f) stateful_strand_step list" assumes "t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" shows "\<exists>a\<in>set S. t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
comp_null_1: fixes f :: "'a" shows "Right_a.comp Right_a.null f = Right_a.null"
diagseq_def: fixes i :: "nat" shows "diagseq i = seqseq i i"
ext_cf_hi: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat" assumes "pol_coeff S c" shows "snd c (fst c) = snd (ext_cf S n c) (n + fst c)"
assms_5: fixes t' :: "'state" assumes "(d, t) \<rightarrow>*t t'"   and "s' \<approx> t'"   and "discr c'" shows "thesis"
MACHIN_TAG_def: fixes a :: "real"   and b :: "real" shows "MACHIN_TAG a b \<equiv> a * b"
monoid: shows "Group.monoid (R\<lparr>carrier := K\<rparr>)"
pred_cong: fixes x :: "'c pmf"   and ya :: "'c pmf"   and P :: "'c \<Rightarrow> bool"   and Pa :: "'c \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set_pmf ya \<Longrightarrow> P z = Pa z" shows "pred_pmf P x = pred_pmf Pa ya"
intro: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "profile_mono_scheme_defined f g"   and "profile_mono\<^sub>d_axioms g" shows "profile_mono\<^sub>d f g"
del_point_blocks_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and p :: "'b" assumes "incidence_system point_set block_collection" shows "incidence_system.del_point_blocks block_collection p \<equiv> {#bl - {p}. bl \<in># block_collection#}"
refine: fixes Xi :: "'b list"   and X :: "'c set"   and ivli :: "real list \<times> real list"   and ivl :: "'c set" assumes "(Xi, X) \<in> appr_rel"   and "(ivli, ivl) \<in> lvivl_rel" shows "(RETURN (subset_spec_ivlc Xi ivli), op_subset $ X $ ivl) \<in> \<langle>bool_rel\<rangle>nres_rel"
x_def: shows "x = [a, a]\<^sub>\<circ>"
pt_fresh_right: fixes pi :: "('b \<times> 'b) list"   and a :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> a \<sharp> x = a \<sharp> rev pi \<bullet> x"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sub>e <\<nu>x>P \<parallel> Q"
iso_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "category.iso (product_category.comp C1 C2) f = (category.iso C1 (fst f) \<and> category.iso C2 (snd f))"
regular_Int: fixes S :: "'b list set"   and T :: "'b list set" assumes "regular S"   and "regular T" shows "regular (S \<inter> T)"
finite_refines_finite: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "finite (A // R)"   and "R \<subseteq> S"   and "equiv A R"   and "equiv A S" shows "finite (A // S)"
tab_simps_5: shows "cod tab = (k\<^sup>* \<star> h) \<star> p\<^sub>0"
fs': shows "of_int_hom.mat_hom (mat_of_cols n fs) \<in> carrier_mat n n"
f: shows "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
Domainp_forall_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "right_total A" shows "rel_fun (rel_fun A (=)) (=) (transfer_bforall (Domainp A)) transfer_forall"
2: shows "is_up_set (\<Up>{y. \<exists>z\<in>F. x \<sqinter> z = y})"
length_tensor_vec_from_lookup: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" shows "length (tensor_vec_from_lookup ds e) = prod_list ds"
simps_12: fixes x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf" shows "Pr x51 x52 x53 \<noteq> Z"
map_comp_Some_iff: fixes m1 :: "'e \<Rightarrow> 'd option"   and m2 :: "'f \<Rightarrow> 'e option"   and k :: "'f"   and v :: "'d" shows "((m1 \<circ>\<^sub>m m2) k = \<lfloor>v\<rfloor>) = (\<exists>k'. m2 k = \<lfloor>k'\<rfloor> \<and> m1 k' = \<lfloor>v\<rfloor>)"
lfp_mono: fixes F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "mono_mono F" shows "mono (lfp F)"
assms_2: shows "Case_Labeling.CTXT IC CT OC1 (ovalidNF P f (\<lambda>_. R))"
pcorres_drefine: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and C :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "pcorres \<phi> G A C"   and "well_def C" shows "drefines \<phi> G A C"
n_eq_INF_v: fixes s :: "'s" assumes "s \<in> S" shows "n s = \<Sqinter> (v ` cfg_on s)"
repcs_bisim: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and x :: "'s \<times> ('c \<Rightarrow> real)"   and x' :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "x \<in> S"   and "x ~ x'"   and "abss x = state cfg" shows "absc (repcs x cfg) = absc (repcs x' cfg)"
isin_simps_2: fixes x :: "'b"   and xs :: "'b list" shows "Sorted_Less.sorted (x # xs) = ((\<forall>y\<in>set xs. x < y) \<and> Sorted_Less.sorted xs)"
disj_conj_abs: fixes P :: "'a upred"   and Q :: "'a upred" shows "(P \<or> P \<and> Q) = P"
OclIf_def: fixes C :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B\<^sub>1 :: "'d state \<times> 'd state \<Rightarrow> 'e"   and B\<^sub>2 :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(if C then B\<^sub>1 else B\<^sub>2 endif) = (\<lambda>\<tau>. if (\<delta> C) \<tau> = true \<tau> then if C \<tau> = true \<tau> then B\<^sub>1 \<tau> else B\<^sub>2 \<tau> else invalid \<tau>)"
linear_imp_has_derivative: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "linear f" shows "(f has_derivative f) net"
local_order_carrier_closed: fixes e1 :: "'msg event"   and i :: "nat"   and e2 :: "'msg event" assumes "e1 \<sqsubset>\<^sup>i e2" shows "{e1, e2} \<subseteq> set (history i)"
isSubobj_isSubobj_isSubobj_rev: 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 C :: "char list"   and D :: "char list"   and Cs :: "char list list" assumes "is_subobj P (C, [D])"   and "is_subobj P (D, D # rev Cs)" shows "is_subobj P (C, D # rev Cs)"
r_le_rtrancl: fixes S :: "('b \<times> 'b) set" shows "S \<subseteq> S\<^sup>*"
elims: fixes x :: "nat \<times> cell list \<times> cell list"   and xa :: "(action \<times> nat) list \<times> nat"   and xb :: "nat"   and y :: "nat \<times> cell list \<times> cell list" assumes "steps x xa xb = y" obtains c :: "nat \<times> cell list \<times> cell list" and p :: "(action \<times> nat) list \<times> nat" where "x = c__" and "xa = p__" and "xb = 0" and "y = c__"   | c :: "nat \<times> cell list \<times> cell list" and p :: "(action \<times> nat) list \<times> nat" and n :: "nat" where "x = c" and "xa = p" and "xb = Suc n" and "y = steps (step c p) p n"
assms_5: shows "\<forall>v\<in>dom G. the (G v) \<in> \<R>\<^sub>+ \<Psi> v"
same_level_path_valid_path_Append: fixes as :: "'edge list"   and as' :: "'edge list" assumes "same_level_path as"   and "valid_path as'" shows "valid_path (as @ as')"
select_result_I2: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and x :: "'ar" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = x"
assms_3: fixes i :: "'c" assumes "i \<in> I" shows "subgroup (S i) G"
preserved: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "P (moebius_pt M u) (moebius_pt M v)" shows "P u v"
prj_in_hom_4: shows "\<guillemotleft>\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1 : HH\<mu>\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<mu>\<nu>.apex\<guillemotright>"
wffp_hdI: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list"   and s :: "('a, 'b, 'c) state" assumes "GM_sec_model.wffp do sl"   and "hd sl \<in> GM_sec_model.\<delta> do s" shows "GM_sec_model.wffp do (s # sl)"
sqrt_nat_or_irrat': fixes a :: "nat" shows "sqrt (real a) \<in> \<nat> \<or> sqrt (real a) \<notin> \<rat>"
v'_4: shows "R' = [v' \<oplus> t]\<^sub>\<R>"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "comparator_of x y = Lt"   and "comparator_of y z = Lt" shows "comparator_of x z = Lt"
bounded_iff: fixes b :: "'a word"   and c :: "'a word"   and a :: "'a word" shows "(signed.max b c \<le>s a) = (b \<le>s a \<and> c \<le>s a)"
D_def: shows "D \<equiv> \<lambda>f. if arr f then mkarr \<^bold>\<lfloor>rep f\<^bold>\<rfloor> else \<F>\<^sub>SC.null"
iso_incl_hom: fixes A :: "'c ring"   and B :: "'c ring" shows "A \<lesssim> B = law_restrict A \<lesssim> law_restrict B"
star_loop_is_least_fixpoint: fixes y :: "'a"   and z :: "'a" shows "is_least_fixpoint (\<lambda>x. y * x \<squnion> z) (y\<^sup>\<star> * z)"
da_elim_cases_11: fixes Env :: "env"   and B :: "lname set"   and l :: "label"   and e :: "expr"   and c :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<guillemotright> A" obtains E :: "assigned" and C :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> B \<union> assigns_if True e \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "nrm A = nrm C \<inter> (B \<union> assigns_if False e)" and "brk A = brk C"
right_neutral: fixes a :: "'c" shows "a + (0::'c) = a"
Compl_Pf: fixes n :: "'trm"   and \<phi> :: "'fmla" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in isTrue (PPf n \<langle>\<phi>\<rangle>) \<longrightarrow> bprv (PPf n \<langle>\<phi>\<rangle>)"
sum_not_inf_int: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a + b \<noteq> \<infinity>"   and "get_const a \<in> \<int>"   and "get_const b \<in> \<int>" shows "get_const (a + b) \<in> \<int>"
sign_S5_thm_1: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) in v]"
nonMT: shows "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S} \<noteq> {}"
oPrec_def: fixes x :: "ordinal" shows "oPrec x = (THE p. x = oSuc p)"
that: fixes A :: "'a list"   and B :: "'a list" shows "f (A @ B) = bind (f A) (\<lambda>x. bind (f B) (\<lambda>y. return (x @ y)))"
rel_sel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a stream"   and b :: "'b stream" shows "stream_all2 R a b = (R (shd a) (shd b) \<and> stream_all2 R (stl a) (stl b))"
bot_eq_sup_iff: fixes x :: "'a"   and y :: "'a" shows "(0 = x + y) = (x = 0 \<and> y = 0)"
ssubst_Pair_rhs: fixes r :: "'b"   and s :: "'c"   and R :: "('b \<times> 'c) set"   and s' :: "'c" assumes "(r, s) \<in> R"   and "s' = s" shows "(r, s') \<in> R"
SUP_empty: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and f :: "'f \<Rightarrow> 'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
Cons_1: fixes s :: "'a"   and ss2 :: "'a \<times> (bool \<times> 'b \<times> 'a) list" assumes "(s, xs) \<in> conv ars"   and "ss2 \<in> conv ars"   and "lst_conv (s, xs) = fst ss2" shows "(s, xs @ snd ss2) \<in> conv ars \<and> lst_conv (s, xs @ snd ss2) = lst_conv ss2"
inject: fixes x1 :: "'a"   and x2 :: "'a rose_tree list"   and y1 :: "'a"   and y2 :: "'a rose_tree list" shows "(RNode x1 x2 = RNode y1 y2) = (x1 = y1 \<and> x2 = y2)"
signed_real_of_word_sum_less_neg_numeral_iff: fixes v :: "'a Word.word"   and w :: "'b Word.word"   and n :: "num" shows "(signed_real_of_word v + signed_real_of_word w < - numeral n) = (sint v + sint w < - numeral n)"
sim_rangeI: fixes M' :: "('b, 'c, 'a) KripkeStructure"   and f :: "'d \<Rightarrow> 'a"   and M :: "('b, 'c, 'd) KripkeStructure" assumes "worlds M' = f ` worlds M"   and "\<And>a. relations M' a \<subseteq> worlds M' \<times> worlds M'" shows "sim_range M M' f"
inj: fixes i :: "'i"   and j :: "'i" assumes "i \<in> I"   and "j \<in> I"   and "G i = G j" shows "\<mu> i = \<mu> j"
eq_fin_end: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u\<^sub>1 :: "'a list"   and u\<^sub>2 :: "'a list"   and v\<^sub>1 :: "'a list"   and v\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* u\<^sub>1 u\<^sub>2"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* (u\<^sub>1 @ v\<^sub>1) (u\<^sub>2 @ v\<^sub>2)" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* v\<^sub>1 v\<^sub>2"
pos_bounded: shows "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
domain: shows "Domainp cr_st_mat = (\<lambda>a. non_neg_mat a \<and> right_stoch_mat a)"
one_before_two_2: fixes r :: "nat \<Rightarrow>f value option"   and x2a :: "(String.literal \<times> value list) stream" shows "alw (\<lambda>x. state_eq (Some 2) (stl x) \<longrightarrow> state_eq (Some 1) x) (make_full_observation drinks (Some 2) r [r $ 1] x2a)"
arr_char: fixes f :: "'a" shows "S.arr f = left \<omega> f"
cone_def: fixes Vs :: "'a vec set" shows "cone Vs = {x. \<exists>Ws. finite Ws \<and> Ws \<subseteq> Vs \<and> x \<in> finite_cone Ws}"
frp_while_algo: fixes T1 :: "('a, 'b) tree_automaton_rec"   and T2 :: "('c, 'b) tree_automaton_rec" assumes "tree_automaton T1"   and "tree_automaton T2" shows "while_algo (frp_algo T1 T2)"
ce_rel_lm_35: fixes r :: "(nat \<times> nat) set"   and s :: "(nat \<times> nat) set" assumes "r \<in> ce_rels"   and "s \<in> ce_rels" shows "r \<inter> s \<in> ce_rels"
inj1: shows "inj_on (\<lambda>p. p(1 := p 1 + 1)) {p. p partitions m \<and> sum p {..m} = k}"
initial_state_cost_def: fixes fs :: "int vec list" shows "initial_state_cost fs = (let (dmus, cost) = d\<mu>_impl_cost fs; ds = IArray.of_fun (\<lambda>i. if i = 0 then 1 else let i1 = i - 1 in dmus !! i1 !! i1) (Suc m); dmus' = IArray.of_fun (\<lambda>i. let row_i = dmus !! i in IArray.of_fun ((!!) row_i) i) m in ((([], fs), dmus', ds), cost))"
strict_trans1: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
MakeFaceFinal_faceListAt_len: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "faceListAt_len (makeFaceFinal f g)"
closedin_contains_derived_set: fixes X :: "'c topology"   and S :: "'c set" shows "closedin X S = (X derived_set_of S \<subseteq> S \<and> S \<subseteq> topspace X)"
gpush_impl_def: fixes v :: "'q"   and s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gpush_impl v s \<equiv> let (a, s) = s in (a @ [acc v], push_impl v s)"
simps_1: fixes f :: "'b \<Rightarrow> 'a Heap" shows "fold_map f [] = return []"
rel_setD1: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d set"   and B :: "'e set"   and x :: "'d" assumes "rel_set R A B"   and "x \<in> A" shows "\<exists>y\<in>B. R x y"
part_UNlift01_UNpart01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "properL dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n) \<and> {} \<notin> F n ` P n" shows "part {..<brnL dl (length dl)} (UNlift01 cl dl P F ` UNpart01 cl dl P)"
comp_def: fixes g :: "'a"   and f :: "'a" shows "L.comp g f = (if left g f \<and> left g g \<and> cod f = local.dom g then g \<cdot> f else null)"
lhsnot0: shows "\<b> * \<k> \<noteq> 0"
interchange_rows_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and b :: "'c" shows "interchange_rows A a b = (\<chi>i j. if i = a then A $ b $ j else if i = b then A $ a $ j else A $ i $ j)"
sset_shift: fixes xs :: "'d list"   and s :: "'d stream" shows "sset (xs @- s) = set xs \<union> sset s"
term_variants_pred_eq_case_Abs: fixes a :: "'c set"   and b :: "'c set"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" assumes "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t s"   and "\<forall>f\<in>funs_term s. \<not> is_Abs f" shows "t = s"
sup1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> k \<squnion> a" shows "A \<squnion> b \<equiv> k \<squnion> (a \<squnion> b)"
res_decompose: fixes c :: "cars" shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1 \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (res ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> len u ts c = ext u \<and> restrict u (res ts) c = lan u \<and> |lan u| = 1)"
create_delete_ids_imply_messages_same: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and ir :: "'a"   and e2 :: "'b" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Create i e1) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
induct: fixes P :: "('b, 'c) RBT_Impl.rbt \<Rightarrow> bool"   and a0 :: "('b, 'c) RBT_Impl.rbt" assumes "\<And>c k v rt. P (Branch c rbt.Empty k v rt)"   and "\<And>c lc llc lk lv lrt k v rt. P (Branch lc llc lk lv lrt) \<Longrightarrow> P (Branch c (Branch lc llc lk lv lrt) k v rt)"   and "P rbt.Empty" shows "P a0"
the_enat_mono: fixes m :: "enat"   and n :: "enat" assumes "m \<noteq> \<infinity>"   and "n \<le> m" shows "the_enat n \<le> the_enat m"
DiffD2: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
fls_pow_nth_below_subdegree: fixes k :: "int"   and n :: "nat"   and f :: "'b fls" assumes "k < int n * fls_subdegree f" shows "f ^ n $$ k = (0::'b)"